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abstract: 'We study the $^{11}\mathrm{Li}$ and $^{22}\mathrm{C}$ nuclei at leading order (LO) in halo effective field theory (Halo EFT). Using the value of the $^{22}\mathrm{C}$ rms matter radius deduced in Ref. [@Tanaka:2010zza] as an input in a LO calculation, we simultaneously constrained the values of the two-neutron (2$n$) separation energy of $^{22}\mathrm{C}$ and the virtual-state energy of the $^{20}\mathrm{C}-$neutron system (hereafter denoted $^{21}$C). The 1$-\sigma$ uncertainty of the input rms matter radius datum, along with the theory error estimated from the anticipated size of the higher-order terms in the Halo EFT expansion, gave an upper bound of about 100 keV for the 2$n$ separation energy. We also study the electric dipole excitation of 2$n$ halo nuclei to a continuum state of two neutrons and the core at LO in Halo EFT. We first compare our results with the $^{11}\mathrm{Li}$ data from a Coulomb dissociation experiment and obtain good agreement within the theoretical uncertainty of a LO calculation. We then obtain the low-energy spectrum of $B(E1)$ of this transition at several different values of the 2$n$ separation energy of $^{22}\mathrm{C}$ and the virtual-state energy of $^{21}\mathrm{C}$. Our predictions can be compared to the outcome of an ongoing experiment on the Coulomb dissociation of $^{22}\mathrm{C}$ to obtain tighter constraints on the two- and three-body energies in the $^{22}\mathrm{C}$ system.'
author:
- Bijaya Acharya
- Daniel Phillips
title: 'Properties of Lithium-11 and Carbon-22 at leading order in halo effective field theory[^1]'
---
Introduction
============
The separation of scales between the size of the core and its distance from the halo nucleons allows the low-energy properites of halo nuclei to be studied using Halo EFT [@Bertulani:2002sz; @Bedaque:2003wa], which is written in terms of the core and halo nucleons as degrees of freedom. Halo EFT yields relations for the low-energy observables as systematic expansions in the ratio of the short-distance scale set by the core size and excitation energies to the long-distance scale associated with the properties of the halo nucleons. At LO, the three-body wavefunction of the 2$n$ halo nucleus is constructed with zero-range two-body interactions, which can be completely characterized by the neutron-neutron ($nn$) and the neutron-core ($nc$) scattering lengths [@Kaplan:1998we]. However, a three-body coupling also enters at LO [@Bedaque:1998kg], necessitating the use of one piece of three-body data as input to render the theory predictive. It is convenient to fix the three-body force by requiring the three-body bound state to lie at $-E_B$, where $E_B$ is the 2$n$ separation energy. The only inputs to the equations that describe a 2$n$ halo are, therefore, $E_B$ together with the energies of the $nc$ virtual/real bound state, $E_{nc}$, and the $nn$ virtual bound state, $E_{nn}$. The effects of interactions that are higher order in the Halo EFT power counting are estimated from the size of the ignored higher-order terms and then included as theory error bands.
In Ref. [@Tanaka:2010zza], Tanaka et al. measured the reaction cross-section of $^{22}\text{C}$ on a hydrogen target and, using Glauber calculations, deduced a $^{22}\text{C}$ rms matter radius of $5.4\pm 0.9$ fm, implying that $^{22}\text{C}$ is an S-wave two-neturon halo nucleus. This conclusion is also supported by data on high-energy two-neutron removal from ${}^{22}$C [@Kobayashi:2011mm]. We used Halo EFT in Ref. [@Acharya:2013aea], to calculate the rms matter radius of $^{22}$C as a model-independent function of $E_B$ and $E_{nc}$. Since the virtual-state energy of the unbound [@Langevin] $^{21}\text{C}$ is not well known [@Mosby:2013bix], we used Halo EFT to find constraints in the $(E_B,E_{nc})$ plane using Tanaka et al.’s value of the rms matter radius.
We have also derived universal relations for the electric dipole excitation of two-neutron halo nuclei into the three-body continuum consisting of the core and the two neutrons in Halo EFT. Our LO calculation of the $B(E1)$ of this transition includes all possible rescatterings with S-wave $nn$ and $nc$ interactions, in both the initial and the final state. We compare our results with the $^{11}\mathrm{Li}$ data from Ref. [@Nakamura:2006zz] and obtain a good agreement within the theoretical uncertainty. We predict the $B(E1)$ spectrum of $^{22}\mathrm{C}$ for selected values of $E_B$ and $E_{nc}$. These findings will be published in Ref. [@Acharya:tobepublished].
Matter radius constraints on binding energy
===========================================
In Fig. \[fig:contourplots\], we plot the sets of ($E_B$, $E_{nc}$) values that give a $^{22}\mathrm{C}$ rms matter radius, $\sqrt{\langle R^2 \rangle}$, of 4.5 fm, 5.4 fm and 6.3 fm, along with the theoretical error bands. All sets of $E_B$ and $E_{nc}$ values in the plotted region that lie within the area bounded by the edges of these bands give an rms matter radius that is consistent with the value Tanaka et al. extracted within the combined ($1-\sigma$) experimental and theoretical errors. The figure shows that, regardless of the value of the $^{21}\text{C}$ virtual energy, Tanaka et al.’s experimental result puts a model-independent upper limit of 100 keV on the 2$n$ separation energy of $^{22}\text{C}$. This is to be compared with another theoretical analysis of the matter radius datum of Ref. [@Tanaka:2010zza] in a three-body model by Ref. [@Yamashita:2011cb], which set an upper bound of 120 keV on $E_B$. Similarly, Ref. [@Fortune:2012zzb] used a correlation between the binding energy and the matter radius derived from a potential model to exclude $E_B>220$ keV. Our constraint is stricter than the ones set by these studies. Although our conclusion is consistent with the experimental value of $-140~(460)$ keV from a direct mass measurement [@Gaudefroy:2012qe], more studies are needed to further reduce the large uncertainty in the 2$n$ separation energy. In this spirit, we study the $E1$ excitation of 2$n$ halo nuclei to the three-body continuum.
![Plots of $\sqrt{\langle R^2 \rangle} $ = 5.4 fm (blue, dashed), 6.3 fm (red, solid), and 4.5 fm (green, dotted), with their theoretical error bands, in the $(E_B,E_{nc})$ plane. (Published in Ref. [@Acharya:2013aea].)[]{data-label="fig:contourplots"}](AcharyaB_fig1.pdf){width="75.00000%"}
The [*B*]{}([*E*]{}1) spectrum
==============================
We first present the result of our LO Halo EFT calculation of the $B(E1)$ for the break up of $^{11}\mathrm{Li}$ into $^{9}\mathrm{Li}$ and two neutrons at energy $E$ in their center of mass frame. Only S-wave $^9\mathrm{Li}-n$ interactions are included. After folding with the detector resolution, we obtain the curve shown in Fig. \[fig:li11\] for $E_B=369.15(65)~\mathrm{keV}$ [@Smith:2008zh] and $E_{nc}=26~\mathrm{keV}$ [@NNDC]. The sensitivity to changes in $E_{nc}$ is much smaller than the EFT error, represented by the purple band. Within the uncertainty of a LO calculation, a good agreement with the RIKEN data [@Nakamura:2006zz] is seen, despite the fact that $^{10}\mathrm{Li}$ has a low-lying P-wave resonance which is not included in this calculation.
![The dipole response spectrum for $^{11}\mathrm{Li}$ after folding with the detector resolution (blue curve) with the theory error (purple band), and data from Ref. [@Nakamura:2006zz].[]{data-label="fig:li11"}](AcharyaB_fig2.pdf){width="75.00000%"}
Figure \[fig:c22\] shows the dipole response spectrum for the break up of $^{22}\mathrm{C}$ into $^{20}\mathrm{C}$ and neutrons for three different combinations of $E_B$ and $E_{nc}$ which lie within the $1-\sigma$ confidence region shown in Fig. \[fig:contourplots\]. These results agree qualitatively with those of a potential model calculation by Ref. [@Ershov:2012fy]. A comparison of Fig. \[fig:c22\] with the forthcoming data [@Nakamura:2013conference] can provide further constraints on the $(E_B,E_{nc})$ plane. However, the individual values of these energies thus extracted will have large error bars because different sets of $(E_B,E_{nc})$ values can give similar curves. This ambiguity can be removed by looking at the neutron-momentum distribution of the Coulomb dissociation cross section [@Acharya:tobepublished].
![The dipole response spectrum for $^{22}\mathrm{C}$ for $E_B=50$ keV, $E_{nc}=10$ keV (blue, dotted); $E_B=50$ keV, $E_{nc}=100$ keV (red, dashed) and $E_B=70$ keV, $E_{nc}=10$ keV (black), with their EFT error bands.[]{data-label="fig:c22"}](AcharyaB_fig3.pdf){width="75.00000%"}
Conclusion
==========
The matter radius and the $E1$ response of S-wave 2$n$ halo nuclei were studied. We put constraints on the $(E_B,E_{nc})$ parameter space using the value of the ${}^{22}$C matter radius. The calculated $B(E1)$ spectrum of $^{11}\mathrm{Li}$ agrees with the experimental result within our theoretical uncertainty. Our $^{22}\mathrm{C}$ result can be tested once the experimental data is available. Further improvements can be made by rigorously calculating the higher-order terms in the EFT expansion and by including higher partial waves.
We thank our collaborators Chen Ji, Hans-Werner Hammer and Philipp Hagen. This work was supported by the US Department of Energy under grant DE-FG02-93ER40756. BA is grateful to the organizers of the conference for the opportunity to present this work and to UT for sponsoring his attendance.
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[^1]: Contribution to the $21^\text{st}$ International Conference on Few-Body Problems in Physics
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---
address:
- |
Deptametno de Matemática - IME, Universidade de São Paulo\
Caixa Postal 66.281 - CEP 05314-970, São Paulo - SP, Brasil
- |
The Institute of Mathematical Sciences\
CIT Campus, Taramani,\
Chennai 600113, India
author:
- 'Daciberg L. Gonçalves'
- Parameswaran Sankaran
title: 'Twisted conjugacy in Richard Thompson’s group $T$'
---
[**Abstract**]{} [*Let $\phi:\Gamma\to \Gamma$ be an automorphism of a group $\Gamma$. We say that $x,y\in \Gamma$ are in the same $\phi$-twisted conjugacy class and write $x\sim_\phi y$ if there exists an element $\gamma\in \Gamma$ such that $y=\gamma x\phi(\gamma^{-1})$. This is an equivalence relation on $\Gamma$ and is called the $\phi$-twisted conjugacy. Let $R(\phi)$ denote the number of $\phi$-twisted conjugacy classes in $\Gamma$. If $R(\phi)$ is infinite for all $\phi\in \aut(\Gamma)$, we say that $\Gamma$ has the $R_\infty$-property. The purpose of this note is to show that the Richard Thompson group $T$ has the $R_\infty$ property.*]{}
Introduction
============
Let $\Gamma$ be a group and let $\phi:\Gamma\to \Gamma$ be an endomorphism. Then $\phi$ determines an action $\Phi$ of $\Gamma$ on itself where, for $\gamma\in \Gamma$ and $x\in \Gamma$, we have $\Phi_\gamma(x)=\gamma x \phi(\gamma^{-1})$. The orbits of this action are called the $\phi$-twisted conjugacy classes. Note that when $\phi$ is the identity automorphism, the orbits are the usual conjugacy classes of $\Gamma$. We denote by $\mathcal{R}(\phi)$ the set of all $\phi$-twisted conjugacy classes and by $R(\phi)$ the cardinality $\#\mathcal{R}(\phi)$ of $\mathcal{R}(\phi)$. We say that $\Gamma$ has the $R_\infty$-property if $R(\phi)=\infty,$ that is if $\mathcal{R}(\phi)$ is infinite, for every automorphism $\phi$ of $\Gamma$.
The problem of determining which groups have the $R_\infty$-property—more briefly the $R_\infty$-problem—has attracted the attention of many researchers after it was discovered that all non-elementary Gromov-hyperbolic groups have the $R_\infty$-property. See [@ll] and [@felshtyn]. It is particularly interesting when the group in question is finitely generated or countable. The notion of twisted conjugacy arises naturally in fixed point theory, representation theory, algebraic geometry and number theory. In the recent years the $R_\infty$-problem has emerged as an active research area. The problem is particularly interesting because there does not seem to be a uniform approach to its resolution. A variety of techniques and ad hoc arguments from several branches of mathematics have been used in the solve this problem depending on the group under consideration. These include (but not restricted to) combinatorial group theory, geometric group theory, homological algebra, $C^*$-algebras, and algebraic groups.
Recall that Richard Thompson introduced three groups $F, T, $ and $V$ in 1965 in an unpublished hand-written manuscript. The group $T$ is the first example of a finitely presented infinite simple group. In this note we give an elementary proof that $T$ is an $R_\infty$-group.
\[main\] The Richard Thompson group $T$ has the $R_\infty$-property.
We shall describe the groups $F$ and $T$ more fully in §2, leaving out $V$. We prove the above theorem in §3. The groups $F$ and $T$ have been generalized by K. S. Brown [@BrownFinite] to obtain families of finitely presented groups $F_{n,\infty}, F_n, T_{n,r}, n\ge 2, r\ge 1$ where $F=F_{2}=F_{2,\infty}$ and $T=T_{2,1}$. Denoting any one of them by $F_n$ the group $F_{n,\infty}$ is isomorphic to a certain subgroup of $F_n$ of index $n-1$. Furthermore $F_{n,r}$ is a subgroup of $T_{n,r}$.
The group $V$ has been generalized by G. Higman [@HigmanFPSG] to obtain an infinite family of finitely presented groups. But we shall not consider them in this paper.
The $R_\infty$-property for $F$ was shown by Bleak, Fel’shtyn and Gonçalves [@BlFeGon]. It has been established by Gonçalves and Kochloukova [@gk Corollary 4.2] that the groups $F_{n,\infty}$ have the $R_\infty$-property. In the last section we make a few comments about the $R_\infty$-property for the groups $F_n$ and $T_{n,r}$.
After this paper was submitted, Collin Bleak had brought our attention to the paper [@bmv] of Burillo, Matucci and Ventura where it is shown, among other things, that $T$ has the $R_\infty$-property. (They also obtain a new proof of the $R_\infty$-property for the group $F$.) Their proof and the proof given here are based on the same idea of constructing elements with specified number of components of fixed point sets. We hope that Lemmas \[finiteorder\] and \[torsion\] which were used in our proof may be useful in other contexts as well.
[**Acknowledgments:**]{} We thank J. Burillo for pointing out a misquote in an earlier version of this paper in the statement of Theorem \[brin\](i), based on which we had erroneously claimed that our proof of Theorem \[main\] also establishes the $R_\infty$-property for $F$. We thank Collin Bleak for bringing to our notice the paper [@bmv]. The first author is indebted to Bleak, A. Fel’shtyn, and J. Taback for fruitful discussions about the Thompson groups. The first author has been partially supported by Fapesp project Temático Topologia Algébrica, Geometrica e Diferencial no 2012/24454-8. This project was initiated during the visit of the second author to the University of São Paulo in August 2012. He thanks the first author for the invitation and the warm hospitality. He is also thankful to the organizers of the XVIII Brazilian Topology Meet (EBT-2012) for the invitation and financial support, making the visit to Brazil possible.
Richard Thompson’s groups $F$ and $T$
=====================================
In this section we give a description of Thompson groups $F$ and $T$. The group $F$ consists of all piecewise linear (PL) homeomorphisms of $\mathbb[0,1]$ with at most a finite set of break points (i.e., points of non-differentiability) which are contained in the dyadic rationals $\mathbb{Z}[1/2]$ and having slopes (at points of differentiability) in the multiplicative group $\langle 2\rangle=\{2^n\mid n\in \mathbb{Z}\}$. Note that elements of $F$ are orientation preserving. It is known that the group $F$ is generated by two elements $A$ and $B$ defined as follows: $$A(x)=\left\{ \begin{array}{lr}
x/2, & 0\le x\le 1/2,\\
x-1/4, & 1/2\le x\le 3/4,\\
2x-1, & 3/4\le x\le 1.\\
\end{array}\right.$$ and $$B(x)=\left\{ \begin{array}{lr}
x, & 0\le x\le 1/2,\\
x/2+1/4, & 1/2\le x\le 3/4,\\
x-1/8, & 3/4\le x\le 7/8,\\
2x-1, & 7/8\le x\le 1.\\
\end{array}\right.$$ Indeed one has a presentation $$F=\langle A, B\mid [AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^{-2} ]\rangle.$$
The group $T$ consists of all PL-homeomorphisms of the circles $\mathbb{S}^1=I/\{0,1\}$ which have at most a finite set of break-points contained in $\mathbb{Z}[1/2]$ and having slopes contained in $\langle 2\rangle$. Again the elements of $T$ preserve the orientation.
Any homeomorphism of $[0,1]$ induces a homeomorphism of $\mathbb{S}^1$ and this allows us to view $F$ as a subgroup of $T$. One has an element $C$ in $T$ which is defined as $$C(x)=\left\{\begin{array}{lr}
x/2+3/4, & 0\le x\le 1/2,\\
2x-1, & 1/2\le x\le 3/4,\\
x-1/4, &3/4\le x\le 1.\\
\end{array}
\right.$$ It is understood that in the above definition $x$ is read modulo $1$.
It is known that $T$ is generated by the elements $A, B, C$ with six relations. Although we will have no need for it here, we list below the relations in the said presentation for the sake of completeness: (Note that the first two are the same as the defining relations of $F$.) See [@CFP] for details.\
(1) $[AB^{-1},A^{-1}BA]=1$,\
(2) $[AB^{-1}, A^{-2}BA^2]=1$,\
(3) $C=BA^{-1}CB$,\
(4) $A^{-1}CB.A^{-1}BA=B.A^{-2}CB^2$\
(5) $CA= (A^{-1} CB)^2$\
(6) $C^3=1$.
Our proof of Theorem \[main\] will crucially make use of the following result of Brin [@BrinCh]. It is easily seen that the reflection map $r$ defined as $r(x)=1-x, x\in [0,1]$, induces an automorphism $\rho: T\to T$ defined as $\rho(f)=r\circ f\circ r^{-1}=r\circ f\circ r$. We denote by the same symbol $\rho$ the restriction $\rho|_F\in \aut(F)$.
([@BrinCh]) \[brin\] (i) The group $\out(F)$ of outer automorphisms of $F$ contains an index two subgroup $\out^+(F)$ isomorphic to $T\times T$. The non-trivial element in the quotient group $\out(F)/\out^+(F)$ is represented $\rho\in \aut(F).$\
(ii) The group of inner automorphisms of $T$ is of index two in $\aut(T)$ and the quotient group $\out(T)$ is generated by $\rho$.
Proof of Theorem \[main\]
=========================
Let $\Gamma$ be a group and let $\phi\in \aut(\Gamma)$. For $\gamma\in \Gamma$, denote by $\iota_\gamma$ the inner automorphism $x\mapsto \gamma x\gamma^{-1}, ~x\in \Gamma$. We first observe that $R(\phi)=R(\phi\circ \iota_\gamma)$. In fact $y=zx\phi(\iota_\gamma(z^{-1}))$ if and only if $y.\phi(\gamma)=z(x\phi(\gamma))\phi(z^{-1})$. Thus elements of $\mathcal{R}(\phi\circ\iota_\gamma)$ are translation on the left by $\phi(\gamma)$ of the elements of $\mathcal{R}(\phi)$. In view of this, to show that $\Gamma$ is an $R_\infty$-group, it suffices to show that $R(\phi)=\infty$ for a set of coset representatives of $\out(\Gamma)$. In the case $\Gamma=T$, in view of Theorem \[brin\] due to Brin, we need only show that $R(\rho)=\infty$ and $R(id)=\infty$. The latter equality is established in Proposition \[conjugacy\] as an easy consequence of Lemma \[support\] below.
[*Let $X$ be a Hausdorff topological space.\
(i) The [*support of*]{} $f\in \homeo(X)$ is the open set $\supp(f):=\{x\in X\mid f(x)\ne x\}$.\
(ii) Let $\sigma:\homeo(X) \to \mathbb{N}\cup \{\infty\}$ be defined as follows: $\sigma(id)=0$, if $f\ne id$, $\sigma(f)$ is the number of connected components of $\supp(f)$ if it is finite, otherwise $\sigma(f)=\infty$.*]{}
\[support\] Let $\Gamma\subset \homeo(X)$ and let $\sigma$ be as defined above. Suppose that $\theta\in \homeo(X)$ normalizes $\Gamma$. Then $\sigma(f)=\sigma (\theta f \theta^{-1})$.
It is clear that the number of connected components of an open set $U\subset X$ remains unchanged under a homeomorphism of $X$. The lemma follows immediately from the observation that $\supp(\theta f \theta^{-1})=\theta(\supp(f))$.
The following proposition is well-known. It follows, for example, from [@BlFeGon Remark 3.1] for the group $F$ and from [@CFP §5] for the group $T$, where certain elements $C_n\in T$ of order $n+2$ for every $n\ge 1$ are explicitly given. However, we give an elementary unified proof for the sake of completeness.
\[conjugacy\][ *The groups $F$ and $T$ have infinitely many conjugacy classes.*]{}
This follows from Lemma \[support\] on observing that $F$ has elements $f$ with $\sigma(f)$ any arbitrary prescribed positive integer. Since $F\subset T$, the same is true of $T$ as well.
We need the following lemma. If $\theta$ is an endomorphism of a group $\Gamma$ we denote by $\fix(\theta)$ the fixed subgroup $\{x\in \Gamma\mid
\theta(x)=x\}$ of $\Gamma$.
\[finiteorder\] Let $\Gamma$ be a group and let $\theta\in \aut(\Gamma)$. Suppose that $\theta^n=\iota_\gamma$. Suppose that $\{x^n\gamma\mid x\in Fix(\theta)\}$ is not contained in the union of finitely many conjugacy classes of $\Gamma$. Then $R(\theta)=\infty$.
Let $x\sim_\theta y$ in $\Gamma$ where $x,y\in \fix(\theta)$. Thus there exists an $z\in \Gamma$ such that $y=z^{-1}x\theta(z)$. Applying $\theta^i$ both sides, we obtain $y=\theta^i(z^{-1})x\theta^{i+1}(z)$ as $x,y\in \fix(\theta)$. Multiplying these equations successively for $0\le i<n$ and using $\theta^n=\iota_\gamma$, we obtain $$y^n=\prod_{0\le i<n} \theta^i(z^{-1})x\theta^{i+1}(z)
=z^{-1}x^n\theta^n(z)=z^{-1}x^n\gamma z \gamma^{-1}.$$ That is, $y^n\sim_{\iota_\gamma} x^n$. Equivalently $y^n\gamma$ and $x^n\gamma$ are in the same conjugacy classes of $\Gamma$. Our hypothesis says that there are infinitely many elements $x_k\in \fix(\theta), k\ge 1$, such that the $x_k^n\gamma$ are in pairwise distinct $\iota_\gamma$-conjugacy classes of $\Gamma$. Hence we conclude that $R(\theta)=\infty$.
We remark that when $\theta\in Aut(\Gamma)$ of order $n$, we may take $\gamma$ to be the identity. Therefore $R(\theta)=\infty$ when $\{x^n \in \Gamma\mid \theta(x)=x\}\subset \Gamma$ is not contained in the union of finitely many conjugacy classes of $\Gamma$. This observation leads to the following.
\[torsion\] Let $\theta\in Aut(\Gamma)$ be of order $n<\infty$. Suppose that the set $\mathcal{T}\subset \mathbb{N}$ of orders of torsion elements of $\fix(\theta)$ is unbounded. Then $R(\theta)=\infty$.
Our hypothesis on $\mathcal{T}$ implies that the $\{o(x^n)\mid x\in \fix(\theta)\}\subset \mathbb{N}$ is unbounded. Therefore elements of $\{x^n\mid x\in \fix(\theta)\}$ represent infinitely many distinct conjugacy classes of $\Gamma$. By Lemma \[finiteorder\] we conclude that $R(\theta)=\infty$.
\[iterates\] Suppose that $h:\mathbb{R}\to \mathbb{R}$ is an orientation preserving homeomorphism. Then $\supp(h)=\supp(h^k)$ for any non-zero integer $k$.
Since $\supp(h)=\supp(h^{-1})$ we may assume that $k>0$. Since $h$ is orientation preserving, it is order preserving. Suppose that $x\in \supp(h)$ so that $h(x)\ne x$. Say, $x<h(x)$. Then applying $h$ to the inequality we obtain $h(x)<h^2(x)$ so that $x<h(x)<h^2(x)$. Repeating this argument yields $x<h(x)<\cdots <h^k(x)$ and so $x\in \supp(h^k)$. The case when $x>h(x)$ is analogous. Thus $\supp(h)\subset \supp(h^k)$. On the other hand, if $x\notin \supp(h)$, then $h(x)=x$ and so $h^k(x)=x$ for all $k$. Therefore equality should hold, completing the proof.
We are now ready to prove our main theorem.
[*Proof of Theorem \[main\]:*]{} By Theorem \[brin\](ii), $\out(T)\cong \mathbb{Z}/2\mathbb{Z}$ generated by $\rho$. By Proposition \[conjugacy\], $R(id)=\infty$. It only remains to verify that $R(\rho)=\infty$. We apply Lemma \[finiteorder\] with $\theta=\rho, n=2, \gamma=1$. It remains to show that $\fix(\rho)$ has infinitely many elements $h$ such that $h^2$ are pairwise non-conjugate.
Let $k\ge 1$. Let $f_k\in F\subset T$ be such that $\supp(f_k)\subset (0,1/2)$ and has exactly $k$ components. Thus, $\sigma(f_k)=k$. (It is easy to construct such an element.) Then $\supp(\rho(f_k))
=\supp(rf_kr^{-1})=r(\supp(f_k))\subset (1/2,1)$ is disjoint from $\supp(f_k)\subset (0,1/2)$. In particular $f_k.\rho(f_k)=\rho(f_k).f_k=:h_k$, $\supp(h_k)=\supp(f_k)\cup r(\supp(f_k))$ and so $\sigma(h_k)=2k$. Moreover, since $\rho^2=1$, we see that $h_k\in Fix(\rho)$. By Lemma \[iterates\], we have $\sigma(h_k^2)=\sigma(h_k)=2k$. It follows that $h_k^2$ are pairwise non-conjugate in $T$, completing the proof. $\Box$.
Generalized Thompson groups
===========================
The group $F$ has been generalized to yield two families of groups $F_{n,\infty}, F_n$, $n\ge 2$, and the group $T$ likewise has been generalized to a family of groups $T_{n, r},n\ge 2, r\ge 1,$ where $F\cong F_2=F_{2,\infty}$ and $T=T_{2,1}$. One has inclusions $F_{n,\infty}\subset F_n\subset T_{n,r}$ for all $n\ge 2, r\ge 1$. Let $\Gamma$ be any one of the groups $F_{n,\infty}, F_n, T_{n,r}$. Then $\Gamma$ is realized as a group of homeomorphisms of $\mathbb{R}$ or $\mathbb{S}^1=\mathbb{R}/r\mathbb{Z}$ according as $\Gamma=F_{n,\infty}, F_n$ or $\Gamma=T_{n,r}$ respectively. More precisely, one has the following description given in [@BrinGuzman Proposition 2.2.6]. The group $F_n$ is the group of all orientation preserving PL-homeomorphisms of $\mathbb{R}$ having only finitely many break points (that is, points of non-differentiability) such that (i) the break-points are all in $\mathbb{Z}[1/n],$ (ii) the slopes at smooth points are all in the set $\{n^k | k \in \mathbb{Z}\}=:\langle n\rangle$, (iii) they map the set $\mathbb{Z}[1/n]$ into itself, and, (iv) they are translations by integers near $-\infty$ and $\infty$. (For the last condition, see Definition \[slopetransl\] below.) The group $F_{n,\infty}$ is the subgroup of $F_n$ which consists of those homeomorphisms $f\in F_n$ which maps $\Delta_n$ into itself where $\Delta_n$ is the kernel of the unique surjective ring homomorphism $\mathbb{Z}[1/n]\to \mathbb{Z}/(n-1)\mathbb{Z}$. The group $T_{n,r}\subset \homeo(\mathbb{R}/r\mathbb{Z})$ consists of those elements which are orientation preserving and lift to PL-homeomorphisms of $\mathbb{R}$ satisfying conditions (i) to (iii) above.
It turns out that $F\cong F_2=F_{2,\infty}$ and $T\cong T_{2,1}$. The groups $F_{n,\infty}, F_n$ and $T_{n,r}$ are referred to as the generalized Thompson groups. See [@BrinGuzman] for a detailed study of these groups and their automorphism groups.
Brin and Guzmán also introduced a family of groups $F_{n,j},n\ge 2, j\in \mathbb{Z}$ each of which is isomorphic to $F_{n,\infty}$. See [@BrinGuzman Lemma 2.1.6 and Corollary 2.3.1.1]. Gonçalves and Kochloukova [@gk] have shown, using the theory of $\Sigma$-invariants in homological algebra, that the groups $F_{n,0},~n \ge 2,$ and hence $F_{n,\infty}$, have the $R_\infty$-property. In this section we show that if $\theta\in \aut(F_n)$ represents a torsion element in the outer automorphism group, then $R(\theta)=\infty$.
It was observed by Brin and Guzmán, using a deep result of McCleary and Rubin [@MR], that every automorphism of a generalized Thompson group is given by conjugation by a homeomorphism of $\mathbb{R}$ or the circle $\mathbb{S}^1$. (Such a homeomorphism is not, in general, a PL-homeomorphism!) Invoking this result, our proof of Proposition \[conjugacy\] applies equally well to the generalized Thompson groups showing that they have infinitely many conjugacy classes.
\[slopetransl\] [ *Let $\gamma$ be a PL-homeomorphism of $\mathbb{R}$ with finitely many break points. Suppose that $\gamma(t)=at+b$ for $t>0$ large. We call $a\in \mathbb{R}$ the [*slope at*]{} $\infty$ and $b\in \mathbb{R}$ [*the translation at*]{} $\infty$ and denote them by $\lambda(\gamma)$ and $\tau(\gamma)$ respectively.* ]{}
We note that $\lambda$ is constant on conjugacy classes of the group of all PL-homeomorphisms of $\mathbb{R}$ with finitely many break points. If $z, h,h'$ are such homeomorphisms and if $\lambda(h)=1=\lambda(h'),$ then $\tau(hh')=\tau(h)+\tau(h')$ and $\tau(zhz^{-1})=\lambda(z).\tau(h)$ as may be verified easily.
By the description of $F_n$ given above, $\tau(\gamma)\in \mathbb{Z}$ and $\lambda(\gamma)=1 $ if $\gamma\in F_n$. Moreover, $\tau(\gamma)\in (n-1)\mathbb{Z}$ if $\gamma\in F_{n,\infty}$.
[*Let $\theta\in \aut(\Gamma)$ represent an outer automorphism $[\theta]$ of $\Gamma$ of finite order where $\Gamma$ is one of the groups $F_{n,\infty}, F_n, n\ge 2$. Then $R(\theta)=\infty$.*]{}
Let $o([\theta])=m$. There exists an $f\in\homeo(\mathbb{R})$ such that $\theta(h)=f h f^{-1}$ for all $h\in \Gamma$. (Cf. [@BrinGuzman Theorem 1.2.4], [@MR].) Since $[\theta]^m=1$ we see that there exists a $\gamma \in \Gamma$ such that $\theta^m(h)=f^m hf^{-m}=\gamma h\gamma^{-1} $ for all $h\in \Gamma$.
Therefore, setting $g:=\gamma^{-1}f^m\in \homeo(\mathbb{R})$ and $ghg^{-1}=h $ for all $h\in \Gamma$. We claim that $g=1$. To see this, assume that $g\ne 1$ and choose an interval $U\subset \mathbb{R}$ such that $g(U)\cap U=\emptyset$. Now let $h\in \Gamma$ be any non-trivial element supported in $U$. Then $ghg^{-1}$ is supported in $g(U)$. This shows that $ghg^{-1}\ne h$, a contradiction. Hence we conclude that $g=1$ and so $\gamma=f^m$. In particular $\gamma^k\in \fix(\theta)$ for all $k\in \mathbb{Z}$.
We may assume that $\gamma\ne 1$. (Otherwise $f\in \homeo(\mathbb{R})$ has order $m$. Hence $m=2$, $\theta(p)=p$ for some $p\in \mathbb{R}$ and $\theta$ interchanges the intervals $(-\infty, p)$ and $(p,\infty)$. We proceed as in the proof of Theorem \[main\] to see that $R(\theta)=\infty$.)
Since $f^m=\gamma\in \Gamma$ and $\gamma\ne 1$, we have that $\supp(f)=\supp(\gamma)$ equals $\mathbb{R}$ or is a union of [*finitely*]{} many open intervals. In particular $\sigma(\gamma)<\infty$.
If $\supp(f)$ is not dense, we merely choose elements $\gamma_k\in \Gamma$ such that $\supp(\gamma_k)$ has exactly $k$ components and $\supp(f)\cap \supp(\gamma_k)=\emptyset $ for all $k\ge 1$. Then $\theta(\gamma_k)=\gamma_k$ and $\sigma(\gamma_k.\gamma)=k+\sigma(\gamma)$ for all $k$. It follows that $\{\gamma_k.\gamma\}_{k\ge 1}$ are in pairwise distinct conjugacy classes.
So assume that $\supp(f)=\supp(\gamma)$ is dense. As remarked above, any element of $\Gamma$ has slope at $\infty$ equals $1$ and translation at infinity an integer, say, $b$. So we have $\gamma(t)=t+b$ for $t>0$ large. Since $\supp(\gamma)$ is dense, we have $b\ne 0$.
Suppose that $\gamma^r=z\gamma^sz^{-1}$ for some $z\in \Gamma$ where $r,s$ are non-zero integers. Applying $\tau$ we obtain $rb=\lambda(z).sb=sb$ as $\lambda(z)=1$. Hence $r=s$. This shows that the elements of $\{\gamma^{mk+1}\mid k\in \mathbb{N}\}$ are in pairwise distinct conjugacy classes of $\Gamma$. By Lemma \[finiteorder\] we conclude that $R(\theta)=\infty$.
\(i) In the case of the generalized Thompson groups $T_{n,r}$, suppose that $\theta\in \aut(T_{n,r})$ represents a torsion element, say of order $m$, in $\out(T_{n,r})$ and that $\theta(x)=fxf^{-1}$ with $f\in \homeo(\mathbb{R}/r\mathbb{Z})$. Suppose $f^m=\gamma\in T_{n,r}$. If $\gamma=1$ our method of proof of Theorem \[main\] can be applied to show that $R(\theta)=\infty$. However, when $\gamma\ne 1$, it is not clear to us how to find elements of $\fix(\theta)$ satisfying the hypotheses of Lemma \[finiteorder\].
\(ii) Our approach yields no information about automorphisms which represent non-torsion elements in the outer automorphism group.
[99]{} C. Bleak, A. Fel’shtyn, and D. L. Gonçalves, *Twisted conjugacy classes in R. Thompson’s group F*, Pacific Journal of Mathematics **238** no.1 (2008), 1–6.
Matthew G. Brin, *The chameleon groups of [R]{}ichard [J]{}. [T]{}hompson: automorphisms and dynamics*, Publication Math. Inst. Hautes Études Sci. [**84**]{} (1997), 5–33.
Matthew G. Brin and Fernando Guzm[á]{}n, *Automorphisms of generalized [T]{}hompson groups*, Journal of Algebra **203** (1998), no. 1, 285–348.
Kenneth S. Brown, *Finiteness properties of groups*, Journal of Pure and Applied Algebra **44** (1987), 45–75.
J. Burillo, F. Matucci, and E. Ventura, [*The conjugacy problem for extensions of the Thompson’s group $F$*]{}, arXiv:1307.6750v2.
J. W. Cannon, W. J. Floyd, and W. R. Parry, *Introductory notes on Richard Thompson’s groups*, Enseignement Mathematique (2), [**42**]{} 3-4 (1996), 215–256.
A. Felshtyn, [*The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite.*]{} Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) [**279**]{} (2001), Geom. i Topol. [**6**]{}, 229–240.
D. L. Gonçalves and D. Kochloukova, [*Sigma theory and twisted conjugacy classes,*]{} Pacific Journal of Mathematics [**247**]{} (2010), 335–352.
Graham Higman, [*Finitely presented infinite simple groups*]{}, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974).
G. Levitt and M. Lustig, [*Most automorphisms of a hyperbolic group have very simply dynamics,*]{} Annales Scient. École Normale Superieur [**33**]{} (2000), 507–517.
S. McCleary and M. Rubin, [*Locally moving groups and the reconstruction problem for chains and circles,*]{} arXiv:math/0510122.
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abstract: 'We study the relation between various notions of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil [@BDS1] with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil [@BDS1].'
title: '<span style="font-variant:small-caps;">Exterior convexity and classical calculus of variations</span> '
---
<span style="font-variant:small-caps;">Saugata Bandyopadhyay</span>
<span style="font-variant:small-caps;">Swarnendu Sil</span>
*Keywords:* calculus of variations, rank one convexity, quasiconvexity, polyconvexity, exterior convexity, exterior form, differential form.\
\
*2010 Mathematics Subject Classification:* 49-XX.
Introduction
============
The notion of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil [@BDS1] is of fundamental importance in calculus of variations on exterior spaces, playing a role similar to what is played by the usual notions of convexity in classical vectorial calculus of variations.
However, the precise connection between these two sets of notions of convexity is a question of somewhat delicate balance. In this article, we explore this connection through the introduction of an appropriate projection map. While this projection map coincides with the canonical alternating projection of the two-tensor fields onto the exterior two-forms, it is non-trivial in the context of higher order forms. Furthermore, the projection map has the crucial property that it projects the tensor product to the exterior product and the gradient to the exterior derivative. It also allows us to express the connection between the notions of exterior convexity and classical notions of convexity in a crisp and explicit way, which is the content of our main theorem stated as follows
\[intro main thm\] Let $2\leq k\leq n$, $f:\Lambda^{k} \rightarrow\mathbb{R}$ and $\pi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow
\Lambda^{k}$ be the projection map. Then the following equivalences hold$$\begin{aligned}
f\text{ ext. one convex }&\Leftrightarrow\text{ }f\circ\pi\text{ rank one
convex. }\\
f\text{ ext. quasiconvex }&\Leftrightarrow\text{ }f\circ\pi\text{ quasiconvex. }\\
f\text{ ext. polyconvex }&\Leftrightarrow\text{ }f\circ\pi\text{ polyconvex. }\end{aligned}$$
The aforementioned result essentially situates the circle of ideas discussed in Bandyopadhyay-Dacorogna-Sil [@BDS1] in its proper place with respect to classical calculus of variations, which is well-developed and by now, standard (cf. Dacorogna [@DCV2]). It allows us to do calculus of variations back and forth between exterior spaces and the space of matrices. In particular, some results which were directly proved in Bandyopadhya-Dacorogna-Sil [@BDS1] turn out be easy corollaries of the theorem aforementioned above, in conjunction with classical results of vectorial calculus of variations. Notable among them is the characterization theorem for ext. quasiaffine functions (compare the proof of Theorem 3.3 in Bandyopadhyay-Dacorogna-Sil [@BDS1] with that of Theorem \[Thm principal quasiaffine\]). While in this process we do sacrifice the intrinsic character and the co-ordinate free advantage of a direct proof in exterior spaces, a simple proof is obtained nonetheless provided we are ready to assume the results of classical calculus of variations which are non-trivial and technical in their own right.
In this article, our main goal is to prove the aforementioned theorem. While proving the first two equivalences in Theorem \[intro main thm\] is easy from the definition of the projection map, proving the third one turns out to be surprizingly difficult and is of our principal concern in this article. One of the obstacles to the proof is the burden of heavy notations. To clarify presentation and to facilitate bookkeeping, we employed a system of notations, which is explained in detail in Section \[notations\] at the end of the article. However, once the cloud of heavy notations is cleared, the proof highlights many intricacies of the algebraic structure of alternating multilinear maps, namely the algebraic structure of determinants and minors and their interrelationship with the algebra of the wedge products which we believe should be of independent interest.
The rest of the paper is organized as follows: In Section 2, we recall the definitions of exterior convexity and introduce the projection map. Section 3 states the main theorem and presents the consequences along with a characterization theorem and a weak lower semicontinuity result. Section 4 explores the algebraic structure of the projection map is greater detail and Section 5 is devoted to the proof of an instrumental lemma, which singles out the crux of the matter. We conclude the proof of the main theorem in Section 6. Finally, the notations used throughout the article is explained in Section 7.
Preliminaries
=============
Notions of exterior convexity
-----------------------------
We start by recalling the notions of exterior convexity as introduced in [@BDS1].
Let $1\leq k\leq n$ and $f:\Lambda^{k}\rightarrow\mathbb{R}.$
**(i)** We say that $f$ is *ext. one convex*, if the function$$g:t\rightarrow g\left( t\right) =f\left( \xi+t\,\alpha\wedge\beta\right)$$ is convex for every $\xi\in\Lambda^{k},$ $\alpha\in\Lambda^{k-1}$ and $\beta\in\Lambda^{1}.$ If the function $g$ is affine we say that $f$ is *ext. one affine.*
**(ii)** A Borel measurable and locally bounded function $f$ is said to be *ext. quasiconvex*, if the inequality$$\int_{\Omega}f\left( \xi+d\omega\right) \geq f\left( \xi\right)
\operatorname*{meas}\Omega$$ holds for every bounded open set $\Omega \subset \mathbb{R}^{n}$, $\xi\in\Lambda^{k}$ and $\omega\in W_{0}^{1,\infty}\left( \Omega;\Lambda^{k-1}\right)$. If equality holds, we say that $f$ is *ext. quasiaffine.*
**(iii)** We say that $f$ is *ext. polyconvex*, if there exists a convex function$$F:\Lambda^{k}\times\cdots\times\Lambda^{\left[ n/k\right]
k}\rightarrow\mathbb{R}$$ such that$$f\left( \xi\right) =F\left( \xi,\cdots,\xi^{\left[ n/k\right]
}\right),\text{ for all }\xi \in\Lambda^{k}.$$ If $F$ is affine, we say that $f$ is *ext. polyaffine.*
There are analogous notions of interior convexity (cf. [@BDS1]). In what follows, we will discuss the case of exterior convexity only. The case of interior convexity can be derived from the case for exterior convexity by means of Hodge duality.
Projection maps
---------------
To study the relationship between the notions introduced in [@BDS1] and the classical notions of the vectorial calculus of variations namely rank one convexity, quasiconvexity and polyconvexity (see [@DCV2]), we will introduce a projection map. We first introduce some notations. As usual, by abuse of notations, we identify $\Lambda^{k}\left( \mathbb{R}^{n}\right) $ with $\mathbb{R}^{\tbinom{n}{k}}.$
Let $2\leq k\leq n.$ We write a matrix $\Xi\in\mathbb{R}^{\tbinom{n}{k-1}\times n},$ the upper indices being ordered alphabetically, as$$\begin{aligned}
\Xi & =\left(
\begin{array}
[c]{ccc}\Xi_{1}^{1\cdots\left( k-1\right) } & \cdots & \Xi_{n}^{1\cdots\left(
k-1\right) }\\
\vdots & \ddots & \vdots\\
\Xi_{1}^{\left( n-k+2\right) \cdots n} & \cdots & \Xi_{n}^{\left(
n-k+2\right) \cdots n}\end{array}
\right) \medskip\\
& =\left( \Xi_{i}^{I}\right) _{i\in\left\{ 1,\cdots,n\right\} }^{I\in\mathcal{T}^{k-1}}=\left(
\begin{array}
[c]{c}\Xi^{1\cdots\left( k-1\right) }\\
\vdots\\
\Xi^{\left( n-k+2\right) \cdots n}\end{array}
\right) =\left( \Xi_{1},\cdots,\Xi_{n}\right).\end{aligned}$$ We define a linear map $\pi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow
\Lambda^{k}\left( \mathbb{R}^{n}\right) $ in the following way$$\begin{aligned}
\pi\left( \Xi\right) &=\sum_{i=1}^{n}\Xi_{i}\wedge
e^{i} ,\end{aligned}$$ where$$\Xi_{i}=\sum_{1\leq i_{1}<\cdots<i_{k-1}\leq n}\Xi_{i}^{i_{1}\cdots i_{k-1}}\,e^{i_{1}}\wedge\cdots\wedge e^{i_{k-1}}=\sum_{I\in\mathcal{T}^{k-1}}\Xi_{i}^{I}\,e^{I}.$$
Observe that this projection map can also be written as, $$\pi(\Xi) = \sum_{I \in \mathcal{T}^k} \left( \sum_{j \in I} \operatorname*{sgn}(j, I_j) \Xi_j^{I_j} \right) e^I,$$ see 3(vii) in Section \[notations\] for the notations.
1. Note that the map $\pi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow
\Lambda^{k}\left( \mathbb{R}^{n}\right) $ is onto.
2. It is easy to see that $\pi:\mathbb{R}^{n\times n}\rightarrow\Lambda
^{2}\left( \mathbb{R}^{n}\right) $ is given by$$\pi\left( \xi\right) =\sum_{i=1}^{n}\xi_{i}\wedge e^{i}=\sum_{1\leq i<j\leq
n}\left( \xi_{j}^{i}-\xi_{i}^{j}\right) e^{i}\wedge e^{j}, $$ so that, with abuse of notation, $$\pi\left( \xi\right) = \xi -\xi^{T} = 2 \left( \frac{\xi -\xi^{T}}{2} \right).$$ So for $k=2$, $\pi$ is just twice the alternating projection for $2$-tensors (or twice the skew-symmetric projection for square matrices).
Main theorem and consequences
=============================
Main theorem
------------
The main result of the article is the following:
\[Prop ext quasi implique quasi\]Let $2\leq k\leq n$, $f:\Lambda^{k} \rightarrow\mathbb{R}$ and $\pi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow
\Lambda^{k}$ be the projection map. Then the following equivalences hold$$\begin{aligned}
f\text{ ext. one convex }&\Leftrightarrow\text{ }f\circ\pi\text{ rank one
convex. }\\
f\text{ ext. quasiconvex }&\Leftrightarrow\text{ }f\circ\pi\text{ quasiconvex. }\\
f\text{ ext. polyconvex }&\Leftrightarrow\text{ }f\circ\pi\text{ polyconvex. }\end{aligned}$$
**(i)** Note that the theorem does not say that any quasiconvex or rank one convex function $\phi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow\mathbb{R}$ is of the form $f\circ\pi$ with $f$ ext. quasiconvex or ext. one convex as the following example shows. We let $n=k=2$ and$$\phi\left( \Xi\right) =\det\Xi$$ which is clearly polyconvex (and thus quasiconvex and rank one convex). But there is no function $f:\Lambda^{k}\rightarrow\mathbb{R}$ such that $\phi=f\circ\pi.$ Indeed if such an $f$ exists, we arrive at a contradiction, since setting
$$X=\left(
\begin{array}
[c]{cc}1 & 0\\
0 & 1
\end{array}
\right) \quad\text{and}\quad Y=\left(
\begin{array}
[c]{cc}0 & 0\\
0 & 0
\end{array}
\right) ,$$ we have $\pi\left( X\right) =\pi\left( Y\right) =0$ and thus$$1=\phi\left( X\right) =f\left( \pi\left( X\right) \right) =f\left(
\pi\left( Y\right) \right) =\phi\left( Y\right) =0.$$
**(ii)** The following equivalence is, of course, trivially true$$f\text{ convex }\Leftrightarrow\text{ }f\circ\pi\text{ convex.}$$
Relations between notions of exterior convexity
-----------------------------------------------
We now list a few simple consequences of the main theorem.
\[Thm general sur poly quasi ...\]Let $1\leq k\leq n$ and $f:\Lambda
^{k} \rightarrow\mathbb{R}.$ Then$$f\text{ convex }\Rightarrow\text{ }f\text{ ext. polyconvex }\Rightarrow\text{
}f\text{ ext. quasiconvex }\Rightarrow\text{ }f\text{ ext. one convex.}$$
The result is immediate from theorem \[Prop ext quasi implique quasi\] and the classical results (cf. [@DCV2]). Another, more direct proof, without using the classical results, can be found in [@BDS1].
\[Thm principal quasiaffine\]Let $1\leq k\leq n$ and $f:\Lambda^{k}\rightarrow\mathbb{R}.$ The following statements are then equivalent.
**(i)** $f$ is ext. polyaffine.
**(ii)** $f$ is ext. quasiaffine.
**(iii)** $f$ is ext. one affine.
**(iv)** For every $0\leq s\leq\left[
n/k\right]$, there exist $c_{s}\in\Lambda^{ks}$ such that, $$f\left( \xi\right) =\sum_{s=0}^{\left[ n/k\right] }\left\langle c_{s};\xi^{s}\right\rangle,\text{ for every }\xi\in\Lambda^{k}.$$
From the definitions of ext. polyaffine functions, it is clear that $$(i)\;\Leftrightarrow\;(iv).$$ The statements$$(i)\;\Leftrightarrow\;(ii)\;\Leftrightarrow\;(iii)$$ follow at once from classical results (cf. Theorem $5.20$ in [@DCV2]) by virtue of Theorem \[Prop ext quasi implique quasi\].
For a more direct proof of the above, see [@BDS1]. See also [@BS2] for yet another proof.
Let $1\leq k\leq n,$ $1 < p < \infty,$ $\Omega
\subset\mathbb{R}^{n}$ be a bounded smooth open set and $f:\Lambda^{k} \rightarrow\mathbb{R}$ be ext. quasiconvex verifying, for every $\xi\in\Lambda^{k},$$$c_{1}\left( \left\vert \xi\right\vert ^{p}-1\right) \leq f\left(
\xi\right) \leq c_{2}\left( \left\vert \xi\right\vert ^{p}+1\right)$$ for some $c_{1}\,,c_{2}>0.$ if$$\alpha_{s}\rightharpoonup\alpha\quad\text{in }W^{1,p}\left( \Omega
;\Lambda^{k-1}\right)$$ then$$\underset{s\rightarrow\infty}{\lim\inf}\int_{\Omega}f\left( d\alpha
_{s}\right) \geq\int_{\Omega}f\left( d\alpha\right) .$$
According to Theorem \[Prop ext quasi implique quasi\], we have that $f\circ\pi$ is quasiconvex. Then classical results (see Theorem 8.4 in [@DCV2]) show that$$\underset{s\rightarrow\infty}{\lim\inf}\int_{\Omega}f\left( d\alpha
_{s}\right) =\underset{s\rightarrow\infty}{\lim\inf}\int_{\Omega}f\left(
\pi\left( \nabla\alpha_{s}\right) \right) \geq\int_{\Omega}f\left(
\pi\left( \nabla\alpha\right) \right) =\int_{\Omega}f\left( d\alpha
\right)$$ as wished.
Algebraic properties of the projection
======================================
We now start exploring the algebraic structure of the projection map in greater detail. The following properties are easily obtained. See [@sil] for a proof.
\[Prop. proj rang un et quasi\] Let $2\leq k\leq n$ and $\pi:\mathbb{R}^{\tbinom{n}{k-1}\times n}\rightarrow\Lambda^{k}\left( \mathbb{R}^{n}\right) $ be the projection map.
\(i) If $\alpha\in\Lambda^{k-1}\left( \mathbb{R}^{n}\right) \sim
\mathbb{R}^{\tbinom{n}{k-1}}$ and $\beta\in\Lambda^{1}\left( \mathbb{R}^{n}\right) \sim\mathbb{R}^{n},$ then,$$\pi\left( \alpha\otimes\beta\right) =\alpha\wedge\beta.$$
\(ii) Let $\omega\in C^{1}\left( \Omega;\Lambda^{k-1}\right) ,$ then, by abuse of notations,$$\pi\left( \nabla\omega\right) =d\omega.$$
The following result is crucial to establish the main theorem in the case of polyconvexity. See section 5.4 of [@DCV2] for the definition of adjugates and section \[notations\] for the notations.
\[formula\] If $k$ is even, then for $2 \leq s \leq \left[ n/k \right]$, $$\left[ \pi (\Xi)\right]^s = (s!) \sum_{I \in
\mathcal{T}^{sk}} \left( \sum\nolimits^I_s
\operatorname*{sgn}(J;\tilde{I}) (\operatorname*{adj}\nolimits_s \Xi)^{\tilde{I}}_{J} \right) e^I ,$$ and $$\left[ \pi (\Xi)\right]^s = 0, \text{ \quad for } \left[ n/k \right] < s \leq \min\left\{ n, \tbinom{n}{k-1} \right\} .$$ If $k$ is odd, $$\left[ \pi (\Xi)\right]^s = 0, \text{ \quad for all } s,\ 2 \leq s \leq \min\left\{ n, \tbinom{n}{k-1} \right\} .$$
We prove only the first equality. Everything else follows by properties of the wedge power. So we prove the case when $k$ is even and $2 \leq s \leq \left[ n/k \right]$. We prove it by induction.\
*Step 1:* To start the induction, we first prove the case when $s=2$.\
We have, $$\pi(\Xi) = \sum_{I \in \mathcal{T}^k} \left( \sum_{j \in I} \operatorname*{sgn}(j, I_j) \Xi_j^{I_j} \right) e^I.$$ So, $$\begin{aligned}
&[\pi(\Xi)]^2 = \pi(\Xi)\wedge \pi(\Xi) \\
&=\sum_{I \in \mathcal{T}^{2k}}\left(
\begin{aligned}
\sum_{\substack{I^1,\ I^2\\
I^1 \cup I^2 = I \\
I^1 \cap I^2 = \emptyset}}\operatorname*{sgn}\left(I^1,I^2\right) \left(\sum_{j_1 \in I^1} \right.
&\left. \operatorname*{sgn}\left(j_1, I^1_{j_1}\right) \Xi_{j_1}^{I^1_{j_1}}\right) \\
& \left(\sum_{j_2 \in I^2}\operatorname*{sgn}\left(j_2, I^2_{j_2}\right)\Xi_{j_2}^{I^2_{j_2}}\right)
\end{aligned} \right) e^I
\end{aligned}$$
Now, since $k$ is even, we have, $$\begin{aligned}
&[\pi(\Xi)]^2 \\
&\begin{aligned} = 2 \sum_{I \in \mathcal{T}^{2k}} \left( \rule{0in}{20pt} \right. \sum\nolimits^I_2
\left( \right. & \operatorname*{sgn}([j_1, I^1_{j_1}],[j_2, I^2_{j_2}])
\operatorname*{sgn}(j_1,I^1_{j_1})\operatorname*{sgn}(j_2,I^2_{j_2}) \Xi_{j_1}^{I^1_{j_1}}\Xi_{j_2}^{I^2_{j_2}}
\\ &+ \operatorname*{sgn}([j_1,I^2_{j_2}],[j_2, I^1_{j_1}])
\operatorname*{sgn}(j_1,I^2_{j_2})\operatorname*{sgn}(j_2,I^1_{j_1}) \Xi_{j_2}^{I^1_{j_1}}\Xi_{j_1}^{I^2_{j_2}}
\left. \right) \left. \rule{0in}{20pt} \right) e^I \end{aligned}\\
&= 2 \sum_{I \in \mathcal{T}^{2k}} \left( \sum\nolimits^I_2 ( \operatorname*{sgn}(j_1,I^1_{j_1},j_2, I^2_{j_2})
\Xi_{j_1}^{I^1_{j_1}}\Xi_{j_2}^{I^2_{j_2}} + \operatorname*{sgn}(j_1,I^2_{j_2},j_2, I^1_{j_1})\Xi_{j_2}^{I^1_{j_1}}\Xi_{j_1}^{I^2_{j_2}} ) \right)e^I \\
&=2\sum_{I \in \mathcal{T}^{2k}} \left( \sum\nolimits^I_2 \operatorname*{sgn} (j_1,I^1_{j_1},j_2, I^2_{j_2})(
\Xi_{j_1}^{I^1_{j_1}}\Xi_{j_2}^{I^2_{j_2}} - \Xi_{j_2}^{I^1_{j_1}}\Xi_{j_1}^{I^2_{j_2}})\right) e^I \\
&= 2\sum_{I \in \mathcal{T}^{2k}} \left( \sum\nolimits^I_2 \operatorname*{sgn} (j_1,I^1_{j_1},j_2, I^2_{j_2})
(\operatorname*{adj}\nolimits_2\Xi)_{j_1j_2}^{I^1_{j_1}I^2_{j_2}}\right) e^I ,
\end{aligned}$$ which proves the case for $s=2$.\
*Step 2:* We assume the result to be true for some $s \geq 2$ and show that it holds for $s+1$, thus completing the induction. Now we know, by Laplace expansion formula for the determinants, $$\begin{aligned}
(\operatorname*{adj}\nolimits_{s+1} \Xi)^{I^1\ldots I^{s+1}}_{j_1\ldots j_{s+1}} &= \sum_{m=1}^{s+1} \Xi_{j_l}^{I^m} (-1)^{l+m}
(\operatorname*{adj}\nolimits_{s} \Xi)^{I^1\ldots \widehat{I^m}\ldots I^{s+1}}_{j_1\ldots \widehat{j_l}\ldots j_{s+1}} , \textrm{ for any } 1 \leq l \leq s+1\\
&= \frac{1}{s+1}\sum_{l=1}^{s+1}\sum_{m=1}^{s+1} \Xi_{j_l}^{I^m} (-1)^{l+m}
(\operatorname*{adj}\nolimits_{s} \Xi)^{I^1\ldots \widehat{I^m}\ldots I^{s+1}}_{j_1\ldots \widehat{j_l}\ldots j_{s+1}} .\end{aligned}$$ Note that, for any $1 \leq l,m \leq s+1$, $$\operatorname*{sgn}(j_1,I^1,\ldots, j_{s+1}, I^{s+1}) = (-1)^{\{ (l-1)+(m-1)(k-1) \}}\operatorname*{sgn}(j_l, I^m,\tilde{I}^{l,m}),$$ where $\tilde{I}^{l,m}$ is a shorthand for the permutation $(\tilde{j}_1, \tilde{I}^1,\ldots, \tilde{j}_s, \tilde{I}^s)$ and
- $\tilde{j}_1 < \ldots < \tilde{j}_s$ and $\{ \tilde{j}_1 , \ldots , \tilde{j}_s\} = \{ j_1, \ldots,
\widehat{j_l}, \ldots, j_{s+1} \}. $
- $\tilde{I}^1 < \ldots < \tilde{I}^s$ and $\{ \tilde{I}^1 , \ldots , \tilde{I}^s\} = \{ I^1, \ldots,
\widehat{I^m}, \ldots, I^{s+1} \}$.
Note that this means $\tilde{j}_r = j_r$ for $1 \leq r < l$ and $\tilde{j}_r = j_{r+1}$ for $l \leq r
\leq s$. Similarly, $\tilde{I}^r = I^r$ for $1 \leq r < m$ and $\tilde{I}^r = I^{r+1}$ for $m \leq r \leq s$. Now since $k$ is even, for any $1 \leq l,m \leq s+1$, $$\begin{aligned}
&\operatorname*{sgn}(j_1, I^1,\ldots, j_{s+1}, I^{s+1}) = (-1)^{l+m}\operatorname*{sgn}(j_l, I^m,\tilde{I}^{l,m}) \\
&=(-1)^{l+m}\operatorname*{sgn}(j_l, I^m)\operatorname*{sgn}(\tilde{I}^{lm})\operatorname*{sgn}([j_l, I^m],[\tilde{I}^{l,m}]).
\end{aligned}$$ Thus, $$\begin{aligned}
&\operatorname*{sgn}(j_1, I^1,\ldots, j_{s+1}, I^{s+1})(\operatorname*{adj}\nolimits_{s+1} \Xi)^{I^1\ldots I^{s+1}}_{j_1\ldots j_{s+1}} \\
&=\frac{1}{(s+1)}\sum_{l,m =1}^{s+1}\operatorname*{sgn}([j_l, I^m],[\tilde{I}^{l,m}])\operatorname*{sgn}(j_l, I^m)\Xi_{j_l}^{I^m}
\operatorname*{sgn}(\tilde{I}^{lm})(\operatorname*{adj}\nolimits_{s} \Xi)^{I^1\ldots \widehat{I^m}\ldots I^{s+1}}_{j_1\ldots \widehat{j_l}\ldots j_{s+1}}.\end{aligned}$$ Hence, [$$\begin{aligned}
& (s+1)! \sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \sum\nolimits^I_{s+1}
\operatorname*{sgn}(j_1,I^1,\ldots , j_{s+1}, I^{s+1}) (\operatorname*{adj}\nolimits_{s+1} \Xi)^{I^1\ldots I^{s+1}}_{j_1\ldots j_{s+1}} \right) e^I\\
&=\frac{(s+1)!}{(s+1)} \sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \rule{0in}{40pt}
\right. \begin{aligned} \sum\nolimits^I_{s+1}\sum_{l,m =1}^{s+1}\operatorname*{sgn}([j_l, I^m],[\tilde{I}^{l,m}])
\operatorname*{sgn}(j_l, I^m)\Xi_{j_l}^{I^m} \\
\operatorname*{sgn}(\tilde{I}^{lm})(\operatorname*{adj}\nolimits_{s} \Xi)^{I^1\ldots
\widehat{I^m}\ldots I^{s+1}}_{j_1\ldots \widehat{j_l}\ldots j_{s+1}} \end{aligned}\left.\rule{0in}{40pt} \right) e^I \\
&=(s!)\sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \rule{0in}{40pt} \right. \begin{aligned} \sum\nolimits^I_{s+1}\sum_{l,m =1}^{s+1}\operatorname*{sgn}([j_l, I^m],[\tilde{I}^{l,m}])
\operatorname*{sgn}(j_l, I^m)\Xi_{j_l}^{I^m} \\
\operatorname*{sgn}(\tilde{I}^{lm})(\operatorname*{adj}\nolimits_{s} \Xi)^{I^1\ldots
\widehat{I^m}\ldots I^{s+1}}_{j_1\ldots \widehat{j_l}\ldots j_{s+1}} \end{aligned}\left. \rule{0in}{40pt} \right) e^I \\
&=\sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \rule{0in}{40pt} \right. \begin{aligned} \sum_{\substack{I' \subset I\\ I' \in \mathcal{T}^k}}& \left( \rule{0in}{20pt} \right. \operatorname*{sgn} (I',[I \backslash I'])
(\sum_{j \in I'} \operatorname*{sgn}(j, I'_j)\Xi_{j}^{I'_j}) \\ & \left( s! \left( \sum\nolimits^{[I\backslash I']}_s
\operatorname*{sgn}(\tilde{j}_1,\tilde{I}^1,\ldots , \tilde{j}_s, \tilde{I}^s)
(\operatorname*{adj}\nolimits_s \Xi)^{\tilde{I}^1\ldots \tilde{I}^s}_{\tilde{j}_1\ldots \tilde{j}_s} )\right) \right) \end{aligned}
\left.\rule{0in}{40pt} \right) e^I ,\end{aligned}$$ ]{}
where the last line is just a rewriting of the penultimate one. Indeed on expanding the sums the map, sending $j_{l}$ to $j$; $I^{m}$ to $I'_{j}$; $I^{1}, \ldots,\widehat{I^{m}}, \ldots , I^{s+1}$ to $\tilde{I}^{1}, \ldots ,\tilde{I}^{s}$ respectively and $j_1, \ldots ,\widehat{j_l} ,\ldots ,j_{s+1}$ to $\tilde{j}_{1}, \ldots , \tilde{j}_{s}$ respectively is a bijection between the terms on the two sides of the last equality.
So, we have, by induction hypothesis, [$$\begin{aligned}
& (s+1)! \sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \sum\nolimits^I_{s+1}
\operatorname*{sgn}(j_1,I^1,\ldots , j_{s+1}, I^{s+1}) (\operatorname*{adj}\nolimits_{s+1} \Xi)^{I^1\ldots I^{s+1}}_{j_1\ldots j_{s+1}} \right) e^I \nonumber \\
&=\sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \rule{0in}{40pt} \right.
\begin{aligned} \sum_{\substack{I' \subset I\\ I' \in \mathcal{T}^k}} &
\operatorname*{sgn} (I',[I \backslash I'])
\left( \rule{0in}{20pt} \text{ coefficient of } e^{I'} \text{ in }
\pi (\Xi) \right) \\
& \times \left( \rule{0in}{20pt} \right. \text{ coefficient of } e^{[I \backslash I']} \text{ in }
\left[ \pi (\Xi)\right]^s \left.\rule{0in}{20pt} \right)
\end{aligned}
\left. \rule{0in}{40pt} \right) e^I \\
&= \sum_{I \in
\mathcal{T}^{(s+1)k}} \left( \text{ coefficient of } e^I \text{ in } \left[ \pi (\Xi)\right]^{s+1} \right) e^I = \left[ \pi (\Xi)\right]^{s+1} .\end{aligned}$$ ]{} This completes the induction proving the desired result.
Since we have seen that $\left[ \pi (\Xi)\right]^s$ depends only on $\operatorname*{adj}\nolimits_s \Xi$, we are now in a position to define a linear projection for every value of $s$. These maps will be useful later.
For every $2 \leq s \leq \min\left\{ n, \tbinom{n}{k-1} \right\}$, we define the linear projection maps $\pi_s:\mathbb{R}^{\tbinom{\tbinom{n}{k-1}}{s}\times \tbinom{n}{s}}\rightarrow \Lambda^{ks}(\mathbb{R}^n)$ by the condition, $$\pi_s(\operatorname*{adj}\nolimits_s(\Xi)) = \left[ \pi (\Xi)\right]^s \text{ for all } \Xi \in \mathbb{R}^{\tbinom{n}{k-1}\times n}.$$
It is clear that this condition uniquely defines the projection maps. For the sake of consistency, we define, $ \pi_1 = \pi $ and $\pi_0 $ is defined to be the identity map from $\mathbb{R}$ to $\mathbb{R}$.
An important lemma
==================
\[polyconvexitylemma\] Let $2 \leq k \leq n$ and $N = \tbinom{n}{k-1} $. Consider the function $$g(X,d)=f(\pi(X)) - \sum_{s=1}^{\min\left\{ N, n \right\}} \left\langle d_s, \operatorname*{adj}\nolimits_s X \right\rangle$$ where $d = ( d_1,\ldots, d_{\min\left\{ N, n \right\}} )$, $d_s \in \mathbb{R}^{\tbinom{N}{s} \times \tbinom{n}{s}}$ for all $1 \leq s \leq \min\left\{ N, n \right\}$ and $X \in \mathbb{R}^{N \times n}$. If for a given vector $d$, the function $X \mapsto g(X,d)$ achieves a minimum over $\mathbb{R}^{N \times n}$, then for all $ 1 \leq s \leq \min\left\{ N, n \right\}$, there exists $\mathcal{D}_{s} \in \Lambda^{ks}$ such that, $$\langle d_s , \operatorname*{adj}\nolimits_{s} Y \rangle = \langle \mathcal{D}_s , \pi_s(\operatorname*{adj}\nolimits_{s} Y) \rangle \text{ \quad for all } Y \in \mathbb{R}^{N \times n}.$$
The lemma is technical and quite heavy in terms of notations. So before proceeding to prove the lemma as stated, it might be helpful to spell out the idea of the proof. The plan is always the same. In short, if the conclusion of the lemma does not hold, we can always choose a matrix $X$ such that $g(X,d)$ can be made to be smaller than any given real number, contradicting the hypothesis that the map $X \mapsto g(X,d)$ assumes a minimum.
Let us fix a vector $d$ and assume that for this $d$, the function $X \mapsto g(X,d)$ achieves a minimum over $\mathbb{R}^{N \times n}$.
We will first show that all adjugates with a common index between subscripts and superscripts must have zero coefficients. More precisely, we claim that,
\[common index have zero coefficients k even\] For any $2 \leq k \leq n$ and for every $1 \leq s \leq \min\left\{ N, n \right\}$, for every $J\in \mathcal{T}^{s}, I = \left\lbrace I^{1}\ldots I^{s} \right\rbrace$ where $ I^{1},\ldots,I^{s} \in \mathcal{T}^{k-1}$,we have, $$\left( d_s \right)^{I}_{J} = 0 \textrm{ whenever }
I\cap J \neq \emptyset.$$
We prove claim \[common index have zero coefficients k even\], using induction over $s$. To start the induction, we first show the case $s=1$. Let $j \in I$, where $I \in \mathcal{T}^{k-1}$. We choose $X = \lambda e^{j}\otimes e^{I} $, then clearly $\pi(X) = 0.$ Also, $g(X,d) = f(0) - \lambda \left(d_1 \right)_{j}^{I}$. By letting $\lambda$ to $+\infty$ and $-\infty$ respectively, we deduce that $\left(d_1 \right)_{j}^{I}= 0$, since otherwise we obtain a contradiction to the fact that $g$ achieves a finite minima.
Now we assume that claim \[common index have zero coefficients k even\] holds for all $ 1 \leq s \leq p $ and prove the result for $s= p+1.$ We consider $ \left( d_{p+1} \right)^{ I^{1} \ldots I^{p+1} }_{ j_{1} \ldots j_{p+1} }$ with $ j_l \in I^m \textrm{ for some } 1 \leq l,m \leq p+1.$
Now we first order the rest of the indices (other than the common index) in subscripts and the rest of the multiindices (other than the one with the common index) in superscripts. Let $\tilde{I}^{1} < \ldots < \tilde{I}^{p}$ and $ \tilde{j}_{1}< \ldots < \tilde{j}_{p}$ represent the multiindices and indices in the sets $\left\lbrace I^{1}, \ldots , I^{p+1} \right\rbrace\setminus \lbrace I^{m}\rbrace $ and $\left\lbrace j_{1}, \ldots , j_{p+1}\right\rbrace\setminus \lbrace j_{l}\rbrace $ respectively.
Now we choose, $$X = \lambda e^{j_{l}}\otimes e^{I^{m}} + \sum_{r=1}^{p} e^{\tilde{j}_r}\otimes e^{\tilde{I}^r}.$$ Since $j_{l} \in I^{m}$, we get $\pi(X)$ is independent of $\lambda$. Also, all lower order non-constant adjugates of $X$ must contain the index $ j_{l}$ both in subscript and in superscript and hence their coefficients are $0$ by the induction hypothesis. Hence, the only non-constant adjugate of $X$ appearing in the expression for $g(X,d)$ is, $$\left( \operatorname*{adj}\nolimits_{p+1} X \right)^{I^{1} \ldots I^{p+1} }_{j_{1} \ldots j_{p+1} } = \left( -1 \right)^{\alpha}\lambda,$$ where $\alpha$ is a fixed integer. Now, $$g(X,d) = \left( -1 \right)^{\alpha+1}\lambda \left( d_{p+1} \right)^{ I^{1} \ldots I^{p+1} }_{ j_{1} \ldots j_{p+1} } + \textrm{ constants }.$$ Again as before, we let $\lambda$ to $+\infty$ and $-\infty$ and we deduce, by the same argument, $\left( d_{p+1} \right)^{ I^{1} \ldots I^{p+1} }_{ j_{1} \ldots j_{p+1} } =0.$ This completes the induction and proves the claim.
At this point we split the proof in two cases, the case when $k$ is an even integer and the case when $k$ is an odd integer.
**Case 1: $\mathbf{ k}$ is even**
Note that, unless $k=2$, it does not follow from above that $d_s = 0$ for all $s \geq [\frac{n}{k}]$. The possibility that two different blocks of multiindices in the superscript have some index in common has not been ruled out. Now we will show that the coefficients of two different adjugates having the same set of indices are related in the following way:
\[interchange of indices k even\] For every $ s \geq 1$, $$\operatorname*{sgn}(J;I)\left( d_s \right)^{I}_{J}
= \operatorname*{sgn}(\tilde{J}; \tilde{I})\left( d_s \right)^{ \tilde{I}}_{ \tilde{J}},$$ whenever $ J \cup I = \tilde{J} \cup \tilde{I} ,$ with $J, \tilde{J} \in \mathcal{T}^{s}$ , $I = \left\lbrace I^{1}\ldots I^{s}\right\rbrace = \left[ I^{1}, \ldots ,I^{s}\right]$, $\tilde{I} = \left\lbrace \tilde{I}^{1}\ldots \tilde{I}^{s}\right\rbrace= [ \tilde{I}^{1}, \ldots , \tilde{I}^{s}]$, $I^{1},\ldots ,I^{s},\tilde{I}^{1},\ldots ,\tilde{I}^{s} \in \mathcal{T}^{k-1}$ and $J\cap I = \emptyset.$ In particular, given any $U \in \mathcal{T}^{ks}$, there exists a constant $\mathcal{D}_{U} \in \mathbb{R}$ such that, $$\label{DI}
\operatorname*{sgn}(J;I)\left( d_s \right)^{I}_{J}= \mathcal{D}_{U},$$ for all $J \cup I = U$ with $J \in \mathcal{T}^{s}$ , $I = \left\lbrace I^{1}\ldots I^{s}\right\rbrace = \left[ I^{1}, \ldots ,I^{s}\right]$, $I^{1},\ldots ,I^{s} \in \mathcal{T}^{k-1}$.
We will prove the claim again by induction over $s$. We first prove it for the case $s=1.$
For the case $s=1$, we just need to prove, for any index $j$, any multindex $I \in \mathcal{T}^{k-1}$ such that $j \cap I = \emptyset$, we have $$\label{s equal to one k even}
\operatorname*{sgn}(j,I)\left( d_1 \right)^{ I}_{ j }
= \operatorname*{sgn}(\tilde{j},\tilde{I})\left( d_1 \right)^{ \tilde{I}}_{\tilde{j}} ,$$ where $[j,I]=[\tilde{j},\tilde{I}].$ We choose $X = \lambda \operatorname*{sgn}(j,I) e^{j}\otimes e^{I} - \lambda \operatorname*{sgn}(\tilde{j},\tilde{I}) e^{\tilde{j}}\otimes e^{\tilde{I}}.$ Clearly, $\pi(X)= 0$ and this gives, $$g(X,d) = f(0) + \lambda \left( \operatorname*{sgn}(j,I) \left( d_1 \right)^{ I}_{ j } - \operatorname*{sgn}(\tilde{j},\tilde{I}) \left( d_1 \right)^{\tilde{I}}_{\tilde{j}} \right),$$ where we have used claim \[common index have zero coefficients k even\] to deduce that $\left(d_2\right)^{[I\tilde{I}]}_{[j\tilde{j}]}=0$. Letting $\lambda$ to $+\infty$ and $-\infty$, we get (\[s equal to one k even\]).
Now we assume the result for all $1 \leq s \leq s_{0}$ and show it for $s= s_{0}+1.$ Let $ [ I^{1} \ldots I^{s_{0}+1} j_{1} \ldots j_{s_{0}+1} ] =
[ \tilde{I}^{1} \ldots \tilde{I}^{s_{0}+1} \tilde{j}_{1} \ldots \tilde{j}_{s_{0}+1}] $. Note that the sets $\left\lbrace I^{1} \ldots I^{s_{0}+1} j_{1} \ldots j_{s_{0}+1}\right\rbrace$ and $\left\lbrace\tilde{I}^{1} \ldots \tilde{I}^{s_{0}+1} \tilde{j}_{1} \ldots \tilde{j}_{s_{0}+1}\right\rbrace$ are permutations of each other, preserving an order relation given by $ j_{1}< \ldots < j_{s_{0}+1}$, $ \tilde{j}_{1}< \ldots < \tilde{j}_{s_{0}+1}$, $I^{1}< \ldots < I^{s_{0}+1}$ and $\tilde{I}^{1} < \ldots < \tilde{I}^{s_{0}+1}$. Thus the aforementioned sets can be related by any permutation (of $k(s_{0}+1)$ indices) that respects this order. Since any such permutation is a product of $k$-flips, it is enough to prove the claim in case of $k$-flips, cf. definition \[kflips\].
We now assume $(J , I)$ and $(\tilde{J} , \tilde{I})$ are related by a $k$-flip interchanging the subscript $j_l$ with one index in the superscript block $I^m$ and keep all the other indices unchanged. Also, we assume that after the interchange, the position of the multiindex containing $j_l$ in the superscript is $p$ and the new position of the index from the multiindex $I^m$ in the subscript is $q$, i.e, $j_l \in \tilde{I}^p$ and $\tilde{j}_q \in I^m.$ We also order the remaining indices and assume , $$\breve{I} = [ \breve{I}^1 , \ldots , \breve{I}^{s_{0}}] = \lbrace \breve{I}^1 \ldots \breve{I}^{s_{0}}\rbrace
= \left\lbrace I^{1} \ldots \widehat{I^{m}} \ldots I^{s_{0}+1} \right\rbrace ,$$ and $$\breve{J}= [\breve{j}_1 \ldots \breve{j}_{s_{0}}]=\lbrace\breve{j}_1 \ldots \breve{j}_{s_{0}}\rbrace
=\left\lbrace j_{1} \ldots \widehat{j_{l}} \ldots j_{s_{0}+1}\right\rbrace$$ respectively. Now we choose, $$X = \lambda \operatorname*{sgn}(j_{l} , I^{m}) e^{j_l}\otimes e^{I^m} - \lambda \operatorname*{sgn}(\tilde{j}_{q} ,
\tilde{I}^{p}) e^{\tilde{j}_{q}}\otimes e^{\tilde{I}^{p}} + \sum_{1 \leq r \leq s_{0}} e^{\breve{j}_r}\otimes e^{\breve{I}_r}.$$
Note that $\pi(X)$ is independent of $\lambda$. Also, all non-constant adjugates of $X$ appearing with possibly non-zero coefficients in the expression for $g(X,d)$ have, either $j_{l}$ in subscript and $I^{m} $ in superscript or has $\tilde{j}_{q} $ as a subscript and $\tilde{I}^{p} $ as a superscript, but never both as then they have zero coefficients by claim \[common index have zero coefficients k even\]. Also, these adjugates occur in pairs. More precisely, for every non-constant adjugate of $X$ appearing with possibly non-zero coefficients in the expression for $g(X,d)$ having $j_{l}$ in subscript and $I^{m} $ in superscript, there is one having $\tilde{j}_{q} $ in subscript and $\tilde{I}^{p} $ in superscript.
Let us show that, for any $1 \leq s \leq s_{0}+1$, any subset $ \bar{J}_{s-1} = \left\lbrace \bar{j}_{1}, \ldots , \bar{j}_{s-1} \right\rbrace
\subset \breve{J} $ of $s$ indices and any choice of of $s-1$ multiindices $ \bar{I}^{1}, \ldots , \bar{I}^{s-1} $ out of $s_{0}$ multiindices $\breve{I}^1 , \ldots , \breve{I}^{s_{0}}$, we have,
$$\label{formula sign adjugate k even}
\frac{\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{ [ j_{l}\bar{J}_{s-1} ] }}
{\operatorname*{sgn}([ j_{l}\bar{J}_{s-1} ] ;[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] ) }
= -\frac{\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{
[ \tilde{j}_{q} \bar{J}_{s-1} ] }}
{\operatorname*{sgn}( [ \tilde{j}_{q} \bar{J}_{s-1} ] ; [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ])} .$$
Let $a_1$ be the position of $j_l$ in $\left[ j_{l}\bar{J}_{s-1}\right]$ , $a_2$ be the position of $ \tilde{j}_{q}$ in $\left[ \tilde{j}_{q} \bar{J}_{s-1} \right]$, $b_1$ be the position of $ I^{m}$ in $[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ]$ and $b_2 $ be the position of $\tilde{I}^{p}$ in $\left[ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} \right]$.
Since $k$ is even, $$\begin{aligned}
\operatorname*{sgn} &([ j_{l}\bar{J}_{s-1} ] ;[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] ) \notag \\
&= (-1)^{\lbrace ( a_{1}-1) +(b_{1}-1) \rbrace}
\operatorname*{sgn}(j_{l},I^{m})\operatorname*{sgn} ( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) \notag\\
&\qquad \qquad \qquad \qquad \operatorname*{sgn}([j_{l}, I^{m}], [( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) ]),\end{aligned}$$ and $$\begin{aligned}
\operatorname*{sgn} &([ \tilde{j}_{q}\bar{J}_{s-1} ] ;[ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] ) \notag \\
&= (-1)^{\lbrace ( a_{2}-1) +(b_{2}-1) \rbrace}
\operatorname*{sgn}(\tilde{j}_{q},\tilde{I}^{p})\operatorname*{sgn} ( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) \notag\\
&\qquad \qquad \qquad \qquad \operatorname*{sgn}([\tilde{j}_{q}, \tilde{I}^{p}], [( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) ]).\end{aligned}$$ We also have, $$\begin{aligned}
\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{ [ j_{l}\bar{J}_{s-1} ] }
= (-1)^{a_{1}+b_{1}}\operatorname*{sgn}(j_{l},I^{m}) \lambda
\left( \operatorname*{adj}\nolimits_{s-1} X \right)^{[ \bar{I}^{1}, \ldots , \bar{I}_{s-1}]}_{[\bar{J}_{s-1}]},\end{aligned}$$ and $$\begin{aligned}
\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{
[ \tilde{j}_{q} \bar{J}_{s-1} ] }
= - (-1)^{a_{2}+b_{2}}\operatorname*{sgn}(\tilde{j}_{q},\tilde{I}^{p}) \lambda
\left( \operatorname*{adj}\nolimits_{s-1} X \right)^{[ \bar{I}^{1}, \ldots , \bar{I}_{s-1}]}_{[\bar{J}_{s-1}]}.\end{aligned}$$ Combining the four equations above, the result follows.
We now finish the proof of claim \[interchange of indices k even\]. Using , we have, $$\begin{aligned}
&g(X,d) = \lambda\left\lbrace \vphantom{\begin{aligned}
\operatorname*{sgn} &(\left[ j_{l}\bar{J}_{s-1}\right];\left[ i_{m}\bar{I}_{s-1} \right])
\left( d_{s}\right)^{\left[ i_{m}\bar{I}_{s-1} \right]}_{\left[ j_{l}\bar{J}_{s-1}\right]} \\
&-\operatorname*{sgn}(\left[ \tilde{j}_{q} \bar{J}_{s-1} \right];\left[\tilde{i}_{p} \bar{I}_{s-1} \right])
\left( d_{s}\right)^{\left[\tilde{i}_{p} \bar{I}_{s-1} \right]}_{\left[ \tilde{j}_{q} \bar{J}_{s-1} \right]}
\end{aligned}} \right.
(-1)^{\alpha} \left( \operatorname*{sgn}(J;I)\left( d_{s_{0}+1} \right)^{I}_{J}
- \operatorname*{sgn}(\tilde{J}; \tilde{I})\left( d_{s_{0}+1} \right)^{ \tilde{I}}_{ \tilde{J}}\right) \\
& + \sum_{s=1}^{s_{0}} \sum\nolimits^{s} k_{s,\gamma}
\left( \begin{aligned}
& \operatorname*{sgn} ([ j_{l}\bar{J}_{s-1} ] ;[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] )
\left( d_{s}\right)^{ [ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ]}_{\left[ j_{l}\bar{J}_{s-1}\right]} \\
&-\operatorname*{sgn} ([ \tilde{j}_{q}\bar{J}_{s-1} ] ;[ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] )
\left( d_{s}\right)^{ [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ]}_{\left[ \tilde{j}_{q} \bar{J}_{s-1} \right]}
\end{aligned} \right)
\left. \vphantom{\begin{aligned}
\operatorname*{sgn} &(\left[ j_{l}\bar{J}_{s-1}\right];\left[ i_{m}\bar{I}_{s-1} \right])
\left( d_{s}\right)^{\left[ i_{m}\bar{I}_{s-1} \right]}_{\left[ j_{l}\bar{J}_{s-1}\right]} \\
&-\operatorname*{sgn}(\left[ \tilde{j}_{q} \bar{J}_{s-1} \right];\left[\tilde{i}_{p} \bar{I}_{s-1} \right])
\left( d_{s}\right)^{\left[\tilde{i}_{p} \bar{I}_{s-1} \right]}_{\left[ \tilde{j}_{q} \bar{J}_{s-1} \right]}
\end{aligned}}\right\rbrace \\
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \textrm{ constants},
\end{aligned}$$
where $\sum\nolimits^{s} $ is a shorthand, for every $1 \leq s \leq s_{0}$, for the sum over all possible such choices of $\bar{J}_{s-1}, \bar{I}^{1}, \bar{I}^{2}, \ldots , \bar{I}^{s-1} $ and $k_{s,\gamma}$ is a generic placeholder for the constants appearing before each term of the sum and $\alpha$ is an integer.
By the induction hypothesis, the sum on the right hand side of the above expression is $0$. Hence, we obtain, $$g(X,d) = (-1)^{\alpha} \lambda \left( \operatorname*{sgn}(J;I)\left( d_{s_{0}+1} \right)^{I}_{J}
- \operatorname*{sgn}(\tilde{J}; \tilde{I})\left( d_{s_{0}+1} \right)^{ \tilde{I}}_{ \tilde{J}}\right) + \textrm{ constants }.$$ Letting $\lambda$ to $+\infty$ and $-\infty$, the claim is proved by induction.
Note that by virtue of claim \[interchange of indices k even\], claim \[common index have zero coefficients k even\] now implies, that for every $1 \leq s \leq \min\left\{ N, n \right\}$ , for every $J\in \mathcal{T}^{s}, I = \left\lbrace I^{1}\ldots I^{s} \right\rbrace$ where $ I^{1},\ldots,I^{s} \in \mathcal{T}^{k-1}$, we have, $$\label{distinct indices}
\left( d_s \right)^{I}_{J} = 0 \textrm{ whenever either }
I\cap J \neq \emptyset \textrm{ or } I^{l}\cap I^{m} \neq \emptyset \textrm{ for some } 1 \leq l < m \leq s.$$ Indeed, if $I \cap J \neq \emptyset$, we are done, using claim \[common index have zero coefficients k even\]. So let us assume $I\cap J = \emptyset$ but $ I^{l}\cap I^{m} \neq \emptyset$ for some $1 \leq l < m \leq s$. Then there exists an index $i$ such that $i \in I^{l}$ and $i \in I^{m}$, we consider the $k$-flip interchanging some index $j$ from subscript with the index $i$ in $I^{l}$. More precisely, let $\tilde{J} \in \mathcal{T}^{s}$ and $\tilde{I}^{l} \in \mathcal{T}^{k-1}$ be such that $i \in \tilde{J}$, $\tilde{J}\setminus \lbrace i \rbrace \subset J$, $I^{l}\setminus \lbrace i \rbrace\subset \tilde{I}^{l}$ and $J\cup I^{l} = \tilde{J}\cup \tilde{I}^{l}$, then by claim \[interchange of indices k even\] we have, $$\operatorname*{sgn}(J;I)\left( d_s \right)^{I}_{J}
= \operatorname*{sgn}\left(\tilde{J}; \left[ \tilde{I}^{l},
I^{1},\ldots,\widehat{I^{l}},\ldots ,I^{s}\right]\right)\left( d_s \right)^{\left[ \tilde{I}^{l}, I^{1},\ldots,\widehat{I^{l}},\ldots , I^{s}\right] }_{ \tilde{J}}.$$ Since, $i \in \tilde{J}$ and $i \in I^{m}$, $\tilde{J}\cap \left[ \tilde{I}^{l},I^{1},\ldots,\widehat{I^{l}},\ldots , I^{s}\right] \neq \emptyset$, the right hand side of above equation is $0$ and so $\left( d_s \right)^{I}_{J} = 0$, which proves . So this now implies, $d_s = 0$ for all $s \geq [\frac{n}{k}]$. Hence we have, using , and proposition \[formula\], [$$\begin{aligned}
\langle d_s , \operatorname*{adj}\nolimits_{s} Y \rangle &= \sum_{I \in
\mathcal{T}^{sk}} \sum\nolimits^I_s (d_s)^{\tilde{I}}_{J} ( \operatorname*{adj}\nolimits_{s} Y )^{\tilde{I}}_{J} \\
&= \sum_{I \in
\mathcal{T}^{sk}} \sum\nolimits^I_s \operatorname*{sgn}(J;\tilde{I}) (d_s)^{\tilde{I}}_{J} \operatorname*{sgn}(J;\tilde{I})
( \operatorname*{adj}\nolimits_{s} Y )^{\tilde{I}}_{J} \\
&= \sum_{I \in
\mathcal{T}^{sk}} \frac{1}{s!} \mathcal{D}_{I} \sum\nolimits^I_s (s!) \operatorname*{sgn}(J;\tilde{I})
( \operatorname*{adj}\nolimits_{s} Y )^{\tilde{I}}_{J} \\
&= \langle \mathcal{D}_{s}, \pi_{s} (\operatorname*{adj}\nolimits_{s} Y ) \rangle ,\end{aligned}$$ ]{} where $ \displaystyle \mathcal{D}_{s} = \frac{1}{s!} \sum_{I \in \mathcal{T}^{sk}} \mathcal{D}_{I} e^{I}$, which finishes the proof when $k$ is even.
**Case 3: $\mathbf{ k}$ is odd**
In this case, by proposition \[formula\], it is enough to show that all coefficients of all terms, except the linear ones must be zero. As in the case above, the plan is to establish a relation between the coefficients of two different adjugates having the same set of indices. But when $k$ is odd, the relationship is not as nice as in the even case and as such there is no general formula. However, we still have a weaker analogue of claim \[interchange of indices k even\] for the case of $k$-flips.
\[interchange of indices k odd\] For $ s \geq 1 $, if $J, \tilde{J} \in \mathcal{T}^{s}$, and $I^{1}\ldots ,I^{s},
\tilde{I}^{1},\ldots ,\tilde{I}^{s} \in \mathcal{T}^{k-1}$, where $J = \lbrace j_{1} \ldots j_{s}\rbrace$, $ \tilde{J} = \lbrace\tilde{j}_{1} \ldots \tilde{j}_{s}\rbrace $, $I = \left\lbrace I^{1}\ldots I^{s}\right\rbrace = \left[ I^{1}, \ldots ,I^{s}\right]$ and $\tilde{I} = \left\lbrace \tilde{I}^{1}\ldots \tilde{I}^{s}\right\rbrace= [ \tilde{I}^{1}, \ldots , \tilde{I}^{s}]$ be such that $J\cap I = \emptyset$ and $(J , I)$ and $ (\tilde{J} , \tilde{I})$ are related by a $k$-flip interchanging an index $j_l$ in the subscript with one from the multiindex $I^m$ in the superscript. Also, we assume that after the interchange, the position of the multiindex containing $j_l$ in the superscript is $p$ and the new position of the index from the multiindex $I^m$ in the subscript is $q$ , i.e , $ j_l \in \tilde{I}^p $ and $ \tilde{j}_q \in I^m.$
Then we have,
$$\operatorname*{sgn}(J;I)\left( d_s \right)^{I}_{J}
= (-1)^{(m-p)}\operatorname*{sgn}(\tilde{J}; \tilde{I})\left( d_s \right)^{ \tilde{I}}_{ \tilde{J}}.$$
Since the proof of claim \[interchange of indices k odd\] is very similar to that of claim \[interchange of indices k even\], we shall indicate only a brief sketch of the proof. Since $k$ is odd, we deduce, $$\begin{gathered}
\operatorname*{sgn} ([ j_{l}\bar{J}_{s-1} ] ;[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] ) \\
= (-1)^{\lbrace ( a_{1}-1) \rbrace}
\operatorname*{sgn}(j_{l},I^{m})\operatorname*{sgn} ( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) \\
\operatorname*{sgn}([j_{l}, I^{m}], [( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) ]) ,\end{gathered}$$ $$\begin{gathered}
\operatorname*{sgn} ([ \tilde{j}_{q}\bar{J}_{s-1} ] ;[ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] ) \\
= (-1)^{\lbrace ( a_{2}-1) \rbrace}
\operatorname*{sgn}(\tilde{j}_{q},\tilde{I}^{p})\operatorname*{sgn} ( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1}\rbrace )\\
\operatorname*{sgn}([\tilde{j}_{q}, \tilde{I}^{p}], [( \bar{J}_{s-1}; \lbrace \bar{I}^{1} \ldots \bar{I}^{s-1} \rbrace ) ]),\end{gathered}$$ and hence, in a manner analogous to the proof of , we have, $$\begin{gathered}
\label{formula sign adjugate k odd}
\frac{\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{ [ j_{l}\bar{J}_{s-1} ] }}
{\operatorname*{sgn}([ j_{l}\bar{J}_{s-1} ] ;[ I^{m}, \bar{I}^{1},\ldots , \bar{I}^{s-1} ] ) } \\
= - (-1)^{(b_{1}-b_{2})}
\frac{\left(\operatorname*{adj}\nolimits_{s} X \right)^{ [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ] }_{
[ \tilde{j}_{q} \bar{J}_{s-1} ] }}
{\operatorname*{sgn}( [ \tilde{j}_{q} \bar{J}_{s-1} ] ; [ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ])} ,\end{gathered}$$ for any $1 \leq s \leq s_{0}+1$, any subset $ \bar{J}_{s-1} = \left\lbrace \bar{j}_{1}, \ldots , \bar{j}_{s-1} \right\rbrace
\subset \breve{J} $ of $s-1$ indices and any choice of of $s$ multiindices $ \bar{I}^{1}, \ldots , \bar{I}^{s-1} $ out of $s_{0}+1$ multiindices, where $a_1$ is the position of $j_l$ in $\left[ j_{l}\bar{J}_{s-1}\right]$ , $a_2$ is the position of $ \tilde{j}_{q}$ in $\left[ \tilde{j}_{q} \bar{J}_{s-1} \right]$, $b_1$ is the position of $ I^{m}$ in $[ I^{m}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} ]$ and $b_2 $ is the position of $\tilde{I}^{p}$ in $\left[ \tilde{I}^{p}, \bar{I}^{1}, \ldots , \bar{I}^{s-1} \right]$. Claim \[interchange of indices k odd\] follows from above.
Note that claim \[interchange of indices k odd\] and claim \[common index have zero coefficients k even\] together now rule out the possibility that an adjugate with non-zero coefficient can have common indices between the blocks of multiindices in the superscript and proves $ d_s = 0 $ for all $s > [\frac{n}{k}]$. Furthermore, by claim \[interchange of indices k odd\], the coefficients of any two adjugates $\left( d_{s} \right)^{I}_{J} ,
\left( d_{s} \right)^{\tilde{I}}_{\tilde{J}} $ such that $ I\cup J =\tilde{I}\cup\tilde{J}$, can differ only by a sign. So clearly, all of them must be $0$ if one of them is. So without loss of generality, we shall restrict our attention to the coefficient of a particularly ordered adjugates, one with all distinct indices in subscript and superscripts , for which $ j_1< \ldots <j_s <i_{1}^{1} <\ldots <i_{k-1}^{1}< \ldots < i_{1}^{s} < < \ldots < i_{k-1}^{s},$ henceforth referred to as the totally ordered adjugate, Hence for a given $s$, $2 \leq s \leq [\frac{n}{k}],$ and given $\mathcal{I} \in \mathcal{T}^{ks}$, we shall show that, $$\label{k odd totally ordered zero}
\left( d_s \right)^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s} = 0,$$
where $ j_1< \ldots <j_s < i_{1}^{1} <\ldots <i_{k-1}^{1}< \ldots < i_{1}^{s} < \ldots < i_{k-1}^{s}.$ To prove (\[k odd totally ordered zero\]), we first need the following:
\[k odd interchnage of r indices\] For any $1 \leq r \leq k-1$, we have, $$\begin{aligned}
\label{interchnage of r indices}
\left( d_s \right)&^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{r}^{1} i_{r+1}^{2} i_{r+2}^{2} \ldots i_{k-1}^{2}\rbrace
\lbrace i_{r+1}^{1} i_{r+2}^{1} \ldots i_{k-1}^{1} i_{1}^{2} i_{2}^{2} \ldots i_{r}^{2} \rbrace \ldots
\lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace }_{j_1j_2 \ldots j_s}\notag \\
& = - \left( d_s \right)^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s}.
\end{aligned}$$
We prove the claim by induction over $r$. The case for $r=1$ follows from repeated applications of claim \[interchange of indices k odd\] as follows.
Using claim \[interchange of indices k odd\] to the $k$-flip interchanging $j_{1}$ and $i^{1}_{1}$, then to the $k$-flip interchanging $i^{1}_{1}$ and $i_{1}^{2}$ and finally to the $k$-flip interchanging $j_{1}$ and $i^{2}_{1}$, we get, $$\begin{aligned}
\left( d_s \right)&^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s}\notag \\
&= (-1)^{s} \left( d_s \right)^{\lbrace j_{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_2 \ldots j_s i_{1}^{1}} \\
&= - (-1)^{s} \left( d_s \right)^{\lbrace j_{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{1} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_2 \ldots j_s i_{1}^{2}} \\
&= - (-1)^{s} (-1)^{s-2} \left( d_s \right)^{\lbrace i_{1}^{1} i_{2}^{2} \ldots i_{k-1}^{2}\rbrace
\lbrace i_{2}^{1} i_{2}^{1} \ldots i_{k-1}^{1} i_{1}^{2}\rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1 j_2 \ldots j_s} .\end{aligned}$$
This proves the case for $r =1$.
We now assume that (\[interchnage of r indices\]) is true for $1 \leq r \leq r_{0}-1$ and show the result for $r = r_{0}.$ To show this, it is enough to prove that for any $2 \leq r_{0} \leq k-1$, $$\begin{aligned}
\label{interchnage induction}
\left( d_s \right)&^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{r_{0}-1}^{1} i_{r_{0}}^{2} i_{r_{0}+1}^{2} \ldots i_{k-1}^{2}\rbrace
\lbrace i_{r_{0}}^{1} i_{r_{0}+1}^{1} \ldots i_{k-1}^{1} i_{1}^{2} i_{2}^{2} \ldots i_{r_{0}-1}^{2} \rbrace \ldots
\lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace }_{j_1j_2 \ldots j_s}\notag \\
& = \left( d_s \right)^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{r_{0}-1}^{1} i_{r_{0}}^{1} i_{r_{0}+1}^{2} \ldots i_{k-1}^{2}\rbrace
\lbrace i_{r_{0}+1}^{1} i_{r_{0}+2}^{1} \ldots i_{k-1}^{1} i_{1}^{2} i_{2}^{2} \ldots i_{r_{0}}^{2} \rbrace \ldots
\lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace }_{j_1j_2 \ldots j_s}.
\end{aligned}$$ Indeed the result for $r = r_{0}$ follows by combining the induction hypothesis and . The proof is similar to the case for $ r = 1$. Indeed, by applying claim \[interchange of indices k odd\] to the $k$-flip interchanging $j_{1}$ and $i^{1}_{r_{0}}$, then to the $k$-flip interchanging $i^{1}_{r_{0}}$ and $i^{2}_{r_{0}}$ and finally to the $k$-flip interchanging $j_{1}$ and $i^{2}_{r_{0}}$, we deduce , $$\begin{aligned}
\left( d_s \right)&^{\lbrace i_{1}^{1} \ldots i_{r_{0}-1}^{1}i_{r_{0}}^{2} \ldots i_{k-1}^{2}\rbrace
\lbrace i_{r_{0}}^{1}i_{r_{0}+1}^{1}\ldots i_{k-1}^{1} i_{1}^{2} \ldots i_{r_{0}-1}^{2} \rbrace
\ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s}\notag \\
&= (-1)^{s-1} \left( d_s \right)^{\lbrace j_{1} i_{r_{0}+1}^{1}\ldots i_{k-1}^{1} i_{1}^{2} \ldots i_{r_{0}-1}^{2}\rbrace
\lbrace i_{1}^{1} \ldots i_{r_{0}-1}^{1}i_{r_{0}}^{2}\ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}
\rbrace}_{j_2 \ldots j_s i_{r_{0}}^{1}}\\
&= - (-1)^{s-1} \left( d_s \right)^{\lbrace j_{1} i_{r_{0}+1}^{1} \ldots i_{k-1}^{1} i_{1}^{2} \ldots i_{r_{0}-1}^{2}\rbrace
\lbrace i_{1}^{1} \ldots i_{r_{0}}^{1} i_{r_{0}+1}^{2} \ldots i_{k-1}^{2} \rbrace
\ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_2 \ldots j_s i_{r_{0}}^{2}} \\
&= - (-1)^{s-1} (-1)^{s-2} \left( d_s \right)^{\lbrace i_{1}^{1} \ldots i_{r_{0}}^{1} i_{r_{0}+1}^{2} \ldots i_{k-1}^{2} \rbrace
\lbrace i_{r_{0}+1}^{1} \ldots i_{k-1}^{1} i_{1}^{2} \ldots i_{r_{0}}^{2} \rbrace \ldots
\lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1 j_2 \ldots j_s} .\end{aligned}$$ This proves ) and establishes claim \[k odd interchnage of r indices\].
Now, using claim \[k odd interchnage of r indices\], in particular for $r = k-1$, we obtain,
$$\begin{aligned}
\left( d_s \right)&^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s}\notag \\
& = - \left( d_s \right)^{\lbrace i_{1}^{1} i_{2}^{1} \ldots i_{k-1}^{1}\rbrace
\lbrace i_{1}^{2} i_{2}^{2} \ldots i_{k-1}^{2} \rbrace \ldots \lbrace i_{1}^{s} i_{2}^{s} \ldots i_{k-1}^{s}\rbrace}_{j_1j_2 \ldots j_s}.
\end{aligned}$$
This proves and finishes the proof of the lemma in the case when $k$ is odd and thereby establishes lemma \[polyconvexitylemma\] in all cases.
Proof of the main theorem
=========================
We start by recalling a result regarding ext. polyconvex functions which we will use later. See [@BDS1] (cf. Proposition 14(ii)) for the proof.
\[Proposition equiv polyconvexite\]Let $f:\Lambda^{k} \rightarrow\mathbb{R}.$ Then $f$ is ext. polyconvex if and only if, for every $\xi
\in\Lambda^{k}$ and $1\leq s\leq\left[ n/k\right] ,$ there exists $c_{s}=c_{s}\left( \xi\right) \in\Lambda^{ks} $ such that$$f\left( \eta\right) \geq f\left( \xi\right) +\sum_{s=1}^{\left[
n/k\right] }\left\langle c_{s}\left( \xi\right) ;\eta^{s}-\xi
^{s}\right\rangle ,\quad\text{for every }\eta\in\Lambda^{k}.$$
Now we are ready to prove the main theorem.
**(i)** Recall (cf. Proposition \[Prop. proj rang un et quasi\]) that$$\pi\left( \alpha\otimes\beta\right) =\alpha\wedge\beta.$$ The rank one convexity of $f\circ\pi$ follows then at once from the ext. one convexity of $f.$ We now prove the converse. Let $\xi\in\Lambda^{k},$ $\alpha\in\Lambda^{k-1}$ and $\beta\in\Lambda^{1};$ we have to show that$$g:t\rightarrow g\left( t\right) =f\left( \xi+t\,\alpha\wedge\beta\right)$$ is convex. Since the map $\pi$ is onto, we can find $\Xi\in\mathbb{R}^{\tbinom{n}{k-1}\times n}$ so that $\pi\left( \Xi\right) =\xi.$ Therefore, $$\begin{aligned}
g\left( t\right) =f\left(\pi\left( \Xi\right) +t\,\pi\left( \alpha\otimes\beta\right) \right)
=f\left( \pi\left( \Xi+t\,\alpha\otimes\beta\right) \right) ,\end{aligned}$$
and the convexity of $g$ follows at once from the rank one convexity of $f\circ\pi.$
**(ii)** Similarly since (cf. Proposition \[Prop. proj rang un et quasi\]) $\pi\left( \nabla\omega\right) =d\omega,$ we immediately infer the quasiconvexity of $f\circ\pi$ from the ext. quasiconvexity of $f.$ The reverse implication follows also in the same manner.
**(iii)** Since $f$ is ext. polyconvex we can find, using proposition \[Proposition equiv polyconvexite\], for every $\alpha
\in\Lambda^{k} $ and $1\leq s\leq [\frac{n}{k}],$ $c_{s}=c_{s}\left( \alpha\right) \in\Lambda^{ks},$ such that$$f\left( \beta\right) \geq f\left( \alpha\right) +\sum_{s=1}^{\left[
n/k\right] }\left\langle c_{s}\left( \alpha\right) ;\beta^{s}-\alpha
^{s}\right\rangle ,\quad\text{for every }\beta\in\Lambda^{k}.$$ Appealing to the proposition \[formula\] we get, for every $\xi \in \mathbb{R}^{\tbinom{n}{k-1}\times n}$,$$\begin{aligned}
f\left( \pi\left( \eta\right) \right)
& \geq f\left( \pi\left(
\xi\right) \right) +\sum_{s=1}^{\left[ n/k\right] }\left\langle
c_{s}\left( \pi\left( \xi\right) \right) ;\left[ \pi\left( \eta\right)
\right] ^{s}-\left[ \pi\left( \xi\right) \right] ^{s}\right\rangle
\smallskip\\
& =f\left( \pi\left( \xi\right) \right) +\sum_{s=1}^{\left[ n/k\right]
}\left\langle \widetilde{c}_{s}\left( \xi\right) ;\operatorname*{adj}\nolimits_{s}\eta-\operatorname*{adj}\nolimits_{s}\xi\right\rangle,\end{aligned}$$ for every $\eta \in \mathbb{R}^{\tbinom{n}{k-1}\times n}$, which shows that $f\circ\pi$ is indeed polyconvex by theorem 5.6 in [@DCV2] .
We now prove the reverse implication. Take $N = \tbinom{n}{k-1} $. Since $f\circ\pi$ is polyconvex, we have, using theorem 5.6 in [@DCV2] again, for every $\xi \in \mathbb{R}^{N\times n}$, there exists $d_s=d_{s}\left( \xi\right) \in \mathbb{R}^{\tbinom{N}{s} \times \tbinom{n}{s}}$ for all $1 \leq s \leq \min\left\{ N, n \right\}$ such that $$\label{fpipolyconvex}
f\left( \pi\left( \eta\right) \right) \geq f\left( \pi\left(
\xi\right) \right) + \sum_{s=1}^{\min\left\{ N, n \right\}} \left\langle
d_{s}\left( \xi\right) ;\operatorname*{adj}\nolimits_s \eta - \operatorname*{adj}\nolimits_s \xi \right\rangle,$$ for every $\eta \in \mathbb{R}^{N\times n}$.
But this means that there exists $d$, given by $d = ( d_1,\ldots, d_{\min\left\{ N, n \right\}} )$ such that the function $X \mapsto g(X,d)$, where $g(X,d)$ is as defined in lemma \[polyconvexitylemma\], achieves a minima at $X=\xi$. Then lemma \[polyconvexitylemma\] implies, for every $1 \leq s \leq \min\left\{ N, n \right\}$, there exists $\mathcal{D}_{s} \in \Lambda^{ks}$ such that $$\left\langle d_s, \operatorname*{adj}\nolimits_s \eta - \operatorname*{adj}\nolimits_s \xi \right\rangle = \left\langle
\mathcal{D}_{s} ;\pi_s(\operatorname*{adj}\nolimits_s \eta )- \pi_s(\operatorname*{adj}\nolimits_s \xi ) \right\rangle ,$$ for every $ \eta \in \mathbb{R}^{N\times n}.$ Hence, we obtain from , for every $\xi \in \mathbb{R}^{ N \times n}$, $$\begin{aligned}
\label{fpipolyconvexityintermediate}
f\left( \pi\left( \eta\right) \right) \geq &\qquad f\left( \pi\left(
\xi\right) \right) + \sum_{s=1}^{\left[ n/k\right]} \left\langle
\mathcal{D}_{s} \left( \xi\right) ;\pi_s(\operatorname*{adj}\nolimits_s \eta )- \pi_s(\operatorname*{adj}\nolimits_s \xi ) \right\rangle ,\end{aligned}$$ for every $\eta \in \mathbb{R}^{ N \times n}$. Since $\pi$ is onto, given any $\alpha, \beta \in\Lambda^{k}$, we can find $\eta, \xi \in \mathbb{R}^{ N \times n}$ such that $\pi(\eta)=\beta$ and $\pi(\xi) = \alpha$. Now using and the definition of $\pi_s$, we have, by defining $c_s (\alpha) = \mathcal{D}_{s} (\xi) $, for every $\alpha
\in\Lambda^{k}$, $$f\left( \beta\right) \geq f\left( \alpha\right) +\sum_{s=1}^{\left[
n/k\right] }\left\langle c_{s}\left( \alpha\right) ;\beta^{s}-\alpha
^{s}\right\rangle ,\quad\text{for every }\beta\in\Lambda^{k}.$$
This proves $f$ is ext. polyconvex and concludes the proof of the theorem.
Notations
=========
We gather here the notations which we will use throughout this article.
1. Let $k$ be a nonnegative integer and $n$ be a positive integer.
- We write $\Lambda^{k}\left( \mathbb{R}^{n}\right) $ (or simply $\Lambda^{k}$) to denote the vector space of all alternating $k-$linear maps $f:\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}_{k-\text{times}}\rightarrow\mathbb{R}.$ For $k=0,$ we set $\Lambda^{0}\left( \mathbb{R}^{n}\right) =\mathbb{R}.$ Note that $\Lambda^{k}\left( \mathbb{R}^{n}\right) =\{0\}$ for $k>n$ and, for $k\leq n,$ $\operatorname{dim}\left(
\Lambda^{k}\left( \mathbb{R}^{n}\right) \right) ={\binom{{n}}{{k}}}.$
- $\wedge,$ $\lrcorner\,,$ $\left\langle\ ;\ \right\rangle $ and $\ast$ denote the exterior product, the interior product, the scalar product and the Hodge star operator respectively.
- If $\left\{ e^{1},\cdots,e^{n}\right\} $ is a basis of $\mathbb{R}^{n},$ then, identifying $\Lambda^{1}$ with $\mathbb{R}^{n},$$$\left\{ e^{i_{1}}\wedge\cdots\wedge e^{i_{k}}:1\leq i_{1}<\cdots<i_{k}\leq
n\right\}$$ is a basis of $\Lambda^{k}.$ An element $\xi\in\Lambda^{k}\left(
\mathbb{R}^{n}\right) $ will therefore be written as$$\xi=\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}\xi_{i_{1}i_{2}\cdots i_{k}}\,e^{i_{1}}\wedge\cdots\wedge e^{i_{k}}=\sum_{I\in\mathcal{T}^{k}}\xi
_{I}\,e^{I}$$ where$$\mathcal{T}^{k}=\left\{ I=\left( i_{1}\,,\cdots,i_{k}\right)
\in\mathbb{N}^{k}:1\leq i_{1}<\cdots<i_{k}\leq n\right\} .$$ An element of $\mathcal{T}^{k}$ will be referred to as a multiindex. We adopt the alphabetical order for comparing two multiindices and we do not reserve a specific symbol for this ordering. The usual ordering symbols, when written in the context of multiindices will denote alphabetical ordering.
- We write$$e^{i_{1}}\wedge\cdots\wedge\widehat{e^{i_{s}}}\wedge\cdots\wedge e^{i_{k}}=e^{i_{1}}\wedge\cdots\wedge e^{i_{s-1}}\wedge e^{i_{s+1}}\wedge\cdots\wedge
e^{i_{k}}.$$ Similarly, $ \widehat{\hphantom{e^{i_{s}}} } $ placed over a string of indices (or multiindices ) will signify the omission of the string under the $ \widehat{\hphantom{e^{i_{s}}} } $ sign.
2. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set.
- The spaces $C^{1}\left( \Omega;\Lambda^{k}\right) ,$ $W^{1,p}\left(
\Omega;\Lambda^{k}\right) $ and $W_{0}^{1,p}\left( \Omega;\Lambda
^{k}\right) ,$ $1\leq p\leq\infty$ are defined in the usual way.
- For any $\omega\in W^{1,p}\left( \Omega;\Lambda^{k}\right) ,$ the exterior derivative $d\omega$ belongs to $L^{p}(\Omega;\Lambda^{k+1})$ and is defined by, for all $1\leq i_{1}<\cdots<i_{k+1}\leq n$, $$(d\omega)_{i_{1}\cdots i_{k+1}}=\sum_{j=1}^{k+1}\left( -1\right) ^{j+1}\frac{\partial\omega_{i_{1}\cdots i_{j-1}i_{j+1}\cdots i_{k+1}}}{\partial
x_{i_{j}}}\,,$$
3. *Notation for indices:* The following system of notations will be employed throughout.
- Single indices will be written as lower case english letters, multiindices will be written as upper case english letters.
- Multiindices will always be indexed by superscripts. The use of a subscript while writing a multiindex is reserved for a special purpose. See (vi) below.
- $\lbrace i_1 \ldots i_r \rbrace$ will represent the string of indices $i_1 \ldots i_r$. In the same way, $\lbrace I^1 \ldots I^r \rbrace$ will represent the string of multiindices obtained by writing out the multiindices in the indicated order.
- $\left( i_1 \ldots i_r \right)$ will stand for the permutation of the $r$ indices that arranges the string $\lbrace i_1 \ldots i_r \rbrace$ of distinct indices in strictly increasing order.
- $[i_1 \ldots i_r]$ will stand for the increasingly ordered string of indices consisting of the distinct indices $i_1, \ldots , i_r$. However, $[I^1, \ldots, I^r]$ will represent the corresponding string of distinct multiindices $I^1, \ldots, I^r$, arranged in the increasing alphabetical order, whereas $[I^1 \ldots I^r]$ will represent the string of indices obtained by arranging all the distinct single indices contained in the multiindices $I^1, \ldots, I^r$ in increasing order.
- For $I \in \mathcal{T}^{k}$ and $j \in I$, $I_{j}$ stands for the multiindex obtained by removing $j$ from $I$.
- The symbol $\left( J; I \right) $, where $J = \lbrace j_1 \ldots j_s\rbrace$ is a string of $s$ single indices and $I = \lbrace I^1 \ldots I^s\rbrace$ is a string of $s$ multiindices, $I^1, \ldots , I^s \in \mathcal{T}^{(k-1)s}$, will be reserved to denote the interlaced string $ \left\lbrace j_{1} I^{1} \ldots j_{s} I^{s} \right\rbrace $.
- The abovementioned system of notations will be in force even when representing indices as subscripts of superscripts of different objects.
4. *Flip:* We shall be employing some particular permutations often.
Let $s \geq 1$, let $J \in \mathcal{T}^{s}$, $I \in \mathcal{T}^{l}$ be written as, $J = \lbrace j_{1} \ldots j_{s} \rbrace $, $I = \lbrace i_{1} \ldots i_{l} \rbrace$ with $J \cap I = \emptyset$. Let $\tilde{J} \in \mathcal{T}^{s}$, $\tilde{I} \in \mathcal{T}^{l}$. We say that $( \tilde{J}, \tilde{I} )$ is obtained from $( J, I )$ by a $1$-flip interchanging $j_{p}$ with $i_{m}$, for some $1 \leq p \leq s$, $1 \leq m \leq l$, if $$\tilde{J} = \left[ j_{1}\ldots j_{p-1} i_{m} i_{p+1} \ldots j_{l} \right] \text{ and }
\tilde{I} = \left[ i_{1}\ldots i_{m-1} j_{p} i_{m+1} \ldots i_{k} \right] .$$
\[kflips\] Let $s \geq 1$, $k \geq 2$. Let $J \in \mathcal{T}^{s},$ $ J = \lbrace j_{1} \ldots j_{s}\rbrace $, $I = \lbrace I^{1} \ldots I^{s}\rbrace
= [ I^{1}, \ldots, I^{s}]$, where $ I^{1}, \ldots, I^{s}
\in \mathcal{T}^{k}$, $I^{r} = \lbrace i^{r}_{1}, \ldots , i^{r}_{k}\rbrace$ for all $1 \leq r \leq s$ and $J \cap I = \emptyset$. We say that $(\tilde{J}, \tilde{I} )$ is obtained from $(J, I)$ by a $k$-flip if there exist integers $1 \leq m, p \leq s $ and $1 \leq q \leq k $ such that, $$\tilde{J}= [j_{1} \ldots j_{p-1} i^{m}_{q} j_{p+1} \ldots j_{s}],$$ and $$\tilde{I}=[ I^{1},
\ldots I^{m-1}, [i^{r}_{1} \ldots i^{r}_{q-1} j_{p} i^{r}_{q+1} \ldots i^{r}_{k} ], I^{m+1}, \ldots , I^{s} ] .$$
Note that a $k$-flip can be seen as a permutation in an obvious way.
5. *Notation for sum:* For $I \in \mathcal{T}^{ks}$, where $1 \leq k \leq n $ and $1 \leq s \leq [\frac{n}{k}]$, the shorthand $\sum\nolimits^I_s $ stands for the sum, $$\sum_{\substack{J, \tilde{I} \\ J = \lbrace j_1\ldots j_s\rbrace = [j_1\ldots j_s],\\
\tilde{I} = \lbrace I^1\ldots I^s \rbrace = [I^1,\ldots , I^s]\\
J\cup\tilde{I} = I}}.$$
**Acknowledgement.** We have benefitted of interesting discussions with Professor Bernard Dacorogna. Part of this work was completed during visits of S. Bandyopadhyay to EPFL, whose hospitality and support is gratefully acknowledged. The research of S. Bandyopadhyay was partially supported by a SERB research project titled “Pullback Equation for Differential Forms".
[1]{} S. Bandyopadhyay, B. Dacorogna and S. Sil, Calculus of variations with differential forms, To appear in [*Journal of European Mathematical Society*]{}.
S. Bandyopadhyay and S. Sil, Characterization of functions affine in the direction of one-divisible forms, Preprint.
B. Dacorogna, , volume 78 of [*Applied Mathematical Sciences*]{}. Springer, New York, second edition, 2008.
J. W. Robbin, R. C. Rogers, and B. Temple, On weak continuity and the [H]{}odge decomposition. , 303(2):609–618, 1987.
S. Sil, PhD Thesis.
|
---
abstract: 'We report on a year-long effort to monitor the central supermassive black hole in M81 in the X-ray and radio bands. Using [*Chandra*]{} and the VLA, we obtained quasi-simultaneous observations of M81\* on seven occasions during 2006. The X-ray and radio luminosity of M81\* are not strongly correlated on the approximately 20-day sampling timescale of our observations, which is commensurate with viscous timescales in the inner flow and orbital timecales in a radially-truncated disk. This suggests that short-term variations in black hole activity may not be rigidly governed by the “fundamental plane”, but rather adhere to the plane in a time-averaged sense. Fits to the X-ray spectra of M81\* with bremsstrahlung models give temperatures that are inconsistent with the outer regions of very simple advection-dominated inflows. However, our results are consistent with the X-ray emission originating in a transition region where a truncated disk and advective flow may overlap. We discuss our results in the context of models for black holes accreting at small fractions of their Eddington limit, and the fundamental plane of black hole accretion.'
author:
- 'J. M. Miller, M. Nowak, S. Markoff, M. P. Rupen, D. Maitra'
title: 'Exploring Accretion and Disk-Jet Connections in the LLAGN M81\*'
---
\[firstpage\]
Introduction
============
Low-luminosity active galactic nuclei (LLAGN) are potentially important transition objects, harboring supermassive black holes that accrete at a rate that is intermediate between Seyfert AGN and quasars, and extremely under-luminous sources such as Sgr A\*. LLAGN may provide clues to jet production: in these systems, compact relativistic radio jets are often detected (Nagar et al. 2002, Anderson & Ulvestad 2005), and the natural time scales are such that the details of jet production can be revealed. Moreover, LLAGN are often “radio-loud” (Ho 2008), meaning that jets are an important part of the overall accretion flow. At a distance of only 3.6 Mpc (Freedman et al. 1994), the accreting supermassive black hole at the center of M81 powers the nearest LLAGN, M81\*.
The nature of the innermost accretion flow in LLAGN is not clear. It is likely that these sources are still fueled partially by an accretion disk - double-peaked optical emission lines are seen in M81\* (Bower et al. 1996; also see Devereux & Shearer 2007) - but relativistic X-ray lines from the inner accretion disk are not clearly detected in these systems (e.g. Dewangan et al. 2004, Reynolds et al. 2009; for a review, see Miller 2007). The inner disk may be truncated, and the innermost flow may be advection–dominated (Narayan & Yi 1994; also see Blandford & Begelman 1999). Theoretical work suggests that thick advective disks and radial flows may help to maintain poloidal magnetic fields and power jets (e.g. Livio 2000; Meier 2001; Reynolds, Garofalo, & Begelman 2006). X-ray observations can test and refine models for advective inflow at low mass accretion rates in a variety of ways. For instance, the inner accretion flow is predicted be extremely hot, with temperatures ranging between $10^{12}$ K centrally to $10^{9-10}$ K in their outermost radii (Narayan & Yi 1994, 1995). Recent observations of X-ray binaries have achieved the sensitivity required to test these predictions (e.g. Bradley et al. 2007), and find emission consistent with much lower temperatures.
X-ray emission is often used as a trace of the accretion inflow (although some X-ray emission could originate in a jet; e.g. Markoff, Falcke, & Fender 2001; also see Miller et al. 2002, Russell et al.2010), and radio emission is used to trace the jet power. In X-ray binaries, X-ray and radio emission follows the relationship $L_{R}
\propto {L}_{ X}^{0.7}$, both in ensemble and in individual sources (Gallo, Fender, & Pooley 2003; however, see Jonker et al. 2009). This relation has been generalized into a fundamental plane of black hole accretion, combining radio luminosity, X-ray luminosity, and black hole mass (Merloni, Heinz, & Di Matteo 2003; Falcke, Kording, & Markoff 2004; Gultekin et al. 2009). If accretion physics scales predictably with black hole mass, then for any individual object of known mass, the relationship between radio and X-ray emission should be fixed, on average.
Prior to discrete jet ejection events in stellar-mass black holes, quasi-periodic oscillations (QPOs) are often observed in the X-ray flux, with a characteristic frequency of $\sim6$ Hz (see, e.g., Nespoli et al. 2003, Ferreira et al. 2006, Klein-Wolt & van der Klis 2008, Fender, Homan, & Belloni 2009). If this frequency is a Keplerian orbital ferequency, it corresponds to a radius of $66~{\rm
GM}/{\rm c}^{2}$ for a black hole with a mass of $10~{\rm M}_{\odot}$. This radius is broadly consistent with lower limits on the inner edge of the accretion disk in M81\* based on the width of the Fe K$\alpha$ emission line (Dewangan et al. 2004, Young et al. 2007). In stellar mass systems, it is not possible to test disk-jet connections on the period defined by the QPO, although the oscillation might be tied to jet production. In supermassive black holes, however, this timescale is accessible. For a black hole of $7\times 10^{7}~{\rm M}_{\odot}$ like that in M81\* (Devereux et al. 2003), monitoring every 2–4 weeks can sample the corresponding timescale.
In this paper, we present contemporaneous X-ray and radio observations of M81\* made using [*Chandra*]{} and the VLA, with visits separated by approximately 20 days. The observations and data reduction methods are described in Section 2. Our analysis and results are presented in Section 3. We do not find a clear correlation between radio and X-ray emission in M81\*, though a small number of simultaneous points were obtained and span a factor of approximately two in X-ray flux. These results are discussed in Section 4.
Observations and Data Reduction
===============================
We observed M81\* on ten occasions using [*Chandra*]{}. Each observation achieved a total exposure of approximately 15 ksec (see Table 1). In order to minimize photon pile-up in the zeroth order ACIS image, the HETGS was inserted into the light path in each case. The ACIS chips were operated in “FAINT” mode.
We used CIAO version 4.0.2 in processing the [*Chandra*]{} data. First-order dispersed spectra from the MEG and HEG were split from the standard “pha2” file, and associated instrument response files were constructed. The first-order MEG spectra and responses were then added using the CIAO tool “add\_grating\_spectra”; the first-order HEG spectra and responses were added in the same way. The zeroth-order ACIS spectra and responses were generated using the CIAO tool “psextract”. In each case, a circular region was used to extract the source flux and a radially–offset annular region was used to extract the background flux. All spectra were grouped to require at least 20 counts per bin using the FTOOL “grppha”, in order to ensure the validity of $\chi^{2}$ statistics.
The VLA also observed M81\* on ten occasions (see Table 1). Useful data were obtained on seven occasions that coincide with the [*Chandra*]{} X-ray observations. All observations were obtained at 8.4 GHz. The first three coincident exposures were obtained in the “A” configuration (achieving a typical angular resolution of approximately 0.3”), while the last four were obtained in the “B” configuration (achieving a typical angular resolution of approximately 1”). Standard compact calibrator sources were used to calibrate phase and amplitude variations, and to set the overall amplitude scale. The average flux density measured in each exposure is reported in Table 1.
Analysis and Results
====================
The [*Chandra*]{} X-ray spectra were analyzed using XSPEC version 12.4 (Arnaud 1996). Spectral fits were made in the 0.5–10.0 keV band. All of the errors reported in this work are $1\sigma$ confidence errors. In calculating luminosity values, distances were assumed to be absolute, and uncertainties in luminosity were derived from the flux uncertainties.
We initially made separate fits to the zeroth-order, combined MEG, and combined HEG spectra. In all direct fits, the equivalent neutral hydrogen column density drifted towards zero, which is unphysical. A value of $4.1\times 10^{20}~{\rm cm}^{-2}$ is expected along this line of sight (Dickey & Lockman 1990), but this value is too low to be constrained directly in the MEG spectra obtained. For consistency, then, the expected value was fixed in all fits. All of the spectra were acceptably fit ($\chi^{2}/\nu \leq 1.0$, where $\nu$ is the number of degrees of freedom in the fit) with a simple power-law model (see Figure 1). The spectrum of M81\* is likely more complex, mostly owing to local diffuse emission (Young et al. 2007), but a simple power-law is an acceptable fit to the modest spectra obtained in our observations.
The zeroth-order spectra suffer from photon pile-up, and are not robust. Particularly in the last two [*Chandra*]{} observations, where the flux is higher, the best-fit power-law photon index was found to be harder. This is consistent with multiple low-energy X-rays being detected as single high energy photons. Moreover, the data/model ratio in each spectrum shows an increasing positive trend with energy. The HEG spectra contain many fewer photons than the MEG spectra, and were found to be of little help in constraning the souce flux or spectral index. We therefore restriced our flux analysis to the combined first-order MEG spectra. The second spectrum listed in Table 1, for instance, has just over 2800 photons, and the penultimate spectrum has 4900 photons.
The results of our spectral analysis of each observation are detailed in Table 1. Figure 2 shows the time evolution of the X-ray flux, X-ray power-law photon index, and radio flux density. Between MJD 53900.0 and MJD 53950.0, the X-ray flux
increases by a factor of approximately two, and the radio flux density increases by slightly less than a factor of two. The X-ray spectral index does not vary significantly during the course of our observations. In each spectrum, the index is consistent with $\Gamma = 1.7$, which is fairly typical of Seyferts (see, e.g., Reynolds 1997) and consistent with prior [*Chandra*]{} obsevations of M81\* (Young et al. 2007). A power-law is not a unique description of the data; bremsstrahlung models also yield acceptable fits with temperatures of ${\rm kT} =
5\pm 2$ keV.
Assuming a flat radio spectrum, we calculated the radio luminosity in the narrow VLA band centered at 8.4 GHz. We also calculated the unabsorbed X-ray flux in the 0.5–10.0 keV band. We assumed a distance of 3.6 Mpc to M81\* (Freedman et al. 1994). This radio luminosity is plotted versus X-ray luminosity in Figure 3. The Spearman’s rank correlation coefficient for the flux values is 0.23, and it is apparent in Figure 3 that there is no strong correlation between the X-ray and radio luminosity.
In Figure 4, the X-ray power-law photon index is plotted versus the X-ray luminosity. Here again, there is no clear correlation visible in the plot. The Spearman’s rank correlation coefficient for these quantities is -0.39, indicating that there is no significant correlation.
The sampling rate of the X-ray light curve is $21\pm 5$ days and that of the radio light curve is $19\pm 5$ days. Therefore, the radio and X-ray peak around MJD 53944.6 are formally consistent with being simultaneous, and any delay between radio and X-rays is $\leq$20 days. Given the sampling rate, it is likely that the radio/X-ray flare is caused by a factor of $\sim$2 change in $\dot M$ between MJD 53929–53960.
Discussion and Conclusions
==========================
The nature of the inner accretion flow in LLAGN is not yet clear. It is possible that LLAGN retain many of the characterstics of Seyferts, perhaps including an accretion disk extending to the ISCO (e.g. Herrnstein et al. 1998, Maoz 2007). It is also possible that LLAGN are more like under-luminous sources, such as Sgr. A\*, and best described in terms of an ADAF or
coupled ADAF-jet system (see, e.g., Nemmen et al. 2010). In this work, we have attempted to explore the the nature of the inner accretion flow in an LLAGN by examining evidence for a disk-jet connection in M81\*. Based on observations of QPOs with a frequency of $\sim 6$ Hz in stellar-mass black holes just prior to jet ejection episodes, we sampled a commensurate timescale in M81\*. If this timescale is an orbital period, it implies a radius that is compatible with lower limits on the inner radius of the accretion disk in M81\* based on modeling of the Fe K$\alpha$ emission line detected in deep obsevations of this source (${\rm R} \geq 50-60~{\rm GM}/{\rm c}^{2}$; Dewangan et al. 2004, Young et al. 2007). If the innermost accretion flow is advective, so that it is geometrically thick but retains some viscosity and angular momentum, then our sampling timescale is also commensurate with the viscous timescale in the very innermost region around the black hole (e.g. $6 {\rm GM}/{\rm
c}^{2}$). Our monitoring observations improve upon many prior investigations of disk-jet connections in many supermassive and stellar-mass black holes in that our radio and X-ray observations were nearly simultaneous.
At low fractions of the Eddington limit, stellar-mass black holes have hard X-ray spectra (see, e.g., Miller et al. 2006, Miller, Homan, & Miniutti 2006, Tomsick et al. 2008, Reis, Fabian, & Miller 2010; for a review, see Remillard & McClintock 2006) and radio emission that is consistent with a compact jet (Fender, Homan, & Belloni 2009). Both in individual sources and in an ensemble, radio and X-ray emission are related by the expression $L_{R} \propto L_{X}^{0.7}$ (Gallo, Fender, & Pooley 2003; see, however, Jonker et al. 2009).
Our data do not strongly confirm nor reject the possibility that M81\* regulates its radio and X-ray output according to $L_{R} \propto
L_{X}^{0.7}$ (see Figure 3). However, it is clear that if M81\* does follow this relation, the regulation of its radiation is not rigid on short time scales. Recent work on Sgr A\* shows that the source approaches the expected relation when it flares, but otherwise falls below it (Markoff 2005). These outcomes suggest that black holes might generally channel a fixed fraction of the matter inflow (traced by X-ray emission) into a jet (traced by radio emission), but not necessarily at every moment. Said differently: the energy channeled into jets at any given time may vary as it is likely to be somewhat stochastic, but on average the expected relationship may hold. In past investigations, scatter in the $L_{R} \propto L_{X}^{0.7}$ relation and the fundamental plane (e.g. Gultekin et al. 2009) could plausibly be explained in terms of non-simultaneous X-ray and radio observations; our results suggest a degree of intrinsic scatter.
Within states where compact, steady jets are produced, spectral hardness and luminosity are positively correlated in stellar-mass black holes (e.g. Tomsick et al. 2001; also see Rykoff et al. 2007). Hardening of this kind has also been observed in Sgr A\* (Baganoff et al. 2001). In constrast, Seyferts appear to become spectrally softer with higher X-ray luminosity (see Vaughan & Fabian 2004). The X-ray spectrum of M81\* does not show a strong trend with luminosity, and we are not able to characterize the variability of M81\* as being more like Sgr A\* or more typical of Seyferts.
Recent modeling of the broad-band spectrum of M81\* suggests that its accretion flow may be very similar to that in the hard state of stellar-mass black holes and Sgr A\*. Markoff et al. (2008) showed that the same jet-dominated broad-band accretion flow model can be applied to stellar-mass black holes, Sgr A\*, and to M81\*. The stellar-mass black hole V404 Cyg may be the source in which X-ray observations permit the best constraints on the inner accretion flow at $10^{-5}~{\rm L}_{\rm Edd.}$. Recent analysis by Bradley et al.(2007) measure a temperature of ${\rm kT} \simeq 5$ keV with bremsstrahlung models. This is much too cold for even the outer parts of a simple advection-dominated accretion flow, for which temperatures of ${\rm kT} \geq 85$ keV are predicted (Narayan & Yi 1995). Similarly, fits to our spectra of M81\* with bremsstrahlung models give ${\rm kT} = 5\pm 2$ keV. In this sense, then, the X-ray spectrum of M81\* is inconsistent with very simple ADAF models because a $\sim$5 keV plasma is too cold to be compatible with such models.
On the other hand, Young et al. (2007) showed that a combined 282ksec [*Chandra*]{} spectrum of M81\* (essentially at the flux level of the first eight observations presented here) could be described by a model that was dominated by emission from collisional plasma with temperatures ranging from 1–100keV. In terms of an emission measure analysis, the peak emission came from $\approx
10$–30 keV plasma. Similarly, one could construct an ADAF type model where $kT \propto R^{-0.5}$, with emission ranging from $\approx
1$–$10^4\, {\rm GM}/{\rm c}^2$. The construction of such models, however, relied upon the detection and measurement of plasma emission line features, which are too weak to detect in any of our short, individual observations. (We have verified, however, that the basic line structure reported by Young et al. is unaltered in the full, combined 450ksec spectra.) In principle, if one could associate line variations with the factor $> 2$ continuum flux level variations shown here, especially for the lower temperature lines that should arise in the less central parts of the system, this would place strong constraints on any ADAF type model.
The large increase in X-ray flux detected in our last two observations occurred on a time scale shorter than two weeks, which corresponds to a light travel distance of $\approx 10^{3.5}~{\rm GM}/{\rm c}^2$. This is smaller than the ADAF emission region postulated by Young et al. (2007), which in any case would respond on time scales longer than the light travel time. Thus, any correlated line/continuum changes would alter the simple ADAF assumptions of emission dominated by a hot, optically thin, flow. For example, in a situation where the hot inner flow and thin disk partially overlap, a transition region with a lower temperature may be expected. Transition regions have been treated in some detail in numerous works, including Blandford & Begelman (1999) and Meyer, Liu, and Meyer-Hofmeister et al. (2000). Emission originating in a transition region can potentially explain our spectral and variability results, and those reported by Young et al.(2007). Moreover, this possibility is qualitatively consistent with evidence of thin disks extending to small radii at low Eddington fractions in LLAGNs and LINERS (Maoz 2007) but still allows for a coupled ADAF plus jet system like that described by Nemmen et al.(2010).
Decisive observations may be feasible with the proposed International X-ray Observatory (IXO): a single 30 ksec observation with IXO will achieve a sensitivity greater than that in the combined 282 ksec [*Chandra*]{} exposure analyzed by Young et al. (2007). The higher spectral resolution of the calorimeter expected to fly aboard IXO will facilitate both the detection of weak lines and the detection of small velocity shifts. If the innermost accretion flow in M81\* is a dynamic environment where X-ray flares help to drive a jet and/or a wind, IXO spectroscopy will be able to detect corresponding variations in the line spectrum discussed by Young et al. (2007). If a weak iron line is produced in the inner disk, the sensitivity of IXO will help to detect and resolve the dynamical information imprinted on any such line.
We thank the anonymous referee for thoughtful comments that improved this paper. J.M.M. gratefully acknowledges funding from the [*Chandra*]{} Guest Observer program. S. M. gratefully acknowledges support from a Netherlands Organization for Scientific Research (NWO) Vidi Fellowship. We wish to thank the [*Chandra*]{} and [*VLA*]{} observatory staff for executing this demanding program.
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Obs. ${\rm T}_{\rm X}$ Exposure $\Gamma$ ${\rm F}_{X}$ ${\rm T}_{\rm R}$ Exposure ${\rm S}_{8.4}$ Configuration
(MJD) (ksec) ($10^{-11}~{\rm erg}~{\rm cm}^{-2}~{\rm s}^{-1}$) (MJD) (ksec) (mJy)
1 53774.9 15.0 1.7(1) 1.6(1) 53775.3 3.4 110.5(1) A
2 53800.0 15.0 1.8(1) 1.7(1) 53800.9 5.3 96.4(1) A
3 53826.4 15.0 1.8(1) 1.6(1) – – – –
4 53849.5 14.7 1.8(1) 1.6(1) – – – –
5 53869.5 15.0 1.8(1) 1.7(1) – – – –
6 53895.8 15.0 1.8(1) 1.5(1) 53892.0 5.2 103.0(1) A
7 53915.0 14.9 1.9(1) 1.7(1) 53915.9 5.3 81.2(1) B
8 53929.6 15.1 1.8(1) 1.8(1) 53928.0 5.2 104.7(1) B
9 53944.5 14.6 1.70(7) 3.1(1) 53944.7 6.3 124.0(1) B
10 53959.7 15.0 1.70(7) 2.8(1) 53959.7 5.2 102.4(1) B
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: X-ray and Radio Observations
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abstract: 'The photonic spin Hall effect (SHE) is generally believed to be a result of an effective spin-orbit coupling, which describes the mutual influence of the spin (polarization) and the trajectory of the light beam. The photonic SHE holds great potential for precision metrology owing to the fact that the spin-dependent splitting in photonic SHE are sensitive to the physical parameter variations of different systems. Remarkably, using the weak measurements, this tiny spin-dependent shifts can be detected with the desirable accuracy so that the corresponding physical parameters can be determined. Here, we will review some of our works on using photonic SHE for precision metrology, such as measuring the thickness of nanometal film, identifying the graphene layers, detecting the strength of axion coupling in topological insulators, and determining the magneto-optical constant of magnetic film.'
author:
- 'Xinxing Zhou, Shizhen Chen, Yachao Liu, Hailu Luo$^{*}$, and Shuangchun Wen$^{\dagger}$ Laboratory for Spin Photonics, College of Physics and Microelectronic Science, Hunan University, Changsha 410082, China'
title: Photonic spin Hall effect for precision metrology
---
INTRODUCTION
============
The photonic spin Hall effect (SHE) manifests itself as spin-dependent splitting of left- and right-handed circularly polarized components when a spatially confined light beam is reflected or transmitted at an interface [@Onoda2004; @Bliokh2006; @Hosten2008]. The photonic SHE can be regarded as a direct optical analogy of SHE in an electronic system, in which the spin photons play the role of the spin charges and a refractive index gradient plays the role of the applied electric field. This interesting phenomenon is generally believed to be a result of an effective spin-orbit coupling, which describes the mutual influence of the spin (polarization) and the trajectory of the light beam. There are two types of geometric phases playing the important role in photonic SHE: the spin redirection Berry phase and the Pancharatnam-Berry phase [@Bliokh2008a; @Bliokh2008b]. The photonic SHE is currently attracting growing attention and has been intensively investigated in different physical systems such as optical physics [@Aiello2008; @Luo2009; @Qin2009; @Hermosa2011; @Luo2011a; @Ling2014], high-energy physics [@Gosselin2007; @Dartora2011], semiconductor physics [@Menard2009; @Menard2010], and plasmonics [@Shitrit2011; @Gorodetski2012; @Shitrit2013; @Yin2013; @Kapitanova2014].
Remarkably, the spin-dependent splitting in photonic SHE are sensitive to the physical parameter variations of different systems, and therefore it holds great potential applications in precision metrology. However, the spin-dependent splitting of photonic SHE in these systems is just a few tens of nanometers so that the actual equipment can not distinguish it directly. We resolve this problem by using the precise signal enhancement technique called quantum weak measurements which has attracted a lot of attention [@Aharonov1988; @Ritchie1991; @Dixon2009; @Lundeen2011; @Kocsis2011; @Dressel2014; @Jordan2014; @Knee2014; @Zhou2014]. The idea of weak measurements can be described as follows: if we initially select the quantum system with a well-defined preselection state, the corresponding large expectation values can be obtained with a suitable postselection state, which makes the eigenvalues to be clearly distinguished [@Jozsa2007; @Dennis2012; @Lorenzo2014]. Using the quantum weak measurement, the photonic SHE can be detected with the desirable accuracy.
In this paper, we will review some of our recent works on using photonic SHE for precision metrology, such as measuring the thickness of nanometal film [@Zhou2012a], identifying the graphene layers [@Zhou2012b], detecting the strength of axion coupling in topological insulators [@Zhou2013], and determining the magneto-optical constant of magnetic film. We find that the physical parameter variations in these systems can effectively change the spin-dependent displacements. We firstly establish the quantitative relationship between the spin-dependent shifts and the physical parameters. After detecting the spin-dependent displacements with weak measurement method, we can accurately determine these physical parameters. The rest of the paper is organized as follows. In Sec. 2, we establish a general propagation model to describe the photonic SHE on the sample. In the Sec. 3, we firstly introduce the weak measurements experimental process. Then, we will briefly review our recent works on using photonic SHE for precision metrology. Finally, a conclusion is given in Sec. 4.
![\[Fig1\] Schematic of photonic SHE on a sample. The sample can be nanometal film, graphene film, topological insulators, and magnetic film. A linearly polarized beam reflects on the sample and then splits into left- and right-handed circularly polarized components, respectively. $\delta^{x}_{|+\rangle}$ and $\delta^{x}_{|-\rangle}$ indicate the in-plane shift of left- and right-handed circularly polarized components. $\delta^{y}_{|+\rangle}$ and $\delta^{y}_{|-\rangle}$ denote the transverse spin-dependent displacements. Here, $\theta_{i}$ is the incident angle. ](Fig1.eps){width="8.5cm"}
GENERAL PROPAGATION MODEL
=========================
Figure \[Fig1\] schematically draws the photonic SHE of light beam reflection from a sample interface. We firstly establish the quantitative relationship between the spin-dependent shifts in photonic SHE and the physical parameters of sample. Here, the physical parameters include the thickness of nanometal film, graphene’s layers, axion angle in topological insulators, and the magneto-optical constant of magnetic film. The incident polarization states are chosen as $|H\rangle$ and $|V\rangle$. In the spin basis, the horizontal and vertical polarization states can be expressed as $|H\rangle=(|+\rangle+|-\rangle)/{\sqrt{2}}$ and $|V\rangle=i(|-\rangle-|+\rangle)/{\sqrt{2}}$. Corresponding, the states of reflected beam can be obtained: $$|H\rangle\rightarrow\frac{r_{p}}{\sqrt{2}}\left[\exp(+ik_{ry}\delta^{H}_{r})|+\rangle+\exp(-ik_{ry}\delta^{H}_{r})|-\rangle\right]\label{H
spectrum},$$ $$|V\rangle\rightarrow\frac{ir_{s}}{\sqrt{2}}\left[-\exp(+ik_{ry}\delta^{V}_{r})|+\rangle+\exp(-ik_{ry}\delta^{V}_{r})|-\rangle\right]\label{V
spectrum}.$$ In the above equations, $\delta^{H}_{r}=(1+r_{s}/r_{p})\cot\theta_{i}/k_{0}$, $\delta^{V}_{r}=(1-r_{p}/r_{s})\cot\theta_{i}/k_{0}$. We should note that, as for the topological insulators, the states of reflected beam have different forms. The detailed discussions can be found in our previous work [@Zhou2013].
The photonic SHE manifests for the spin-dependent splitting of left- and right-handed circularly polarized components. We consider the spin separation in the x direction (in-plane shift) and y direction (transverse shift). In the following, we calculate the shifts of these two spin components. The wavefunction of reflected photons is composed of the packet spatial extent $\phi(k_{ry})$ and the polarization description $|H,V\rangle$: $$|\Phi^{H,V}\rangle=\int dk_{ry}
\phi(k_{ry})|k_{ry}\rangle|H,V\rangle\label{inital}.$$ After photons reflection from the sample interface, the initial state $|\Phi_{inital}^{H,V}\rangle$ evolve into the final state $|\Phi_{final}^{H,V}\rangle$. As a result of spin-orbit coupling, the shifts of the two spin components compared to the geometrical-optics prediction are given by $$\delta_{|\pm\rangle}^{H,V}=\frac{\langle
\Phi^{H,V}|i\partial_{\mathbf{k_\perp}}|{\Phi^{H,V}}\rangle}{\langle
\Phi^{H,V}|\Phi^{H,V}\rangle}.\label{BCII}$$ Here, we suppose the $\phi(k_{ry})$ is a Gaussian wave function. Calculating the reflected shifts of photonic SHE requires the explicit solution of the boundary conditions at the sample interfaces. As for the nanometal film and graphene film, we need to deal with the multilayer structure model. Thus, we need to know the generalized Fresnel reflection of the sample, $$\begin{aligned}
r_{A}=\frac{R_{A}+R_{A}^{'}\exp(2ik_{0}\sqrt{n^{2}-\sin^{2}\theta_{i}}d)}{1+R_{A}R_{A}^{'}\exp(2ik_{0}\sqrt{n^{2}-\sin^{2}\theta_{i}}d)}.\end{aligned}$$ Here, $A\in\{p,s\}$, $R_{A}$ and $R_{A}^{'}$is the Fresnel reflection coefficients at the first interface and second interface, respectively. $n$ and $d$ represent the refractive index and the thickness of the nanometal film and graphene film, respectively.
![\[Fig2\] The experimental setup in weak measurements. The sample is a BK7 prism prepared with the nanometal film, graphene film, topological insulators, and magnetic film. L1 and L2, lenses with effective focal length $50\mathrm{mm}$ and $250\mathrm{mm}$, respectively. HWP, half-wave plate (for adjusting the intensity). P1 and P2, Glan Laser polarizers. CCD, charge-coupled device (Coherent LaserCam HR). The light source is a $21\mathrm{mW}$ linearly polarized He-Ne laser at $632.8\mathrm{nm}$ (Thorlabs HNL210L-EC). The inset shows the states of preselection and postselection. Here, the preselection state is prepared in $|V\rangle$. ](Fig2.eps){width="10cm"}
PHOTONIC SPIN HALL EFFECT FOR PRECISION METROLOGY
=================================================
We have established the relationship between the physical parameters of sample and the spin-dependent displacements induced by photonic SHE. Next, we will use the weak measurements method to detect the this tiny shifts. After detecting the spin-dependent displacements, we can accurately determine these physical parameters. As shown in Fig. \[Fig2\], our experimental setup is similar to that in Ref [@Luo2011]. Our samples are the usual BK7 prism prepared with the nanometal film, graphene film, topological insulators, and magnetic film. A Gauss beam generated by He-Ne laser is firstly focused by the lens (L1) and experiences preselection in the state $|\psi_{1}\rangle$=$|H\rangle$ or $|V\rangle$ with the polarizer P1. When the light beam reflects from the sample interface, the photonic SHE happens allowing for the left- and right-handed circularly polarized components splitting in the x and y directions corresponding to the in-plane and transverse displacements. This process can be seen as the weak interaction allowing for the coupling between the observable and the meter. And then the beam passes through the second polarizer P2 preparing for the postselection state $|\psi_{2}\rangle=|V\pm\Delta\rangle$ or $|H\pm\Delta\rangle$. At the surface of the second polarizer, the two spin components experience destructive interference making the enhanced shift in the meter much larger than the initial one. Calculating the reflected field distribution yields the amplified shifts of photonic SHE. After passing through the second lens (L2), a CCD is used to capture the optical signal and measure the amplified shifts. The process discussed above is called the weak value amplification and $\Delta$ is the postselection angle. We should note that the imaginary weak value also corresponds to a shift of the meter in momentum space, which leads to the possibility of even larger enhancements following the beam free evolution. This process can be seen as propagation amplification that produces the amplified factor F. In the following, we will review some of our recent works on using photonic SHE for precision metrology.
![\[Fig3\] In the case of horizontal polarized (left column) and vertical polarized (right column), the amplified displacements of light beam reflection on Ag film with different thicknesses: \[(a),(b)\] $10\mathrm{nm}$ and $12\mathrm{nm}$, \[(c),(d)\] $30\mathrm{nm}$, and \[(e),(f)\] $60\mathrm{nm}$. The lines show the theoretical value and the dots denote the experimental results. The inset shows the experimental sample: a BK7 prism coated with Ag film. [@Zhou2012a] ](Fig3.eps){width="10cm"}
In the fist work, we have used the photonic SHE to measure the thickness of nanometal film [@Zhou2012a]. We establish a general propagation model to describe the photonic SHE on a nanometal film and reveal the impact of the corresponding physical parameters on the spin-dependent splitting in photonic SHE. It is well known that the photonic SHE manifests itself as the spin-orbit coupling. We find that the spin-orbit coupling in the photonic SHE can be effectively modulated by adjusting the thickness of the metal film. A similar effect can also be observed in layered nanostructures, in which the transverse displacement changes periodically with the air gap increasing or decreasing. Additionally, the transverse displacement is sensitive to the thickness of metal film in certain range for horizontal polarization light beam. We also note tha a large negative transverse shift can be observed.
Next, we focus our attention on the weak measurements experiment. Here, the BK7 glass substrate coated Ag film is chosen as our sample (with three different thickness $10\mathrm{nm}$, $30\mathrm{nm}$ and $60\mathrm{nm}$). The experimental setup is described in the above contents. We measure the displacements of photonic SHE on the nanometal film every $5^{\circ}$ from $30^{\circ}$ to $85^{\circ}$ in the case of horizontal and vertical polarization, respectively. Limited by the large holders of the lens, polarizers and He-Ne laser, displacements at small incident angles were not measured. It should be noted that the experimental results are in good agreement with the theoretical ones when the film thicknesses are $30\mathrm{nm}$ and $60\mathrm{nm}$. However, we observe a small deviation when the thickness is $10\mathrm{nm}$. Note that the thickness of the nanometal film has an error limited by the experimental condition. When the thickness reaches to $10\mathrm{nm}$, the SHE of light is very sensitive to the error. It is the reason why there is a small deviation between the experimental and the theoretical data. From the experimental results, we can conclude that the actual thickness of the film is about $12\mathrm{nm}$ (Fig. \[Fig3\]). These findings provide a pathway for modulating the photonic SHE and thereby open the possibility of developing nanophotonic applications.
We also propose using the photonic SHE to identify the graphene layers [@Zhou2012b]. The quick and convenient technique for identifying the layer numbers of graphene film is important for accelerating the study and exploration of graphene material. There have many methods for determining the layer numbers of graphene film, yet existing limitation. For example, atomic force microscopy technique is the straight way to determine the layer numbers of graphene. But this method shows a slow throughput and may induce damage to the sample. Unconventional quantum Hall effects [@Zhang2005] are usually used to distinguish one layer and two layers graphene from multiple layers. Raman spectroscopy [@Gupta2006] shows characteristic for quick and nondestructive measuring the layer numbers of graphene. However, it is not obvious to tell the differences between bilayer and a few layers of graphene films [@Ni2007]. We find that the photonic SHE can serve as a useful metrological tool for characterizing the structure parameters’ variations of nanostructure due to their sensitive dependence. So, the photonic SHE may have a potential to determine the layer numbers of graphene.
![\[Fig4\] Experimental results for determining the layer numbers of graphene. (a) shows the theoretical spin-dependent shifts in the case of graphene layer numbers changing from one to five. (b) describes the experimental results for determining the layer numbers of graphene. The lines represent the theoretical results. The circle, square and triangle show the experimental data obtained from three different areas of the graphene sample. (c) Raman spectroscopy of the sample. The inset shows the graphene sample. [@Zhou2012b] ](Fig4.eps){width="17cm"}
We establish the relationship between the spin-dependent displacements and the graphene layer numbers. The weak measurements method has been used to detect the transverse shifts, and so the graphene layer numbers can be obtained. However, there exists two unknown parameters (refractive index and layer numbers of graphene) to be identified. Before identifying the graphene layers, we need to choose the suitable refractive index parameter of graphene. We choose one suitable refractive index according from the work of Bruna and Borini [@Bruna2009]. Here, the refractive index of graphene is about $3.0+1.149i$ at 633 nm. Through measuring the spin-dependent shifts of photonic SHE on the graphene film, we prove that this refractive index is suitable for actual situation. Using the suitable refractive index n=$3.0+1.149i$ at 633 nm, we can identify the layer numbers of an unknown graphene film. It should be noted that we cannot fabricate the graphene film with the precise layer numbers when the graphene film has more than two layers. We just know the approximate layer numbers ranging from three to five layers. We want to determine the actual layer numbers of this graphene film. As shown in Fig. \[Fig4\], we measure the transverse displacements with the incident angle changing from $56^{\circ}$ to $62^{\circ}$. To avoid the influence of impurities and other surface quality factors of graphene film, we carried out the experiments for three different areas of the graphene sample. It is concluded that the actual layer numbers of the film is three.
![\[Fig5\] The photonic SHE and magneto-optical Kerr effect induced by axion coupling at air-TI interfaces in the case of horizontal polarization (a-d) and the corresponding weak measurements (e). Here the parameters are the refractive index of TIs n=10 (appropriate for the TIs such as $Bi_{1-x}Se_{x}$). The beam waist is selected as $w_{0}=20\lambda$. In the weak measurements process, the incident angle is chosen as $\theta_{i}=84^{\circ}$. [@Zhou2013] ](Fig5.eps){width="10cm"}
Recently, the topological insulators (TIs) material has aroused tremendous interest [@Qi2010; @Moore2009]. It has gapless helical surface states owing to the topological protection of the time-reversal symmetry and represents a full energy gap in the bulk [@Fu2007; @Maciejko2010]. In a recent paper, we theoretically investigate the photonic SHE of a Gaussian beam reflected from the interface between air and topological insulators (TIs) [@Zhou2013]. We reveal that the spin-orbit coupling effect in TIs can be routed by adjusting the axion angle variations. It is shown that the magneto-optical Kerr effect can be significantly altered due to the axion coupling and shows close relationship with spin-dependent splitting in photonic SHE \[Fig. \[Fig5\](a) and \[Fig5\](b)\]. We find that, unlike the transverse spin-dependent splitting, the in-plane one is sensitive to the axion angle \[Fig. \[Fig5\](c) and \[Fig5\](d)\]. Due to the the limitation of experimental condition, we theoretically propose a weak measurement method to determine the strength of axion coupling by probing the in-plane splitting of the photonic SHE.
![\[Fig6\] The preliminary results for determining the magneto-optical constants. (a) and (b) describe the initial spin-dependent shifts of Fe and Co materials in the case of H and V polarizations. The corresponding amplified displacements under the condition of H and V polarizations are shown in (c) and (d). ](Fig6.eps){width="10cm"}
The incident beam is focused by the lens and is preselected in the horizontal polarization, and then it is postselected in the polarization state with $|V\pm\Delta\rangle$ ($\Delta$ is the amplified angle). The relevant amplitude of the reflected field at a plane can be obtained, allowing for calculation of amplified displacement. The theoretical amplified shifts are shownin Fig. \[Fig5\](e). Here the incident angle is fixed to $84^{\circ}$. Then we obtain the amplified displacements varying with axion angle and amplified angle. For a fixed angle $\Delta$, the amplified in-plane shifts change clearly with the different axion angles, and so we can measure the axion coupling effect by determining the in-plane displacements with weak measurements. These findings offer us potential methods for determining the strength of the axion coupling and provide new insight into the interaction of light with TIs.
We have also used the photonic SHE for determining the magneto-optical constant of magnetic film and the preliminary results can be seen in Fig. \[Fig6\]. The magneto-optical constant is an important parameter for the study and exploration of magnetic material. The relationship between the spin-dependent splitting in photonic SHE and the magneto-optical constant of magnetic film is established. Here, we choose the Fe and Co as our samples, which have different magneto-optical constants ($Q_{Fe}$=$0.0215-0.0016i$ [@Johnson1974; @Yang1993] and $Q_{Co}$=$0.0189-0.0043i$ [@Osgood1997] at 633 nm). From the Fig. \[Fig6\](a) and \[Fig6\](b), we can see that the spin-dependent displacements are sensitive to the magneto-optical constants of different magnetic materials. So, we can determine the magneto-optical constants by measuring the spin-dependent splitting of photonic SHE. In our experiment, the weak measurements has been used to detect this tiny shifts \[the preliminary theoretical results are shown in Fig. \[Fig6\](c) and \[Fig6\](d)\]. We also find that the amplified spin shifts are sensitive to the variations of magneto-optical constants (Fe and Co). Importantly, the Kerr rotation angle in magneto-optical Kerr effect can also be detected by using this way, which shows higher accuracy than the normal extinction method. In fact, our experiment is in progress and the manuscript is in preparation.
CONCLUSIONS
===========
In summary, we have reviewed our recent works on using photonic SHE for precision metrology. After establishing the quantitative relationship between the spin-dependent shifts in photonic SHE and the physical parameters in different systems, we have used the weak measurements methods for measuring the thickness of nanometal film, identifying the graphene layers, detecting the strength of axion coupling in topological insulators, and determining the magneto-optical constant of magnetic film. These findings provide a practical application for photonic SHE and thereby open the possibility of developing spin-based nanophotonic device.
This research was partially supported by the National Natural Science Foundation of China (Grants Nos. 61025024 and 11274106) and Hunan Provincial Innovation Foundation for Postgraduate (Grant No. CX2013B130).
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---
abstract: 'The aim of this paper is to develop a general framework for training neural networks (NNs) in a distributed environment, where training data is partitioned over a set of agents that communicate with each other through a sparse, possibly time-varying, connectivity pattern. In such distributed scenario, the training problem can be formulated as the (regularized) optimization of a non-convex social cost function, given by the sum of local (non-convex) costs, where each agent contributes with a single error term defined with respect to its local dataset. To devise a flexible and efficient solution, we customize a recently proposed framework for non-convex optimization over networks, which hinges on a (primal) convexification-decomposition technique to handle non-convexity, and a dynamic consensus procedure to diffuse information among the agents. Several typical choices for the training criterion (e.g., squared loss, cross entropy, etc.) and regularization (e.g., $\ell_2$ norm, sparsity inducing penalties, etc.) are included in the framework and explored along the paper. Convergence to a stationary solution of the social non-convex problem is guaranteed under mild assumptions. Additionally, we show a principled way allowing each agent to exploit a possible multi-core architecture (e.g., a local cloud) in order to parallelize its local optimization step, resulting in strategies that are both distributed (across the agents) and parallel (inside each agent) in nature. A comprehensive set of experimental results validate the proposed approach.'
address:
- 'Department of Information Engineering, Electronics and Telecommunications, “Sapienza” University of Rome, Via Eudossiana 18, 00184 Rome, Italy'
- 'Department of Engineering, University of Perugia, Via G. Duranti 93, 06125, Perugia, Italy'
author:
- Simone Scardapane
- Paolo Di Lorenzo
bibliography:
- 'biblio.bib'
title: A Framework for Parallel and Distributed Training of Neural Networks
---
Neural network, distributed learning, parallel computing, networks.
[ Published on **Neural Networks**, doi 10.1016/j.neunet.2017.04.004.]{}
Introduction {#sec:introduction}
============
We consider the problem of training a Neural Network (NN) model, when training data is distributed over different agents that are connected by a sparse, possibly time-varying, communication network. To grasp the main motivation, let us consider a ‘smart’ environment, wherein thousands of low-power sensors (e.g., cameras, wearables, etc.) are embedded to provide context-aware assistance, security provisioning, and so forth [@pottie2000wireless; @boric2002wireless]. If the amount of produced data is small and we can count on a very reliable communication network, we may think of a centralized approach where all the data are transmitted to one (or more) fusion center that performs the learning task. However, in big data applications, sharing local information with a central processor might be either unfeasible or not economical/efficient, owing to the large size of the network and volume of data, time-varying network topology, energy constraints, robustness and/or privacy concerns. Performing the computation in a centralized fashion may raise robustness concerns as well, since the central processor represents a bottleneck and an isolated point of failure. For these reasons, effective learning methods must necessarily exploit distributed computation/learning architectures (with possibly parallelized multi-core processors), while keeping into account the distributed large-scale storage of data over the network and communication constraints. Very often, the implementation of such learning schemes requires the training of a *shared* predictive function, i.e., a common model accessible independently by each of them. Considering the previous example, suppose that a set of embedded cameras is taking multiple high-resolution photos of a possible security threat. In this case, if the threat needs to be recognized quickly in the near future, the sensors have to train a shared classifier that must leverage on all the currently acquired photos, in order to obtain a sufficiently high accuracy. These problems are ubiquitous in the real world, and appear in many practical systems such as, e.g., wireless sensor networks [@predd2006distributed], smart grids, distributed databases [@lazarevic2002boosting], robotic swarms, just to name a few.
If a predictive behavior is needed, however, the designer of the distributed system has to answer a necessary question: What kind of model should be chosen as a classifier/regressor? Since deep NNs are currently obtaining state-of-the-art results in several fields [@schmidhuber2015deep; @lecun2015deep], employing them appears as a reasonable choice. Nevertheless, somewhat surprisingly, the literature on distributed training algorithms for NNs satisfying all the above requirements is extremely scarce. Most authors resort either to an ensemble of models trained independently by the agents [@lazarevic2002boosting; @zhang2013privacy], or to strategies requiring the sum of the gradients’ contributions for all agents at every single iteration [@samet2012privacy; @georgopoulos2014distributed], exploiting the additivity of the gradients updates. Both these approaches can be easily shown to be unsatisfactory in general. In the former case, we have no guarantee that the ensemble of models will perform as good as a single model trained on the collection of all local datasets. In the latter case, instead, a global sum at every iteration might be infeasible due to an excessive amount of communication, particularly for large models comprising several hundred thousands parameters. It is also worth mentioning that a lot of research has been devoted recently to the design of parallel, asynchronous versions of stochastic gradient descent for training NNs on large clusters of commodity hardware [@dean2012large; @sak2014squence; @abadi2016tensorflow]. However, all these previous methods require the presence of at least one central server node, which coordinates the learning process; thus, they are not applicable in our context.
One of the reasons for the lack of distributed training methods for NNs is that, in principle, these methods require the solution of a distributed *non-convex* optimization problem, which was tackled only in a few papers even in the optimization literature [@bianchi2013convergence; @di2016next]. On the other side, if we turn our attention to methods for convex learning problems, the literature on their distributed training is vast, including algorithms for decentralized optimization of linear predictors [@xiao2007distributed; @sayed2014adaptive; @sayed2014adaptation], sparse linear models [@mateos2010distributed; @di2013sparse], kernel ridge regression [@predd2006distributed; @predd2009collaborative], random-weights networks [@huang2015distributed; @scardapane2015distributed; @scardapane2016decentralized], support vector machines [@navia2006distributed; @lu2008distributed; @forero2010consensus; @scardapane2016distributed], and kernel filtering [@perez2010robust; @gao2015diffusion].
*Contribution:* In this paper, we propose an algorithmic framework for training general NN models in a fully distributed scenario, which encompasses several common loss functions and regularization terms.[^1] In particular, we build upon the in-network nonconvex optimization (NEXT) algorithm proposed in [@di2016next], and recently extended in [@sun2016distributed] to handle general time-varying topologies. NEXT is one of the first methods to solve distributed non-convex optimization problems over networks of agents. The algorithm, which leverages on the so-called successive convex approximation (SCA) family of methods [@facchinei2015parallel], is built upon two foundational ideas. First and foremost, at every iteration, the original non-convex problem is replaced with a strongly convex approximation, which is solved *locally* at every agent. As we will illustrate along the paper, several kinds of convexification are possible, resulting in different trade-offs in terms of computational complexity and speed of convergence. Second, the framework exploits a dynamic consensus procedure [@zhu2010discrete], so that each agent can recover the information relative to all the other agents, which typically is not available at its local side. The resulting algorithms are shown to be convergent to a stationary solution of the social non-convex problem under loose requirements relative to the agents’ communication topology, the choice of the algorithm’s parameters, and the structure of the optimization problem. A further interesting aspect of the framework presented here is that the local optimization problems can be easily parallelized in a principled way (up to one NN parameter per available processor), without loosing the convergence properties of the framework. Consider, for example, the case of multiple medical institutions requiring the training of a common NN (e.g., for diagnosis purposes) leveraging on all historical clinical information [@vieira2006secure]. In this case, a decentralized algorithm is required due to strong privacy concerns on the release of medical, sensible information about the patients. Nonetheless, each institution may have access to an internal private cloud infrastructure. Using the framework outlined in this paper, privacy is guaranteed via the use of a distributed protocol, while each institution can parallelize its optimization steps using local cloud computing hardware. In this way, the resulting algorithms are both distributed (across the nodes) and parallel (inside each node) in nature. At the end of the (distributed) training process, each agent has access to the optimal set of NN’s parameters, and it can apply the resulting model to newly arriving data (e.g., new photos taken from the camera) independently of the other agents. A comprehensive set of experimental results validate the proposed approach.
*Outline of the paper:* The rest of the paper is organized as follows. In Section \[sec:problem\_formulation\], we formalize the problem of distributed NN training. In Section \[sec:next\] we describe the general framework for distributed NN training built upon the NEXT algorithm. Then, in Section \[sec:practical\], we consider the customization of the framework to different loss functions (squared loss, cross entropy, etc.) and regularization terms ($\ell_2$ norm, sparsity inducing penalties, etc.). Section \[sec:parallelizing\_surrogate\_optimization\] describes a principled way to parallelize the optimization phase. In Section \[sec:experimental\_validation\], we perform a large set of experiments aimed at assessing the performance of the proposed framework. Finally, Section \[sec:conclusion\] draws some conclusions and future lines of research.
*Notation:* We denote vectors using boldface lowercase letters, e.g., ${\mathbf{a}}$; matrices are denoted by boldface uppercase letters, e.g., ${\mathbf{A}}$. All vectors are assumed to be column vectors. The operator ${\left\lVert \cdot \right\rVert_{p}}$ is the standard $\ell_p$ norm on an Euclidean space. For $p=2$, it coincides with the Euclidean norm, while for $p=1$ we obtain the Manhattan (or taxicab) norm defined for a generic vector ${\mathbf{v}} \in \R^B$ as ${\left\lVert {\mathbf{v}} \right\rVert_{1}} = \sum_{k=1}^B |v_k|$. The notation $a[n]$ denotes the dependence of $a$ on the time-index $n$. Other notation is introduced along the paper when required.
Problem Formulation {#sec:problem_formulation}
===================
Let us consider the problem of training a generic NN model $f({\mathbf{w}}; {\mathbf{x}})$, where ${\mathbf{x}} \in \R^d$ denotes the $d$-dimensional input vector of the network, whereas ${\mathbf{w}} \in \R^Q$ is the vector collecting all the adaptable parameters that we aim to optimize. Note that we are considering the NN as a function of its parameters, as this will make the following derivation simpler. We are not concerned with the specific structure of the NN $f(\cdot)$ (i.e., number of hidden layers, choice of the activation functions, etc.), as long as the following assumptions are satisfied for any possible input vector ${\mathbf{x}} \in \R^d$.
**Assumption A \[On the NN model\]:**
(A1)
: $f$ is in $C^1$, i.e., it is continuously differentiable with respect to ${\mathbf{w}}$;
(A2)
: $f$ has Lipschitz continuous gradient, with respect to ${\mathbf{w}}$, for some Lipschitz constant $L$, i.e.: $$\left\lVert\nabla_{{\mathbf{w}}} f({\mathbf{w}}_1; {\mathbf{x}}) - \nabla_{{\mathbf{w}}}f({\mathbf{w}}_2; {\mathbf{x}})\right\rVert_2 \le L \bigl\lVert{\mathbf{w}}_1 - {\mathbf{w}}_2\bigr\rVert_2 \,.
\label{eq:lip_gradient}$$
Assumption A is satisfied by most NN models commonly used in the literature, with the only notable exception of NN having non-differentiable activation functions such as ReLu neurons [@glorot2011deep], maxout neurons [@goodfellow2013maxout], and a few others. Nonetheless, convergence guarantees for these architectures are relatively uncommon even in the centralized case. In this paper, we are concerned with distributed architectures, where the data required to train the NN is not available on a centralized location, but is instead partitioned among $I$ interconnected agents. Prototypical examples of agents can be sensors in a wireless sensor networks (WSN), peers in a P2P network, power units in a smart grid, or mobile robots in a robotic swarm. At every specific time instant $n$, the communication network enabling interaction among the agents is modeled as a directed graph (digraph) $\mathcal{G}[n]=(\mathcal{V,E}[n])$, where $\mathcal{V}=\{1,\ldots,I\}$ is the vertex set (i.e., the set of agents), and $\mathcal{E}[n]$ is the set of (possibly) time-varying directed edges. The in-neighborhood of agent $i$ at time $n$ (including node $i$) is defined as $\mathcal{N}_i^{\rm in}[n]=\{j|(j,i)\in\mathcal{E}[n]\}\cup\{i\}$: node $i$ can receive information from node $j\neq i$ at time instant $n$ only if $j\in\mathcal{N}_i^{\rm in}[n]$. By assuming only single-hop communication, the resulting framework can be applied to the broadest possible class of problems.[^2] Due to this, each agent has a limited view and knowledge about the overall (possibly time-varying) network. Also, we assume that there is no agent (or finite number of them) that is able to collect all the data and coordinate the overall learning process. Associated with each graph $\mathcal{G}[n]$, we introduce (possibly) time-varying weights $c_{ij}[n]$ matching $\mathcal{G}[n]$: $$\begin{aligned}
\label{weights}
c_{ij}[n]=\left\{
\begin{array}{ll}
\theta_{ij}\in[\vartheta,1] & \hbox{if $j\in \mathcal{N}_i^{\rm in}[n]$;} \vspace{.2cm}\\
0 & \hbox{otherwise,}
\end{array}
\right.\end{aligned}$$ for some $\vartheta\in (0,1)$, and define the matrix ${\mathbf{C}}[n]\triangleq (c_{ij}[n])_{i,j=1}^I$. These weights are used in the definition of the proposed algorithm in order to locally combine the information diffused over every neighborhood, i.e., $c_{ij}$ represents the weight given by agent $i$ to the information coming from agent $j$. The weights are given and are required to respect some properties listed later on in . Many choices are possible, and a brief overview can be found in @di2016next. Clearly, different setups for the weights may influence the convergence speed. Roughly speaking, simple choices like the one we detail in Section \[sec:experimental\_setup\] can be implemented immediately with no knowledge of the graph topology far from each neighborhood. On the contrary, more sophisticated weights can speedup convergence, while requiring global knowledge of the network and/or the solution to some optimization problem, e.g. see the strategies detailed in @xiao2004fast.
For the purpose of training the NN, we assume that the $i$th agent has access to a local training dataset of $N_i$ examples, denoted as $\mathcal{S}_i = \left\{ {\mathbf{x}}_{i,m}, d_{i,m} \right\}_{m=1}^{N_i}$, where we consider a single-output problem with $d_{i,m} \in \R$ for simplicity of overall notation. The output of the NN is an integer or a real value, depending on whether we are facing a classification task or a regression task, respectively. Given all the previous definitions, a general formulation for the distributed training of NNs can be cast as the minimization of a social cost function $G$ plus a regularization term $r(\cdot)$, which writes as: $$\underset{{\mathbf{w}}}{\min} \;\; U({\mathbf{w}}) = G({\mathbf{w}}) + r({\mathbf{w}}) = \sum_{i=1}^I g_i({\mathbf{w}}) + r({\mathbf{w}}) \,,
\label{Dist_NN_training}$$ where $g_i(\cdot)$ is the error term relative to the $i$th local dataset: $$g_i({\mathbf{w}}) = \sum_{m\in \mathcal{S}_i} l\Bigl(d_{i,m}, f({\mathbf{w}};{\mathbf{x}}_{i,m})\Bigr) \,,
\label{eq:local_NN_cost}$$ with $l(\cdot, \cdot)$ denoting a generic (convex) loss function, while $r({\mathbf{w}})$ is a regularization term. Due to the nonlinearity of the NN model $f({\mathbf{w}};{\mathbf{x}})$, problem (\[Dist\_NN\_training\]) is typically *non-convex*. In this work, we consider the following assumptions on the functions involved in (\[Dist\_NN\_training\])-(\[eq:local\_NN\_cost\]).
**Assumption B \[On Problem (\[Dist\_NN\_training\])\]:**
(B1)
: $l$ is convex and $C^1$, with Lipschitz continuous gradient;
(B2)
: [$r$ satisfies (B1), or it is a nondifferentiable convex function with bounded subgradients;]{}
(B3)
: [$U$ is coercive, i.e., $\displaystyle\lim_{\|{\mathbf{w}}\|\rightarrow \infty} U({\mathbf{w}})=+\infty$.]{}
The structure of the function $l$ in (\[eq:local\_NN\_cost\]) depends on the learning task (i.e., regression, classification, etc.). Typical choices are the squared loss for regression problems, and the cross-entropy for classification tasks [@haykin2009neural]. The regularization function $r({\mathbf{w}})$ in (\[Dist\_NN\_training\]) is commonly chosen to avoid overfitted solutions and/or impose a specific structure in the solution, e.g., sparsity or group sparsity. Typical choices are the $\ell_2$ and $\ell_1$ norms. All these functions satisfy Assumption B, and will be discussed in detail in the sequel. In view of the distributed nature of the problem, the $i$th agent knows its own cost function $g_i$ and the common regularization term $r$, but it does not have access to $g_j$ for $j \neq i$, nor can it exchange freely its own dataset $\mathcal{S}_i$ due to a variety of reasons, including privacy, data volume, and communication constraints. This aspect, combined with the non-convexity of $\eqref{Dist_NN_training}$, makes optimizing $\eqref{Dist_NN_training}$ in a distributed fashion a challenging problem, which has no ready-to-use solution available in the literature. The design of such algorithmic framework is the topic of the next three sections.
NEXT: In-Network Successive Convex Approximation {#sec:next}
================================================
In this section, we review the basics of the NEXT framework proposed in [@di2016next], which was designed to solve general nonconvex distributed problems of the form . The next section will then focus on how to customize the framework to the NN distributed training problem considered in this paper. Due to lack of space, we provide only a very brief introduction to the NEXT framework, and we refer the interested readers to [@di2016next; @sun2016distributed] for a full treatment, which also includes a proof of the convergence results. NEXT combines SCA techniques (Step 1) with dynamic consensus mechanisms (Steps 2 and 3), as described next.
**Step 1 (local SCA optimization):** Each agent $i$ maintains a local estimate ${\mathbf{w}}_i[n]$ of the optimization variable ${\mathbf{w}}$ that is iteratively updated. Solving directly Problem (\[Dist\_NN\_training\]) may be too costly (due to the nonconvexity of $G$) and is not even feasible in a distributed setting. One may then prefer to approximate Problem (\[Dist\_NN\_training\]), in some suitable sense, in order to permit each agent to compute *locally* and *efficiently* the new iteration. In particular, writing $G({\mathbf{w}}_i)=g_i({\mathbf{w}}_i)+\sum_{j\neq i}g_j({\mathbf{w}}_i)$, we consider a convexification of ${G}$ having the following form: i) at every iteration $n$, the (possibly) nonconvex $g_i({\mathbf{w}}_i)$ is replaced by a strongly convex surrogate, say $\widetilde{g}_i(\cdot;{\mathbf{w}}_i[n]):\mathbb{R}^Q \rightarrow \mathbb{R}$, which may depend on the current iterate ${\mathbf{w}}_i[n]$; and ii) $\sum_{j\neq i}g_j({\mathbf{w}}_i)$ is linearized around ${\mathbf{w}}_i[n]$. More formally, the proposed updating scheme reads: at every iteration $n$, given the local estimate ${\mathbf{w}}_i[n]$, each agent $i$ solves the *strongly convex* optimization problem: $$\begin{aligned}
\label{best_resp_x_hat_2}
&\widetilde{{\mathbf{w}}}_i[n]\,=\,\underset{{\mathbf{w}}_i}{\arg\min} \;\;\widetilde{U}_i\left({\mathbf{w}}_i;{\mathbf{w}}_i[n],\boldsymbol{\pi}_{i}[n]\right) \\ &=\,\underset{{\mathbf{w}}_i}{\arg\min} \;\; \widetilde{g}_i({\mathbf{w}}_i;{\mathbf{w}}_i[n])+\boldsymbol{\pi}_{i}[n]^{T}({\mathbf{w}}_i-{\mathbf{w}}_i[n])+r({\mathbf{w}}_i), \nonumber\end{aligned}$$ where $$\label{pi}
\boldsymbol{\pi}_i[n]\triangleq\sum_{j\neq i}\nabla_{{\mathbf{w}}}\;g_j({\mathbf{w}}_i[n]). \vspace{-0.1cm}$$ The evaluation of (\[pi\]) would require the knowledge of all $\nabla g_j({\mathbf{w}}_i[n])$, $j\neq i$ at node $i$. This information is not directly available at node $i$; we will cope with this local lack of global knowledge later on in step 3. Once the surrogate problem (\[best\_resp\_x\_hat\_2\]) is solved, each agent computes an auxiliary variable, say ${\mathbf{z}}_i[n]$, as the convex combination: $${\mathbf{z}}_i[n] = {\mathbf{w}}_i[n] + \alpha[n]\left( \widetilde{{\mathbf{w}}}_i[n] - {\mathbf{w}}_i[n] \right) \,,
\label{eq:z_update_next}$$ where $\alpha[n]$ is a possibly time-varying step-size sequence. This concludes the optimization phase of the algorithm. An appropriate choice of the surrogate function $\widetilde{g}_i(\cdot;{\mathbf{w}}_i[n])$ guarantees the coincidence between the fixed-points of $\widetilde{{\mathbf{w}}}_i[n]$ and the stationary solutions of Problem (\[Dist\_NN\_training\]). The main results are given in the following proposition [@facchinei2015parallel]:
Given Problem (\[Dist\_NN\_training\]) under A1-A2 and B1-B3, suppose that $\widetilde{g}_i$ satisfies the following conditions:
(F1)
: $\widetilde{g}_{i} (\mathbf{\cdot};{\mathbf{w}})$ is uniformly strongly convex with $\tau_i>0$;
(F2)
: $\nabla \widetilde{g}_{i} ({\mathbf{w}};{\mathbf{w}}) = \nabla g_i({\mathbf{w}})$ for all ${\mathbf{w}}$;
(F3)
: $\nabla \widetilde{g}_{i} ({\mathbf{w}};\mathbf{\cdot})$ is uniformly Lipschitz continuous.
Then, the set of fixed-point of $\widetilde{{\mathbf{w}}}_i[n]$ in (\[best\_resp\_x\_hat\_2\]) coincides with that of the stationary solutions of (\[Dist\_NN\_training\]).
Conditions F1-F3 state that $\widetilde{g}_{i}$ should be regarded as a strongly convex approximation of $g_i$ at the point ${\mathbf{w}}$, which preserves the first order properties of $g_i$. Several feasible choices are possible for a given $g_i$; the appropriate one depends on computational and communication requirements. The goal of the next section will be to illustrate some possible choices for the local surrogate cost $\widetilde{g}_{i}$ properly customized to our distributed NN training problem.
**Step 2 (agreement update):** To force the asymptotic agreement among the ${\mathbf{w}}_i$’s, a consensus-based step is employed on the auxiliary variables ${\mathbf{z}}_i[n]$’s. Each agent $i$ updates its local variable ${\mathbf{w}}_i[n]$ as: $$\label{consensus_update}
{\mathbf{w}}_i[n+1]= \sum_{j\in \mathcal{N}_i^{\rm in}[n]} c_{ij}[n]\, {\mathbf{z}}_i[n],\vspace{-0.1cm}$$ where ${\mathbf{C}}[n]=(c_{ij}[n])_{ij}$ is defined in (\[weights\]), and satisfies $$\begin{aligned}
\label{double_stochastic}
{\mathbf{C}}[n]\,\mathbf{1}=\mathbf{1} \quad \text{and}\quad \mathbf{1}^T {\mathbf{C}}[n]=\mathbf{1}^T \quad \forall n.\end{aligned}$$ Since the weights are constrained by the network topology, can be implemented via local message exchanges: agent $i$ updates its estimate ${\mathbf{w}}_i$ by averaging over the current solutions ${\mathbf{z}}_j[n]$ received from its neighbors. The double stochasticity condition in (\[double\_stochastic\]) can be achieved according to a variety of predefined strategies, including the Metropolis-Hastings criterion [@xiao2007distributed], or by optimizing a cost function with respect to the spectral properties of the graph [@xiao2004fast].
**Step 3 (diffusion of information over the network):** The computation of $\widetilde{{\mathbf{w}}}_{i}[n]$ in (\[best\_resp\_x\_hat\_2\]) is not fully distributed yet, because the evaluation of $\boldsymbol{\pi}_{i}[n]$ in (\[pi\]) would require the knowledge of all $\nabla g_j({\mathbf{w}}_i[n])$, $j\neq i$, which is a global information that is not available locally at node $i$. To cope with this issue, as proposed in [@di2016next], we replace $\boldsymbol{\pi}_i[n]$ in (\[best\_resp\_x\_hat\_2\]) with a *local* estimate, say $\widetilde{\boldsymbol{\pi}}_{i}[n]$, asymptotically converging to $\boldsymbol{\pi}_{i}[n]$. Thus, we can update the local estimate $\widetilde{\boldsymbol{\pi}}_{i}[n]$ in a fully distributed manner as: $$\label{pi3}
\widetilde{\boldsymbol{\pi}}_i[n]\triangleq I\cdot {\mathbf{y}}_i[n]-\nabla g_i({\mathbf{w}}_i[n]),$$ where ${\mathbf{y}}_i[n]$ is a local auxiliary variable (controlled by agent $i$) that aims to asymptotically track the average of the gradients. This can be done updating ${\mathbf{y}}_i[n]$ according to the following dynamic consensus recursion: $$\label{y2}
\hspace{-0.04cm}{\mathbf{y}}_i[n+1]\triangleq\sum_{j=1}^I c_{ij}[n]{\mathbf{y}}_j[n] \hspace{-0.02cm} + \hspace{-0.02cm}\left(\nabla g_i({\mathbf{w}}_i[n+1])\hspace{-0.02cm}-\hspace{-0.02cm}\nabla g_i({\mathbf{w}}_i[n])\right) \hspace{-0.2cm}\vspace{-0.05cm}$$ where ${\mathbf{y}}_i[0]\triangleq\nabla_{{\mathbf{w}}_i}g_i({\mathbf{w}}_i[0])$, and can be computed locally by every agent. Note that the update of ${\mathbf{y}}_i[n]$ and thus $\widetilde{\boldsymbol{\pi}}_i[n]$ can be now performed locally with message exchanges with the agents in the neighborhood.
The overall procedure is summarized in Algorithm \[alg:general\], where $\nabla {\mathbf{g}}_i[n]$ is used as a simplified notation for $\nabla_{{\mathbf{w}}_i} g_i({\mathbf{w}}_i[n])$. Its convergence properties are reported in the following Proposition.
[ Let $\{{\mathbf{w}}[n]\}_n\triangleq \{({\mathbf{w}}_i[n])_{i=1}^I\}_n$ be the sequence generated by Algorithm 1, and let $\{\overline{{\mathbf{w}}}[n]\}_n\triangleq \{(1/I)\,\sum_{i=1}^I{\mathbf{w}}_i[n]\}_n$ be its average. Suppose that i) Assumptions A and B hold; ii) the sequence of graphs describing the network is $B$-strongly connected[^3]; iii) condition (\[double\_stochastic\]) holds; and iv) the step-size sequence $\{\alpha[n]\}_n$ is chosen so that $\alpha[n]\in (0,1]$ for all $n$ and $\sum_{n=0}^{\infty}\alpha[n]=\infty$. Then, (a) all the limit points of the sequence $\{\overline{{\mathbf{w}}}[n]\}_n$ are stationary solutions of (\[Dist\_NN\_training\]); (b) all the sequences $\{{\mathbf{w}}_i[n]\}_n$ asymptotically agree, i.e., $\|{\mathbf{w}}_{i}[n]-\overline{{\mathbf{w}}}[n]\|_2\underset{n\rightarrow\infty}{\longrightarrow}0
$, for all $i$.]{} \[convergence\_th\]
Algorithm 1 is a special case of an extension of the NEXT framework proposed in [@sun2016distributed] (i.e., the SONATA algorithm). Then, under the above assumptions on the NN model in (\[Dist\_NN\_training\]), the network among agents, and the algorithm’s parameters, all conditions of Theorem 1 in [@sun2016distributed] are satisfied, and the convergence result follows.
$\textbf{Data}:$ ${\mathbf{w}}_{i}[0]$, ${\mathbf{y}}_i[0]= \nabla g_i[0]$, $\boldsymbol{\pi}_{i}[0]=I{\mathbf{y}}_i[0]-\nabla g_i[0]$, $\forall i=1,\ldots ,I$, and $\{{\mathbf{C}}[n]\}_n$. Set $n=0$.
`\mbox{(S.1)}`$\,\,$If $\mathbf{w}_i[{n}]$ satisfies a global termination criterion: STOP;
`\mbox{(S.2)}` `Local Optimization`: Each agent $i$
\(a) computes $\widetilde{{\mathbf{w}}}_{i}[n]$ as: $$\begin{aligned}
\label{opt_prob_alg}
\widetilde{{\mathbf{w}}}_{i}[n]\,=&\,\underset{{\mathbf{w}}_{i}}{\arg\min} \;\;\widetilde{U}_{i}\left({\mathbf{w}}_{i};{\mathbf{w}}_{i}[n],\widetilde{\boldsymbol{\pi}}_{i}[n]\right) \,,\end{aligned}$$
\(b) updates its local variable ${\mathbf{z}}_i[n]$: $${\mathbf{z}}_i[n]={\mathbf{w}}_i[n]+\alpha[n]\left(\widetilde{{\mathbf{w}}}_{i}[n]-{\mathbf{w}}_i[n]\right) \,. \nonumber$$
`\mbox{(S.3)}` `Consensus update`: Each agent $i$
\(a) collects ${\mathbf{z}}_j[n]$ and ${\mathbf{y}}_j[n]$ from neighbors,
\(b) updates ${\mathbf{w}}_i[n]$ as: $${\mathbf{w}}_i[n+1] = \sum_{j=1}^I c_{ij}[n]\, {\mathbf{z}}_j[n] \nonumber \,,$$
\(c) updates ${\mathbf{y}}_i[n]$ as: $$\displaystyle {\mathbf{y}}_i[n+1]=\sum_{j=1}^I c_{ij}[n]\,{\mathbf{y}}_j[n]+\left(\nabla g_i[n+1]-\nabla g_i[n]\right) \nonumber\,,$$
\(d) updates $\widetilde{\boldsymbol{\pi}}_{i}[n]$ : $$\widetilde{\boldsymbol{\pi}}_{i}[n+1]=I\cdot {\mathbf{y}}_i[n+1]-\nabla g_i[n+1] \nonumber \,.$$
`\mbox{(S.4)}` $n\leftarrow n+1$, and go to `\mbox{(S.1)}.`
It is interesting to notice that convergence conditions are particularly loose. With respect to the network connecting the agents, it is enough to ensure connectivity over a finite (but arbitrary) *union* of time instants.
Step-size sequences satisfying the conditions can be derived easily, either fixed (and sufficiently small) as remarked in [@sun2016distributed], or diminishing, e.g., using the following quadratically decreasing rule that was found particularly effective in our experiments: $$\alpha[n] = \alpha[n-1]\left( 1 - \varepsilon\alpha[n-1] \right) \,,
\label{eq:step_size_sequence_example}$$ where $\alpha[0], \varepsilon \in \left(0, 1\right]$ must be chosen by the user.
The per-iteration cost of the algorithm is clearly dominated by the solution of the surrogate optimization problem in (\[opt\_prob\_alg\]). As we will see in the next section, the flexibility of the framework allows to select different choices of surrogate functions, typically impacting the complexity/performance tradeoff of the algorithm. The framework can be accelerated in two ways. First, we can parallelize the surrogate optimization in (\[opt\_prob\_alg\]); this point will be discussed in Section \[sec:parallelizing\_surrogate\_optimization\]. Second, at each iteration $n$, we can consider an *inexact* solution of the surrogate problems in (\[opt\_prob\_alg\]) within a user-specified error bound $\epsilon_i[n]$. In this case, it can be shown that convergence is still guaranteed, as long as the following condition is satisfied: $\sum_{n=0}^{\infty} \alpha[n]\epsilon_i[n] < \infty$, $\forall i \in 1, \ldots, I$, which establish a decaying rate of the error sequence over time. For further details, we refer to [@di2016next Theorem 4].
Strategies for Distributed NN Training {#sec:practical}
======================================
In this section, we customize the NEXT framework for the solution of several distributed NN training problems. In particular, we focus on the choice of the surrogate functions $\widetilde{g}_i$ in (\[best\_resp\_x\_hat\_2\]). From Proposition 1, we know that they must be chosen to satisfy F1-F3. Thus, we explore two general-purpose strategies that can be used to this end, before analyzing some practical algorithms resulting from the combination of these two strategies with common choices of the loss function and the regularization term. Essentially, the aim of $\widetilde{g}_i(\cdot)$ is to provide a strongly convex approximation of (the non-convex) $g_i$ around the current point, preserving (at least) the first-order information of the original function. Then, the most basic idea is to linearize the entire $g_i$, irrespectively of the actual choice of loss function $l$, as: $$\begin{aligned}
\widetilde{g}_i^{\text{FL}}({\mathbf{w}}_i; {\mathbf{w}}_i[n]) \;=\; & g_i({\mathbf{w}}_i[n]) + \nabla g_i({\mathbf{w}}_i[n])^T({\mathbf{w}}_i - {\mathbf{w}}_i[n]) \, \nonumber \\ & +\frac{\tau}{2}{\left\lVert {\mathbf{w}}_i - {\mathbf{w}}_i[n] \right\rVert_{2}}^2 \,,
\label{eq:full_linearization}\end{aligned}$$ where the last term in is a proximal regularization term (with $\tau \geq 0$) used to ensure strong convexity; in what follows, we will refer to as the full linearization strategy (FL). In general, the use of the FL strategy leads to the formulation of surrogate problems in allowing for a simple, closed-form solution for most choices of regularization. At the same time, this strategy is throwing away most information of $g_i(\cdot)$, by only keeping first-order information on its gradient. For this reason, the resulting family of algorithms can possess a slow convergence speed, similarly to what happens with the use of (centralized) steepest descent optimization procedures.
To implement a more sophisticated approximation aimed at preserving the hidden convexity in the problem, we start noticing that the loss function in is composed of the summation of terms, each one given by the composition of an exterior convex function (i.e., the loss function $l$), and an interior nonlinear function (i.e., the NN model $f$). Then, a possible choice for $\widetilde{g}_i$ is to preserve the convexity of $l$, while linearizing $f$ around the current estimate ${\mathbf{w}}_i[n]$, and a generic input point ${\mathbf{x}}_{i,m}$, as: $$\begin{aligned}
\widetilde{f}({\mathbf{w}}_i;{\mathbf{w}}_i[n],{\mathbf{x}}_{i,m}) = f({\mathbf{w}}_i[n]; {\mathbf{x}}_{i,m}) + \nabla f({\mathbf{w}}_i[n];{\mathbf{x}}_{i,m})^T({\mathbf{w}}_i-{\mathbf{w}}_i[n]) \,.
\label{eq:f_tilde}\end{aligned}$$ Then, the surrogate $\widetilde{g}_i$ is obtained as: $$\begin{aligned}
\widetilde{g}_i^{\text{PL}}({\mathbf{w}}_i; {\mathbf{w}}_i[n]) =\; \sum_{m\in \mathcal{S}_i} l(d_{i,m}, \widetilde{f}({\mathbf{w}}_i;{\mathbf{w}}_i[n],{\mathbf{x}}_{i,m})) +\dfrac{\tau}{2} \|{\mathbf{w}}_{i} - {\mathbf{w}}_{i}[n]\|^2,
\label{eq:partial_linearization}\end{aligned}$$ with $\tau\geq0$. We will refer to (\[eq:partial\_linearization\]) as the partial linearization (PL) strategy. It is straightforward to check that the surrogate $\widetilde{g}_i^{\text{PL}}$ in (\[eq:partial\_linearization\]) satisfies the properties F1-F3 required by Proposition 1.
In the remainder of the section, we consider a set of practical examples resulting from the use of our general framework.
Case 1: ridge regression cost {#sec:ridge_regression}
-----------------------------
As a first example, we consider the use of a squared loss function combined with a classical $\ell_2$ norm regularization on the weights (also known as weight decay in the NN literature [@moody1995simple]): $$l(a, b) \triangleq \left( a - b \right)^2 , \;\; r({\mathbf{w}}) \triangleq \frac{\lambda}{2} {\left\lVert {\mathbf{w}} \right\rVert_{2}}^2 \,,
\label{eq:ridge_regression}$$ where $\lambda$ is a positive regularization parameter. Historically, this is the most common training criterion for NNs, and it is still widely used today for regression problems. Being equivalent to a nonlinear ridge regression, we borrow this terminology here.
Let us begin with the FL strategy in (\[eq:full\_linearization\]). Note that, thanks to the specific form of the regularizer, the resulting optimization problem in (\[opt\_prob\_alg\]) is already strongly convex, so that we can set $\tau=0$. Then, using (\[eq:full\_linearization\]) and (\[eq:ridge\_regression\]), the surrogate problem in (\[opt\_prob\_alg\]) reduces to the minimization of a positive definite quadratic function, which admits a simple closed form solution, given by:
[equation]{} \_i\[n\] = -( \_i\[n\] + \_i\[n\] ) , \[eq:ridge\_FL\]
where as before $\nabla {\mathbf{g}}_i[n]$ is used as a simplified notation for $\nabla_{{\mathbf{w}}_i} g_i({\mathbf{w}}_i[n])$. Eq. represents the first practical implementation of the framework in Algorithm 1 for distributed NN training. As we can notice from , the FL strategy discards all information on the global cost function $U$ in (\[Dist\_NN\_training\]), except for a first-order approximation. Thus, the descent direction in will be proportional to the opposite of the gradient of $U$, thanks to the current estimate $\widetilde{\boldsymbol{\pi}}_i[n]$ of that is locally available at node $i$. As we will see in the numerical results, the performance of the resulting distributed scheme is similar to a centralized gradient method, sharing its advantages (low computational complexity) and its drawbacks (possible slow convergence speed).
We now proceed considering the PL strategy in (\[eq:partial\_linearization\]). To this aim, let us introduce the following ‘residual’ terms: $$r_{i,m}[n] = d_{i,m} - f({\mathbf{w}}_i[n]; {\mathbf{x}}_{i,m}) + {\mathbf{J}}_{i,m}[n]^T {\mathbf{w}}_i[n] \,,
\label{eq:residual}$$ where ${\mathbf{J}}_{i,m}[n] = \nabla_{{\mathbf{w}}_i} f({\mathbf{w}}_i[n];{\mathbf{x}}_{i,m})$ is a $Q$-dimensional vector containing the derivatives of the NN output with respect to any single weight parameter. In the general case, it will be a matrix with one column per NN output. This quantity is sometimes denoted as the weight Jacobian [@blackwell2012neural], since it measures the influence of a small parameter change on the output of the neural network.[^4] Now, using (\[eq:partial\_linearization\]) in (\[opt\_prob\_alg\]), it is easy to show that the surrogate problem can be written again as the minimization of a positive definite quadratic form, given by: $$\widetilde{{\mathbf{w}}}_i[n] = \underset{{\mathbf{w}}_i}{\arg\min}\;\; {\mathbf{w}}_i^T \Bigl( {\mathbf{A}}_i[n] + \frac{\lambda}{2}{\mathbf{I}} \Bigr){\mathbf{w}}_i - 2{\mathbf{b}}_i[n]^T{\mathbf{w}}_i \,,
\label{eq:surrogate_ridge_pl}$$ where $$\begin{aligned}
{\mathbf{A}}_i[n] &= \sum_{m\in S_i} {\mathbf{J}}_{i,m}[n]{\mathbf{J}}_{i,m}[n]^T \,, \label{eq:Ai}\\
{\mathbf{b}}_i[n] &= \sum_{m\in S_i} {\mathbf{J}}_{i,m}[n]r_{i,m}[n] - 0.5\,\widetilde{\boldsymbol{\pi}}_i[n] \label{eq:bi}\,.\end{aligned}$$ As an interesting side note, in the NN literature the matrix is known as an outer product approximation to the Hessian matrix of $g_i(\cdot)$ (i.e., the error function local to agent $i$), which is obtained by assuming that the error is uncorrelated with the second derivative of the network’s output [@bishop2006pattern Section 5.4.2]. Finally, solving the resulting minimization problem in (\[eq:surrogate\_ridge\_pl\]), the solution $\widetilde{{\mathbf{w}}}_i[n]$ of the surrogate problem, to be used in (\[opt\_prob\_alg\]), is given by:
[equation]{} \_i\[n\] = ( \_i\[n\] + )\^[-1]{}\_i\[n\] . \[eq:ridge\_PL\]
Differently from the FL strategy, whose computational complexity is linear in the number of parameters, in this case solving the surrogate problem is of the order $\mathcal{O}(Q^3)$, where $Q$ is the number of adaptable NN parameters, due to the matrix inversion step. Nevertheless, as we will see in the numerical results, the resulting descent direction provides a very large improvement in terms of convergence speed. Additionally, this strategy can benefit from a larger relative speedup when employing the parallelization strategy described in Section \[sec:parallelizing\_surrogate\_optimization\].
Case 2a: squared error with weight sparsity {#sec:lasso}
-------------------------------------------
As a second example, let us consider again the use of a squared loss term $l$ in (\[eq:ridge\_regression\]), combined this time with a sparsity promoting term given by the $\ell_1$ norm on the weight vector, i.e., $$\label{eq:ell1}
r({\mathbf{w}}) \triangleq \lambda {\left\lVert {\mathbf{w}} \right\rVert_{1}} = \lambda \sum_{k=1}^Q | w_k | \,.$$ The $\ell_1$ norm promotes sparsity of the weight vector, acting as a convex approximation of the non-convex, non-differentiable $\ell_0$ norm [@tibshirani1996regression]. While there exists customized algorithms to solve non-convex $\ell_1$ regularized problems [@ochs2015iteratively], it is common in the NN literature to apply first-order procedures (e.g., stochastic descent with momentum) followed by a thresholding step to obtain sparse solutions [@bengio2012practical]. In what follows, we illustrate the customization of the NEXT framework to this use case, using both FL and PL strategies in (\[eq:full\_linearization\]) and (\[eq:partial\_linearization\]), respectively. In the FL case, using (\[eq:full\_linearization\]) and (\[eq:ell1\]), with $\tau>0$ to ensure strong convexity, the problem in (\[opt\_prob\_alg\]) can be written as the minimization of the sum of $q$ independent functions, as follows: $$\begin{aligned}
\widetilde{{\mathbf{w}}}_i[n] =\, & \underset{{\mathbf{w}}_i}{\arg\min}\; \sum_{k=1}^q \Bigl\{ \left( \nabla g_{ik}[n] + \widetilde{\pi}_{ik}[n] - \tau w_{ik}[n] \right) w_{ik} \Bigr. \, \nonumber \\ & \Bigl. +\frac{\tau}{2} w_{ik}^2 + \lambda |w_{ik}| \Bigr\} \,.
\label{eq:surrogate_lasso_fl}\end{aligned}$$ After some easy calculations, the solution of the optimization problem in is given by the closed form expression:
[equation]{} \_i\[n\] = \_[/]{} ( \_[i]{}\[n\] - \_i\[n\] - \_i\[n\] ) , \[eq:minimizer\_surrogate\_LASSO\_parallel2\]
where $$\begin{aligned}
\mathcal{S}_{\gamma}(z)={\rm sign}(z)\max(0,|z|-\gamma),
\label{eq:soft_thresh}\end{aligned}$$ is the (component-wise) soft thresholding function.
In the PL case, using (\[eq:partial\_linearization\]) and (\[eq:ell1\]), the problem in (\[opt\_prob\_alg\]) can be cast as an $\ell_1$ regularized quadratic program, given by:
[align]{} \_i\[n\] & = { \_i\^T ( \_i\[n\] + ) \_i - .\
& . 2( \_i\[n\] + 0.5\_i\[n\] )\^T \_i + [\_i \_[1]{}]{} } , \[eq:surrogate\_lasso\_pl\]
where ${\mathbf{A}}_i[n]$ and ${\mathbf{b}}_i[n]$ are given by (\[eq:Ai\]) and (\[eq:bi\]), respectively. This is the first case we encounter where the solution of the optimization step cannot be expressed immediately in a closed form. Nevertheless, problem is given by the sum of a strongly convex function and an $\ell_1$ term, and many efficient strategies can be used for its approximate solution, including FISTA [@beck2009fast], coordinate descent methods [@cevher2014convex], and several others.
Case 2b: group sparse penalization {#sec:group_sparse}
----------------------------------
The formulation introduced in Sec. \[sec:lasso\] can be easily extended to handle a *group sparse* penalization, which allows the selective removal of entire neurons during the training process, see, e.g., [@scardapane2017group]. The basic idea is to force all the outgoing weights from a neuron to be *simultaneously* either non-zero or zero; the latter resulting in the direct removal of the neuron itself. Note that a neuron here can correspond to an input neuron, to a neuron in a hidden layer, or to a bias term, thus allowing the removal of input features, hidden neurons, and bias terms from the trained network (see [@scardapane2017group] for details). To this aim, let us suppose that the neurons are ordered and indexed as $1, \ldots, P$. Also, let us denote by ${\mathbf{w}}_{i,p}, \, p=1,\ldots,P$, the subset of weights of ${\mathbf{w}}_i$ collecting all connections between the $p$th neuron with all the neurons in the following layer. Group sparsity can then be imposed by choosing in (\[Dist\_NN\_training\]) the following regularization term: $$r({\mathbf{w}}) \triangleq \lambda \displaystyle \sum_{p=1}^P \rho_p {\left\lVert {\mathbf{w}}_{p} \right\rVert_{2}} \,,
\label{eq:reg_group_lasso}$$ where $\rho_p = \sqrt{r_p}$ are positive constants, $p=1,\ldots,P$, with $r_p$ denoting the dimensionality of ${\mathbf{w}}_p$.
Let us now analyze the customization of our framework when the FL strategy in (\[eq:full\_linearization\]) is applied. Then, let us define ${\mathbf{a}}_i[n] = \nabla {\mathbf{g}}_{i}[n] + \widetilde{\boldsymbol{\pi}}_{i}[n] - \tau {\mathbf{w}}_{i}[n]$, denoting with ${\mathbf{a}}_{i,p}$ the restriction of ${\mathbf{a}}_i$ to the indexes associated with the $p$th group. Thus, the surrogate problem in (\[opt\_prob\_alg\]) writes as:
[align]{} \_i\[n\] = \_[p=1]{}\^P { \_[i,p]{}\^T\_[i,p]{} + . . [\_[i,p]{} \_[2]{}]{}\^2 + \_p [\_[i,p]{} \_[2]{}]{} } . \[eq:group\_lasso\_fl\]
As we can notice from (\[eq:group\_lasso\_fl\]), the cost function is given by a summation of costs, each one dependent on a single neuron. Also in this case, even if problem (\[eq:group\_lasso\_fl\]) cannot be solved in closed form, it is possible to implement very fast and efficient algorithms for its solution, see, e.g., [@schmidt2010graphical; @cevher2014convex]. Furthermore, in the case each agent has a multi-core architecture, the structure of (\[eq:group\_lasso\_fl\]) makes straightforward the parallelization of computation, where each local processor can take care only of a subset of neurons. Finally, considering the PL strategy, the resulting formulation is equivalent to , with the only difference that replaces the $\ell_1$ norm. Again, many of the techniques mentioned before can be used to solve also the resulting (group sparse) strongly convex problem.
Case 3: cross-entropy loss {#sec:cross_entropy}
--------------------------
As an additional example, let us consider the case of binary classification, i.e., $d_{i,m} \in \left\{0,1\right\}$. Then, assuming the output of the NN is limited between $0$ and $1$, a standard optimization criterion involves the cross-entropy loss function in (\[Dist\_NN\_training\]), i.e.: $$l(a,b) \triangleq - a\log(b) - (1-a)\log(1-b) \,.
\label{eq:cross_entropy_loss}$$ In this case, using the FL strategy in , we obtain the same closed form solution as in (or ) by using the $\ell_2$ (or $\ell_1$) regularization, with the only difference being that each function $g_i$ in (\[eq:local\_NN\_cost\]) now depends on the cross-entropy loss in . The PL case, instead, requires some additional care. In particular, although the NN output is bounded, the same is not true for its linear approximation . Simply substituting in might result in undefined values, since the argument of the logarithm must be positive. To tackle this issue, let us notice that in this case the NN model can be written as: $$f({\mathbf{w}}; {\mathbf{x}}) = \sigma\left(f_L({\mathbf{w}}; {\mathbf{x}})\right) \,,$$ where $\sigma(\cdot)$ is a squashing function (without loss of generality, we assume it to be a sigmoid), and $f_L$ is the NN output up to (but not including) the activation function of the output neuron. The sigmoid $\sigma(z)$ is non-convex, but its internal composition with the cross-entropy loss in is convex, see, e.g., [@boyd2004convex]. Exploiting such hidden convexity, we can write the surrogate problem in (\[opt\_prob\_alg\]), while satisfying the conditions F1-F3 in Proposition 1, as follows:
[align]{} \[n\] = & { l(d\_[i,m]{}, (\_L(\_i;\_i\[n\],\_[i,m]{}))) .\
& . +\_i\[n\]\^T \_i + r(\_i) + [\_i - \_i\[n\] \_[2]{}]{}\^2 } , \[eq:surrogate\_cross\_entropy\_pl\]
where $\widetilde{f}_L(\cdot)$ is the first-order linearization of $f_L$ defined as in . Also in this case we cannot make any further simplifications although, once again, the strong convexity of the problem makes it relatively easy to be solved (roughly equivalent to a traditional logistic regression).
Parallelizing the Local Optimization {#sec:parallelizing_surrogate_optimization}
====================================
In this section, we explore how each agent can parallelize the local optimization in (\[opt\_prob\_alg\]), when having access to $C$ separate computing machines (e.g., cores, or computers in a cloud). As we stated in the introduction, this effectively gives rise to algorithms that are both distributed (across agents) and parallel (inside each agent) in nature. To this end, suppose that the weight vector ${\mathbf{w}}_i$ is partitioned in $C$ non-overlapping blocks ${\mathbf{w}}_{i,1}, \ldots, {\mathbf{w}}_{i,C}$, so that ${\mathbf{w}}_i = \bigcup_{c=1}^C {\mathbf{w}}_{i,c}$ (assuming that the union keeps the original order). Note that we use a similar notation as in Section \[sec:group\_sparse\] to identify a single group, i.e., using an additional subscript under the variable. For convenience, we also define ${\mathbf{w}}_{i,-c}[n]\triangleq ({\mathbf{w}}_{i,p}[n])_{1=p\neq c}^C$ as the tuple of all blocks excepts the $c$-th one, and similarly for all other variables. Additionally, we assume that the regularization term $r$ is block separable, i.e., $r({\mathbf{w}}_i)=\sum_{c=1}^C r_{i,c}({\mathbf{w}}_{i,c})$ for some $r_{i,c}$. This is true for the $\ell_2$ and $\ell_1$ norms, and it holds true also for the group sparse norm in (\[eq:reg\_group\_lasso\]) if we choose the groups in a consistent way. Then, the key idea is to decompose (\[opt\_prob\_alg\]) on a per-core-basis, and solve a sequence of (strongly) convex low-complexity subproblems, whereby all processors of agent $i$ update their blocks in parallel. To this aim, we build a surrogate function $\widetilde{g}_i$ that additively decomposes over the different cores, i.e.: $$\widetilde{g}_i({\mathbf{w}}_i; {\mathbf{w}}_i[n])=\sum_{c=1}^C \widetilde{g}_{i,c}({\mathbf{w}}_{i,c};{\mathbf{w}}_{i,-c}[n]) \,,
\label{eq:parallel_surrogate_function}$$ where each $\widetilde{g}_{i,c}(\cdot; {\mathbf{w}}_{i,-c}[n])$ is any surrogate function satisfying conditions F1-F3 on the variable ${\mathbf{w}}_{i,c}$. It is easy to check that the surrogate $\widetilde{g}_i$ in (\[eq:parallel\_surrogate\_function\]) satisfies F1-F3 on the variable ${\mathbf{w}}_i$. Given (\[eq:parallel\_surrogate\_function\]), each core $c$ can then minimize its corresponding term independently of the others, and their solutions can be aggregated to form the final solution vector. In the case of the FL strategy, parallelization is not particularly effective. In fact, the final solution is given by simple aggregation of vectors as in , whose computation has linear complexity with respect to the size of ${\mathbf{w}}_i$, eventually with a point-wise application of the thresholding operator in . However, the (linear) cost of solving the surrogate problems at each core is easily overshadowed by the need of computing gradients via a backpropagation step. On the other side, parallelization can largely reduce computational complexity when using the PL strategy. To give an example of application of the proposed methodology, in the sequel we illustrate how to parallelize the local optimization in the case of a ridge regression cost as in Sec. \[sec:ridge\_regression\]. In particular, let us consider the surrogate function in . To obtain the surrogate function associated to each core $c$, we fix in all the variables ${\mathbf{w}}_{i,-c}[n]$, such that the resulting function depends only on ${\mathbf{w}}_{i,c}$. The surrogate associated to core $c$ is then given by: $$\begin{aligned}
\label{eq:rige_parallel}
\widetilde{U}_{i,c}({\mathbf{w}}_{i,c}; {\mathbf{w}}_{i,-c}[n]) & = {\mathbf{w}}_{i,c}^T \Bigl( {\mathbf{A}}_{i,c,c}[n] + \frac{\lambda}{2}{\mathbf{I}} \Bigr){\mathbf{w}}_{i,c} \, - \nonumber \\ & 2\left({\mathbf{b}}_{i,c}[n] - {\mathbf{A}}_{i,c,-c}[n]{\mathbf{w}}_{i,-c}[n] \right)^T{\mathbf{w}}_i \,,\end{aligned}$$ where ${\mathbf{A}}_{i,c,c}[n]$ is the block (rows and columns) of the matrix ${\mathbf{A}}_i[n]$ in (\[eq:Ai\]) corresponding to the $c$-th partition, whereas ${\mathbf{A}}_{i,c,-c}[n]$ takes the rows corresponding to the $c$-th partition and all the columns not associated to $c$. The minimum of (\[eq:rige\_parallel\]) is:
[equation]{}\[eq:rige\_parallel\_solution\] \_[i,c]{}\[n\] = ( \_[i,c,c]{}\[n\] + )\^[-1]{}(\_[i,c]{}\[n\] - \_[i,c,-c]{}\[n\]\_[i,-c]{}\[n\] ) ,
$c=1,\ldots,C$, and the overall solution is given by $\widetilde{{\mathbf{w}}}_{i}[n]=(\widetilde{{\mathbf{w}}}_{i,c}[n])_{c=1}^C$. As we can see from (\[eq:rige\_parallel\_solution\]), the effect of the parallelization is evident: At each iteration $n$, each core has to invert a matrix having (approximately) size $\frac{1}{C}$ of the original one in (\[eq:ridge\_PL\]), thus remarkably reducing the overall computational burden. Similar arguments can be used also to parallelize the formulations in , , and .
Experimental Validation {#sec:experimental_validation}
=======================
In this section, we assess the performance of the proposed method via numerical simulations. We begin by analyzing the test error of the solutions obtained by the algorithms for some representative regression and classification datasets in Sections \[sec:results\_regression\_datasets\] and \[sec:results\_classification\_datasets\], respectively. Then, we consider the convergence behaviors of the proposed framework, comparing it to centralized and distributed counterparts, in Section \[sec:analysis\_of\_convergence\]. In Section \[sec:exploiting\_parallelization\], we describe the speed-up achieved thanks to the parallelization strategy outlined before. Finally, we consider large-scale inference in Section \[sec:large\_scale\_experiment\]. Python code to repeat the experiments is available under open-source license on the web.[^5] The code is built upon the Theano [@bergstra2010theano] and Lasagne[^6] libraries.
Experimental setup {#sec:experimental_setup}
------------------
In all experiments, the original dataset is normalized so that both inputs and outputs lie in the $\left[0, 1\right]$ range. Then, the dataset is partitioned as follows. First, $20\%$ of the dataset is kept separate to test the algorithms. The remaining $80\%$ is partitioned evenly among a randomly generated network of $10$ agents. For simplicity, we consider networks with time-invariant, symmetric connectivity, such that every pair of agents have a $20\%$ probability of being connected, with the only requirement that the overall network is connected. An example of such connectivity is shown in Fig. \[fig:network\_example\].
![Example of communication network with $10$ agents (represented by red dots), possessing a sparse, time-invariant, symmetric connectivity.[]{data-label="fig:network_example"}](images/network_example)
We have selected the weight coefficients in (\[weights\]) using the *Metropolis-Hastings* strategy [@lopes2008diffusion]: $$C_{ij} =
\begin{cases}
\frac{1}{\max\{\delta_i, \delta_j\} + 1} & \; i \neq j,\, j \in \mathcal{N}_i \\
1 - \sum_{j \in \mathcal{N}_i} \frac{1}{\max\{\delta_i, \delta_j\} + 1} & \; i = j \\
0 & \; \text{otherwise}
\end{cases}\,$$ where $\delta_i$ is the degree of node $i$. It it easy to check that this choice of the weight matrix satisfies the convergence conditions of the framework. Missing data is handled by removing the corresponding example. All experiments are repeated $25$ times by varying the data partitioning and the NN initialization.
Regarding the NN structure, we use hyperbolic tangent nonlinearities in all neurons, except for classification problems, where we use a sigmoid nonlinearity in the output neuron. The weights of the NN are initialized independently at every agent using the normalized strategy described by @glorot2010understanding. All algorithms run for a maximum of $1000$ epochs. In all the figures illustrating the results of the distributed strategies, whenever not explicitly stated, we consider the evolution of the average weight vector $\overline{{\mathbf{w}}}[n]$ as defined in Proposition $2$.
Results with regression datasets {#sec:results_regression_datasets}
--------------------------------
**Dataset** **Features** **Samples** **NN Topology** $\boldsymbol{\lambda}$ **Source**
------------- -------------- ------------- ----------------- ------------------------ ------------
Boston $13$ $506$ $10$ $10^{-1}$ UCI
Kin8nm $7$ $8192$ $8/5$ $10^{-2}$ DELVE
Wine $10$ $4898$ $12$ $10^{-2}$ UCI
: Schematic description of the datasets used for regression. For the NN topology, $x/y$ denotes a NN with two hidden layers of dimensions $x$ and $y$ respectively.[]{data-label="tab:datasets_regression"}
We start considering three representative regression datasets, whose characteristics are summarized in Table \[tab:datasets\_regression\]. Boston (also known as the Housing dataset) is the task of predicting the median value of a house based on a set of features describing it [@quinlan1993combining]. Kin8nm is a member of the kinematics family of datasets[^7], having high non-linearity and a medium amount of additive noise. Finally, Wine concerns predicting the subjective quality of a (white) wine based on a wide set of chemical features [@cortez2009modeling]. The fourth and fifth columns in Table \[tab:datasets\_regression\] describe the parameters of the NN in terms of hidden neurons and regularization coefficients. These parameters are chosen based on an analysis of previous literature in order to obtain state-of-the-art results. However, we underline that our aim is to compare different solvers for the same NN optimization problem, and for this reason only relative differences in accuracy are of concern.
In particular, we compare the results of our algorithms with respect to five state-of-the-art *centralized* solvers, in terms of mean-squared error (MSE) over the test data, when solving the global optimization problem with the ridge regression cost in . Note that these solvers would not be available in a distributed scenario, and are only used for comparison purposes as optimal benchmarks. Specifically, we consider the following algorithms:
**Gradient descent** (GD)
: : this is a simple first-order steepest descent procedure with fixed step-size.
**AdaGrad**
: [@duchi2011adaptive] : differently from GD, this algorithm employs different adaptive step-sizes per weight, which evolve according to the relative values of the gradients’ updates.
RMSProp
: : equivalent to an AdaDelta variant [@zeiler2012adadelta], it also considers adaptive independent step-sizes; however they are adapted based on shorter time windows in order to avoid exponentially decreasing schedules.
Conjugate gradient (CG)
: : this is the Polak-Ribiere variant of the nonlinear conjugate gradient algorithm [@nocedal2006numerical], implemented in the SciPy library.[^8]
L-BFGS
: : a low-memory version of the second-order BFGS algorithm [@byrd1995limited], keeping track of an approximation to the full Hessian matrix, also implemented in the SciPy library.
In addition, we consider the behavior of a centralized implementation of the PL strategy in , denoted as PL-SCA, resulting in a novel centralized algorithm. In particular, assuming all data is available on a single location, we can consider a centralized equivalent of and as: $$\begin{aligned}
{\mathbf{A}}[n] & = \sum_{i=1}^I {\mathbf{A}}_i[n] \,, \label{eq:A} \\
{\mathbf{b}}[n] & = \sum_{i=1}^I \sum_{m \in \mathcal{S}_i} {\mathbf{J}}_{i,m}[n]r_{i,m}[n] \,.\end{aligned}$$ Following similar arguments as in Sections \[sec:next\] and \[sec:ridge\_regression\], PL-SCA is defined by the iterative application of the following recursion: $$\begin{aligned}
\widetilde{{\mathbf{w}}}[n] & = \Bigl( {\mathbf{A}}[n] + \frac{\lambda}{2}{\mathbf{I}} \Bigr)^{-1}{\mathbf{b}}[n] \,, \\
{\mathbf{w}}[n+1] & = {\mathbf{w}}[n] + \alpha[n]\left( \widetilde{{\mathbf{w}}}[n] - {\mathbf{w}}[n] \right) \,.\end{aligned}$$ The centralized implementation of the FL strategy is almost equivalent to GD, so we do not consider it separately. For the distributed algorithms, we consider both the PL strategy in , denoted as PL-NEXT, and the FL strategy in , denoted as FL-NEXT. For PL-SCA, PL-NEXT and FL-NEXT we use the quadratically decreasing step-size sequence defined in . To have a fair comparison, the parameters of the step-size sequence in (\[eq:step\_size\_sequence\_example\]) were tuned at hand in order to select the fastest convergence behavior for all algorithms.
------------------------- ----------------------- ----------------------- ------------------- ------------------------------ ---------------------------------- ---------------------------------- ----------------------- ----------------------------------
**Dataset**
(lr)[2-7]{} (lr)[8-9]{} GD AdaGrad RMSProp CG L-BFGS PL-SCA FL-NEXT PL-NEXT
Boston $0.010 \pm 0.001$ $0.009 \pm 0.001$ $0.009 \pm 0.001$ ${\mathbf{0.007 \pm 0.001}}$ ${\mathbf{0.007 \pm 0.001}}$ ${\mathbf{0.007 \pm 0.001}}$ $0.010 \pm 0.001$ ${\mathbf{0.007 \pm 0.001}}$
Kin8nm $0.019 \pm \approx 0$ $0.018 \pm \approx 0$ $0.015 \pm 0.001$ $0.011 \pm 0.001$ ${\mathbf{0.009 \pm 0.001}}$ ${\mathbf{0.009 \pm \approx 0}}$ $0.019 \pm \approx 0$ ${\mathbf{0.009 \pm \approx 0}}$
Wine $0.018 \pm 0.001$ $0.016 \pm 0.001$ $0.017 \pm 0.001$ $0.038 \pm 0.019$ ${\mathbf{0.014 \pm \approx 0}}$ ${\mathbf{0.014 \pm 0.001}}$ $0.017 \pm 0.001$ ${\mathbf{0.014 \pm 0.001}}$
------------------------- ----------------------- ----------------------- ------------------- ------------------------------ ---------------------------------- ---------------------------------- ----------------------- ----------------------------------
The results on this set of experiments are provided in Table \[tab:results\_regression\_problems\], both in terms of the mean and the standard deviation. Several conclusions can be drawn from the table. For the centralized algorithms, L-BFGS, being a second-order algorithm, is able to obtain the best accuracies, and it is matched only by CG in the Boston case (in the next section we will show some plots of the convergence behavior of the different algorithms). Interestingly, PL-SCA is able to match L-BFGS in all cases, complementing our previous observation that the matrix in acts as an approximation of the Hessian matrix. For the distributed algorithms, we see similar distinctions between FL-NEXT and PL-NEXT. Specifically, PL-NEXT has comparable accuracies with respect to L-BFGS, while FL-NEXT obtains errors comparable to GD and AdaGrad. Clearly, the improved convergence comes at the cost of a higher computational burden (due to the need of inverting a matrix in ), in line with the equivalent difference in the centralized case. Summarizing, we see that FL-NEXT and PL-NEXT represent viable algorithms for distributed scenarios, providing a relative trade-off with respect to convergence and computational requirements, and matching the respective centralized implementations that are not viable in the distributed setting treated in this paper. Importantly, this is also achieved with a minimal (or non-existent) increase in term of variance. We defer a statistical analysis of the results to the next section, in order to consider also the classification datasets.
Results with classification datasets {#sec:results_classification_datasets}
------------------------------------
In this section, we analyze the performance of the distributed algorithms when applied to two classification problems, whose characteristics are briefly summarized in Table \[tab:datasets\_classification\]. Wisconsin is a medical classification task, aimed at separating cancerous cells from non-cancerous ones from several features describing the cell nucleus.[^9] The Cardiotocography (CGT) dataset is another clinical problem, where we wish to infer suspect/pathological fetuses from several biometric signals.[^10] In this case, we solve the global optimization problem with the cross-entropy loss in and a squared regularization term. We analyze the behavior of both the FL strategy and the PL strategy when compared to the state-of-the-art solvers described in the previous section. For PL-NEXT, the local surrogate problem in is solved with AdaGrad, run for a maximum of $50$ iterations (with an initial step-size of $0.1$), or until the gradient norm is below a fixed threshold of $10^{-6}$. For the local optimization at each agent, we perform a ‘warm start’ from the current estimate ${\mathbf{w}}_i[n]$.
**Dataset** **Features** **Samples** **NN Topology** $\boldsymbol{\lambda}$ **Source**
------------- -------------- ------------- ----------------- ------------------------ ------------
Wisconsin $9$ $689$ $10$ $10^{-0.5}$ UCI
CTG $28$ $2126$ $15/8$ $10$ UCI
: Schematic description of the datasets used for classification. See Table \[tab:datasets\_regression\] and the text for details on the NN topology.[]{data-label="tab:datasets_classification"}
------------------------- ------------------------------ ------------------- ------------------- ------------------------------ ------------------------------ ------------------- ------------------------------
**Dataset**
(lr)[2-6]{} (lr)[7-8]{} GD AdaGrad RMSprop CG L-BFGS FL-NEXT PL-NEXT
Wisconsin ${\mathbf{0.025 \pm 0.009}}$ $0.027 \pm 0.006$ $0.027 \pm 0.005$ $0.028 \pm 0.006$ $0.028 \pm 0.006$ $0.027 \pm 0.007$ ${\mathbf{0.025 \pm 0.009}}$
CTG $0.084 \pm 0.010$ $0.082 \pm 0.010$ $0.087 \pm 0.014$ ${\mathbf{0.083 \pm 0.011}}$ ${\mathbf{0.083 \pm 0.010}}$ $0.087 \pm 0.007$ ${\mathbf{0.084 \pm 0.009}}$
------------------------- ------------------------------ ------------------- ------------------- ------------------------------ ------------------------------ ------------------- ------------------------------
The overall results are given in Table \[tab:results\_classification\_problems\] in terms of misclassification rate. We see that, in this case, first-order algorithms are generally competitive, with the GD solver obtaining the best accuracy among the centralized solvers for the Wisconsin dataset, and CG/L-BFGS obtaining a slightly better result in the CTG case. Nevertheless, the distributed strategies are again able to obtain state-of-the-art results, with PL-NEXT consistently obtaining the lowest misclassification rate, and FL-NEXT ranging close to AdaGrad and RMSProp. In order to formalize the intuition that PL-NEXT is generally converging to a better minimum than FL-NEXT, we perform a Wilcoxon signed-rank test [@demsar2006statistical] on the results over both regression and classification datasets. The difference is found to be significant with a $p=0.05$ confidence value (although the number of datasets under consideration is relatively small). We can reasonably conclude that PL-NEXT seems a better choice in terms of accuracy, if it is possible for the agents to cope with the increased computational cost.
Analysis of convergence {#sec:analysis_of_convergence}
-----------------------
In a distributed setting, the final accuracy is not the only parameter of interest. We are also concerned on how fast this accuracy is obtained, because the convergence speed has a direct impact on the communication burden over the network of agents. As we mentioned in the introduction, in the case of general non-differentiable regularizers $r$, there is no ready-to-use alternative for comparing our proposed algorithms. However, in the specific case where the regularization function $r$ satisfies assumption B1, we can easily adapt the framework introduced in @bianchi2013convergence, resulting in a simple method that we denote as the distributed gradient (DistGrad) algorithm. Similarly to the NEXT framework, DistGrad alternates between a local optimization phase and a communication phase. In the optimization phase, each agent iteratively updates its own estimate according to a local gradient descent step as follows: $${\mathbf{z}}_{i}[n] = {\mathbf{w}}_i[n] - \eta[n]\left( \nabla {\mathbf{g}}_i[n] + \frac{1}{I} \nabla r({\mathbf{w}}_i[n]) \right) \,,
\label{eq:distgrad}$$ where $\eta[n]$ is the step-size sequence. In the communication phase, the local estimates ${\mathbf{z}}_{i}[n]$ are combined similarly to . DistGrad can be seen as a simplified version of the FL strategy, where we do not consider the dynamic consensus step (i.e., Step 3 of NEXT). For fairness of comparison, we use the step-size rule in , and the same strategy for selecting the combination coefficients in (\[weights\]).
In Figs. \[fig:boston\_obj\]-\[fig:kin8nm\_obj\] we plot the evolution of the global cost function in for FL-NEXT, PL-NEXT, DistGrad and a few representative centralized solvers for two different datasets. For improved readability, the behavior of centralized solvers is depicted using dashed lines, while the distributed algorithms are shown with solid lines. We see that the results are similar to what we have already discussed previously for the final test error: PL-NEXT is able to track consistently the convergence rate of L-BFGS, while FL-NEXT achieves results comparable to (centralized) first-order procedures. Differently, the DistGrad algorithm is slower and, for a given number of epochs, has a very large gap compared to other methods.
Another performance metric of interest is the transient behavior of the test error in terms of the amount of scalar values that are exchanged among agents in the network. We plot this metric for the three distributed algorithms in Figs. \[fig:boston\_test\_error\]-\[fig:kin8nm\_test\_error\], where the $y$-axis is shown with a logarithmic scale. We notice that DistGrad requires exactly half as many scalars to be exchanged at every iteration (since it does not rely on the dynamic consensus to track the average gradient). Nevertheless, from Figs. \[fig:boston\_test\_error\]-\[fig:kin8nm\_test\_error\], we can see that both PL-NEXT and FL-NEXT can reach better errors with respect to DistGrad for any given amount of scalars exchanged, showing their better efficiency in terms of overall communication burden. PL-NEXT is particularly well performing, with only a very small amount of communication required for achieving an error close to the optimal one.
A final metric of interest is the average disagreement among the agents, which is computed as: $$\label{disagreement}
D[n] \triangleq \frac{1}{I} {\left\lVert {\mathbf{w}}_i[n] - \overline{{\mathbf{w}}}[n] \right\rVert_{\infty}} \,.$$ We plot the behavior of (\[disagreement\]) for PL-NEXT and FL-NEXT in Figs. \[fig:boston\_dis\]-\[fig:kin8nm\_dis\], where we can see that both algorithms rapidly tend to reach a consensus among the different agents in the network.
Exploiting parallelization {#sec:exploiting_parallelization}
--------------------------
Next, we investigate the speed-up obtained by parallelizing the local optimization at each agent. We consider again the Boston and Kin8nm datasets, but we vary the number of (local) processors available at every agent in the range $2^j$, with $j=0, 1, \ldots, 4$. The relative speedup with respect to the baseline $C=1$ is shown in Figs. \[fig:boston\_parallel\_time\]-\[fig:kin8nm\_parallel\_time\]. We see that the speedup is roughly linear with respect to the amount of available processors, so that in the case $C=16$ we only need $\approx \frac{1}{3}$ of the time for Boston, and $\approx \frac{1}{2}$ for Kin8nm. Additionally, in Figs. \[fig:boston\_parallel\_obj\]-\[fig:kin8nm\_parallel\_obj\], we can visualize the evolution of the overall cost function for $C=1$, $C=4$ and $C=16$. From the figures, we can notice that the improvement in training time is obtained with only a limited effect on the convergence behavior, where in the worst case we obtain only a very small (or null) decrease.
Experiment on a large-scale dataset {#sec:large_scale_experiment}
-----------------------------------
Before concluding this experimental section, we briefly discuss the important point of large-scale distributed learning, i.e., performing distributed inference whenever $N_k$ is very large for the majority of the agents. To this end, we consider the YearPredictionMSD dataset [@bertin2011million], which is one of the largest regression datasets available on the UCI repository. The task is to predict the year of release of a song starting from $90$ audio features. There are $463,715$ examples for training, and $51,630$ examples for testing (of different authors). Similarly to before, we preprocess the input and output values in $\left[0, 1\right]$, and we consider a NN with two hidden layers having, respectively, $40$ and $20$ neurons. We partition the training data among $10$ different agents, and we compare PL-NEXT with AdaGrad. We choose AdaGrad for two main reasons, i.e., it was found to be extremely fast in the previous section, and we can use it together with stochastic updates over small batches of data in order to handle the large-scale dataset. Specifically, for every iteration AdaGrad is updated with mini-batches of $500$ elements, and accuracy is computed after a complete pass over the training dataset. The regularization is chosen as $\lambda = 10$. Step-sizes are chosen in order to guarantee a smooth convergence behavior.
![Average evolution of the loss on the MSD dataset (see the text for a full description). For AdaGrad, one epoch corresponds to an entire pass over the training data.[]{data-label="fig:msd_loss_evolution"}](images/msd_loss_evolution)
The evolution of the global loss function in is shown in Fig. \[fig:msd\_loss\_evolution\]. Despite AdaGrad making several stochastic update steps at every iteration, PL-NEXT is able to achieve a comparable convergence behavior, with a minimum loss value which is slightly better due to the unnoisy gradient evaluations. Both algorithms also achieve a similar mean squared error on the independent test set, which is around $0.011$. For comparison, the average MSE of a support vector algorithm is around $0.013/0.014$ [@ho2012large].
This example shows two important aspects of large-scale inference over networks. First of all, what is considered a challenging benchmark in a centralized environment might be relatively simpler in a distributed experiment, since the training data must be partitioned over several agents. In this case, for example, the original half a million training points must be partitioned over $10$ agents, so that each agent only has to deal with $\approx 50,000$ training points. Thus, there is the need of designing more elaborate benchmarks to test the capabilities of the algorithms is larger situations. Secondly, properly handling these datasets will require the development of stochastic updates at every agent, paralleling the stochastic algorithms used in the centralized case and commonly used in the deep learning literature. Having such stochastic algorithms for distributed, non-convex problems remain an open problem in the literature, and we remark it here as the main line of research for future investigations.
Conclusions and Future Works {#sec:conclusion}
============================
In this paper, we have investigated the problem of training a NN model in a distributed scenario, where multiple agents have a limited knowledge of the training dataset. We have proposed a provably convergent procedure to this aim, which builds exclusively on local optimization steps and one-hop communication steps. The method can be customized to several typical error functions and regularization terms. We have also described an immediate way to parallelize the local optimization phase across multiple processors/machines, available at each agent, with a limited impact on the convergence behavior.
One immediate extension of the framework presented here is to handle non-convex regularization terms, which are generally considered too challenging in practice. One example is the sample variance penalization [@maurer2009empirical], which is defined in terms of the NN output. Additional extensions can consider the presence of non-differentiable points in the NN model (e.g., by using ReLu activation functions), stochastic updates of the surrogate functions, and online formulations where new data arrives in a streaming fashion, like in distributed filtering [@sayed2014adaptive]. Some interesting results can derive by considering the literature on distributed constraint optimization problems (DCOP), which deals with distributed decision making problems where the decision variables are separated among the different agents [@modi2005adopt; @rogers2011bounded]. Finally, we are interested in testing our framework on real-world applications such as, e.g., multimedia classification and chaotic prediction tasks.
[^1]: A preliminary version of this work, focusing only on the squared loss function, was presented in [@di2016neuralnetworks].
[^2]: More in general, $\mathcal{G}[n]$ corresponds to all feasible communication links between two agents. A multi-hop network can be described with an equivalent single-hop network by considering all possible paths as a direct link in the equivalent graph.
[^3]: Formally, there exists an integer $B > 0$ such that the graph $\mathcal{G}[k]=(\mathcal{V},\mathcal{E}_B[k])$, with $\mathcal{E}_B[k]=\bigcup_{n=kB}^{(k+1)B-1}\mathcal{E}[n]$ is strongly connected, for all $k\geq0$.
[^4]: Note that a single back-propagation step per iteration is needed to build the weight Jacobian, as discussed in @bishop2006pattern [Section 5.3.4].
[^5]: <https://bitbucket.org/ispamm/parallel-and-distributed-neural-networks>
[^6]: <https://github.com/Lasagne/Lasagne>
[^7]: <http://www.cs.toronto.edu/~delve/data/kin/desc.html>
[^8]: <https://scipy.org/>
[^9]: <https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+(Diagnostic)>
[^10]: <https://archive.ics.uci.edu/ml/datasets/Cardiotocography>
|
---
abstract: 'In this paper, we study the coexistence of critical and non-critical Internet of Things (IoT) services on a grant-free channel consisting of radio access and backhaul segments. On the radio access segment, IoT devices send packets to access points (APs) over an erasure collision channel using the slotted ALOHA protocol. Then, the APs forward correctly received messages to a base station (BS) over a shared wireless backhaul segment, modeled as an erasure collision channel. The APs hence play the role of uncoordinated relays that provide space diversity and may reduce performance losses caused by collisions. Both non-orthogonal and inter-service orthogonal resource sharing are considered and compared. Throughput and reliability metrics are analyzed, and numerical results are provided to assess the performance trade-offs between critical and non-critical IoT services.'
author:
- '\'
bibliography:
- 'Biblio.bib'
title: |
Space Diversity-Based Grant-Free Random Access for Critical and Non-Critical IoT Services\
[^1]
---
Beyond 5G, IoT, Grant-Free, Radio Access
Introduction
============
Future generations of cellular and satellite networks, starting with 5G, will cater to heterogeneous services with vastly different performance requirements [@5Goverview][@opportunistic_coexistence]. Among these services are Internet of Things (IoT) networks characterized by short and sporadic packet transmissions, which will support applications with critical or non-critical requirements in terms of reliability.
In the presence of a large number of IoT devices such as in massive Machine Type Communications (mMTC) scenarios [@mmtcsaad], conventional grant-based radio access protocols can cause a significant overhead on the access network due to the large number of handshakes to be established. A potentially more efficient solution is given by grant-free radio access protocols, which are used by many commercial solutions both in the terrestrial domain, e.g. Sigfox [@sigfox] and LoRa [@lora] and in the satellite domain, using constellations of Low-Earth Orbit (LEO) satellites to collect information, e.g., Orbcomm [@orbcomm] and Myriota [@myriota]. Under grant-free access, devices transmit whenever they have a packet to deliver without any prior handshake [@grant_free_popovski; @rahif_grant_free; @grant_free_cavdar]. This is typically done via some variants of the classical ALOHA random access scheme [@abramson1970aloha].
In the presence of different IoT services and devices, orthogonal resource allocation schemes such as inter-service Time Division Multiple Access (TDMA) are used [@3gpp_nbiot]. Orthogonal schemes may cause an inefficient use of resources in future IoT scenarios due to limited spectral resources and inherent inefficiency when traffic patterns become unpredictable. Recent work has hence proposed to apply non-orthogonal resource allocation to heterogeneous services [@rahif_access_2018][@popovski2018slicing]. In order to mitigate interference in non-orthogonal schemes, one can leverage successive interference cancellation (SIC) [@aloha_noma], time diversity [@coded_slotted_aloha], and/or space diversity [@munari_multiple_aloha][@vladimir_cooperative_ALOHA]. The latter is provided by multiple Access Points (APs) that play the role of relays between the devices and the Base Station (BS), as illustrated in Fig. \[fig:system\_model\].
In this work, we study the space diversity-based model introduced in Fig. \[fig:system\_model\] that provides grant-free access to both critical and non-critical services. We assume uncoordinated APs, so that both radio access and backhaul channels are operating using ALOHA. The lack of coordination among APs can be considered as a worst-case analysis for dense low-cost cellular deployments [@het_networks_no_coordination] [@vladimir_cooperative_ALOHA]. It also may account for the scenario in Fig. \[fig:system\_model\], where a constellation of LEO satellites act as relays between ground terminals and a central ground station, since the presence of inter-satellite links is too costly to be deployed. For the system in Fig. \[fig:system\_model\], we derive throughput and reliability measures for critical and non-critical services as a function of key parameters such as the number of APs and traffic loads. The analysis accounts for orthogonal and non-orthogonal inter-service protocols and considers two receiver models, namely, *superposition* and *collision models* as detailed in the next section. The most related prior work is [@frederico2019modern], in which a simplified collision model with only a single service was considered for the same space-diversity model.
The rest of the paper is organized as follows. In Sec. \[sec:system\_model\_performance\_metrics\] we describe the system model used and the performance metrics. In Sec. \[sec:throughput\_reliability\], we derive throughput and reliability under the erasure channels model. Numerical results are provided in Sec. \[sec:numerical\_results\], and conclusions are drawn in Sec. \[sec:conclusions\].
System model and Performance metrics {#sec:system_model_performance_metrics}
====================================
System Model
------------
We consider the system illustrated in Fig. \[fig:system\_model\], in which $L$ APs, e.g., LEO satellites, provide connectivity to IoT devices. The APs are in turn connected to a BS, e.g., a ground station, through a shared wireless backhaul channel. We assume that time over both access and backhaul channels is divided into frames and each frame contains $T$ time-slots. At the beginning of each frame, a random number of IoT devices are active. The number of active IoT devices that generate critical and non-critical messages at the begining of the frame follow independent Poisson distributions with average loads $\gamma_c G$ and $(1-\gamma_c)G\ \mathrm{[packet/slot]}$, respectively, for some parameter $\gamma_c \in [0,1]$ and total system load $G$. Users select a time-slot $t$ uniformly at random among the $T$ time-slots in the frame and independently from each other. By the Poisson thinning property [@billingsley2008probability], the random number $N_c(t)$ of critical messages transmitted in a time-slot $t$ follows a Poisson distribution with average $G_c = \gamma_c G /T$, while the random number $N_{\bar{c}}(t)$ of non-critical messages transmitted in slot $t$ follows a Poisson distribution with average $G_{\Bar{c}}=(1-\gamma_c)G/T$.
*Radio Access Model:* As in, e.g., [@frederico2019modern; @azimi2017content; @calderbank_erasure], we model the access links between any device and an AP as an independent interfering erasure channel with erasure probability $\epsilon_1$. Specifically, a packet sent by a user is independently erased at each receiver with probability $\epsilon_1$, causing no interference, or is received with full power with probability $1-\epsilon_1$. The erasure channels are independent and identically distributed (i.i.d.) across all slots and frames. Interference from messages of the same type received at an AP is assumed to cause a destructive collision. Furthermore, critical messages are assumed transmitted with a higher power than non-critical messages so as to improve their reliability, hence creating significant interference on non-critical messages. As a result, in each time-slot, an AP can be in three possible states:\
$\bullet$ a critical message is retrieved successfully if the AP receives only one critical message. Critical messages are assumed to be immune to non-critical transmissions due to their large transmission power;\
$\bullet$ a non-critical message is retrieved if the AP receives only one non-critical message and zero critical messages;\
$\bullet$ no message is retrieved if multiple critical messages and/or non-critical messages are received at the AP, or if no messages are received due to channel erasures, or also if no messages were transmitted (i.e., none of the devices is active).
*Backhaul model:* The APs share a wireless out-of-band backhaul that operates in a full-duplex mode and in an uncoordinated fashion as in [@frederico2019modern]. In each time-slot $t+1$, an AP sends a message retrieved on the radio access channel in the corresponding time-slot $t$ to the BS over the backhaul channel. APs with no message retrieved in slot $t$ remain silent in the corresponding backhaul time-slot $t+1$. The link between each AP and the BS is modeled as an erasure channel with erasure probability $\epsilon_2$, and destructive collisions occur at the BS if two or more messages of the same type are received. As for the radio access case, erasure channels are i.i.d. across APs, slots and frames.
In order to model interference between APs, we consider two scenarios. The first, referred to as *collision model*, assumes that multiple messages from the same device cause destructive collision. Under this model, in each time-slot, the BS’s receiver can be in three possible states:\
$\bullet$ a critical message is retrieved successfully at the BS is only one critical message is received. As in the radio access scenario, critical messages are not affected by non-critical messages due to their larger transmission power;\
$\bullet$ a non-critical message is retrieved successfully if no other critical or non-critical message is received;\
$\bullet$ no message is retrieved at the BS if multiple critical or non-critical messages are received at the BS or no messages are received due to channel erasures or also no messages were transmitted.
In the second model, referred to as *superposition model*, the BS is able to decode from the superposition of multiple instances of the same packet that are relayed by different APs on the same backhaul slot, assuming no other transmission occured on it. In practice, this can be accomplished by ensuring that the time asynchronism between APs is no larger that the cyclic prefix in a multicarrier modulation implementation. This can be done, for example, by having a central master clock at the BS against which the local time bases of APs are synchronized [@timesynchro_patent_AP]. Note that this model is valid for uncoordinated APs. Hence, the BS’s receiver can be in three possible states:\
$\bullet$ a critical message is retrieved successfully at the BS in a given time-slot if no *different* critical message is received by the BS;\
$\bullet$ a non-critical message is retrieved successfully if no critical messages and no *different* non-critical messages are received in the same slot;\
$\bullet$ no message is retrieved at the BS if multiple different critical or non-critical messages are received at the BS or no message is received due to channel erasures, or also if no messages were transmitted.
In addition to non-orthogonal resource allocation whereby devices from both services share the entire frame of $T$ time-slots, we also consider orthogonal resource allocation, namely *inter-service time division multiple access* (TDMA) where a fraction $\alpha T$ of the frame’s time-slots are reserved to critical devices and the remaining $(1-\alpha)T$ for non-critical devices. Inter-service contention in each allocated fraction follows a slotted ALOHA protocol as discussed above. In the following, we derive the performance metrics under the more general non-orthogonal scheme described above. The performance metrics under TDMA for each service can be obtained by replacing $T$ with the corresponding fraction of resources in the performance metrics equations and taking the interference from the other service to zero.
Performance Metrics {#sec:performance_metrics}
-------------------
We are interested in computing the throughputs $R_c$ and $R_{\bar{c}}\ [\mathrm{packet/slot}]$ and the reliability levels $\Gamma_c$ and $\Gamma_{\bar{c}}\ [\mathrm{packet/frame}]$ for critical and non-critical messages respectively. The throughputs are defined as the average number of packets decoded correctly in any given time-slot at the BS for each type of service. The reliability levels are defined by the average fraction of critical and non-critical packets generated in a frame that are retrieved by the BS by the end of the frame.
Throughput and Reliability Analysis {#sec:throughput_reliability}
===================================
In this section, we derive the throughputs and reliability levels for both types of messages under the collision and superposition models described above. Throughout the discussion, we denote as $X\sim \operatorname{Bin}(n,p)$ a Binomial random variable (RV) with $n$ trials and probability of success $p$; as $X ~\sim \operatorname{Poiss}(\lambda)$ a Poisson RV with parameter $\lambda$. We also write $(X,Y)\sim f \cdot g$ for two independent RVs $X$ and $Y$ with respective probability density functions $f$ and $g$.
Collision Model
---------------
Under the collision model, two messages received at the BS in the same time-slot and generated from the same devices undergo a destructive collision. We start by introducing RVs $B_i(t)$ for the state of the $i$-th AP, with $i=1,\ldots,L$ and RV $B(t)$ for the state of the BS in any time-slot $t=1,\ldots,T$. Since all RVs are i.i.d. across time-slots, the index $t$ is dropped for simplicity of notation whenever no confusion may arise. These RVs take values as $$B_i \sim B = \begin{cases}
c & \text{if a critical message is retrieved} \\
\bar{c} & \text{if a non-critical message is retrieved} \\
0 & \text{if no message is retrieved due to erasures} \\ & \text{or collisions or no transmitted messages}
\end{cases} \label{eq:B_collision}$$ in the given time-slot and for $i=1,\ldots,L$. Furthermore, we denote by $M_c$ and $M_{\Bar{c}}$ the RVs representing the overall number of received critical and non-critical messages, respectively, at all the APs in a given time-slot. Accordingly, RVs $M_c$ and $M_{\bar{c}}$ can be written as $$M_c = \sum_{i=1}^{L} \mathbbm{1}_{ \{ B_i = c \}}
\ \ \text{and}\ \
M_{\bar{c}} = \sum_{i=1}^{L} \mathbbm{1}_{ \{ B_i = \bar{c} \}}.$$ where $\mathbbm{1}_{\{ a\}}$ is the indicator function of an event $a$. Conditioned on the number of transmitted messages $N_c$ and $N_{\Bar{c}}$, RVs $M_c$ and $M_{\bar{c}}$ are distributed as $$M_{\bar{c}}|N_c,N_{\bar{c}} \sim \operatorname{Bin}(L,p_{\bar{c}}) \label{eq:dis_Mbarc_collision}$$ and $$M_c|M_{\bar{c}},N_c,N_{\bar{c}} \sim \operatorname{Bin}(L-M_{\bar{c}},p_c), \label{eq:dis_Mc_collision}$$ with the corresponding parameters given as
$$\begin{aligned}
{1}
& p_{c}\! =\! \mathrm{Pr}[B_i = c|N_c\! = \!n_c, N_{\Bar{c}}\!=\!n_{\bar{c}} ]= n_c (1\!-\!\epsilon_1)\epsilon_1^{n_c -1} \label{eq:pc}\\
\mathrm{and}\ & p_{\Bar{c}}\! =\!\mathrm{Pr}[B_i \!=\!\Bar{c}| N_c=n_c,N_{\Bar{c}}=n_{\bar{c}}] = n_{\Bar{c}} (1 \! - \! \epsilon_1) \epsilon_1^{n_{\Bar{c}}-1} \epsilon_1^{n_c}. \label{eq:pbarc}\end{aligned}$$
\[eq:pc\_and\_pbarc\]
The expression is the probability of an AP receiving a critical message from any of the $N_c=n_c$ active critical devices in the slot. The expression is the probability of an AP receiving a non-critical message from any of the $N_{\bar{c}}=n_{\bar{c}}$ active non-critical devices. Note that the latter requires all critical messages to be erased which is represented by the probability term $\epsilon_{1}^{n_c}$.
Following a similar reasoning, given $M_c$, $M_{\bar{c}}$, $N_c$ and $N_{\bar{c}}$, the probability of retrieving successfully a critical message at the BS in a given time-slot can be written as $$\begin{aligned}
q_c& =\mathrm{Pr}[B=c|N_c = n_c , N_{\bar{c}}=n_{\bar{c}} , M_c = m_c, M_{\bar{c}}=m_{\bar{c}}] \\ & = m_c (1 - \epsilon_2)\epsilon_2^{m_c - 1}. \label{eq:m_c_Bin}
\end{aligned}$$ The probability of retrieving a non-critical message at the BS is given as $$\begin{aligned}
q_{{\bar{c}}} & = \mathrm{Pr}[B=\bar{c} | N_c = n_c , N_{\bar{c}}=n_{\bar{c}} , M_c = m_c, M_{\bar{c}} = m_{\bar{c}}] \\ & =m_{\bar{c}} (1 - \epsilon_2) \epsilon_2^{m_c} \epsilon_2^{m_{\Bar{c}}- 1} .
\end{aligned}$$ Removing the conditioning on $M_c,M_{\Bar{c}},N_c$ and $N_{\Bar{c}}$ and using the distributions and , the throughputs can be directly computed as the expectations $$R_c=\mathbb{E}[q_c]\ \ \text{and}\ \
R_{\bar{c}}=\mathbb{E}[ q_{\Bar{c}}] \label{eq:expectation_rate}$$ where averages are taken over RVs $N_c, N_{\bar{c}}, M_c$ and $M_{\bar{c}}$. These expectations can be derived in closed form as detailed in [@rahif_uncoordinated].
Given the above definitions, the reliability levels of critical and non-critical messages can be written respectively as
$$\begin{aligned}
{1}
& \Gamma_c = \mathbb{E}\Bigg[ \frac{\sum_{t=1}^{T} \mathbbm{1}_{\{B(t)=c \}}}{\sum_{t=1}^{T} N_c (t)} \bigg| \sum_{t=1}^{T} N_c(t) \geq 1 \Bigg] \label{eq:reliability_collision_a} \\ &\text{and}\ \ \Gamma_{\bar{c}}= \mathbb{E}\Bigg[ \frac{\sum_{t=1}^{T} \mathbbm{1}_{\{B(t)=\bar{c} \}}}{\sum_{t=1}^{T} N_{\bar{c}}(t)} \bigg| \sum_{t=1}^{T} N_{\bar{c}}(t) \geq 1 \Bigg],
\label{eq:reliability_collision_b}\end{aligned}$$
\[eq:reliability\_collision\]
with expectations taken over RVs $N_c(t), N_{\bar{c}}(t), M_c(t)$, $M_{\bar{c}}(t)$, and $B(t)$ across all slots $t=1, \ldots, T$. The conditioning in ensures that at least one packet of the given type is transmitted in the given frame. The conditional joint distributions needed to compute are defined through the chain rule by the distributions
$$\begin{aligned}
{1}
& \{N_c(t),N_{\bar{c}}(t) \}_{t=1}^{T} \bigg| \sum_{t=1}^{T} N_c(t) \geq 1 \sim \nonumber \\ & \ \ \ \Big( \prod_{t=1}^{T} \text{Poiss}(n_c | g_c) \Big) \Big( \frac{1}{Z} \prod_{t=1}^{T}\text{Poiss}(n_{\bar{c}} | g_{\bar{c}}) \mathbbm{1}_{\{\sum_{t=1}^{T} N_{\bar{c}}(t) \geq 1 \}} \Big) \\
&\ \mathrm{and}\ \ \{M_c(t), M_{\bar{c}}(t) \}_{t=1}^{T} \bigg| \{N_c(t),N_{\bar{c}}(t) \}_{t=1}^{T} \sim \nonumber \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \prod_{t=1}^{T} \operatorname{Bin}(L, p_{\bar{c}}(t)) \operatorname{Bin}(L-M_{\bar{c}} (t), p_{{c}}(t)),\end{aligned}$$
where $Z = 1 - \mathrm{Pr}(\sum_{t=1}^{T} N_{\bar{c}} (t)= 0)$ is a normalizing factor; and $p_c(t)$ and $p_{\bar{c}}(t)$ are defined as in with $N_c(t)$ and $N_{\bar{c}}(t)$ in lieu of $n_c$ and $n_{\bar{c}}$, respectively. Similar expressions apply for . Note that, conditioned on there being at least one non-critical message transmitted in the frame, the RVs $\{N_{\bar{c}}(t) \}_{t=1}^{T}$ are not i.i.d.
Superposition Model
-------------------
In this subsection, we derive the throughput and reliability measures of critical and non-critical messages under the superposition model. To this end, unlike for the collision model, one needs to keep track of the index of the messages decoded by the APs in order to be able to detect when multiple instances of the same message (i.e., sent by the same device) are received at the BS. We start by defining the RVs $B_i$ to denote the index of the message received at AP $i$ and RV $B$ for the BS at any time-slot. Accordingly, for given values $N_c=n_c$ and $N_{\bar{c}}=n_{\bar{c}}$ of transmitted messages, RVs $\{B_i\}$ can take values $$\begin{cases}
0 & \text{if no message is retrieved due to}\\ & \text{erasures or collisions }\\
1 \leq m \leq n_c & \text{if the $m$-th critical message is }\\ & \text{retrieved} \\
n_c + 1 \leq m \leq n_c + n_{\bar{c}} & \text{if the $(m-n_c)$-th non-critical }\\ &\text{message is retrieved.}
\label{eq:Bi_superposition}
\end{cases}$$ Note that we have indexed critical messages from $1$ to $n_c$ and non-critical messages from $n_c + 1$ to $n_c + n_{\bar{c}}$. $B$ is defined as in . Furthermore, we define as $M_m = \sum_{i=1}^{L} \mathbbm{1}_{\{B_i=m\}}$ the RVs denoting the number of APs that have message of index $m \in \{0,1, \ldots, n_c,n_c+1, \ldots , n_c + n_{\bar{c}} \}$. The joint distribution of RVs $\{M_m \}_{m=0}^{n_c+n_{\bar{c}}}$ given $N_c$ and $N_{\bar{c}}$ is multinomial and can be written as follows $$\begin{aligned}
& \{M_m \}_{m=0}^{n_c+n_{\bar{c}}}|N_c,N_{\bar{c}} \sim \\&\operatorname{Multinomial}\Big(L ,\overbrace{1-p_c-p_{\bar{c}}}^{0},\overbrace{\frac{p_c}{n_c}, \ldots,\frac{p_c}{n_c}}^{n_c}, \overbrace{\frac{p_{\bar{c}}}{n_{\bar{c}}},\ldots, \frac{p_{\bar{c}}}{n_{\bar{c}}}}^{n_{\bar{c}}} \Big), \label{eq:multinomial}
\end{aligned}$$ where we used the the probabilities in that one of the critical or non-critical message is received at an AP respectively in a given time-slot. The probability of retrieving a critical message in a given time-slot at the BS conditioned on $N_c$, $N_{\bar{c}}$ and $\{M_{m\prime} \}_{m\prime=0}^{n_c + n_{\bar{c}}}$ can be then written as $$\begin{aligned}
q_c& =\mathrm{Pr}[B=c | N_c=n_c , N_{\bar{c}}=n_{\bar{c}}, \{M_{m^\prime} \}_{m^\prime=0}^{n_c + n_{\bar{c}}}] \\ & = \sum_{m=1}^{n_c} \sum_{j=1}^{M_m} {M_m \choose j} (1-\epsilon_2)^j \epsilon_2^{\delta_1}, \label{eq:proba_c_superposition}
\end{aligned}$$ where $\delta_1$ is defined as follows $$\delta_1= \sum_{ \substack{m^\prime = 0 \\ m^\prime \neq m}}^{n_c} M_{m^\prime} + M_m - j. \label{eq:delta_1}$$ The first sum in is over all possible critical messages and the second sum is over all combinations of APs that have the critical message $m$. The sum in is over all APs that have a critical message $m^{\prime} \neq m$. The throughput of critical messages can be computed by averaging over all conditioning variables as $$R_c = \mathbb{E}_{N_c , N_{\bar{c}} , \{M_m\}_{m=0}^{N_c+N_{\bar{c}}}} [q_c].$$ In a similar manner, the conditional probability of receiving a non-critical message at the BS can be written as $$\begin{aligned}
& q_{\bar{c}} =\mathrm{Pr}[B = \bar{c} | N_c=n_c , N_{\bar{c}}=n_{\bar{c}} , \{M_{m^\prime} \}_{m^\prime = 0}^{n_c + n_{\bar{c}}} ] \\ &= \sum_{m=n_c + 1}^{n_c + n_{\bar{c}}} \sum_{j=1}^{M_m} {M_m \choose j} (1-\epsilon_2)^j \epsilon_2^{\delta_2},
\end{aligned} \label{eq:proba_cbar_one}$$ where $\delta_2$ is written as $$\delta_2 = \sum_{ \substack{m^{\prime\prime}=n_c + 1 \\ m^{\prime\prime} \neq m } }^{n_c + n_{\bar{c}}} M_{m^{\prime \prime}} + M_m - j +\sum_{m^\prime = 1}^{n_c} M_{m^\prime}. \label{eq:delta_2}$$ The first sum in is over all possible non-critical messages $m$ while the second sum is over all possible combinations of APs that have message $m$. The first and second sums in are over all APs that have a different non-critical message and a critical message respectively. The throughput of non-critical messages can be then obtained by averaging over the conditioning RVs as $$R_{\bar{c}} = \mathbb{E}_{N_c, N_{\bar{c}}, \{ M_m \}_{m= 0}^{N_c + N_{\bar{c}}} } [q_{\bar{c}}].$$
The reliability levels under the superposition model can be defined as in with the caveat that one needs to average over the RVs $M_m(t)$, for $m\in \{0,1,\ldots , n_c + n_{\bar{c}} \}$ and $t = 1 , \ldots, T$ instead of $M_c (t)$, by using the distribution in .
numerical Results {#sec:numerical_results}
=================
In this section, we numerically evaluate performance trade-offs in terms of throughput and reliability level for both services as function of key system parameters such as the channel erasure probabilities $\epsilon_1$ and $\epsilon_2$, number of APs $L$, and frame duration $T$. Unless specified otherwise, we assume throughout this section that we have $\epsilon_1=\epsilon_2=\epsilon$.
We start by plotting the region of achievable throughputs for critical and non-critical messages for both collision and superposition models in Fig. \[fig:rate\_region\] for $\epsilon = 0.5$, total load $G=16\ [\mathrm{packet/frame}]$, $T=4\ \mathrm{[time\textrm{-}slot/frame]}$, and $L = 3$ APs. The region includes all throughput pairs that are achievable for some value of the fraction $\gamma_c$ of critical messages, as well as all throughput pairs that are dominated by an achievable throughput pair (i.e., for which both critical and non-critical throughputs are smaller than for an achievable pair). For reference, we also plot the throughput region for a conventional inter-service TDMA protocol, whereby a fraction $\alpha T$ for $\alpha \in [0,1]$ of the $T$ time-slots is allocated for critical messages and the remaining time-slots to non-critical messages. For TDMA, the throughput region includes all throughput pairs that are achievable for some value of $\alpha$, as well as of $\gamma_c$. A first observation from the figure is that non-orthogonal resource allocation can accommodate a significant non-critical throughput without affecting the critical throughput, while TDMA causes a reduction in the critical throughput for any increase in the non-critical throughput. This is due to the need in TDMA to allocate orthogonal time resources to non-critical messages in order to increase the corresponding throughput. However, with non-orthogonal resource allocation, the maximum non-critical throughput is generally penalized by the interference caused by the collisions from critical messages, while this is not the case for TDMA. In brief, TDMA is preferable when one wishes to guarantee a large non-critical throughput and the critical throughput requirements are loose; otherwise, non-orthogonal resource allocation outperforms TDMA in terms of throughput. Finally, we observe that significant gains can be obtained under the superposition model, leveraging as useful the superposition of multiple packets containing the same message.
\[fig:rate\_L\]
In Fig. \[fig:rate\_L\], we explore the effect of the number of APs $L$ on the throughputs of both type of messages. To capture separately the effects of the radio access and the backhaul channel erasures, we consider here different values of the channel erasure probabilities $\epsilon_1$ and $\epsilon_2$. We highlight two different regimes: the first is when $\epsilon_1$ is large and $\epsilon_2$ is small, and hence larger erasures occur on the access channel; while the second covers the complementary case where $\epsilon_1$ is small and $\epsilon_2$ is large. In the first regime, increasing the number of APs is initially beneficial to both critical and non-critical messages in order to provide additional spatial diversity for the radio access given the large value of $\epsilon_1$; but larger values of $L$ eventually increase the probability of collisions at the BS on the backhaul due to the low value of $\epsilon_2$. In the second regime, when $\epsilon_1=0.1$ and $\epsilon_2=0.8$ much lower throughputs are obtained due to the significant losses on the backhaul channel. This can be mitigated by increasing the number of APs, which increases the probability of receiving a packet at the BS.
[.5]{}
[.5]{}
Finally, we consider the interplay between the throughputs and reliability levels for both non-orthogonal resource allocation and TDMA as function of the number of time-slots $T$. These are plotted in Fig. \[fig:throughput\_reliability\] for $G=15\ [\mathrm{packet/frame}]$, $\epsilon_1 = \epsilon_2 = 0.5, L=3$ APs, $\alpha=0.5$ and $\gamma_c = 0.5$. For both services, we observe that the reliability level under both allocation schemes increases as function of $T$. This is because larger value of $T$ decrease chances of packet collisions. However, this not the case for the throughput, since large values of $T$ may cause some time-slots to be left unused, which penalizes the throughput. For the critical service in Fig. \[fig:throughput\_reliability\_critical\_messages\], it is seen that non-orthogonal resource allocation outperforms TDMA in both throughput and reliability level due to the larger number of available resources. Moving to the non-critical service in Fig. \[fig:throughput\_reliability\_Noncritical\_messages\], we observe that TDMA provides better throughput and reliability level than non-orthogonal resource allocation. The main reason for this is that the lower number of resources in TDMA is compensated by the absence of inter-service interference from critical messages.
Conclusions {#sec:conclusions}
===========
This paper studies grant-free random access for coexisting critical and non-critical services in IoT systems with shared wireless backhaul and uncoordinated access points (APs). A non-orthogonal resource sharing scheme based on random access is considered, whereby critical messages are transmitted with a larger power. From the critical service perspective, it was found that non-orthogonal sharing is preferable to a standard inter-service TDMA protocol in terms of both throughput and reliability level. In contrast, this is not the case for the non-critical service, since inter-service orthogonal resource allocation eliminates interference from the larger-power critical service. Finally, we have identified different regimes in terms of channels erasure probabilities for which increasing the number of APs may be beneficial, thanks to additional space diversity, or harmful, due to the increased inter-AP interference. Among possible extensions of this work, we mention the consideration of a more general collision model in which critical messages can tolerate no more than a given number of interfering non-critical messages [@rahif_uncoordinated].
[^1]: The work of Rahif Kassab and Osvaldo Simeone has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement 725731).
|
---
abstract: 'We consider the quenched localisation of the Bouchaud trap model on the positive integers in the case that the trap distribution has a slowly varying tail at infinity. Our main result is that for each $N \in \{2, 3, \ldots\}$ there exists a slowly varying tail such that quenched localisation occurs on exactly $N$ sites. As far as we are aware, this is the first example of a model in which the exact number of localisation sites are able to be ‘tuned’ according to the model parameters. Key intuition for this result is provided by an observation about the sum-max ratio for sequences of independent and identically distributed random variables with a slowly varying distributional tail, which is of independent interest.'
address:
- '$^1$Department of Statistics, University of Warwick'
- '$^2$Department of Mathematics, University College London (current address: Mathematical Institute, University of Oxford)'
author:
- 'David A. Croydon$^1$'
- Stephen Muirhead$^2$
bibliography:
- 'paper.bib'
title: |
Quenched localisation in the Bouchaud trap\
model with slowly varying traps
---
[^1]
Introduction
============
This article studies localisation properties of the Bouchaud trap model (BTM) on the positive integers in the case of slowly varying traps. To define the BTM, we first introduce a trapping landscape $\sigma=(\sigma_x)_{x \in \mathbb{Z}^+}$, which is a collection of independent and identically distributed (i.i.d.) strictly-positive random variables built on a probability space with probability measure $\mathbf{P}$. Conditional on $\sigma$, the dynamics of the BTM are given by a continuous-time $\mathbb{Z}^+$-valued Markov chain $X=(X_t)_{t\geq 0}$, started from the origin, with transition rates $$\label{trates}
w_{x \to y} = \begin{cases}
\frac{1}{2 \sigma_x}, & \text{if } y \sim x, \\
0, & \text{otherwise,}
\end{cases}$$ where $y \sim x$ means that $x$ and $y$ are nearest neighbours in $\mathbb{Z}^+$. We denote the law of $X$ conditional on $\sigma$, the so-called ‘quenched’ law of the BTM, by $P_\sigma$. Our focus is on the case in which the trap distribution $\sigma_0$ has a slowly varying tail at infinity, i.e. when the (non-decreasing, unbounded, càdlàg) function $$L(u) := \frac{1}{\mathbf{P}(\sigma_0 > u)}$$ satisfies the assumption $$\label{eq:slow}
\lim_{u \to \infty} \frac{L(u v)}{L(u)} = 1 \quad \text{for all} \ v > 0.$$
Slowly varying trap models arise naturally in the study of certain random walks in random media, such as biased random walks on critical Galton-Watson trees [@Croydon13], and spin-glass dynamics on subexponential time scales [@BenArous12; @Bovier13]. They also have parallels with Sinai’s random walk [@Sinai82], as reflected in the logarithmic rate of escape to infinity and strong localisation properties of that model. With regards to the BTM with slowly varying traps in particular, recent work has studied localisation [@Muirhead15], ageing [@Gun13], and scaling limits [@Croydon15], which are qualitatively different from the equivalent phenomena in the case of integrable or regularly varying traps.
In this work we continue the study of the BTM with slowly varying traps by considering more delicate localisation properties of the model, namely those that hold under the quenched law for typical realisations of the trapping landscape. We expect that similar quenched localisation properties hold throughout the class of general slowly varying trap models. For simplicity we have chosen to work in the one-sided case (i.e. on the positive integers, rather than on the integers); this avoids some of the technical difficulties present in the two-side case, yet still exhibits the phenomena that interest us. We make some remarks about quenched localisation in the BTM on the integers below.
Localisation in the BTM
-----------------------
It was recently shown in [@Muirhead15] that the BTM on the integers with slowly varying traps exhibits two-site localisation in probability, that is, there exists a ($\mathbf{P}$-measurable) set-valued process $\Gamma_t$ such that $|\Gamma_t|=2$ and, as $t \to \infty$, $$\begin{aligned}
\label{eq:twosite}
P_\sigma(X_t \in \Gamma_t) \to 1 \qquad \text{in } \mathbf{P} \text{-probability,}
\end{aligned}$$ and, moreover, no set-valued process $\Gamma_t$ with $|\Gamma_t| = 1$ satisfies equation ; note that here the probability measures $\mathbf{P}$ and $P_\sigma$ refer to the BTM on the integers. The basic fact underlying this localisation result is that the cumulative sum of i.i.d. sequences of slowly varying random variables is asymptotically dominated, with high probability, by the maximal term. Translated to the setting of the BTM, this property implies that the BTM with slowly varying traps is highly likely to be located, at any sufficient large time, on the largest traps on the positive and negative half-lines that are ‘within reach’ of the BTM by this time. Note that two-site localisation is not exhibited in the BTM with regularly varying or integrable traps.
We seek to establish an almost-sure analogue of the above result, that is, to determine the smallest ($\mathbf{P}$-measurable) set-valued process $\Gamma_t$ such that, as $t \to \infty$, $$P_\sigma(X_t \in \Gamma_t) \to 1 \qquad \mathbf{P}\text{-almost-surely};$$ whenever we refer to ‘quenched localisation’, it is a limit such as this that we mean. In the one-sided case, it is possible to check that the analogue to (\[eq:twosite\]) holds with $|\Gamma_t|=1$ (see Theorem \[thm:maincl\] below). From this, simple heuristics suggest that the strongest form of quenched localisation that one might hope to observe is that there exists a localisation set $\Gamma_t$ such that $|\Gamma_t| = 2$; this follows from the fact that the probability mass function eventually moves out to infinity and so, by continuity, must be spread over at least two sites at arbitrarily large times.
In this paper we prove that this strongest form of quenched localisation is actually attained for certain examples of the BTM with slowly varying traps; in other words, we prove that there exists a certain class of slowly varying tails for which the BTM localises on two sites eventually almost-surely, and for which one localisation site would be insufficient.
More surprisingly perhaps, for each $N \in \{2, 3, \ldots \}$ we show that there exists a slowly varying tail such that quenched localisation occurs on exactly $N$ sites. As such, the BTM with slowly varying traps is an example of a model that exhibits quenched localisation on a finite number of sites, with the exact number of localisation sites able to be tuned by adjusting the parameters of the model. As far as we are aware, this is the first known example of such a model.
Main results
------------
In this section, we describe our main results on localisation in the BTM on the positive integers. We assume throughout that the trap distribution satisfies the slow-variation assumption (\[eq:slow\]). As a preliminary, we first state the one-sided analogue of equation ; that is, we establish the complete localisation in probability of the BTM on the positive integers.
\[thm:maincl\] It is possible to define a ($\mathbf{P}$-measurable) process $Z_t$ such that, as $t \to \infty$, $$\begin{aligned}
P_\sigma(X_t = Z_t) \to 1 \quad \text{in } \mathbf{P}\text{-probability.}\nonumber\end{aligned}$$
The main focus of the paper is to establish quenched analogues of Theorem \[thm:maincl\]. Interestingly, quenched localisation in the BTM turns out to depend on rather fine properties of the trap distribution $\sigma_0$. To introduce these properties, we first recall that a function $L$ is said to be *second-order slowly varying with rate* $g$, if there exist functions $g, k$ such that $g(u) \to 0$ as $u \to \infty$ and $$\begin{aligned}
\label{sosv}
\lim_{u \to \infty} \frac{\frac{L(u v)}{L(u)} - 1}{ g(u)} = k(v), \quad \text{for any } v > 0,\end{aligned}$$ and where there exists a $v$ such that $k(v)\neq 0$ and $k(uv)\neq k(u)$ for all $u>0$; as discussed in [@Croydon15], in our setting it follows from this assumption that $g$ itself is slowly varying and (possibly after multiplying by a constant) we can take $k(v)=\log v$. Second-order slow-variation is a natural strengthening of the slow-variation property , giving more precise information about the fluctuations of $L$ at infinity; see [@Bingham87 Chapter 3] for an overview of second-order slow-variation.
In particular, each of our main results will depend on the assumption that $L$ is second-order slowly varying. In addition, it will be convenient to assume certain extra regularity condition on $L$ and $g$, namely that $L$ is continuous (which avoids complications in our treatment of records of the sequence $\sigma$) and that the decay of $g$ is eventually monotone (which allow us to control the decay of $g(x)$ through bounds on $x$).
\[assumpt:sosv\] The function $L$ is continuous, and satisfies with second-order slow-variation rate $g$ that is eventually monotone decreasing.
We shall now explain how the quenched localisation behaviour of the BTM depends on the precise asymptotic decay of the second-order slow-variation rate $g$. For the rest of this section we work under Assumption \[assumpt:sosv\], and abbreviate the function $$d(u) := g(L^{-1}(u)) ,$$ where $L^{-1}$ denotes the right-continuous inverse of $L$, noting that $d(u) \to 0$ as $u \to \infty$. Define the integer $$\begin{aligned}
\label{eq:N}
N := \min \left\{\ell \in \{2, 3, \ldots\} : \sum_{n \in \mathbb{N}} \left(d(e^n) \log n\right)^{\ell-1} < \infty \right\} ,\end{aligned}$$ setting $N = \infty$ if no such $\ell$ exists. Our first main theorem (Theorem \[thm:main1\]) identifies $N$ as the number of quenched localisation sites of the BTM. Before stating this result, we first need to introduce an additional assumption that is necessary for certain aspects of our results to hold. This assumption acts to exclude boundary cases, in which the number of localisation sites in a sense falls intermediate between two integers.
\[assumpt:g22\] It is the case that $N < \infty$, and\
$$\text{(a)} \quad \sum_{n \in \mathbb{N}} d(e^n)^{N-2} = \infty \quad, \qquad \text{(b)} \quad \sum_{n \in \mathbb{N}} d(e^n)^{N-1} (\log n)^N < \infty .$$
We note that Assumptions \[assumpt:sosv\] and \[assumpt:g22\] are satisfied for a wide range of slowly varying distributions $\sigma_0$. The main examples we have in mind are distributions satisfying
$$\begin{aligned}
\label{eq:L}
L(u) := \exp \{ (\log (1+ u))^\gamma \} , \quad \gamma \in (0, 1) ,\end{aligned}$$
for which $g(u) =\gamma (\log (1+ u))^{\gamma-1}$, $d(e^n)\sim\gamma n^{-\frac{1-\gamma}{\gamma}}$, and $$N = 2 + \left \lfloor \frac{\gamma}{1-\gamma} \right \rfloor .$$ In this example, we observe that $N = 2$ if and only if $\gamma < 1/2$, that $N \to \infty$ as $\gamma \to 1$, and moreover that any $N \in \{2, 3, \ldots\}$ is attainable by selecting an appropriate $\gamma \in (0, 1)$. Other classes of slowly varying distribution for which our results hold are those with logarithmic decay ($L(u) = (1+\log (1+ u))^\gamma, \gamma > 0$), or double logarithmic decay ($L(u) = (1+\log (1 + \log (1+ u)))^\gamma, \gamma > 0$); in both cases $L$ satisfies Assumptions \[assumpt:sosv\] and \[assumpt:g22\] with $N = 2$.
Our first main result establishes the property of $N$-site localisation almost-surely.
\[thm:main1\] If Assumption \[assumpt:sosv\] holds, then there exists a ($\mathbf{P}$-measurable) set-valued process $\Gamma_t$ satisfying $|\Gamma_t| \le N$ such that, as $t \to \infty$, $$\begin{aligned}
\label{eq:asloc}
P_\sigma(X_t \in \Gamma_t) \to 1 \quad \mathbf{P}\text{-almost-surely.}\end{aligned}$$ Moreover if Assumption \[assumpt:g22\] also holds, then there is no set-valued process $\Gamma_t$ satisfying $|\Gamma_t| < N$ such that holds.
The first claim of Theorem \[thm:main1\] states that the probability mass of the BTM is asymptotically supported by a certain ($\mathbf{P}$-measurable) collection of $N$ sites, where $N \ge 2$. To deduce the second claim that no smaller set will do, we study the most favoured site of the BTM. In particular, our second main result (Theorem \[thm:main2\]) shows that the associated probability mass fluctuates infinitely often between the bounds of $1/N$ and $1$; Figure \[trajpic\] shows a sketch of a typical trajectory of this probability mass. An immediate implication of this is that there exist arbitrary large times such that the probability mass of the BTM is approximately uniform across each of the $N$ localisation sites, which is sufficient to complete the proof of Theorem \[thm:main1\].
\[thm:main2\] If Assumption \[assumpt:sosv\] holds, then $\mathbf{P}$-almost-surely, $$\liminf_{t \to \infty} \, \sup_{x \in \mathbb{Z}^+} P_\sigma(X_t = x) \ge 1/N \quad \text{and} \quad \limsup_{t \to \infty} \, \sup_{x \in \mathbb{Z}^+}P_\sigma(X_t = x) = 1 .$$ Moreover if Assumption \[assumpt:g22\] also holds, then $\mathbf{P}$-almost-surely, $$\liminf_{t \to \infty} \, \sup_{x \in \mathbb{Z}^+} P_\sigma(X_t = x) = 1/N .$$
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To give some intuition as to why the number of localisation sites depends on the second-order slow-variation rate $g$ in the way determined by , consider that, for $v > 1$, $$\mathbf{P} \left(\sigma_0 \in (u, uv] \:|\: \sigma_0 > u \right) = \frac{1/L(u) - 1/L(uv)}{1/L(u)} = 1 - \frac{L(u)}{L(uv)} \sim g(u) k(v) ,$$ and so $g(u)$ gives an approximate measure of how likely records, or near records, of the sequence $(\sigma_i)_{i \in \mathbb{N}}$ are to cluster on the same scale $u$. In particular, $g(u)^k$ gives the approximate probability that such a cluster consists of at least $k$ sites. Next, consider that the height of the $n^{\text{th}}$ record of the sequence $(\sigma_i)_{i \in \mathbb{N}}$ is approximately $L^{-1}(e^n)$ (see, e.g. Lemma \[lem:probrecord\]). Hence, by a Borel-Cantelli argument, the summability of the sequence $$\left( d(e^n)^{k} \right)_{n \in \mathbb{N}}:=\left( g \left( L^{-1}(e^n) \right)^k \right)_{n \in \mathbb{N}}$$ determines whether a cluster of $k$ records, or near records, occurs around the $n^{\text{th}}$ record eventually almost-surely. From here, notice that a cluster of records, or near records, on the same scale naturally gives rise to a division of the probability mass function of the BTM across this cluster. Counting the site of the $n^{\text{th}}$ record and the site from which the BTM eventually escapes after leaving the associated cluster of records or near records, this line of argument suggests that, under Assumptions \[assumpt:sosv\] and \[assumpt:g22\], quenched localisation will occur on $N$ sites. With regard to the extra logarithmic factors appearing in the definition of $N$ in and in Assumption \[assumpt:g22\], it is possible that these are artifacts of our proof which could be removed (or at least relaxed).
The above heuristics allow us to conjecture the quenched localisation behaviour of the BTM on the integers. In particular, we expect that quenched localisation in the BTM on the integers occurs on a set of cardinality $N+1$, i.e. one larger than for the positive integers. The intuition is that the clustering argument described above is valid across the whole positive and negative half-lines. The extra localisation site takes into account the fact that the BTM can now escape, after leaving the cluster, in two directions. Nevertheless, formalising this heuristic presents additional technical challenges not present in the one-sided case, and we do not pursue this here.
Let us draw a comparison with the BTM with regularly varying traps. In this case, it was recently shown in [@Croydon16] that $$\limsup_{t \to \infty} \sup_{x \in \mathbb{Z}} P_\sigma(X_t = x) = 1 \quad \mathbf{P}\text{-almost-surely},$$ refining the original observation of [@FIN99] that the above $\limsup$ expression is strictly positive; note that here the probability measures $\mathbf{P}$ and $P_\sigma$ refer to the BTM on the integers. In other words, just as in the slowly varying case, there exist arbitrarily large times at which the probability mass of the BTM is, up to any specified error, completely localised. On the other hand, in [@Croydon16] it was also shown that in the regularly varying case $$\liminf_{t \to \infty} \sup_{x \in \mathbb{Z}} P_\sigma(X_t = x) = 0 \quad \mathbf{P}\text{-almost-surely},$$ in other words, there are also arbitrarily large times at which the BTM is completely delocalised.
In light of Theorems \[thm:main1\] and \[thm:main2\], it is natural to expect that the localisation set of the BTM is related to the set of ‘record traps’ $$\mathcal{R} := \left\{ x \in \mathbb{Z}^+ : \sigma_x > \max_{0 \le y < x} \sigma_y \right\} .$$ The following result shows that the localisation set can actually be chosen to be a subset of $\mathcal{R}$ if and only if $N = 2$. It will become clear from our proofs that for $N > 2$ the localisation set will, at arbitrarily large times, also include certain ‘near records’.
\[thm:main3\] Suppose Assumptions \[assumpt:sosv\] and \[assumpt:g22\] both hold. Then $$P_\sigma(X_t \in \mathcal{R} ) \to 1 \quad \mathbf{P}\text{-almost-surely}$$ if and only if $N = 2$.
The regimes in which two-site localisation occurs almost-surely, and indeed occurs on record traps, are precisely the regimes identified in [@Croydon15] and [@Kasahara86] in which simplified scaling limit theorems are available (see [@Croydon15 Remark 1.5] and [@Kasahara86 Remark 2.4]).
Finally, we observe that each of our main results on localisation in the BTM with slowly varying traps is underpinned by the analogous result regarding the sum-max ratio for sequences of i.i.d. slowly varying random variables. As we noted above, with high probability, the partial sums of such sequences are asymptotically dominated by the maximum. However, it turns out that in general this is not the case almost-surely, as is demonstrated by the following theorem. This provides a crucial ingredient in our arguments for the BTM, and to the best of our knowledge has not appeared in the literature before.
\[thm:main4\] Let $(\sigma_i)_{i \in \mathbb{N}}$ be an i.i.d. sequence of copies of $\sigma_0$. Let $m_i$ and $S_i$ denote the maximum and sum respectively of the partial sequence $(\sigma_j)_{j \le i}$ . If Assumption \[assumpt:sosv\] holds, then, almost-surely, $$\liminf_{i \to \infty} \frac{S_i}{m_i} = 1 \quad \text{and} \quad \limsup_{i \to \infty} \frac{S_i}{m_i} \le N-1 .$$ Moreover, if Assumption \[assumpt:g22\](a) (with $N < \infty$) also holds, then $$\limsup_{i \to \infty} \frac{S_i}{m_i} = N-1 .$$
In the special case of $L$ satisfying , this implies that $$\liminf_{i \to \infty} \frac{S_i}{m_i} = 1 \quad \text{and} \quad \limsup_{i \to \infty} \frac{S_i}{m_i} = 1+ \left \lfloor \frac{\gamma}{1-\gamma} \right \rfloor ,$$ almost-surely. Hence, in this case, $ \lim_{i \to \infty} {S_i}/{m_i} = 1$ almost-surely if and only if $\gamma < 1/2$. For comparison, we recall that the latter limit holds in probability for all slowly varying tails [@Darling52]; an observation which (together with Fatou’s lemma) already yields the $\liminf$ part of the previous result.
Outline of the paper
--------------------
The rest of this paper is organised as follows. In Section \[sec:prelim\] and Section \[sec:svseq\] we collect preliminary results that will be crucial in proving the main theorems. The results in Section \[sec:prelim\] relate to random walks in inhomogeneous trapping landscapes; the results in Section \[sec:svseq\] consider general properties of sequences of slowly varying random variables, including the key almost-sure bound on the ratio of the cumulative sum to the maximum stated as Theorem \[thm:main4\] above.
In Section \[sec:loc\] we study upper bounds on the size of the quenched localisation set. We first give an explicit description of the localisation set $\Gamma_t$, and show that localisation does occur on this set eventually almost-surely. Here we also show that $|\Gamma_t| = 1$ with overwhelming probability, establishing the complete localisation in probability of Theorem \[thm:maincl\]. We next consider the almost-sure cardinality of $\Gamma_t$, showing that $|\Gamma_t| \le N$ eventually almost-surely, establishing the first claim of Theorem \[thm:main1\] above. As a corollary, we show that, if $N = 2$, the localisation set $\Gamma_t$ is contained in the set of record traps $\mathcal{R}$, establishing one direction of Theorem \[thm:main3\].
Finally, in Section \[sec:fav\] we study the most favoured site (Theorem \[thm:main2\] above). Since Theorem \[thm:maincl\], and the first part of Theorem \[thm:main1\], will already have been proved at this point, it will be sufficient to show that there exist arbitrarily large times at which the BTM is evenly balanced across a certain set of $N$ localisation sites. Since for $N \ge 3$, the above-mentioned $N$ sites are not all contained in the record traps $\mathcal{R}$, this will also establish the converse direction of Theorem \[thm:main3\].
Random walks in inhomogeneous trapping landscapes {#sec:prelim}
=================================================
In this section we collect preliminary results on random walks in inhomogeneous trapping landscapes, in particular relating to hitting times and localisation. Note that in this section the trapping landscape $\sigma$ will always be assumed to be completely arbitrary and deterministic.
Throughout this section, fix $a, b \in \mathbb{Z}$ such that $a < b$ and let $X^{a, b}=(X^{a, b}_t)_{t \ge 0}$ denote the inhomogeneous continuous-time random walk (CTRW) in an arbitrary (deterministic) trapping landscape $\sigma=(\sigma_x)_{x \in [a, b]\cap \mathbb{Z}}$, with reflected boundary conditions at $a$ and $b$. More precisely, $X^{a, b}$ is the continuous-time Markov chain on $[a, b] \cap \mathbb{Z}$ with transition rates as at (\[trates\]), where $y \sim x$ here means that $x$ and $y$ are nearest neighbours in $[a, b] \cap \mathbb{Z}$. Let $P^{a, b}_x$ denote the law of $X^{a, b}$ when started from the site $x \in [a, b] \cap \mathbb{Z}$. Henceforth, and throughout the rest of the paper, we shall refer to $X^{a, b}$ as the ‘inhomogeneous CTRW on $[a, b] \cap \mathbb{Z}$ in the trapping landscape $\sigma$’.
Hitting times for inhomogeneous CTRWs
-------------------------------------
We start by considering upper and lower bounds on the hitting times of the boundary by $X^{a, b}$. We let $\tau_a$ and $\tau_b$ denote the hitting time of the boundary at $a$ and $b$ respectively, i.e.$$\tau_a := \inf\{ s: X_s^{a, b} = a \} \quad \text{and} \quad \tau_b := \inf\{ s: X_s^{a, b} = b \} ,$$ and let $\tau$ be the hitting time of either boundary, i.e.$$\tau := \min\{ \tau_a, \tau_b\}.$$
\[prop:hittingtimeub\] For each $x \in [a, b] \cap \mathbb{Z}$ and $t > 0$, $$P^{a, b}_x \left( \tau_b \geq t \right) \leq 2 t^{-1} \, (b-x) \sum_{a \le z < b} \sigma_z .$$ Moreover, if ${S} \subseteq [x, b) \cap \mathbb{Z}$, then $$P^{a, b}_x ( \tau_b \geq t ) \leq 2 t^{-1} \bigg( (b-x) \sum_{ \{a \le z < b\} \setminus {S} } \sigma_z + |{S}| \max_{z \in {S}}\{b-z\} \max_{z \in {S}} \{ \sigma_z \} \bigg) .$$
By basic properties of simple random walks, the expected number of times the process $X^{a, b}$, started from $x$, visits a site $z \in (a, b) \cap \mathbb{Z}$ prior to $\tau_b$ is equal to $2 \min \{ b-z, b-x \}$, with the mean holding time at each visit to $z$ being $\sigma_z$. Similarly, the expected number of times $X^{a, b}$ hits the site $a$ prior to $\tau_b$ is equal to $(b-x)$ with the mean holding time at each visit being $2 \sigma_a$. Hence $$E_x^{a, b} \left[ \tau_b \right] = \sum_{a \le z < b} 2 \min \{ b-z, b-x \} \, \sigma_z .$$ For the first statement, we bound this expectation simply by $$E_x^{a, b} \left[ \tau_b \right] \le 2 ( b-x) \, \sum_{a \le z < b} \sigma_z ,$$ and apply Markov’s inequality. For the second statement, we instead bound the expectation by $$E_x^{a, b} \left[ \tau_b \right] \le 2 ( b-x) \, \sum_{\{a \le z < b\} \setminus {S} } \sigma_z + 2 |{S}| \max_{z \in {S}}\{b-z\} \max_{z \in {S}} \{ \sigma_z \} ,$$ and again apply Markov’s inequality.
\[prop:hittingtimelb\] For each $x \in [a, b) \cap \mathbb{Z}$, $z \in [x, b) \cap \mathbb{Z}$ and $t > 0$, $$P^{a, b}_x ( \tau_b \le t) \leq \frac{t}{2 (b-z) \sigma_z } .$$ Moreover, for each $x \in (a, b) \cap \mathbb{Z}$ and $t > 0$, $$P^{a, b}_x ( \tau \le t) \leq \frac{t}{\min\{x-a,b-x\} \sigma_x } .$$
Consider the first statement, and note that $\tau_b$ is bounded below by the time spent by $X^{a, b}$ at the site $z$ prior to the time $\tau_b$. Assume for the moment that $x \neq a$. By basic properties of simple random walks, the number of times $X^{a, b}$, started from $x$, visits a site $z \in [x, b) \cap \mathbb{Z}$ prior to $\tau_b$ is distributed as a geometric random variable (supported on $\{1, 2, \ldots\}$) with mean $2(b-z)$. Moreover, the time spent at each visit is an independent exponential random variable with mean $\sigma_z$. Hence, the time spent at $z$ prior to $\tau_b$ is exponentially distributed with mean $2(b-z)\sigma_z$. This implies that $$P^{a, b}_x \left( \tau_b \le t \right) \le 1 - \exp\left\{ - \frac{t}{2 (b-z) \sigma_z } \right\} \leq\frac{t}{2 (b-z) \sigma_z } .$$ If $x = a$ the proof is identical, since the extra factors of two in the means of the number of visits and holding time distributions exactly cancel each other out.
The proof of the second statement is similar. This time, consider that the number of times $X^{a, b}$ visits $x$ prior to the time $\tau$ is distributed as a geometric random variable (supported on $\{1, 2, \ldots\}$) with mean $$2 \left( \frac{1}{x-a} + \frac{1}{b-x} \right)^{-1} \ge \min \{ x-a, b-x \} ,$$ and the result follows as before.
Localisation of inhomogeneous CTRWs
-----------------------------------
To finish the section, we state a simple localisation property of $X^{a, b}$, expressed in terms of the trapping landscape $\sigma$.
\[prop:local\] For each $x \in [a, b] \cap \mathbb{Z}$ and ${S} \subseteq [a, b]\cap \mathbb{Z}$ such that $x \notin {S}$, $$\sup_{t \geq 0} P^{a, b}_x \left( X^{a, b}_t \in {S} \right) \leq \frac{\sum_{z \in {S}} \sigma_z}{\sigma_x } .$$
First note that, by standard continuous-time Markov chain theory, the process $X^{a, b}$ has a unique equilibrium distribution $\pi$. From the detailed balance equations, it is straightforward to check that $\pi \propto \sigma$, i.e. for each $z \in [a, b] \cap \mathbb{Z}$, $$\pi(z) =\frac{\sigma_z}{\sum_{a \le y \le b}\sigma_y} .$$ Hence, using the reversibility of the process, for each $z \in [a, b] \cap \mathbb{Z}$ and time $t \ge 0$, $$P^{a, b}_x \left( X_t = z \right) = P^{a, b}_z \left( X_t = x \right) \frac{\pi(z)}{\pi(x)} \le \frac{\pi(z)}{\pi(x)} = \frac{\sigma_z}{\sigma_x} .$$ Summing over ${S}$ yields the result.
Sequences of slowly varying random variables {#sec:svseq}
============================================
In this section we collect results on i.i.d. sequences of copies of $\sigma_0$. Throughout this section we shall assume that $L$ is slowly varying, but we shall not necessarily assume that $L$ satisfies the second-order slow-variation in Assumption \[assumpt:sosv\] or, indeed Assumption \[assumpt:g22\], unless we specify this explicitly. We first prove preliminary results on general i.i.d. sequences of slowly varying random variable; these relate to exceedences, records, and partial sums of such sequences. We then apply these results to establish the almost-sure bound on the ratio of the sum to the maximum of the sequence $(\sigma_i)_{i \in \mathbb{N}}$ of Theorem \[thm:main4\]. Finally, we study certain types of ‘hyperbolic’ exceedence; the relevance of these exceedences to the BTM will be made clear in Section \[sec:loc\].
Before proceeding, we shall first recall a useful consequence of second-order slow-variation Assumption \[assumpt:sosv\] for certain expectations involving $\sigma_0$; the spirit is similar to that of de Haan’s theorem (see [@Bingham87 Section 3.7]). We also state a weaker version of the result which holds for general slowly varying tails; the spirit is similar to that of Karamata’s theorem.
\[prop:sosv\] Assume $L$ satisfies Assumption \[assumpt:sosv\]. Let $f:(0,\infty) \to \mathbb{R}^+$ be a continuously differentiable function and $I \subseteq (0,\infty)$ an interval (which may be unbounded), and suppose there exists a $\delta > 0$ for which: (i) $f(t)\mathbf{1}_{\{t\in I\}} = o(t^\delta)$ as $t \rightarrow 0$; and (ii) both $f'(t) t^\delta$ and $f'(t) t^{-\delta}$ are integrable over the interval $I$. Then the function $$\Gamma(n) := \mathbf{E} \left[ f(\sigma_0 / n) \mathbf{1}_{ \{ \sigma_0/n \in I \}} \right]$$ satisfies $$\lim_{n \to \infty} \frac{L(n) \Gamma(n)}{g(n)} =- \lambda \int_I f'(t) \log t \, dt$$ for some constant $\lambda > 0$ that only depends on $L$ and $g$. Moreover, even if $L$ does not satisfy Assumption \[assumpt:sosv\] but is still slowly varying, we have that $$\lim_{n \to \infty} L(n) \Gamma(n) = 0 .$$
Note that the first statement of this result is a very slight generalisation of [@Croydon15 Proposition 5.1], which is recovered by setting $I := (0,\infty)$. It is proved in an identical manner. The second statement is also proved in an identical manner.
Preliminary results: Exceedences, records, and partial sums {#subsec:prelim}
-----------------------------------------------------------
Our preliminary results on general sequences of slowly varying random variables are split into three categories, containing results on: (i) first exceedences of levels; (ii) records of the sequence; and (iii) partial sums. Note that some of the results in this section hold only if Assumption \[assumpt:sosv\] is satisfied; Assumption \[assumpt:g22\] will not be relevant to this section.
Note that when we describe a collection $(X_i)_{i\in I}$ of non-negative random variables as being bounded above or bounded below in probability we mean that $(X_i)_{i\in I}$ or $(1/X_i)_{i\in I}$, respectively, is tight.
### First exceedences of levels
For a level $x > 0$, let $i_x$ denote the index of the first exceedence of $x$ in the sequence $(\sigma_i)_{i \in \mathbb{N}}$, that is $ i_x := \min \{ i : \sigma_i > x \}$. Further, denote by $ i_x^- := {\rm{argmax}} \{ \sigma_i : i < i_x \}$.
\[lem:iprob\] As $x \to \infty$, $$\frac{i_x}{ L(x) } ,\quad \frac{L(\sigma_{i_x})}{L(x)} - 1,\quad \frac{L(\sigma_{i^-_x})}{L(x)} \quad \text{and} \quad \ 1-\frac{L(\sigma_{i^-_x})}{L(x)}$$ are all bounded above and below in probability.
We first note that $$\sum_{i\in\mathbb{N}}\delta_{(\frac{i}{n},\frac{L(\sigma_i)}{n})}\rightarrow \nu=\sum_{i}\delta_{(u_i,v_i)},$$ in distribution as random measures on $\mathbb{R}^+\times\mathbb{R}^+$, where $\nu$ is a Poisson random measure with intensity $v^{-2}dudv$. (Here $\delta_{(u,v)}$ is the probability measure placing all its mass at $(u,v)$.) It follows that $$\label{poisson}
\min\left\{\frac{i}{n}:\:\frac{L(\sigma_i)}{n}>1\right\}\rightarrow \inf\left\{u_i:\:v_i>1\right\}$$ in distribution, where the limit is a $(0,\infty)$-valued random variable. Taking $n=L(x)$ in the above yields $i_x/L(x)$ converges in distribution, and so is bounded above and below in probability. Moreover, ${L(\sigma_{i_x})}/{L(x)}$ converges in distribution to the $v_i$ such that $(u_i,v_i)$ is an atom of $\nu$ and $u_i$ obtains the infimum in (\[poisson\]). Since the latter is a $(1,\infty)$-valued random variable, the second claim holds. Similarly, ${L(\sigma_{i^-_x})}/{L(x)}$ converges to maximum $v_j$ such that $(u_j,v_j)$ is an atom of $\nu$ and $u_j$ is strictly less than the infimum on the right-hand side of (\[poisson\]). Since this is a $(0,1)$-valued random variable, the proof is complete.
\[lem:ex\] Assume $L$ satisfies Assumption \[assumpt:sosv\]. Then there exists a $c > 0$ such that, as $x \to \infty$, $$\mathbf{E} \left[ \sigma_{i_x}^{-1} \right] < c \, x^{-1} g(x)$$ eventually.
We compute as follows $$\mathbf{E} \left[ \sigma_{i_x}^{-1} \right] = L(x) \, \mathbf{E} \left[ \sigma_0^{-1} \mathbf{1}_{\{\sigma_0 > x \} } \right]
= x^{-1} L(x) \, \mathbf{E} \left[ f(\sigma_0/x)\mathbf{1}_{\{\sigma_0/x > 1 \}} \right] ,$$ where $f(x) := x^{-1} $. Applying the first statement of Proposition \[prop:sosv\] we deduce that $$\mathbf{E} \left[ f(\sigma_0/x) \mathbf{1}_{\{\sigma_0/x > 1 \}} \right] \sim \frac{ \lambda g(x) }{L(x)} \int_1^\infty t^{-2} \log t \, dt ,$$ which yields the result.
### Records of the sequence
For $n \in \mathbb{N}$, let $r_n$ indicate the index of the $n^{\rm{th}}$ record of the sequence $(\sigma_i)_{i \in \mathbb{N}}$, and abbreviate $\sigma_{(n)} := \sigma_{r_n}$.
\[lem:probrecord\] Assume $L$ satisfies Assumption \[assumpt:sosv\]. Then for each $\varepsilon > 0$ there exists a $c > 0$ such that $$\mathbf{P} \left( \log L(\sigma_{(n)}) \notin n(1-\varepsilon, 1+\varepsilon) \right) < c n^{-2}$$ and $$\mathbf{P} \left( r_n < L(\sigma_{(n-1)}) / n^2 \quad \text{or} \quad r_n > 2 L(\sigma_{(n-1)}) \log n \right) < c n^{-2}$$ hold for all $n$. In particular, as $n \to \infty$, $$\log L(\sigma_{(n)}) \sim n \quad \text{and} \quad L(\sigma_{(n-1)}) / n^2 \leq r_n\leq 2 L(\sigma_{(n-1)}) \log n$$ eventually almost-surely.
For the first statement, note that the continuity of $L$ (guaranteed by Assumption \[assumpt:sosv\]) ensures that the sequence $(\log L(\sigma_i))_{i \in \mathbb{N}}$ consists of unit mean exponentially distributed random variables. By the memoryless property of the exponential distribution, the gaps in the sequence $(\log L(\sigma_{(i)}))_{i\in\mathbb{N}}$ are therefore also unit mean exponentially distributed random variables. Hence the statement is just a standard large deviation bound for exponentially distributed random variables. For the second statement, note that $$\begin{aligned}
\mathbf{P} ( r_n < L(\sigma_{(n-1)}) / n^2\:|\:\sigma_{(n-1)}) &\le& \mathbf{P} ( r_n - r_{n-1} < L(\sigma_{(n-1)}) / n^2\:|\:\sigma_{(n-1)})\\
& \le& \frac{L(\sigma_{(n-1)}) }{n^2 }\frac{1}{L(\sigma_{(n-1)})} = n^{-2}
\end{aligned}$$ by the union bound, and $$\label{tb}
\mathbf{P} ( r_n > 2 L(\sigma_{(n-1)}) \log n\:|\:\sigma_{(n-1)}) = \left( 1 - \frac{1}{L(\sigma_{(n-1)}) } \right)^{2 L(\sigma_{(n-1)}) \log n }\leq n^{-2} .$$ Hence, taking expectations, we have that $$\mathbf{P} ( r_n < L(\sigma_{(n-1)}) / n^2 \quad \text{or} \quad r_n > 2 L(\sigma_{(n-1)}) \log n) \leq 2n^{-2},$$ which yields the result. Finally, the last statement is just an application of the Borel-Cantelli lemma.
\[lem:record\] For each $k \in \mathbb{N}$ and $n \in \mathbb{N}$, $$\mathbf{E} \left[ \left( \frac{r_{n}}{L(\sigma_{(n-1)})} \right)^{k} \right] < k! .$$
Similarly to (\[tb\]), for each $\lambda > 0$ we have $$\mathbf{P}\left(r_n>\lambda L(\sigma_{(n-1)})|\: \sigma_{(n-1)}\right) = \left(1-\frac{1}{L(\sigma_{(n-1)})}\right)^{\lambda L(\sigma_{(n-1)})} < e^{-\lambda}.$$ Hence $\mathbf{P}(r_n>\lambda L(\sigma_{(n-1)})) < e^{-\lambda}$, in other words, the random variable $r_n/L(\sigma_{(n-1)})$ is stochastically dominated by a mean one exponential random variable. The $k$th moment of the latter is precisely $k!$, and so we are done.
\[momentcomparison\] Assume $L$ satisfies Assumption \[assumpt:sosv\]. Then there exists a $c > 0$ and $x_0<\infty$ such that, almost-surely: if $x\geq x_0$ and $n\geq m$, then $$\mathbf{E}\left(\frac{\sigma_{(m)}}{\sigma_{(n)}}\:\vline\:\sigma_{(m)}\right)\mathbf{1}_{\{\sigma_{(m)}\geq x\}}
< c^{n-m} \, g(x)^{n-m}.$$
Choose $x_0$ large enough so that the bound of Lemma \[lem:ex\] holds for $x\geq x_0$, and let $c$ be the constant of that lemma. Then, applying Lemma \[lem:ex\] repeatedly, we find that, almost-surely, if $x\geq x_0$ and $m\leq n$, then $$\begin{aligned}
\mathbf{E}\left(\frac{\sigma_{(m)}}{\sigma_{(n)}}\:\vline\:\sigma_{(m)}\right)\mathbf{1}_{\{\sigma_{(m)}\geq x\}}
&=&\mathbf{E}\left(\frac{\sigma_{(m)}}{\sigma_{(n-1)}}\mathbf{E}\left(\frac{\sigma_{(n-1)}}{\sigma_{(n)}}\:\vline\:\sigma_{(n-1)}\right)
\:\vline\:\sigma_{(m)}\right)\mathbf{1}_{\{\sigma_{(m)}\geq x\}}\\
& < &\mathbf{E}\left(\frac{\sigma_{(m)}}{\sigma_{(n-1)}} \, c \, g( \sigma_{(n-1)}) \:\vline\:\sigma_{(m)}\right)\mathbf{1}_{\{\sigma_{(m)}\geq x\}}\\
& < & c \, g( x) \, \mathbf{E}\left(\frac{\sigma_{(m)}}{\sigma_{(n-1)}} \:\vline\:\sigma_{(m)}\right)\mathbf{1}_{\{\sigma_{(m)}\geq x\}}\\
&&\vdots\\
&\leq & c^{n-m} g(x)^{n-m},\end{aligned}$$ as desired. Note that we use the monotonicity of $g$ (guaranteed by Assumption \[assumpt:sosv\]) to deduce that $g( \sigma_{(n-k)})\leq g(x)$ for $k=1,\dots, n-m$.
\[sumsofrecords\] Assume $L$ satisfies Assumption \[assumpt:sosv\], and recall the definition $d(u) := g(L^{-1}(u))$. Then for each $\varepsilon>0$ sufficiently small, integer $k\geq1$ and positive sequence $\delta_n$, there exists a $c > 0$ such that $$\mathbf{P}\left(\sum_{i=1}^{n-1}\sigma_{(i)} \geq (k-1+ \delta_n)\sigma_{(n)}\right) < c n^{-2} + c \, \delta_n^{-1} \, d(e^{n(1-\varepsilon)})^k$$ for all $n$.
Define the event $\mathcal{A}_n :=\{ \sigma_{(n-k)} \geq L^{-1}(e^{n(1-\varepsilon)})\}$. Then, by Lemma \[lem:probrecord\] and Markov’s inequality, there exists a $c > 0$ such that $$\begin{aligned}
\mathbf{P} \left(\sum_{i=1}^{n-1}\sigma_{(i)} \geq (k-1+ \delta_n)\sigma_{(n)}\right) & < & c n^{-2}+\mathbf{P}\left(\sum_{i=1}^{n-k}\sigma_{(i)} \geq \delta_n \sigma_{(n)}, \, \mathcal{A}_n \right)\\
& < &cn^{-2}+ \delta_n^{-1}\sum_{i=1}^{n-k}\mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n)}} \mathbf{1}_{\{\mathcal{A}_n \}} \right).\end{aligned}$$ The lower bound for $\sigma_{(n-k)}$ that holds on $\mathcal{A}_n$ allows us to apply Lemma \[momentcomparison\] to deduce that this is eventually bounded above, for some $c_1 > 0$, by $$\begin{aligned}
& \qquad \qquad < c_1 n^{-2} + c_1 \, \delta_n^{-1} \, d(e^{n(1-\varepsilon)})^k \, \sum_{i=1}^{n-k}
\mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n-k)}} \mathbf{1}_{\{\mathcal{A}_n \}} \right).\end{aligned}$$ Next, we note that the summands are bounded as follows $$\mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n-k)}} \mathbf{1}_{\{\mathcal{A}_n \}} \right)\leq \mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n-k)}} \right) \leq \mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n-k)}} \mathbf{1}_{\{\sigma_{(i)}\geq x\}}\right)+\mathbf{P}\left(\sigma_{(i)}\leq x\right).$$ For the first term, we again apply Lemma \[momentcomparison\] to deduce that, for large enough $x$, $$\mathbf{E}\left(\frac{\sigma_{(i)}}{\sigma_{(n-k)}} \mathbf{1}_{\{\sigma_{(i)}\geq x\}}\right)\leq (c g(x))^{n-k-i}.$$ In particular, by the fact that $g(x)\rightarrow 0$, we can choose $x$ such that $cg(x)<1$. Moreover, it is clear that, for each fixed $x$, there exists a constant $c_x$ such that $$\mathbf{P}\left(\sigma_{(i)}\leq x\right)\leq\mathbf{P}\left(\log L(\sigma_{(i)})\leq \log L(x)\right) = \mathbf{P}\left(\mathrm{Po}( \log L(x))\geq i\right)\leq c_xe^{-i},$$ where we use that $(\log L(\sigma_{(i)}))_{i\geq 1}$ simply represents the points of a unit rate Poisson process, and denote by $\mathrm{Po}(\lambda)$ a Poisson random variable with parameter $\lambda$. Hence, we conclude that $$\mathbf{P}\left(\sum_{i=1}^{n-1}\sigma_{(i)} < (k-1+ \delta_n)\sigma_{(n)}\right) < c n^{-2} + c \, \delta_n^{-1} \, d(e^{n(1-\varepsilon)})^k \sum_{i=1}^{n-k}\left((c g(x))^{n-k-i} +c_xe^{-i}\right),$$ and since the sum is bounded in $n$, this completes the proof.
### Partial sums
For $i$ an index, and let $(\sigma_{i}^{(1)}, \sigma_{i}^{(2)}, \ldots, \sigma_{i}^{(i)})$ be the (descending) order statistics of the subsequence $\{\sigma_j\}_{1 \le j \le i}$. Moreover, let $S_{i}^{(k)}$ denote the sum of the collection $(\sigma_{i}^{(j)})_{k \le j \le i}$.
\[lem:sumlevel\] Fix a $k \in \mathbb{N}$, and let $\ell_i, \delta_i$ be positive sequences such that $\ell_i \to \infty$ as $i \to \infty$. Then, there exists a $c > 0$ such that, as $i \to \infty$, $$\textbf{P} \left( S_{i}^{(k)} > \ell_i \delta_i\:\vline\:\sigma_1,\dots,\sigma_i\leq l_i \right) < c \, \delta_i^{-1} i^k \mathbf{E}\left(f_k\left(\frac{\sigma_0}{\ell_i}\right)\mathbf{1}_{\{\sigma_0/\ell_i\leq 1\}}\right)^k$$ eventually, where $f_k(x):=x^{1/k}$.
Let $i\geq k$. By symmetry and Markov’s inequality we have that $$\begin{aligned}
\lefteqn{\mathbf{P}\left(S_{i}^{(k)}> \ell_i \delta_i,\:\sigma_1,\dots,\sigma_i \leq \ell_i \right)}\\
&\leq & i^{k-1}\mathbf{P}\left(S_{i}^{(k)} > \ell_i \delta_i,\:\sigma_{i}^{(j)}=\sigma_j\mbox{ for }j=1,\dots,k-1,\:\sigma_{i}^{(1)}\leq \ell_i \right)\\
& = & i^{k-1}\mathbf{P}\left(\sum_{j = k}^i \sigma_j > \ell_i \delta_i,\: \sigma_{k-1} \leq \sigma_{k-2}\leq\dots\leq\sigma_1\leq \ell_i, \: \sigma_j \le \sigma_{k-1} \mbox{ for }j=k,\dots,i \right)\\
&\leq & i^{k-1} \delta_i^{-1} \mathbf{E}\left(\sum_{j=k}^i \frac{\sigma_j}{\ell_i} \mathbf{1}_{\{\sigma_{k-1}\leq\sigma_{k-2}\leq\dots\leq\sigma_1\leq \ell_i\}} \mathbf{1}_{\{\sigma_j\leq \sigma_{k-1} \text{ for }j=k,\dots,i \}}\right) \\
& \leq & i^{k} \delta_i^{-1} \mathbf{E}\left( \frac{\sigma_k}{\ell_i} \mathbf{1}_{\{\sigma_k \le \sigma_{k-1}\leq\sigma_{k-2}\leq\dots\leq\sigma_1\leq \ell_i \}} \mathbf{1}_{\{ \sigma_j\leq\ell_i \text{ for }j=k+1,\dots,i \}}\right) .\end{aligned}$$ Using the independence of the $(\sigma_j)_{j\geq 1}$, the above expectation is equal to $$\begin{aligned}
& \mathbf{E}\left( \frac{ \sigma_k }{\ell_i} \mathbf{1}_{\{\sigma_k \le \sigma_{k-1}\leq\sigma_{k-2}\leq\dots\leq\sigma_1\leq \ell_i\}} \right) \mathbf{P}\left(\sigma_0\leq \ell_i \right)^{i-k} \\
& \qquad \qquad \leq \mathbf{E}\left(\prod_{j=1}^k\left(\frac{\sigma_j}{\ell_i}\right)^{1/k}\mathbf{1}_{\{\sigma_j\leq \ell_i\}}\right) \mathbf{P}\left(\sigma_0\leq \ell_i \right)^{i-k} \\
& \qquad \qquad = \mathbf{E}\left(f_k\left(\frac{\sigma_0}{\ell_i}\right)\mathbf{1}_{\{\sigma_0/\ell_i\leq 1\}}\right)^k \mathbf{P}\left(\sigma_0\leq \ell_i \right)^{i-k} .\end{aligned}$$ On dividing through by $\mathbf{P}(\sigma_1,\dots,\sigma_i\leq \ell_i )=(1-L(\ell_i)^{-1})^{i}$, we thus obtain $$\mathbf{P}\left(S_{i}^{(k)}>\ell_i \delta_i \:\vline\:\sigma_1,\dots,\sigma_i\leq \ell_i \right)\leq {i^{k} \delta_i^{-1}} \mathbf{E}\left(f_k\left(\frac{\sigma_0}{\ell_i}\right)\mathbf{1}_{\{\sigma_0/\ell_i\leq 1\}}\right)^k\left(1-L(\ell_i)^{-1}\right)^{-k},$$ which proves the result.
\[cor:sumlevel\] Assume $L$ satisfies Assumption \[assumpt:sosv\]. Fix a $k \in \mathbb{N}$, and let $\ell_i, \delta_i$ be positive sequences such that $\ell_i \to \infty$ as $i \to \infty$. Then there exists a $c > 0$ such that, as $i \to \infty$, $$\textbf{P} \left( S_{i}^{(k)} > \ell_i \delta_i\:\vline\:\sigma_1,\dots,\sigma_i \le \ell_i \right) < c \, \delta_i^{-1} \left( \frac{i}{L(\ell_i)} g(\ell_i) \right)^k$$ eventually.
By the first statement of Proposition \[prop:sosv\] we deduce that, for some $\lambda > 0$, $$\mathbf{E}\left(f_k\left(\frac{\sigma_0}{\ell_i}\right)\mathbf{1}_{\{\sigma_0/\ell_i\leq 1\}}\right) \sim -\frac{\lambda \, g(\ell_i)}{L({\ell_i})} \int_{0}^1 f_k'(t)\log t \, dt,$$ from which the result follows by applying Lemma \[lem:sumlevel\].
The ratio of the sum to the maximum
-----------------------------------
In this section we prove the key almost-sure bound on the ratio of the sum to the maximum of the sequence $(\sigma_i)_{i \in \mathbb{N}}$ in Theorem \[thm:main4\]. Throughout this section, we assume that $L$ satisfies Assumption \[assumpt:sosv\], and define $N$ as in .
Observe that Theorem \[thm:main4\] is a consequence of the following two results, which we prove in the next two subsections. Recall that $m_i$ and $S_i$ denote the maximum and partial sum of the partial sequence $(\sigma_j)_{j \le i}$, and $S_{i}^{(k)}$ is the sum from the $k$th largest term.
\[prop:ub\] Suppose Assumption \[assumpt:sosv\] holds. For each $\varepsilon > 0$, as $i \to \infty$, $$\frac{S_i}{m_i} < N - 1 + \varepsilon$$ eventually almost-surely. Moreover, if Assumption \[assumpt:g22\](b) (with $N < \infty$) also holds, then additionally, as $i \to \infty$, $$\frac{S_i^{(N)}}{m_i} < \frac{\varepsilon}{\log i}$$ eventually almost-surely.
\[prop:lb\] If Assumptions \[assumpt:sosv\] and \[assumpt:g22\](a) (with $N < \infty$) hold, then for each $\varepsilon > 0$ sufficiently small, as $i \to \infty$, $$\frac{S_i}{m_i} > N - 1 - \varepsilon$$ holds infinitely often.
Of course, the control on the rate of convergence in the second part of Proposition \[prop:ub\] under Assumption \[assumpt:g22\](b) is stronger than we need for Theorem \[thm:main4\], but this extra control will be useful in Section \[sec:fav\].
### Upper bound on sum/max ratio
Recall, for each $n \in \mathbb{N}$, the notation $r_n$ and $\sigma_{(n)}$ for the location and magnitude of the $n$th record from Section \[subsec:prelim\], and let $S_{(n)}^-$ denote the sum of the collection $\{\sigma_i\}_{i < r_n}$. Further, for each $\varepsilon > 0$ and $n\in\mathbb{N}$ define the events $$\mathcal{A}^\varepsilon_n := \left\{ \frac{ S_{(n)}^- }{ \sigma_{(n-1)} } > N-1 + \varepsilon \right\},\qquad \bar{\mathcal{A}}^\varepsilon_n := \left\{ \frac{ S^{(N)}_{r_n-1}}{ \sigma_{(n-1)} } > \frac{ \varepsilon} {\log n} \right\} .$$ Since the ratio of the sum to the max is increasing up until new records of the sequence, to establish Proposition \[prop:ub\], by the Borel-Cantelli lemma it is sufficient to prove the following.
\[anlem\] Suppose Assumption \[assumpt:sosv\] holds. For each $\varepsilon > 0$ we have that $$\sum_{n \in \mathbb{N}} \mathbf{P} \left( \mathcal{A}_n^\varepsilon \right) < \infty .$$ Moreover, if Assumption \[assumpt:g22\](b) (with $N < \infty$) also holds, then additionally $$\sum_{n \in \mathbb{N}} \mathbf{P} \left( \bar{\mathcal{A}}_n^\varepsilon \right) < \infty .$$
By definition, we have that $$S_{(n)}^- =\sum_{i=1}^{n-1}\sigma_{(i)}+\sum_{i=1}^{r_n-1}\sigma_i\mathbf{1}_{\{i\not\in\mathcal{R}\}},$$ where we recall that $\mathcal{R}$ is the collection of record traps $(r_n)_{n\geq 1}$. Now, conditional on $\{(r_i,\sigma_{(i)}):\:i\leq n\}$, the traps that contribute to the second sum are independent. Moreover, for $i\in (r_{m-1},r_{m})$, $m\in\{1,\dots,n\}$, we have that the traps are distributed as $\sigma_0|\{\sigma_0\leq \sigma_{(m-1)}\}$, and so are stochastically dominated by $\sigma_0|\{\sigma_0\leq \sigma_{(n-1)}\}$. It follows that $$\mathbf{P}\left(\sum_{i=1}^{r_n-1}\sigma_i\mathbf{1}_{\{i\not\in\mathcal{R}\}}\geq \lambda\sigma_{(n-1)}\:\vline\:(r_i,\sigma_{(i)}):\:i\leq n\right) \leq F \left(r_n, \sigma_{(n-1)}, \floor{\lambda}+1, \lambda - \floor{\lambda} \right),$$ where, recalling the notation for $S^{(k)}_r$ from Section \[subsec:prelim\], $$\label{Fdef}
F(r, l, k, \delta):=\mathbf{P}({S}^{(k)}_{r}\geq l \delta | \sigma_1, \ldots, \sigma_r \le l) .$$ Applying the above reasoning, we have $$\begin{aligned}
\lefteqn{\mathbf{P} \left( \mathcal{A}_n^\varepsilon \right)}\\
& \leq &
\sum_{k=0}^{N-1}\mathbf{P} \left(\sum_{i=1}^{n-1}\sigma_{(i)}\geq \left(k + \varepsilon/2 \right) \sigma_{(n-1)} , \:\sum_{i=1}^{r_n-1} \sigma_i\mathbf{1}_{\{i\not\in\mathcal{R}\}}\geq
\left(N-2-k + \varepsilon/2 \right)\sigma_{(n-1)}\right)\\
&\leq &\sum_{k=0}^{N-1}\mathbf{E}\left(\mathbf{1}_{\{\sum_{i=1}^{n-2}\sigma_{(i)}\geq (k-1+\varepsilon/2 )\sigma_{(n-1)}\}} F \left((r_n,\sigma_{(n-1)},N-1-k, \varepsilon/2 \right) \right).\end{aligned}$$ Recall that, by Lemma \[lem:probrecord\] and the union bound, there exists a $c > 0$ such that, for sufficiently large $n$, $$\mathbf{P} \left(L(\sigma_{(n-1)}) < e^{n(1- \varepsilon)} \, , \ r_n > 2 L(\sigma_{(n-1)})\log n \right) < c n^{-2} .$$ Applying Lemma \[sumsofrecords\] and Corollary \[cor:sumlevel\] (with $\delta_i = \varepsilon/2$), it follows that there exists a $c > 0$ such that, for sufficiently large $n$, $$\begin{aligned}
\mathbf{P} \left( \mathcal{A}_n^\varepsilon \right)
&< & c n^{-2} + c \, \sum_{k=0}^{N-1}\left( d( e^{n(1-\varepsilon)}) \log n \right)^{N-1-k}
\mathbf{P}\left(\sum_{i=1}^{n-2}\sigma_{(i)}\geq (k-1+\varepsilon/2)\sigma_{(n-1)}\right)\\
& < &c n^{-2}+ c \left( d(e^{n(1-\varepsilon)}) \log n \right)^{N-1} .\end{aligned}$$ Considering the definition of $N$ in , and noting that a monotonic sequence $a_n$ is summable if and only if $a_{\floor{(1-\varepsilon) n}}$ is summable, this completes the proof of the first statement.
For the second statement, a similar argument holds. In particular, we start by noting that $$S_{r_n-1}^{(N)}=\min_{k=1,\dots,N-1}\left\{\sum_{i=1}^{(n-1)-(N-k)}\sigma_{(i)}+\Sigma^{(k)}\right\},$$ where $\Sigma^{(k)}$ is the sum $\sum_{i=1}^{r_n-1}\sigma_i\mathbf{1}_{\{i\not\in\mathcal{R}\}}$ with the largest $k-1$ terms excluded. Since $$\left\{S_{r_n-1}^{(N)}>\varepsilon\sigma_{(n-1)}/\log n,\:\Sigma^{(k)}<\varepsilon\sigma_{(n-1)}/2\log n\right\}\subseteq\left\{\sum_{i=1}^{(n-1)-(N-k)}\sigma_{(i)}>\varepsilon\sigma_{(n-1)}/2\log n\right\},$$ decomposing the probability space into regions where $\Sigma^{(k)}>\varepsilon\sigma_{(n-1)}/2\log n\geq\Sigma^{(k+1)}$ thus yields $$\begin{aligned}
{\mathbf{P} \left( \bar{\mathcal{A}}_n^\varepsilon \right)}&\leq &\mathbf{P} \left(\sum_{i=1}^{(n-1)-(N-1)}\sigma_{(i)}>\varepsilon\sigma_{(n-1)}/2\log n \right)\\
&&+\sum_{k=1}^{N-1}\mathbf{E}\left(\mathbf{1}_{\{\sum_{i=1}^{(n-1)-(N-k-1)}\sigma_{(i)}\geq \varepsilon\sigma_{(n-1)}/2\log n\}} F \left(r_n,\sigma_{(n-1)},k, \varepsilon/2\log n \right) \right).\end{aligned}$$ Noting that the proof of Lemma \[sumsofrecords\] shows that $$\mathbf{P} \left(\sum_{i=1}^{(n-1)-j}\sigma_{(i)}>\varepsilon\sigma_{(n-1)}/2\log n \right) < cn^{-2}+cd(e^{n(1-\varepsilon)})^j\log n,$$ uniformly for $j=1,\dots,N-1$, and applying Corollary \[cor:sumlevel\] with $\delta_i = \varepsilon/2\log i$ similarly to above, it follows that $$\mathbf{P} \left( \bar{\mathcal{A}}_n^\varepsilon \right) < cn^{-2}+
cd(e^{n(1-\varepsilon)})^{N-1}\left(\log n\right)^{N},$$ which is summable on Assumption \[assumpt:g22\](b).
### Lower bound on sum/max ratio
Again recall, for each $n \in \mathbb{N}$, the notation $r_n$ and $\sigma_{(n)}$ from Section \[subsec:prelim\], and let $\{ r_n^{(1)}, r_n^{(2)}, \ldots , r_n^{(N-2)} \}$ denote the indices of the largest $N-2$ terms of the sequence lying between $r_n$ and $r_{n+1}$ in increasing order of index (if there are insufficient terms, set the undefined terms to be equal to $r_{n+1}$), abbreviating $\sigma_{(n)}^{(i)} := \sigma_{r_n^{(i)}}$. To establish the lower bound we consider the following event, defined for each $\varepsilon \in (0, 1)$ and $n\in\mathbb{N}$, $$\mathcal{A}^{\varepsilon}_n := \bigcap_{1 \le i \le N-2} \left\{ \sigma^{(i)}_{(n)} / \sigma_{(n)} \in (1 - \varepsilon, 1) \right\} .$$ Clearly on the event $\mathcal{A}^\varepsilon_n$ we have that $$\frac{ S_{r_n^{(N-2)}} }{m_{r_n^{(N-2)}}} > (N-1)(1-\varepsilon),$$ and so in order to establish Proposition \[prop:lb\] it is sufficient to prove that for each $\varepsilon \in (0, 1)$ the events $\mathcal{A}_n^\varepsilon$ hold infinitely often. That this is true can be deduced by applying the following lemma in conjunction with the conditional Borel-Cantelli lemma.
Let $\mathcal{F}_n$ denote the $\sigma$-algebra generated by $\sigma_1,\sigma_2,\dots,\sigma_{r_{n+1}}$. Then $\mathcal{A}_n^\varepsilon\in \mathcal{F}_n$, and, on Assumptions \[assumpt:sosv\] and \[assumpt:g22\](a) (with $N < \infty$), $$\sum_n \mathbf{P}(\mathcal{A}^\varepsilon_{n} | \mathcal{F}_{n-1}) = \infty$$ almost-surely.
That $\mathcal{A}_n^\varepsilon\in \mathcal{F}_n$ is clear by definition. For the second claim, we note that the event $\mathcal{A}^\varepsilon_n$ is just the event that the first $N-2$ exceedences of the level $\sigma_{(n)} (1 - \varepsilon)$ after $r_n$ do not also exceed the level $\sigma_{(n)}$. Hence $$\begin{aligned}
\mathbf{P}(\mathcal{A}^\varepsilon_{n}\: |\: \mathcal{F}_{n-1}) &=&\mathbf{P}\left(\sigma_0<u\:|\:\sigma_0>(1-\varepsilon)u\right)^{N-2}|_{u=\sigma_{(n)}}\\
&=&\left(1-\frac{L((1-\varepsilon)\sigma_{(n)})}{L(\sigma_{(n)})}\right)^{N-2}.\end{aligned}$$ Combining with the bounds on $\sigma_{(n)}$ from Lemma \[lem:probrecord\] and the definition of second-order slow-variation, it follows that there exists a $c > 0$ such that $$\mathbf{P}(\mathcal{A}^\varepsilon_{n}\: |\: \mathcal{F}_{n-1})
\sim \left(-k(1-\varepsilon)g(\sigma_{(n)})\right)^{N-2} \ge c d(e^{n(1+\varepsilon)} )^{N-2}$$ eventually, where we have used the eventual monotonicity of $g$ guaranteed by Assumption \[assumpt:sosv\]. Again noting that a monotone sequence $a_n$ is summable if and only if $a_{\floor{(1+\varepsilon) n}}$ is summable, the above sequence is not summable on Assumption \[assumpt:g22\](a), completing the proof.
Hyperbolic exceedences
----------------------
In this section we study certain ‘hyperbolic’ exceedences; the relevance of these exceedences to the localisation of the BTM will be made clear in Section \[sec:loc\]. Note that we do not apply Assumptions \[assumpt:sosv\] or \[assumpt:g22\] here.
Before defining these hyperbolic exceedences, recall that $i_x$ indicates the first exceedence of a level $x > 0$ of the sequence $(\sigma_i)_{i \in \mathbb{N}}$, and $S_i$ denotes the cumulative sum of the sequence up to an index $i$. Define an auxiliary function $h_t$ such that $h_t \to \infty$ as $t \to \infty$ sufficiently slowly such that $$\begin{aligned}
\label{eq:h}
\frac{L(t h^3_t)}{L(t)} < 1 + \frac{1}{h_t} \quad \text{and} \quad \frac{L(t /h_t^3)}{L(t)} > 1 - \frac{1}{h_t}\end{aligned}$$ eventually. We note that the choice of such a $h_t$ is possible for any slowly varying function $L$; see [@Croydon15] for an explicit construction. We define the *hyperbolic exceedences* of the level $t$ to be the sites $$\label{jtdef}
j_t:= \min \{ i : i S_i>t/h_t\} \quad \text{and} \quad j_t^- := {\rm{argmax}} \{ \sigma_i : i \leq j_t \} .$$
\[lem:jprob\] For each $t \ge 0$, denote $$\label{elltdef}
\ell_t := \min \{s \ge 0: s L(s) \ge t \} ,$$ which is well-defined since $L$ is càdlàg. Then, as $t \to \infty$, $$\mathbf{P} \left( i_{\ell_t}=j_t=j_t^- \right) \to 1 .$$
First note that, by our choice of $h_t$ in and applying Lemma \[lem:iprob\], we have that, as $t \to \infty$, $i_{\ell_t} > L(\ell_t) h^{-1}_{\ell_t}$ and $\sigma_{i_{\ell_t}} > \ell_t h^3_{\ell_t}$ both hold with high probability. Then it is clear that $j_t \le i_{\ell_t}$, since then, with high probability, $$i_{\ell_t} S_{i_{\ell_t}} > i_{\ell_t} \sigma_{i_{\ell_t}} > L(\ell_t) h^{-1}_{\ell_t} \times \ell_t h^3_{\ell_t} > t.$$ For the other direction, consider the proceeding record site $r_{n_t}$ prior to $i_{\ell_t}$. Applying Lemma \[lem:iprob\] again, with high probability we have that $$i_{\ell_t} < L(\ell_t) h_{\ell_t}^{1/2}, \qquad \sigma_{r_{n_t}}=\sigma_{i_{\ell_t}^-} < \ell_t / h^3_{\ell_t} \qquad \text{and}\qquad L(\sigma_{i_{\ell_t}^-} )>L(\ell_t)h_{\ell_t}^{-1/3}.$$ Thus, using the notation $F$ from (\[Fdef\]) and writing $\Gamma(x):=\mathbf{E}(\frac{\sigma_0}{x}\mathbf{1}_{{\sigma_0}\leq{x}})$, $$\begin{aligned}
\mathbf{P}\left(S_{i_{\ell_t} - 1} / \sigma_{r_{n_t}}>h_{\ell_t}\right)&=& \mathbf{E}\left(F(i_{\ell_t}-2,\sigma_{i_{\ell_t}^-},1,h_{\ell_t}-1)\right)\\
&\leq& \mathbf{E}\left(\min\left\{\frac{cL(\ell_t)\Gamma(\sigma_{i_{\ell_t}^-})}{\sqrt{h_{\ell_t}}},1\right\}\right)+o(1), \end{aligned}$$ where to deduce the inequality we have applied Lemma \[lem:sumlevel\]. Now, by Proposition \[prop:sosv\], $\Gamma(x)=o(L(x)^{-1})$. Hence it follows that $$\mathbf{P}\left(S_{i_{\ell_t} - 1} / \sigma_{r_{n_t}}>h_{\ell_t}\right)\leq
\mathbf{E}\left(\min\left\{\frac{cL(\ell_t)}{\sqrt{h_{\ell_t}}L(\sigma_{i_{\ell_t}^-})},1\right\}\right)+o(1)\leq c h_{\ell_t}^{-1/6}+o(1)\rightarrow 0.$$ This implies that with high probability eventually $$(i_{\ell_t}-1) S_{i_{\ell_t}-1} < h_{\ell_t}i_{\ell_t} \sigma_{r_{n_t}} < L(\ell_t) h_{\ell_t}^{3/2} \times \ell_t / h_{\ell_t}^3 = t/h_{\ell_t}^{3/2} < t/h_{t}.$$ Thus we have shown that $\mathbf{P}(i_{\ell_t}=j_t)\to 1$ as $t\to \infty$. To complete the proof, we simply note that $i_{\ell_t}=j_t$ implies $j_t$ is a record, and therefore $j_t=j_t^-$.
Quenched localisation on $N$ sites {#sec:loc}
==================================
In this section we establish that localisation takes place on no more than $N$-sites almost-surely, that is, we prove the first claim of Theorem \[thm:main1\]. Note that this claim only has content if $N$ is finite, so wherever we work under Assumption \[assumpt:sosv\] in this section we shall always assume that $N < \infty$. On the other hand, Assumption \[assumpt:g22\] will not play a role in this section.
We begin in Section \[subsec:def\] by giving an explicit construction of the localisation set $\Gamma_t$. We note that there are several different possible approaches to defining this set; one way, for instance, would be to select a certain set of $N$ sites, whereby the cardinality of $\Gamma_t$ would be bounded by construction. We choose a construction that makes no explicit reference to cardinality; instead $\Gamma_t$ is defined to include all points lying in a certain region of $(x, \sigma_x)$-space. The advantage of this definition is that the subsequent proof that the BTM localises on $\Gamma_t$ is straightforward; we do this in Section \[subsec:loc\]. The trade-off is that bounding the cardinality of $\Gamma_t$ is no longer straightforward, and requires a somewhat lengthy computation; we undertake this computation in Section \[subsec:card\]. Along the way, we also establish that the localisation set $\Gamma_t$ consists of a single site with overwhelming probability, and moreover that $\Gamma_t$ is contained on the record traps eventually almost-surely if $N = 2$, hence completing the proof of Theorem \[thm:maincl\] and the forward direction of Theorem \[thm:main3\].
Defining the localisation set {#subsec:def}
-----------------------------
We shall define the localisation set $\Gamma_t$ by first considering a certain region $\mathcal{G}_t \subseteq \mathbb{Z}^+ \times \mathbb{R}^+$ defined by an outer boundary $O_t$ and a lower boundary $D_t$; the localisation set will then consist of all points $(x, \sigma_x)$ that lie inside this region. Before we define these explicitly, we first motivate our construction, which is based around certain record traps $z^{I}_t$ and $z^{O}_t$ such that $0 < z^{I}_t \le z^{O}_t$. First, we construct the site $z^{I}_t$ to be the furthest record site from the origin such that the BTM is overwhelmingly likely to have visited this site by time $t$, essentially because this site is not too far from the origin and the traps lying between it and the origin are sufficiently shallow. Naturally this construction demands that we use an upper bound on hitting times of the BTM, and indeed we approximate the time until the BTM exits a certain region by the product of the sum of the traps in the region and the length of the region. This suggests that the site $z^{I}_t$ should be defined via the notion of [hyberbolic exceedences]{} (see Section \[sec:svseq\]).
Second, we construct the site $z^{O}_t$ to be the furthest record site from the origin such that it is [possible]{} for the BTM to reach this site by time $t$. This time we use a lower bound for the hitting times of the BTM, approximating the time until the BTM exits a certain region by the product of the length of the region and the depth of the deepest trap in the region. We optimise our construction of $z^{O}_t$ by a process of ‘chaining’ from the initial site $z^{I}_t$, using the observation that each new record trap that is visited by the BTM by time $t$ [reduces]{} the successive distance that the BTM is able to travel by time $t$, since the BTM now has to overcome the holding time associated with this new record trap. Note that it is possible (and indeed will turn out to be likely) that the site $z^{O}_t$ is actually the same as the site $z^{I}_t$; this occurs if the chaining terminates at the first stage.
Once we have defined the sites $z^{I}_t$ and $z^{O}_t$, we construct the outer boundary $O_t$ to appropriately contain these sites, i.e. such that $0 < z^{I}_t \le z^{O}_t < O_t$. This is done in such a way to guarantee that the BTM is located within this boundary at time $t$; again we use a lower bound for hitting times of the BTM. Our construction also guarantees that $z^{O}_t$ is the largest trap located in the region $[0,O_t)$. Finally, we make use of the observation that the BTM, at any given time $t$, is more likely to be located in a deep trap than a shallow trap; the lower boundary $D_t$ represents the trap depth above which we know that the BTM is located with high probability.
We now formalise the above heuristics to give an explicit definition of the localisation set $\Gamma_t$. In the following definitions we make reference to a certain auxiliary scaling function $h_u \to \infty$ as $u \to \infty$. We think of $h_u$ as being arbitrarily slowly growing, and indeed we shall insist that $h_u$ grows sufficiently slowly to satisfy certain conditions. First, similarly to , we shall require that, for any $k \in \mathbb{N}$, $$\begin{aligned}
\label{eq:h2}
\frac{L(t h^k_t)}{L(t)} < 1 + \frac{1}{h_t} \quad \text{and} \quad \frac{L(t /h_t^k)}{L(t)} > 1 - \frac{1}{h_t}\end{aligned}$$ for large $t$. Further, we shall also require that $$\begin{aligned}
\label{eq:h3}
{h^4_t}{L(t h_t^2)} \mathbb{E}\left[ \frac{\sigma_0}{t h_t^2} \mathbf{1}_{\{ \sigma_0 < t h_t^2 \}} \right] \to 0,\end{aligned}$$ remarking that this is possible by the second statement of Proposition \[prop:sosv\]. When working under Assumption \[assumpt:sosv\] (with $N < \infty$), defining $\hat{h}_n := h_{2 N e^{2n} L(e^{2n}) \log n}$, we shall additionally require that $$\begin{aligned}
\label{eq:h4}
\sum_{n \in \mathbb{N}} e^{-n/2} \hat{h}_n < \infty \quad \text{and} \quad \sum_{n \in \mathbb{N}} \left( d(e^n) (\log n) (\hat{h}_n)^5 \right)^{N-1} < \infty.\end{aligned}$$ Finally, for technical reasons, we shall also require that $h_u$ is continuous. The relevance of these conditions will become clear later in the section; for now, note simply that it is always possible to choose such an $h$ (see [@Croydon15] for remarks as to an explicit construction).
We first define the sites $z^{I}_t$ and $z^{O}_t$. The site $z^{I}_t$ is taken to be the record site $j_t^-$, as introduced at . To define the site $z^{O}_t$, define iteratively $$y_t^1:= z^{I}_t \ , \quad y_t^{i+1} := \min \{ z\in (y_t^{i},y_t^{i} + h_t \max\{t / \sigma_{y_t^i}, 1\}) :\: \sigma_z > \sigma_{y_t^{i}} \}$$ until this chain terminates. The site $z^{O}_t$ is defined to be the last site so-defined by the chaining (which is possibly, and indeed probably, the same as the site $z^{I}_t$). Note that this method of ‘chaining’ can be thought of as a general procedure that starts from a certain site $z^{I}_t$ and occurs at a certain time $t$; we will refer back to this general method of ‘chaining’ in Section \[subsec:card\]. We also note that in the process of chaining we make use of the lower bound $\max\{ \cdot, 1\}$ when extending the outer boundary from $y_t^{i}$ to $y_t^{i+1}$; this is done for technical reasons, essentially to ensure that the regions we are considering are always growing (albeit arbitrarily slowly) with $t$. This will allow us to successfully apply our holding time estimates to these regions.
We now define the localisation set $\Gamma_t$, by specifying an outer boundary $O_t$ and a lower boundary $D_t$ for the localisation region $\mathcal{G}_t$. First, define the outer boundary $O_t$ to be $$O_t := z^{O}_t + h_t \max\{ t / \sigma_{z^{O}_t}, 1 \} \, .$$ To define the lower boundary $D_t$, for a level $\ell > 0$ denote the quantity $S^\ell_i := \sum_{ z < i: \sigma_z < \ell } \sigma_z $, and set $$D_t := \max\{ \ell \ge 0 : S^\ell_{O_t} < \sigma_{z^{I}_t} / h_t \} .$$ Note that this construction of the lower boundary can be thought of as a general procedure that is given by a certain boundary $O_t$ and level $\sigma_{z^{I}_t} / h_t$; we will refer back to this general procedure in Section \[subsec:card\]. We can now define the localisation set to be the point set $$\Gamma_t := \{ x \in \mathbb{Z} : (x, \sigma_x) \in \mathcal{G}_t \}$$ where $\mathcal{G}_t := \{ (x, \sigma_x) : x < O_t , \sigma_x \ge D_t \}$. Figure \[gtpic\] shows typical and atypical configurations of this set.
(0,0) – (2.1, 0) node\[anchor=north\] [$O_t$]{} – (4.5,0) node\[anchor=north west\] [$\mathbb{Z}$]{}; (0,0) – (0,1.6) node\[anchor=east\] [$\ell_t$]{} – (0,2) node\[anchor=east\] [$D_t$]{} – (0,4.5) node\[anchor=south\] [$L(\sigma_z)$]{}; (0,1.6) – (-0.1,1.6); (0,2) – (-0.1,2); (1.6,0) – (1.6,-0.1); (2.1,0) – (2.1,-0.1); (0.1,4) .. controls (0.15,1.5) and (0.2,1.5) .. (4.3,1.5) ; (2.1, 2) – (2.1, 4); (0, 2) – (2.1, 2); in [(0.5,0.08), (1.1,0.76), (1.6,1.96), (2.6,2.76), (3.3,3.46)]{}[ at ; ]{} (3, 1.7) node ; (0,0) – (0.5,0) – (0.5, 0.1) – (1.1, 0.1) – (1.1, 0.8) – (1.6, 0.8) – (1.6, 2) – (2.6, 2) – (2.6, 2.8) – (3.3, 2.8) – (3.3, 3.5) – (4.0, 3.5) ; (1, 3) node\[anchor=south\] [$\mathcal{G}_t$]{}; (1.05, 2.2) node\[anchor=south\] [$\Gamma_t$]{}; (1.27, 2.38) – (1.53, 2.13) ; (4.5, 1.55) node\[anchor=north\] [$\text{hyp}_t$]{}; (0.9, 0.1) node\[anchor=north\] [$z_t^I = z_t^O$]{} ;
(0,0) – (4.5,0) node\[anchor=north west\] [$\mathbb{Z}$]{}; (0,0) – (0,2) node\[anchor=east\] [$\ell_t$]{} – (0,4.5) node\[anchor=south\] [$L(\sigma_z)$]{}; (0,1.7) – (-0.1,1.7); (0,2) – (-0.1,2); (1.6,0) – (1.6,-0.1); (3.3,0) – (3.3,-0.1); (3.4,0) – (3.4,-0.1); (0.1,4) .. controls (0.15,2) and (0.2,2) .. (4.3,2) ; (3.4, 1.7) – (3.4, 4); (0, 1.7) – (3.4, 1.7); in [(0.5,0.08), (1.1,0.76), (1.6,1.96), (2.6,2.76), (3.3,3.46)]{}[ at ; ]{} (3, 1.7) node ; (0,0) – (0.5,0) – (0.5, 0.1) – (1.1, 0.1) – (1.1, 0.8) – (1.6, 0.8) – (1.6, 2) – (2.6, 2) – (2.6, 2.8) – (3.3, 2.8) – (3.3, 3.5) – (4.0, 3.5) ; (1.6, 3) node\[anchor=south\] [$\mathcal{G}_t$]{}; (3.8, 2.3) node\[anchor=south\] [$\Gamma_t$]{}; (3.7, 2.85) – (3.5, 3.25) ; (3.5, 2.6) – (3, 2.65) ; (3.5, 2.5) – (2.2, 2.13) ; (3.7, 2.4) – (3.3, 2.1) ; (4.5, 2.05) node\[anchor=north\] [$\text{hyp}_t$]{}; (1.6, 0.1) node\[anchor=north\] [$z_t^I$]{} ; (3, 0.1) node\[anchor=north\] [$z_t^O$]{} ; (3.6, 0) node\[anchor=north\] [$O_t$]{} ; (0,1.7) node\[anchor=east\] [$D_t$]{};
To complete this section, we prove that the localisation site consists of a single site with overwhelming probability; the precise description of this single site closely mirrors the construction in [@Muirhead15]. Note that this result does not require Assumption \[assumpt:sosv\], and holds for any slowly varying $L$.
For each $t \ge 0$ define the level $ \ell_t$ as at (\[elltdef\]), and denote the site $Z_t :=i_{\ell_t}$ (in the notation of Section \[subsec:prelim\]). Then, as $t \to \infty$, $$\mathbf{P} \left( \Gamma_t = Z_t \right) \to 1 .$$
Since $Z_t=i_{\ell_t}$ and $z_t^{I}=j^-_t$ we have by Lemma \[lem:jprob\] that $\mathbf{P} ( Z_t = z_t^{I})\rightarrow 1$. From this it immediately follows that $\mathbf{P} ( Z_t \subseteq \Gamma_t ) \to 1$. Hence, it is sufficient to show that, as $t \to \infty$, $$\mathbf{P} ( | \Gamma_t | = 1) \to 1.$$ To this end, we need to show that with high probability neither of the two disjoint regions $$R_1 := \{ x:\: x < O_t,\: \sigma_x > \sigma_{z^{I}_t} \} \quad \text{and} \quad R_2 := \{x: x < O_t,\: D_t \le \sigma_x \le \sigma_{z^{I}_t} \} \setminus \{(z_t^I, \sigma_{z_t^I})\} .$$ contains a point.
Observe that $$\mathbf{P}\left(\exists x\in (Z_t, Z_t + h_t \max\{t / \sigma_{Z_t},1\})\mbox{ such that }\sigma_x>\sigma_{Z_t}|\:\sigma_{Z_t}\right)\leq \frac{h_t \max\{t / \sigma_{Z_t},1\}}{L(\sigma_{Z_t})}.$$ If $\sigma_{Z_t} > \ell_t h_{\ell_t}^2$, which by Lemma \[lem:iprob\] and our assumptions on $h_t$ in holds with high probability, then the right-hand side is bounded above by $$\max\left\{\frac{h_t}{L(\ell_t h_{\ell_t}^2)},\frac{1}{h_t}\right\},$$ and, possibly choosing an even more slowly growing $h$ than already required by our assumptions in , one can check that this converges to 0 as $t\rightarrow\infty$. Since we also know from the previous paragraph that $Z_t = z_t^{I}$ with high probability, it follows that with high probability the chaining procedure terminates at the first step and we have $\mathbf{P}(Z_t = z_t^{I}=z_t^{O})\rightarrow 1$ as $t\rightarrow\infty$. In particular, this implies $\mathbf{P}(R_1=\emptyset)\rightarrow1$.
We now deal with $R_2$. For this, note that $Z_t = z_t^{I}=z_t^{O}$ implies $$(0 ,O_t)\subseteq (0, Z_t + h_t \max\{t / \sigma_{Z_t},1\}) .$$ Moreover, note that if $\sigma_{Z_t} > \ell_t h_{\ell_t}^2$ and $L(\ell_t) / h_{\ell_t}< Z_t$, then $$\begin{aligned}
h_t \max\{t / \sigma_{Z_t},1\} & \leq h_t\max\left\{t/\ell_th_{\ell_t}^2,1\right\} \le
h_t\max\left\{L(\ell_t)/h_{\ell_t}^2,1\right\} \\& \leq h_t\max\left\{Z_t/h_{\ell_t},1\right\}\leq 2Z_t,
\end{aligned}$$ where for the purposes of the final inequality, we suppose that $h$ is so slowly varying as to satisfy $h_t\sim h_{\ell_t}$.
Since the assumptions hold with high probability (by applying our previous observations and Lemma \[lem:iprob\] again), it follows that $$\label{decay1}
\mathbf{P}\left(Z_t = z_t^{I},\:(0 ,O_t)\subseteq (0,3 Z_t)\right)\to 1$$ as $t\to\infty$. Now, define $$\tilde{S}_t := \sum_{\{x < 3Z_t\} \setminus \{z_t^I\} } \sigma_x\mathbf{1}_{\{\sigma_x \le \sigma_{Z_t}\}}.$$
Similarly to the proof of Lemma \[anlem\], applying Lemma \[lem:sumlevel\] we have that eventually $$\mathbf{P}\left(\tilde{S}_t > h_{\ell_t}^{-1}\sigma_{Z_t}|\:(Z_t,\sigma_{Z_t})\right) < h_{\ell_t}^2 Z_t\Gamma(\sigma_{Z_t}),$$ where $\Gamma(x):=\mathbf{E}(\frac{\sigma_0}{x} \mathbf{1}_{ \{ \sigma_0 < x\} })$. (For this it is useful to note that, conditional on $(Z_t,\sigma_{Z_t})$, if $x< Z_t$, then $\sigma_x$ is distributed as $\sigma_0|\{\sigma_0 \le \ell_t\}\prec \sigma_0|\{\sigma_0 \le\sigma_{Z_t}\}$, whereas if $x> Z_t$, then $\sigma_x$ is simply a copy of $\sigma_0$ and so $\sigma_x\mathbf{1}_{\{\sigma_x \le \sigma_{Z_t}\}}\prec \sigma_0|\{\sigma_0 \le \sigma_{Z_t}\}$, and all the relevant traps are independent.) Considering our assumptions on $h_t$ in , along with the fact that $\sigma_{Z_t} > \ell_t h_{\ell_t}^2$ and $Z_t < L(\ell_t) h_t$ with high probability by Lemma \[lem:iprob\], this implies that with high probability $$\mathbf{P}\left(\tilde{S}_t > h_t^{-1}\sigma_{Z_t}|\:(Z_t,\sigma_{Z_t})\right) < \frac{L(\ell_t) }{h_{\ell_t}L(\ell_t h_{\ell_t}^2)}.$$ Since the upper bound here converges to $0$ as $t \to \infty$, we thus deduce $\mathbf{P}(\tilde{S}_t > h_{\ell_t}^{-1}\sigma_{Z_t})\to 0$. Combining this with (\[decay1\]), we find that $$\mathbf{P}\left(R_2\neq \emptyset\right) \rightarrow 1,$$ which completes the proof.
In conjunction with Proposition \[prop:loc\] (established in the next section), the previous result yields Theorem \[thm:maincl\].
Localisation on the localisation set {#subsec:loc}
------------------------------------
In this section we prove that localisation occurs on the set $\Gamma_t$ eventually almost-surely. The argument follows a similar structure to that used to show localisation of one-dimensional random walk in random environments in [@Zeitouni Theorem 2.5.3], for example, with the distinction that in our case the localisation set can consist of more than one point. In particular, we prove the following.
\[prop:loc\] As $t \to \infty$, $$P_\sigma(X_t \in \Gamma_t) \to 1 \quad \mathbf{P}\text{-almost-surely.}$$
Before we prove Proposition \[prop:loc\], we first define some notation. For each $t > 0$ and trapping landscape $\sigma$, define the random times $$\tau_t^1 := \min \{s : X_s = z^{I}_t \} \quad \text{and} \quad \tau_t^2 := \min \{ s > \tau_t^1 : X_s \ge O_t \} .$$ Proposition \[prop:loc\] is then an easy consequence of the following three lemmas.
As $t \to \infty$, $$P_\sigma(\tau_t^1 \le t ) \to 1 \quad \mathbf{P}\text{-almost-surely} .$$
Applying the upper bound on hitting times in the first statement of Proposition \[prop:hittingtimeub\] (with $a = x = 0$ and $b = z_t^I$), $$P_\sigma \left( \tau^1_t \le t \right) > 1 - 2 t^{-1} z_t^I \sum_{z < z_t^I} \sigma_z .$$ By the definition of the site $z^{I}_t$, the sum of traps here is less than ${t}/{(z^{I}_t-1)h_t}$. It is a simple exercise to check that $z_t^{I}\geq 2$ for large $t$ almost-surely, and thus the result follows.
\[lem:stay\] As $t \to \infty$, $$P_\sigma(\tau_t^2 - \tau^1_t > t) \to 1 \quad \mathbf{P}\text{-almost-surely} .$$
First, define the hitting time $\tau^{O}_t := \min\{s : X_s= z^{O}_t \}$, which satisfies $\tau^O_t \ge \tau_t^1$. Applying the lower bound on hitting times in the first statement of Proposition \[prop:hittingtimelb\] (with $a = 0$, $b = O_t$ and $x = z_t^O$) yields that $$P_\sigma(\tau_t^2 - \tau^1_t \le t) \le P_\sigma \left(\tau^2_t-\tau^{O}_t \le t \right) < \frac{t}{2 (O_t - z_t^O) \sigma_{z_t^O}} ,$$ By the definition of the localisation set $\Gamma_t$ $$O_t - z^{O}_t = h_t \max \{ t/ \sigma_{z^{O}_t}, 1 \} \ge h_t \, t / \sigma_{z^{O}_t},$$ and the result follows.
As $t \to \infty$, $$P_\sigma\left( X_t \in \Gamma_t | \:\tau_t^1 < t < \tau_t^2-\tau_t^1 \right) \to 1 \quad \mathbf{P}\text{-almost-surely}.$$
For a given $t$, let $(\hat X^t_s)_{s\geq 0}$ be the inhomogeneous CTRW on $[0 ,O_t]$ in the trapping landscape $\sigma$ (see Section \[sec:prelim\] for the definition of this Markov process), started from $z^{I}_t$, and let $\hat{P}$ denote its law. Then we have that $(X_{(s+\tau_t^1)\wedge \tau_t^2})_{s\geq 0}$ has the same distribution as $(\hat X^t_{s\wedge \tau_t^2})_{s\geq 0}$. In particular, applying the Markov property at $\tau_t^1$, we have $$\begin{aligned}
{P_\sigma\left(X_t\not\in\Gamma_t|\:\tau_t^1 < t < \tau_t^2-\tau_t^1 \right)}
&\leq&\sup_{s\leq t} \hat{P} \left(\hat{X}^t_s\not\in \Gamma_t|\: \tau_t^2>t\right)\\
&\leq&\frac{\sup_{s\leq t} \hat{P} \left(\hat{X}^t_s\not\in\Gamma_t\right)}{P_\sigma\left({\tau}_t^2-\tau_t^1>t \right)}.\end{aligned}$$ From Lemma \[lem:stay\], we know the denominator converges to one almost-surely. Moreover, applying the localisation result of Proposition \[prop:local\] (and the definition of the lower boundary $D_t$), we have that the numerator converges to zero, so we are done.
The cardinality of the localisation set {#subsec:card}
---------------------------------------
In this section we establish that $|\Gamma_t| \le N$ eventually almost-surely. The fact that $X_t$ is contained on the record traps $\mathcal{R}$ eventually almost-surely if $N = 2$ will follow as an easy corollary. Throughout this section we shall work on Assumption \[assumpt:sosv\] and additionally assume that $N < \infty$.
Similarly to the proof of Theorem \[thm:main4\] in Section \[sec:svseq\] above, we prove that $|\Gamma_t| \le N$ by defining a certain sequence of events for the countable sequence $\mathcal{R}$ of record sites. Broadly speaking, this event is whether there are more than $N$ sites in $\Gamma_t$ at the times when $|\Gamma_t|$ is at a local maximum. Considering the construction of $\Gamma_t$, it can be seen that $|\Gamma_t|$ is at a local maximum precisely at the ‘relocalisation times’ between successive record traps. So let us define this event.
Recall the sequence $\mathcal{R} := (r_n)_{n\in\mathbb{N}}$ of record traps, and that $S_{(n)}^-$ denotes the sum of the traps prior to $r_n$. Set $t_0=0$, and for $n\geq 1$, define the time $t_n$ to satisfy the equation $$t_n / h_{t_n} = S_{(n)}^- (r_n-1) ,$$ which is well-defined since we insisted that $h_t$ be continuous. For simplicity, we abbreviate $h_{(n)} := h_{t_n}$. Note that for $t\in[t_{n-1},t_n)$, we have that $z^{I}_t=r_{n-1}$. Hence the time $t_n$ represents the ‘relocalisation time’ between the record traps at $r_{n-1}$ and $r_n$.
Observe further that for $t\in[t_{n-1},t_n)$, the boundary $O_t$ is strictly increasing. In particular, it will be sufficient to check that $|\Gamma_t| \le N $ at the instants immediately prior to $t \in \mathcal{T}:=\cup_{n\geq 1}\{ t_n \}$ (at least for large $n$, almost-surely). So for each $n$ define $O_{(n)}$ to be the outer boundary obtained by starting at site $r_{n-1}$ and chaining according to the procedure introduced in Section \[subsec:def\] at time $t_{n}$. Moreover, define the lower boundary $D_{(n)}$ using the general construction from the boundary $O_{(n)}$ and level $\sigma_{(n-1)}$. Then it is clear that for all such times $t \in [t_{n-1}, t_n)$ we have $\Gamma_t\subseteq \{r_{n-1},r_n\}\cup(\cup_{i=1}^2R_i)$, where $(R_i)_{i=1}^2$ are given by the disjoint sets (depicted in Figure \[gtpic2\]) $$\begin{aligned}
R_1&:=&\left\{x:\:x\in (r_n,O_{(n)}),\:\sigma_x > \sigma_{(n-1)}\right\},\\
R_2&:=&\left\{x:\:x < O_{(n)},\: D_{(n)} \le \sigma_x < \sigma_{(n-1)}\right\} .\end{aligned}$$ Note that here we are assuming $L$ is continuous, by Assumption \[assumpt:sosv\]. Hence, to show that $|\Gamma_t|\leq N$ for large $t$ almost-surely, it will suffice to show that $|R_1| + |R_2| \leq N-2$ for large $n$ almost-surely. We begin by studying the cardinality of $R_1$.
(0,0) – (4.5,0) node\[anchor=north west\] [$\mathbb{Z}$]{}; (0,0) – (0,4.5) node\[anchor=south\] [$L(\sigma_z)$]{}; (0,1.75) – (-0.1,1.75); (0,2) – (-0.1,2); (1.6,0) – (1.6,-0.1); (2.6,0) – (2.6,-0.1); (3.4,0) – (3.4,-0.1); (3.4, 1.8) – (3.4, 4); (0, 1.75) – (3.4, 1.75); (0, 2) – (3.4, 2); (2.6, 2) – (2.6, 4); in [(0.5,0.06), (1.1,0.76), (1.6,1.96), (2.6,2.76), (3.3,3.46)]{}[ at ; ]{} (3, 1.7) node ; (2.2, 3.3) – (2.5, 3.3) ; (1.1, 2.4) – (1.1, 2.13) ; (3.8, 2.3) node\[anchor=south\] [$\Gamma_t$]{}; (3.7, 2.85) – (3.5, 3.25) ; (3.5, 2.6) – (3, 2.65) ; (3.5, 2.5) – (2.2, 2.13) ; (3.7, 2.4) – (3.3, 2.1) ; (1.1, 2.4) node\[anchor=south\] [$R_1$]{}; (1.8, 3) node\[anchor=south\] [$R_2$]{}; (1.6, 0) node\[anchor=north\] [$r_{n-1}$]{} ; (2.6, 0) node\[anchor=north\] [$r_{n}$]{} ; (3.6, 0) node\[anchor=north\] [$O_{(n)}$]{} ; (0,1.55) node\[anchor=east\] [$L(D_{(n)})$]{}; (0,2.1) node\[anchor=east\] [$L(\sigma_{(n-1)})$]{};
\[r1lem\] Suppose Assumption \[assumpt:sosv\] holds. Then for each $k \in \mathbb{N}$ and $\varepsilon > 0$ there exists a constant $c >0$ and a sequence $(a_n)_{n\geq 1}$ satisfying $\sum_{n \in \mathbb{N}} a_n < \infty$ such that, as $n \to \infty$, $$\mathbf{P}(|R_1|\geq k) < a_n + c \left( d(e^{n(1-\varepsilon)}) \hat{h}_n^2 \right)^k$$ eventually. In particular, $|R_1|\leq N-2$ eventually almost-surely.
By definition, the chaining window from $r_n$ is given by $$h_{(n)}\max \left\{ \frac{ t_n }{\sigma_{(n)} } , 1 \right \} \le h_{(n)}^2\max \left\{ \frac{r_n S_{(n)}^-}{\sigma_{(n)}}, 1 \right\} .$$ Thus, by the union bound we find that $$\mathbf{P}\left(|R_1|\geq 1|\: \sigma_{(n-1)}, \sigma_{(n)},r_n,S_{(n)}^-\right)\leq\frac{h^2_{(n)}}{L(\sigma_{(n-1)})}\max \left\{ \frac{r_n S_{(n)}^-}{\sigma_{(n)}},1\right\}.$$ Similarly, to bound the probability of there being at least $k$ sites in this region given that there is at least one, we can condition on the height of the first site, which is an i.i.d. copy of $\sigma_{(n)}$, and repeat the process. In particular, $$\mathbf{P}\left(|R_1|\geq k|\: \sigma_{(n-1)},r_n,S_{(n)}^-\right)\leq \mathbf{E}\left(\prod_{i = 1, \ldots , k}\frac{h^2_{(n)}}{L(\sigma_{(n-1)})}\max \left\{ \frac{r_n S_{(n)}^-}{\sigma^i_{(n)}},1\right\} \bigg|\: \sigma_{(n-1)},r_n,S_{(n)}^-\right),$$ where, under the conditioned law, $(\sigma^i_{(n)})_{i\geq 2}$ are i.i.d. copies of $\sigma_{(n)}$ (i.e. first exceedences of the level $\sigma_{(n-1)}$). Applying the independence and integrating out the $\sigma_{(n)}$ using Lemma \[lem:ex\], it follows that, for some $c > 0$, as $\sigma_{(n-1)} \to \infty$ eventually $$\begin{aligned}
\mathbf{P}\left(|R_1|\geq k|\: \sigma_{(n-1)},r_n,S_{(n)}^-\right)&\leq& \frac{(h_{(n)})^{2k}}{L(\sigma_{(n-1)})^k}\mathbf{E}\left( \frac{r_n S_{(n)}^-}{\sigma_{(n)}} + 1 \bigg|\: \sigma_{(n-1)},r_n,S_{(n)}^-\right)^k\nonumber\\
& < &
\frac{(h_{(n)})^{2k}}{L(\sigma_{(n-1)})^k} \left(\frac{ c \, r_n S_{(n)}^- \, g(\sigma_{(n-1)}) }{\sigma_{(n-1)}}+1\right)^k\nonumber\\
& < & \frac{(2c)^{k} \, (h_{(n)})^{2k}}{L(\sigma_{(n-1)})^k} \left(
\left(\frac{r_n S_{(n)}^- g(\sigma_{(n-1)})}{\sigma_{(n-1)}}\right)^k+1\right) .\label{condestr1}\end{aligned}$$ Define the event $$\label{bnest}
\mathcal{B}_n := \left\{ \log L( \sigma_{(n-1)}) \in (1- \varepsilon, 1+\varepsilon)n \, , \ r_n < 2 L(\sigma_{(n-1)}) \log n \, , \ S_{(n)}^- < N \sigma_{(n-1)} \right \} ,$$ and recall that, by Lemmas \[lem:probrecord\] and \[anlem\] and the union bound, this event satisfies $\mathbf{P}(\mathcal{B}^c_n) < a_n$ for some sequence satisfying $\sum_n a_n < \infty$. Further, by the definition of $\hat{h}_n$, note that on the event $\mathcal{B}_n$ we have $h_{(n)} < \hat{h}_n$. Consequently, by the choice of $\hat{h}_n$ in , we have that, for some constant $c_1>0$, as $n \to \infty$ eventually $$\mathbf{P}(|R_1|\geq k) < a_n + c_1 \, \left( d( e^{n(1-\varepsilon)}) \hat{h}_n^2 \right)^k \,
\mathbf{E}\left(\left(\frac{r_n}{L(\sigma_{(n-1)})}\right)^k\right) .$$ Applying the moment estimate of Lemma \[lem:record\], this provides the probability estimate. The almost-sure claim follows by considering the definition of $N$ and the properties of $\hat{h}_n$ in , and then by applying a Borel-Cantelli argument.
We now include into the analysis the set $R_2$. To increase our ability to exploit independence, we shall initially substitute the set $R_2$ for different set $\tilde{R}_2$, which contains $R_2$ with high probability but will be simpler to work with. To this end, recall the notation $S_t^\ell$ used to define the lower boundary $D_t$, and define $$\tilde{D}_{(n)} := \max \{ \ell \ge 0: S^\ell_{r_n \hat{h}^3_{n}} < \sigma_{(n-1)} / \hat{h}_{n} \}$$ and $$\begin{aligned}
\tilde{R}_2&:=&\left\{x:\:x\in(0, r_n \, \hat{h}^3_{n}),\: \tilde{D}_{(n)} \le \sigma_x < \sigma_{(n-1)} \right\} .\end{aligned}$$
Before analysing the cardinality of $\tilde{R}_2$, we make the link between the sets $R_2$ and $\tilde{R}_2$.
\[lem:r2r2\] Suppose Assumption \[assumpt:sosv\] holds. Then we have that $$\sum_{n \in \mathbb{N}} \mathbf{P}( R_2 \nsubseteq \tilde{R}_2 ) < \infty .$$ In particular, $R_2 \subseteq \tilde{R}_2$ eventually almost-surely.
Denote by $\mathcal{B}_n$ the event $$\mathcal{B}_n := \left\{ S_{(n)}^- < N \sigma_{(n-1)} , \: |R_1| < N ,\:h_{(n)}\leq\hat{h}_n \right\} .$$ By considering the chaining that defines the outer boundary $O_t$, we observe that on $\mathcal{B}_n$ we know that $O_t < r_n \hat{h}_n^3$. Given the respective definitions of $\tilde{D}_{(n)}$ and $D_{(n)}$, this in turn implies $\tilde{D}_{(n)} \le D_{(n)}$ and hence $R_2 \subseteq \tilde{R}_2$. Hence we infer that $$\sum_{n \in \mathbb{N}} \mathbf{P}( R_2 \nsubseteq \tilde{R}_2 ) \le \sum_{n \in \mathbb{N}} \mathbf{P}(\mathcal{B}^c_n ),$$ and the result follows by Lemmas \[anlem\] and \[r1lem\] (as well as the observation from the proof of the latter lemma that $h_{(n)}\leq\hat{h}_n$ holds on an event of probability greater than $1-cn^2$).
To complete our analysis of the cardinality of $\Gamma_t$ we bound $\mathbf{P}(|R_1| + |\tilde{R}_2| \ge N-1)$, from where the fact that $|\Gamma_t| \le N$ eventually almost-surely follows by a Borel-Cantelli argument.
\[r2lem\] Suppose Assumption \[assumpt:sosv\] holds. We have that $$\sum_{n \in \mathbb{N}} \mathbf{P}(|R_1| + |\tilde{R}_2| \geq N-1) < \infty.$$ In particular, $|R_1| + |\tilde{R}_2| \leq N-2$ eventually almost-surely.
First observe that $$\begin{aligned}
\mathbf{P}(|R_1| + & |\tilde{R}_2| \geq N-1) \nonumber\\
& = \sum_{k = 0}^{N-1} \mathbf{E} \left[ \mathbf{1}_{\{|R_1| \ge k \}} \mathbf{P}(|\tilde{R}_2| \ge N-1-k \:|\: \sigma_{(n-1)}, r_{n-1}, \{(x,\sigma_x):\:\sigma_x\geq \sigma_{(n-1)}\}) \right] .\label{bound}\end{aligned}$$ To control the conditional probability in (\[bound\]), note that $|\tilde{R}_2| \ge N-1-k$ implies that if we exclude the largest $N-2-k$ terms from sum $$\sum_{i=1}^{r_n\hat{h}_n^3}\sigma_i\mathbf{1}_{\{\sigma_i< \sigma_{(n-1)}\}}=
\sum_{i=1}^{n-2}\sigma_{(i)}+\sum_{i=1}^{r_n\hat{h}_n^3}\sigma_i\mathbf{1}_{\{\sigma_i< \sigma_{(n-1)},\:\sigma_i\not\in\mathcal{R}\}},$$ then the result is still greater than $\sigma_{(n-1)}/\hat{h}_n$. Hence, by following the same argument as in the proof of the second part of Lemma \[anlem\] we have that $$\begin{aligned}
\lefteqn{\mathbf{P}(|\tilde{R}_2| \ge N-1-k \:|\: \sigma_{(n-1)}, r_{n-1}, \{(x,\sigma_x):\:\sigma_x\geq \sigma_{(n-1)}\})}\\
&\leq &\mathbf{P} \left(\sum_{i=1}^{(n-2)-(N-k-2)}\sigma_{(i)}>\hat{h}_n^{-1}\sigma_{(n-1)} \:\vline\:\sigma_{(n-1)}\right)\\
&&+\sum_{l=1}^{N-k-2}\mathbf{P}\left(\sum_{i=1}^{(n-2)-(N-k-2-l)}\sigma_{(i)}\geq \hat{h}_n^{-1}\sigma_{(n-1)} \:\vline\:\sigma_{(n-1)}\right) F \left(r_n\hat{h}_n^3,\sigma_{(n-1)},l,\hat{h}_n^{-1} \right).\end{aligned}$$ Note that, in contrast to the proof of Lemma \[anlem\], we have also included the traps in $(\sigma_{(n)}, r_n \hat{h}_{n}^3)$ into the analysis, but this causes no problem since under the relevant conditioning the terms $\sigma_i\mathbf{1}_{\{\sigma_i< \sigma_{(n-1)},\:\sigma_i\not\in\mathcal{R}\}}$ are either identically zero, or have the distribution $\sigma_0|\sigma_0<\sigma_{(n-1)}$. Now, conditioning on the event $\mathcal{B}_n$ defined at (\[bnest\]) (whose probability is estimated in Lemmas \[lem:probrecord\] and \[anlem\]), applying the conditional estimate for the tail of $|R_1|$ of (\[condestr1\]), and the estimate for $F$ from Corollary \[cor:sumlevel\], one deduces that $$\begin{aligned}
\lefteqn{\mathbf{P}(|R_1| + |\tilde{R}_2| \geq N-1) }\\
& \leq & a_n +c \sum_{k = 0}^{N-1} (d(e^{n(1-\varepsilon)})\hat{h}_n^2)^k \times\left[\mathbf{P} \left(\sum_{i=1}^{(n-2)-(N-k-2)}\sigma_{(i)}>\hat{h}_n^{-1}\sigma_{(n-1)} \right)\right.\\
&&+\left.\sum_{l=1}^{N-k-2}\mathbf{P}\left(\sum_{i=1}^{(n-2)-(N-k-2-l)}\sigma_{(i)}\geq \hat{h}_n^{-1}\sigma_{(n-1)} \right)\hat{h}_n(d(e^{n(1-\varepsilon)})\hat{h}_n^3\log n)^l\right].\end{aligned}$$ for some sequence $(a_n)_{n\geq 1}$ with $\sum_n a_n < \infty$. To estimate the remaining probabilities, one can apply the argument of Lemma \[sumsofrecords\]. The result is summable by the choice of $\hat{h}_n$ in , which completes the proof of the first claim of the lemma. The almost-surely statement then follows from a Borel-Cantelli argument.
From Lemmas \[lem:r2r2\] and \[r2lem\], we have the following corollary.
\[cor:N\] Suppose Assumption \[assumpt:sosv\] holds. We have that $$\sum_{n \in \mathbb{N}} \mathbf{P}( |R_1| + |\tilde{R}_2| \ge N-1 \mbox{ or } \ R_2 \nsubseteq \tilde{R}_2 ) < \infty \, .$$ In particular, $|R_1| + |{R}_2| \leq N-2$ eventually almost-surely.
Recalling that, by construction, $\limsup_{t\rightarrow\infty}|\Gamma_t|\leq 2+\limsup_{n\rightarrow\infty}(|R_1| + |{R}_2|)$, the previous result (together with Proposition \[prop:loc\]) completes the proof of the first claim of Theorem \[thm:main1\]. To complete the section, we point out a second easy corollary of the above, which confirms one implication of Theorem \[thm:main3\] holds.
Suppose Assumption \[assumpt:sosv\] holds. Assume $N =2$. Then, as $t \to \infty$, $$\Gamma_t \subseteq \mathcal{R}$$ eventually almost-surely.
As noted above, $\Gamma_t\subseteq \{r_{n-1},r_n\}\cup R_1\cup R_2$ for $t\in[t_{n-1},t_n)$. However, we have by Corollary \[cor:N\] that $|R_1| + |{R}_2|\leq N-2=0 $ eventually almost-surely. Thus $\Gamma_t\subseteq \{r_{n-1},r_n\}\subseteq \mathcal{R}$ eventually almost-surely.
Most favoured site {#sec:fav}
==================
In this section we consider the most favoured site; that is, we complete the proof of Theorem \[thm:main2\]. We note that by the first part of Theorem \[thm:main1\], as was proved in the previous section, the most favoured site must have at least $1/N$ proportion of the probability mass at all sufficiently large times. Moreover, by Theorem \[thm:maincl\], also proved in the previous section, and combining with Fatou’s lemma, one readily deduces that $$\limsup_{t \to \infty} \sup_{x \in \mathbb{Z}^+} P_\sigma(X_t = x) = 1 \, .$$ Hence it is sufficient to show that there exist arbitrarily large times at which the probability mass of the BTM is evenly balanced across exactly $N$ sites. Proving this will also finish the proof of Theorem \[thm:main1\]. Furthermore, to prove the converse direction of Theorem \[thm:main3\], we just need to show that if $N \ge 3$ then additionally the $N$ sites referred to above are not all record traps.
We proceed in two steps. First, we establish the above ‘balanced localisation’ result on the assumption that a certain favourable event $\mathcal{E}_{n}$ involving the trapping landscape holds infinitely often. Second, we prove that this favourable event does indeed hold infinitely often almost-surely; it is here that we will need to work under Assumption \[assumpt:g22\].
Defining the favourable event
-----------------------------
In this section, we define the favourable event $\mathcal{E}_n$. The definition of this involves a certain $n$-dependent collection of sites $(z_i)_{1 \le i \le N}$, (we drop the explicit dependence on $n$ for brevity). In particular, we fix $\varepsilon_0\in(0,1)$, define $z_1:=r_{n-1}$ and set, for $i=2,\dots, N$, $$z_{i}=\min\left\{z>z_{i-1}:\:\sigma_z>(1-\varepsilon_0)\sigma_{(n-1)}\right\}.$$ We also introduce the notation $$\Lambda_n:= L((1-\varepsilon_0)\sigma_{(n)}).$$ For $\varepsilon_1,\varepsilon_2,\varepsilon_3,\varepsilon_4,\varepsilon_5,\varepsilon_6,\varepsilon_7\in(0,1)$, satisfying $\varepsilon_1 < \varepsilon_2$, we then suppose $\mathcal{E}_n$ is defined to be the event $$\begin{aligned}
&&\left\{z_N=r_n,\:\frac{z_N-z_{N-1}}{\Lambda_{n-1}}\in(\varepsilon^{-1}_1,\varepsilon^{-1}_2),
\:\frac{z_{N-1}-z_1}{\Lambda_{n-1}}<\varepsilon_3^{-1},\:\frac{z_1}{\Lambda_{n-1}}<\varepsilon_4^{-1}\right\}\\
&&\cap\left\{\sum_{\substack{z<z_N:\\z\not\in\{z_1,\dots,z_{N-1}\}}}\sigma_z<\varepsilon_4\sigma_{(n-1)},\:\sigma_{(n)}>\varepsilon_5^{-1}\sigma_{(n-1)},
\:\sum_{z_N<z\leq z_N+\varepsilon_6^{-1}\Lambda_{n-1}}\sigma_z<\varepsilon_7\sigma_{(n-1)}
\right\}.\end{aligned}$$ Constraints on $(\varepsilon_i)_{i=0}^7$ will be imposed later as they become necessary for the argument. We will also allow these parameters to depend on $n$ where needed. (Actually we only need to do this for $\varepsilon_4$.) See Figure \[enpic\] for a typical configuration on this event.
(0,0) – (10,0) node\[anchor=north west\] ; (0,0) – (0,4.5) node\[anchor=south\] [$\sigma_z / \sigma_{(n-1)}$]{}; (3, 0) – (3, -0.1); (3, 0) node\[anchor=north\] [$z_1/ \Lambda_{n-1}$]{}; (4.5, 0) – (4.5, -0.1); (4.5, 0) node\[anchor=north\] [$z_2/ \Lambda_{n-1}$]{}; (7.2, 0) – (7.2, -0.1); (7.2, 0) node\[anchor=north\] [$z_{N-1}/ \Lambda_{n-1}$]{}; (9, 0) – (9, -0.1); (9, 0) node\[anchor=north\] [$z_N/ \Lambda_{n-1}$]{}; (0, 0.4) – (-0.1, 0.4); (0, 0.4) node\[anchor=east\] [$\min\{\varepsilon_4, \varepsilon_7\}$]{}; (0, 0.4) – (10, 0.4); (0, 2) – (-0.1, 2); (0, 2) node\[anchor=east\] [$1$]{}; (0, 4) – (-0.1, 4); (0, 4) node\[anchor=east\] [$\sigma_{(n)} / \sigma_{(n-1)}$]{}; in [(1, 0.2), (5.3, 0.1), (9.6, 0.2), (3,2), (4.5,1.7), (5.2, 1.8), (6, 1.9), (7.2, 1.8), (9, 4)]{}[ at ; ]{} (0,2.5) – (1.5, 2.5) node\[anchor=south\] [$< \varepsilon_4^{-1}$]{} – (3, 2.5) ; (3,3) – (5.1, 3) node\[anchor=south\] [$< \varepsilon_3^{-1}$]{} – (7.2, 3) ; (7.2,4.5) – (8.1, 4.5) node\[anchor=south\] [$\in ( \varepsilon_1^{-1}, \varepsilon_2^{-1})$]{} – (9, 4.5) ; (9, 0.6) – (9.6, 0.6) node\[anchor=south\] [$ \varepsilon_6^{-1}$]{} – (10, 0.6) ; (7.5,1.5) – (7.5, 1.75) node\[anchor=west\] [$< \varepsilon_0$]{} – (7.5, 2) ; (9.2,2) – (9.2, 3) node\[anchor=west\] [$> \varepsilon_5^{-1} $]{} – (9.2, 4) ;
Balanced localisation assuming the favourable event holds
---------------------------------------------------------
In this section, we prove the remaining part of Theorem \[thm:main2\], as well as the converse direction of Theorem \[thm:main3\], under the following assumption.
\[assump\] The event $\mathcal{E}_n$ holds infinitely often almost-surely whenever $(\varepsilon_i)_{i=0}^7$ satisfy $\varepsilon_1^{-1}<\varepsilon_2^{-1}$ and $\varepsilon_4=1/4\log n$.
To prove that balanced localisation occurs, the idea is to show that, under $\mathcal{E}_n$, the chain mixes quickly on $\{z_1,\dots,z_{N-1}\}$, meaning that on ‘short’ time scales the mass is approximately evenly distributed on these sites, whereas on a suitably selected ‘long’ time scale an appropriate amount of mass, approximately $1/N$, has seeped onto $z_N$. From this it is possible to conclude that, at this latter time, the process is approximately uniformly distributed over $\{z_1,\dots,z_{N}\}$.
We start by defining the long time scale. In particular, we choose $t_n$ to be the unique time such that $$P_\sigma\left(\tau_{z_N}\leq t_n\right)=\frac{1}{N},$$ where $\tau_{x}$ is the hitting time of $x$ by $X$ (started from 0). (Note that for $x\geq 1$, $\tau_x$ has a continuous distribution with full support on $(0,\infty)$, and so $t_n$ is well-defined.) The following lemma is a ready consequence of this definition.
\[lem:tnbounds\] Suppose $\mathcal{E}_n$ holds, then $$\frac{2(1-\varepsilon_0)}{\varepsilon_1N} < \frac{t_n}{\Lambda_{(n-1)} \sigma_{(n-1)}} < 6 (\varepsilon_2^{-1}+\varepsilon_3^{-1})N^2 .$$
For the lower bound, applying the lower bound on hitting times in the first statement of Proposition \[prop:hittingtimelb\] (with $a = x =0$, $b = z_N$ and $z = z_1$) and the definition of $\mathcal{E}_n$ we have that $$P_{\sigma}\left(\tau_{z_N}\leq t\right) < \frac{t}{2(z_N - z_1) \sigma_{z_1}} < \frac{t}{2(1-\varepsilon_0) \varepsilon_1^{-1} \Lambda_{(n-1)}\sigma_{(n-1)}} < \frac{t \varepsilon_1}{2(1-\varepsilon_0)\Lambda_{(n-1)}\sigma_{(n-1)}} .$$ Given the definition of $t_n$, taking $ t= {2(1-\varepsilon_0)\Lambda_{(n-1)}\sigma_{(n-1)}}/{\varepsilon_1 N}$ in the above establishes the result. Similarly for the upper bound, applying the upper bound on hitting times in the second statement of Proposition \[prop:hittingtimeub\] (with $a = x = 0$, $b = z_N$ and ${S} = \{z_1, \ldots, z_{N-1} \}$) and the definition of $\mathcal{E}_n$ we have that $$\begin{aligned}
\lefteqn{P_{\sigma} \left( \tau_{z_N} \le t \right)} \\
& >& 1- 2 t^{-1} \left( z_N \sum_{ \{z < z_N\} \setminus \{z_1, \ldots, z_{N-1}\} } \sigma_z + (N-1)(z_N - z_1) \sigma_{{(n-1)} } \right) \\
& > &1 - 2 t^{-1} \left( (\varepsilon_2^{-1}+\varepsilon_3^{-1}+\varepsilon_4^{-1}) \varepsilon_4 \Lambda_{(n-1)} \sigma_{(n-1)} + (\varepsilon_2^{-1}+\varepsilon_3^{-1})(N-1)\Lambda_{n-1}\sigma_{(n-1)} \right) \\
& > &1 - 3 t^{-1} (\varepsilon_2^{-1}+\varepsilon_3^{-1})N\Lambda_{n-1}\sigma_{(n-1)} .\end{aligned}$$ Taking $t = 6 (\varepsilon_2^{-1}+\varepsilon_3^{-1})N^2\Lambda_{n-1}\sigma_{(n-1)}$, we have $P_{\sigma}( \tau_{z_N} \le t) > 1 - \frac{1}{2N} > \frac{1}{N}$ (since $N \ge 2$), which establishes the result.
We now bound the probability mass on the final site at times given by the long time scale, showing that it is very nearly $1/N$.
\[rnbal\] Suppose $\mathcal{E}_n$ holds, then $$P_\sigma\left(X_{t_n}=z_N\right) > \frac{1}{N}\left(1-
\varepsilon_5(\varepsilon_4+\varepsilon_7)-
6\varepsilon_5\max\{\varepsilon_1,\varepsilon_6\} (\varepsilon_2^{-1}+\varepsilon_3^{-1})N^2
\right).$$
By the Markov property and the definition of $t_n$, we have that $$\begin{aligned}
P_\sigma\left(X_{t_n}=z_N\right)&=&P_\sigma\left(X_{t_n}=z_N,\:\tau_{z_N}\leq t_n\right)\\
&\geq &\frac{1}{N}\min_{t\leq t_n}P_\sigma\left(X_t=z_N\:|\:X_0=z_N\right).\end{aligned}$$ To estimate the probability here, we consider the inhomogeneous CTRW on $$\Omega_n:= [z_N-\varepsilon_{1}^{-1}\Lambda_{n-1}, z_N+\varepsilon_{6}^{-1}\Lambda_{n-1}] \cap \mathbb{Z^+}$$ in the trapping landscape $(\sigma_x)_{x\in \Omega_n}$ (see Section \[sec:prelim\] for the definition of this Markov chain).
Note that on $\mathcal{E}_n$ we have $\Omega_n \subseteq \mathbb{Z}^+$, and hence, in particular, if $X$ and $X^n$ are both started from $z_N$, then their distributions are the same up to the hitting time of the endpoints of $\Omega_n$. Denoting the latter stopping time by $\tau$, we thus obtain $$\begin{aligned}
P_\sigma\left(X_{t}=z_N\:|\:x_0=z_N\right)&\geq & P_\sigma\left(X_{t}=z_N,\:\tau>t\:|\:X_0=z_N\right)\\
&=&P^n_{z_N}\left(X^n_{t}=z_N,\:\tau>t\right)\\
&\geq & 1-P^n_{z_N}\left(X^n_{t}\neq z_N\right)-P^n_{z_N}\left(\tau\leq t\right),\end{aligned}$$ where ${P}^n_x$ is the law of ${X}^n$ started from $x$.
To bound the first term, applying the localisation result in Proposition \[prop:local\] (with $z = z_N$ and ${S} = \Omega_n \setminus \{z_N\}$) and the definition of $\mathcal{E}_n$ yields that, for $t \ge 0$, $$P^n_{z_N}\left(X^n_{t}\neq z_N\right) \leq \frac{ \sum_{z \in \Omega_n \backslash\{z_N\} } \sigma_z }{\sigma_{z_N}} < \varepsilon_5(\varepsilon_4+\varepsilon_7) .$$ To bound the second term, applying the lower bound on hitting times in the second statement of Proposition \[prop:hittingtimelb\] (with $x = z_N$) and the definition of $\mathcal{E}_n$ yields that, for $t \ge 0$, $$P^n_{z_N}\left(\tau\leq t\right) < \frac{ t\varepsilon_5 } {\min\{ \varepsilon_1^{-1}, \varepsilon_6^{-1} \} \Lambda_{n-1} \sigma_{z_N} } < \frac{ t \varepsilon_5 \max\{\varepsilon_1,\varepsilon_6\} }{ \Lambda_{n-1}\sigma_{(n-1)} }.$$ Combining with the upper bound on $t_n$ in Lemma \[lem:tnbounds\], we have that $$P^n_{z_N}\left(\tau\leq t_n \right) < 6 \varepsilon_5 \max\{\varepsilon_1,\varepsilon_6\}( \varepsilon_2^{-1} + \varepsilon_3^{-1}) N^2 .$$ and the result follows.
We proceed now to investigate the short time scale. To this end, let $\tilde{X}^n$ be the inhomogeneous CTRW on $[0, z_N] \cap \mathbb{Z}$ in the trapping landscape $(\sigma^n_x)_{x \in \mathbb{Z}^+}$ consisting of a copy of $\sigma$ but with $\sigma_{z_N}$ replaced by $\varepsilon_4\sigma_{(n-1)}$. In particular, the processes $X$ and $\tilde{X}^n$ have the same distribution up to hitting $z_N$. For $\varepsilon\in(0,1)$, we define $$t_{\rm mix}^n(\varepsilon):=\inf\left\{t\geq 0:\:\max_{i=1,\dots,N-1}\sum_{0\leq y\leq z_N}\left|\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y\right)-\tilde{\pi}^n(y)\right|\leq \varepsilon\right\},$$ where $\tilde{P}^n_x$ is the law of $\tilde{X}^n$ started from $x$, and $\tilde{\pi}^n$ is its invariant probability measure. This is a version of the ($\varepsilon$-)mixing time of $\tilde{X}^n$, and will provide the short time scale for our argument. Note in particular that the maximum is only taken over starting points from $(z_i)_{i=1}^{N-1}$ rather than the entire interval $[0, z_N] \cap \mathbb{Z}$, as would be the case in the usual definition of a mixing time. In this setting, for suitable values of $\varepsilon$ this results in a quantity of a much lower order, since the majority of the mass is explored quickly when $\tilde{X}^n$ is started from one of the sites in the collection $(z_i)_{i=1}^{N-1}$. (Conversely, if the process is started from a site close to $z_N$, it would take a relatively long time to find the vertices $(z_i)_{i=1}^{N-1}$.) The following lemma provides a key estimate on this quantity. The proof is based on an argument for upper bounding mixing times presented in [@Aldous82] that involves considering a stopping time at which the random walk hits a stationary random vertex.
\[mixest\] Suppose $\mathcal{E}_n$ holds, and that $2\varepsilon_4\leq \varepsilon_0\leq N^{-1}$. Then $$\frac{ t_{\rm mix}^n\left(8{\varepsilon}_0^{1/2}\right) }{\Lambda_{n-1}\sigma_{(n-1)}} < \frac{N}{\varepsilon_0^2\varepsilon_3}.$$
We start by defining a randomised stopping time $T$, which will be the time taken to hit an almost stationary random vertex in $\{z_i:\:i=1,\dots,N-1\}$. In particular, let $Z$ be a random vertex in $\{z_i:\:i=1,\dots,N-1\}$ that is independent of $\tilde{X}^n$ and satisfies $$\mathbf{P}\left(Z=z_i\right)=\frac{\tilde{\pi}^n(z_i)}{\sum_{j=1}^{N-1}\tilde{\pi}^n(z_j)},$$ and set $$T:=\inf\{t\geq 0:\:\tilde{X}^n_t=Z\}.$$ Clearly $\tilde{P}^n_x(X_T=z_i)=\mathbf{P}(Z=z_i)$ for each $x\in \{0,\dots,z_N\}$, $i=1,\dots,N-1$. (For simplicity of notation, we suppose the law $\tilde{P}^n_x$ is the joint law of $\tilde{X}^n$ and $Z$.) Moreover, we have that $$\tilde{P}^n_{z_i}(T > t) \le \max\left\{ \tilde{P}^n_{z_1}(\tau_{z_{N-1}} > t) \, , \ \tilde{P}^n_{z_{N-1}}(\tau_{z_{1}} > t) \right\} .$$ Applying the upper bound on hitting times in the first statement of Proposition \[prop:hittingtimeub\] (first with $a = 0, b = z_{N-1}$ and $x = z_1$, and then with $a = z_N, b = z_1$ and $x = z_{N-1}$, by symmetry) and the definition of $\mathcal{E}_n$, yields that (recalling that $\sigma^n_{z_N} = \varepsilon_4 \sigma_{(n-1)}$) $$\begin{aligned}
\label{tbound}
\tilde{P}^n_{z_i}(T > t) & \le 2t^{-1} (z_{N-1} - z_1) \bigg( \sum_{0 \le z \le z_{N}-1} \sigma_z + \varepsilon_4 \sigma_{(n-1)} \bigg) \\
\nonumber & < 2t^{-1} \varepsilon_3^{-1} \Lambda_{n-1} \sigma_{(n-1)} \left( N - 1 + 2\varepsilon_4 \right) < 4 N t^{-1} \varepsilon_3^{-1} \Lambda_{n-1} \sigma_{(n-1)}.\end{aligned}$$
The remainder of the proof closely follows [@Aldous82]. Specifically, for $i=1,\dots,N-1$ and $t,t_0>0$, we can write $$\begin{aligned}
\sum_{0\leq y\leq z_N}\left|\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y\right)-\tilde{\pi}^n(y)\right|
& \! \! \! \! \leq& \! \! \! \! \! \sum_{0\leq y\leq z_N}\left|\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y\right)-\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y,\:T\leq t_0\right)\right|\label{bbb}\\
&&+\sum_{0\leq y\leq z_N}\left|\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y,\:T\leq t_0\right)-\tilde{\pi}^n(y)\right|.\nonumber\end{aligned}$$ Let us denote the two sums on the right-hand side by $S_1$ and $S_2$. To deal with the first of these, we simply note that $$\label{bbb1}
S_1=\tilde{P}^n_{z_i}\left(T> t_0\right) < \frac{4N\varepsilon_3^{-1}\Lambda_{n-1}\sigma_{(n-1)}}{t_0},$$ where we have applied (\[tbound\]) to deduce the inequality. For the second term, we start by applying Cauchy-Schwarz to deduce $$\begin{aligned}
S_2^2&\leq& \sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}\left(\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y,\:T\leq t_0\right)-\tilde{\pi}^n(y)\right)^2\nonumber\\
&=&\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}\tilde{P}^n_{z_i}\left(\tilde{X}^n_t=y,\:T\leq t_0\right)^2-2\tilde{P}^n_{z_i}\left(T\leq t_0\right)+1.\label{s2}\end{aligned}$$ Now, define a measure $\nu$ on $\{0,\dots,z_N\}\times [0,t_0]$ by setting $$\nu\left(\cdot,\cdot\right):=\tilde{P}^n_{z_i}\left(T\leq t_0,\:\left(\tilde{X}^n_T,T\right)\in\left(\cdot,\cdot\right)\right),$$ and a function $f$ on $\{0,\dots,z_N\}\times \mathbb{R}^+$ by $$f(x,s):=\tilde{P}^n_{z_i}\left(T\leq t_0,\:\tilde{X}^n_{t_0+s/2}=x\right)\equiv\int_{\{0,\dots,z_N\}\times [0,t_0]}\tilde{P}^n_{y}\left(\tilde{X}^n_{t_0+s/2-r}=x\right)\nu(dy,dr).$$ By the definition of $f$ and reversibility, it is possible to deduce that $$\begin{aligned}
\lefteqn{\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}f(y,s)^2}\\
&=&\int\int\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}\tilde{P}^n_{y_1}\left(\tilde{X}^n_{t_0+s/2-r_1}=y\right)\tilde{P}^n_{y_2}\left(\tilde{X}^n_{t_0+s/2-r_2}=y\right)\nu(dy_1,dr_1)\nu(dy_2,dr_2)\\
&=&\int\int\frac{1}{\tilde{\pi}^n(y_2)}\tilde{P}^n_{y_1}\left(\tilde{X}^n_{2t_0+s-r_1-r_2}=y_2\right)\nu(dy_1,dr_1)\nu(dy_2,dr_2),\end{aligned}$$ where each of the integrals above is over $\{0,\dots,z_N\}\times [0,t_0]$. Hence, for any $s_0$, $$\begin{aligned}
\lefteqn{\frac{1}{s_0}\int_0^{s_0}\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}f(y,s)^2ds}\\
&\leq& \frac{1}{s_0}\int\int\frac{1}{\tilde{\pi}^n(y_2)}\int_0^{2t_0+s_0}\tilde{P}^n_{y_1}\left(\tilde{X}^n_{s}=y_2\right)ds\nu(dy_1,dr_1)\nu(dy_2,dr_2)\\
&\leq &\frac{1}{s_0}\sum_{j_1,j_2=1}^{N-1}
\frac{1}{\tilde{\pi}^n(z_{j_2})}\int_0^{2t_0+s_0}\tilde{P}^n_{z_{j_1}}\left(\tilde{X}^n_{s}=z_{j_2}\right)ds
\frac{\tilde{\pi}^n(z_{j_1})\tilde{\pi}^n(z_{j_2})}{\left(\sum_{k=1}^{N-1}\tilde{\pi}^n(z_k)\right)^2},\end{aligned}$$ where for the second inequality we note that the inner integral does not depend on $r_1$ or $r_2$, and apply the definition of $\nu$ and the stopping time $T$. To estimate the right-hand side here, we note that $\tilde{\pi}^n(x)$ is proportional to $\sigma^n_x$, from which it is elementary to check that (recalling that $\sigma^n_{z_N} = \varepsilon_4 \sigma_{(n-1)}$) $$\label{pinlower}
\tilde{\pi}^n\left(\{z_k:\:k=1,\dots,N-1\}\right) > \frac{(N-1)(1-\varepsilon_0)}{N-1+2\varepsilon_4}.$$ It follows that $$\frac{1}{s_0}\int_0^{s_0}\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}f(y,s)^2ds < \frac{2t_0+s_0}{s_0}\times\frac{N-1+2\varepsilon_4}{(N-1)(1-\varepsilon_0)}.$$ For $\varepsilon_0,\varepsilon_4$ satisfying the assumptions of the lemma, and $s_0:=t_0/2\varepsilon_0$, it is straightforward to check that this implies $$\frac{1}{s_0}\int_0^{s_0}\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}f(y,s)^2ds < \left(1+4\varepsilon_0\right)^2 < 1+16\varepsilon_0.$$ In particular, there must exist an $s\leq s_0$ such that $\sum_{0\leq y\leq z_N}\frac{1}{\tilde{\pi}^n(y)}f(y,s)^2 < 1+16\varepsilon_0$, and, returning to (\[s2\]), $$\label{bbb2}
S_2^2 < 16\varepsilon_0+2\tilde{P}^n_{z_i}\left(T> t_0\right)$$ for some $t\leq t_0+s_0/2$. Since the left-hand side of (\[bbb\]) is decreasing, if we choose $t_0:=2N\Lambda_{n-1}\sigma_{(n-1)}/\varepsilon_0\varepsilon_3$, then, by (\[bbb1\]) and , $$\sum_{0\leq y\leq z_N}\left|\tilde{P}^n_{z_i}\left(\tilde{X}^n_{t_0+s_0/2}=y\right)-\tilde{\pi}^n(y)\right|\leq S_1+S_2 < 8\varepsilon^{1/2}_0.$$ The result follows.
We are now ready to put the pieces together to establish that under $\mathcal{E}_n$ the process $X$ is approximately balanced on the sites $\{z_1,\dots,z_{N-1}\}$ at time $t_n$. Since the result for $i=N$ was already established as Lemma \[rnbal\], this will be enough to conclude the relevant part of Theorem \[thm:main2\].
\[balance\] Suppose $\mathcal{E}_n$ holds, that $2\varepsilon_4\leq \varepsilon_0\leq N^{-1}$ and also that $N^2\varepsilon_1\leq (1-\varepsilon_0)\varepsilon_0^2\varepsilon_3$. Then, there exists a constant $c$ depending only on $N$ such that, for $i=1,\dots,{N-1}$, $$P_\sigma\left(X_{t_n}=z_i\right) > \frac{1}{N}\left(1-c\left(\varepsilon_0^{1/2}+\frac{\varepsilon_1}{\varepsilon_0^2\varepsilon_3}\right)\right).$$
We first note that under the assumptions on the $(\varepsilon_i)_{i=0}^7$, the lower bound in Lemma \[lem:tnbounds\] and Lemma \[mixest\] imply that $t_n > 2t_{\rm mix}^n(\tilde{\varepsilon})$, where $\tilde{\varepsilon}:=8\varepsilon_0^{1/2}$. So, we have that $s_n:=t_n-t_{\rm mix}^n(\tilde{\varepsilon}) > t_{\rm mix}^n(\tilde{\varepsilon})>0$, and we can apply the Markov property at time $s_n$ to obtain, for $i\in\{1,\dots, N-1\}$, $$\begin{aligned}
P_\sigma\left(X_{t_n}=z_i\right)&\geq &
P_\sigma\left(X_{t_n}=z_i,\:\tau_{z_N}>t_n\right)\nonumber\\
&\geq&\sum_{j=1}^{N-1}P_\sigma\left({X}_{s_n}=z_j,\:\tau_{z_N}>s_n\right)\tilde{P}^n_{z_j}\left(\tilde{X}^n_{t_{\rm mix}^n(\tilde{\varepsilon})}=z_i,\:\tau_{z_N}>t_{\rm mix}^n(\tilde{\varepsilon})\right).\label{twoterms}\end{aligned}$$ For the second probability in the above expression, we have that $$\begin{aligned}
\lefteqn{\tilde{P}^n_{z_j}\left(\tilde{X}^n_{t_{\rm mix}^n(\tilde{\varepsilon})}=z_i,\:\tau_{z_N}>t_{\rm mix}^n(\tilde{\varepsilon})\right)}\\
&\geq& \tilde{P}^n_{z_j}\left(\tilde{X}^n_{t_{\rm mix}^n(\tilde{\varepsilon})}=z_i\right)-\tilde{P}^n_{z_j}\left(\tau_{z_N}\leq t_{\rm mix}^n(\tilde{\varepsilon})\right)\\
& > & \tilde{\pi}^n(z_i)-\tilde{\varepsilon}-\frac{ t_{\rm mix}^n(\tilde{\varepsilon})}{(1-\varepsilon_0)\varepsilon_1^{-1}\Lambda_{(n-1)}\sigma_{(n-1)}},\end{aligned}$$ where, to deduce the final inequality, we have applied the definition of the mixing time, and bounded $\tilde{P}^n_{z_j}(\tau_{z_N}\leq t_{\rm mix}^n(\tilde{\varepsilon}))$ using the lower bound on hitting times in the first statement of Proposition \[prop:hittingtimelb\] (with $a = 0$, $b = z_N$ and $x = z = z_j$). Plugging in the definition of $\tilde{\varepsilon}$, noting the bound of Lemma \[mixest\], and estimating the measure similarly to (\[pinlower\]), we thus find that $\tilde{P}^n_{z_j}(\tilde{X}^n_{t_{\rm mix}^n(\tilde{\varepsilon})}=z_i,\:\tau_{z_N}>t_{\rm mix}^n(\tilde{\varepsilon}))$ is bounded below by $$\label{lowerfirst}
\frac{1}{N-1}-\frac{3\varepsilon_0}{(1-\varepsilon_0)}-8\varepsilon_0^{1/2}-
\frac{N\varepsilon_1}{(1-\varepsilon_0)\varepsilon_0^2\varepsilon_3}.$$ For the remaining part of the sum at (\[twoterms\]), we have that $$\begin{aligned}
\sum_{j=1}^{N-1}P_\sigma\left({X}_{s_n}=z_j,\:\tau_{z_N}>s_n\right)&\geq&
P_\sigma\left(\tau_{z_N}>s_n\right)-\tilde{P}^n_0\left(\tilde{X}_{s_n}\not\in\{z_j:\:j=1,\dots,{N-1}\}\right).\end{aligned}$$ Clearly $P_\sigma(\tau_{z_N}>s_n)>(N-1)/N$, by the definition of $t_n$. Furthermore $$\begin{aligned}
\lefteqn{\tilde{P}^n_0\left(\tilde{X}_{s_n}\not\in\{z_j:\:j=1,\dots,{N-1}\}\right)}\\
&\leq &P_\sigma\left(\tau_{z_1}>s_n\right)+\sup_{t\geq 0}\tilde{P}^n_{z_1}\left(\tilde{X}_t \not\in\{z_j:\:j=1,\dots,{N-1}\}\right)\\
& < & N\varepsilon_1+ \frac{\varepsilon_4}{(N-1)(1-\varepsilon_0)},\end{aligned}$$ where in the second inequality we used the upper bound on hitting times in the first statement of Proposition \[prop:hittingtimeub\] (with $a = x = 0$ and $b = z_1$), the lower bound on $s_n >t_n/2$ of Lemma \[lem:tnbounds\], and the localisation result in Proposition \[prop:local\] (with ${S} = [0, z_N] \cap \mathbb{Z}^+ \setminus \{z_1, \ldots, z_N \}$).
In particular, we conclude that $$\sum_{j=1}^{N-1}P_\sigma\left({X}_{s_n}=z_j,\:\tau_{r_n}>s_n\right) >
\frac{N-1}{N}- N\varepsilon_1-\frac{\varepsilon_4}{(N-1)(1-\varepsilon_0)}.$$ Putting this together with (\[lowerfirst\]), we obtain the result.
\[cor:main2\] Suppose Assumption \[assump\] holds. Then $$\liminf_{t\rightarrow\infty}\sup_{x\in\mathbb{Z}^+}P_\sigma\left(X_t=x\right)=\frac{1}{N} \qquad \mathbf{P}\text{-almost-surely.}$$
Fix $\varepsilon\in(0,1)$. By Assumption \[assump\], we can suppose that almost-surely there exists an infinite sequence $(n_i)_{i\geq 1}$ such that $\cap_{i\geq1}\mathcal{E}_{n_i}$ holds for $\varepsilon_0=\varepsilon_5=\varepsilon^2$, $\varepsilon_1=\varepsilon_6=\varepsilon^6$, $\varepsilon_3=\varepsilon_7=\varepsilon$, $\varepsilon_4=\frac{1}{4\log n_i}$ and $\varepsilon_2=\varepsilon^7$. Now, by Lemma \[rnbal\] and Proposition \[balance\], on $\cap_{i\geq1}\mathcal{E}_{n_i}$ we have that $$\limsup_{i\rightarrow \infty}\sup_{z\in\mathbb{Z}^+}P_\sigma\left(X_{t_{n_i}}=z\right)\leq \frac1N\left(1+c\varepsilon\right),$$ where $c$ is some deterministic constant. Thus to complete the proof, it will suffice to show that on $\cap_{i\geq1}\mathcal{E}_{n_i}$ we also have $t_{n_i}\rightarrow\infty$ almost-surely. By Lemmas \[lem:probrecord\] and \[lem:tnbounds\], we have that on $\cap_{i\geq1}\mathcal{E}_{n_i}$, $t_{n_i} > N^{-1}\varepsilon_1^{-1}\Lambda_{n_i-1}\sigma_{(n_i-1)}\rightarrow\infty$ almost-surely, as desired.
\[cor:main3\] Suppose Assumption \[assump\] holds and that $N \ge 3$. Then $$\limsup_{t \to \infty} P_\sigma(X_t\not \in \mathcal{R})
\ge \frac{N-2}{N} \qquad \mathbf{P}\text{-almost-surely.}$$
As in the proof of Corollary \[cor:main2\], by Assumption \[assump\], Proposition \[balance\] and Lemmas \[lem:probrecord\] and \[lem:tnbounds\], there exists a sequence $(n_i)_{i \ge i}$ such that $\cap_{i\geq1}\mathcal{E}_{n_i}$ holds for a certain choice of $(\varepsilon_i)_{0 \le i \le 7}$, and such that, on $\cap_{i\geq1}\mathcal{E}_{n_i}$, both $$\limsup_{i \to \infty} P_\sigma \left( X_{t_{n_i}} \in \{z_2, \ldots, z_{N-1} \} \right) \ge \frac{N-2}{N}$$ and $t_{n_i} \to \infty$ hold almost-surely. To complete the proof, note simply that on the event $\mathcal{E}_n$ the sites $z_2, \ldots , z_{N-1}$ are not contained in the record traps $\mathcal{R}$ by definition.
The favourable event occurs infinitely often
--------------------------------------------
In this section, we establish that the event $\mathcal{E}_n$ occurs infinitely often; throughout we shall work under Assumptions \[assumpt:sosv\] and \[assumpt:g22\]. Our proof breaks down into two parts. We start by considering the part involving the sum over vertices $z<z_N$. In particular, note that, for any $\varepsilon_0\in(0,1)$, $$\left\{z_N=r_n,\:\sum_{\substack{z<z_N:\\z\not\in\{z_1,\dots,z_{N-1}\}}}\sigma_z>\varepsilon_4\sigma_{(n-1)}\right\}\subseteq
\left\{S_{r_n-1}^{(N)}>\varepsilon_4\sigma_{(n-1)}\right\},$$ and we recall from Lemma \[anlem\] that if $\varepsilon_4:=1/4\log n$ then the right-hand side only occurs for finitely many $n$ almost-surely (recall that we are working under Assumptions \[assumpt:sosv\] and \[assumpt:g22\]). Moreover, for the same choice of $\varepsilon_4$, we have from Lemma \[lem:probrecord\] and the slow-variation of $L$ that, for any $\varepsilon_0\in(0,1)$, $z_1=r_{n-1}<2\log n L(\sigma_{(n-2)}) <\varepsilon_4^{-1}\Lambda_{n-1}$ eventually almost-surely. Hence to show that $\mathcal{E}_n$ occurs infinitely often with $\varepsilon_4=1/4\log n$, it will be enough to show that $$\begin{aligned}
\tilde{\mathcal{E}}_n&:=&\left\{z_N=r_n,\:\frac{z_N-z_{N-1}}{\Lambda_{n-1}}\in(\varepsilon^{-1}_1,\varepsilon^{-1}_2),
\:\frac{z_{N-1}-z_1}{\Lambda_{n-1}}<\varepsilon_3^{-1}\right\}\\
&&\cap\left\{\sigma_{(n)}>\varepsilon_5^{-1}\sigma_{(n-1)},
\:\sum_{z_N<z<z_N+\varepsilon_6^{-1}\Lambda_{n-1}}\sigma_z<\varepsilon_7\sigma_{(n-1)}
\right\}.\end{aligned}$$ holds infinitely often. Together with the conditional Borel-Cantelli lemma, the following lemma establishes that this is indeed the case.
Suppose $(\varepsilon_i)_{i = 0}^7$ are such that $\varepsilon_i \in (0, 1)$, $\varepsilon_1^{-1}<\varepsilon_{2}^{-1}$ and $\varepsilon_4:=1/4\log n$. Let $\mathcal{F}_n$ denote the filtration generated by $\{ \sigma_z: z \leq r_{n+1} \}$. Then $\tilde{\mathcal{E}}_n\in\mathcal{F}_n$, and, under Assumptions \[assumpt:sosv\] and \[assumpt:g22\], $$\sum_n \mathbf{P}\left( \tilde{\mathcal{E}}_{2(n+1)} | \mathcal{F}_{2n} \right) = \infty$$ almost-surely.
It is clear that $$\left\{z_N=r_n,\:\frac{z_N-z_{N-1}}{\Lambda_{n-1}}\in(\varepsilon^{-1}_1,\varepsilon_2^{-1}),
\:\frac{z_{N-1}-z_1}{\Lambda_{n-1}}<\varepsilon_3^{-1},\:\sigma_{(n)}>\varepsilon_5^{-1}\sigma_{(n-1)}\right\}\in \mathcal{F}_n.$$ Moreover, on the event $$\left\{\sum_{z_N<z<z_N+\varepsilon_6^{-1}\Lambda_{n-1}}\sigma_z<\varepsilon_7\sigma_{(n-1)}\right\},$$ we have that $\sigma_z<\varepsilon_7{\sigma_{(n-1)}}<\sigma_{(n)}$ for all $z\in \{z_N+1,\dots,z_N+\varepsilon_6^{-1}\Lambda_{n-1}-1\}$, and so it must be the case that $r_{n+1}\geq z_N+\varepsilon_6^{-1}\Lambda_{n-1}$. In particular, the sum is $\mathcal{F}_n$-measurable. Thus we conclude that $\tilde{\mathcal{E}}_{n}\in\mathcal{F}_n$, as desired.
For the remainder of the proof, it is useful to note that $\tilde{\mathcal{E}}_{n}$ contains the following event: $$\left\{\frac{z_{i+1}-z_i}{\Lambda_{n-1}}<\frac{\varepsilon_3^{-1}}{N-2},\:i=1,\dots,N-2\right\}\cap
\left\{\sigma_{z_i}\leq \sigma_{(n-1)},\:i=2,\dots,N-1\right\}$$ $$\cap\left\{\sigma_{(n)}>\varepsilon_5^{-1}\sigma_{(n-1)}\right\}\cap
\left\{\frac{z_N-z_{N-1}}{\Lambda_{n-1}}\in(\varepsilon_1^{-1},\varepsilon_2^{-1})\right\}\cap\left\{\sum_{z_N<z<z_N+\varepsilon_6^{-1}\Lambda_{n-1}} \sigma_z<\varepsilon_7\sigma_{(n-1)}
\right\}.$$ Hence, it is easy to see using the independence properties of $(\sigma_x)_{x\geq 0}$ that $$\mathbf{P}\left( \tilde{\mathcal{E}}_{2(n+1)} | \mathcal{F}_{2n} \right)\geq p_1^{N-2}p_2^{N-2}p_3p_4p_5,$$ where $$\begin{aligned}
p_1&=&\mathbf{P}\left(z_{2}-z_1<\frac{\Lambda_{2n+1}\varepsilon_3^{-1}}{N-2}\:\vline\:\sigma_{(2n+1)}\right),\\
p_2&=&\mathbf{P}\left(\sigma_{z_2}\leq \sigma_{(2n+1)}\:\vline\:\sigma_{(2n+1)}\right),\\
p_3&=&\mathbf{P}\left(\sigma_{(2n+2)}>\varepsilon_5^{-1}\sigma_{(2n+1)}
\:\vline\:\sigma_{(2n+1)}\right),\\
p_4&=&\mathbf{P}\left(\frac{z_N-z_{N-1}}{\Lambda_{2n+1}}\in(\varepsilon_1^{-1},\varepsilon_2^{-1})\:\vline\:\sigma_{(2n+1)}\right),\\
p_5&=&\mathbf{P}\left(\sum_{z_N<z<z_N+\varepsilon_6^{-1}\Lambda_{2n+1}}\sigma_z<\varepsilon_7\sigma_{(2n+1)}\:\vline\:\sigma_{(2n+1)}\right).\end{aligned}$$ (Note that the indices relate to $\tilde{\mathcal{E}}_{2(n+1)}$, and so $z_1=r_{2n+1}$ and $z_N=r_{2n+2}$.) In particular, it is straightforward to check that, almost-surely, $$p_1= 1-\left(1-\frac{1}{\Lambda_{2n+1}}\right)^{\frac{\varepsilon_3^{-1}\Lambda_{2n+1}}{N-2}}\geq 1-e^{-\frac{\varepsilon_3^{-1}}{N-2}}>0,$$ $$p_2=1-\frac{L((1-\varepsilon_0)\sigma_{(2n+1)})}{L(\sigma_{(2n+1)})}
\sim-\log(1-\varepsilon_0)g(\sigma_{(2n+1)})\geq cd(e^{(1+\varepsilon)(2n+1)}),$$ where we used the eventual monotonicity of $g$ guaranteed by Assumption \[assumpt:sosv\] and the almost-sure bounds on records of Lemma \[lem:probrecord\], $$p_3= \frac{L(\sigma_{(2n+1)})}{L(\varepsilon_5^{-1}\sigma_{(2n+1)})}\rightarrow 1,$$ $$\begin{aligned}
p_4&=&\mathbf{P}\left(\frac{z_N-z_{N-1}}{\Lambda_{2n+1}}>\varepsilon_1^{-1}
\:\vline\:\sigma_{(2n+1)}\right)-
\mathbf{P}\left(\frac{z_N-z_{N-1}}{\Lambda_{2n+1}}>\varepsilon_2^{-1}\:\vline\:\sigma_{(2n+1)}\right)\\
&=&\left(1-\frac{1}{\Lambda_{2n+1}}\right)^{\Lambda_{2n+1}\varepsilon_1^{-1}}-
\left(1-\frac{1}{\Lambda_{2n+1}}\right)^{\Lambda_{2n+1}\varepsilon_2^{-1}}
\sim e^{-\varepsilon_1^{-1}}-e^{-\varepsilon_2^{-1}},\end{aligned}$$ which is strictly positive, and finally, $$p_5=\mathbf{P}\left(\frac{L((1-\varepsilon_0)\varepsilon_7^{-1}S_m)}{m}<\varepsilon_6\right)\vline_{m=\varepsilon_6^{-1}\Lambda_{2n+1}}\rightarrow e^{-\varepsilon_6^{-1}}>0.$$ Hence we deduce that $$\mathbf{P}\left( \tilde{\mathcal{E}}_{2(n+1)} | \mathcal{F}_{2n} \right)\geq cd(e^{(1+\varepsilon)(2n+1)})^{N-2}.$$ Noting that a monotone sequence $a_n$ is summable if and only if $a_{\floor{(1+\varepsilon)(2n+1)}}$ is summable, the above sequence is not summable under Assumption \[assumpt:g22\], completing the proof.
Note that the previous lemma shows that, under the same conditions, Assumption \[assump\] holds. Thus, together with Corollary \[cor:main2\], this completes the proofs of Theorems \[thm:main1\] and \[thm:main2\]. Moreover, together with Corollary \[cor:main3\], it completes the proof of Theorem \[thm:main3\].
[^1]: Part of this article was written whilst the first author was a Visiting Associate Professor at Kyoto University, Research Institute for Mathematical Sciences. He would like to thank Takashi Kumagai and Ryoki Fukushima for their kind and generous hospitality. The second author was partially supported by a Graduate Research Scholarship from University College London and the Leverhulme Research Grant RPG-2012-608 held by Nadia Sidorova, and partially supported by the Engineering & Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1 held by Dmitry Belyaev. We would like to thank an anonymous referee for their thoughtful comments.
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---
abstract: 'We construct new examples of cubic polynomials with a parabolic fixed point that cannot be approximated by Misiurewicz polynomials. In particular, such parameters admit maximal bifurcations, but do not belong to the support of the bifurcation measure.'
address:
- 'Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan'
- 'Institute for Mathematical Sciences, Stony Brook University, NY, 11794, USA'
author:
- Hiroyuki Inou
- Sabyasachi Mukherjee
bibliography:
- 'bifurcation.bib'
title: On The Support of The Bifurcation Measure Of Cubic Polynomials
---
[^1]
Introduction {#sec_intro}
============
The connectedness locus ${\mathcal{C}}$ of cubic polynomials is the set of parameters such that the corresponding Julia sets are connected. It is a compact set in the parameter space ${\mathbb{C}}^2$ of all cubic polynomials [@BH1]. For suitable parametrizations, the two critical points of a cubic polynomial can be holomorphically followed throughout the parameter space (see [@BH1; @DF; @Du1] for various related parametrizations).
In [@Du1], cubic polynomials were parametrized as[^2] $$f_{c,v}(z)=z^3- 3c^2z+ 2c^3 +v,$$ where $c,v\in{\mathbb{C}}$. The two critical points of $f_{c,v}$ are $\pm c$. The critical point $\pm c$ is said to be *passive* near the parameter $(c_0,v_0)$ if the sequence of functions $(c,v)\mapsto f_{c,v}^{\circ n}(\pm c)$ forms a normal family in a neighborhood of $(c_0,v_0)$ in ${\mathbb{C}}^2$. Otherwise, $\pm c$ is said to be *active* near $(c_0,v_0)$. According to [@Du1], the critical point $\pm c$ is active precisely on the set $\partial{\mathcal{C}}^{\pm}$, where ${\mathcal{C}}^{\pm}$ is the set of parameters for which $\pm c$ has bounded orbit. Note that ${\mathcal{C}}={\mathcal{C}}^+\cap{\mathcal{C}}^-$.
The *bifurcation locus* ${{\mathcal{C}}^{\mathrm{bif}}}$ of cubic polynomials is defined as the complement of the set of all $J$-stable parameters; i.e. ${{\mathcal{C}}^{\mathrm{bif}}}$ consists of parameters for which at least one critical point is active (see [@Mcm1 Theorem 4.2] for several equivalent conditions for $J$-stability). Clearly, we have that ${{\mathcal{C}}^{\mathrm{bif}}}=\partial{\mathcal{C}}^+\cup\partial{\mathcal{C}}^-\supset\partial{\mathcal{C}}$.
We denote by ${\mathcal{C}}^*$ the intersection of the activity loci of the two critical points; i.e. ${\mathcal{C}}^*:=\partial{\mathcal{C}}^+\cap\partial{\mathcal{C}}^-\subset\partial{\mathcal{C}}$. Since ${\mathcal{C}}^*$ is the set of parameters for which both critical points are active, it is called the *bi-activity locus*.
DeMarco introduced a natural $(1,1)$-current supported exactly on the bifurcation locus [@DeM01; @DeM03]. In [@BB], Bassanelli and Berteloot constructed a natural probability measure supported on the boundary of the connectedness locus $\partial{\mathcal{C}}$ (which is contained in the bifurcation locus), by taking a power of the bifurcation current. This measure is called the *bifurcation measure*, and is denoted by $\mu_{\mathrm{bif}}$. It has several interesting dynamical properties, and can be thought of as the correct generalization of the harmonic measure of the Mandelbrot set. Dujardin and Favre [@DF] showed that the support of the bifurcation measure is equal to the closure of Misiurewicz parameters (in fact, Misiurewicz parameters are equidistributed by $\mu_{\mathrm{bif}}$), which is a subset of ${\mathcal{C}}^*$.
The bifurcation measure is designed to capture maximal bifurcations in the family. In this vein, one may ask if $\mathrm{Supp}(\mu_{\mathrm{bif}})$ is equal to ${\mathcal{C}}^*$. However, it was pointed out by Douady that there are parabolic parameters in ${\mathcal{C}}^*$ that cannot be approximated by Misiurewicz parameters [@DF Example 6.13]. These maps have a parabolic fixed point which attracts the forward orbits of both critical points. Consequently, they have a complex one-dimensional (quasi-conformal) deformation space. Moreover, any small perturbation of such a map is either parabolic (and quasi-conformally conjugate to the original map) or has at least one attracting fixed point. In other words, these parabolic parameters are *parabolic-attracting* in the language of [@Adam1] (or *virtually attracting* in the language of [@Buff1]).
This naturally leads to a study of the set ${\mathcal{C}}^*\setminus\mathrm{Supp}(\mu_{\mathrm{bif}})$. The principal goal of this paper is to construct examples of parabolic-repelling parameters in ${\mathcal{C}}^*\setminus\mathrm{Supp}(\mu_{\mathrm{bif}})$ (at a first glance, it is much less obvious that such parameters may lie outside the support of the bifurcation measure), which shows that the gap between the bi-activity locus and the support of the bifurcation measure is bigger than what was known previously.
\[main\_thm\] There exists an interval $I$ of parabolic-repelling parameters such that $\displaystyle I~\subset~{\mathcal{C}}^*~\setminus~\mathrm{Supp}(\mu_{\mathrm{bif}})$. More precisely, if $a\in I$, then any sufficiently small perturbation of $f_a$ (in the cubic parameter space) is either in the escape locus, or has an attracting/parabolic fixed point.
The interval $I$ satisfying the statement of Theorem \[main\_thm\] consists of parameters having a parabolic fixed point of multiplier $1$. In other words, $I$ is contained in the slice $\mathrm{Per}_1(1)$ (see the definition below).
Note that every parabolic-attracting parameter is contained in ${\mathcal{C}}^*~\setminus~\mathrm{Supp}(\mu_{\mathrm{bif}})$. On the other hand, by employing parabolic perturbation techniques, we show that a (suitably chosen) parabolic-repelling parameter can be approximated by Misiurewicz parameters only if two dynamically defined conformal invariants associated with the map satisfy a certain condition. The interval $I$ in Theorem \[main\_thm\] is concocted so that the two conformal invariants of the corresponding maps violate this condition.
Every parameter in $\partial{\mathcal{C}}$ near $I$ has an attracting or parabolic cycle. Moreover, each of these nearby parameters admits a disk of quasi-conformal deformations. On the other hand, parameters outside ${\mathcal{C}}$ close to $I$ have at least one escaping critical point. Hence, by the usual wringing deformation (see [@BH1]), such parameters also admit quasi-conformal deformations. It follows that the interval $I$ does not intersect the closure of quasi-conformally rigid parameters.
According to [@DF Proposition 6.14], the support $\mathrm{Supp}(\mu_{\mathrm{bif}})$ of the bifurcation measure is the Shilov boundary of ${\mathcal{C}}$. Heuristically speaking, the interval $I$ that we construct in Theorem \[main\_thm\] does not lie in the Shilov boundary of ${\mathcal{C}}$ since $\partial{\mathcal{C}}$ is foliated by holomorphic disks (coming from quasi-conformal deformations described above) locally near $I$.
Let us now describe the organization of the paper and the key ideas of the proof of the main theorem.
In Section \[slice\], we recall some basic properties of cubic polynomials with a parabolic fixed point of multiplier $1$. These maps form a complex one-dimensional slice (of the parameter space ${\mathbb{C}}^2$ of all cubic polynomials) which is denoted by $\mathrm{Per}_1(1)$. We focus on ‘real’ maps in $\mathrm{Per}_1(1)$ for which the two critical points are complex conjugates of each other and both critical orbits converge to the unique parabolic fixed point. Both critical points of the maps under consideration are active; i.e. these maps belong to the bi-activity locus ${\mathcal{C}}^*$. We associate a ‘global’ conformal conjugacy invariant (called the *critical Ecalle height*) with these maps, that can be used as a local parameter for the quasi-conformal deformation space of these maps. We also recall the notion of the residue fixed point index of a parabolic map, which is a ‘local’ conformal conjugacy invariant of parabolic dynamics. These two invariants play a crucial role in the proof of the main theorem.
Section \[para\_perturb\] contains a brief discussion of perturbation of parabolic points. We go over some basic properties of persistent Fatou coordinates, horn maps, and lifted phase for perturbations of parabolic maps.
The final Section \[proof\_main\_thm\] is devoted to the proof of Theorem \[main\_thm\]. In Lemma \[height\_para\_rep\], we show that if the critical Ecalle height of a real map in $\mathrm{Per}_1(1)$ (introduced in Section \[slice\]) is not too large, then the map is parabolic-repelling. Finally, a careful analysis of the lifted phase of the perturbed maps shows that if the critical Ecalle height of a parabolic-repelling parameter is not too small (i.e. bounded below by a function of the residue fixed point index), every perturbation of such a map either has an attracting fixed point or has an escaping critical point. This yields an interval of real parabolic-repelling parameters in $\mathrm{Per}_1(1)$ that cannot be approximated by Misiurewicz parameters, and completes the proof of the theorem.
The Slice $\mathrm{Per}_1(1)$ {#slice}
=============================
The family of cubic polynomials with a parabolic fixed point of multiplier $1$ is denoted by $\mathrm{Per}_1(1)$, and can be parametrized as $$\mathrm{Per}_1(1):=\{f_a(z)=z+az^2+z^3:a\in{\mathbb{C}}\}.$$ This family has been studied in [@Pas]. We only recall some basic facts about $\mathrm{Per}_1(1)$ that we will need in this paper.
If $a\in{\mathbb{R}}$, then $f_a$ commutes with the complex conjugation map. It is easy to see that for $a\in(-\sqrt{3},\sqrt{3})$, the two critical points of $f_a$ are complex conjugate. We denote the critical point in the lower (respectively, upper) half plane by $c_{-}(a)$ (respectively, $c_{+}(a)$).
For $a=0$, the corresponding map has a double parabolic fixed point at the origin (i.e. it has two attracting petals), while for $a\neq0$, the corresponding map has a simple parabolic fixed point at the origin. The two immediate basins of $f_0$ are real-symmetric, and each basin contains a critical point.
It follows by real symmetry that for $a\in(-\sqrt{3},\sqrt{3})\setminus\{0\}$, both critical points of $f_a$ lie in the unique immediate basin of the parabolic fixed point. In particular, the parabolic basin is connected, and the Julia set is a Jordan curve.
For $a=\pm\sqrt{3}$, the two critical points of $f_a$ coalesce to form a double critical point. Finally, if $a\in(-\infty,-\sqrt{3})\cup(\sqrt{3},+\infty)$, then the two critical points of $f_a$ are real.
Let $\mathcal{I}:=(0,\sqrt{3})$. For all $a\in\mathcal{I}$, we will normalize the attracting (respectively, repelling) Fatou coordinate $\psi^{\mathrm{att}}_a$ (respectively, $\psi^{\mathrm{rep}}_a$) of $f_a$ at the parabolic fixed point $0$ such that $\psi^{\mathrm{att/rep}}_a$ commutes with complex conjugation. Since Fatou coordinates are unique up to addition of a complex constant, the above normalization implies that $\psi^{\mathrm{att/rep}}_a$ is unique up to horizontal translations. Therefore, the imaginary part of $\psi^{\mathrm{att/rep}}_a$ is well-defined. We will refer to $\operatorname{Im}(\psi^{\mathrm{att/rep}}_a)$ as the attracting/repelling Ecalle height.
In particular, we have that $$\operatorname{Re}(\psi^{\mathrm{att}}_a(c_{+}(a)))=\operatorname{Re}(\psi^{\mathrm{att}}_a(c_{-}(a))),\ \mathrm{and}\ \operatorname{Im}(\psi^{\mathrm{att}}_a(c_{\pm}(a)))=\pm h_a/2,$$ for some $h_a>0$. Moreover, $$h_a=\psi^{\mathrm{att}}_a(c_{+}(a))-\psi^{\mathrm{att}}_a(c_{-}(a))$$ is a conformal conjugacy invariant of $f_a$. We call $h_a$ the *critical Ecalle height* of $f_a$.
By changing $h_a$ using a quasi-conformal deformation argument, we will show that all maps on $\mathcal{I}$ are quasi-conformally conjugate.
\[para\_arc\] All cubic polynomials $f_a$, where $a\in\mathcal{I}$, are quasi-conformally conjugate. Moreover, $\mathcal{I}$ admits a real-analytic parametrization $a:(0,+\infty)\to~\mathcal{I}$ such that $h_{a(t)}=t$.
The proof is similar to [@MNS Theorem 3.2].
Pick $a_0\in\mathcal{I}$ such that $h_{a_0}=t_0$. We can choose the attracting Fatou coordinate $\psi_{a_0}^{\mathrm{att}}$ so that both the critical points $c_{\pm}(a_0)$ have real part $1/2$ within the Ecalle cylinder.
Let $\zeta=x+iy$. Now, for every $t\in(0,+\infty)$, the map $$\ell_t : (x,y)\longmapsto \left(x,\frac{t}{t_0}y\right)$$ is a quasi-conformal homeomorphism of ${\mathbb{C}}/{\mathbb{Z}}$ commuting with complex conjugation. Note that $\ell_t(1/2,\pm t_0/2) = (1/2,\pm t/2)$. Translating the map $\ell_t$ by positive integers, we obtain a quasi-conformal map $\ell_t$ commuting with $\zeta\mapsto\zeta+1$ on a right half plane.
By the coordinate change $\psi_{a_0}^{\mathrm{att}}:z\mapsto \zeta$, we can transport this Beltrami form (defined by the quasi-conformal homeomorphism $\ell_t$) into the attracting petal at $0$ such that it is forward invariant under $f_{a_0}$. Pulling it back by the dynamics, we can spread the Beltrami form to the entire parabolic basin. Extending it by the zero Beltrami form outside of the parabolic basin, we obtain an $f_{a_0}$-invariant Beltrami form. Moreover, this Beltrami form respects the complex conjugation map.
The Measurable Riemann Mapping Theorem (with parameters) now supplies a quasi-conformal map ${\varphi}_t$ integrating this Beltrami form such that ${\varphi}_t$ commutes with complex conjugation. We can normalize ${\varphi}_t$ such that it fixes $0$ and $\infty$. Then, ${\varphi}_t$ conjugates $f_{a_0}$ to a cubic polynomial fixing the origin. We can further require that the conjugated polynomial ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}$ is monic. By [@Na], ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}$ has a simple parabolic fixed point of multiplier $1$ at the origin. Hence, ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}\in\mathrm{Per}_1(1)$.
Since ${\varphi}_t$ commutes with complex conjugation, it follows that ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}$ is a real cubic polynomial. Furthermore, since the complex conjugation map exchanges the two distinct critical points of $f_{a_0}$, the same must be true for ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}$ as well. It follows that ${\varphi}_t\circ f_{a_0}\circ{\varphi}_t^{-1}=f_{a(t)}$, for some $a(t)\in\mathcal{I}$.
The attracting Fatou coordinate of $f_{a(t)}$ is given by $\psi_{a(t)}^{\mathrm{att}}= \ell_t\circ\psi_{a_0}^{\mathrm{att}}\circ{\varphi}_t^{-1}$. Thus, $\operatorname{Im}\psi^{\mathrm{att}}_{a(t)}(c_{\pm}(a(t)))=\pm t/2$, and hence $h_{a(t)}=t$.
Note that the Beltrami form constructed above depends real-analytically on $t$, so the parameter $a(t)$ depends real-analytically on $t$ as well. Therefore, we obtain a real-analytic map $a:(0,+\infty)\to\mathcal{I}$. Since the critical points of all $f_{a(t)}$ have different Ecalle heights, which is a conformal invariant, this map is injective.
It remains to show that $a((0,+\infty))=\mathcal{I}$. As $t\to0$, the two critical points of $f_{a(t)}$ tend to merge together. It follows that $\displaystyle\lim_{t\to0^+}a(t)=\sqrt{3}$. On the other hand, any accumulation point of $a(t)$ as $t\to+\infty$ must be a double parabolic parameter. Hence, $\displaystyle\lim_{t\to+\infty}a(t)=0$. Since $a((0,+\infty))$ is connected, it follows that $a((0,+\infty))=(0,\sqrt{3})=\mathcal{I}$.
The next lemma shows that the arc $\mathcal{I}$ is contained in the bi-activity locus ${\mathcal{C}}^\mathrm{bif}$.
\[para\_in\_bif\_locus\] $\mathcal{I}\subset{\mathcal{C}}^\mathrm{bif}$.
Let $\tilde{a}\in\mathcal{I}$. For cubic polynomials $g$ close to $f_{\tilde{a}}$, we mark the two critical points of $g$ by $c_\pm(g)$.
We will prove the lemma by contradiction. To this end, let us assume that there is an open neighborhood $U$ of $\tilde{a}$ (in the full cubic parameters space ${\mathbb{C}}^2$) such that the sequence of holomorphic functions $\{U\ni g\mapsto g^{\circ n}(c_+(g))\}_{n\in{\mathbb{N}}}$ forms a normal family. Note that $U$ intersects a hyperbolic component of period one non-trivially such that for these hyperbolic polynomials, both critical orbits converge to a common attracting fixed point. By normality of the above family of functions, it follows that the forward orbit of $c_+(g)$ converges to a fixed point $w(g)$ (of $g$) for every map $g$ in $U$. Moreover, $w(g)$ is a holomorphic function of $g$ on $U$ (as it is a limit of holomorphic functions). Therefore, the multiplier $g'(w(g))$ of the fixed point is also a holomorphic function of $g$ in $U$. Since the multiplier of the fixed point of $f_{\tilde{a}}$ is $1$, it follows by the maximum modulus principle that $w(g)$ must be a repelling fixed point for an open set of maps in $U$. However, this is impossible as an orbit cannot non-trivially converge to a repelling fixed point.
Since $\tilde{a}$ is real and $c_\pm(f_{\tilde{a}})$ are complex conjugate, $c_+(f_{\tilde{a}})$ is active if and only if $c_-(f_{\tilde{a}})$ is. Hence, both critical points of $f_{\tilde{a}}$ are active. Thus, $\mathcal{I}\subset{\mathcal{C}}^\mathrm{bif}$.
The *residue fixed point index* of $f_a$ at the parabolic fixed point $0$ is defined to be the complex number $$\begin{aligned}
\displaystyle \iota(f_a,0) &= \frac{1}{2\pi i} \oint \frac{dz}{z-f_a(z)},\end{aligned}$$ where we integrate in a small loop in the positive direction around $0$. A simple computation shows that $\iota(f_a,0) =1/a^2$ (when $a\neq0$). The fixed point index is a conformal conjugacy invariant (see [@M1new §12] for a general discussion on the concept of residue fixed point index).
The *r[é]{}sidu it[é]{}ratif* of the parabolic fixed point $0$ of $f_a$ is defined as $1-\iota(f_a,0)=1-1/a^2$. It is denoted by $\operatorname{r\acute esit}(f_a)$. This quantity plays an important role in the study of perturbation of parabolic germs.
The origin is called a parabolic-attracting (respectively, parabolic-repelling) fixed point of $f_a$ if $\operatorname{Re}(\operatorname{r\acute esit}(f_a))<0$ (respectively, if $\operatorname{Re}(\operatorname{r\acute esit}(f_a))>0$). Clearly, for $a\in(0,1)$, the origin is a parabolic-attracting fixed point of $f_a$. On the other hand, for $a\in(1,\sqrt{3})$, the origin is a parabolic-repelling fixed point of $f_a$.
\[upper\_bound\] For all $a\in\mathcal{I}$, we have that $\operatorname{r\acute esit}(f_a)<2/3$.
If $a\in\mathcal{I}=(0,\sqrt{3})$, then $1/a^2>1/3$. Hence, $\operatorname{r\acute esit}(f_a)=(1-1/a^2)<2/3$.
Perturbation of Parabolic Points {#para_perturb}
================================
In this section, we will recall some basic facts on perturbation of parabolic points, and fix the terminologies for the rest of the paper. We will only be concerned with perturbations creating eggbeater dynamics (other perturbations always create an attracting fixed point, and are uninteresting from our point of view, see [@Shi] for further details).
If a map $f_a$ (with $a\in\mathcal{I}$) is perturbed outside of $\mathrm{Per}_1(1)$ creating an eggbeater dynamics, then the simple parabolic fixed point $0$ splits into two simple fixed points. In the dynamical plane of such a perturbed map, there is a curve joining these two fixed points, which is called the *gate*. Moreover, there exist an attracting domain $V^\mathrm{att}$, and a repelling domain $V^\mathrm{rep}$ having the two simple fixed points on their boundaries. The points in the attracting domain eventually transit through the gate, and escape to the repelling domain. Moreover, there are injective holomorphic maps $\psi^\mathrm{att/rep}$ defined on $V^\mathrm{att/rep}$ conjugating the dynamics to translation by $+1$ (as long as the orbit stays in the domain of definition of the maps). The maps are referred to as persistent Fatou coordinates. The quotient of $V^\mathrm{att/rep}$ by the dynamics is a bi-infinite cylinder, which is denoted by $C^\mathrm{att/rep}$.
There exists an open set $U$ in the cubic parameter space with $\mathcal{I}\subset\partial U$ such that every map in $U$ exhibits eggbeater dynamics and admits persistent Fatou coordinates as above (compare [@Shi Proposition 3.2.2, Proposition 3.2.3]). We will only consider perturbations of $f_a$ in $U$. It makes sense to label the critical points of the perturbed maps as $c_\pm$ so that $\operatorname{Im}(c_+)>0$, and $\operatorname{Im}(c_-)<0$.
The *lifted horn maps* of the parabolic fixed point of $f_a$ are defined as $H_a^\pm=\psi_a^{\mathrm{att}}\circ\left(\psi_a^{\mathrm{rep}}\right)^{-1}$ on regions with sufficiently large imaginary part in the repelling Fatou coordinates. More precisely, $H_a^+$ (respectively, $H_a^-$) is defined on $\lbrace\operatorname{Im}(\zeta)>M\rbrace$ (respectively, on $\lbrace\operatorname{Im}(\zeta)<-M\rbrace$) for some sufficiently large positive $M$.
By our normalization of Fatou coordinates, we have that $$\psi_{a}^{\mathrm{att}}(z)-\psi_{a}^{\mathrm{rep}}(z)\approx\mp i\pi \operatorname{r\acute esit}(f_a),$$ as $z$ tends to the upper/lower end of the cylinders. It follows that $$H_a^\pm(\zeta)\approx \zeta\mp i\pi \operatorname{r\acute esit}(f_a)$$ as $\operatorname{Im}(\zeta)\to\pm\infty$. The map $\mathrm{exp}:\zeta\mapsto e^{2\pi i\zeta}$ conjugates the lifted horn maps $H_a^\pm$ to a pair of germs $\mathfrak{h}_a^\pm$ fixing $0$ and $\infty$ respectively. These maps are called the *horn maps* of the parabolic fixed point of $f_a$. They satisfy $$\left(\mathfrak{h}_a^+\right)'(0)=e^{2\pi^2\operatorname{r\acute esit}(f_a)}= \left(\mathfrak{h}_a^-\right)'(\infty).$$
For perturbed maps, one can still define horn maps. Points in $V^\mathrm{rep}$ with large imaginary part in the repelling Fatou coordinate eventually land in $V^\mathrm{att}$. This defines a map from the ends of $\psi^\mathrm{rep}(V^\mathrm{rep})$ to $\psi^\mathrm{att}(V^\mathrm{att})$, which is called the lifted horn map of the perturbed map. The persistent Fatou coordinates can be normalized so that they depend continuously on the parameters. With such normalizations, the horn maps depend continuously on the parameters.
In the perturbed situation, points in $V^\mathrm{att}$ are mapped to $V^\mathrm{rep}$ by some large iterate of the dynamics (where the required number of iterations tends to $+\infty$ as the perturbation goes to zero). This *transit map* induces an isomorphism of the cylinder ${\mathbb{C}}/{\mathbb{Z}}$ (via the Fatou coordinates). Hence, the transit map can be written as $\left(\psi^\mathrm{rep}\right)^{-1}\circ T_\sigma\circ\psi^{\mathrm{att}}$, where $T_\sigma$ is translation by some complex number $\sigma$. The complex number $\sigma$ is called the *lifted phase* of the perturbed map.
Finally, one can define a *return map* from the top and bottom ends of $V^\mathrm{rep}$ to $V^\mathrm{rep}$ for the perturbed maps. In the Fatou coordinates, this map can be expressed as the composition of the lifted horn map and the translation $T_\sigma$ (for some $\sigma\in{\mathbb{C}}$). For sufficiently small perturbations, $\mathrm{exp}:\zeta\mapsto e^{2\pi i\zeta}$ conjugates these return maps to germs $\mathcal{R}^\pm$ that are close to $e^{2\pi i\sigma}\mathfrak{h}_a^\pm$.
Proof of Theorem \[main\_thm\] {#proof_main_thm}
==============================
Let $a\in(0,1)$. Then $0$ is a parabolic-attracting fixed point of $f_a$. By [@Buff1 Theorem 1] (also compare [@M1new Theorem 12.10]), every cubic polynomial sufficiently close to $f_a$ has at least one non-repelling fixed point. Thus, $f_a$ cannot be approximated by Misiurewicz maps. Following [@DF Example 6.13], one can conclude that the parabolic-attracting map $f_a$ lies in ${\mathcal{C}}^*\setminus\mathrm{Supp}(\mu_{\mathrm{bif}})$.
The goal of this section is to prove Theorem \[main\_thm\], which asserts that ${\mathcal{C}}^*\setminus\mathrm{Supp}(\mu_{\mathrm{bif}})$ does not consist only of parabolic-attracting maps, it contains parabolic-repelling maps as well (which is perhaps more surprising). To this end, we need to study the geometry of the dynamical plane of maps in $\mathcal{I}$.
Let $a\in\mathcal{I}$. In the dynamical plane of $f_a$, the projection of the basin of infinity into the repelling Ecalle cylinder is an annulus of modulus $\frac{\pi}{\ln 3}$ (compare [@H1 Proposition 3.5]). Since $\frac{\pi}{\ln 3}>\frac{1}{2}$, it follows by [@BDH Theorem I] that this conformal annulus contains a round annulus of modulus at least $m=\frac{\pi}{\ln 3}-\frac{1}{2}\approx 2.3596$ (centered at the origin). Hence, due to real symmetry, there is an interval $(-m/2,m/2)$ of repelling Ecalle heights such that in the repelling Ecalle cylinder, the round cylinder ${\mathbb{R}}/{\mathbb{Z}}\times(-m/2,m/2)$ is contained in the projection of the basin of infinity (see Figure \[cylinders\_pic\]).
The proof of the main theorem makes essential use of the above geometric property of the basin of infinity (of the maps in $\mathcal{I}$) and bi-criticality of cubic polynomials. The rough idea of the proof is as follows. We construct a suitable sub-interval $I\subset\mathcal{I}$ (consisting of parabolic-repelling parameters) such that if the lifted phase of a nearby map has small imaginary part, then at least one critical point escapes to infinity (which is a consequence of the fact that the basin of infinity occupies a definite annulus). On the other hand, if the imaginary part of the lifted phase is large enough to prohibit a critical point from escaping, then there is an attracting fixed point. Thus, no map close to $I$ can be Misiurewicz.
![Left: The attracting cylinder of $f_{a(t)}$ with the critical points marked. Right: The projection of the basin of infinity of $f_{a(t)}$ into repelling Ecalle cylinder contains the round cylinder ${\mathbb{R}}/{\mathbb{Z}}\times(-m/2,m/2)$.[]{data-label="cylinders_pic"}](attract.png "fig:") ![Left: The attracting cylinder of $f_{a(t)}$ with the critical points marked. Right: The projection of the basin of infinity of $f_{a(t)}$ into repelling Ecalle cylinder contains the round cylinder ${\mathbb{R}}/{\mathbb{Z}}\times(-m/2,m/2)$.[]{data-label="cylinders_pic"}](repel.png "fig:")
For the rest of the paper, we fix an ${\varepsilon}$ sufficiently small; for instance with $0<{\varepsilon}<0.01$. Recall that the *shift locus* is the set of maps with both critical points escaping.
\[height\_para\_rep\] Let $0<t<m-2{\varepsilon}$. Then $f_{a(t)}$ lies on the boundary of the shift locus. In particular, $f_{a(t)}$ is parabolic-repelling; i.e. $a(t)\in(1,\sqrt{3})$.
Let $0<t<m-2{\varepsilon}$. We consider a small open set $U$ with $a(t)\in\partial U$ such that perturbing $a(t)$ in $U$ creates eggbeater dynamics.
Note that the critical points and the Fatou coordinates depend continuously on the parameter throughout $U$. Hence, by shrinking $U$, we can assume that the imaginary part of $\psi^{\mathrm{att}}(c_+)$ is less than $(t/2+{\varepsilon}/3)<(m/2-2{\varepsilon}/3)$, and the imaginary part of $\psi^{\mathrm{att}}(c_-)$ is greater than $(-t/2-{\varepsilon}/3)> (-m/2+2{\varepsilon}/3)$.
Moreover, the basin of infinity can not get too small when $a(t)$ is slightly perturbed (compare [@D2 Theorem 5.1(a)]). Hence, by shrinking $U$ further, we can also assume that for every parameter in $U$, the round cylinder ${\mathbb{R}}/{\mathbb{Z}}\times\left(-m/2+{\varepsilon}/3, m/2-{\varepsilon}/3 \right)$ is contained in the projection of the basin of infinity into the repelling cylinder (note that in the repelling Ecalle cylinder of $a(t)$, the round cylinder ${\mathbb{R}}/{\mathbb{Z}}\times\left(-m/2, m/2 \right)$ is contained in the projection of the basin of infinity).
Now consider the following perturbation of $f_{a(t)}$: $$g_\delta(z)= f_{a(t)}(z)+\delta \quad (\delta>0).$$ Since $g_\delta$ is real, the transit map is a horizontal translation. In the dynamical plane of such a perturbed map, the critical orbits “transit” from the attracting Ecalle cylinder to the repelling cylinder and the imaginary parts of the Fatou coordinates are preserved in the process. By our construction, this would provide with points of the two critical orbits with repelling Ecalle height in $\left(-m/2+2{\varepsilon}/3, m/2-2{\varepsilon}/3 \right)$ in the repelling Ecalle cylinder. But since the perturbed map is in $U$, any point in the repelling cylinder with repelling Ecalle height in $\left(-m/2+{\varepsilon}/3, m/2-{\varepsilon}/3 \right)$ is contained in the projection of the basin of infinity. Therefore, for such perturbations, both critical points lie in the basin of infinity. Hence, the perturbed maps lie in the shift locus, and both fixed points of the perturbed maps are repelling.
Therefore, $f_{a(t)}$ lies on the boundary of the shift locus and is parabolic-repelling; i.e. $a(t)\in(1,\sqrt{3})$.
Suppose that $a(t)\in\mathcal{I}$ with $0<t<m-2{\varepsilon}$. Consider a small (eggbeater-type) perturbation of $f_{a(t)}$ with associated transit map $T_\sigma$. It follows from the proof of Lemma \[height\_para\_rep\] that if the lifted phase $\sigma$ of the perturbed map is real, then both of its critical points escape to infinity. In the next lemma, we look at the other side of the story. More precisely, we study perturbed maps whose lifted phase $\sigma$ has a large imaginary part.
\[phase\_restriction\] If $\operatorname{Im}(\sigma)>\pi\operatorname{r\acute esit}(f_{a(t)})$, then the perturbed map has an attracting fixed point.
For sufficiently small perturbations, the absolute value of the multiplier of the ‘return map’ $\mathcal{R}^+$ at the origin is close to $$\vert e^{2\pi i\sigma}\cdot\left(\mathfrak{h}_{a(t)}^+\right)'(0)\vert=\vert e^{2\pi i\sigma}\cdot e^{2\pi^2\operatorname{r\acute esit}(f_{a(t)})}\vert=e^{-2\pi(\operatorname{Im}(\sigma)-\pi\operatorname{r\acute esit}(f_{a(t)}))}.$$ Therefore, if $\operatorname{Im}(\sigma)>\pi\operatorname{r\acute esit}(f_{a(t)})$, then $0$ is an attracting fixed point of $\mathcal{R}^+$. It follows that one of the simple fixed points of the perturbed map is attracting.
Choose $a(t)\in\mathcal{I}$ with $$4\pi/3-m+2{\varepsilon}<t<m-2{\varepsilon}.$$ This is possible because $4\pi/3-m+2{\varepsilon}\approx1.8292+2{\varepsilon}<1.85$, and $m-2{\varepsilon}\approx2.3596-2{\varepsilon}>2.33$. By Lemma \[height\_para\_rep\], $f_{a(t)}$ is parabolic-repelling.
Consider a perturbation of $f_{a(t)}$ in the connectedness locus. If it is not eggbeater-type, then we have an attracting or parabolic fixed point. So we may assume this is an eggbeater-type perturbation. Let the transit map be $T_\sigma$, for some $\sigma\in{\mathbb{C}}$. We can assume that $\operatorname{Im}(\sigma)>0$ (the other case is symmetric).
Note that $\psi^{\mathrm{att}}_{a(t)}(c_{-}(a(t)))=- t/2$. Under small perturbation, the imaginary part of the attracting Fatou coordinate of $\psi^{\mathrm{att}}(c_{-})$ lies in $(-t/2-{\varepsilon}/2,-t/2+{\varepsilon}/2)$. Since the perturbed map is in the connectedness locus, the critical point must not land in the basin of infinity after exiting through the gate. But for a small perturbation, the basin of infinity occupies at least the cylinder ${\mathbb{R}}/{\mathbb{Z}}\times(-m/2+{\varepsilon}/2,m/2-{\varepsilon}/2)$ in the repelling cylinder. Note that $(-t/2-{\varepsilon}/2)>(-m/2+{\varepsilon}/2)$. Hence, the imaginary part of the lifted phase must be large enough to push the ‘lower’ critical point sufficiently up so that it avoids the basin of infinity; i.e. $$- t/2+{\varepsilon}/2+\operatorname{Im}(\sigma)\geq m/2-{\varepsilon}/2;$$ $$\mathrm{or},\quad \operatorname{Im}(\sigma)\geq m/2+t/2-{\varepsilon}.$$
By our choice of $t$ and Lemma \[upper\_bound\], we have that $$m/2+t/2-{\varepsilon}>2\pi/3>\pi\operatorname{r\acute esit}(f_{a(t)}).$$ This means that the imaginary part of $\sigma$ is larger than $\pi\operatorname{r\acute esit}(f_{a(t)})$; i.e. $\operatorname{Im}(\sigma)>\pi\operatorname{r\acute esit}(f_{a(t)})$. But then Lemma \[phase\_restriction\] forces the perturbed map to have an attracting fixed point.
Hence $I:=a((4\pi/3-m+2{\varepsilon},m-2{\varepsilon}))\subset\mathcal{I}$ consists of parabolic-repelling parameters that are not contained in $\mathrm{Supp}(\mu_\mathrm{bif})$. By Lemma \[para\_in\_bif\_locus\], we conclude that $I\subset{\mathcal{C}}^*\setminus\mathrm{Supp}(\mu_\mathrm{bif})$.
1. Similar techniques can be used to prove the existence of parabolic-repelling biquadratic polynomials (lying on the parabolic arcs of period one of the tricorn) outside the support of the bifurcation measure, compare [@IM1 Theorem 1.2].
2. If a cubic polynomial has a Siegel disk containing a post-critical point, then the corresponding map belongs to the bi-activity locus, and admits a disk of quasi-conformal deformations in the parameter space. It will be interesting to know if such parameters always lie in the support of the bifurcation measure.
3. Our proof works only when the modulus of the basin of infinity in the repelling Ecalle cylinder is sufficiently large. It seems unlikely to have such an interval when the modulus is small, i.e., if the degree of the map or the period of the parabolic periodic point is large.
[^1]: The first author was supported by JSPS KAKENHI Grant Numbers 26400115 and 26287016.
[^2]: The chosen parametrization plays no special role in the current paper as the results of this paper are mostly coordinate-free.
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---
abstract: 'We describe a realistic model for a focused high-intensity laser pulse in three dimensions. Relativistic dynamics of an electron submitted to such pulse is described by equations of motion with ponderomotive potential depending on a single free parameter in the problem, which we refer to as the “asymmetry parameter”. It is shown that the asymmetry parameter can be chosen to provide quantitative agreement of the developed theory with experimental results of Malka *et al.* \[Phys. Rev. Lett. [**78**]{}, 3314 (1997)\] who detected angular asymmetry in the spatial pattern of electrons accelerated in vacuum by a high-intensity laser pulse.'
address: 'Moscow State Engineering Physics Institute, 115409 Moscow, Russia'
author:
- 'N.B. Narozhny[^1], M.S. Fofanov[^2]'
title: 'Anisotropy of Electrons Accelerated by a High-Intensity Laser Pulse'
---
In their recent paper, Malka *et al.* [@MLM] have reported experimental observation of electrons accelerated to relativistic energies by a high-intensity linearly polarized subpicosecond laser pulse in vacuum (see also comments on the paper [@MLM] and the author’s reply in Ref. [@McD; @MQ; @MLM2]). This effect, known as high-intensity ponderomotive scattering, was discussed in detail in Ref. [@H]. It occurs when the quiver amplitude imparted by the laser field to an electron becomes comparable to the focal spot radius of the laser beam. If the beam is Gaussian, the radial restoring force acting on the electron decays exponentially and the electron can be scattered out of the pulse.
The data of Malka *et al.* [@MLM] show that the energies gained by the scattered electrons are in good quantitative agreement with calculations of electron trajectories in the polarization plane made with the first-order paraxial model for the laser field [@MLM; @MQ]. Nevertheless, the first order paraxial model predicts isotropic electron scattering [@MQ] that is not supported by experimental results. Indeed, accelerated electrons were detected by Malka *et al.* only in the $\left({\bf E,k}\right)$ plane, while no significant signal was detected after rotating the laser polarization direction by $90^{o}$ [@MLM]. In our opinion, this discrepancy between the theory and experiment is due to the following. In the first-order paraxial model focusing of a plane monochromatic wave only leads to an appearance of nonvanishing longitudinal components of electromagnetic fields in the focal region. However, the focusing is known to affect transverse components of the fields also (see, e.g. Ref. [@B]). In particular [@B], a plane monochromatic wave, polarized linearly along the $x$ axis and propagating along the $z$ axis, is converted by an aplantic system to a converging spherical wave with non-vanishing $y$ and $x$ components of electric and magnetic fields, respectively.
In this Letter, we show that experimentally observed [@MLM] anisotropy of ponderomotive electron scattering can be explained in the framework of a realistic model for the laser field developed in our recent paper [@NF]. Our model is based on an exact solution of Maxwell equations in three dimensions (3D), which can serve to describe a stationary, focused monochromatic laser beam with characteristic frequency $\omega$ and arbitrary intensity. Amplitudes of electric and magnetic fields in the model depend on radial coordinates as well as the coordinate along the direction of the beam propagation. These amplitudes are characterized by parameters $R$ and $L=\omega R^{2}$, which can be interpreted as the focal spot radius and the Rayleigh length of the laser beam, respectively. The model admits different field configurations, which are determined by two coordinate functions satisfying certain second-order partial differential equations. Some special choice of these functions describes the Gaussian beams, which are widely used in optics. The model can be generalized by introducing temporal amplitude envelope $g(\varphi/\omega\tau)$, where $\varphi$ is the relativistically invariant phase of the traveling wave, to describe a laser pulse with finite duration $\tau$. (It is assumed that the function $g(\varphi/\omega\tau)$ is equal to unity at the point $\varphi=0$ and decreases exponentially at the periphery of the pulse for $|\varphi|\gg\omega\tau$.) In this case the electric and magnetic fields of the model constitute an approximate solution of Maxwell equations with the second-order accuracy with respect to small parameters $\Delta$ and $\Delta^\prime$, defined as
$$\label{1} \Delta^\prime =1/\omega\tau\lesssim\Delta = 1/\omega R\ll 1.$$
For a pulse propagating along the $z$ direction, $x$ and $y$ components of the electric field oscillate with phase difference, which depends on $z$ and values of $x$ and $y$ coordinates of a point in the plain $z=const$. Moreover, the aforementioned phase difference depends also on $\varphi$. Therefore, one cannot ascribe some definite type of polarization to a tightly focused laser pulse. Nevertheless, for a weakly focused pulse ($\Delta \ll 1$), there always exists a region near the axis of the beam $r\ll R$, where the field properties are very close to those of a plane wave field. This region we call “the plane wave zone”. It is reasonable to ascribe polarization of the field in this region to the beam as a whole. Hereafter we refer to the field of the pulse as linearly polarized in this sense only. For a tightly focused beam the focal spot radius is of the order of the wavelength and the plane wave zone doesn’t exist. Therefore, only polarization of the parental beam incident on the focusing optical system can be ascribed to the focused beam in this case.
An arbitrary field, linearly polarized along the $x$ axis, may be represented [@NF] as a superposition of $E$- and $H$-polarized waves (i.e. waves with the vectors $\vec{E}$ and $\vec{H}$ being perpendicular to the direction of the pulse propagation, compare with Ref. [@BW]). Relative contributions of $E$- and $H$-polarized waves to the resulting field are characterized by the “asymmetry parameter” $\mu$
$$\label{2} \mu =
\frac{E_{x0}^{h}-E_{x0}^{e}}{E_{x0}^{h}+E_{x0}^{e}},\quad\quad-\infty<\mu<\infty,$$
where $E_{x0}^{e,h}$ are the $x$ components of the electric field for $E$- and $H$-polarized waves at the focal point ${\bf r}=0$ for $\varphi =0$. Note, that in contrast to the amplitude, the quantities $E_{x0}^{e,h}$ can take both positive and negative values.
In the lowest approximation in $\Delta$ and $\Delta'$, the averaged equations of motion of electrons (ponderomotive equations) in the field of the linearly polarized laser pulse take the form [@NF]
$$\label{10}
\begin{array}{rclrcl} {\displaystyle}{ \frac{d {\vec{q}_\perp}}{d\varphi}
}&=&{\displaystyle}{ -\Delta\frac{m}{q_{-}}\frac{\partial U}{\partial
{\vec{\rho}_\perp}}}, &\qquad {\displaystyle}{ \frac{d\vec{ \rho}_{\perp}}{d\varphi}}&
=&{\displaystyle}{ \Delta\frac{\vec{q}_{\perp}}{q_{-}} },
\\{}\\ {\displaystyle}{ \frac{d q_-}{d\varphi}}&=&0,
&\qquad {\displaystyle}{ \frac{d \zeta}{d\varphi} }&=&{\displaystyle}{ \Delta^2\frac{q_{z}}{q_{-}}
}.
\end{array}$$
Here $q^{\mu}= \langle p^{\mu}\rangle$, where $p^{\mu}$ is the 4-momentum of the electron, $q_{-}=q_{0}-q_{z}$, $\vec{\rho}_\perp=
\langle \vec{r}/R\rangle$, $\zeta=\langle z/R\rangle$, brackets $\langle\rangle$ mean averaging over fast oscillations, and the ponderomotive potential $U$ is defined by the expression
$$\label{3} U=\frac{m\eta_0^2}{2}g^2(\varphi/\omega\tau) \left\{ \left|
F_1\right|^2 +\mu^2\left| F_2\right|^2 + \mu\cos 2\psi \left(
F_1F_2^*+F_1^*F_2\right)\right\},$$
where $\tan\psi=\varrho_{x}/\varrho_{y}$, and $\eta_{0}$ is the value of the dimensionless field intensity parameter
$$\label{4} \eta^{2}=\frac{e^{2}\langle
\textbf{E}^{2}\rangle}{m^{2}\omega^{2}},$$
at the focal point at the moment $\varphi=0$ ($\eta_0=a/\sqrt{2}$, where $a$ is the parameter of Malka [*et al.*]{}). Functions $F_{i}(\vec{\varrho}_{\perp},\zeta;\Delta)$ are chosen in the form corresponding to the Gaussian beam [@NF]
$$\label{5}
\begin{array}{c} {\displaystyle}{ F_{1} =
(1+2i\zeta)^{-2} \left\{1-\frac{{\varrho_{\perp}}^{2}}{1+2i\zeta}\right\}
\exp\left\{-\frac{{\varrho_{\perp}}^{2}}{1+2i\zeta}\right\}},
\\{}\\
{\displaystyle}{F_{2}=-{\varrho_{\perp}}^{2}(1+2i\zeta)^{-3}\exp\left\{-\frac{{\varrho_{\perp}}
^{2}} {1+2i\zeta}\right\}}.
\end{array}$$
The equation (\[3\]) shows that the ponderomotive potential $U$ depends on the azimuthal angle $\psi$, and hence is generally speaking asymmetric. The potential $U$ is symmetric only for the case $\mu=0$. The shape of the ponderomotive potential in the plane $z=0$ for $\varphi=0$ is shown in Fig.\[fig:1\] for the cases $\mu=0$ and $\mu=-1.55$. Figure \[fig:1\][*a*]{} represents the ponderomotive potential for the standard case of Gaussian beam commonly used in literature, while Figure \[fig:1\][*b*]{} illustrates the dramatic difference between the cases $\mu=0$ and $\mu\neq 0$. For $\mu\neq 0$, the ponderomotive potential, possesses (besides the central peak) two extra maxima, which are located in the polarization plane. They arise as a result of the non-uniform intensity distribution in the plane $z=0$ for $\mu\neq 0$. Locations of the additional maxima, as well as their amplitudes, are determined by the value of $\mu$. It is noteworthy, that the case $\mu=0$ is the only one when $E_{y}$- and $H_{x}$-components of the electric and magnetic fields remain to be equal to zero outside the plane wave zone for the pulse polarized along the $x$ axis.
We use Eqs. (\[10\]) and (\[3\]) for our analysis of free electron acceleration by a co-propagating intense laser pulse in vacuum under conditions close to those used in the experiments of Malka [*et al.*]{} [@MLM]. The initial electron energy is taken to be $\varepsilon=10$ keV ($v_0=0.2c$). The laser field parameters are: $\lambda=1$ $\mu$m, $\eta_0=2.12$ (corresponds to the parameter $a=3$ of Malka [*et al.*]{}), $R=10$ $\mu$m and $\omega\tau=480$. For the temporal envelope of the pulse $ g(\varphi/\omega\tau)$ a sine-squared shape is taken. The asymmetry parameter is not determined experimentally and remains a free parameter of the problem. Its value, $\mu=-1.55$, has been chosen for better fitting of our computational results to the results of the experiment.
It is clear that the maximum energy will be gained by electrons that initially propagate exactly along the axis of the laser beam. However, the ponderomotive equations yield zero net energy transfer for such electrons, since they can “feel” the spatial gradient of the Gaussian laser field, and hence can be scattered out of the pulse, only due to the quiver motion which is absent for the average trajectory described by the ponderomotive equations. Nevertheless, there exist a family of trajectories with nonzero initial distances from the beam axis, for which the gained energy is close to its maximum value. The energy and scattering angle of the electrons depend also on the position at which the particle is overtaken by the pulse [@MLM]. Indeed, the maximum energy is obtained for the electrons that experience the peak field of the laser and therefore meet the laser pulse at some distance before the focus. As a result, we obtain a 3D domain of injection positions of electrons obtaining the final kinetic energy $W\geq
0.9$ MeV at the scattering angle $39.5^{o}$.
Different cross-sections of this domain for the potential with $\mu=-1.55$ (Fig. \[fig:1\][*b*]{}) are shown in Fig. \[fig:2\]. The longitudinal size of the domain is of the order of the Rayleigh length for the laser beam $L$, whereas its transverse size is much less than the focal spot radius $R$. The cross sections in Fig. \[fig:2\] display high degree of radial anisotropy. Their shape, of course, is essentially determined by the type of the ponderomotive potential or by the value of $\mu$. In particular, the domain of injection positions for the case (not presented here for the sake of compactness) of the potential shown in Fig. \[fig:1\][*a*]{} with $\mu=0$ is purely radial, in agreement with [@MQ].
To obtain the angular distribution of scattered electrons we have calculated their trajectories numerically, applying the fourth-order Runge-Kutta method to the ponderomotive equations (\[10\]). Initial positions of the electrons were taken from the domain shown in Fig. \[fig:2\]. Since the shape of the cross-sections $z=const$ varies very slowly at intervals $\delta z \sim R$, we have considered equidistant planes $z=nR$ with $n=-27,-26,\ldots,5$. In each of these cross-sections, the electron injection positions were chosen randomly under condition that their density was constant and equal to $3\times 10^{15}/R^2$. Physically, such procedure corresponds to uniformity of the initial electron beam. The total number of electron trajectories considered in such a way was more than $2\times 10^{7}$. We were interested only in those trajectories, which crossed the plane $z=11.66$ cm at the points contained inside the ring with radii $r_{1}=8.99$ cm and $r_{2}=9.89$ cm. The latter conditions were determined by the position and angular size of the detector in the experiment [@MLM].
The results of the calculations are presented in Fig. \[fig:3\]. We plot the normalized number of scattered electrons $\langle n\rangle$ with final energies $W\geq 0.9$ MeV as a function of the azimuthal scattering angle $\alpha$. The values of $\langle n\rangle$ at any given $\alpha$ were obtained by averaging the number of scattered electrons over the range of $\alpha$ equal to the angular size of the detector used in the experiments [@MLM]. It is easily seen, that the angular distribution of scattered electrons essentially depends on the parameter of asymmetry $\mu$. At $\mu=0$, which corresponds to the symmetrical Gaussian ponderomotive potential shown in Fig. \[fig:1\][*a*]{}, the distribution is isotropic and purely radial (compare to the result of Ref. [@MQ]). At the same time, if $\mu=-1.55$, the number of scattered electrons detected in the $\left({\bf E,k}\right)$ plane ($\alpha=0$) is about 30 times higher than that in the $\left({\bf H,k}\right)$ plane ($\alpha=\pi/2$). This result is in good quantitative agreement with observations of Malka *et al.* [@MLM].
The 30-fold anisotropy of accelerated electrons is clearly explained by asymmetry of the ponderomotive potential (\[3\]). The cross term in (\[3\]), besides asymmetric corrections to the radial force, gives rise to a tangential force which is responsible for pushing electrons out of the plain perpendicular to the polarization plane. The latter corresponds to minimum of the ponderomotive potential as function of azimuthal angle $\psi$, while the perpendicular plane to its maximum. Therefore one could be surprised that the number of electrons scattered at the angle $\alpha=\pi/2$ is not equal to zero. Certainly it is explained by complicated structure of the ponderomotive potential (\[3\]), namely by the fact that the cross term can change its sign at the periphery of the focus.
The asymmetry of the ponderomotive potential itself is determined by non-zero value of the parameter $\mu$ which characterizes relative contributions of E- and H-polarized waves to the resulting field. As far as we know, nobody has never controlled the parameter $\mu$ in experiments. The reason is evident. Before the work of Malka [*et al*]{} [@MLM] there were no experiments where the three-dimensional intensity distribution of a laser pulse influenced physical results. Therefore even qualitative coincidence of our calculations with the results of the experiment [@MLM] would give a deeper insight into physics of the electron-laser interaction. However, it appeared that the model describes the experiment quantitatively. Of course, the quantitative agreement between the developed theory and the experiment is based on fitting of a single free parameter $\mu$ in the problem, which has not been measured experimentally. Therefore, from the standpoint of our model, the experiment of Malka *et al.* [@MLM] could be considered as a probe for 3D field distribution in the laser pulse. The correctness of our approach could be verified by an independent experiment for another physical situation performed with the same laser system. Measurements of angular distribution of ATI electrons could serve as a good example of such experiment.
We thank M.V. Fedorov and V.D. Mur for fruitful discussions. This work was supported by the Russian Foundation for Basic Research under projects 00-02-16354 and 00-02-17078.
G. Malka, E. Lefebvre, and J.L. Miquel, Phys. Rev. Lett. [**78**]{}, 3314 (1997) K.T. McDonald, Phys. Rev. Lett. [**80**]{}, 1350 (1998) P. Mora and B. Quesnel, Phys. Rev. Lett. [**80**]{}, 1351 (1998) G. Malka, E. Lefebvre, and J.L. Miquel, Phys. Rev. Lett. [**80**]{}, 1352 (1998) F.V. Hartemann *et al.*, Phys. Rev. E [**51**]{}, 4833 (1995) A. Boivin and E. Wolf, Phys. Rev. [**138**]{}, B1561 (1965) N.B. Narozhny, M.S. Fofanov, JETP, [**90**]{}, 753 (2000) M. Born and E. Wolf, [*[Principles of Optics]{}*]{} (Pergamon Press, New York, 1964)
[^1]: E-mail: narozhny@theor.mephi.ru
[^2]: E-mail: fofanov@theor.mephi.ru
|
---
abstract: 'To investigate the potential connection between the intense X-ray emission from young, low-mass stars and the lifetimes of their circumstellar, planet-forming disks, we have compiled the X-ray luminosities ($L_X$) of M stars in the $\sim$8 Myr-old TW Hya Association (TWA) for which X-ray data are presently available. Our investigation includes analysis of archival [*Chandra*]{} data for the TWA binary systems TWA 8, 9, and 13. Although our study suffers from poor statistics for stars later than M3, we find a trend of decreasing $L_X/L_{bol}$ with decreasing $T_{eff}$ for TWA M stars wherein the earliest-type (M0–M2) stars cluster near $\log{(L_X/L_{bol})} \approx -3.0$ and then $\log{(L_X/L_{bol})}$ decreases, and its distribution broadens, for types M4 and later. The fraction of TWA stars that display evidence for residual primordial disk material also sharply increases in this same (mid-M) spectral type regime. This apparent anticorrelation between the relative X-ray luminosities of low-mass TWA stars and the longevities of their circumstellar disks suggests that primordial disks orbiting early-type M stars in the TWA have dispersed rapidly as a consequence of their persistent large X-ray fluxes. Conversely, the disks orbiting the very lowest-mass pre-MS stars and pre-MS brown dwarfs in the Association may have survived because their X-ray luminosities and, hence, disk photoevaporation rates are very low to begin with, and then further decline relatively early in their pre-MS evolution.'
author:
- 'Joel H. Kastner, David A. Principe, Kristina Punzi, Beate Stelzer, Uma Gorti, Ilaria Pascucci, Costanza Argiroffi'
title: |
M Stars in the TW Hya Association:\
Stellar X-rays and Disk Dissipation
---
Introduction
============
Thanks to their low luminosities and close-in habitable zones — i.e., the range of exoplanet orbital semimajor axes where water may exist in liquid form and, as a result, life may eventually flourish — the lowest-mass (M-type) stars represent the best targets for future direct (imaging) giant planet searches and indirect (transiting) discovery and characterization of potentially habitable exoplanets. It is hence essential to establish, on both theoretical and observational grounds, whether planets are expected to be common around M stars. Based on the data available thus far, giant planets seem to be rare around M dwarfs, but terrestrial planets and super-Earths may be quite common [e.g., @Mulders2015a; @Mulders2015b]. Indeed, the occurrence rate of 1–4 $R_{\rm Earth}$ planets around M dwarfs appears to be higher than that around solar-mass stars [@Howard2012; @Mulders2015a; @Mulders2015b].
Meanwhile, the answer to the corresponding, fundamental theoretical questions — why should giant planets be rare and terrestrial planets common around M stars? — requires understanding the harsh conditions out of which such planets form. Low-mass, pre-main sequence (pre-MS) stars are characterized by intense high-energy radiation fields [e.g., @Preibisch2005; @Guedel2007]. This strong UV and X-ray emission has its origins in a combination of stellar magnetic and accretion activity. As low-mass pre-MS stars descend to the main sequence, their deep convective envelopes combine with differential rotation to produce strong magnetic dynamos and, hence, high levels of chromospheric and coronal activity; the former is a source of bright UV emission, while the latter generates strong X-ray emission [e.g., @Stelzer2013 and refs. therein]. In the case of T Tauri stars that are actively accreting from circumstellar disks, shocks at the bases of accretion columns can also significantly contribute to UV and soft X-ray emission [@Guenther2007; @Sacco2010].
The resulting irradiation of protoplanetary disks by high-energy stellar photons likely regulates exoplanet formation and evolution processes. Various theoretical studies have shown that EUV and X-ray radiation from young stars can drive disk dissipation and disk chemistry, thereby determining the timescale over which, and the conditions out of which, exoplanets and their atmospheres emerge [e.g., @Gorti2009; @Ercolano2009; @Glassgold2012; @Owen2012; @Walsh2012; @Cleeves2013; @Gorti2015]. In particular, the EUV and soft ($\stackrel{<}{\sim}$1 keV) X-ray radiation field of the central star should be a major source of disk surface heating and, as a result, potentially represents an important driver of slow, photoevaporative disk winds [@Gorti2009; @Ercolano2009; @Owen2012]. There is observational evidence for the presence of such stellar EUV/X-ray-generated photoevaporative disk winds in the case of relatively evolved pre-main sequence stars of roughly solar mass [@Pascucci2009; @Sacco2012; @ClarkeOwen2015]. Whether protoplanetary disks orbiting ultra-low-mass stars and brown dwarfs are similarly irradiated and actively photoevaporating remains to be determined [see, e.g., @Pascucci2013].
To improve our understanding of the potential effects of X-rays on planet formation in disks orbiting M stars, we must characterize the X-ray emission properties of the lowest-mass (mid- to late-type M type) pre-MS stars of age $>$3 Myr, i.e., the epoch during or just after giant planet building and just preceding terrestrial planet building. The TW Hya Association [TWA; @Kastner1997; @Webb1999] affords just such an opportunity, thanks to its mean distance of just $\sim$50 pc and age $\sim$8 Myr [@Torres2008; @Ducourant2014; @HerHill2015 and refs. therein]. Here, we present an analysis of all published and archival X-ray observations of M-type stars in the TWA in light of the presence or absence of evidence for gas and dust in circumstellar disks around these same stars, so as to further constrain the potential relationship between X-rays and disk dispersal.
TW Hya Association M stars: X-ray luminosities and disk detection rates
=======================================================================
The sample considered here (Table \[tbl:GenData\]) consists of those high-probability M-type members of the TWA for which archival or published X-ray data from ROSAT (All-Sky Survey; RASS), Chandra, or XMM-Newton were available at the time this paper was written, based on a search of the HEASARC `browse` search utility[^1] and the literature. Appendix A contains notes concerning the sample’s inclusion and exclusion of specific TWA members and candidate members. For each sample star, we list in Table \[tbl:GenData\] the spectral type and distance we have adopted from the literature, as well as 2MASS $J$ magnitude and bolometric luminosity $L_{bol}$. For known binaries that are unresolved in the available X-ray data (TWA 2AB, 3A, and 5A), we list composite spectral types and total $L_{bol}$ values. The listed spectral types are obtained from @Schneider2012 [and refs. therein], @Manara2013, and @HerHill2014. The spectral types of TWA 14 (M0.5), TWA 13AB (both M1), TWA 2A (M2), TWA 8A (M3) and TWA 8B (M5) listed in @Schneider2012 were confirmed in one or both of the @Manara2013 and @HerHill2014 studies, while one or both of these studies determined later spectral types for TWA 7 (M3), TWA 9B (M3), TWA 15A (M3.5), TWA 15B (M3), and TWA 25 (M0.5). We adopt these revised classifications here. Except where noted, the $L_{bol}$ values were obtained from the listed spectral types, stellar distances, and $J$ band data based on the (spectral-type-dependent) $J$ band bolometric corrections determined by @PecautMamajek2013, assuming no reddening.
X-ray luminosities
------------------
Table \[tbl:XrayData\] lists X-ray luminosities ($L_X$) for the sample stars. The values of $L_X$ in columns 3, 4 and 5 of Table \[tbl:XrayData\] are all calculated over the energy range 0.3–8.0 keV from available archival (HEASARC database) ROSAT All-sky Survey (RASS), XMM-Newton, and Chandra count rates, respectively, as described below. Note that whereas X-ray count rates (hence $L_X$ values) are available for all but one of the early-M (M0 to M2) TWA members from the RASS, only a few TWA stars of type M3 or later were detected in the RASS, as a consequence of its limited sensitivity. We hence do not consider RASS count rate upper limits in the present study, as these upper limits do not place meaningful constraints on $L_X$ for young mid- to late-M stars at the distance of the TWA [@Rodriguez2013].
To obtain the values of $L_X$ in columns 3–5, we converted the count rates to 0.3–8.0 keV X-ray fluxes ($F_X$) via the `webpimms` tool[^2], which accounts for the different energy sensitivities of the three missions (the Chandra and XMM-Newton X-ray Observatories and their back-illuminated CCD sensors cover the entire 0.3–8.0 keV range, whereas the Position Sensitive Proportional Counter aboard ROSAT was sensitive in the range $\sim$0.1-2.0 keV). We assumed a single-component absorbed thermal plasma model whose parameters are characteristic plasma temperature ($T_X$), metallicity relative to solar, and intervening absorbing column ($N_H$). We adopted parameter values of $kT_X = 1.0$ keV ($T_X \approx 12$ MK), $N_H = 10^{19}$ cm$^{-2}$, and a metallicity of 0.2. These choices for parameter values are based on the results of model fitting to the XMM-Newton spectra of TWA 11B [@Kastner2008 and refs. therein] and TWA 30A [@Principe2015], as well as the results of Chandra X-ray spectral fitting presented in Appendix B for the individual components of the binary systems TWA 8AB, 9AB, and 13AB. For these model parameter values, the count rate to $F_X$ conversion factors are $6.3\times10^{-12}$, $3.2\times10^{-12}$, and $4.1\times10^{-12}$ erg cm$^{-2}$ count$^{-1}$ for ROSAT, XMM-Newton, and Chandra, respectively. For binaries that are unresolved by ROSAT (TWA 3AB, 8AB, 9AB, and 13AB), we have split the ROSAT-based $F_X$ (column 3) equally between components. We then calculated $L_X$ from the values of $F_X$, assuming the distances listed in Table \[tbl:GenData\].
It is evident from Table \[tbl:XrayData\] that all of the stars for which data is available from multiple observatories (i.e., multiple epochs) show variability at the level of a factor $\sim$2–3, as is typical for low-mass pre-MS stars [e.g., @Guedel2007; @Principe2014]. The most extreme case in Table \[tbl:XrayData\] is the M5 star TWA 30A, which displayed at least a factor $\sim$10 change (decrease) in $L_X$ between the (1990) RASS and (2011) XMM-Newton observing epochs [@Principe2015].
In columns 6 and 7 of the Table we list the values of $L_X$ and $\log{(L_X/L_{bol})}$, respectively, that we have adopted for the analysis described in §3. Wherever possible, these adopted values of $L_X$ were obtained from analyses of Chandra or XMM-Newton spectra. Specifically, for the binary systems TWA 8, 9, and 13, the $L_X$ values were determined from our analysis of archival Chandra data (Appendix B), while for the stars TWA 5A, 5B, 7, 11C, 26, 27, 28, 30A, and 30B, the adopted $L_X$ values in column 6 were obtained from the literature (see references listed in column 8 of Table \[tbl:XrayData\]). In adopting a value of $L_X$ for each star, Chandra results have been given priority, since Chandra most reliably resolves the X-ray counterparts to TWA binary systems. In cases where both XMM-Newton and RASS observations (but no Chandra data) are available, we adopt the XMM-based $L_X$. Comparing the values of $L_X$ calculated for TWA 5A, 7, 13A and 13B based on the single-component `webpimms` model (column 5) with values obtained from spectral modeling of the same (Chandra) data (column 6), it appears that our use of the `webpimms` model may systematically overestimate $F_X$ by $\sim$30%. This potential level of systematic error does not affect the results described in §3.
Presence or absence of circumstellar disks
------------------------------------------
In the last column of Table \[tbl:GenData\] we indicate whether each sample star displays a mid-infrared excess indicative of the presence of warm ($T \sim$ 100–300 K) dust in a circumstellar disk. These assessments of the presence of absence of dusty disks are drawn from the analysis of WISE data by @Schneider2012, with the exception of TWA 14, 15AB, and 21 (which were omitted from their sample; see Appendix A). The WISE point source catalog $W1-W4$ colors of these three stars are 0.25, 0.20, and 0.08, respectively (where the TWA 15AB binary is unresolved by WISE), all of which lie well within the locus of TWA star colors for stars that lack IR excesses [@Schneider2012]. The mid-IR excess stars TWA 3A, 7, 30A, 30B, and 31 also display relatively strong far-IR fluxes [@Riviere2013; @Liu2015]. Reanalysis of archival HST imaging has yielded a direct detection of the TWA 7 dust disk via scattered starlight; TWA 25 also displays a compact, nearly edge-on dust disk in HST imaging despite its lack of detectable mid-IR excess [@Choquet2015].
Of the seven Table \[tbl:GenData\] stars that display mid-IR excesses, all but one [TWA 7; @Manara2013] also show evidence for active accretion of gas from their disks in the form of unusually strong, broad H$\alpha$ emission [see §3.2 and @Muzerolle2000; @Stelzer2007; @Looper2010; @Shkolnik2011]. The variable, weak accretors TWA 27 and 28 also display He [i]{} line emission [@Herczeg2009]. In the case of the $\sim$1.4$''$ separation binary TWA 3AB, Chandra X-ray spectroscopy (and large UV excess) also indicate the presence of accretion shocks in this system, with the X-ray signature of accretion more evident at the dusty binary component [TWA 3A; @Huenemoerder2007]. But the most extreme examples of gas-rich disks among the TWA M stars with mid-IR excesses are the two components of the wide binary TWA 30A and 30B. These star/disk systems, both of which are evidently viewed nearly edge-on, display evidence for the presence of circumstellar gas in the form of forbidden emission lines detected via optical spectroscopy [@Looper2010] and, in the case of TWA 30A, in the form of attenuation of stellar X-rays [@Principe2015]. The relatively weak H$\alpha$ emission seen toward TWA 30A and 30B (emission-line equivalent widths $<10$Å) can be ascribed to their viewing geometry, which in each case at least partially obscures the regions of active accretion onto the star [@Looper2010; @Looper2010a]. Furthermore, during a photometric monitoring campaign, TWA 30A was seen to display quasi-periodic dips in its multi-band optical and near-IR lightcurves, bolstering the @Looper2010a model invoking disk structures that rotate in and out of the line of sight, temporarily obscuring the star [@Stelzer2015].
Knowledge of the presence or absence of cold ($T < 100$ K) circumstellar dust and gas is scant for the Table \[tbl:GenData\] stars. Only three of these stars — TWA 30A, 30B, and 31 — have been the subject of sensitive mm- and submm-wave continuum and CO observations [with the Atacama Large Millimeter Array; @Rodriguez2015]. All three display mid- to far-IR excesses indicative of warm dust [@Schneider2012; @Liu2015]. However, only TWA 30B was detected as a submm continuum source, and none of the three were detected as CO sources, in the @Rodriguez2015 ALMA survey of ultra-low-mass members and candidate members of the TWA. The CO nondetections imply the gas disks orbiting these three stars have very low masses and/or small radii [$\stackrel{<}{\sim}$0.1 Earth masses and/or $\stackrel{<}{\sim}$10 AU, respectively; @Rodriguez2015].
Trends in X-ray luminosity and disk fraction
--------------------------------------------
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In Fig. \[fig:lxlbolsptype\], we plot the log of the ratio $L_X/L_{bol}$ as a function of M spectral subtype for the TWA M stars. A trend is apparent, in which the earliest-type (M0–M2) stars cluster near $\log{(L_X/L_{bol})} \approx -3.0$, and then $\log{(L_X/L_{bol})}$ decreases, and its distribution broadens, for spectral types M4 and later. As Fig. \[fig:lxlbolsptype\] demonstrates, such a trend of decreasing $\log{(L_X/L_{bol})}$ with M subtype is also apparent for the younger pre-MS M stars in the Taurus molecular clouds [median age $\sim$0.5 Myr; @Grosso2007] and for M stars in the young cluster IC 348 [age $\sim$3 Myr; @Stelzer2012]. The decline of $\log{(L_X/L_{bol})}$ with decreasing M star effective temperature appears to be somewhat steeper in the TWA than in Taurus, and is similar to IC 348 in the range M0–M5. The $L_X/L_{bol}$ ratios of early-M TWA stars appear to be systematically larger than those of their counterparts in Taurus and IC 348 (but see below).
In Fig. \[fig:lxlbol\], we plot $L_X$ vs. $L_{bol}$ for the TWA M stars, and overlay the empirical relationships determined for pre-MS low-mass stars and brown dwarf candidates in Taurus [@Grosso2007] and IC 348 [@Stelzer2012]. As was also clear in Fig. \[fig:lxlbolsptype\] (top panel), it seems that the more luminous early-M type TWA stars (i.e., those with $L_{bol} > 0.1$ $L_\odot$) are overluminous in X-rays, relative to Taurus TTS in this same range of bolometric luminosity. The mid- to late-M type (lowest-luminosity) TWA stars (those with $L_{bol} < 0.02$ $L_\odot$) also appear underluminous in X-rays relative to their Taurus counterparts, although the comparison in this $L_{bol}$ range suffers both from the small number statistics of our TWA sample and a lack of X-ray detections for the Taurus sample [@Grosso2007].
The latter caveat notwithstanding, we find the slope of $\log{L_X}$ vs. $\log{L_{bol}}$ for the X-ray-detected TWA M stars, $1.41 \pm 0.14$, is significantly steeper than that determined for the $\sim$0.5 Myr-old M stars in Taurus [0.98$\pm$0.06 for X-ray-detected stars; @Grosso2007], and that the zero-$\log{(L_{bol}/L_\odot)}$ intercept for the TWA sample, $\log{L_{X,0}} = 30.73$, is larger than found for Taurus [$\log{L_{X,0}} = 30.06$; @Grosso2007]. The latter comparison may reflect the fact that there are no known examples of actively accreting stars in the TWA in the M0–M2 spectral type range (see below and §3.1). Indeed, the X-ray luminosities of early-type M stars in the TWA are similar to those of Class III (diskless) early-M stars in IC 348 (Fig. \[fig:lxlbolsptype\], bottom panel). As illustrated in Fig. \[fig:lxlbol\], the slope of $\log{L_X}$ vs. $\log{L_{bol}}$ we find for the TWA — which includes stars with and without disks, and spans the range $L_{bol} =$ 0.015...0.3 $L_\odot$ — is intermediate between the slopes determined for Class II and Class III pre-MS stars in IC 348 in the range $L_{bol} =$ 0.03...3.0 $L_\odot$, i.e., 1.13$\pm$0.11 and 1.65$\pm$0.22, respectively [where these determinations take into account upper limits on $L_X$; @Stelzer2012].
The overall disk fraction among M-type TWA members with available X-ray data (Table \[tbl:GenData\]) is $\sim$25% (7 of 28, excluding the two stars for which the presence or absence of disks cannot be established due to the proximity of IR-bright companions). This disk fraction is similar to but somewhat smaller than that inferred for TWA members overall [@Schneider2012]. However, Figs. \[fig:lxlbolsptype\] and \[fig:lxlbol\] also make apparent that the fraction of TWA M stars with evidence for circumstellar disks increases as stellar mass decreases. Specifically, 6 of 11 stars with spectral types M3.5 and later ($\sim$50%) have detectable warm circumstellar dust. All 6 of these stars also display evidence for circumstellar gas (§2.2). In contrast, the disk fraction among stars of type M3 and earlier drops to $\sim$5%, i.e., one in 17, with the lone exception being the debris disk orbiting the M3 star TWA 7. The apparent jump in disk fraction hence occurs very near the same (M3/M4) spectral type boundary where $L_X/L_{bol}$ appears to decline (Fig. \[fig:lxlbolsptype\]). @Schneider2012 noted a similar trend in which candidate substellar members of the TWA are more likely to display evidence for disks than are stellar members.
Discussion
==========
Implications for the early evolution of X-ray emission in M stars
-----------------------------------------------------------------
The comparison of the $\log{(L_X/L_{bol})}$ and $L_X$ distributions of the TWA M stars with the corresponding distributions for the much younger pre-MS M stars in Taurus (Figs. \[fig:lxlbolsptype\], \[fig:lxlbol\]) suggests that over the first $\sim$8 Myr of the lifetime of an early-M pre-MS star — an epoch during which $L_{bol}$ is monotonically decreasing — coronal X-ray luminosity remains roughly constant or perhaps even increases. In contrast, based on the sparse data presently available for mid- to late-M stars in the TWA, it appears that X-ray luminosity has decreased at least as fast as — and, in at least some cases, much faster than — bolometric luminosity for these stars (Fig. \[fig:lxlbolsptype\]). It is possible that gas in the disks orbiting the mid- to late-type M type TWA stars is attenuating their X-ray emission. However, such an explanation for the smaller values of $\log{(L_X/L_{bol})}$ observed for these stars appears unlikely, since the majority of the TWA M stars with disks do not show evidence for extinction of their photospheres by disk dust — the notable exceptions being TWA 30A and 30B [@Looper2010; @Looper2010a; @Principe2015; @Stelzer2015].
The X-ray data presently available for the TWA, in combination with similar studies for Orion, Taurus, and IC 348 [@PreibischEtal2005; @Grosso2007; @Stelzer2012], thereby potentially shed further light on the age at which coronal activity declines for stellar masses near the main sequence H-burning limit of $\sim$0.08 $M_\odot$. Such a decline is apparent when comparing the X-ray luminosities of very young (age $\stackrel{<}{\sim}$1 Myr) ultra-low-mass stars and brown dwarfs with those of (old) late-M and L-type field stars. Specifically, X-ray-detected pre-MS brown dwarf candidates in Orion and Taurus have typical X-ray luminosities $L_X \sim 10^{28}$ erg s$^{-1}$ [@PreibischEtal2005; @Grosso2007], whereas $L_X \stackrel{<}{\sim} 10^{26}$ erg s$^{-1}$ for ultra-low-mass stars and brown dwarfs in the field [e.g., @Stelzer2006; @Williams2014 and refs. therein].
We note that the spectral type boundary where the ($\sim$8 Myr-old) TWA stars appear to display a decline in $L_X/L_{bol}$ — i.e., near M4 — approximately corresponds to the dividing line between pre-MS stars that will and will not undergo core H burning once on the main sequence, according to evolutionary models [e.g., @DAntona1997]. However, this is likely just a coincidence. In very low mass field stars (with ages of order Gyr), a drop in magnetic activity is observed around M8 [e.g., @West2004; @Berger2006; @Stelzer2006; @Berger2010], i.e., at an effective temperature well above the H-burning limit. The likely cause of this decrease of X-ray (and H$\alpha$) activity is the low level of ionization within the cool atmospheres of the latest-type M stars [e.g., @Mohanty2002].
The same (low ionization) effect also might play a role in the coolest (i.e., M8–M9) objects among our TWA sample, but would not explain the apparent low X-ray activity levels of three of the six mid-M stars (TWA 30A, 30B, and 31; Table \[tbl:XrayData\]). In that regard, it is interesting that all three of these stars are actively accreting from their dusty disks (§ 2.3). The apparent bifurcation of $L_X/L_{bol}$ in this spectral type range that is hinted at by Fig. \[fig:lxlbolsptype\] may therefore reflect the suppression of coronal X-ray emission by accretion. Alternatively, the internal (e.g., convective and/or rotational) structures of young, ultra-low-mass stars may differ fundamentally from those of higher-mass young stars [see discussion in @Stelzer2012 and refs. therein]. X-ray observations of additional mid- to late-type M stars of age $\sim$10 Myr, combined with spectroscopic diagnostics of mass accretion rates, will be necessary to distinguish between these different scenarios for the apparent overall drop in $L_X/L_{bol}$ for such stars at this age.
Implications for disk dispersal via photoevaporation
----------------------------------------------------
It is difficult to establish whether the dust responsible for the mid-IR excesses associated with M stars in the TWA resides in primordial (gas-rich) disks or (gas-poor) debris disks. Due to their small radial extent and low masses, the gaseous constituents of these disks are exceedingly difficult to detect in (sub)mm-wave molecular line emission. Indeed, as noted in §2.2, gas has been detected in the disks orbiting TWA 30A and 30B in the form of optical forbidden-line emission from disk winds and jets [@Looper2010a; @Looper2010] as well as X-ray absorption by the (nearly edge-on) TWA 30A disk itself [@Principe2015]; however, a sensitive search for CO in these same disks with ALMA yielded negative results [@Rodriguez2015].
Nonetheless, there is reason to conclude that almost all of the TWA M stars with mid-IR excesses due to warm dust in fact host primordial disks that, given the $\sim$8 Myr age of the TWA, are highly evolved and in the process of dispersing. Specifically, a majority of these stars display H$\alpha$ emission indicative of active accretion of disk gas (Table \[tbl:accretion\]) and \[O [i]{}\] $\lambda$6300 emission indicative of residual disk gas that is either orbiting or flowing out in a disk wind or jets. The components of the TWA 30 binary represent extreme cases in terms of \[O [i]{}\] equivalent width [@Looper2010], but \[O [i]{}\] emission has also been detected from TWA 3A, 27, and 28 [@Herczeg2009]. Among the TWA M stars with mid-IR excesses, only TWA 7 appears to harbor a gas-poor debris disk [@Riviere2013; @Choquet2015].
If most of the disks orbiting M stars in the TWA are indeed primordial, then the anticorrelation between X-ray luminosity and disk frequency noted in §2.3 (and apparent in Figs. \[fig:lxlbolsptype\], \[fig:lxlbol\]) may have implications for models describing how high-energy photons from low-mass stars can play a central role in driving disk dispersal via photoevaporation [@Gorti2009; @Ercolano2009; @Ercolano2010; @Owen2012; @Gorti2015 and refs. therein]. Specifically, models formulated by @Owen2012 [and refs. therein] predict that the rate of stellar-X-ray-induced disk gas photoevaporation $\dot{M}_X$ is directly proportional to the stellar X-ray luminosity $L_X$, $$\dot{M}_X = 8\times10^{-9} L_{X30} \: M_\odot \; {\rm yr}^{-1},$$ where $L_{X30}$ is X-ray luminosity integrated over the energy range 0.1–10 keV in units of $10^{30}$ erg s$^{-1}$ assuming a relatively hard stellar X-ray spectral energy distribution (see below).
Eq. 1 predicts that, given their present X-ray luminosities ($L_{X30} \approx 1.0$; Fig. \[fig:lxlbol\]), $\dot{M}_X\approx10^{-8}$ $M_\odot$ yr$^{-1}$ for early-M stars in the TWA. The predicted present-day photoevaporative mass loss rates for stars of type M4 and later, for which $L_{X30} \stackrel{<}{\sim} 0.1$, are at least an order of magnitude smaller. Viewed in this context, the results in Figs. \[fig:lxlbolsptype\], \[fig:lxlbol\] could be taken to suggest that primordial disks orbiting early-type M stars have dispersed rapidly as a consequence of their persistent large X-ray fluxes, while disks orbiting the very lowest-mass pre-MS stars and pre-MS brown dwarfs can survive to ages $\sim$10 Myr because their X-ray luminosities — and, hence, disk photoevaporation rates — are very low to begin with, and then further decline relatively early in their pre-MS evolution. Such an interpretation of Fig. \[fig:lxlbol\] would be consistent with the suggestion that the well-established anticorrelation of $L_X$ and accretion rate in T Tauri stars [e.g., @Stelzer2012 and refs. therein] is indicative of “photoevaporation-starved T Tauri accretion” [@Drake2009].
This simple interpretation is subject to several caveats. First, it requires that the values of $L_{X}$ (hence $\dot{M}_X$) presently measured for TWA M stars are, at least in relative terms, representative of these values over most of the lifetime of the TWA. The comparisons with pre-MS M stars in Taurus (age $\stackrel{<}{\sim}$1 Myr) and IC 348 (age $\sim$3 Myr) presented in Figs. \[fig:lxlbolsptype\], \[fig:lxlbol\] and discussed in §2.3 would appear to support such an assertion.
Second, because soft photons dominate the disk surface heating, Eq. 1 would yield inaccurate estimates of $\dot{M}_X$ for those stars whose X-ray spectral energy distributions (SEDs) differ significantly from the irradiating spectrum assumed in the @Owen2012 models. This model X-ray spectrum spans a range of plasma temperatures, peaking at $\log{T_X ({\rm K})} \approx7.3$. It is hence somewhat harder than the observed X-ray spectral distributions of TWA M stars (see §2.2 and Appendix B). If so, the predicted mass loss rates obtained via Eq. 1 would be systematically underestimated. On the other hand, the X-ray spectral fitting results obtained for the three TWA M star binaries analyzed here (see Appendix B) hint at the likelihood that $T_X$ is correlated with stellar mass [see also @Johnstone2015], and that the plasma emission model adopted by @Owen2012 more closely resembles the plasma parameters that are characteristic of early-M stars. If so, then Eq. 1 would be most directly applicable to these (higher-mass) M stars, but would tend to underestimate $\dot{M}_X$ for mid- to late-M stars.
Third, comparison of the theoretically predicted disk mass loss rates and mass loss timescales for accretion vs. EUV- and X-ray-driven photoevaporation indicates that photoevaporation should only become the dominant disk gas dispersal mechanism very late in disk evolution, after the bulk of the initial disk mass has already been lost via accretion [@Owen2012; @Gorti2015 and references therein]. As a preliminary evaluation of the relative importance of accretion vs. photoevaporation for the TWA stars that appear to retain residual gaseous disks, we compare in Table \[tbl:accretion\] mass accretion diagnostics (i.e., the 10% width of H$\alpha$ emission) and inferred mass accretion rates ($\dot{M}_{acc}$) with predicted values of (or upper limits on) $\dot{M}_X$ as obtained from Eq. 1. Based on the predictions for $\dot{M}_X$, it seems that the mass loss rate due to photoevaporation may equal or perhaps exceed that due to accretion in the cases of TWA 3A and 30A. In contrast, the inferred accretion rate of TWA 31 is at least two orders of magnitude larger than its predicted X-ray-driven photoevaporative mass loss rate. The comparisons of $\dot{M}_{acc}$ with $\dot{M}_X$ for the remaining three stars, all of which are undetected in X-rays, are inconclusive.
Summary and Conclusions
=======================
To investigate the potential connection between the intense X-ray emission from young, low-mass stars and the lifetimes of their circumstellar, planet-forming disks, we have compiled a list of the X-ray luminosities ($L_X$) of all presently known M-type members of the 8 Myr-old TW Hya Association (TWA). The values of $L_X$ are mainly drawn from available archival data and published values in the recent literature. We have obtained $L_X$ for the individual components of the binary star systems TWA 8, 9, and 13 via analysis of archival [*Chandra*]{} data (Appendix B).
Although our study suffers from poor statistics for stars later than M3, we find a trend of decreasing $L_X/L_{bol}$ with decreasing $T_{eff}$ for TWA M stars wherein the earliest-type (M0–M2) stars cluster near $\log{(L_X/L_{bol})} \approx -3.0$ and then $\log{(L_X/L_{bol})}$ decreases, and its distribution broadens, for types M4 and later. These mid- to late-M TWA stars generally appear underluminous in X-rays relative to very young pre-main sequence stars of similar spectral type and luminosity, consistent with previous studies indicating that mean $L_X$ declines more rapidly for ultra-low-mass stars and brown dwarfs than for early-M stars [e.g., @Stelzer2012]. This apparent decline of $L_X/L_{bol}$ near a spectral type of M4, if real, may reflect either the suppression of coronal X-ray emission by accretion or a fundamental difference between the internal structures of ultra-low-mass pre-MS stars and earlier-type pre-MS M stars.
Notably, the fraction of TWA stars with evidence for residual primordial disk material also sharply increases for subtypes of M4 and later, i.e., near the stellar effective temperature where $L_X$ decreases. Furthermore, most of the newly discovered TWA candidates that are of late-M and L-type also display evidence for disks [@Rodriguez2015]. The extant data for the TWA hence suggest that disk survival times may be longer for ultra-low mass stars and brown dwarfs than for higher-mass (early-type) M stars — a result that would be consistent with studies of the Upper Sco region [age $\sim$10 Myr; @Scholz2007; @Luhman2012 and refs. therein]. These observations have interesting implications for models of disk evolution, which generally predict that any dependence of disk lifetime on stellar mass should be very weak [e.g., @Gorti2009; @Owen2012].
The apparent anticorrelation between the X-ray luminosities of low-mass TWA stars and the longevities of their circumstellar disks (Figs. \[fig:lxlbolsptype\], \[fig:lxlbol\]) could be interpreted to indicate that the persistent large X-ray fluxes from early-type M stars in the Association have contributed to the rapid dispersal of their primordial disks. Conversely, disks orbiting the lowest-mass pre-MS stars and pre-MS brown dwarfs in the TWA may have survived because their relatively small X-ray luminosities have resulted in overall low disk photoevaporation rates.
In the specific cases of TWA 3A and 30A, we infer that the rate of mass loss due to X-ray-driven photoevaporation may exceed that due to accretion. The dominant role of photoevaporation in the dispersal of these disks, if confirmed, would be consistent with their advanced ages. However, the (X-ray-undetected) star TWA 31 appears to be accreting disk mass far more rapidly than it could be losing disk mass loss via photoevaporation.
X-ray observations of additional mid- to late-type M stars in the TWA [e.g., @Rodriguez2015] and M-type members of similarly nearby young stellar groups, combined with acquisition of spectroscopic diagnostics of the disk gas masses and mass accretion rates of these same stars, are required to verify and further investigate the apparent coincidence of a dropoff of $L_X/L_{bol}$ and increase in primordial disk fraction for ultra-low-mass stars and brown dwarfs at an age of $\sim$10 Myr. High-quality X-ray spectra of selected stars spanning the full range of M spectral types would also better inform photoevaporation models, by constraining the hardness of the X-ray radiation that is incident on protoplanetary disks orbiting the lowest-mass stars.
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Appendix A: Notes on Inclusion and Exclusion of Individual Stars {#appendix-a-notes-on-inclusion-and-exclusion-of-individual-stars .unnumbered}
================================================================
TWA 6:
: This star was classified as K7 by @Webb1999 and @Manara2013 and as M0 by @Torres2006 and @HerHill2014. We include TWA 6 in the sample, though we note that it appears to be at the K/M spectral type borderline.
TWA 9:
: As @Weinberger2013 and @PecautMamajek2013 have noted, the [*Hipparcos*]{} parallax of TWA 9 casts some doubt on its membership in the TWA. Based on other considerations (e.g., Li line strength, proper motion, kinematic distance, location relative to other TWA members), @PecautMamajek2013 conclude that TWA 9 is indeed likely a member of the Association, and that the parallax measurement is spurious. Hence, we retain TWA 9B in our TWA M star sample. As the higher-mass component TWA 9A is a mid-K star [@Webb1999], our consideration of this component is restricted to the X-ray spectral analysis presented in Appendix B.
TWA 14, 15, 21:
: @Schneider2012 [and refs. therein] conclude these M stars are unlikely to be TWA members, and do not include them in their WISE-based assessment of the presence or absence of IR excesses. However, @Ducourant2014 find these systems are indeed likely TWA members, based on convergent point analysis, so we include them in our sample.
TWA 18:
: Several studies have concluded that this system is unlikely to be a TWA member [@Torres2008; @Schneider2012; @Ducourant2014], so we have excluded it from our sample.
TWA 22:
: @Mamajek2005 finds a low probability that TWA 22 (M5) is a TWA member; @Teixeira2009 concluded this star is a member of the $\beta$ Pic Moving Group (rather than the TWA); and @Ducourant2014 find that TWA 22 fails their convergence point membership analysis. Although @Schneider2012 list this system as a possible member of the TWA, we exclude TWA 22 from our analysis.
TWA 31:
: @Schneider2012 list this system as a possible member of the TWA, while @Ducourant2014 find that it fails their convergence point membership analysis. While this star could hence be unrelated to the Association (at 110 pc, it is one of the more distant candidates), it displays strong, broad H$\alpha$ emission indicative of ongoing accretion [@Shkolnik2011], suggesting it is not merely a young field star that happens to lie in the direction of the TWA. We retain TWA 31 in our sample.
Appendix B: Chandra X-ray Spectral Analysis for TWA 8, 9 and 13 {#appendix-b-chandra-x-ray-spectral-analysis-for-twa-8-9-and-13 .unnumbered}
===============================================================
We have performed analysis of archival Chandra X-ray data for the M binary systems TWA 8, 9, and 13 [@Webb1999; @Sterzik1999]. Five of the six components of these systems are M-type stars (Table 1), the lone exception being the K5 star TWA 9A. The binary separations of TWA 8, 9 and 13 are $\sim$13$''$, $6''$ and 5$''$, respectively. A summary of the available Chandra observations of the three systems is listed in Table \[tbl:XRayObs\]. All of these data were obtained using CCD S3 of the Advanced CCD Imaging Spectrometer (ACIS) array. The resulting Chandra/ACIS-S3 images are displayed, alongside 2MASS J band images of the systems, in Fig. \[fig:TWAimages\].
The pipeline-processed data files provided by the Chandra X-Ray Center were analyzed using standard science threads with CIAO version 4.7[^3]. The CIAO processing used calibration data from CALDB version 4.6.5. Spectra (and associated calibration data) were extracted within circular regions with diameters of $\sim$3–$8''$ centered on the stellar X-ray sources (see Fig. \[fig:TWAimages\]). Background spectra were extracted within circular regions from nearby, source-free regions. The background-subtracted source count rates are listed in Table \[tbl:XRayObs\].
Spectral fitting was performed with the HEASOFT *Xanadu*[^4] software package (version 6.16) using XSPEC[^5] version 12.8.2. We adopted the XSPEC optically thin thermal plasma model `vapec`, whose parameters are the plasma abundances, temperature, and emission measure (via the model normalization). The potential effects of intervening absorption were included via XSPEC’s `wabs` absorption model, although we found the absorbing column $N_H$ to be negligible in all cases (with the possible exception of TWA 13A, for which we find $N_H \sim 1.5\times10^{18}$ cm$^{-2}$). We found that two temperature components are required to obtain acceptable fits to the spectra of four of the six stars, based on $\chi^{2}$ statistics (the exceptions being the relatively faint sources TWA 8B and 9B). Initially, we fixed the parameters for the plasma abundances to values that have been determined to be typical of T Tauri stars in Taurus [@Skinner2013 and references therein]. These abundance values (relative to solar) are H = 1.0, He = 1.0, C = 0.45, N = 0.79, O = 0.43, Ne = 0.83, Mg = 0.26, Al = 0.50, Si = 0.31, S = 0.42, Ar = 0.55, Ca = 0.195, Fe = 0.195, and Ni = 0.195. We then allowed the abundances of Ne and Fe to be free parameters, since emission lines of these elements (plus O) tend to dominate the X-ray spectra of T Tauri stars in the $\sim$0.5–2.0 keV energy range [@Kastner2002]. Apart from the cases of TWA 8B and 9B, we find that leaving the Ne and Fe abundances free marginally improves the fit (slightly lowers $\chi^{2}$). In all cases, however, leaving Ne and Fe free only affects the results for X-ray flux (hence $L_X$) at the level of a few percent.
The results of the spectral analysis are presented in Table \[tbl:SpecAnalysis\] and Fig. \[fig:TWAfits\]. The values for $L_X$ over the energy range 0.3–8.0 keV obtained from the fits are within $\sim$30% of those determined by @Brown2015 in all cases. We find that characteristic plasma temperature is correlated with $L_X$ for this small sample of TWA member stars. Specifically, the weakest X-ray sources (TWA 8B and 9B) have best-fit plasma temperatures $T_X\sim8$ MK, while the brighter sources (TWA 8A, 9A, 13A, and 13B) have characteristic (emission-measure-weighted) plasma temperatures $T_X\sim15$ MK. The results also indicate that TWA 8A may depart from the “standard” T Tauri star abundance pattern, in that it appears deficient in Ne and significantly overabundant in Fe. The other three stars for which we can extract plasma abundance results with some confidence — TWA 9A, 13A, and 13B — show marginal enhancements of Ne relative to the “standard” T Tauri star abundance.
{height="2.5in"} {height="2.5in"} {height="2.5in"}
{width="2.5in"} {width="2.5in"} {width="2.5in"} {width="2.5in"} {width="2.5in"} {width="2.5in"}
[^1]: https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl
[^2]: https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl
[^3]: http://asc.harvard.edu/ciao.
[^4]: http://heasarc.gsfc.nasa.gov/docs/xanadu/xanadu.html.
[^5]: http://heasarc.gsfc.nasa.gov/xanadu/xspec.
|
---
abstract: 'Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensates are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, results in very good qualitative and also reasonable quantitative agreement with the numerical findings.'
author:
- Wenlong Wang
- 'P.G. Kevrekidis'
- 'R. Carretero-Gonz[á]{}lez'
- 'D. J. Frantzeskakis'
- 'Tasso J. Kaper'
- Manjun Ma
title: '**Stabilization of ring dark solitons in Bose-Einstein condensates**'
---
Introduction
============
Over the past few years, there has been an intense research interest, not only theoretically, but also experimentally, in the physics of atomic Bose-Einstein condensates (BECs) [@book1; @book2], and particularly in the study of nonlinear waves [@emergent]. Bright [@expb1; @expb2; @expb3], dark [@djf] and gap [@gap] matter-wave solitons, as well as vortices [@emergent; @fetter1; @fetter2], solitonic vortices and vortex rings [@komineas_rev] are only some among the many structures studied (including more exotic ones such as Skyrmions [@bigelow] or Dirac monopoles [@david]).
One of the most prototypical excitations that have been intensely studied in experiments are dark solitons [@djf]. While the early experiments in this theme were significantly limited by dynamical instabilities and thermal effects [@han1; @nist; @dutton; @han2; @nsbec], more recent efforts have been significantly more successful in generating and exploring these structures. By now, the substantial control of the generation, and dynamical interactions of such structures has led to a wide range of experimental works monitoring their evolution in different settings [@engels; @Becker:Nature:2008; @hambcol; @kip; @andreas; @jeffs].
The instability of dark solitons in higher dimensions (towards bending [@nist] and eventual snaking towards vortices/vortex rings [@nsbec; @beckerus]) has been one of the key reasons for the inability to study such states in higher dimensions. Although external stabilization mechanisms, e.g., utilizing a blue-detuned laser beam [@manjun], have been proposed, importantly also variants of such dark solitons have been explored in higher dimensions in the form of [*ring dark solitons*]{} (RDSs). These efforts were at least in part motivated by works in nonlinear optics, where they initially were introduced in Ref. [@KYR], and studied in detail, both theoretically (in conservative [@thring; @djfbam; @ektoras] and —more recently— in dissipative [@dum] settings) and experimentally [@expring1]. In turn, RDSs in BECs were originally proposed in Ref. [@rds2003] and their dynamics was analyzed by means of the perturbation theory of dark matter-wave solitons [@djf]. In other works, RDSs were studied by different approaches, e.g., in a radial box [@ldc1], by using a quasi-particle approach [@kamch], or by considering them as exact solutions in certain versions of the Gross-Pitaevskii equation (GPE) [@toik1]. Proposals for the creation of RDS, e.g., by means of BEC self-interference [@ch1] or by employing the phase-imprinting method [@song], as well as generalizations of such radial states (including multi-nodal ones) [@ldc1; @herring] have also been considered. Moreover, generalizations of RDSs were studied in multi-component settings, in the form of dark-bright ring solitons [@stockhofe] (emulating the intensely studied context of multi-component one-dimensional (1D) dark-bright solitons [@buschanglin; @sengdb; @peter1; @peterprl]), or in the form of vector RDS in spinor $F=1$ BECs [@song]. Importantly, structures of the form of radially symmetric dark solitons, closely connected to RDSs exist also in three-dimensions with a spherical rather than cylindrical symmetry (so-called “spherical shell solitons” [@ldc1]). Nevertheless, in none of these contexts (either one- or multi-component), was it possible to achieve complete stabilization of the RDSs. In particular, stabilization mechanisms that have been proposed, e.g., by “filling” the RDS by a bright soliton component [@stockhofe] or by employing the nonlinearity, management (alias “Feshbach resonance management” [@FRM]) technique [@ch2], were only able to prolong the RDSs’ life time. In fact, it was illustrated that the instabilities of the ring-shaped solitons were connected, bifurcation-wise, to the existence of vortex “multi-poles”, such as vortex squares (which are generically stable in evolutionary dynamics), vortex hexagons, octagons, decagons etc.; all of these states are progressively more unstable. This picture has been corroborated by detailed numerical computations in Ref. [@middelphysd]. It is our aim in this work to revisit the RDSs and their destabilization mechanisms and, indeed, to propose a technique for their complete dynamical stabilization. Our technique is reminiscent of that of Ref. [@manjun] in that we introduce a potential induced by a [*radial*]{} blue-detuned laser beam. Radial potentials of a similar form have been intensely used in recent experiments, e.g., by the groups of [@gretchen] and [@boshier] and are hence accessible to state-of-the-art experimental settings.
Our presentation of this effort to stabilize the RDS in the form of a dynamically robust state of quasi-two-dimensional BECs can be summarized as follows: we introduce, in Sec. \[setup\], the mathematical model and our specific proposal towards a potential stabilization of the RDS. We also incorporate in this Section theoretical attempts to explore the coherent dynamics of the ring soliton by means of a particle model. Our numerical results are presented in Sec. \[results\], initially revisiting (for reasons of completeness and to facilitate the exposition) the case without the external radial barrier potential and subsequently incorporating it in the picture. Finally, our concluding remarks are presented in Sec. \[conclusion\], and a number of important open future directions is also highlighted.
Model and mathematical set-up {#setup}
=============================
The Gross-Pitaevskii equation
-----------------------------
In the framework of lowest-order mean-field theory, and for sufficiently low-temperatures, the dynamics of a quasi-2D (pancake-shaped) BEC confined in a time-independent trap $V(r)$ is described by the following dimensionless GPE [@emergent]: $$i \psi_t=-\frac{1}{2} \nabla^2 \psi+V(r) \psi +| \psi |^2 \psi-\mu \psi,
\label{GPE}$$ where $\psi(x,y,t)$ is the macroscopic wavefunction of the BEC, $\mu$ is the chemical potential, and $V(r)$ (with $r=\sqrt{x^2+y^2})$ is the external potential. The latter, is assumed to be a combination of a standard parabolic (e.g., magnetic) trap, $V_{\rm MT}(r)$, and a localized radial “perturbation potential”, $V_{\rm pert}(r)$, namely: $$V(r) = V_{\rm MT}(r) + V_{\rm pert}(r) = \frac{1}{2} \Omega^2 r^2 + V_{\rm pert}(r),
\label{eq:VMT+Vpert}$$ with $\Omega$ being the effective strength of the magnetic trap. For the numerical results in this work we chose a nominal value of $\Omega=1$ unless stated otherwise. As will be evident from the scaling of our findings below, the particular value of $\Omega$ will not play a crucial role in our conclusions. The GPE in the Thomas-Fermi (TF) limit of large $\mu$ has a well known ground state $\psi_{\rm{TF}}=\sqrt{\max(\mu-V,0)}$. The other interesting limit is the linear one where the self-interaction term can effectively be ignored. In this limit, the GPE reduces to the 2D harmonic oscillator problem. Both limits are particularly useful for our considerations: the former enables the consideration of the ring-shaped soliton as an effective particle, the latter enables the construction of the ring as an exact solution in the linear limit, which is continued in the nonlinear regime.
Here, we will focus on the single RDS which, in the linear limit, can be viewed as a superposition of the $|2,0\rangle$ and $|0,2\rangle$ quantum harmonic oscillator states, namely: $$|\psi_{\rm{RDS}}\rangle_{\rm{linear}}=\frac{|2,0\rangle+|0,2\rangle}{\sqrt{2}}
\propto (r^2-1) e^{-\Omega r^2/2}.$$ This linear state, which exists for $\mu > 3 \Omega$ (i.e., beyond the corresponding linear limit of the above degenerate $n+m=2$ states, where $n$ and $m$ are the respective indices along the $x$- and $y$-directions, characterizing the quantum harmonic oscillator sate $|n,m\rangle$), can be continued to higher chemical potentials. However, the RDS is known to be inherently unstable for all values of $\mu$ beyond the linear limit [@ldc1; @herring; @toddric]. This instability breaks the original radially symmetric state into vortex multi-poles, as originally shown in Ref. [@rds2003] and subsequently examined from a bifurcation perspective in Ref. [@middelphysd]. Our scope is to provide a systematic understanding of the RDS instability modes and how to suppress them, so as to potentially enable its experimental realization. Similar considerations in the context of exciton-polariton condensates (where a larger range of tunable parameters exists due to the open nature of the system and the presence of gain and loss) have led both to the theoretical analysis [@rodr] and to the experimental observation [@sanv] of [*stable*]{} RDSs. Following the motivation of the earlier work of Ref. [@manjun] on planar dark solitons, in conjunction with the recent experimental developments in the context of radial [@gretchen] and more broadly, in principle arbitrary, so-called painted [@boshier] potentials, we propose the following form for $V_{\rm pert}(r)$: $$V_{\rm{pert}}(r)=Ae^{-(r-r_c)^2/(2\sigma^2)},$$ where $r_c$, $A$ and $\sigma$ represent, respectively, the radius, the amplitude and the width of this ring-shaped potential. Since RDSs feature radial symmetry, we first express Eq. (\[GPE\]) in the form: $$i \psi_t=-\frac{1}{2}\left(\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr}\right)\psi+V(r) \psi +| \psi |^2 \psi-\mu \psi.
\label{GPE1D}$$ We also assume that a stationary RDS state, $\psi=\psi(r,t)$, governed by the effectively 1D model (\[GPE1D\]), is characterized by a radius $r_c$. In other words, we will hereafter opt to locate the perturbation potential at the fixed equilibrium position of the RDS. For our analysis, the control parameters will be the strength $A$ of the perturbation potential and the nonlinearity strength (characterized by the chemical potential $\mu$); as concerns the width $\sigma$ of $V_{\rm{pert}}(r)$, it will be fixed (unless otherwise stated) to the value $\sigma=1$, which is of the order of the soliton width —i.e., of the healing length.
Below, we proceed with the study of the effect of the perturbation potential on the existence and stability of the RDS. Stability will be studied from both the spectral perspective, through a Bogolyubov-de Gennes (BdG) analysis, and from a dynamical time evolution perspective. The latter, will involve direct numerical integration of Eq. (\[GPE\]), whereby a (potentially perturbed) RDS is initialized and its evolution is monitored at later times. On the other hand, BdG analysis for a stationary RDS, $\psi_0(r)$, will involve the study of the eigenvalue problem stemming from the linearization of Eq. (\[GPE\]), upon using the perturbation ansatz: $$\begin{aligned}
\psi(x,y,t)=\psi_0(r) + \delta \left(u(x,y) e^{\lambda t}
+ \upsilon^{\ast}(x,y) e^{\lambda^{\ast} t} \right),~~
\label{bdg}\end{aligned}$$ where $[\lambda,(u,\upsilon)^T]$ is the eigenvalue-eigenvector pair, $\delta$ is a formal small parameter, and the asterisk denotes complex conjugation. Then, the existence of eigenvalues with non-vanishing real part signals the presence of dynamical instabilities. These come in two possible forms: (a) genuinely real eigenvalue pairs, which are associated with an exponential instability; and (b) complex eigenvalue quartets that denote an oscillatory instability, where growth is coupled with oscillation. The above symmetry of the eigenvalue pairs (i.e., the fact that they only arise in pairs or quartets) stems from the Hamiltonian nature of the problem.
The particle picture for the ring dark soliton {#sec:sub:part}
----------------------------------------------
A natural way to obtain a reduced dynamical description of the RDS is to adopt a particle picture and use a variational approximation discussed in detail in Ref. [@oberthaler]. According to this approach, in the TF limit (i.e., for sufficiently large chemical potential), the RDS state can be approximated by a product of the TF ground state, $\psi_{\rm{TF}}=\sqrt{\max(\mu-V,0)}$, and a (potentially traveling) dark soliton of radial symmetry, of the form: $$\psi_{\rm DS}(r,t)= b(t) \tanh[\sqrt{\mu}\,b(t)(r-r_c(t))]+ia(t),
\label{pe}$$ where $b$ and $a$ (with $a^2+b^2=1$) set, respectively, the depth and velocity of the soliton, while $r_c$ is the RDS radius. Then, the Euler-Lagrange equations for the two independent effective variational parameters $r_c$ and $a$, stemming from the averaged renormalized Lagrangian of the system, take the following form [@oberthaler]: $$\begin{aligned}
\label{a}
\dot{a}&=&-\frac{b^2}{\sqrt{\mu}}
\left\{\left(\frac{V'}{2}-\frac{\mu}{3r_c}\right)+\frac{VV'}{3\mu} \right. \nonumber \\
&&\left. +V'\left[\frac{V^2}{3\mu^2}+\frac{1}{4}\left(\frac{2}{3}-\frac{\pi^2}{9}\right)\frac{V'^2}{\mu^2}\right]\right\}, \\
\label{b}
\dot{r}_c&=&\sqrt{\mu}\left[a\left(1-\frac{V}{2\mu}\right) \right. \nonumber \\
&&
\left.-\frac{a}{4b^2}\left(\frac{5}{3}-\frac{\pi^2}{9}\right)\frac{V'^2}{\mu}\left(1-\frac{2V}{\mu}\right)\right].\end{aligned}$$ The above system suggests the existence of stationary RDSs, due to the interplay (to the leading-order approximation in $\Omega$) of an effective attractive trapping potential and an effective curvature-induced repulsive logarithmic potential —see first and second terms in the right-hand side of Eq. (\[a\]), respectively. A more systematic analysis, that takes into regard higher-order terms in $\Omega$, shows that the critical radius for which a stationary ring exists is given by [@oberthaler] $$r_c=\frac{\sqrt{0.5616 \mu}}{\Omega}.
\label{appr1}$$ Notice that, according to the discussion of Ref. [@oberthaler] and in accordance with the computational analysis presented below (see Sec. \[sec:results:particle\]), the numerical results strongly suggest an asymptotic critical radius $r_c=\sqrt{\mu/2}/\Omega$ (see also the discussion in Refs. [@oberthaler; @rds2003; @kamch]).
This discrepancy suggests the consideration of alternative ways of determining the stationary RDS’ radius. Here, for reasons of completeness, we will present such an alternative approach, based on the earlier work of Ref. [@kaper] for a different system (namely, ring-like steady state solutions of coupled reaction-diffusion equations). More specifically, our starting point will be the steady state problem associated with Eq. (\[GPE1D\]), where we will “lump” the potential terms as $V(r)=V_{\rm MT} + V_{\rm pert}(r)$. Using the ansatz $\psi(r)=\psi_{\rm TF}(r) q(r)$, we obtain the steady state problem: $$\begin{aligned}
\frac{1}{2} q'' + \mu q (1-q^2) = P(r),
\label{neweq}\end{aligned}$$ where $$P(r)= V q (1-q^2) -\frac{q'}{2 r}
-\frac{\psi_{\rm TF}''}{2 \psi_{\rm TF}} q
- \frac{\psi_{\rm TF}'}{\psi_{\rm TF}} q'
- \frac{1}{r} \frac{\psi_{\rm TF}'}{2 \psi_{\rm TF}} q,$$ and primes denote derivatives with respect to $r$. Then, seeking a stationary RDS solution in the form of $q(r)=\tanh(\sqrt{\mu}
(r-r_c))$ and multiplying both sides by $q'$ in Eq. (\[neweq\]), we find that the left-hand side is simply $dH/dr$, where $H$ is the effective Hamiltonian $H=q'^2/4 - \mu (1-q^2)^2/4$. Hence, upon integrating in $r$ from $-\infty$ to $\infty$, bearing in mind that the error between $r=0$ and $r \rightarrow -\infty$ is exponentially small, we obtain the explicit solvability, Melnikov-type, condition [@GH]: $$\begin{aligned}
\int_{-\infty}^{\infty} P(r) q'(r) dr=0.
\label{neweq2}\end{aligned}$$ Upon evaluating the integrals of all five terms associated with $P(r)$ within Eq. (\[neweq2\]), we should obtain an algebraic equation for the equilibrium position of the RDS. Indeed, evaluating the first potential term (for $A=0$), through a series of rescalings and integrations by parts, leads to $\Omega^2 r_c/(3\sqrt{ \mu})$. In turn, the second term yields $-2 \sqrt{\mu}/(3 r_c)$ and the fourth term yields $2 \Omega^2 r_c \sqrt{\mu}/[3 (\mu-\Omega^2 r_c^2/2)]$, while the other terms contribute at higher order. Putting all the terms together in the case of $A=0$ yields the prediction $$\begin{aligned}
r_c=\frac{\sqrt{\alpha\mu}}{\Omega},
\label{neweq3}\end{aligned}$$ where $\alpha=4-2\sqrt{3} \approx 0.5359$; this result is more accurate than the one of Eq. (\[appr1\]), as will be discussed in more detail in Sec. \[sec:results:particle\].
Finally, we proceed to give a third method, based on the analysis of Ref. [@kamch], that will prove to be the most accurate one in connection to our computations of not only statics but also dynamics of RDS states in the numerical section that will follow. In the latter approach, it is argued that the equation of motion can be derived by a local conservation law (i.e., an adiabatic invariant) in the form of the energy of a dark soliton under the effect of curvature and of the density variation associated with it. More specifically, knowing that the energy of the one-dimensional dark soliton is given by [@djf] ${\cal E}_{\rm DS}=(4/3) (\mu - \dot{x_c} )^{3/2}$, where $x_c$ is the dark soliton position, the generalization of the relevant quantity in a two-dimensional domain bearing density modulations reads: $${\cal E}_{\rm RDS}(r) = 2 \pi r \left[ \frac{4}{3} (\mu-V(r)-\dot{r}^2)^{3/2} \right].
%= 2 \pi r_c \left[ \frac{4}{3} (\mu-V(r_c)-\dot{r}_c^2)^{3/2} \right].
\label{PPnew1}$$ Thus, by assuming this quantity is constant, namely ${\cal E}_{\rm RDS}(r)={\cal E}_{\rm RDS}(r_c)$, where $r_c$ is the equilibrium location of the ring, we obtain an equation for $\dot{r}^2$. Taking another time derivative on both sides, we finally obtain Newtonian particle dynamics for the ring in the form: $$\ddot{r}=-\frac{1}{2}\frac{\partial V}{\partial r}
+\frac{1}{3 r}\left(\frac{r_c}{r}\right)^{2/3}\left[\mu-V(r_c)\right].
\label{PPextra}$$ When $A=0$, this equation of motion for the RDS position yields the equilibrium $r_c=\sqrt{{\mu}/{2}}/\Omega$, a result which, as highlighted also above and as will be demonstrated below, is the one most consistent with the numerical observations. This, in turn, motivates us to use the above approach of Ref. [@kamch] not only for the statics, but also for the dynamics in the following section and additionally, not only for the case without the radial defect of $A=0$, but also for that bearing the radial defect i.e., for $A \neq 0$.
![(Color online) Top panels: the RDS’ real-valued profile (left) and the corresponding density plot (right) for $\mu=16$. Middle left: a radial profile of the relevant state. Middle right: number of particles $N=\int |\psi|^2dr$ as a function of chemical potential $\mu$, showing the continuation of states from the linear limit to the nonlinear regime. Bottom panels: the imaginary (left) and real parts (right) of the spectrum; showcased is the generic instability of the RDS, and the emergence of additional unstable eigenmodes thereof as $\mu$ is increased. The value for the trap strength in this figure and all remaining figures is $\Omega=1$ (unless stated otherwise). []{data-label="lc"}](\rootfig LCnewNNN.jpg.ps){width="8.5cm"}
We now proceed to test these predictions, as well as to examine the BdG stability analysis and the dynamical evolution of the RDS, both in the absence (initially, for comparison and guidance) and then in the presence of the radial perturbation potential.
Results
=======
First, we briefly summarize the numerical techniques used in this work. Stationary states in both 1D (i.e., in a radial form) and 2D were identified using a centered finite-difference scheme within Newton’s method. The spectrum of the stationary states (i.e., the result of the BdG analysis) was calculated using the eigenvalue problem derived from Eq. (\[bdg\]). Finally, for the dynamics of the system, we used direct integration employing second order finite differences in space and fourth-order Runge-Kutta in time.
Basic properties of the ring dark soliton
-----------------------------------------
Let us start by summarizing some of the basic properties of the RDS *without the perturbation potential.* A typical RDS state in the TF limit of large chemical potential $\mu$ is shown in the top and middle left panels of Fig. \[lc\]; the top right panel shows the corresponding density. As indicated in the previous section, the RDS has a linear limit (built out of the eigenstates of the 2D quantum harmonic oscillator). The continuation of such a state in the nonlinear regime is shown in the middle right panel of Fig. \[lc\]. The imaginary and real parts of the spectrum of the RDS are shown in the bottom panels of Fig. \[lc\]. Note that the RDS is unstable for any value of $\mu$ beyond the linear limit. More importantly, in line with what was also presented in Ref. [@middelphysd], as $\mu$ increases, more unstable modes keep emerging, through eigenvalue pairs that cross through the origin. These signal pitchfork bifurcations, to which we now turn.
![(Color online) The most unstable modes at a few representative chemical potential values $\mu=4$, 6, 9, 11, 14, and 16 (from left to right, top to bottom) associated with the instability of the RDS. Left and right subpanels correspond, respectively, to the absolute value and phase of the modes. []{data-label="evecs"}](\rootfig evecsN.jpg.ps){width="8.5cm"}
![(Color online) Dynamics ensuing from the unstable RDS for $\mu=16$. Note that the RDS first deforms into seven pairs of vortices (in accordance with the most unstable mode for these parameters values; see the bottom right panel of Fig. \[evecs\]), and then eventually turns into a dynamical evolving vortex cluster for longer times. During evolution, some of the vortices are “absorbed” by the BEC periphery and the system is eventually left with four interacting vortices.[]{data-label="RK4A0"}](\rootfig RK4A0N.jpg.ps){width="8.5cm"}
Studies of RDS in atomic BECs have illustrated their dynamical breakup into vortex-antivortex pairs (see, e.g., Refs. [@rds2003; @herring]). To complement this picture, we now discuss the most unstable modes of the BdG analysis. Some representative eigenmodes at $\mu=4,$ 6, 9, 11, 14 and 16 are shown in Fig. \[evecs\]. It is interesting to observe that the identified modes indicate a clear connection to an increasing number of pairs of vortices. The first unstable mode appears to be connected to two-pairs, i.e., to a vortex quadrupole. Indeed, the vortex quadrupole exists as a state [@mottonen] for any value of $\mu$ beyond the linear limit of $\mu=3 \Omega$, being constructed as: $$|\psi_{\rm{Q}}\rangle_{\rm{linear}}=\frac{|2,0\rangle + i |0,2\rangle}{\sqrt{2}}.$$ Subsequent destabilization modes reveal a three-fold symmetry (leading to the bifurcation of vortex hexagons [@middelphysd]), a four-fold symmetry (leading to vortex octagons), then a five-fold (decagons), a six-fold (dodecagons), and so on. These different eigenvectors are clearly illustrated in Fig. \[evecs\] and the existence and stability of the corresponding emerging (from the pitchfork bifurcation) vortex $n$-gon cluster states is discussed in Ref. [@annab].
A dynamical study of the states shows that the evolution initially results in vortex pairs, in agreement with Fig. \[evecs\]. However, gradually some vortices may move out of the BEC and get lost in the background, leaving behind a complex, interacting cloud of vortices, as shown for $\mu=16$ in Fig. \[RK4A0\]. The resulting interaction dynamics between vortices in the cluster, and the associated transfer of energies between different scales, may represent a very interesting setting for exploring turbulence phenomena and associated cascades in line, e.g., with recent experimental efforts of Ref. [@bpa_turb].
![(Color online) Max$(|\Psi|)$ as a function of $(A,\mu)$. Note that $N$ decreases as $A$ increases when holding $\mu$ fixed until some critical set of values of $A$ (depicted by the purple line) beyond which the RDS will cease to exist. In the linear limit $\mu=3 \Omega$, even a very small positive perturbation of $A$ will destroy the RDS state.[]{data-label="surface2"}](\rootfig Surface2NN.jpg.ps){width="8.5cm"}
![(Color online) Instability growth rate max(Re($\lambda$)) as a function of $(A,\mu)$. The area between the left (green) and right (purple) curves corresponds to the region where the RDS exists and with vanishing max(Re($\lambda$)), i.e. the RDS is completely stable. The rightmost purple line is also the boundary of the critical values of $A$ beyond which no RDS solution exists. []{data-label="surface3"}](\rootfig Surface3NN.jpg.ps){width="8.5cm"}
Adding the perturbation potential
---------------------------------
Having analyzed the unperturbed case, we now examine the case with the radial Gaussian potential. The existence of the RDS structure in the latter case can be captured as a function of $(A,\mu)$ —see Fig. \[surface2\]. We used max$(|\Psi|)$ (i.e., the max root density) as a diagnostic instead of $N$ for practical visualization purposes, in this case. We can see that for a fixed value of $\mu$, the density decreases as $A$ increases (a natural feature, given the repulsive nature of the perturbation potential) until a critical value of $A$ —shown as a purple line— is reached, beyond which the RDS will cease to exist. In the linear limit of $\mu=3 \Omega$, even a very small positive perturbation of $A$ will destroy the RDS state. The monotonic dependence of $\mu$ on the critical $A$ appears to be approximately linear.
We proceed now with the central theme of this study, which is the dynamical stabilization of the RDS. To characterize the stability of the RDS in the $(A,\mu)$ plane, in Fig. \[surface3\] we show a plot of the max(Re($\lambda$)) as a function of $(A,\mu)$. The right most purple line, as before, depicts the critical values of $A$ beyond which no RDS solution exists. The region enclosed between the green and purple lines corresponds to the regimes where RDS exists [*with vanishing max(Re($\lambda$))*]{}, i.e., the RDS is completely stabilized by the presence of the external Gaussian ring perturbation potential. One interesting feature is that the relevant stability landscape is rather complex with potential sequences of destabilization and restabilization for values of $\mu \geq 3.6$ (we will return to this point below). However, the principal conclusion obtained from Fig. \[surface3\] is that the RDS is generically subject to full dynamical stabilization for any value of the chemical potential and for suitable intervals of the perturbation potential strength $A$ in the vicinity of the linear limit. The feature that the stabilization is enabled near the linear limit is rather natural to expect also on the basis of our earlier results for $A=0$ in Fig. \[lc\]. Given that the RDS is progressively more and more unstable (with a higher number of destabilizing modes) as $\mu$ increases suggests that the perturbation potential may be unable to suppress this multitude of unstable modes, especially far from the linear limit.
![(Color online) Cross section of the instability growth rate max(Re($\lambda$)) at $\mu=4$. The right most point of the curve corresponds to the critical value of $A$ beyond which no RDS solution exists. The two blue squares are two points in two different instability regimes but with similar instability rates whose full spectrum is shown in Fig. \[css1\]. []{data-label="css"}](\rootfig cssN.ps){width="8.5cm"}
![(Color online) Full stability spectrum corresponding to the two blue squares in the two different instability regimes in Fig. \[css\] for $\mu=4$. Left and right panels correspond to $A=1.07$ and $A=1.28$, respectively. Note that the two regimes do not share the same nature of instability. The large amplitude case (left) has the instability on the real axis (i.e., exponential instability) while the small amplitude case (right) has the instability in the form of a complex quartet (oscillatory instability).[]{data-label="css1"}](\rootfig css1.ps){width="8.5cm"}
To gain further insight on this stability plane, let us now study a typical cross section of Fig. \[surface3\] at $\mu=4$. The cross section is shown in Fig. \[css\]. A detailed study of the full spectrum shows the existence of two intervals of instability which are not of the same nature. The leftmost interval (including $A=0$ in the absence of a defect) corresponds to a typically large(r) growth rate. Here, the instability derives from real eigenvalue pairs. Connecting with Fig. \[lc\] and the case of $A=0$, we recognize that this unstable mode is associated with the breakup to vortex quadrupoles. As $A$ becomes increasingly more negative to the left of the figure, other modes may, in turn, dominate the instability dynamics (the “bend” in the stability diagram represents such a “take-over” of the dominant instability by a different mode; cf. Fig. \[lc\]). However, it is observed that as $A$ increases on the positive side, the unstable real pair(s) decrease in their real part and eventually cross through the origin of the spectral plane, becoming imaginary and hence stabilizing the RDS state. This is, once again, a key finding of our work, representing the RDS stabilization. However, as the (formerly unstable) eigenvalues bear a so-called ‘negative energy’, upon climbing up the imaginary axis, they may collide with eigenvalues associated with ‘positive energy’ modes (see, e.g., the discussion in pp. 56–58 of Ref. [@book2]). This type of collision gives rise to a complex eigenvalue quartet and a different (weak) oscillatory dynamical instability, or a so-called Hamiltonian Hopf bifurcation; see, e.g., the discussion of Ref. [@goodman]. The latter scenario leads to small instability bubbles, as the quartet may form, but subsequently the eigenvalues may return to the imaginary axis, splitting anew into two imaginary pairs.
![(Color online) The most unstable modes of Fig. \[css1\]. Top and bottom row of panels correspond, respectively, to the absolute value (left subpanels) and phase (right subpanels) of the solutions for the left and right cases depicted in Fig. \[css1\]. []{data-label="css2"}](\rootfig css2N.jpg.ps){width="8.5cm"}
![(Color online) Dynamics of the state in the top panels of Fig. \[css2\]. The odd panels depict the absolute value of the field while the even panels depict its phase. The state is oscillating between the vortex quadrupole and the RDS, but very weakly. []{data-label="RK4A107"}](\rootfig RK4A107N.jpg.ps){width="8.5cm"}
![(Color online) The same plot as Fig. \[RK4A107\] but for the state in the bottom panels of Fig. \[css2\]. This state has a different nature of instability from the one in Fig. \[RK4A107\]. The instability is like a vibrational mode. []{data-label="RK4A128"}](\rootfig RK4A128N.jpg.ps){width="8.5cm"}
The two (exponential and oscillatory) instability scenarios are illustrated in the two panels of Fig. \[css1\] for smaller and larger values of $A$, respectively. The most unstable mode of each state is shown in Fig. \[css2\], illustrating the distinct nature of the instability in the different scenarios. The state at $A=1.07$ is in the same branch of $A=-1,0$ and 1, and its instability leads to a deformation towards a vortex quadrupole state in a way similar as the first plot of Fig. \[evecs\]. On the other hand, the state at $A=1.28$ appears to have a different type of instability that instead resembles a vibrational mode (the type of mode that could be captured through a ring particle model). The time dynamics of the two states are shown in Fig. \[RK4A107\] and Fig. \[RK4A128\] respectively. In the former case, we observe the recurrent formation of a vortex quadrupole (this is not immediately discernible in the density but distinctly visible in the phase pattern), in accordance with the identified unstable mode. In the latter, indeed unstable vibrational dynamical characteristics can be seen in the motion of the ring, which, however, appears to maintain its radial structure.
A different cross section of the stability plane of Fig. \[surface3\] is given in Fig. \[LCA\], now for the case of $A=0.5$, and varying the chemical potential $\mu$. From this perspective, we observe that $A$ delays the onset of instabilities as $\mu$ is increased. Another way to look at the effects of $A$ and $\mu$ is that $A$ plays effectively the opposite role to that of $\mu$: the increase of $A$ (for fixed $\mu$) drives the eigenmodes away from the real axis and into the imaginary axis while the increase of the chemical potential for fixed $A$ drives the eigenmodes away from the imaginary axis and into the real axis, causing instability. We believe that this discussion provides a unified perspective on the sources of destabilization and the potential for re-stabilization of the RDS.
![(Color online) Cross section of max(Re($\lambda$)) at $A=0.5$. The solution starts to exist around $\mu=3.35$. Note that $A$ delays the set in of instabilities as $\mu$ is increased. []{data-label="LCA"}](\rootfig LCAN.ps){width="8.5cm"}
![(Color online) Time evolution of states at $\mu=4$ with $A=-1$, 0 and 1 (top to bottom rows of panels). Note that the state of $A=1$ is significantly less unstable than those of $A=-1$ and $A=0$, which have roughly the same instability growth rate. Note also that all three states deform toward the vortex quadrupole state initially, although the third one maintains an oscillatory pattern between a recurring ring and a vortex quadrupole. []{data-label="RK4"}](\rootfig RK4N.jpg.ps){width="8.5cm"}
![(Color online) Time evolution of states at $\mu=4$ with $A=1.14$ which is in a completely stable parametric interval. The state is shown to be stable up to t=1000, in agreement with our spectral results. []{data-label="RK4s"}](\rootfig RK4sN.jpg.ps){width="8.5cm"}
In all the cases considered, the stability conclusions were also found to be consonant with the corresponding dynamics, of which we now present a few additional case examples. In particular, we study the dynamical evolution of states at $\mu=4$ for different values of $A=-1,0,1$ (see Fig. \[RK4\]) and $1.14$ (see Fig. \[RK4s\]) to probe the effects of the variation of $A$. Note that the cases of $A=-1$ and $0$ are about equally unstable at $\mu=4$ with $A=-1$ bearing a slightly larger growth rate. The case of $A=1$, however, is very close to, albeit not within the stabilization regime. On the other hand, the case of $A=1.14$ is fully stabilized. We add a random perturbation to the states, ensuring that the number of atoms in each case is, upon perturbation, 1.0013 times of the unperturbed one. The results of the dynamical evolution of the former three cases are shown in Fig. \[RK4\]. Note that both states for $A=-1$ and $A=0$ are relatively quickly deformed around $t=25$ while the state for $A=1$ deforms only much later around $t=70$, due to its weaker growth rate. In all three cases, the states evolve initially towards the vortex quadrupole waveform. While the former two states will quickly deform afterwards and lose their radial structure, the third state can oscillate between the RDS state and the vortex quadrupole state for a much longer time at least up to $t=1000$. A dynamical evolution of states at $A=1.14$, which is in a completely stable parametric interval, is shown in Fig. \[RK4s\]. The state is shown to be stable at least up to $t=1000$, in agreement with the spectral findings and corroborating the full stabilization achieved by the presence of the Gaussian repulsive impurity.
![(Color online) The location of the RDS scaled by $\sqrt{\mu}/\Omega$ as a function of $\mu$ (thick solid blue line). Note that the numerical values reach a limiting value of $1/\sqrt{2}$ (thin horizontal solid red line) when $\mu$ is large. The particle picture can approximately describe the $\sqrt{\mu}$ behavior and over estimates $r_c$, but nevertheless is still an interesting approximate description of the RDS. The particle approach using the perturbed Lagrangian method \[see Eqs. (\[a\]) and (\[b\])\] corresponds to the thin dotted-dash green line while the solvability condition for the steady state problem method \[see Eq. (\[neweq3\])\] is depicted by the thin dashed black line. []{data-label="rc1"}](\rootfig rc2N.ps){width="7.5cm"}
![(Color online) Dark soliton width $w$ as a function of the chemical potential $\mu$. The blue solid line corresponds to fitting a profile $\psi_{\rm TF}(r)\times\tanh(\sqrt{w}(r-r_c))$ to the PDE steady state for $\Omega=0.2$. The red dashed line corresponds to the approximate value of the background at the location of the RDS, see Eq. (\[eq:mu0\]). []{data-label="fig:width"}](\rootfig width.ps){width="7.5cm"}
The particle picture of the ring dark soliton {#sec:results:particle}
---------------------------------------------
We first study how the equilibrium location $r_c$ of the RDS changes with chemical potential $\mu$, especially in the large density limit without the perturbation potential, and compare the numerical results and the particle picture predictions. Numerical results (for $\Omega=1$) suggest that $r_c=\sqrt{\mu/2}$ (see thin horizontal red line) in the large $\mu$ limit as shown in Fig. \[rc1\]. As mentioned in Sec. \[sec:sub:part\], a systematic analysis of Eqs. (\[a\]) and (\[b\]) yields the estimate $r_c=\sqrt{0.5616\mu}/\Omega$ (see horizontal thin dotted-dash green line). On the other hand, using the solvability condition for the steady state problem described in Sec. \[sec:sub:part\], one obtains the better prediction of the RDS position $r_c=
\sqrt{0.5359\mu}/\Omega$; see Eq. (\[neweq3\]) and thin horizontal dashed black line in Fig. \[rc1\]. It is important to mention that, although the above two particle approaches are able to capture the $\sqrt{\mu}/\Omega$ behavior of $r_c$, they do not lead to the precise numerical prefactor. This may be attributed to the choice of the ansatz (\[pe\]), where the width of the stationary dark soliton is chosen to be $\sqrt{\mu}$. This selection corresponds to the width of a dark soliton in a homogeneous background of density $\mu$. However, due to the non-homogeneity of the BEC background, the RDS placed at $r_c$ experiences a background density $\mu_0$ which can be approximated using the TF regime (valid for large $\mu$) to be $$\mu_0\approx\psi_{\rm TF}^2(r_c)=\mu-V(r_c)=\mu-\frac{1}{2}\Omega^2r_c^2.
\label{eq:mu0}$$ For instance, in Fig. \[fig:width\] we show an example where we extracted the width of the dark soliton for $\Omega=0.2$ as a function of $\mu$. As it is clear from the figure, as $\mu$ increases, the width of the dark soliton converges to $\sqrt{\mu_0}$ as prescribed in Eq. (\[eq:mu0\]). Lastly, it is relevant to point out that, remarkably, the adiabatic invariant theory of Ref. [@kamch] properly captures the asymptotic growth of the radius of the RDS as $\mu$ increases. It is for that reason that we will hereafter utilize the particle picture of Eq. (\[PPextra\]) and Ref. [@kamch] for our further static and dynamics considerations.
We now study the effect of $A$ on $r_c$. Figure \[rc2\] depicts $r_c$ as a function of $A$ for $\mu=24$. It is clear that the particle picture can capture the effect of $A$ fairly accurately. It is also observed that the critical radius decreases in comparison to the $A=0$ limit, in the presence of a repulsive defect, while the opposite is true in the case of an attractive defect.
![ Position of the RDS as a function of $A$ for $\mu=24$. The (red) circles correspond to the full PDE dynamics and the (green) triangles to the particle picture (PP) described in Sec. \[sec:sub:part\].[]{data-label="rc2"}](\rootfig rcc2.ps){width="8.5cm"}
Finally, we study the radial oscillatory motion of the RDS in both the case bearing and in that without the perturbation potential. We initialize our displaced RDS state by superposing a suitable hyperbolic tangent profile to (i.e., multiplying it with) the numerically exact ground state at the same chemical potential $\mu=24$. Note that the RDS is unstable at such a high chemical potential, therefore, we can only simulate the PDE dynamics for a limited amount of time, before an instability leading to a polygonal cluster of vortices ensues. The comparison of the PDE and the particle picture dynamics for the cases of $A=0$ and $A=1$ are shown, respectively, in the top and bottom panels of Fig. \[rctime1\]. We see that the particle picture is able to capture the essential PDE radial oscillation dynamics both with and without the Gaussian barrier.
![Radial oscillatory motion of the RDS with $\mu=24$ for $A=0$ and $A=1$. The central radius of the RDS is extracted from the PDE dynamics (green dots) and compared to the ODE evolution of the particle picture (PP, red line) according to Eq. (\[PPextra\]).[]{data-label="rctime1"}](\rootfig rctimenew3.ps "fig:"){width="8.5cm"} ![Radial oscillatory motion of the RDS with $\mu=24$ for $A=0$ and $A=1$. The central radius of the RDS is extracted from the PDE dynamics (green dots) and compared to the ODE evolution of the particle picture (PP, red line) according to Eq. (\[PPextra\]).[]{data-label="rctime1"}](\rootfig rctimenew4.ps "fig:"){width="8.5cm"}
Conclusions and Future Challenges {#conclusion}
=================================
In this work, we studied the existence and stability of ring dark soliton states, initially in the absence and subsequently in the presence of a radially localized Gaussian perturbation potential. We have systematically shown, via a combination of spectral analysis and direct numerical simulations, that the ring dark soliton can be stabilized by adding the perturbation potential with a suitable strength, for all values of the chemical potential that we have considered herein. Our systematic spectral analysis has also revealed why this stabilization mechanism can only be effective near the linear limit of the system. It has also revealed the potential for secondary instabilities (due to pair collisions on the imaginary axis and complex eigenvalue quartets emerging from them) due to the excited state nature of the ring.
An additional effort, significantly motivated by the potential of the above method to lead to stable RDS vibrations, was that of deriving dynamical equations for their motion. We evaluated different techniques to this effect, showcasing the fact that although all approaches gave fairly similar results, the adiabatic invariant method of Ref. [@kamch] presented a distinct advantage in capturing the radius of stationary rings. A self-consistent perturbative technique (based on earlier work in reaction-diffusion systems) was also adopted to that effect and was shown to give reasonably accurate results in its comparison with the full numerical results. Going beyond the “stationary particle” approach, allowing motion along the radial direction, an intriguing goal for the future may be to examine the ring soliton as a filamentary pattern embedded in 2D, which, in addition to radial internal modes, may possess bending ones (but without breaking). Such studies may in turn enable the observation of collisions and deformations of rings upon interactions, a topic that has been of interest also in nonlinear optics [@ektoras].
Finally, it may well be relevant to explore settings beyond the realm of two spatial dimensions, extending the present considerations to the case of 3D solitonic or vortex rings and other such patterns. Earlier work established how to construct such states in isotropic and anisotropic 3D limits starting from linear eigenstates [@craso2]. It is then of particular interest to continue such states in the nonlinear realm and explore their spectral and dynamical stability using tools similar to the ones proposed herein. Efforts along these directions are currently in progress and will be presented in future publications.
W.W. acknowledges support from NSF (grant No. DMR-1208046). P.G.K. gratefully acknowledges the support of NSF-DMS-1312856, as well as from the US-AFOSR under grant FA9550-12-1-0332, and the ERC under FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-605096). P.G.K.’s work at Los Alamos is supported in part by the U.S. Department of Energy. R.C.G. gratefully acknowledges the support of NSF-DMS-1309035. The work of D.J.F. was partially supported by the Special Account for Research Grants of the University of Athens. The work of T.J.K. was supported in part by NSF grant DMS-1109587. M.M. gratefully acknowledges support from the provincial Natural Science Foundation of Zhejiang (LY15A010017) and the National Natural Science Foundation of China (No. 11271342).
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---
abstract: 'Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.'
author:
- Barbara Baumeister
- 'Kai-Uwe Bux'
- Friedrich Götze
- Dawid Kielak
- Henning Krause
title: 'Non-crossing partitions'
---
The poset of non-crossing partitions
====================================
A *partition* $p$ of a set $U$ is a decomposition of $U$ into pairwise disjoint subsets $B_{i}$: $$U = \biguplus_{i} B_{i}$$ The subsets $B_{i}$ are called the *blocks* of the partition $p$. Another way to look at this is to consider $p$ as an equivalence relation on $U$. In this perspective, the subsets $B_{i}$ are the equivalence classes. Let $q$ be another partition of the same set $U$. We say that $q$ is a *refinement* of $p$ if each block of $q$ is contained in a block of $p$. In terms of equivalence relations, if two elements of $U$ are $q$-equivalent, they are also $p$-equivalent. We also say that $q$ is *finer* than $p$ or that $p$ is *coarser* than $q$; and we write $
q\preceq p
$.
Let $\operatorname{P}(U)$ be the set of all partitions on the underlying set $U$. The refinement relation $\preceq$ is a partial order on the set $\operatorname{P}(U)$, which is therefore a *poset*. Moreover, it is a *lattice*, i.e., every non-empty finite subset ${\mathcal{P}}\subseteq\operatorname{P}(U)$ has a least upper bound and a greatest lower bound. We remark that the partition lattice is *complete*, i.e., even arbitrary infinite subsets have least upper and greatest lower bounds.
\[a4-c3-c13-half-a-lattice-is-a-lattice\] It is interesting that the definition of a complete lattice can be weakened by breaking the symmetry between upper and lower bounds. If a poset has upper bounds and greatest lower bounds, it is already a complete lattice (i.e. it also has lowest upper bounds).
Let ${\mathcal{P}}$ be a non-empty subset of the poset. We consider the the set $ B^+({\mathcal{P}}) $ of all common upper bounds for the non-empty subset ${\mathcal{P}}$. Since the poset has upper bounds, $B^+({\mathcal{P}})$ is non-empty. Hence it has a greatest lower bound, which turns out to be the lowest upper bound of ${\mathcal{P}}$.
Consider the following reflexive and symmetric relations on $U$: $$\begin{aligned}
x &\sim y
\quad:\Leftrightarrow\quad
\exists p\in{\mathcal{P}}\,:\,\,
x \text{\ and\ } y \text{\ are\ }p\text{-equivalent}
\\
x &\approx y
\quad:\Leftrightarrow\quad
\forall p\in{\mathcal{P}}\,:\,\,
x \text{\ and\ } y \text{\ are\ }p\text{-equivalent}
\end{aligned}$$ It is clear that $\approx$ is itself an equivalence relation. It corresponds to the *meet* $\bigwedge {\mathcal{P}}$ of the partitions in ${\mathcal{P}}$, i.e., the greatest lower bound of ${\mathcal{P}}$. The transitive closure of $\sim$ is an equivalence relation, which corresponds to the *join* $\bigvee {\mathcal{P}}$ of the partitions in ${\mathcal{P}}$.
Now, we restrict our consideration to finite sets. For a natural number $m\in{\mathbb{N}}$, let us denote by $[m]$ the set $\{\,1,2,3,\ldots,m\,\}$. We fix the natural cyclic ordering on $[m]$ and represent its elements as the vertices $ v_{1},\ldots,v_{m} $ of a regular $m$-gon inscribed in the unit circle. Let $p$ be a partition of $[m]$. We say that two blocks $B$ and $B'$ of the partition $p$ *cross* if their convex hulls intersect. The partition $p$ is called *non-crossing* if its blocks pairwise do not cross. A non-crossing partition can thus be depicted by colouring the convex hulls of its blocks. For blocks of size one or two, we fatten up the convex hull.
It is clear from the visualisation that the complements of the coloured regions also are pairwise disjoint. This gives rise to the *Kreweras complement*. Here, we put *dual vertices* $w_{1},\ldots,w_{m}$ within the arcs $v_{i}-v_{i+1}$. There is no natural numbering, and we choose to place $w_{1}$ within the arc from $v_{1}$ to $v_{2}$. Let $p$ be a non-crossing partition. Two dual vertices lie in the same block of the complement $p^\mathrm{c}$ if they lie within the same complementary region of the convex hulls of blocks of $p$.
The set $\operatorname{NC}(m)$ of all non-crossing partitions of $[m]$ is partially ordered with respect to refinement. It is thus a subposet of the set of all partitions of $[m]$. It turns out that $\operatorname{NC}(m)$ is also a lattice. This is clear from Remark \[a4-c3-c13-half-a-lattice-is-a-lattice\] since greatest lower bounds are inherited from the partition lattice and upper bounds exist trivially since the trivial partition with a single block is noncrossing.
However, the noncrossing partition lattice is not a sublattice of the whole partition lattice: the join operation in both structures differ, i.e., the finest partition coarser than some given non-crossing partitions does not need to be non-crossing; see Remark \[a4-c3-c13-not-a-sublattice\] for a counterexample.
The complement map $$\begin{aligned}
\operatorname{NC}(m) & \longrightarrow \operatorname{NC}(m) \\
p & \longmapsto p^\mathrm{c}
\end{aligned}$$ is an anti-automorphism of the lattice $\operatorname{NC}(m)$: it reverses the refinement relation and interchanges the roles of meet and join. It is, however, not an involution. In the picture, taking the Kreweras complement twice seems to get you back to the original partition. This is true; however, the indexing of the vertices shifts by one. Thus, the square of the Kreweras complement is given by cyclically rotating the element of the underlying set $\{\,1,\ldots,m\,\}$.
The *bottom* (finest) element $\bot$ of $\operatorname{NC}(m)$ is the partition with $m$ blocks, each of size one. The *top* (coarsest) element $\top$ of $\operatorname{NC}(m)$ is the partition with a single block. For each non-crossing partition $p$, we define its *rank* $\operatorname{rk}(p)$ in terms of its number of blocks: $$\operatorname{rk}(p) := m - \# \{\,\text{blocks of\ }p\,\}$$ For any non-crossing partition $p$, all maximal chains from the bottom element $\bot$ to $p$ have the same length, which coincides with the rank $\operatorname{rk}(p)$. Let us summarise the properties and non-properties of the poset of non-crossing partitions:
The set $\operatorname{NC}(m)$ of non-crossing partitions of an $m$-element is partially ordered by refinement. This poset is a lattice and self-dual with respect to the Kreweras complement, i.e., $$\begin{aligned}
(p\wedge q)^\mathrm{c}
&= p^\mathrm{c}\vee q^\mathrm{c}\\
(p\vee q)^\mathrm{c}
&= p^\mathrm{c}\wedge q^\mathrm{c}\\
\end{aligned}$$ for any two $p,q\in\operatorname{NC}(m)$.
The automorphism $p\mapsto(p^\mathrm{c})^\mathrm{c}$ has order $m$.
All maximal chains from bottom to top have length $m-1$. For any non-crossing partition $p$, there is a maximal chain from bottom to top going through $p$. The non-crossing partition lattice is graded and one has $$m-1 = \operatorname{rk}(p)+\operatorname{rk}(p^\mathrm{c})$$ for any $p$.
\[a4-c3-c13-not-a-sublattice\] For $m\geqslant 4$, the non-crossing partition lattice $\operatorname{NC}(m)$ is not a sub-lattice of the partition lattice: the join operations do not coincide. A counterexample for $m=4$ is $ p=\{\, \{\,1,3\,\},\{\,2\,\},\{\,4\,\} \,\} $ and $ q=\{\, \{\,1\,\},\{\,2,4\,\},\{\,3\,\} \,\} $. The join of these partitions in the partition lattice is $\{\,\{\,1,3\,\},\{\,2,4\,\}\,\}$ whereas the join in $\operatorname{NC}(4)$ is the top element. These two partitions also show that the non-crossing partition lattice $\operatorname{NC}(m)$ is not *semi-modular*, i.e., the following inequality does not hold for all partitions $p$ and $q$, $$\operatorname{rk}(p)+\operatorname{rk}(q)
\geqslant
\operatorname{rk}(p\vee q)
+
\operatorname{rk}(p\wedge q)
.$$
Enumerative properties of the noncrosing partitition lattice are well understood. Kreweras counted the number of non-crossing partitions.
For any $m$, we have $$\left|\,
\operatorname{NC}(m)
\, \right|
=
C_{m}$$ where $
C_{m}
= \frac{1}{m+1}
{\binom {2m} m}
= \frac{(2m)!}{m!(m+1)!}
$ is the $m^{\text{th}}$ *Catalan number*.
Kreweras also determined the M[ö]{}bius function for the lattice of non-crossing partitions. Recall that, for a finite poset $P$, the *M[ö]{}bius function* $$\mu :
\{\,\,
(u,v)
\in
P \times P
\,\,|\,\,
u {\leqslant}v
\,\,\}
\longrightarrow {\mathbb{Z}{\hspace{0.5pt}}}$$ is defined by the following recursion: $$\begin{aligned}
\mu(u,u) &= 1, \\
\mu(u,v)
& = - \sum_{u{\leqslant}w<v}
\mu(u,w) .
\end{aligned}$$ Note that the value $\mu(u,v)$ is completely determined by the isomorphism type (as a poset) of the interval $
[u,v]
:=
\{\,\, w \in P \,\,|\,\, u{\leqslant}w{\leqslant}v \,\,\}
$.
For the non-crossing partition poset $\operatorname{NC}(m)$, the M[ö]{}bius function satisfies $$\label{a4-c3-c13-moebius}
\mu(\bot,\top)
=
(-1)^{m-1}C_{m-1}
= (-1)^{m-1} \frac{(2m-2)!}{(m-1)!m!}$$
Let $p$ be a non-crossing partition, and consider a non-crossing partition $q\preceq p$. Let $B$ be a block of $p$. The blocks of $q$ contained in $B$ may be thought of as a non-crossing partition of $B$. Thus, we have the following:
\[a4-c3-c13-intervall-structure\] Let $p\in\operatorname{NC}(m)$ be a non-crossing partition, and let $B_{1},\ldots,B_{k}$ be its blocks. Then the *order ideal* $
p_{\preceq}
:=
\{\,\,
q\in\operatorname{NC}(m)
\,\,|\,\,
q\preceq p
\,\,\}
$ is isomorphic as a poset to the cartesian product $
\operatorname{NC}(B_{1})
\times \cdots \times
\operatorname{NC}(B_{k})
$.
Let $B'_{1},\ldots,B'_{m-k+1}$ be the blocks of the Kreweras complement $p^\mathrm{c}$. Since the complement is an antiautomorphism of the non-crossing partition lattice, the *filter* $ p_{\succeq} := \{\,\, q\in\operatorname{NC}(m) \,\,|\,\,
q\succeq p \,\,\} $ is isomorphic as a poset to the cartesian product $ \operatorname{NC}(B'_{1}) \times \cdots \times
\operatorname{NC}(B'_{m-k+1}) $.
For non-crossing partitions $p\preceq q$, the interval $[p,q]$ is the filter for $p$ within the order ideal of $q$. Hence, by combining the previous isomorphisms, we see that $
[p,q]
$ is isomorphic to the product $
\prod_{B} \operatorname{NC}(B)
$ where $B$ ranges over the blocks of the “blockwise Kreweras complement” of $p$ in $q$.
Since the M[ö]{}bius function is multiplicative with respect to cartesian products of posets, Observation \[a4-c3-c13-intervall-structure\] allows one to derive the values of $\mu(p,q)$ in terms of the blockwise complement of $p$ in $q$ from Kreweras’ formula (\[a4-c3-c13-moebius\]).
\[a4-c3-c13-reversing-order\] To every poset $(P,{\leqslant})$, one associates the *order complex* . This is the simplicial complex $\Delta(P,{\leqslant})$ whose vertices are the elements of $P$ and whose simplices are *chains* in $P$, i.e., non-empty subsets of $P$ on which ${\leqslant}$ is a total order. By a theorem of P. Hall, one can interpret the M[ö]{}bius function as the Euler characteristic of order complexes [@a4-c3-c13-Stanley Prop. 3.8.6], $$\mu(u,v)
=
\chi( \Delta( (u,v) ) ),
\qquad\text{for\ }u<v.$$ Here $
(u,v)
:=
\{\,\, w \in P \,\,|\,\,
u < w < v
\,\,\}
$ is the *open interval* from $u$ to $v$.
A significant implication is that the M[ö]{}bius function is invariant with respect to reversing the order relation: let $\mu_{{\leqslant}}$ be the M[ö]{}bius function of $(P,{\leqslant})$ and let $\mu_{{\geqslant}}$ be the M[ö]{}bius function of the reversed poset $(P,{\geqslant})$; then, we have $$\mu_{{\leqslant}}(u,v)
=
\mu_{{\geqslant}}(v,u).$$
Non-crossing partitions in free probability
===========================================
Classical probability spaces $(\Omega, {\mathcal{F}}, \mathbf{P})$ can be reformulated using the commutative $C^*$-algebra ${\mathcal{A}}= L^{\infty}(\Omega, {\mathcal{F}}, \mathbf{P})$ as follows. Real valued (bounded) random variables correspond to elements of ${\mathcal{A}}$ and their expectations are given by evaluation of the linear functional $\varphi(a):= \int_{\Omega} a d \mathbf{P} $. The ’distribution’ of a random variable $a$ is the induced distribution $\mu_a(A):= \mathbf{P}(a^{-1}(A))$ and its $k$th moment is given by $\varphi(a^k) =\int_{\Omega} a^k d\mathbf{P}= \int_{\mathbb{R}}x^k\,\mu_a(dx) = \int_{\mathbb{R}}x
\,\mu_{a^k}(dx)$. This construction admits the following non commutative extension. Denote by $(M_d({\mathbb{C}}), \operatorname{tr}) $ the space of $d\times d$ complex matrices, together with the normalised trace and the usual matrix conjugation. Consider now the algebra of *random matrices* ${\mathcal{A}}:= M_d(L^{\infty}(\Omega, {\mathcal{F}}, \mathbf{P}))$ together with the linear functional $\varphi(a) := \int_{\Omega}\operatorname{tr}(a)
d\mathbf{P}$.
This represents a genuine *non-commutative $C^*$-probability space* $({\mathcal{A}}, \varphi )$, which is a unital $C^*$-algebra over ${\mathbb{C}}$ together with a unital and tracial positive linear functional $\varphi : {\mathcal{A}}\to {\mathbb{C}}$, that is $$\varphi(1) = 1, \qquad \varphi(a^*a){\geqslant}0, \quad \varphi(ab) = \varphi(ba), \qquad \text{for all } a,b \in {\mathcal{A}}.$$ Furthermore, we shall assume that $\varphi$ is faithful, that is $\varphi(a^*a)=0$ is equivalent to $a=0$. See the survey [@a4-c3-c13-Voiculescu:2000].
Many constructions in non commutative probability are parallel to those in classical probability, and this is also reflected in the notation: If $a$ is a self-adjoint element in ${\mathcal{A}}$, i.e. $a^*=a$, the value $\varphi(a)$ is sometimes called the *expectation* of $a$, the values $\varphi(a^k)$, $k \in {\mathbb{N}}$, are called the *moments* of $a$, and the compactly supported probability measure $\mu_a$ on ${\mathbb{R}}$ with $\int x^k \mu_a(dx) = \varphi(a^k)$, $k \in {\mathbb{N}}$, is also called the *distribution* of $a$ which always exists for self-adjoint elements in a $C^*$-probability space. If the measure $\mu_a$ admits a density $f_a$, the latter is also called the *density* of $a$. Similarly, given two self-adjoint elements $a$ and $b$ in ${\mathcal{A}}$, the *joint moments* of $a$ and $b$ are given by the values $\varphi(w)$, $w$ being a “word” in $a$ and $b$.\
Recall that a compactly supported Borel measure $\mu$ on ${\mathbb{R}}$ (and more generally any $\mu$ with $\int e^{zx}\mu(dx)$ locally analytic around $z=0$) is *uniquely* characterised by its moments $\int x^k \mu(dx)$ since then the Fourier transform of $\mu$ is a convergent power series with coefficients given by the moment sequence.
In order to define a corresponding notion of independence for self-adjoint elements (like that for random variables in classical probability theory), recall that two random variables $a,b\in L^{\infty}(\Omega, {\mathcal{F}}, \mathbf{P})$ endowed with expectation $\varphi$ as above are *independent*, if $\varphi(a^k b^l) = \varphi(a^k) \varphi(b^l)$ or equivalently $$\varphi\Big((a^k-\varphi(a^k))(b^l-\varphi(b^l))\Big)=0
\label{a4-c3-c13-eq:indep}$$ for all $k,l\in \mathbf{N}_{0}$.
Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ denote unital sub-algebras in ${\mathcal{A}}$, for instance generated by elements $a$ and $b$ respectively. They are called ‘free’ if the expectations of all products with factors alternating between elements from ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ vanish whenever the expectations of all factors vanish. Hence the elements $a,b \in {\mathcal{A}}$ are called *free* if $$\begin{gathered}
\label{a4-c3-c13-eq:free}
\varphi\big(
(a^{j_1}-\varphi(a^{j_1}))
(b^{k_1}-\varphi(b^{k_1}))\cdots
(a^{j_m}-\varphi(a^{j_m}))
(b^{k_m}-\varphi(b^{k_m})) \big) = 0 \end{gathered}$$ for all $m \in {\mathbb{N}}$ and all $j_1,\hdots,j_m,k_1,\hdots,k_m \in
{\mathbb{N}}$. Hence for $m=1$ this rule for the evaluation of joint moments coincides with the classical rule $\varphi(ab)=\varphi(a)\varphi(b)$ but is apparently different for $m>1$. The rules as well as allow to reduce by induction the evaluation of joint moments $\varphi(a^{j_1} b^{k_1} \cdots a^{j_m} b^{k_m})$ of these free or independent elements to the moments $\varphi(a^j)$ and $\varphi(b^k)$, which determine the marginal distribution of $a$ resp. $b$. Thus freeness may be regarded as a (non-commutative) analogue of the notion of independence in classical probability theory, allowing the development of a *free probability theory*. In particular allows to to compute the expectation of $\varphi((a+b)^n)$ for any $n\in {\mathbb{N}}$, $a\in {\mathcal{A}}_1$ and $b \in {\mathcal{A}}_2$, thus determining the distribution in the sense described above of the ‘free’ sum of $a$ and $b$ via the moments of $a$ and $b$ only. Hence, this assigns to compactly supported measures $\mu, \nu$ (with moments given by those of $a,b$) a *free additive convolution* $\mu\boxplus \nu$, see the survey [@a4-c3-c13-Voiculescu:2000]. This notion may be considered as an asymptotic limit of a corresponding notion for sequences of random matrices with independent entries of increasing dimension and their limiting spectral measures, [@sfb701 Chapter 1].
More generally, a set of unital sub-algebras ${\mathcal{A}}_j \subset {\mathcal{A}}, \,j\in I $, indexed by a set $I$, is called free if for any integer $k$ and $a_j \in {\mathcal{A}}_{i_j}, j=1, \ldots, k, i_j \in I$, $$\begin{aligned}
\varphi(a_1\ldots a_k)&=0 \quad \text{provided that}\quad \varphi(a_j) =0, \quad j=1, \ldots , k, \notag \\
&\quad \text{and} \quad i_1 \ne i_2, i_2 \ne i_3,\dots, i_{k-1} \ne i_k,\label{a4-c3-c13-eq:freek}
\end{aligned}$$ that is, all adjacent elements in $a_1 \ldots a_k$ belong to different sub-algebras $A_{j_i}$. This notion has similar properties as classical independence. For instance, polynomials $P(a_j)$ of free self-adjoint elements $a_j$ (generating a sub-algebra) are free again.
The density $\psi(x)=\frac 1{\sqrt{2\pi}} \exp(-x^2/2)$ defines the standard Gaussian distribution. Hence, the *classical central limit theorem (CLT)* may be stated for independent random elements $a_i, i \in {\mathbb{N}}$ from a commutative $C^*$-probability space $({\mathcal{A}}, \varphi)$ with *identical* distribution such that $\varphi(a_i)=0 ,\, \varphi(a_i^2)=1$ (such variables are called *standardised*).
The moments of the normalised sum $S_N:=\frac{a_1+\ldots +a_N} {\sqrt{N}}$ satisfy $$\lim_{N \to \infty} \varphi(S_N^k) = \int x^k \psi(x) \, dx, \quad k \in {\mathbb{N}}.$$
Consider *free* random elements $a_i$ from a (non-commutative) $C^*$-probability space $({\mathcal{A}}, \varphi)$ , standardized via $\varphi(a_i)=0 , \varphi(a_i^2)=1$ with *identical distribution*, that is $\varphi(a_j^l)$ depends on $l$ only. In order to describe a corresponding *free* ‘central limit theorem’ for this setup we have to determine the asymptotic behaviour of moments of type $\varphi(a_{i_1}\ldots a_{i_k})$ subject to the assumption of freeness .
Note that by freeness all mixed moments vanish provided an element $a_j$ occurs only once in the product vanish. (Note that this holds as well for mixed moments of independent random variables). Thus, we only need to consider mixed moments with factors occurring at least twice. For a product $a_{i_1}\cdots a_{i_k}$ of $k$ factors, such that $s$ of them, say $b_1, \ldots, b_s$, are different, let ${ p}=\{B_1,\ldots,B_s\}$ denote the corresponding partition of the set $\{1,\ldots, k\}$ into $|{ p}|:=s$ nonempty blocks $B_j$ of the positions of $b_j$ in $1{\leqslant}j {\leqslant}k$. One can show by induction that all mixed moments of free or independent elements $\varphi(a_{i_1}a_{i_2} \cdots a_{i_k})$ where $1{\leqslant}i_j {\leqslant}N$, can be computed via resp. as above also for $s {\geqslant}2$ in terms of moments $c_l=\varphi(b_j^l)$ for $j=1,\ldots, s$ which depend on $l$ only by the assumption of identical distribution. Thus these mixed moments depend *on the partition scheme of* $i_1, \dots, i_k$, say ${{ p}}$, only and will be denoted by $m_{{ p}}$. The number of such mixed moments in $a_1, \ldots, a_N$ corresponding to a given partition scheme depends on $|{ p}|$ only and is given by $A_{N,{ p}}=N(N-1)\cdots (N-|{ p}|+1)$ . Thus $$\label{a4-c3-c13-eq:moment}
\varphi(S_N^k) = \sum_{{ p}} m_{{ p}} A_{N,{ p}}N^{-k/2}.$$ For a partition ${ p}$ we have $A_{N,{ p}} < N^{|{ p}|}$. If all parts of ${ p}$ satisfy $|B_j|{\geqslant}2 $ and one block is of size at least three, the corresponding contribution in is of order $|m_{{ p}}| A_{N,{ p}}N^{-k/2}{\leqslant}|m_{{ p}}| N^{-1/2}$, that is all these terms are asymptotically negligible as $N$ tends to infinity.
Hence, computing the asymptotic limit of $\varphi(S_N^k)$ reduces to considering all mixed moments of $k$ factors with each random element occurring precisely *twice*, a consequence being that $\lim_{N\to\infty}\varphi(S_N^k) = 0$ for $k$ odd.
Recall that $ \operatorname{NC}(n)$ denoted the lattice of all non-crossing partitions on the set $[n]=\{
1,\ldots,n\}$. Furthermore, let $ \operatorname{NC}_{2}(2k)$ denote the subset of non-crossing partitions with blocks of size 2 only, called ’non-crossing pair partitions’ on a set of $2k$ elements.
Now consider as an example three free standardised variables $a,b, c$. Then the product $abc^2ab$ corresponds to a pair partition with a *crossing*, that is ${
p}=\{\{1,5\},\{3,4\},\{2,6\}\}$. Hence $\varphi(abc^2ab)=\varphi(abab) \varphi(c^2)= 0$ by freeness, that is . Otherwise for a non-crossing pair partition like $ca^2b^2c$ we have $\varphi(ca^2b^2c)=\varphi(cb^2c)\varphi(a^2)=\varphi(cc)\varphi(b^2)=1$. These simple observations can be generalised by induction in the following Lemma to determine the values of joint moments $m_{p}= \varphi(a_{i_1}a_{i_2} \cdots a_{i_k})$ for pair partitions $p$ of free variables.
For *any* pair partition ${ p}$, $$m_{p}=\begin{cases}0 & \text{if $p$ has a crossing }\\
1 & \text{if $p$ is non-crossing.} \end{cases}$$
Thus, we conclude from and the previous results that $$\lim_{N\to \infty}\varphi(S_N^{2k})=\lim_{N\to \infty} \sum_{{ p} \in \operatorname{NC}_{2}(2k)} \frac{A_{N, { p}}}{N^{k/2}}= | \operatorname{NC}_{2}(2k)|.$$ Furthermore, one shows that $$C_k:=| \operatorname{NC}_{2}(2k)|=| \operatorname{NC}(k)|,$$ where $C_k=\frac{1}{k+1} \binom{2k}{k}$ is the $k$th Catalan number. Among its numerous interpretations, it represents as well the $2k$th moment of a compactly supported measure with density $w(x):=\frac 1{2 \pi} (4-x^2)^{1/2}, |x| {\leqslant}2$. This is the so-called *Wigner* measure or semi-circular distribution. See [@a4-c3-c13-Nica-Speicher:2006 Rem. 9.5].
Now the *free central limit theorem* for a sequence of free variables $a_j, j \in {\mathbb{N}}$, which are standardised via $\varphi(a_j)=0, \, \varphi(a_j^2)=1$, and $S_N:=\frac{a_1+\ldots +a_N} {\sqrt{N}}$ may be stated as follows.
$S_N$ converges in distribution to $w$ which serves as the Gaussian distribution in free probability, i.e. $$\label{a4-c3-c13-eq:momentfree}
\lim_{N \to \infty} \varphi(S_N^k) = \int x^k w(x) dx, \quad k \in {\mathbb{N}}.$$
This means e.g. that the rescaled sum $(a_1+a_2)/\sqrt2$ of two free elements $a_1,a_2$ of a non-commutative probability space $({\mathcal{A}}, \varphi)$ which both have density $w(x)$ again has a Wigner distribution. In free probability an element $s$ of $({\mathcal{A}}, \varphi) $ with density $w(x)$ is called *semi-circular* and its moments are given by $$\label{a4-c3-c13-eq:smoment}
\varphi(s^n) = \begin{cases} \frac{1}{k+1} \binom{2k}{k}, &\mbox{if}\quad n=2k,\\
0, & \mbox{if} \quad n \text{ odd}. \end{cases}$$
Recall that $a\in ({\mathcal{A}}, \varphi)$ is called *positive* if there exists an $c \in ({\mathcal{A}}, \varphi)$ with $a=c^* c$ . Thus $a$ is self-adjoint. Define the *free multiplicative convolution* of two compactly supported measures $\mu_a,\mu_b$, of *positive* free elements $a, b \in ({\mathcal{A}}, \varphi)$, say $\mu_a\boxtimes \mu_b$, as follows by specifying its moments. Since in a $C^*$-probability space ${\mathcal{A}}$ positive square roots $ a^{1/2}$ resp. $b^{1/2}$ of $a$ resp. $b$ as well as the positive element $p_{a,b}:= a^{1/2}ba^{1/2}$ are again in ${\mathcal{A}}$, we may define $\mu_a\boxtimes \mu_b$ by: $$\int x^k d \mu_a\boxtimes \mu_b (x) := \varphi(p_{a,b}^k), \qquad k \in {\mathbb{N}}.$$ Since $\varphi(p_{a,b}^k)= \varphi(p_{b,a}^k), \, k \in {\mathbb{N}}$, because $\varphi$ is tracial, i.e. $\varphi(ba)=\varphi(ab)$, we conclude that the free convolution $\boxtimes$ is commutative. By the same tracial property and the relation of freeness, we show that $\varphi(p_{a,b}^k)= \varphi((ab)^k)$ and this implies the associativity of $\boxtimes$. Moreover it follows from this representation that the multiplicative convolution measure $\mu_a\boxtimes \mu_b$ is uniquely determined by the distributions of $\mu_a$ and $\mu_b$.
In order to effectively compute both additive and multiplicative convolution of measures, one needs more properties of the lattice of partitions of $1, \ldots, n$ into blocks and the subset of non-crossing partitions together with the notion of multi-linear cumulant functionals. As above let $B_j, j=1,\ldots s$ denote the blocks of a partition ${ p}\in \operatorname{NC}(n)$ of $1, \ldots, n$.
For ${p}\in\operatorname{NC}(n)$, the *free mixed cumulants* are multi-linear functionals $\kappa_{{p}}: {\mathcal{A}}^n \to
{\mathbb{C}}$ defined in terms of a moment decomposition using the M[ö]{}bius function $ \mu({ q}, { p})$ of the lattice of non-crossing partitions $ \operatorname{NC}(n)$. We define the general mixed cumulant functionals $\kappa_{{ p}}$ as follows: $$\begin{aligned}
\label{a4-c3-c13-eq:cumdef}
\kappa_{{ p}}[a_1, \ldots ,a_n] &
=
\sum_{ q \in \operatorname{NC}(n), { p}\preceq{ q}}
\varphi_{{ q}}[a_1, \ldots, a_n] \,
\mu({ p}, { q}), \quad \text{where} \\
\varphi_{ q}[a_1, \ldots, a_n] &
:=
\varphi\left(\prod_{k\in B_{1}} a_k\right)
\cdots \varphi\left(\prod_{k\in B_{s}} a_k\right),\notag
\end{aligned}$$ and the products $\prod_{k\in B_{j}} a_k$ repeat the order of indices within the block $B_j$. Note that by Hall’s theorem, the coefficient $ \mu( p, q)$ can also be written as $ \mu_{\succeq}( q, p)$ using the relation of reversed refinement (see Remark \[a4-c3-c13-reversing-order\]).
Then one shows, see [@a4-c3-c13-Nica-Speicher:2006 Prop. 11.4], that $$\label{a4-c3-c13-eq:momcum}
\varphi(a_1 \cdots a_n) = \sum_{{ p}\in \operatorname{NC}(n)} \kappa_{{ p}}[a_1,\ldots, a_n].$$
In the special case $p={1_n}$ we write $\kappa_n$ instead of $\kappa_{1_n}$. The following lemma is proved by induction on $n$.
\[a4-c3-c13-freechar\] The elements $a_1, \ldots, a_n \in {\mathcal{A}}$ are free if and only if all mixed cumulants satisfy $$\kappa_n[a_{j_1}, \ldots ,a_{j_k}] =0,$$ whenever $ a_{j_1}, \ldots a_{j_k}$, $1{\leqslant}j_l{\leqslant}n, 1{\leqslant}k{\leqslant}n$ contains at least two different elements.
In contrast to , this characterisation of freeness holds even if the $\varphi(a_j)$ are non-zero.
For a partition ${ p} \in \operatorname{NC}(n)$, recall that ${ p}^\mathrm{c}$ denotes its *Kreweras* complement in $ \operatorname{NC}(n)$. Then, one shows that for free elements $a,b$ the following recursion involving the Kreweras complement holds: $$\label{a4-c3-c13-eq:recursion}
\kappa_n[ab, \ldots, ab]= \sum_{{ p} \in \operatorname{NC}(n)} \kappa_{{ p}}[a,\ldots,a] \kappa_{{ p}^\mathrm{c}}[b,\ldots, b].$$ See [@a4-c3-c13-Nica-Speicher:2006 Rem. 14.5]. This entails that the cumulants of $ab$ and thus by the moments of $ab$ are indeed determined by multi-linear functionals of $a$ and $b$ alone which again by virtue of are determined by the moments of $a$ together with the moments of $b$. The recursive equation and the definition of cumulants may be conveniently encoded as algebraic relations between the following formal generating series. For $a \in {\mathcal{A}}$ let $M_a(z)= \sum_{n=1}^\infty \varphi(a^n)z^n$ denote the moment generating series and with $\kappa_n(a):=\kappa_n[a,\ldots,a]$ let $R_a(z):= \sum_{n=1}^{\infty} \kappa_n(a)z^n$ and ${\mathcal{R}}_a(z):= z^{-1}R_a(z)$ denote cumulant generating series. In particular, for free self-adjoint $a,b \in {\mathcal{A}}$ we get by binomial expansion of $\kappa_n(a+b)$ and Lemma \[a4-c3-c13-freechar\] that $\kappa_n(a+b)=\kappa_n(a)+\kappa_n(b)$ and furthermore, as shown in [@a4-c3-c13-Nica-Speicher:2006 Lect. 12],
One has the following identities: $$\begin{aligned}
R_{a+b}(z) &= R_a(z)+R_b(z),\label{a4-c3-c13-eq:rhom} \\
R_a(zM_a(z)+z) &= M_a(z),\quad G_a\Big(\frac{1+R_a(z)}{z}\Big) =z,\label{a4-c3-c13-eq:rdef}
\end{aligned}$$ where $$G_a(z):=\frac{1}{z}+ \sum_{n=1}^\infty
\frac{\varphi(a^n)}{z^{n+1}}=\frac1 z \Big(1+M_a(\frac
1 z)\Big),$$ can be identified with the Cauchy transform of the corresponding spectral measure $\mu_a$, that is $$G_a(z)=\int_{\mathbb{R}}\frac{d\mu_a(t)}{z-t}.$$
Hence the so-called R-transform ${\mathcal{R}}$ of a spectral measure $\mu_a$, introduced by Voiculescu in [@a4-c3-c13-Voiculescu:2000], is determined analytically by the inverse function of the Cauchy transform of $\mu_a$ on the complex plane which is the starting point of the complex analytic theory of the asymptotic approximations of free additive convolution as developed in [@a4-c3-c13-CG11; @a4-c3-c13-CG08a; @a4-c3-c13-CG08b; @A4-CG13]. Assuming that $\kappa_1=m_1 \ne 0$, $R_{\mu_a}(z):=R_a(z)$ admits a formal inverse power series $R_a^{(-1)}(z)$. This may be defined via the inverse function of the Cauchy transform of $\mu_a$, which is well defined in a certain region in ${\mathbb{C}}$.
The so-called *$S$-transform* $$\label{a4-c3-c13-eq:rsrel}
S_a(z):=\frac1 zR_a^{(-1)}(z) =\frac{1+z}{z} M_a^{(-1)}(z),$$ of Voiculescu is a multiplicative homomorphism for free multiplicative convolution. That is, see [@a4-c3-c13-Nica-Speicher:2006 Lect. 18], one has the following result.
For two free self-adjoint positive elements $a,b\in {\mathcal{A}}$, one has $$S_{ab}(z)= S_a(z) S_b(z)$$
Since $S_a$ is determined by the spectral measure of $a$, this means with $S_{\mu_a}:=S_a$ for measures $\mu =\mu_a$, $\nu=\mu_b$ we have $ S_{\mu\boxtimes \nu}(z)= S_{\mu}(z) S_{\nu}(z)$, which uniquely determines the multiplicative free convolution $\mu\boxtimes \nu$ in terms of the measures $\mu$ and $\nu$ on the positive reals via the characterising property of the $S$-transform.
Note that by , Let $s$ be a semi-circular element as in . Then the moment generating functions of $s$ and $s^2$ are given by $M_{s}=f(z^2)$ and $M_{s^2}(z)=f(z)$ respectively, where $f(z) = (1- \sqrt{1- 4z})/(2z) -1$. The corresponding distribution of $s^2$ is called Marchenko-Pastur or free Poisson law; it is given by the density $p(x):=\frac 1 {2\pi} \sqrt{4/x-1}$ on the interval $[0,4]$. Via the inverse function $f^{(-1)}(z)=z(1 + z)^{-2}$ of $f$ we obtain in view of , $$\label{a4-c3-c13-eq:smp}
S_{s^2}(z)=f^{(-1)}(z)\frac{1+z}{z}= \frac 1{1+z}$$ and hence in view of again $R_{s^2}^{(-1)}(z)=\frac{z}{1+z}$ or $R_{s^2}(z)= \frac{z}{1-z}$, whereas from we deduce with $g(z):=z(1+M_a(z))$ and $g^{(-1)}(z)=\frac{z}{1+z^2}$ and hence $R_s(z)= \frac{z} {g^{(-1)}(z)} -1=z^2$.
From here, we obtain for free variables $t_1, \ldots, t_l$ with identical distribution given by $s^2$, the so-called Marchenko–Pastur distribution, in view of $$S_{t_1\ldots t_l}(z) = S_{t_1}(z)^l = \frac{1} {(1+z)^l},$$ which determines the so-called free Bessel distributions, $\mu_l$ with support in $[0,K_l]$, $K_l=(l+1)^{l+1}/l^l$. Their moments are given by the so called Fuss–Catalan numbers, that is, if an element $a\in {\mathcal{A}}$ has $S$-transform $S_a(z)=\frac{1}{(1+z)^l}$ we have $$\varphi(a^k)=\frac{1}{lk+1}\binom{lk+1}{k}=:C_{k,l}, \qquad \text{for all } \, k {\geqslant}1.$$ The proof is based on combinatorial properties of non crossing partitions, see [@a4-c3-c13-Banica-Belinschi-Capitaine-Collins:2011].
For a sequence of $N\times N$ independent non-Hermitian random matrices, $G_1, \ldots G_l$, with independent Gaussian centered entries with variance $1/N$, let $W:=G_1\cdots G_l$. Consider the normalised moments of $W W^*$. As $N\to \infty$ they converge as follows: $$\label{a4-c3-c13-eq:rmtmom}
\lim_{N\to \infty}\frac 1 N \int_{\Omega} \operatorname{tr}(W W^*)^l \, d\mathbf{P} = \int_0^{K_l} x^k d\mu_l = C_{k,l}$$
This can be shown by induction, using $$\operatorname{tr}(WW^*)^k= \operatorname{tr}(G_1(G_2\cdots G_l G_l^* \cdots G^*_1G_1)^{l-1}G_2\cdots G_l\cdots G^*_l \cdots G^*_1),$$ which by moving $G_1$ to the right yields $$\begin{aligned}
& \operatorname{tr}((G_2\cdots G_l G_l^* \cdots G^*_1G_1)^{l-1}
G_2\cdots G_l\cdots G^*_l \cdots G^*_1G_1)
\\
& \qquad\qquad\qquad\qquad\qquad\qquad = \operatorname{tr}((G_2\cdots G_l G_l^* \cdots G^*_1G_1)^l)
\\
& \qquad\qquad\qquad\qquad\qquad\qquad = \operatorname{tr}\Bigl((G_2\cdots G_l G_l^* \cdots G_2^*)(G_1^*G_1)\Bigr)^l.\end{aligned}$$ Since $(G_2\cdots G_l G_l^* \cdots G_2^*)$ and $G_1^*G_1$ are asymptotically free of this volume, we get by induction for the asymptotic distribution of $\pi_l$ the recursion $\pi_l = \pi_{l-1}\boxtimes\pi_1$, where $\pi_1$ can be identified with the limiting Marchenko–Pastur distribution of $G_1G_1^*$. For arbitrary $N\times N$ independent Wigner matrices (which are Hermitian matrices with entries which are independent random variables unless restricted by symmetry) the relation has been shown by combinatorial techniques after an appropriate regularization in [@a4-c3-c13-AGT10]. For more details on the asymptotic spectral distribution of products of so-called Girko–Ginibre matrices (having independent and identically distributed random entries) and their inverses using the free probability calculus, see [@a4-c3-c13-GKT15]. Strictly speaking one needs to extend the non-commutative $C^*$-probability spaces to spaces of unbounded operators to include distributions with non-compact support like those of Gaussian matrices see e.g. [@a4-c3-c13-CG11].
Remarkably, the same results hold for powers instead of products. Since $G_1^{l-1} (G_1^{l-1})^*$ and $G_1^* G_1$ are also asymptotically free, a similar argument as above shows that the asymptotic distribution of $(G_1^l) (G_1^l)^*$ is also given by $\pi_l$. Similarly as above, these results also extend to powers of non-Gaussian random matrices.
The calculus of $S$-transforms may even be used to describe the asymptotic spectral measure of $W W^*$ when some of the factors in $W = G_1 \cdots G_l$ are inverted, after appropriate regularisation of the inverse matrices [@a4-c3-c13-GKT15]. For instance, for $W = G_1 G_2^{-1}$, the limiting distribution of $W W^*$ is given by the square of a Cauchy distribution.
Moreover, the calculus of $R$-transforms makes it possible, at least in principle, to deal with the case where $W$ is a sum of independent products as above [@a4-c3-c13-KT15]. For instance, for $W = G_1 G_2^{-1} + G_3 G_4^{-1}$, the limiting distribution of $W W^*$ is also given by the square of a Cauchy distribution. This is related to the Cauchy distribution being “stable” under free additive convolution.
Braid groups {#a4-c3-c13-braid-groups}
============
Let $\mathbb{D}$ be the unit disk. The *braid group* ${\mathcal{B}}_{n}$ on $n$ strands can be defined as the fundamental group of the *configuration space* $$X_{n}:=
\{\,\,
\{\,z_{1},\ldots,z_{n}\,\}
\subset
\mathbb{D}
\,\,|\,\,
z_{i}\neq z_{j}
\text{\ for\ }
i\neq j
\,\,\}$$ of unordered $n$-point-subsets in $\mathbb{D}$. One can visualize a path in $X_{n}$ as a collection of $n$ distinct points moving continuously in $\mathbb{D}$ subject only to the restriction that points are not allowed to collide. Since $X_{n}$ is connected, the braid group (up to isomorphism) does not depend on the choice of a base point.
We find it convenient to choose as the base point a set $
S
=
\{\,
v_{1},\ldots,v_{n}
\,\}
$ of $n$ points on the boundary circle $\partial\, \mathbb{D}$ numbered in counter-clockwise order.
Then, we regard $\operatorname{NC}(n)$ as the poset of non-crossing partitions of the set $S$, i.e., for any two distinct blocks of the partition, their convex hulls do not intersect. A non-crossing partition $p\in\operatorname{NC}(n)$ can be interpreted as a braid on $n$ strands as follows: for each block $
B=
\{\,
v_{\alpha_{1}},\ldots,v_{\alpha_{k}}
\,\}
$, consider the counter-clockwise rotation of the block by one step: $$\varrho_{B} :
v_{\alpha_{1}} \mapsto
v_{{\alpha_{2}}} \mapsto \cdots \mapsto
v_{\alpha_{k}} \mapsto v_{\alpha_{1}}$$ The product $$\sigma_{p}
:=
\prod_{B\,:\text{\ block of\ }p}
\varrho_{B}$$ describes a loop in the configuration space $X_{n}$, which does not depend (up to homotopy relative to the basepoint) on the order of factors. We identify it with the corresponding element of the fundamental group ${\mathcal{B}}_{n}$.
\[a4-c3-c13-artin-presentation-braid-groups\] The braid group ${\mathcal{B}}_{n}$ is generated by the braids $\sigma_{i}$ corresponding to the counter-clockwise rotations $
v_{i}\mapsto v_{i+1}\mapsto v_{i}
$ for $i=1,\ldots,n-1$.
In terms of these generators, the braid group ${\mathcal{B}}_{n}$ admits the following presentation: $${\mathcal{B}}_{n}
=
\left\langle\,{
\sigma_{1},\ldots,\sigma_{n-1}
}\,\,\,\vrule\,\,
\begin{array}{ll}
\sigma_{i}\sigma_{j}
=
\sigma_{j}\sigma_{i}
&
\text{\ for\ }
|i-j| \geqslant 2
\\
\sigma_{i}\sigma_{j}\sigma_{i}
=
\sigma_{j}\sigma_{i}\sigma_{j}
&
\text{\ for\ }
|i-j| =1
\end{array}
\,\right\rangle$$
There is an obvious homomorphism $$\pi:
{\mathcal{B}}_{n}\longrightarrow S_{n}$$ from the braid group on $n$ strands to the symmetric group on $n$ letters. A braid corresponds to a motion of the $n$ points $v_{1},\ldots,v_{n}$, and at the end of this motion, the dots may have changed positions. This way, each braid induces a permutation.
The homomorphism $
\pi:
{\mathcal{B}}_{n}\longrightarrow S_{n}
$ is onto. On the level of presentations, it amounts to making the generators $\sigma_{i}$ involutions. Formally: the symmetric group has the presentation $$S_{n} =
\left\langle\,{
s_{1},\ldots,s_{n-1}
}\,\,\,\vrule\,\,
\begin{array}{ll}
s_{i}s_{j}
=
s_{j}s_{i}
&
\text{\ for\ }
|i-j| \geqslant 2
\\
s_{i}s_{j}s_{i}
=
s_{j}s_{i}s_{j}
&
\text{\ for\ }
|i-j| =1
\\
s_{i}=s_{i}^{-1}
&
\text{\ for all\ }i
\end{array}
\,\right\rangle$$ and the homomorphism $\pi$ is sending $\sigma_{i}$ to $s_{i}$.
*Strand diagrams* are another frequently used visual representation of braids. Recall that a braid is given by a path in configuration space, i.e. the simultaneous motion of $n$ points in the disk $\mathbb{D}$. Parametrizing time by a real number in $[0,1]$, each of those moving points traces out a “strand” in $\mathbb{D}\times[0,1]$. The diagrams we have used so far can be regarded as a “top view” onto the cylinder $\mathbb{D}\times[0,1]$. A strand diagram is a view from the front. Here, it is useful to put the initial configuration $U$ with the hemicircle fully visible from the front. Figure \[a4-c3-c13-fig-strand-diagram\] shows the two representations of the generator $\sigma_{2}$ in ${\mathcal{B}}_{5}$. Here, the generator $\sigma_{i}$ corresponds to a *crossing* of the $i^\text{th}$ and the $(i+1)^\text{th}$ strands. The left strand runs over the right strand. We call such a crossing *positive*. The inverses of the generators correspond to *negative* crossings.
A classifying space for the braid group {#a4-c3-c13-classifingy-space-for-braid-group}
---------------------------------------
Tom Brady [@a4-c3-c13-Brady2001] has given a construction of a classifying space for braid groups that is strongly related to non-crossing partitions and has found some interesting applications.
Recall that the *Cayley graph* $
\operatorname{CG}_{\Sigma}(G)
$ of a group $G$ relative to a specified generating set $\Sigma$ is the graph with vertex set $G$ and edges connecting $g$ to $gx$ for any $g\in G$ and $
x\in\Sigma\setminus\{\,1\,\}
$. Note that the requirement $x\neq 1$ rules out loops. Obviously, there is more structure here: the edge is oriented from $g$ to $gx$ and should be regarded as labeled by the generator $x$.
Since $\Sigma$ is a generating set for $G$, the Cayley graph $\operatorname{CG}_{\Sigma}(G)$ is connected: if we can write an element $g$ as a word $$g
=
x_{1}^{\varepsilon_{1}}
\cdots
x_{k}^{\varepsilon_{k}}$$ in the generators and their inverses, then $$1
\,\,-\negthinspace\negthinspace\negthinspace-\,\,
x_{1}^{\varepsilon_{1}}
\,\,-\negthinspace\negthinspace\negthinspace-\,\,
x_{1}^{\varepsilon_{1}}
x_{2}^{{\varepsilon_{2}}}
\,\,-\negthinspace\negthinspace\negthinspace-\,\,
x_{1}^{\varepsilon_{1}}
x_{2}^{{\varepsilon_{2}}}
x_{3}^{{\varepsilon_{3}}}
\,\,-\negthinspace\negthinspace\negthinspace-\,\, \cdots \,\,-\negthinspace\negthinspace\negthinspace-\,\,
g$$ is an edge path connecting the identity element $1$ to $g$. Note that the exponents of the generators tell us whether to traverse edges with or against their orientation.
There are two generating sets for the braid group (and the symmetric group) of particular interest to us. First, we consider the digon generators $
\sigma_{ij}
$ corresponding to the counter-clockwise rotation $
v_{i}\mapsto v_{j}\mapsto v_{i}
$. Let ${\mathcal{B}}^{*}_{n}$ be the Birman–Ko–Lee-monoid [@a4-c3-c13-BKL Section 2], i.e., the monoid generated by all the $\sigma_{ij}$. We remark that ${\mathcal{B}}^{*}_{n}$ is strictly larger than the submonoid of positive braids (those that can be drawn using positive crossings only), which is the monoid generated by the $\sigma_{i}$. We define a partial order on the braid group by: $$\beta{\leqslant}\beta'
\quad:\Longleftrightarrow\quad
\beta^{-1}\beta'\in{\mathcal{B}}^{*}_{n}$$
The image $s_{ij}\in S_{n}$ of $\sigma_{ij}$ in the symmetric group is a transposition. Consider the Cayley graph of the symmetric group $S_{n}$ with respect to the generating set $T\subseteq S_{n}$ of all transpositions. We define a partial order, called the *absolute order*, on $S_{n}$ as follows: For permutations $\xi,\psi\in S_{n}$ we declare $\xi{\leqslant}_{T}\psi$ if there is a geodesic (i.e., shortest possible) path in the Cayley graph connecting the identity 1 to $\psi$ and passing through $\xi$.
Our largest generating set is: $$\Gamma_{n} :=
\{\,\,{
\sigma_{p}
}\,\,|\,\,
p\in\operatorname{NC}(n)
\,\,\}
\subseteq{\mathcal{B}}_{n}$$ which is in $1$-$1$ correspondence to the non-crossing partition lattice. Let $s_{p}$ denote the image of $\sigma_{p}$ in the symmetric group $S_{n}$. It turns out that the subset $
\{\,\,{ s_{p} }\,\,|\,\,p\in\operatorname{NC}(n)\,\,\}
\subseteq
S_{n}
$ is the order ideal of the $n$-cycle $
1\mapsto2\mapsto\cdots\mapsto n\mapsto 1
$ with respect to the partial order ${\leqslant}_{T}$ just defined, that is the subset consists of all elements in $S_{n}$ bounded above by the $n$-cycle. In fact, we have isomorphisms of various posets:
\[a4-c3-c13-set\] Let $p,q\in\operatorname{NC}(n)$. Then the following are equivalent:
1. \[a4-c3-c13-set-a\] In $\operatorname{NC}(n)$, we have $p\preceq q$.
2. \[a4-c3-c13-set-b\] In $\Gamma_{n}$, the element $\sigma_{p}$ is a left-divisor of $\sigma_{q}$, i.e., there exists $r\in\operatorname{NC}(n)$ such that $$\sigma_{q}=\sigma_{p}\sigma_{r}$$
3. \[a4-c3-c13-set-c\] In $\Gamma_{n}$, the element $\sigma_{p}$ is a right-divisor of $\sigma_{q}$, i.e., there exists $r\in\operatorname{NC}(n)$ such that $$\sigma_{q}=\sigma_{r}\sigma_{p}$$
4. \[a4-c3-c13-set-d\] In the braid group ${\mathcal{B}}_{n}$, we have $\sigma_{p}{\leqslant}\sigma_{q}$.
5. \[a4-c3-c13-set-e\] In the symmetric group $S_{n}$, we have $s_{p}{\leqslant}_{T}s_{q}$.
Thus, on $\Gamma_{n}$ the three partial orderings given by left-divisibility, right-divisibility, and the partial order ${\leqslant}$ from ${\mathcal{B}}_{n}$ coincide. Moreover, we have isomorphisms $$\operatorname{NC}(n)
\cong
\{\,\,{ \sigma_{p} }\,\,|\,\,p\in\operatorname{NC}(n)\,\,\}
\cong
\{\,\,{ s_{p} }\,\,|\,\,p\in\operatorname{NC}(n)\,\,\}$$ of posets.
$$\begin{aligned}
\begin{array}{c}
\raisebox{0pt}{\includegraphics{a4_c3_c13_figure-7-eps-converted-to.pdf}}
\end{array}
&=
\begin{array}{c}
\raisebox{0pt}{\includegraphics{a4_c3_c13_figure-8-eps-converted-to.pdf}}
\end{array}
\circ
\begin{array}{c}
\raisebox{0pt}{\includegraphics{a4_c3_c13_figure-9-eps-converted-to.pdf}}
\end{array}
\\
&=
\begin{array}{c}
\raisebox{0pt}{\includegraphics{a4_c3_c13_figure-10-eps-converted-to.pdf}}
\end{array}
\circ
\begin{array}{c}
\raisebox{0pt}{\includegraphics{a4_c3_c13_figure-11-eps-converted-to.pdf}}
\end{array}
\end{aligned}$$
Consider the non-crossing partititions $$p :=
\{\,
\{\,1,2,8\,\},
\{\,3,5\,\},
\{\,4\,\},
\{\,6\,\},
\{\,7\,\}
\,\}
\quad\text{and}\quad
q :=
\{\,
\{\,1,2,6,7,8\,\},
\{\,3,5\,\},
\{\,4\,\}
\,\}$$ in $\operatorname{NC}(8)$. Here, $p{\leqslant}q$ holds and we expect $\sigma_{p}$ to be a left- and right-divisor of $q$ within $\Gamma_{8}$. Figure \[a4-c3-c13-division\] shows the corresponding factorisations. One can interpret the complementary divisors as the blockwise Kreweras complements. In particular, the Kreweras complement yields factorisations of the maximal element in $\Gamma_{n}$.
The braid group ${\mathcal{B}}_{n}$ has a particularly nice presentation over the generating set $\Gamma_{n}$:
\[a4-c3-c13-triangular-presentation\] The valid equations $$\label{a4-c3-c13-triangular-relations}
\sigma_{1}\sigma_{2}=\sigma_{3}
\text{\ for\ }
\sigma_{1},\sigma_{2},\sigma_{3}\in\Gamma_{n}\setminus\{\,1\,\}$$ are a defining set of triangular relations for the braid group $
{\mathcal{B}}_{n}
$ with respect to the generating set $
\Gamma_{n}\setminus\{\,1\,\}
$.
Let $\tilde{\Gamma}_{n}$ be the Cayley graph of the braid group ${\mathcal{B}}_{n}$ with respect to the generating set $\Gamma_{n}\setminus\{\,1\,\}$. A *clique* in $\tilde{\Gamma}_{n}$ is a set of vertices that are pairwise connected via an edge. As a directed graph, $\tilde{\Gamma}_{n}$ does not have oriented cycles and each clique is totally ordered by the orientation of edges. Thus, a clique is of the form $$\{\,
\beta,
\beta\sigma_{{p_{1}}},
\beta\sigma_{{p_{2}}},
\ldots,
\beta\sigma_{{p_{k}}}
\,\}$$ where $
p_{1}\prec p_{2}
\prec\cdots\prec
p_{k}
$ is an ascending chain in $\operatorname{NC}(n)$, and $\beta \in {\mathcal{B}}_{n}$ is some element. We denote by $\tilde{Y}_{n}$ the simplicial complex of cliques (also known as the *flag complex* induced by the graph) in $\tilde{\Gamma}_{n}$. In particular, $\tilde{\Gamma}_{n}$ is the $1$-skeleton of $\tilde{Y}_{n}$.
\[a4-c3-c13-dimension-of-brady-complex\] All maximal chains in $\operatorname{NC}(n)$ have length $n$. Hence, all maximal simplices in $\tilde{Y}_{n}$ have dimension $n$.
The most important fact about $\tilde{Y}_{n}$ is its contractibilty.
\[a4-c3-c13-brady-main\] The clique complex $
\tilde{Y}_{n}
$ is contractible, and the braid group ${\mathcal{B}}_{n}$ acts freely on it. Consequently, the orbit space $$Y_{n} :=
{\mathcal{B}}_{n}\setminus\tilde{Y}_{n}$$ is a classifying space for the braid group ${\mathcal{B}}_{n}$.
Higher generation by subgroups
------------------------------
For a subset $
I\subseteq
\{\,
1,\ldots,n
\,\}
$ let ${\mathcal{B}}_{n}^{I}$ be the subgroup of ${\mathcal{B}}_{n}=\pi_1(X_{n})$ given by those paths, where the points in $
\{\,\,{v_{i}}\,\,|\,\,i\in I\,\,\}
$ do not move at all. For $k\in\{\,1,\ldots,n\,\}$, we put $
{\mathcal{B}}_{n}^{k}
:=
{\mathcal{B}}_{n}^{\{\,v_{k}\,\}}
$, i.e., ${\mathcal{B}}_{n}^{k}$ is the group of braids where the $k^{\text{th}}$ strand is rigid. It is, one might say, a group on $n-1$ strands and one rod. However, since $v_{k}$ is a point on the boundary $\partial\, \mathbb{D}$, braiding with the rod is impossible. Thus, ${\mathcal{B}}_{n}^{k}$ really is just an isomorphic copy of ${\mathcal{B}}_{n-1}$ inside of ${\mathcal{B}}_{n}$. Similarly, $
{\mathcal{B}}_{n}^{I}
=
\bigcap_{{
k\in I
}}
{\mathcal{B}}_{n}^{k}
$ is isomorphic to ${\mathcal{B}}_{n-\# I}$.
Let $\operatorname{NC}^{k}(n)$ be the lattice of those non-crossing partitions in $\operatorname{NC}(n)$ where the singleton $\{\,k\,\}$ is a block. For a subset $
I\subseteq
\{\,
1,\ldots,n
\,\}
$, put $
\operatorname{NC}^{I}(n)
:=
\bigcap_{k\in I}
\operatorname{NC}^{k}(n)
$. Then, $
\Gamma_{n}^{I}
:=
\{\,\,{
\sigma_{p}
}\,\,|\,\,
p\in\operatorname{NC}^{I}(n)
\,\,\}
$ is a generating set for ${\mathcal{B}}_{n}^{I}$.
Note that the inclusion $
{\mathcal{B}}_{n}^{I}
\hookrightarrow
{\mathcal{B}}_{n}
$ induces a bijection $
\Gamma_{n-\# I}
\cong
\Gamma_{n}^{I}
$. Recall that $\Gamma_{n-\# I}$ is a poset with respect to divisibility. A priory, there are two poset structures on $\Gamma_{n}^{I}$: one from intrinsic divisibility with quotients again in $\Gamma_{n}^{I}$ and one induced from the ambient poset $\Gamma_{n}$, i.e., divisibility where quotients are allowed to be anywhere in $\Gamma_{n}$. However, since $
\Gamma_{n}^{I}
=
\Gamma_{n}
\cap
{\mathcal{B}}_{n}^{I}
$ , the two poset structures coincide. Then, $
\Gamma_{n-\# I}
\cong
\Gamma_{n}^{I}
$ is an isomorphism of posets.
Moreover, the order preserving bijection $
\{\,1,\ldots,n-\# I\,\}
\rightarrow
\{\,1,\ldots,n\,\}\setminus I
$ induces an isomorphism $
\operatorname{NC}(n-\# I)
\cong
\operatorname{NC}^{I}(n)
$. This isomorphism is compatible with the poset isomorphism from Fact \[a4-c3-c13-set\], and we have a commutative square of poset isomorphisms: $$\begin{CD}
{\Gamma_{n-\# I}} @= {\Gamma_{n}^{I}} \\
@| @| \\
{\operatorname{NC}(n-\# I)} @= {\operatorname{NC}^{I}(n)}
\end{CD}$$
The identity $
\Gamma_{n}^{I}
=
\Gamma_{n}
\cap
{\mathcal{B}}_{n}^{I}
$ has another consequence:
\[a4-c3-c13-contractible-pieces\] Let $\tilde{Y}_{n}^{I}$ be the full subcomplex spanned by ${\mathcal{B}}_{n}^{I}$ as a set of vertices in $\tilde{Y}_{n}$. Then, $\tilde{Y}_{n}^{I}$ is isomorphic to $\tilde{Y}_{n-\# I}$, whence it is contractible by Theorem \[a4-c3-c13-brady-main\]. For any coset $\beta{\mathcal{B}}_{n}^{I}$, regarded as a set of vertices in $\tilde{Y}_{n}$, the full subcomplex spanned by $\beta{\mathcal{B}}_{n}^{I}$ is the translate $\beta\tilde{Y}_{n}^{I}$ and also contractible.
\[a4-c3-c13-intersection-pattern\] Assume that two coset complexes $\beta\tilde{Y}_{n}^{I}$ and $\beta'\tilde{Y}_{n}^{J}$ intersect, say in $\bar{\beta}$. Then $
\beta\tilde{Y}_{n}^{I}
=
\bar{\beta}\tilde{Y}_{n}^{I}
$ and $
\beta'\tilde{Y}_{n}^{J}
=
\bar{\beta}\tilde{Y}_{n}^{J}
$. In this case, the intersection $$\bar{\beta}\tilde{Y}_{n}^{I}
\cap
\bar{\beta}\tilde{Y}_{n}^{J}
=
\bar{\beta}\tilde{Y}_{n}^{I\cup J}$$ is contractible.
Let ${{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}}:=(U_{\alpha})_{\alpha\in A}$ be a family of sets. For a subset $\sigma\subseteq A$ let $$U_{\sigma} :=
\bigcap_{\alpha\in\sigma}U_{\alpha}$$ denote the associated intersection. The simplicial complex $$N({{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}})
:=
\{\,\, \sigma\subseteq A \,\,|\,\,
\varnothing \neq U_{\sigma}
\,\,\}$$ of all index sets whose associated intersection is non-empty is called the *nerve* of the family ${{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}}$. If ${{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}}$ is a family of subcomplexes in a CW complex, one has the following:
\[a4-c3-c13-nerve-cover\] Suppose ${{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}}=(U_{\alpha})_{\alpha\in A}$ is a covering of a simplicial complex $X$ by a family of contractible subcomplexes. Suppose further that, for each $\sigma\in N({{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}})$, the intersection $U_{\sigma}$ is contractible. Then, the nerve $N({{\hspace{0.5pt}}{\hspace{0.5pt}}\mathcal{U}})$ is homotopy equivalent to $X$.
According to Observation \[a4-c3-c13-intersection-pattern\], the Nerve Theorem applies in particular to the union: $$\tilde{X}_{n}
:=
\bigcup_{k}
\bigcup_{{
\beta\in{\mathcal{B}}_{n}
}}
\beta\tilde{Y}_{n}^{k}$$ We deduce:
\[a4-c3-c13-BradySubcomplex\] The complex $\tilde{X}_{n}$ is homotopy equivalent to the nerve $N$ of the family $$\{\,\,{ \beta{\mathcal{B}}_{n}^{k} }\,\,|\,\,
\beta\in{\mathcal{B}}_{n},\,
1\leqslant k\leqslant n
\,\,\}$$ of cosets.
This relates to higher generation by subgroups as defined by Abels and Holz.
Let $G$ be a group and let $\mathfrak{H}$ be a family of subgroups. We say that $\mathfrak{H}$ is *$m$-generating* for $G$ if the *coset nerve* $$N_{G}(\mathfrak{H})
:=
N(
\{\,\,gH\,\,|\,\,
g\in G,\,
H\in\mathfrak{H}
\,\,\}
)$$ is $(m-1)$-connected.
From Proposition \[a4-c3-c13-BradySubcomplex\], we conclude immediately:
\[a4-c3-c13-first-generation-criterion\] The family $
\mathfrak{B}_{n}
:=
\{\,
{\mathcal{B}}_{n}^{1},\ldots,
{\mathcal{B}}_{n}^{n}
\,\}
$ is $m$-generating for the braid group $
{\mathcal{B}}_{n}
$ if and only if $\tilde{X}_{n}$ is $(m-1)$-connected.
Recall that ${\mathcal{B}}_{n}$ acts freely on the simplicial complex $\tilde{Y}_{n}$. The projection $\tilde{Y}_{n}\rightarrow Y_{n}$ is a covering space map. In fact, $\tilde{Y}_{n}$ is the universal cover of $Y_{n}$ and the braid group ${\mathcal{B}}_{n}$ acts as the group of deck transformations. The subcomplex $\tilde{X}_{n}$ is ${\mathcal{B}}_{n}$-invariant. Let $X_{n}$ be its image in $Y_{n}$.
\[a4-c3-c13-second-generation-criterion\] The family $
\mathfrak{B}_{n}
:=
\{\,
{\mathcal{B}}_{n}^{1},\ldots,
{\mathcal{B}}_{n}^{n}
\,\}
$ is $m$-generating for the braid group $
{\mathcal{B}}_{n}
$ if and only if the pair $(Y_{n},X_{n})$ is $m$-connected.
First, consider the long exact sequence of homotopy groups for the inclusion $\tilde{X}_{n}{\leqslant}\tilde{Y}_{n}$: $$\cdots\longrightarrow
\pi_{1}(\tilde{X}_{n}) \longrightarrow
\pi_{1}(\tilde{Y}_{n}) \longrightarrow
\pi_{1}(\tilde{Y}_{n},\tilde{X}_{n})
\longrightarrow
\pi_{0}(\tilde{X}_{n}) \longrightarrow
\pi_{0}(\tilde{Y}_{n})$$ Since $\tilde{Y}_{n}$ is contractible, we obtain isomorphisms: $$\pi_{d+1}(\tilde{Y}_{n},\tilde{X}_{n})
\cong
\pi_{d}(\tilde{X}_{n})$$ On the other hand, $\tilde{Y}_{n}\rightarrow Y_{n}$ is a covering space projection and therefore enjoys the homotopy lifting property. Moreover, $\tilde{X}_{n}$ is the full preimage of $X_{n}$. Therefore any map $$\left(
\mathbb{B}^{d+1},
{\mathbb{S}}^{d},
* \right)
\longrightarrow
\left(
Y_{n},
X_{n},
1 \right)$$ lifts uniquely to a map $$\left(
\mathbb{B}^{d+1},
{\mathbb{S}}^{d},
* \right)
\longrightarrow
\left(
\tilde{Y}_{n},
\tilde{X}_{n},
1 \right)$$ inducing a map $$\pi_{d+1}(
Y_{n},
X_{n}
)
\longrightarrow
\pi_{d+1}(
\tilde{Y}_{n},
\tilde{X}_{n}
)$$ which is inverse to the map $$\pi_{d+1}(
\tilde{Y}_{n},
\tilde{X}_{n}
)
\longrightarrow
\pi_{d+1}(
Y_{n},
X_{n}
)$$ coming from the covering space projection. Thus, we have isomorphisms $$\pi_{d+1}(
Y_{n},
X_{n}
)
\cong
\pi_{d+1}(\tilde{Y}_{n},\tilde{X}_{n})
\cong
\pi_{d}(\tilde{X}_{n})$$ and the claim follows from Corollary \[a4-c3-c13-first-generation-criterion\].
We can detect $1$-generating and $2$-generating families by hand.
\[a4-c3-c13-small-gen\] For $n\geqslant3 $, the family $\mathfrak{B}_{n}$ is $1$-generating for ${\mathcal{B}}_{n}$, and for $n\geqslant 4$, it is $2$-generating.
A family $\mathfrak{H}$ is $1$-generating for $G$ if and only if $\bigcup_{H\in\mathfrak{H}}H$ generates $G$. It is $2$-generating for $G$ if $G$ is the product of the $H\in\mathfrak{H}$ amalgamated along their intersections [@a4-c3-c13-Abels.Holz 2.4].
Note that the braid group ${\mathcal{B}}_{n}$ is generated by counter-clockwise rotations $$\beta_{ij}
:=
v_{i} \mapsto v_{j}
\mapsto v_{i}$$ around digons. Thus, $
\mathfrak{B}_{n}
:=
\{\,
{\mathcal{B}}_{n}^{1},\ldots,
{\mathcal{B}}_{n}^{n}
\,\}
$ generates as long as $n\geqslant 3$ since then each digon-generator is contained in some ${\mathcal{B}}_{n}^{k}$.
Considering the digon-generators for ${\mathcal{B}}_{n}$, defining relations are given by braid relations, visible in isomorphic copies of ${\mathcal{B}}_{3}$ inside ${\mathcal{B}}_{n}$, and commutator relations, visible in isomorphic copies of ${\mathcal{B}}_{4}$ inside ${\mathcal{B}}_{n}$. Hence all necessary defining relations are visible in the amalgamated product of the ${\mathcal{B}}_{n}^{k}\cong{\mathcal{B}}_{n-1}$ provided $n\geqslant 5$.
For $n=4$, the challenge is to derive the commutator relations: $$\beta_{12}\beta_{34}=\beta_{34}\beta_{12}
\qquad\text{and}\qquad
\beta_{23}\beta_{41}=\beta_{41}\beta_{23}$$ We do the first, the second is done analogously. Calculating with only three strands at a time, we find: $$\begin{aligned}
\beta_{12}\beta_{34}\beta_{24}
&=
\beta_{12}\beta_{23}\beta_{34}
=
\beta_{23}\beta_{13}\beta_{34}
=
\\
&=
\beta_{23}\beta_{34}\beta_{14}
=
\beta_{34}\beta_{24}\beta_{14}
=
\beta_{34}\beta_{12}\beta_{24}
\end{aligned}$$ The desired commutator relation follows.
The little computation at the end of the preceeding proof shows that the commutator relations are redundant in the braid group presentation given in [@a4-c3-c13-Brady2001 Lem. 4.2]. Accordingly, they are also redundant in the analoguous presentation from [@a4-c3-c13-BKL Prop. 2.1].
\[a4-c3-c13-criterion-acyclic\] For $n\geqslant 4$, the family $\mathfrak{B}_{n}$ is $m$-generating for ${\mathcal{B}}_{n}$ if and only if the homology groups $
\operatorname{H}_{d}(
Y_{n},X_{n}
)
$ are trivial for $1\leqslant d\leqslant m$.
As $n\geqslant 4$, the pair $(
Y_{n},X_{n}
)$ is $1$-connected by Propositions \[a4-c3-c13-second-generation-criterion\] and \[a4-c3-c13-small-gen\]. Thus, it follows from the relative Hurewicz theorem that $m$-connectivity of the pair is equivalent to $m$-acyclicity. By Proposition \[a4-c3-c13-second-generation-criterion\], this translates into higher generation of ${\mathcal{B}}_{n}$ by $\mathfrak{B}_{n}$.
As the pair $(
Y_{n},X_{n}
)$ consists of finite complexes that can be described explicitly, Theorem \[a4-c3-c13-criterion-acyclic\] implies that is it a *finite problem* to determine the higher connectivity properties of ${\mathcal{B}}_{n}$ relative to the family $\mathfrak{B}_{n}$. In particular, the question whether the bounds derived in [@sfb701 Example 15.5.4] for higher generation in braid groups are sharp becomes amenable to empirical investigation.
Curvature in braid groups {#a4-c3-c13-curvature-in-braid-groups}
-------------------------
\[a4-c3-c13-artin-group\] For an $n \times n$ symmetric matrix $(m_{ij})$ with entries in $\{ 2,3, \dots \} \cup \{ \infty\}$ we define the associated *Artin group* to be $$\left\langle\,{
s_{1},\ldots,s_{n}
}\,\,\,\vrule\,\,
\underbrace{
s_{i}s_{j}s_{i}\cdots
}_{m_{ij}\text{\ factors}}
=
\underbrace{
s_{j}s_{i}s_{j}\cdots
}_{m_{ij}\text{\ factors}}
\,\right\rangle$$ Here, $m_{ij}=\infty$ indicates that there is no defining relation for $s_{i}$ and $s_{j}$. We will refer to the relations appearing above as *braid relations* (even though some authors reserve this term for the relation with $m_{ij}=3$).
If one additionally forces the generators $s_{i}$ into being involutions, one obtains the associated *Coxeter group*. A pair consisting of a Coxeter group together with the generating set $\{ s_{1},\ldots,s_{n} \}$ is called a *Coxeter system*; its *rank* is defined to be the cardinality of the generating set. If the Coxeter group is spherical, the Coxeter system is said to be *spherical* as well.
A Coxeter group is *spherical* if it is finite; an Artin group is *spherical* if the corresponding Coxeter group is spherical.
Note that the braid group ${\mathcal{B}}_{n}$ is an Artin group and the symmetric group $S_{n}$ is the associated Coxeter group. Here, $m_{ij}=3$ for $|i-j|=1$ and $m_{ij}=2$ otherwise. See Fact \[a4-c3-c13-artin-presentation-braid-groups\]
Artin groups form a rich class of groups of importance in geometric group theory and beyond. From geometric group theory perspective they remain in focus largely due to the following conjecture.
\[a4-c3-c13-conj charney\] Every Artin group is , i.e. it acts properly and cocompactly on a space.
A space is a metric space with curvature bounded from above by 0; for details see the book by Bridson–Haefliger [@a4-c3-c13-BridsonHaefliger1999]. From the current perspective let us list some properties of groups: algorithmically, such groups have quadratic Dehn functions and hence soluble word problem; geometrically, all free-abelian subgroups thereof are undistorted; algebraically, the centralisers of infinite cyclic subgroups thereof split; topologically, the space witnessing -ness of a group $G$ is a finite model for $\underline{E}G$ and thus, for example, allows to compute the K-theory of the reduced $C^{*}$-algebra $C^{*}_{\text{r}}(G)$ provided the Baum–Connes conjecture is known for $G$.
Conjecture \[a4-c3-c13-conj charney\] has been verified by Charney–Davis for *right-angled Artin groups (RAAGs)*, that is for Artin groups with each $m_{ij}$ equal to 2 or $\infty$. Outside of this class, the conjecture is mostly open. In particular, it is open (in general) for the braid groups ${\mathcal{B}}_{n}$.
To prove that a group $G$ is , one has to first construct a space $X$ on which $G$ acts properly and cocompactly, and then prove that the space is indeed . We shall use the space $\tilde{Y}_{n}$ from above, on which ${\mathcal{B}}_{n}$ acts freely and with compact quotient.
What is missing, however, is a metric structure on $\tilde{Y}_{n}$. Such a metric can be specified by realising the simplices in euclidean space, i.e., by endowing each simplex in $\tilde{Y}_{n}$ with the metric of a euclidean polytope. Instead of the standard one, we will follow Brady–McCammond [@a4-c3-c13-BradyMcCammond2010].
Let $e_{1},\ldots,e_{m}$ denote the standard basis of ${\mathbb{R}}^{m}$. The *$m$-orthoscheme* is the convex hull of $
\{\,
0,
e_{1},
e_{1}+e_{2},
\ldots,
e_{1}+e_{2}+\cdots+e_{m}
\,\}
$. The orthoscheme has the structure of an $m$-simplex and the vertices come with a grading: the vertex $
e_{1} + \cdots + e_{k}
$ is declared to be of *rank* $k$.
We now endow each maximal simplex in $\tilde{Y}_{n}$ with the orthoscheme metric. Let $$\Sigma
=
\{\,
\beta, \beta\sigma_{1}, \ldots,
\beta\sigma_{n}
\,\}$$ be a maximal simplex. Here, $\beta$ is a braid in ${\mathcal{B}}_{n}$ and $
1 < \sigma_{1}
<\sigma_{2}
<\cdots<
\sigma_{n}
$ is a maximal chain in $\Gamma_{n}\cong\operatorname{NC}(n)$, which has length $n$ by Observation \[a4-c3-c13-dimension-of-brady-complex\]. We endow $\Sigma$ with the metric of the standard $n$-orthoscheme by identifying $\beta\sigma_{k}$ with the vertex of rank $k$ in the orthoscheme. It is easy to see that if two maximal simplices intersect, they induce identical metric on their common face. Thus we have turned $\tilde{Y}_{n}$ into a metric simplicial complex.
Note that $\tilde{Y}_{n}$ is obtained by gluing copies of a single shape, the $n$-orthoscheme, and so $\tilde{Y}_{n}$ is a geodesic metric space by a result of Bridson (finitely many shapes of cells would suffice). Since the shape is euclidean, we may use Gromov’s link condition and deduce the following:
$\tilde{Y}_{n}$ is if and only if the link of each vertex in $\tilde{Y}_{n}$ is .
Here means that the curvature of the space is bounded above by that of the unit sphere; again, for details see [@a4-c3-c13-BridsonHaefliger1999].
The poset $\Gamma_{n}\cong\operatorname{NC}(n)$ has a unique maximal element, which is the braid $\gamma$ corresponding to the full counter-clockwise rotation: $$v_{1}\mapsto v_{2}\mapsto\cdots\mapsto
v_{m}\mapsto v_{1}$$ The $n^{\textit{th}}$ power $\gamma^{n}$ is central in the braid group ${\mathcal{B}}_{n}$. In fact, it generates the infinite cyclic center of ${\mathcal{B}}_{n}$. Brady–McCammond observed in [@a4-c3-c13-BradyMcCammond2010] that this algebraic fact has a geometric counterpart: $\tilde{Y}_{n}$ splits as a cartesian product of the real line ${\mathbb{R}}$ and another metric space. The ${\mathbb{R}}$-factor inside $\tilde{Y}_{n}$ points in the direction of the edges labelled by $\gamma$.
Because of this, instead of looking at the link of a vertex $u$ in $\tilde{Y}_{n}$, one can look at the link of a midpoint of the (long) edge $(u,u\gamma)$; every two such links are isometric (since ${\mathcal{B}}_{n}$ acts transitively on the vertices of $\tilde{Y}_{n}$), and so let $L$ denote any such link.
To compute the curvature of $L$, it is enough to study the subcomplex of $\tilde{Y}_{n}$ spanned by all simplices containing the edge $(u,u\gamma)$. Clearly, this is the subcomplex spanned by $L$ and $u\sigma$ with $
1{\leqslant}\sigma{\leqslant}\gamma
$, with simplices defined by the chain condition as before. Thus, such a link is isomorphic as a simplicial complex to the realisation of $\operatorname{NC}(n)$; the subcomplex also comes with a metric, and it is clear that this coincides with the realisation of $\operatorname{NC}(n)$ being endowed with its own orthoscheme metric defined as before by identifying each maximal simplex with the $n$-orthoscheme. We will refer to the realisation of $\operatorname{NC}(n)$ with this metric simply as the *orthoscheme complex* of $\operatorname{NC}(n)$.
Note that if the orthoscheme complex of $\operatorname{NC}(n)$ is , then $L$, isometric to the link of the midpoint of the main diagonal, is $\mathrm{CAT(1)}$, which implies that $\tilde{Y}_{n}$, and so ${\mathcal{B}}_{n}$, is .
In view of the above, Brady–McCammond formulate the following conjecture.
For every $n$, the orthoscheme complex of $\operatorname{NC}(n)$ is , and so the braid group ${\mathcal{B}}_{n}$ is .
For $n\leqslant 4$, the conjecture is easily seen to be true.
If we know that the orthoscheme complexes of $\operatorname{NC}(m)$ are for each $m<n$, then in fact the orthoscheme complex of $\operatorname{NC}(n)$ is if and only if the link $L$ is . Thus, for $n=5$, it is enough to study $L$, which is the realisation of the poset obtained from $\operatorname{NC}(n)$ by removing the trivial and improper partitions, and endowing the realisation with the *spherical orthoscheme metric*. Knowing that the conjecture is true for all $m<5$ tells us that $L$ is locally . Thus, using the work of Bowditch [@a4-c3-c13-Bowditch1995], it is enough to check whether any loop in $L$ of length less than $2\pi$ can be *shrunk*, i.e., homotoped to the trivial loop without increasing its length in the process.
Brady–McCammond use a computer to analyse all loops in $L$ shorter than $2\pi$, and show that they are indeed shrinkable, thus establishing:
For $n\leqslant 5$, the braid group ${\mathcal{B}}_{n}$ is .
Haettel, Kielak and Schwer go beyond that, proving
\[a4-c3-c13-thm c3-HKS\] For $n\leqslant 6$, the braid group ${\mathcal{B}}_{n}$ is .
Note that their proof is not computer assisted. The crucial improvement in the work of Haettel–Kielak–Schwer is to use the observation (present already in [@a4-c3-c13-BradyMcCammond2010]), that the link $L$ can be embedded into a spherical building, in the following way.
First observe that the vertices of $L$ are non-trivial proper partitions; let $p$ be such a partition with blocks $
B_{1},\ldots,B_{k}
$. Let ${\mathbb{F}}$ be the field of two elements; we associate to $p$ the subspace of $
{\mathbb{F}}^{n}
=
\langle
\boldsymbol{b}_{1},\ldots,\boldsymbol{b}_{n}
\rangle
$ which is the intersections of the kernels of the characters $$\sum_{{
j \in B_{i}
}}
\boldsymbol{b}^{*}_{j}
= 0$$ where $1 \leqslant i \leqslant k$, and $\boldsymbol{b}^{*}_{j}$ is the $j$-th character in the basis dual to the $\boldsymbol{b}_{j}$.
It is easy to see that this gives a map sending each vertex of $L$ to a proper non-trivial subspace of $
V
:=
\ker\big(
\sum_{j=1}^{n}
\boldsymbol{b}^{*}_{j}
\big)
$. But these subspaces are precisely the vertices of the spherical building of $
\operatorname{SL}_{n-1}({\mathbb{F}})
$, and it turns out that our bijection extends to a map sending each maximal simplex in $L$ onto a chamber (i.e. maximal simplex) in the building in an isometric way. Thus we may view $L$ as a subcomplex of the building.
The spherical building is , and this information gives the extra leverage used to prove Theorem \[a4-c3-c13-thm c3-HKS\].
Non-crossing partitions in Coxeter groups
=========================================
In this section, we introduce the general theory of non-crossing partitions and explain how non-crossing partitions appear in group theory. As already observed in the beginning of Section \[a4-c3-c13-curvature-in-braid-groups\], the symmetric group $S_n$ is a Coxeter group and $(S_n,S_{\rm tr})$ is a Coxeter system of rank $n-1$ where $$S_{\rm tr}:= \{(i,i+1)~|~ 1 {\leqslant}i {\leqslant}n-1\}$$ is the set of neighbouring transpositions.
Every Coxeter system $(W,S)$ acts faithfully on a real vector space that is equipped with a symmetric bilinear form $(- , -)$ such that for every $s \in S$ there is a vector $\alpha_s \in V$ so that $s$ acts as the reflection $$r_{\alpha_s}: v \mapsto v - 2\frac{(v, \alpha_s)}{(\alpha_s,
\alpha_s)}\alpha_s$$ on $V$. Thus every Coxeter group is a reflection group that is a group generated by a set of reflections on a vector space $(V,(- , -))$.
The vectors $\alpha_s$ can be chosen so that the subset $\Phi = \{w(\alpha_s) ~|~s \in S, w \in W\}$ of $V$ is a so called *root system*. For a spherical Coxeter system a *root system* $\Phi$ is characterised by the following three axioms
- $\Phi$ generates $V$;
- $\Phi \cap {\mathbb{R}}\alpha = \{\pm \alpha\}$ for all $\alpha \in \Phi$;
- $s_\alpha(\beta)$ is in $\Phi$ for all $\alpha, \beta \in \Phi$.
The spherical Coxeter groups $W$ are precisely the finite real reflection groups.
Coxeter classified the finite root systems which then also gives a classification of the spherical Coxeter systems: there are the infinite families of type $A_n, B_n, C_n$ and $D_n$ and some exceptional groups. For instance $(S_n,S_{\rm tr})$ is of type $A_{n-1}$. Note that the groups of type $B_n$ and $C_n$ are isomorphic; and also that the root systems of type $A_n, B_n, C_n$ and $D_n$ are all *crystallographic* that is $$\frac{(\alpha,\beta)}{(\alpha, \alpha)} \in {\mathbb{Z}{\hspace{0.5pt}}}~\mbox{for all}~
\alpha, \beta \in \Phi.$$
We call $T:= \cup_{w \in W}w^{-1}Sw$ the set of reflections of the Coxeter system $(W,S)$. If the system is spherical, then $T$ is indeed the set of all reflections.
For instance in the symmetric group $S_n$ the set $T$ is the conjugacy class of transpositions, see also Section \[a4-c3-c13-braid-groups\]. There the so called *absolute order* ${\leqslant}_T$ on $S_n$ has been introduced. Let $[id, (1,2,\ldots ,n)]_{{\leqslant}_T}$ be the closed intervall in $S_n$ with respect to ${\leqslant}_T$. In Fact 1.3.4 it has been stated that $(\operatorname{NC}(n),\subseteq)$ and $([id, (1,2,\ldots ,n)]_{{\leqslant}_T}, {\leqslant}_T)$ are posets that are isomorphic. Therefore $\operatorname{NC}(n)$ can be thought of being of type $A_{n-1}$.
Out of combinatorial interest, Reiner generalised the concept of non-crossing partitions to the infinite series of type $B_n$ and $D_n$ geometrically [@a4-c3-c13-Rei]. Independently of his work and of each other Brady and Watt [@a4-c3-c13-BrWa] as well as Bessis [@a4-c3-c13-Dual] generalised the concept of non-crossing partitions to all the finite Coxeter systems. Their approach agrees with Reiner’s in type $B_n$ [@a4-c3-c13-Arm].
Brady and Watt as well as Bessis started independently the study of the dual Coxeter system $(W,T)$ instead of $(W,S)$. A *dual Coxeter system* $(W,T)$ of finite rank $n$ has the property that there is a subset $S$ of $T$ such that $(W,S)$ is a Coxeter system [@a4-c3-c13-Dual]. It then follows that $T$ is the set of reflections in $(W,S)$. This concept is called by Bessis *dual approach to Coxeter and Artin groups*.
A *(parabolic) standard Coxeter element* in $(W, S)$ is the product of all the elements in (a subset of) $S$ in some order and a *(parabolic) Coxeter element* in $(W,T)$ is a (parabolic) standard Coxeter element in $(W,S)$ for some simple system $S$ in $T$ for $W$.
For instance in type $A_{n-1}$, so in the symmetric group $S_n$, the standard Coxeter elements with respect to $S= S_{\rm tr}$ are precisely those $n$-cycles in $S_n$ that can be written as a first increasing and then decreasing cycle. All the $n$-cycles in $S_n$ are the Coxeter elements in the dual system $(S_n, T)$ where $T$ is the set of reflections, that is the conjugacy class of transpositions.
The partial order ${\leqslant}_T$ on the symmetric group $S_n$ presented in Section \[a4-c3-c13-braid-groups\] can be generalized to all the dual Coxeter systems $(W,T)$. We consider the Cayley graph ${\rm CG}_T(W)$ of the group $W$ with respect to the generating set $T$. For $u,v \in W$ we declare $u {\leqslant}_T v$ if there is a geodesic path in the Cayley graph connecting the identity to $v$ and passing through $u$. This partial order is also called the *absolute order* on $W$.
We also introduce a length function $l_T$ on $W$: for $u \in W$ we define $l_T(u) = k$ if there is a geodesic path from the identity to $u$ of length $k$ in the Cayley graph. Notice, if $l_T(u) = m$ then $u$ is the product of $m$ reflections, that is $u = t_1\cdots t_m$ with $t_i \in T$, and there is no shorter factorisation of $u$ in a product of reflections. In this case we say that $u = t_1\cdots t_m$ is a *$T$-reduced factorisation* of $u$. In particular, if $u {\leqslant}_T v$, then there are $k, m \in {\mathbb{N}}$ with $k {\leqslant}m$ and reflections $t_1, \ldots , t_m$ in $T$ such that $u = t_1 \cdots t_k$ and $v= t_1 \cdots t_m$. Thus $$u {\leqslant}_T v~\mbox{if and only if}~l_T(u)+l_T(u^{-1}v) = l_T(v).$$
For a dual Coxeter system $(W,T)$ and a Coxeter element $c$ in $W$ the set of *non-crossing partitions* is $$\operatorname{NC}(W,c) = \{u \in W~|~u {\leqslant}_T c\}.$$
This definition is conform with the definition in type $A_n$, see Fact \[a4-c3-c13-set\].
The length function $l_T$ yields a grading on $\operatorname{NC}(W,c)$ and the map $$d: \operatorname{NC}(W,c) \rightarrow \operatorname{NC}(W,c),~ x \mapsto x^{-1}c$$ a duality on $\operatorname{NC}(W,c)$ that inverses the order relation.
This implies the following.
$\operatorname{NC}(W,c)$ is a poset that is
- graded
- selfdual
- [@a4-c3-a4-c3-c13-BrWa2; @a4-c3-c13-Dual] a lattice if $W$ is spherical.
The number of elements in $\operatorname{NC}(W,c)$ in a finite dual Coxeter system of type $X$ is the generalised Catalan number of type $X$. In types $B_n$ and $D_n$ there are also nice geometric models for the posets of non-crossing partitions.
Note that in a spherical Coxeter system always $T \subseteq \operatorname{NC}(W,c)$.
There is also a presentation of $W$ with generating set $T$ [@a4-c3-c13-Dual]. The relations are the so called *dual braid relations* with respect to a Coxeter element $c \in W$: $$\mbox{ for every}~ s,t,t^\prime \in T~\mbox{set}~ st = t^\prime s~\mbox{ whenever}$$ $$\mbox{ the relation }~st = t^\prime s ~\mbox{holds in $W$ and}~ st {\leqslant}_T c.$$
The Matsumoto property means if we have for some $w \in W$ two shortest factorisations as products of elements of $S$, or equivalently two geodesic paths from $id$ to $w$ in the Cayley graph ${\rm CG}_S(W)$, then we can transform one factorisation or path into the other one just by applying braid relations; that is $W$ has a group presentation as given in Definition \[a4-c3-c13-artin-group\].
The *dual Matsumoto property* for a Coxeter element $c \in W$ is the statement that if we have two shortest factorisations $$c = t_1 \cdots t_m = u_1 \cdots u_m ~\mbox{with}~ t_i, u_i \in T$$ as products of elements of $T$, that is two $T$-reduced factorisations of $c$ in $W$, then one factorisation can be transformed into the other one just by applying dual braid relations. It follows that the dual Matsumoto property holds for $c$, since $$\langle T~|~~\mbox{dual braid relations}\rangle$$ is a presentation of $W$.
We obtain the dual Matsumoto property for an arbitrary element $w \in W$ by replacing $c$ by $w$ in the definition of the dual braid relations and of the dual Matsumoto property above.
For an element $w \in W$, let $$\operatorname{Red}_T(w) = \{(t_1, \ldots, t_m)~|~t_i \in T~\mbox{and}~ w = t_1 \cdots t_m~\mbox{is $T$-reduced}\}.$$
The dual Matsumoto property for $w \in W$ is equivalent to the transitive *Hurwitz action* of the braid group ${\mathcal{B}}_{l_T(w)}$ on the set of $T$-reduced factorisations $\operatorname{Red}_T(w)$ of $w$. For the braid $\sigma_i \in {\mathcal{B}}_{l_T(w)}$, see Fact \[a4-c3-c13-artin-presentation-braid-groups\], the action is given by $$\sigma_i(t_1, \ldots , t_n) = (t_1, \ldots ,t_{i-1}, t_i^{-1}
t_{i+1}{t_i}, t_i, t_{i+2}, \ldots , t_n).$$ We will discuss this action in more detail in the next section.
The dual approach can also be applied to Artin groups; given a Coxeter system $(W,S)$, we will denote the corresponding Artin group by ${\mathcal{A}}(W,S)$. If in the following the Coxeter system $(W,S)$ is of type $X$, then we abbreviate ${\mathcal{A}}(W,S)$ either by ${\mathcal{A}}(W)$ or by ${\mathcal{A}}_X$. Further we take a copy $S_a$ of $S$ in ${\mathcal{A}}(W,S)$ and write $${\mathcal{A}}(W,S) := \langle S^{}_a~|~ (s^{}_1)^{}_a(s^{}_2)^{}_a (s^{}_1)^{}_a
\cdots = (s^{}_2)^{}_a (s^{}_1)^{}_a (s^{}_2)^{}_a
\cdots ~\mbox{for}~s^{}_1, s^{}_2 \in S \rangle$$ in order to distinguish between $W$ and ${\mathcal{A}}(W)$. We call an Artin group ${\mathcal{A}}(W)$ *spherical* if the Coxeter group is spherical. And in the rest of this section, we always consider spherical Artin groups.
Notice that the Matsumoto property implies that one can lift every $w \in W$ to an element in ${\mathcal{A}}(W)$ just by mapping $w$ to $(s_1)_a \cdots (s_k)_a \in {\mathcal{A}}_W$ whenever $w = s_1 \cdots s_k$ is a reduced factorisation of $w$ into elements of $S$. We denote this section of $W$ in ${\mathcal{A}}(W)$ by ${\mathcal{W}}$.
The non-crossing partitions are a good tool for the better understanding of the spherical Artin groups; for instance they can be used to construct a finite simplicial classifying space for the spherical Artin groups (see Section \[a4-c3-c13-classifingy-space-for-braid-group\]), or to solve the word or the conjugacy problem in them, see [@a4-c3-c13-BrWa; @a4-c3-c13-Dual].
The basic idea of this solution of the word and the conjugacy problem in the spherical Artin group ${\mathcal{A}}(W)$ is to give a new presentation of ${\mathcal{A}}(W)$ as follows. Let $\operatorname{NC}(W,c)_a$ be a copy of the set of non-crossing partitions $\operatorname{NC}(W,c)$ with respect to a standard Coxeter element $c$, that is there is a bijection $$a: \operatorname{NC}(W,c) \rightarrow \operatorname{NC}(W,c)_a.$$ Then the new generating set is $\operatorname{NC}(W,c)_a$; and the new relations are the expressions $(w_1)_a \cdots (w_r)_a$ whenever $w_1, w_2, \ldots , w_r$ are the vertices of a circuit in $$[id, c]_{{\leqslant}_T} \subseteq {\rm CG}_{\operatorname{NC}(W,c)}(W).$$ Then this presentation can be used to obtain a new normal form for the elements in ${\mathcal{A}}(W)$ [@a4-c3-c13-Dual]. Notice that this presentation generalises the presentation of the braid group given by Birman, Ko and Lee [@a4-c3-c13-BKL] to all the spherical Artin groups, see also Fact \[a4-c3-c13-triangular-presentation\] in Section \[a4-c3-c13-classifingy-space-for-braid-group\].
Next, we explain this new presentation. Denote the group given by the presentation above by ${\mathcal{A}}(W,c)$. The strategy to prove that ${\mathcal{A}}(W,c)$ and ${\mathcal{A}}(W)$ are isomorphic is to use Garside theory. As a first step the presentation above can be transformed into a presentation with set of generators a copy $T_a = \{t_a~|~t \in T\}$ of $T$ and set of relations the dual braid relations with respect to $c$. The next step is to consider the monoid ${\mathcal{A}}(W,c)^*$ generated by $T_a$ and the dual braid relations, and to show that this is a Garside monoid. Then using Garside theory one shows that the group of fractions $\mathrm{Frac}({\mathcal{A}}(W,c)^*)$ of ${\mathcal{A}}(W,c)^*$ equals ${\mathcal{A}}(W,c)$. The last step is to prove that the group of fractions $\mathrm{Frac}({\mathcal{A}}(W,c)^*)$ and the Artin group ${\mathcal{A}}(W)$ are isomorphic.
\[a4-c3-c13-NewPresentation\] Let ${\mathcal{A}}_W$ be a spherical Artin group. Then, $${\mathcal{A}}_W \cong \langle T_a~ |~ t_a t'_a= (tt't)_a t_a~\text{if } t, t' \in T~\mbox{and}~ tt'{\leqslant}_T c\rangle.$$
Note also that a basic ingredient in the proof of Theorem \[a4-c3-c13-NewPresentation\] is the dual Matsumoto property for $c$, that is the transitivity of the Hurwitz action of the braid group ${\mathcal{B}}_{l_T(c)}$ on $\operatorname{Red}_T(c)$.
The isomorphism between ${\mathcal{A}}(W,c)$ and ${\mathcal{A}}_W$ given by Bessis is difficult to understand explicitly. So an immediate question is what the elements of $\operatorname{NC}(W,c)_a$ are expressed in the generating set $S_a$?
The rational permutation braids, that is, the elements ${ x}{y}^{-1}$ where ${x}, {y} \in {\mathcal{W}}$, are also called *Mikado braids* as they satisfy in type $A_{n-1}$ a topological condition and are therefore easy to recognise. This condition on an element in the Artin group ${\mathcal{A}}(W)$ of type $A_{n-1}$, that is on a braid in the braid group ${\mathcal{B}}_n$, is that we can lift and remove continuously one strand after the next of the braid without disturbing the remaining strands until we reach an empty braid [@a4-c3-c13-DG].
\[a4-c3-c13-Mikado\] If ${\mathcal{A}}_W$ is spherical Artin group and $c \in W$ a standard Coxeter element, then the dual generators of ${\mathcal{A}}(W,c)$, that is the elements of $\operatorname{NC}(W,c)_a$, are Mikado braids in ${\mathcal{A}}_W$.
This is [@a4-c3-c13-DG] for those groups of type different from $D_n$ and [@a4-c3-c13-BG] for those of type $D_n$.
Notice that Licata and Queffelec [@a4-c3-c13-LQ] have a proof of Theorem \[a4-c3-c13-Mikado\] in types A,D,E with a different approach using categorification.
In order to be able to find a topological property that characterises the Mikado braids as in type $A_{n-1}$ topological models for the series of spherical Artin groups ${\mathcal{A}}_W$ are needed. There is an embedding of Artin groups of type $B_n$ into those of type $A_{2n-1}$. The situation in type $D_n$ is as follows [@a4-c3-c13-BG]: The root system of type $D_n$ embeds into the root system of type $B_n$, which implies that the Coxeter system of type $D_n$ is a subsystem of that one of type $B_n$. But there is not an embedding of the Artin group of type $D_n$ into that one of type $B_n$ that satisfies a certain natural condition. Let $(W, S)$ be a Coxeter system of type $B_n$. Then there is precisely one element $s \in S$ that is a reflection corresponding to a short root. Let $$\overline{{\mathcal{A}}_{B_n}} := {\mathcal{A}}_{B_n}/\ll s^2 \gg,$$ where $\ll s^2 \gg$ is the normal closure of $s^2$ in ${\mathcal{A}}_{B_n}$. Then the following holds.
There is a natural embedding of ${\mathcal{A}}_{D_n}$ onto an index-$2$ subgroup of $\overline{{\mathcal{A}}_{B_n}}$. More precisely, there is the following commutative diagram $$\begin{CD}
{{\mathcal{A}}_{B_n}} @>\pi>> {\overline{{\mathcal{A}}_{B_n}}} @<<< {\langle t_1, \dots, t_{n}\rangle} @<\cong<< {{\mathcal{A}}_{D_n}}\\
@| @VV\pi_{\overline{\mathcal{B}}}V @V{\pi_{\mathcal{D}}}VV \\
{{\mathcal{A}}_{B_n}} @>>\pi_{\mathcal{B}}> {W_{B_n}} @<<< {W_{D_n}}
\end{CD}$$
The embedding of ${\mathcal{A}}_{D_n}$ into $\overline{{\mathcal{A}}_{B_n}}$ makes it possible to associate braid pictures to the ${\mathcal{A}}_{D_n}$-elements and to characterise Mikado braids in type $D_n$ geometrically.
A reader familiar with Hecke algebras will find it interesting that the Mikado braids satisfy a positivity property involving the canonical Kazhdan-Lusztig basis ${\mathcal{C}}:= \{C_w~|~w \in W\}$ of the Iwahori–Hecke algebra $H(W)$ related to the Coxeter system $(W,S)$, see [@a4-c3-c13-KL; @a4-c3-c13-DG]. There is a natural group homomorphism $a: {\mathcal{A}}_W\longrightarrow H(W)^\times$ from ${\mathcal{A}}_W$ into the multiplicative group $H(W)^\times$ of $H(W)$. The image of a Mikado braid, that is of a rational permutation braid, in $H(W)^\times$ has as coefficients Laurent polynomials with non-negative coefficients when expressed in the canonical basis ${\mathcal{C}}$ by a result by Dyer and Lehrer (see [@a4-c3-c13-DL; @a4-c3-c13-DG]).
The Hurwitz action
==================
*Hurwitz action in Coxeter systems.* Deligne showed the dual Matsumoto property in spherical Coxeter systems, that is he showed that the Hurwitz action of the braid group ${\mathcal{B}}_{l_T(c)}$ on $\operatorname{Red}_T(c)$ is transitive for every Coxeter element $c$ in $(W,S)$ [@a4-c3-c13-Del2]; and Igusa and Schiffler proved it for arbitrary Coxeter systems [@a4-c3-c13-IS]. In [@a4-c3-c13-BDSW] a new, more general and first of all constructive proof of this property is given:
\[a4-c3-c13-HurwitzTrans\] Let $(W,T)$ be a (finite or infinite) dual Coxeter system of finite rank $n$ and let $c = s_1 \cdots s_m$ be a parabolic Coxeter element in W. The Hurwitz action on $\operatorname{Red}_T(c)$ is transitive.
Theorem \[a4-c3-c13-HurwitzTrans\] is also more general then Theorem 1.4 in [@a4-c3-c13-IS], as in [@a4-c3-c13-BDSW] dual Coxeter systems are considered while in [@a4-c3-c13-IS] Coxeter systems, and in general the set of Coxeter elements is in a dual system larger than that one in a Coxeter system.
The proof of Thereom \[a4-c3-c13-HurwitzTrans\] is based on a study of the Cayley graphs ${\rm CG}_S(W)$ and ${\rm CG}_T(W)$. Using the same methods one can also show that every reflection occurring in a reduced $T$-factorisation of an element of a parabolic subgroup $P$ of $W$ is already contained in that parabolic subgroup.
\[a4-c3-c13-Factor\] Let $(W,S)$ be a (finite or infinite) Coxeter system, $P$ a parabolic subgroup and $w \in P$. Then $\operatorname{Red}_T(w) = \operatorname{Red}_{T \cap P}(w)$.
This basic fact was not known before and can be seen as a founding stone towards a general theory for ‘dual’ Coxeter systems.
Hurwitz action in the spherical Coxeter systems and quasi-Coxeter elements. {#hurwitz-action-in-the-spherical-coxeter-systems-and-quasi-coxeter-elements. .unnumbered}
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In the rest of the section, $(W,T)$ is a finite dual Coxeter system.
In order to understand the dual Coxeter systems $(W,T)$ one also needs to know for which elements in $W$ the Hurwitz action is transitive. The answer to that question is as follows [@a4-c3-a4-c3-c13-BGRW].
A *parabolic quasi-Coxeter element* is an element $w \in W$ that has a reduced factorisation into reflections such that these reflections generate a parabolic subgroup of $W$.
Note if one reduced $T$-factorisation of $w \in W$ generates a parabolic subgroup $P$ then every reduced $T$-factorisation of $w$ is in $P$ by Theorem \[a4-c3-c13-Factor\]. It also follows that every such factorisation generates $P$ [@a4-c3-a4-c3-c13-BGRW Thm. 1.2].
If a factorisation of $w$ generates the whole group $W$, it is a *quasi-Coxeter element*. Clearly every Coxeter element is a quasi-Coxeter element. In type $A_n$ and $B_n$ every quasi-Coxeter element is already a Coxeter element. The smallest Coxeter system containing a proper quasi-Coxeter element is of type $D_4$.
Now we can answer the question above.
\[a4-c3-c13-Quasi-Cox-Lattice\] Let $(W,S)$ be a spherical Coxeter system and let $w \in W$. The Hurwitz action is transitive on $\operatorname{Red}_T(w)$ if and only if $w$ is a parabolic quasi-Coxeter element.
Recently, Wegener showed that the dual Matsumoto property holds for quasi-Coxeter elements in affine Coxeter systems as well [@a4-c3-c13-We]. These two results have the following consequence.
Let $(W,T)$ be a dual Coxeter system, $w\in W$ and $w = t_1 \cdots t_m$ a reduced $T$-factorisation, then the Hurwitz action is transitive on $\operatorname{Red}_T(w)$ in the Coxeter group $W^\prime:= \langle t_1, \ldots , t_m\rangle$ whenever $W^\prime$ is a spherical or an affine Coxeter group.
According to Theorem 3.3 of [@a4-c3-c13-Dy], $W^\prime:= \langle t_1, \ldots , t_m\rangle$ is a Coxeter group. Theorem \[a4-c3-c13-Quasi-Cox-Lattice\] and the main result in [@a4-c3-c13-We] then yield the statement.
The (parabolic) quasi-Coxeter elements are interesting for more reasons; for instance also for the following. Let $\Phi$ be the root system related to $(W,S)$ and let $L(\Phi) := {\mathbb{Z}{\hspace{0.5pt}}}\Phi$ and $L(\Phi^{\vee}):= {\mathbb{Z}{\hspace{0.5pt}}}\Phi^{\vee}$ where $\alpha^{\vee}:= 2\alpha/(\alpha, \alpha)$ be the root and the coroot lattices, respectively. Quasi-Coxeter elements are also intrinsic in the dual Coxeter systems as they generate the root as well as the coroot lattice: Let $w = t_1 \cdots t_n$ be a reduced $T$-factorisation of $w \in W$ and let $\alpha_i \in \Phi$ be the root related to the reflection $t_i$ for $1 {\leqslant}i {\leqslant}n$.
Let $\Phi$ be a finite crystallographic root system of rank $n$. Then $w$ is a quasi-Coxeter element if and only if
1. $\{\alpha_i~|~1 {\leqslant}i {\leqslant}n\}$ is a ${\mathbb{Z}{\hspace{0.5pt}}}$-basis of the root lattice $L(\Phi)$, and
2. $\{ \alpha_i^{\vee}~|~1 {\leqslant}i {\leqslant}n\}$ is a ${\mathbb{Z}{\hspace{0.5pt}}}$-basis of the coroot lattice $L(\Phi^{\vee})$.
Thus if all the roots in $\Phi$ are of the same length, then $L(\Phi) = L(\Phi^{\vee})$ and the quasi-Coxeter elements correspond precisely to the basis of the root lattice.
Quasi-Coxeter elements and Coxeter elements share further important properties beyond Hurwitz transitivity.
An element $x \in W$ is a parabolic quasi-Coxeter element if and only if $x {\leqslant}_T w$ for a quasi-Coxeter element $w$.
Finally, Gobet observed that, in a spherical Coxeter system, every parabolic quasi-Coxeter element can be uniquely written as a product of commuting parabolic quasi-Coxeter elements [@a4-c3-c13-Go]. This factorisation of a quasi-Coxeter element can be thought of as a generalisation of the unique disjoint cycle decomposition of a permutation.
Non-crossing partitions arising in representation theory
========================================================
In this section, we explain how non-crossing partitions arise naturally in representation theory. For any finite dimensional algebra $A$ over a field $k$ we consider the category $\operatorname{mod} A$ of finite dimensional (right) $A$-modules and denote by $K_0(A)$ its *Grothendieck group*. This group is free abelian of finite rank, and a representative set of simple $A$-modules $S_1,\ldots, S_n$ provides a basis $e_1,\ldots,e_n$ if one sets $e_i=[S_i]$ for all $i$. As usual, we denote for any $A$-module $X$ by $[X]$ the corresponding class in $K_0(A)$. The Grothendieck group comes equipped with the *Euler form* $K_0(A)\times K_0(A)\to{\mathbb{Z}{\hspace{0.5pt}}}$ given by $$\langle[X],[Y]\rangle=\sum_{n{\geqslant}0}(-1)^n\dim_k\operatorname{Ext}^n_A(X,Y)$$ which is bilinear and non-degenerate (assuming that $A$ is of finite global dimension). The corresponding symmetrised form is given by $(x,y)=\langle x,y\rangle +\langle y,x\rangle$. For a class $x=[X]$ given by a module $X$, one defines the reflection $$\label{eq:defn-reflection}
s_x\colon K_0(A)\longrightarrow K_0(A),\quad a\mapsto
a-2\frac{(a,x)}{(x,x)}x,$$ assuming that $(x,x)\neq 0$ divides $(e_i,x)$ for all $i$. Let us denote by $W(A)$ the group of automorphisms of $K_0(A)$ that is generated by the set of simple reflections $S(A)=\{s_{e_1},\ldots,s_{e_n}\}$; it is called the *Weyl group* of $A$.
From now on, assume that $A$ is *hereditary*, that is, of global dimension at most one. Then, one can show that the Weyl group $W(A)$ is actually a Coxeter group. For example, the path algebra $kQ$ of any quiver $Q$ is hereditary and in that case $kQ$-modules identify with $k$-linear representations of $Q$.
A Coxeter system $(W,S)$ is of the form $(W(A),S(A))$ for some finite dimensional hereditary algebra $A$ if and only if it is crystallographic in the following sense:
1. $m_{st}\in\{2,3,4,6,\infty\}$ for all $s\neq t$ in $S$, and
2. in each circuit of the Coxeter graph not containing the edge label $\infty$, the number of edges labelled $4$ (resp. $6$) is even.
We may assume that the simple $A$-modules are numbered in such a way that $\langle e_i,e_j\rangle=0$ for $i >j$, and we set $c=s_{e_1}\cdots s_{e_n}$. Note that $c=c(A)$ is a *Coxeter element* which is determined by the formula $$\langle x,y\rangle=-\langle y,c(x)\rangle\qquad\text{for}\qquad x,y\in K_0(A).$$
We are now in a position to formulate a theorem which provides an explicit bijection between certain subcategories of $\operatorname{mod} A$ and the non-crossing partitions in $\operatorname{NC}(W(A),c)$. Call a full subcategory ${\mathcal{C}}\subseteq \operatorname{mod} A$ *thick* if it is closed under direct summands and satisfies the following two-out-of-three property: any exact sequence $0\to X\to Y\to Z\to 0$ of $A$-modules lies in ${\mathcal{C}}$ if two of $\{X,Y,Z\}$ are in ${\mathcal{C}}$. A subcategory is *coreflective* if the inclusion functor admits a right adjoint.
\[th:algebras-main\] Let $A$ be a hereditary finite dimensional algebra. Then, there is an order preserving bijection between the set of thick and coreflective subcategories of $\operatorname{mod} A$ (ordered by inclusion) and the partially ordered set of non-crossing partitions $\operatorname{NC}(W(A),c)$. The map sends a subcategory which is generated by an exceptional sequence $E=(E_1,\ldots, E_r)$ to the product of reflections $s_E=s_{E_1}\cdots s_{E_r}$.
The rest of this article is devoted to explaining this result. In particular, the crucial notion of an exceptional sequence will be discussed.
This result goes back to beautiful work of Ingalls and Thomas [@a4-c3-c13-IT2009]. It was then established for arbitary path algebras by Igusa, Schiffler, and Thomas [@a4-c3-c13-IS], and we refer to [@a4-c3-c13-HK2016] for the general case. Observe that path algebras of quivers cover only the Coxeter groups of simply laced type (via the correspondence $A\mapsto W(A)$); so there are further hereditary algebras.
We may think of Theorem \[th:algebras-main\] as a *categorification* of the poset of non-crossing partitions. There is an immediate (and easy) consequence which is not obvious at all from the original definition of non-crossing partitions; the first (combinatorial) proof required a case by case analysis.
For a finite crystallographic Coxeter group, the corresponding poset of non-crossing partitions is a lattice.
Any finite Coxeter group can be realised as the the Weyl group $W(A)$ of a hereditary algebra of finite representation type. In that case any thick subcategory is coreflective. On the other hand, it is clear from the definition that the intersection of any collection of thick subcategories is again thick. This yields the join, but also the meet operation; so the poset of thick and coreflective subcategories is actually a lattice; see Remark \[a4-c3-c13-half-a-lattice-is-a-lattice\]
This categorification provides some further insight into the *collection* of all posets of non-crossing partitions. This is based on the simple observation that any thick and coreflective subcategory ${\mathcal{C}}\subseteq\operatorname{mod} A$ (given by an exceptional sequence $E=(E_1,\ldots,E_r)$) is again the module category of a finite dimensional hereditary algebra, say ${\mathcal{C}}=\operatorname{mod} B$. Then the inclusion $\operatorname{mod} B\to \operatorname{mod} A$ induces not only an inclusion $K_0(B)\to K_0(A)$, but also an inclusion $W(B)\to W(A)$ for the corresponding Weyl groups, which identifies $W(B)$ with the subgroup of $W(A)$ generated by $s_{E_1},\ldots,s_{E_r}$, and identifies the Coxeter element $c(B)$ with the non-crossing partition $s_E$ in $W(A)$. Moreover, the inclusion $W(B)\to W(A)$ induces an isomorphism $$\operatorname{NC}(W(B),c(B))\stackrel{\sim}\to \{x\in \operatorname{NC}(W(A),c(A))\mid x{\leqslant}s_E\}.$$
The following result summarises this discussion; it reflects the fact that there is a *category of non-crossing partitions*. This means that we consider a poset of non-crossing partitions not as a single object but look instead at the relation with other posets of non-crossing partitions.
Let $\operatorname{NC}(W,c)$ be the poset of non-crossing partitions given by a crystallographic Coxeter group $W$. Then, any element $x\in \operatorname{NC}(W,c)$ is the Coxeter element of a subgroup $W'{\leqslant}W$ that is again a crystallographic Coxeter group. Moreover, $$\operatorname{NC}(W',x)= \{y\in \operatorname{NC}(W,c)\mid y{\leqslant}x\}.\qedhere$$
Generalised Cartan lattices
===========================
Coxeter groups and non-crossing partitions are closely related to root systems. The approach via representation theory provides a natural setting, because the Grothendieck group equipped with the Euler form determines a root system; we call this a *generalised Cartan lattice* and refer to [@a4-c3-c13-HK2016] for a detailed study.
The following definition formalises the properties of the Grothendieck group $K_0(A)$. A *generalised Cartan lattice* is a free abelian group ${\Gamma}\cong{\mathbb{Z}{\hspace{0.5pt}}}^n$ with an ordered standard basis $e_1,\ldots,e_n$ and a bilinear form $\langle -,-\rangle\colon{\Gamma}\times{\Gamma}\to{\mathbb{Z}{\hspace{0.5pt}}}$ satisfying the following conditions.
1. $\langle e_i,e_i\rangle>0$ and $\langle e_i,e_i\rangle$ divides $\langle e_i,e_j\rangle$ for all $i,j$.
2. $\langle e_i,e_j\rangle =0$ for all $i>j$.
3. $\langle e_i,e_j\rangle {\leqslant}0$ for all $i<j$.
The corresponding *symmetrised form* is $$(x,y) = \langle x,y\rangle+\langle y,x\rangle\quad\text{for }
x,y\in {\Gamma}.$$ The ordering of the basis yields the *Coxeter element* $$\operatorname{cox}({\Gamma}):=s_{e_1}\cdots s_{e_n}.$$ We can define reflections $s_x$ as in and denote by $W=W({\Gamma})$ the corresponding *Weyl group*, which is the subgroup of $\operatorname{Aut}({\Gamma})$ generated by the simple reflections $s_{e_1},\ldots, s_{e_n}$. We write $\operatorname{NC}({\Gamma})=\operatorname{NC}(W,c)$ with $c=\operatorname{cox}({\Gamma})$ for the poset of non-crossing partitions, and the set of *real roots* is $$\Phi({\Gamma}) := \{ w(e_i) \mid w\in W({\Gamma}),\,1{\leqslant}i{\leqslant}n \}
\subseteq{\Gamma}.$$ A *real exceptional sequence* of ${\Gamma}$ is a sequence $(x_1,\ldots,x_r)$ of elements that can be extended to a basis $x_1,\ldots,x_n$ of ${\Gamma}$ consisting of real roots and satisfying $\langle x_i,x_j\rangle =0$ for all $i>j$. A *morphisms* ${\Gamma}'\to{\Gamma}$ of generalised Cartan lattices is given by an isometry (morphism of abelian groups preserving the bilinear form $\langle-,-\rangle$) that maps the standard basis of ${\Gamma}'$ to a real exceptional sequence of ${\Gamma}$. This yields a *category of generalised Cartan lattices*.
What is this category good for? One of the basic principles of category theory is *Yoneda’s lemma* which tells us that we understand an object ${\Gamma}$ by looking at the representable functor $\operatorname{Hom}(-,{\Gamma})$ which records all morphisms that are received by ${\Gamma}$. In our category all morphisms are monomorphisms, so $\operatorname{Hom}(-,{\Gamma})$ amounts to the poset of subobjects (equivalence classes of monomorphisms ${\Gamma}'\to{\Gamma}$).
The poset of subobjects of a generalised Cartan lattice ${\Gamma}$ is isomorphic to the poset of non-crossing partitions $\operatorname{NC}({\Gamma})$. The isomorphism sends a monomorphism $\phi\colon{\Gamma}'\to{\Gamma}$ to $s_{\phi(e_1)}\cdots s_{\phi(e_r)}$ where $\operatorname{cox}({\Gamma}')=s_{e_1}\cdots s_{e_r}$. Moreover, the assignment $w\mapsto w|_{{\Gamma}'}$ induces an isomorphism $$W({\Gamma})\supseteq\langle s_{\phi(e_1)},\ldots,
s_{\phi(e_r)}\rangle\stackrel{\sim}\longrightarrow W({\Gamma}').\qedhere$$
Braid group actions on exceptional sequences
============================================
The link between representation theory and non-crossing partitions is based on the notion of an exceptional sequence and the action of the braid group on the collection of complete exceptional sequences. This will be explained in the following section.
There are two sorts of abelian categories that we need to consider. This follows from a theorem of Happel [@a4-c3-c13-Ha; @a4-c3-c13-HR2002] which we now explain. Fix a field $k$ and consider a connected hereditary abelian category ${\mathcal{A}}$ that is $k$-linear with finite dimensional Hom and Ext spaces. Suppose in addition that ${\mathcal{A}}$ admits a *tilting object*. This is by definition an object $T$ in ${\mathcal{A}}$ with $\operatorname{Ext}^1_{\mathcal{A}}(T,T)=0$ such that $\operatorname{Hom}_{\mathcal{A}}(T,A)=0$ and $\operatorname{Ext}^1_{\mathcal{A}}(T,A)=0$ imply $A=0$. Thus the functor $\operatorname{Hom}_{\mathcal{A}}(T,-)\colon{\mathcal{A}}\to\operatorname{mod}\Lambda$ into the category of modules over the endomorphism algebra $\Lambda=\operatorname{End}_{\mathcal{A}}(T)$ induces an equivalence $$\mathbf{D}^b({\mathcal{A}})\xrightarrow{\sim}\mathbf{D}^b(\operatorname{mod}\Lambda)$$ of derived categories [@a4-c3-c13-Ac3-HK2007]. There are two important classes of such hereditary abelian categories admitting a tilting object: module categories over hereditary algebras, and categories of coherent sheaves on weighted projective lines in the sense of Geigle and Lenzing [@a4-c3-c13-GL1987]. Happel’s theorem then states that there are no further classes.
A hereditary abelian category with a tilting object is, up to a derived equivalence, either of the form $\operatorname{mod} A$ for some finite dimensional hereditary algebra $A$ or of the form $\operatorname{coh}{\mathbb{X}}$ for some weighted projective line ${\mathbb{X}}$.
It is interesting to observe that these abelian categories form a category: Any thick and coreflective subcategory is again an abelian category of that type; so the morphisms are given by such inclusion functors.
Now, fix an abelian category ${\mathcal{A}}$ which is either of the form ${\mathcal{A}}=\operatorname{mod} A$ or ${\mathcal{A}}=\operatorname{coh}{\mathbb{X}}$, as above. Note that in both cases the Grothendieck group $K_0({\mathcal{A}})$ is free of finite rank and equipped with an Euler form, as explained before. An object $X$ in ${\mathcal{A}}$ is called *exceptional* if it is indecomposable and $\operatorname{Ext}_{\mathcal{A}}^1(X,X)=0$. A sequence $(X_1,\ldots,X_r)$ of objects is called *exceptional* if each $X_i$ is exceptional and $\operatorname{Hom}_{\mathcal{A}}(X_i,X_j)=0=\operatorname{Ext}_{\mathcal{A}}^1(X_i,X_j)$ for all $i>j$. Such a sequence is *complete* if $r$ equals the rank of the Grothendieck group $K_0({\mathcal{A}})$. Let $n$ denote rank of $K_0({\mathcal{A}})$. Then, the braid group ${\mathcal{B}}_n$ on $n$ strands is acting on the collection of isomorphism classes of complete exceptional sequences in ${\mathcal{A}}$ via mutations, and it is an important theorem that this action is transitive (due to Crawley-Boevey [@a4-c3-c13-CB1992] and Ringel [@a4-c3-c13-Ri1994] for module categories, and Kussin–Meltzer [@a4-c3-c13-KM2002] for coherent sheaves).
Any tilting object $T$ admits a decomposition $T=\bigoplus_{i=1}^nT_i$ such that $(T_1,\ldots,T_n)$ is a complete exceptional sequence. We denote by $W({\mathcal{A}})$ the group of automorphisms of $K_0({\mathcal{A}})$ that is generated by the corresponding reflections $s_{T_1},\ldots,s_{T_n}$; it is the *Weyl group* with Coxeter element $c=s_{T_1}\cdots s_{T_n}$ and does not depend on the choice of $T$. Thus we can consider the poset of non-crossing partitions and we have the Hurwitz action on factorisations of the Coxeter element as product of reflections. But it is important to note that $W({\mathcal{A}})$ is not always a Coxeter group when ${\mathcal{A}}=\operatorname{coh}{\mathbb{X}}$, and it is an open question whether the Hurwitz action is transitive.
The key observation is now the following.
The map $$(E_1,\ldots, E_r)\longmapsto s_{E_1}\cdots s_{E_r}$$ which assigns to an exceptional sequence in ${\mathcal{A}}$ the product of reflections in $W({\mathcal{A}})$ is equivariant for the action of the braid group ${\mathcal{B}}_r$.
The proof is straightforward. But a priori it is not clear that the product $s_{E_1}\cdots s_{E_r}$ is a non-crossing partition. In fact, the proof of Theorem \[th:algebras-main\] hinges on the transitivity of the Hurwitz action on factorisations of the Coxeter element. So the analogue of Theorem \[th:algebras-main\] for categories of type ${\mathcal{A}}=\operatorname{coh}{\mathbb{X}}$ remains open. A proof would provide an interesting extension of the theory of crystallograpic Coxeter groups and non-crossing partitions, which seems very natural in view of Happel’s theorem since the Grothendieck group $K_0({\mathcal{A}})$ is a derived invariant.
Partial results were obtained recently by Wegener in his thesis [@a4-c3-a4-c3-c13-We2017]. In fact, when a weighted projective line ${\mathbb{X}}$ is of tubular type (that is, the weight sequence is up to permutation of the form $(2,2,2,2), (3,3,3), (2,4,4)$ or $(2,3,6)$), then the Grothendieck group gives rise to a tubular elliptic root system [@a4-c3-c13-Sai; @a4-c3-c13-STW]. Wegener showed the transitivity of the Hurwitz action in this case. Thus, one has in particular the analogue of Theorem \[th:algebras-main\] for $\operatorname{coh}{\mathbb{X}}$ in the tubular case.
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abstract: 'We make a further study of the very deep *Chandra* observation of the X-ray brightest galaxy cluster, A426 in Perseus. We examine the radial distribution of energy flux inferred by the quasi-concentric ripples in surface brightness, assuming they are due to sound waves, and show that it is a significant fraction of the energy lost by radiative cooling within the inner 75-100 kpc, where the cooling time is 4–5 Gyr, respectively. The wave flux decreases outward with radius, consistent with energy being dissipated. Some newly discovered large ripples beyond 100 kpc, and a possible intact bubble at 170 kpc radius, may indicate a larger level of activity by the nucleus a few 100 Myr ago. The distribution of metals in the intracluster gas peaks at a radius of about 40 kpc and is significantly clumpy on scales of 5 kpc. The temperature distribution of the soft X-ray filaments and the hard X-ray emission component found within the inner 50 kpc are analysed in detail. The pressure due to the nonthermal electrons, responsible for a spectral component interpreted as inverse Compton emission, is high within 40 kpc of the centre and boosts the power in sound waves there; it drops steeply beyond 40 kpc. We find no thermal emission from the radio bubbles; in order for any thermal gas to have a filling factor within the bubbles exceeding 50 per cent, the temperature of that gas has to exceed 50 keV.'
author:
- |
J.S. Sanders[^1] and A.C. Fabian\
Institute of Astronomy, Madingley Road, Cambridge. CB3 0HA
bibliography:
- 'refs.bib'
title: 'A deeper X-ray study of the core of the Perseus galaxy cluster: the power of sound waves and the distribution of metals and cosmic rays'
---
X-rays: galaxies — galaxies: clusters: individual: Perseus — intergalactic medium — cooling flows
Introduction
============
The Perseus cluster, Abell 426, is the brightest galaxy cluster in the sky when viewed in the X-ray band. The cluster contains a bright radio source 3C 84 [@PedlarPerseus90], the radio lobes of which are displacing the X-ray emitting thermal gas of the cluster [@BohringerPer93; @FabianPer00]. The X-ray emission from the intracluster medium (ICM) is highly peaked in the centre and the radiative cooling time of the hot gas is less than 5 Gyr within a radius of 100 kpc, decreasing to $3\times 10^8\yr$ within the central 10 kpc [@SandersPer04]. A cooling flow of several $100\Msunpyr$ would take place if radiative energy losses from the inner ICM are not balanced by some form of energy injection. As expected from cooling, the gas temperature does drop from the outer value of 7 keV within the central 100 kpc but only down to about 2.5 keV with little X-ray emitting gas found at lower temperatures, except in coincidence with line-emitting filaments seen at optical wavelengths [@FabianPer03; @FabianPer06]. Heating by the central radio source is widely considered responsible for balancing the radiative cooling, although the exact mechanisms by which the energy is transported and dissipated and a heating/cooling balance established have been unclear.
Similar behaviour is found in many X-ray peaked, cool-core clusters (see @PetersonFabian06 for a review). The central galaxy in such clusters could grow considerably larger than observed if radiative cooling was unchecked. X-ray observations of their innermost regions provide an excellent means to study the feedback of an Active Galactic Nucleus (AGN) on its host galaxy in action. Here we use a long (900 ks) Chandra observations of the X-ray brightest cluster core to examine the energy balance and metallicity in considerable detail. The quality of the data, in terms of counts per arcsec, are much higher than available with any other cluster. We examine the power propagating through the core in terms of pressure ripples, or sound waves, and the cosmic-ray implications of a hard X-ray component. We also use the metallicity distribution, the temperature profile of the X-ray emitting gas across an optical filament and assess the thermal gas content of the radio bubbles.
No X-ray emission has been detected from within the radio bubbles [@SchmidtPer02; @SandersPer04]. Two depressions in the X-ray surface brightness are also found, to the north-west and south, not associated with high-frequency radio emission. These are likely to be ‘ghost’ radio bubbles which have detached from the nucleus and buoyantly risen in the gravitational field [@ChurazovPer00; @FabianPer00], an idea supported by weak low frequency radio spurs seen pointing towards their direction [@FabianCelottiPer02].
Surrounding the inner radio lobes are X-ray bright rims [@FabianPer00] at a higher pressure than the outer regions, and separated from them by a weak isothermal shock [@FabianPer06]. Extending further into the clusters are concentric fluctuations in surface brightness. These are plausibly pressure and density ripples, in which case they are sound waves generated by the inflation of the radio bubbles [@FabianPer03; @FabianPer06]. The period of the waves is about 10 Myr, close to the expected age of the bubbles due to buoyancy, and is not plausibly related to any other source of disturbance. Ripples due to sound waves have since been found in simulations [@Ruszkowski04; @Sijacki06]. Such sound waves can transport significant energy in a roughly isotropic manner and so balance radiative cooling of the intracluster gas if they dissipate their energy over the surrounding 50–100 kpc [@FabianPer03; @Ruszkowski04; @FabianReynolds05]. Concentric ripple-like features are also found around M87 in the Virgo cluster, and are interpreted there as weak shocks [@FormanM8706]. More powerful outbursts include those discovered in MS 0735.6+7421 [@McNamara05], Hercules A [@NulsenHerc05] and Hydra A [@NulsenHydra05].
Around the central galaxy in the Perseus cluster, NGC 1275, lies a giant line-emitting nebula [@Lynds1970; @Conselice01], associated with cool $\sim 0.7$ keV X-ray emitting filaments [@FabianPerFilament03] of $\sim 10^{9}\Msun$ mass [@FabianPer06]. The H$\alpha$ emitting filaments appear to be have been drawn out of the central galaxy by the rising bubbles [@HatchPer06].
There is spectral evidence for hard X-ray emission from the central region, found either with hot thermal [@SandersPer04] or non-thermal [@SandersNonTherm05] models. The 2-10 keV luminosity of this emission is $\sim 5 \times 10^{43} \ergps$.
The iron metallicity structure in the core of the cluster is inhomogeneous and complex [@SchmidtPer02; @SandersPer04; @SandersNonTherm05] with the abundance dropping in the very core. There are also structures such as high-metallicity ridge and blobs, which may be associated with the bubbles.
We use a Perseus cluster redshift of 0.0183 here, which gives an angular scale of 0.37 kpc per arcsec, assuming $H_0 = 70
\kmpspMpc$.
Data preparation
================
The datasets analysed in this paper are those that were examined in [@FabianPer06]. However here we used the standard <span style="font-variant:small-caps;">ciao</span> data preparation tools on the event files. We used the datasets from the *Chandra* archive which had gone through Reprocessing III (reprocessed using version 7.6.7.1 of the pipeline). Datasets 03209 and 04289 were not reprocessed at the time of the analysis, so we therefore went through each of the steps in the Science Threads to reprocess the data manually for these. The reprocessing was done prior to CALDB 3.3.0 (using gainfile acisD2000-01-29gain\_ctiN0005.fits).
We filtered the level 2 event files for flares as in [@FabianPer06]. We then reprojected each dataset to match the 04952 observation in sky coordinates. We also reprocessed a combined 980-ks blank sky observation file to use the same calibration files as used by the foreground observations. We randomised the order of the events in the background file in order to remove any potential spectral variability. The background file was split into sections, to provide a background for each foreground observation. The length of each section was chosen to have the same ratio to the total as the ratio of its respective foreground dataset to the total foreground. The exposure time of each section was adjusted to give the same count rate in the 9-12 keV band as its respective foreground (where there is no source). The background sections were then reprojected to the original level 2 event file for the matching observation, and then reprojected to the 04952 observation.
In addition we constructed separate background event files for each dataset to account for out-of-time events, where photons hit the detector while it is being read out. We used the <span style="font-variant:small-caps;">make\_readout\_bg</span> script (written by M. Markevitch) to construct these files from the original level 1 event files after filtering bad time periods. The readout backgrounds were reprojected to match the 04952 observation.
For the spectral analysis, for a particular region we extracted foreground spectra from each of the foreground event files relevant for the region in question. Similarly we made spectra from the background event files and and readout background files. The foreground spectra were added together to make a total foreground spectrum with a total exposure time. Background spectra were added similarly to make a total background, and so were readout background spectra.
We also created responses and ancillary responses for each of the foreground datasets, weighting the responses according to the number of counts in each spatial region between 0.5 and 7 keV. A response and ancillary response was made for the total foreground spectrum by adding the responses and ancillary responses for the individual observations, weighting according to the number of counts between 0.5 and 7 keV.
To analyse the spectra, we fit them in <span style="font-variant:small-caps;">xspec</span> version 11.3.2 [@ArnaudXspec]. The energy range 0.6 to 8 keV was used during fitting, and the spectra were grouped to have at least 20 counts per spectral bin. The spectra extracted from the appropriate region from the background files was used as a background spectrum, and the spectra from the region from the readout background files was used as a ‘correction file’.
In this paper we use the <span style="font-variant:small-caps;">apec</span> [@SmithApec01] and <span style="font-variant:small-caps;">mekal</span> [@MeweMekal85; @MeweMekal86; @KaastraMekal92; @LiedahlMekal95] thermal spectral models. To model photoelectric absorption we use the <span style="font-variant:small-caps;">phabs</span> model [@BalucinskaChurchPhabs92].
X-ray surface brightness
========================
Surface brightness images {#sect:sb}
-------------------------
{width="\textwidth"}
[@FabianPer03] and [@FabianPer06] used unsharp-masking techniques to reveal the surface brightness fluctuations in the intracluster medium. Unsharp masking increases the noise in the outer parts of the image where there are relatively few counts. We have experimented with several techniques to improve on simple unsharp masking. We split the exposure-map-corrected image (Fig. \[fig:sb\] left panel) into 40 sectors (centred on the central nucleus), and fitted a King model to each of the sectors outside of a radius of 13 arcsec (to avoid the central source). We constructed a model surface brightness image by iterating over each pixel, using the value obtained by interpolating in angle between the model surface brightness profiles of the two neighbouring sectors. This model image was then subtracted from the original image, resulting in Fig. \[fig:sb\] (centre panel). The image very clearly highlights the surface brightness increase associated with the low-temperature spiral in the cluster [@FabianPer00; @ChurazovPer00]. A number of other features can be seen, including the possible cold front to the south, the ‘bay’ and the ‘arc’ (figure 2 in @FabianPer06). It does not show the ripples particularly clearly (except perhaps by the southern bubble) as the cool swirl is dominant.
The ripples are more clearly highlighted using a Fourier high-pass filter technique. A two-dimensional fast Fourier transform of the exposure-map-corrected image was made. We removed the low frequency components with a wavelength greater than 75 arcsec. Frequency components between wavelengths of 75 and 38 arcsec were allowed through using with a linear filter increasing from 0 to 1 between these wavelengths. All shorter wavelengths were left to remain. The Fourier-transformed image was then transformed back to give Fig. \[fig:sb\] (right panel) after light smoothing. This technique removes the cool swirl and the underlying cluster emission. It clearly reveals the ripples, presumably sound waves generated by the inflation of the bubbles, discovered by [@FabianPer03]. It shows a previously unseen ripple near the edge of the image to the east.
Sector across eastern ripples
-----------------------------
![Full band X-ray image showing the sector examined for surface brightness fluctuations (Fig. \[fig:sbripple\]). The region is centred on the inner NE radio bubble to better match the surface brightness contours.[]{data-label="fig:sbregions"}](fig02.jpg.eps){width="\columnwidth"}
![Surface brightness profile to the ESE of the cluster core (0.3 to 7 keV), also showing a King model fit, fractional residuals from the fit, a temperature profile with fit, and fractional residuals from the temperature fit. The dotted lines on the temperature residual plot show the expected variation associated with a 3 per cent surface brightness fluctuation. Note that the profile is not centred on the nucleus.[]{data-label="fig:sbripple"}](fig03.jpg.eps){width="\columnwidth"}
To quantitatively examine the ripples, we have examined the surface brightness in a sector shown in Fig. \[fig:sbregions\]. We show in Fig. \[fig:sbripple\] (top panel) a surface brightness profile (in the 0.3 to 7 keV band) made to the east-south-east (ESE) of the cluster core. Also shown in the top panel is a King model fit to the profile. In the second panel we show the fractional residuals to the fit, clearly showing the oscillations in surface brightness observed in Fig. \[fig:sb\] (right panel) at the few percent level (the first peak in this figure corresponds to the bright rim of the radio bubbles).
To look for any temperature structure associated with the oscillations we fitted <span style="font-variant:small-caps;">phabs</span> absorbed <span style="font-variant:small-caps;">apec</span> thermal spectral models to spectra extracted in annuli from the spectra. The fitting procedure allowed the temperature, metallicity, normalisation and absorption to vary, and minimised the C-statistic [@Cash79] in each fit. The projected temperature profile is shown in the third panel of Fig. \[fig:sbripple\]. We also show the results from fitting the same model (using the $\chi^2$ statistic) to deprojected spectra in larger bins. We used the deprojection method in Appendix \[appendix:deproj\] to create the deprojected spectra.
We finally show the residuals from a simple ‘$\eta$ model’ [@AllenSchmidtFabian01] fits to the projected and deprojected temperature profiles (Fig. \[fig:sbripple\] fourth panel) superficially. There is no obvious temperature structure associated with the ripples in this location of the cluster. The amount of temperature variation expected from the surface brightness fluctuations can be estimated. Simulations of thermal spectra in <span style="font-variant:small-caps;">xspec</span> shows that the surface brightness is independent of temperature at constant density in the temperature range 3–6 keV. If the adiabatic index of the ICM is $\gamma=5/3$, assuming the ideal gas law and that the X-ray surface brightness is proportional to the density squared, the fractional temperature ($T$) fluctuations associated with surface brightness ($I$) changes should be of magnitude $$\frac{\delta T}{T} = \frac{1}{3} \frac{\delta I}{I}.$$ We plot on the lower panel of Fig. \[fig:sbripple\] dotted lines showing the range of temperature variation expected to be associated with 3 per cent variations in surface brightness (which are the maximum observed here). These include a scaling factor of 5 to convert from projected surface brightness fluctuations to intrinsic emissivity fluctuations (see Section \[sect:wavepower\]). The deprojected temperatures are comparable to those expected but we caution that there is significant noise in the results.
Wave power {#sect:wavepower}
----------
We now estimate the power implied by the surface brightness fluctuations observed in the cluster (Fig. \[fig:sb\] right), assuming that they are sound waves. This is then compared with the power radiated within the inner regions of the cluster core where the radiative cooling time is shorter than its expected age of a few Gyr (by age in this context we mean the time since the last major merger).
The instantaneous power, $\mathcal P$, transmitted in a spherical sound wave is given by [@LandauLifshitzFluids] $$\mathcal P = 4 \pi r^2 \frac{\delta P^2}{\rho c},
\label{eqn:wavepower}$$ at a radius $r$, where the sound wave pressure amplitude is $\delta
P$, the mass density of the medium is $\rho$ and the sound speed is $c$. The sound wave pressure amplitude can be computed from the density amplitude with $$\delta P = \frac{5}{3} \, n_\mathrm{e} \, kT \,
\frac{\delta n_\mathrm{e}}{n_\mathrm{e}}\alpha,$$ assuming $\gamma=5/3$, where $\alpha$ is a factor to convert from the electron number density $n_\mathrm{e}$ to the total number density.
The fractional variation in density over the wave $\delta
n_\mathrm{e}/n_\mathrm{e}$ is estimated from the surface brightness fractional change. As discussed above, there is little variation of surface brightness with temperature at constant density in this temperature range. Assuming bremsstrahlung emission, the fractional variation in density should be half that of the surface brightness seen. This is not the case in reality, as projection effects are dominant. We constructed simple numerical simulations of 10 to 20 kpc wavelength density waves in a cluster density profile (we tried the profiles in @ChurazovPer03 and @SandersPer04). The typical conversion factor from a fractional surface brightness to density perturbation is 2–3, with more suppression for smaller wavelength waves.
We take the surface brightness image, and filter it with a high-pass filter as in Section \[sect:sb\]. We use a looser filtering here as some of the observed 20 kpc waves are otherwise suppressed (not filtering anything below a wavelength 62 arcsec, increasingly filtering up to 124 arcsec and discarding everything longer wavelength). The original surface brightness image was binned using the contour binning algorithm [@SandersBin06] to have $10^4$ counts per region. We applied the same binning to the filtered image, and divided it by the binned surface brightness image. This created a fractional surface brightness variation map. The contour-binning routine follows the surface brightness closely, so bins are also aligned well with the ripples.
![Power in surface brightness fluctuations, assuming they are due to spherical waves. The units are $\mathrm{log}_{10}
\ergps$. The rectangles show the regions examined in Fig. \[fig:wavepowcuml\] and \[fig:wavepowinst\]. The circles show the excluded inner bubble and NW bubble regions. We assume a factor of 2.5 to convert from surface brightness to density variations.[]{data-label="fig:wavepowerimg"}](fig04.jpg.eps){width="\columnwidth"}
For each pixel on the fractional surface brightness variation map, we compute the power in a spherical wave from Equation \[eqn:wavepower\], assuming that the radius of the wave is the projected radius on the sky. Deprojected density and temperature values at each radius were calculated from a fit to the average profiles in [@SandersPer04]. In Fig. \[fig:wavepowerimg\] we show a map of computed power values for each pixel, assuming a factor of 2 to convert from surface brightness variations to density variations. Many of the ripple features in Fig. \[fig:sb\] (right panel) are seen in this image, plus the radio bubbles (which are not themselves sound waves). We bin the data in this image rather than use simple smoothing, as noise in the outer regions means that the power is overestimated.
![Plot of mean power in sound waves and the cluster cumulative X-ray luminosity inwards of where the cooling time is 4 or 5 Gyr. The vertical line shows the inner radius of our measurements due to the radio lobes. The total power in the lobes approaches $10^{45}\ergps$ [@DunnFabian04].[]{data-label="fig:wavepowcuml"}](fig05.eps){width="\columnwidth"}
{width="70.00000%"}
In Fig. \[fig:wavepowcuml\] we show the average values at each radius, masking out the central bubbles and north-west bubbles and the edges of the CCD to avoid filtering artifacts (using the regions in Fig. \[fig:wavepowerimg\]). On the plot we also display the cumulative luminosity from the cluster calculated from the deprojected density, temperature and abundance values from [@SandersPer04]. We accumulate the luminosity inwards from a radius of 75 or 100 kpc, the radii corresponding to where the mean radiative cooling time of the gas [@SandersPer04] is $\sim 4$ or $5 \Gyr$, the likely age of the cluster. Note that if the sound speed is a function of azimuth in the cluster (the temperature map indicates that this is so), then the phasing of the waves depends on azimuth. This will lead to smearing of the ripples in Fig. \[fig:wavepowcuml\].
We plot the wave power out to larger radii in Fig. \[fig:wavepowinst\]. In this graph we also show the variation in wave power at each radius as a shaded region. This was computed by repeating the calculation of the average power in six equal sectors, and shading the region between the minimum and maximum values at each radius. At large radii only a couple of sectors were used, due to the position of the source on the detector. We show the luminosity of the cluster per unit length of radius in this plot to compare to the wave power.
Fig. \[fig:wavepowcuml\] shows that the net power in the ripples is a few times $10^{44}\ergps$ and sufficient to offset a significant part of the radiative cooling within the innermost 70 kpc or so. The power implied by the analysis drops off with radius out to 120 kpc, with an e-folding length of about 50 kpc, consistent with models of viscous dissipation [@FabianPer03], which is required if heating by sound waves offsets radiative cooling.
Larger power is seen near the edge of Fig. \[fig:wavepowinst\] around 105, 115 and 125 kpc radius. These show possible evidence that the source was more powerful several $10^8\yr$ ago. Such powerful shocks could have been created as several individual bursts, or from a single burst producing multiple sound waves [@Brueggen07]. We caution that most of the signal comes from a small angular region to the extreme SW of the main detector. Further observations covering a wider region are required to accurately determine the wave power at this radius.
![Further possible surface brightness discontinuities to the north of the cluster. This is an unsharp-masked image, subtracting images smoothed with a Gaussian by 4 and 16 arcsec and dividing by the 16 arcsec map. Point sources were excluded from the smoothing. The apparent features are at radii of $\sim 130$ and 170 kpc (indicated by solid lines).[]{data-label="fig:unsharp_outer"}](fig07.jpg.eps){width="\columnwidth"}
![Fractional residuals from a $\beta$ model fit to the surface brightness in sectors to the north at large radii. The full sector shows a profile between 19 and $75^\circ$ from west towards north, while the narrow sector is between 59 and $75^\circ$.[]{data-label="fig:betafit_outer"}](fig08.eps){width="\columnwidth"}
We also looked for structure in images of the cluster combining all of the ACIS (Advanced CCD Imaging Spectrometer) CCDs. Fig. \[fig:unsharp\_outer\] shows an unsharp-masked image of the north of the cluster to large radii. We note that there appears to be a sharp edge at around 130 kpc radius in this direction (although it varies in radius in the northern sector). This is at approximately the same radius as the ripple seen in Fig. \[fig:wavepowinst\]. The edge can be seen in residuals from $\beta$ model plus background fits to a wide and narrow sector (Fig. \[fig:betafit\_outer\]). This discontinuity can also be seen in an *XMM-Newton* observation of Perseus (Fig. 7 in @ChurazovPer03).
At a radius of around 170 kpc is another feature. This appears to be a dip in surface brightness followed by a rise. This feature is particularly sharp in fits to the surface brightness in a narrow sector (Fig. \[fig:betafit\_outer\]). A possible interpretation is an ancient radio bubble which is still intact. The thermal gas pressure is likely to be around 4 times less at this radius, so if the bubble remains intact it will be around 4 times larger than it was originally (if it retains its original energy). Rough scaling with the rising bubble NW of the nucleus suggests that it would have to have at least twice as much energy. The direction from the centre in which the dip is most visible (the longer solid line in Fig. \[fig:unsharp\_outer\]) is also directly along the northern H$\alpha$ filament and fountain [@FabianPer06]. If it is indeed a bubble then it shows that bubbles remain intact to very large radii in galaxy clusters. Bubble-like low pressure regions were also seen to the south of the core [@FabianPer06].
To constrain better the power in the feature at 130 kpc and confirm the radio bubble near 170 kpc radius, requires further deep observations by *Chandra* offset from the cluster core to improve the point spread function (the average radius of which is 10 arcsec at 10 arcmin off axis, compared with $\sim 1$ arcsec on axis).
Metallicity map
===============
![Metallicity map of the core of the cluster relative to solar. Regions contain greater than $4\times 10^{4}$ counts. Fits assume solar ratios of elements, but the results mainly depend on the iron abundance. Uncertainties for each spectral fit range smoothly from $0.025$ in the centre to $0.07\Zsun$ to the extreme bottom left of this image. The circle shows the approximate position of the ancient bubble, and the diamond shows the high metallicity blob in its apparent wake.[]{data-label="fig:Zmap"}](fig09.jpg.eps){width="\columnwidth"}
Fig. \[fig:Zmap\] shows a metallicity map of the core of the cluster. This was generated by extracting spectra from contour-binned regions [@SandersBin06] containing $\sim 4 \times 10^4$ counts. The spectra were fit by a <span style="font-variant:small-caps;">phabs</span> absorbed <span style="font-variant:small-caps;">mekal</span> model with the absorption, temperature, emission-measure and metallicity free. Note that the plot does not clearly show the high abundance shell (possibly marking the location of an ancient bubble) found by [@SandersNonTherm05] as there were no new data in that region in these observations, and the spatial regions we use here are larger.
![Radius versus projected temperature and metallicity. Values were measured from spectral fitting a <span style="font-variant:small-caps;">mekal</span> model to spectra from bins with $\sim 2.5 \times 10^5$ counts. Also plotted is the projected entropy as a function of radius (see Section \[sect:metalrelations\]).[]{data-label="fig:rad_Z_T"}](fig10.jpg.eps){width="\columnwidth"}
To examine the variation more quantitatively, we have repeated the spectral fitting using regions containing $\sim 2.5 \times 10^5$ counts to decrease the size of the uncertainties. The radial temperature and metallicity variation are plotted in Fig. \[fig:rad\_Z\_T\], generated by plotting the average radius of each bin against the value obtained from that region. At each radius there is a large spread in temperature and abundance.
Metallicity relations {#sect:metalrelations}
---------------------
![Temperature versus metallicity and pseudo-entropy versus metallicity plots. All values are projected. The bins used are the same as in Fig. \[fig:rad\_Z\_T\].[]{data-label="fig:Z_entropy_T"}](fig11.jpg.eps){width="\columnwidth"}
In [@SandersPer04] we plotted the temperature of regions against their metallicity. As the temperature of the gas declines towards the centre, the metallicity reaches a maximum at a radius of 40 kpc, then decreases again. We plot the metallicity-temperature relation from the new data in Fig. \[fig:Z\_entropy\_T\] (top panel). Although we see a similar relation, the reduced errors bars show that there is significant metallicity scatter at each temperature. Indeed the scatter appears similar to that in the radial plot of the metallicity (Fig. \[fig:rad\_Z\_T\]). The temperature of the gas does not appear to correlate better with the metallicity than the radius.
Another interesting physical quantity is the entropy of the gas. Entropy in clusters is usually defined as $K = kT n_{e}^{-2/3}$. Using the <span style="font-variant:small-caps;">xspec</span> normalisation[^2] per unit area on the sky, $N \propto
n_{e}^{2} d$, where $d$ is a depth in the cluster, which we assume to be constant, we can calculate a pseduo-projected entropy quantity $N^{-1/3} kT$. The plot of this quantity against the metallicity (Fig. \[fig:Z\_entropy\_T\] bottom panel) looks similar to the temperature-metallicity plot, with a great deal of scatter at each entropy value. The values use $N$ in <span style="font-variant:small-caps;">xspec</span> normalisation units per square arcsec and $kT$ in keV.
The scatter in the abundance seems unrelated to the temperature, radius, or entropy. Higher metallicity regions will have shorter mean radiative cooling times as the line emission is stronger, but this effect is not strong at temperatures of $\sim 3-7 \keV$, so it is perhaps not surprising that metallicity and temperature or entropy (and therefore cooling time) are unrelated.
Central abundance drop
----------------------
There is also a drop in metallicity in the central regions (as can be seen in the metallicity map in Fig. \[fig:Zmap\]). This feature appears to be unaffected by including other components, such as extra temperature components, or powerlaw models.
![Iron metallicity profile to the NW of the cluster between position angles 290.0 and $4.7^\circ$ from 3C 84. This was produced by fitting <span style="font-variant:small-caps;">vmekal</span> models to projected and deprojected spectra.[]{data-label="fig:Z_deproj"}](fig12.eps){width="\columnwidth"}
We have investigated whether the central drop in metallicity could be caused by projection effects. We use the spectral deprojection method outlined in Appendix \[appendix:deproj\] to construct a set of deprojected spectra. Fig. \[fig:Z\_deproj\] shows the iron metallicity profile computed by fitting isothermal <span style="font-variant:small-caps;">vmekal</span> models to the spectra before and after deprojection. In the fit the elemental abundances of O, Ne, Mg, Si, S, Ar, Ca, Fe and Ni were allowed to vary, with the gas temperature, the model emission measure, and the absorbing column density. The analysis shows the peak at around 40 kpc radius is enhanced by accounting for projection and the central drop remains.
The drop in metallicity in the central regions appears to be robust. The entropy of the gas in the central regions is much lower than further out, which means it is difficult for the source of the low metallicity gas to be from larger radius.
Small scale variation
---------------------
![Abundance and temperature profiles across the high metallicity blob near 31940, $+41^\circ29'15''$. The profile crosses the blob in the SE to NW direction, inclined by an angle of $40^\circ$ from the west northwards.[]{data-label="fig:blob_profile"}](fig13.eps){width="\columnwidth"}
Much of the structure in the metallicity map is real. To demonstrate this, we show in Fig. \[fig:blob\_profile\] a metallicity and temperature profile across the high metallicity blob apparently in the wake of a possible ancient bubble down to the extreme SW (marked on Fig. \[fig:Zmap\]; @SandersNonTherm05). The metallicity profile shows a significant peak with a width of $\sim 5 \kpc$ at the position of the blob (we also see $\sim 5 \kpc$ metallicity features near the filaments in Section \[sect:proffilament\]). There is no obvious correlation between the temperature and metallicity profiles. Taking the global diffusion coefficient of $2 \times 10^{29}
\cm^{2} \s^{-1}$ measured by [@RebuscoDiff05] in Perseus and a lengthscale of 5 kpc, the lifetime of such a feature is only $\sim
40$ Myr. This is roughly comparable to the buoyancy timescale estimated for the ancient bubble of 100 Myr [@DunnPer06], given the large uncertainties, but may require that diffusion is suppressed on small scales as it appears unlikely that the metals could have been injected in-situ. The sharp edges of the feature would imply diffusion times of only a few Myr, requiring significant suppression.
The blob has a metallicity approximately $0.2 \Zsun$ higher than its surroundings (which are at approximately $0.6 \Zsun$). This is probably a lower limit because of projection effects. Assuming the blob is sphere with diameter 5 kpc, and a local electron density of around $0.037 \pcmcu$ [@SandersPer04], it represents an enhancement of around $2.6 \times 10^4 \Msun$ of Fe. If most of this enrichment is due to Type Ia supernova, then this corresponds to around $3.7 \times 10^4$ supernova (assuming $0.7 \Msun$ of Fe produced per supernova). Taking the timescale from diffusion, this corresponds to 0.1 supernova type Ia per century for this 5 kpc radius region, which is a significant fraction of the rate expected from a single galaxy.
The high velocity system {#sect:hvs}
========================
[@GillmonPer04] studied the High Velocity System (HVS), a distinct emission-line system at a higher velocity than NGC 1275 [@Minkowski57], using an earlier 200-ks *Chandra* observation of the system. The system is observed in absorption in X-rays [@FabianPer00], as it lies between most or all of the cluster and the observer. [@GillmonPer04] mapped the absorbing column density and placed a lower limit of 57 kpc of the distance between the nucleus of NGC 1275 and the HVS. They obtained a total absorbing gas mass of $1.3 \times 10^9 \Msun$, assuming Solar metallicities.
![(Top panel) Image of the high-velocity system in the 0.3 to 0.7 keV band smoothed by a Gaussian of 0.49 arcsec. (Bottom panel) Measurements of the equivalent Hydrogen column density over the high-velocity system region, in units of $10^{22}\psqcm$ (subtracting Galactic absorption of $0.12 \times 10^{22}\psqcm$). The values were measured from spectral fitting in $0.74 \times
0.74$ arcsec pixels. The central nucleus of 3C 84 shows up as a point-like object near 31948, $+41^\circ30'40''$. The uncertainty on the column density on individual pixels is around 0.1 for the regions of highest absorption and 0.03 outside of the HVS.[]{data-label="fig:NHmap"}](fig14.jpg.eps){width="\columnwidth"}
Here we repeat the mapping analysis with the new 900-ks dataset. Fig. \[fig:NHmap\] (top panel) shows an image in the 0.3-0.7 keV band, clearly showing the absorbing material. To quantify the amount of material we fitted an <span style="font-variant:small-caps;">apec</span> thermal spectral model for the cluster emission, absorbed by a <span style="font-variant:small-caps;">phabs</span> photoelectric absorber to spectra extracted from a grid of $0.74 \times 0.74$ arcsec regions around the HVS. In the fits the temperature of the thermal emission, its normalisation, and the absorbing column density were free. The abundance of the thermal gas was fixed to $0.7 \Zsun$ (based on the results of spectral fitting to larger regions near the core of the cluster; the results are very similar if this is allowed to be free). Spectra were binned to have a minimum of 5 counts per spectral channel. We minimised the C-statistic to find the best fitting parameters (note that no background or correction spectra were used in the fits here, as the cluster is much brighter than the background here).
In Fig. \[fig:NHmap\] (bottom panel) we show the resulting Hydrogen column density map over the HVS region, including the Galactic contribution. We stress that these values are *equivalent* Hydrogen column density. The measurements are most sensitive to the abundance of Oxygen in the absorber. The Hydrogen column density is calculated assuming the Solar abundance ratios of [@AndersGrevesse89], which gives an Oxygen to Hydrogen number density ratio of $8.51 \times 10^{-4}$. The column density is flat in this region beyond the HVS. We obtain an average value of this Galactic component of $0.12 \times 10^{22} \psqcm$, consistently from several surrounding regions.
The total number of absorbing atoms can be calculated using the mean absorption over the HVS, the Galactic contribution, and the distance to Perseus. We converted this into a total absorbing mass of $1.1
\times 10^9 \Msun$, assuming Solar metallicity ratios. This is lower than the value of $(1.32 \pm 0.05) \times 10^9 \Msun$ quoted by @GillmonPer04. We repeated our analysis using the 1.96 arcsec binning factor used in that paper, but this did not significantly change our result. The twenty per cent difference may be due to the different calibration used in the earlier analysis, particularly the lack of correction for the variation in the contaminant on the detector, as the Galactic value is important to determine the total mass.
To place an improved lower limit of the distance of the HVS from the cluster nucleus, we examined a 0.3 to 0.7 keV image, where the absorption is strongest. Taking 1-1.5 arcsec diameter regions (with total area 3.9 arcsec$^2$) in the three highest absorption regions we find 17 counts in total in this band. Exposure-map correcting this surface brightness, and comparing it to the exposure-map corrected image of the cluster, we find that the surface brightness from the cluster only goes down to this level at a radius of approximately 110 kpc. This is a lower limit of the distance of the HVS from the cluster centre, as if the HVS were closer, the cluster emission should be ‘filling in’ the decrement in count rate caused by the absorption [@GillmonPer04]. This lower limit is a significantly better the previous value of 57 kpc.
As we discuss in Section \[sect:hvsthermal\], the HVS could have a shock cone behind it if it is travelling exactly along our line of sight towards the cluster. If this is the case, the shocked material would have a significantly lower density than the cluster at the same radius. This would mean that the lower limit we compute above would be overestimated.
Profiles across filaments {#sect:proffilament}
=========================
![Location of the regions used to measure the thermal properties across the filaments, displayed on a 0.3 to 1 keV image. Each of the 80 regions is $1\times27.3$ arcsec in size.[]{data-label="fig:filamentregions"}](fig15.jpg.eps){width="\columnwidth"}
To investigate the thermal structure of the X-ray filaments associated with the H$\alpha$ nebula in detail, we extracted spectra from small $1 \times 27.3$ arcsec boxes in a profile across the filament (the regions we used are shown in Fig. \[fig:filamentregions\]). We created background spectra, responses, ancillary responses, and out of time background spectra for each region.
![Emission measure profiles across the filaments in the different temperature components. The top panel shows an unsharp-masked 0.5-7 keV X-ray image rotated so that the bins lie across it. The second panel shows a similar H$\alpha$ image. The next panels show the 0.5, 1, 2 and 4 keV temperature component <span style="font-variant:small-caps;">xspec</span> normalisations, measured from the 1 arcsec wide bins. The final panels show the best fitting absorbing Hydrogen column density and the metallicity of thermal components.[]{data-label="fig:filamentmulti"}](fig16.jpg.eps){width="\columnwidth"}
Multiphase model
----------------
Our first model was to fit the spectra with a multiphase model consisting of <span style="font-variant:small-caps;">apec</span> thermal components at fixed temperatures of 0.5, 1, 2, 4 and 8 keV. The normalisations were allowed to vary and the metallicities of each of the components were tied together. They were absorbed with a <span style="font-variant:small-caps;">phabs</span> photoelectric absorber which was allowed to vary. We assume each component has the same metallicity as we cannot measure them independently. The measurement is likely to be driven by the cooler components as they are line-dominated. The model is similar to that used to produce figure 12 in [@FabianPer06], mapping the multiphase gas. We show the emission measure of each temperature component in Fig. \[fig:filamentmulti\], with the absorbing column density and metallicity. Also shown is an unsharp-masked image of the region examined in the 0.5-7 keV X-ray band (the X-ray image has been rotated so that the 1-arcsec bins lie horizontally across the region) and the H$\alpha$ image from [@Conselice01] taken using the WIYN telescope. We chose the rotation angle so that the bins were lined up across the central X-ray features. The H$\alpha$ filaments are not quite aligned in angle.
The plot shows that the filaments do not contain gas at temperatures of $\sim 4$ keV. At 2 keV we start to see the filaments, although their signal is fairly weak against that from nearby gas. The filaments are very strong near 1 keV, and still visible at 0.5 keV. There are some point-to-point differences however. The strong filament at 39 arcsec distance is strong in 1 and 0.5 keV, but the one nearby 46 arcsec does not appear at 1 keV, but does at 2 and 0.5 keV.
The column density profile is fairly flat. A linear model fitted to the column density profile gives a reduced $\chi^2$ of $86/78=1.10$. We see no evidence for additional X-ray absorption associated with the filaments.
The metallicity profile shows several features, which matches the complex structure seen in the metallicity map (Fig. \[fig:Zmap\]). A simple linear fit gives a reduced $\chi^2$ of $105/78=1.34$. There are quite strong peaks at the 59 arcsec, and at 12 arcsec. Neither is exactly at the position of a filament, though the one at 59 arcsec is offset by an arcsec or two from some gas from 2 to 0.5 keV. It is possible that some of the metallicity variation is caused by incomplete modelling of the multiphase gas within the filaments, but we observe almost identical variation with single phase and cooling flow (below) models. All models assume that the gas at each temperature has the same metallicity, though the cooler gas will dominate in the measurement of the metallicity.
![Results of fitting a cooling flow model to spectra extracted across the filaments in Fig. \[fig:filamentregions\]. The top panel shows an image of the region. In the second is plotted the mass deposition rates per bin obtained by fixing the lower temperature of the cooling flow models to certain values. The bottom panel shows the best fitting lower temperature if it is allowed to be free.[]{data-label="fig:filamentcflow"}](fig17.jpg.eps){width="\columnwidth"}
Cooling flow model
------------------
Given the range in the temperature in the filaments, we try fitting a cooling flow model to the spectra, as the gas may be cooling within the filament. We used the isobaric <span style="font-variant:small-caps;">mkcflow</span> model, which models cooling between two temperatures at a certain metallicity and gives the normalisation as a mass deposition rate, plus a <span style="font-variant:small-caps;">mekal</span> single phase thermal component. The upper temperature of the cooling flow component was tied to the temperature of the thermal component, and the metallicities were also tied. Both components were absorbed by a <span style="font-variant:small-caps;">phabs</span> absorber. Fig. \[fig:filamentcflow\] shows the mass deposition rates obtained with fixed minimum temperatures. The 0.25 keV rate is equivalent to a traditional cooling flow where the gas cools out of the X-ray band, while the others are truncated cooling flows. Also plotted is the best fitting minimum temperature if it is allowed to vary.
The results show that within the filaments, the model adds increasing amounts of the cooling flow component, indicating a range of temperatures. Using a cooling flow component with a 0.25 keV minimum temperature, the emission measure of the single phase thermal component varies relatively smoothly over the region. There is less apparent cooling as the minimum temperature decreases (as with the results of @Peterson03), until below $\sim 0.8 \keV$ where the results are approximately consistent. If the gas is actually cooling in these filaments, these results indicate a rate of $\sim
0.25 \Msunpyr$ in the central filament in this small region examined. All of the $\sim 0.5 \keV$ X-ray emitting gas observed in the Perseus cluster appears to be associated with the H$\alpha$ emitting filaments or is close to the nucleus [@FabianPer06]. The lower gas temperature measured here is consistent with zero in the centres of the filaments, but appears to increase by a few arcsec. We note that the best fitting absorbing column density is almost identical to the values from the multiphase model in Fig. \[fig:filamentmulti\], so no additional absorption is required.
![Comparison of H$\alpha$ surface brightness flux profile across the filaments (using data from @Conselice01) versus unabsorbed 0.001 to 30 keV surface brightness flux from the cooling flow model cooling down to 0.25 keV. Cooling to 0.0808 keV, the minimum available, increases the results by about 5 per cent. Both of these fluxes are in the same units.[]{data-label="fig:filament_cflowflux"}](fig18.eps){width="\columnwidth"}
In Fig. \[fig:filament\_cflowflux\] we compare the flux from the cooling flow component cooling down to 0.25 keV (above) against the flux in the continuum-subtracted H$\alpha$ waveband from the data from [@Conselice01] (using the fluxes given in several regions to calibrate the count to flux scale). The H$\alpha$ filter used had a central wavelength of 6690 A and a FWHM of 77 A, and the fluxes have been corrected for Galactic extinction, and includes N <span style="font-variant:small-caps;">ii</span> emission. The plot shows that the X-ray flux of the filaments, assuming a cooling flow model, is very close to the H$\alpha$+N \[<span style="font-variant:small-caps;">ii</span>\] flux for most of the filaments. The flux from the multitemperature model (Fig. \[fig:filamentmulti\]) is somewhat similar to that from the cooling flow model. If the 2, 4 and 8 keV components are ignored, the flux is smaller by a factor of $\sim 4$ than the cooling flow flux, but including the 2 keV flux boosts this to well above this value.
Filament geometry
-----------------
If the gas in a filament is in pressure equilibrium with its surroundings, the pressure is known and thus is the temperature of the gas in the filament, so we can estimate the volume of the emitting region.
We fitted a simple absorbed two-temperature <span style="font-variant:small-caps;">mekal</span> spectral model to the $1\times27.3$ arcsec bin at 38 arcsec offset where the filaments are strongest (see Fig. \[fig:filamentmulti\]). The best fitting temperatures from the spectral fit are $0.69 \pm 0.05$ and $3.36 \pm 0.09$ keV, corresponding to the temperature of the filament and its surroundings, respectively, under the assumption that the gas has a single temperature in the filament. Taking the emission measure of the filament component and an electron pressure of $0.128 \keV
\cm^{-3}$ (from the deprojected values in @SandersPer04, Fig. 19), then the volume of the emitting region is $1.0 \times
10^{63} \cm^3$. The extracted region on the sky is $0.37 \times 10.2
\kpc$. If we assume the filament is a cuboid with the facing side of the dimensions given, we calculate a depth of 9 pc. If the filament is a single long and thin tube with length 10.2 kpc, it would have dimensions of $\sim 60 \pc$ across.
The lack of depth shows the filament is unlikely to be a sheet viewed from the side, and the disparity with the other dimensions suggests that the filament is in fact made up of small unresolved knots of X-ray emission. If there were $10^3$ such blobs, they would have dimensions of $\sim 30$ pc.
Flux emitted from the filament
------------------------------
Taking the volume above for the filament in the 38 arcsec bin and assuming pressure equilibrium implies there are $\sim 1.9 \times
10^{62}$ electrons in the cooler filamentary component. The difference in temperature of the cooling X-ray emitting gas from the surrounding gas is $\sim 2.7 \keV$. This implies there would be $\sim 2.5 \times
10^{54} \erg$ released if the X-ray emitting filament was to cool out of the surrounding intracluster medium (assuming twice the number of particles as electrons and $3/2 \: kT$ energy per particle, which does not include any work done by the surroundings on the cooling filament).
The luminosity of the bin in the H$\alpha$ waveband is $6 \times
10^{40} \ergps$. This can be multiplied by a factor of 20 to account for other line emission, for example Ly$\alpha$. Therefore if the line emission is powered by cooling out of the intracluster medium, the timescale for this process is $\sim 7 \times 10^4$ yr. The dynamical timescale for a 30 pc region, assuming a sound speed of $300 \kmps$, would be $\sim 10^5 \yr$, so it is approximately possible for the cooling medium to be replenished.
Conduction of heat from the surrounding intracluster medium may also be able to power the filaments. For a 2 keV plasma, the [@Spitzer62] conductivity is $\sim 9 \times 10^{11} \ergps
\cm^{-1} \K^{-1}$. If we assume a geometry of a 30 pc wide tube which is 10.2 kpc long and if the heat is being conducted between 3.4 and 0.7 keV over 30 pc, this corresponds to a heat flux of $\sim 5 \times
10^{42} \ergps$. Such a flux is sufficient to fuel the line emission. It however depends greatly on the assumed conductivity (which depends on temperature to the 5/2 power) and geometry [@BohringerFabian89; @NipotiBinney04]. To provide the energy for the line emission, the heat must be able to travel to presumably smaller and cooler regions than that of the the 0.7 keV gas, which conductivity makes it increasingly hard to do. Note that any conduction model requires that the surrounding soft X-ray emitting gas and the filament have a low relative velocity. If conduction is too efficient at transporting heat the filament will evaporate, leading to a certain critical minimum size for growth [@BohringerFabian89], which depends on how much conduction is suppressed and the temperature and density of the surrounding ICM.
Hard X-ray emission {#sect:hxray}
===================
As discussed in the introduction, [@SandersPer04] found evidence for a distributed hard component surrounding the core of the cluster by fitting a multitemperature model with a 16 keV component. The total 2-10 keV luminosity of this hard component is $\sim 5 \times 10^{43}
\ergps$. Later [@SandersNonTherm05] used a thermal plus powerlaw model to fit the data giving a similar flux. This hard component has been confirmed with *XMM-Newton* data (Molendi, private communication). Assuming this emission is the result of inverse Compton scattering of IR and CMB photons by the population of electrons which also emit the observed radio, the magnetic field over the core of the cluster was mapped.
Here we examine the hard flux from the cluster using this very deep set of observations, and investigate the effect of different spectral models. We binned the data using the contour binning algorithm with a signal to noise of 500 ($\sim 2.5 \times 10^{5}$ counts in each bin).
{width="\textwidth"}
Our first model is a multitemperature model made of different temperature components plus a powerlaw to account for the hard emission. This is a more complex model than that used by [@SandersNonTherm05], as we wished to account for the known cool gas in the cluster which may affect the powerlaw signal if not modelled correctly. Hotter gas projected from larger radii in the cluster can also give a false signal. The data were fitted using a model made up of <span style="font-variant:small-caps;">apec</span> thermal components at fixed temperatures of 0.5, 1, 2, 3, 4 and 8 keV, plus a $\Gamma=2$ powerlaw, all absorbed by a <span style="font-variant:small-caps;">phabs</span> photoelectric absorber. The normalisations of the components were allowed to vary, the metallicities tied to the same free value, and the absorption was free. We used a $\Gamma=2$ powerlaw here as this is close to the best fitting photon index of the radio emission [@SijbringThesis93], and the best fitting X-ray spectral index in the core [@SandersNonTherm05]. Fig. \[fig:plawnorms\] shows the emission measure per unit area for each of the thermal components, the powerlaw normalisation per unit area, and the broad band *Chandra* image of the same region.
![Flux per unit area on the sky for multicomponent models plus powerlaws or plus a hot 16 keV component. Also plotted are the background-subtracted results from an earlier analysis [@SandersNonTherm05].[]{data-label="fig:plawprofile"}](fig20.eps){width="\columnwidth"}
The thermal gas maps show similar distributions to the earlier analyses with smaller uncertainties [@SandersPer04; @FabianPer06]. We plot the radial profile of the 2-10 keV flux of the powerlaw component in Fig. \[fig:plawprofile\]. Also shown is the radial profile for a $\Gamma=1.5$ powerlaw instead of the $\Gamma=2$ powerlaw, or a 16 keV thermal component, and the older results from [@SandersNonTherm05] (after subtracting the estimated ‘background’ from hot projected thermal gas). The total emitted flux from each of the models is fairly consistent, except at large radii where the signal is low. Hot thermal gas, as expected, gives similar results to a $\Gamma=1.5$ powerlaw. The best fitting powerlaw index looks similar to the results in [@SandersNonTherm05], with a transition of $\Gamma \sim 2.2$ powerlaw in the centre to $\Gamma \sim 1.4$ in the outskirts.
![Fluxes for the powerlaw or hot thermal components for various models. The units are $\mathrm{log}_{10} \, \ergpcmsqps$, in the 2–10 keV band. The panels show, from left to right, top to bottom, (1) fitting a thermal <span style="font-variant:small-caps;">mekal</span> plus a $\Gamma=2$ powerlaw from 0.6 to 8 keV, (2) only fitting it between 2.3 and 8 keV, (3) allowing the $\Gamma$ to vary from 1.4 to 2.4, (4) multicomponent thermal <span style="font-variant:small-caps;">apec</span> model plus a $\Gamma=2$ powerlaw, (5) using $\Gamma=1.8$, (6) using $\Gamma=1.5$, (7) allowing $\Gamma$ to vary between 1.4 and 2.4, and (8) multicomponent thermal <span style="font-variant:small-caps;">apec</span> model plus 16 keV component.[]{data-label="fig:hardmaps"}](fig21.jpg.eps){width="\columnwidth"}
We map the distribution of hard flux per unit area for a variety of different models in Fig. \[fig:hardmaps\]. We show the variation of flux (top left) just using a single thermal model plus powerlaw (as in @SandersNonTherm05), (top right) fitting that model just in the high energy band, and (second row left) allowing the powerlaw index to vary. We also show the results (second row right, and third row) from multitemperature plus powerlaw models with $\Gamma = 2, 1.8$ and 1.5, and (bottom left) fitting for the index. Finally we show the result (bottom right) using a multitemperature model including a 16 keV hard component.
There are problems with steep powerlaw components as they predict significant extra flux at low X-ray energies. Such flux is not observed, and therefore the best fitting photoelectric absorption increases in the central regions where the powerlaw is strong. It is possible that there is excess absorption in the central regions, but the flux of the powerlaw is very closely correlated with the absorption, so some physical connection between the two would be required. Presumably the non-thermal emission process would need to be dependent on absorbing material, which appears unlikely.
![$N_\mathrm{H}$ column density maps in units of $10^{22}
\psqcm$ generated by fitting different spectral models to regions containing a signal to noise ratio of 500. Also shown is an X-ray image of the same area. ‘Single’ shows the results of fitting a single <span style="font-variant:small-caps;">phabs</span> absorbed <span style="font-variant:small-caps;">mekal</span>. ‘Gamma=2, 1.8 and 1.5’ shows the results fitting a multitemperature model plus a powerlaw of the photon index given. ‘16 keV’ shows the results using a multitemperature model including a 16 keV hot thermal component.[]{data-label="fig:nhmaps"}](fig22.jpg.eps){width="\columnwidth"}
Fig. \[fig:nhmaps\] shows the absorption distribution for different models. The top-left panel shows the distribution from fitting a single <span style="font-variant:small-caps;">mekal</span> plasma to the projected spectra. There is obvious variation across the image. Some of this variation may be due to the buildup of contaminant on the ACIS detector, but the current calibration should account for this in the creation of ancillary response matrices. Probably most of the variation is because the cluster lies close to the Galactic plane ($b \sim
-13^\circ$). If the image is aligned to Galactic coordinates, the variation is mostly in Galactic latitude.
Fitting using a multitemperature model plus a $\Gamma=1.5$ powerlaw or 16 keV thermal component produces absorption maps very similar to the single temperature map. These models require no additional absorption. The $\Gamma=1.8$ requires moderate additional absorption, and $\Gamma=2$ produces absorption clearly correlated to the powerlaw flux (Fig. \[fig:hardmaps\] centre-top row, right column).
To demonstrate the effect of the hard component on the spectral fit, we show in Fig. \[fig:hardspec\] the contribution of the hard component to the best fitting model to a spectrum from a region around 1.8 arcmin north of the nucleus. For a 16 keV component the effect is around 10 per cent at low energies, increasing to fifty per cent at high energies: the hard component is about one third or more flux above 4 keV.
Origin of the hard emission
---------------------------
If we assume that the hard component is real and not an instrumental or modelling artifact, there are two sets of possible emission mechanisms, thermal or non-thermal. If it is non-thermal emission it is likely to be inverse Compton emission from CMB or IR photons being scattered by relativistic electrons in the ICM [@SandersNonTherm05]. Another possible origin is hot thermal gas. The probable origin of such material in the cluster is from a shock, and the obvious candidate is the HVS.
### Thermal origin: merger of HVS {#sect:hvsthermal}
From Section \[sect:hvs\], the HVS lies at minimum distance from the cluster core of 110 kpc, where the electron density of the gas is $\sim 5 \times 10^{-3} \pcmcu$ [@SandersPer04]. The HVS is moving at $3000\kmps$ relative to the main NGC1275 system along the line of sight to the observer [@Minkowski57]. Assuming a 5 keV plasma for the cluster, and that the galaxy is travelling along the line of sight, the Mach number of the HVS is around 2.3.
The collision should produce a shocked Mach cone around the HVS. If the HVS lies at its minimum distance from the cluster, using the Rankine-Hugoniot jump conditions, the density of the post-shocked material should be a factor of 2.1 times greater than its surroundings ($n_e \sim 10^{-2} \pcmcu$) and its temperature around 15 keV.
If the origin of the hard component is the shocked material in the cone, and it lies at the minimum distance from the cluster, we can estimate the depth of the cone from the emission measure of the 16 keV component (close to the expected 15 keV temperature) using the density above. The depth we derive from the peak of the emission is between 200-300 kpc. If this is the correct interpretation, the layer of shocked material is extremely large. This number can be reduced significantly if the preshocked material has a higher density, as the depth is inversely proportional to the square of the density. At the previous minimum distance of around $60\kpc$ from the HVS from the core of the cluster [@GillmonPer04], the minimum depth is only a few kpc, as the density was increased by a factor of 10. If the HVS lies further away than our minimum distance from the cluster core, it becomes much harder for the thermal interpretation to be a plausible explanation. Whether a thermal origin for the hard component is plausible depends on whether our lower limit of the distance from the cluster to the HVS is overestimated (see Section \[sect:hvs\]).
### Non-thermal origin: inverse Compton processes
If the origin of the flux is from inverse Compton emission, there is a large population of relativistic electrons scattering CMB or IR photons. The bright radio emission from the mini radiohalo in the cluster core indicates that there are relativistic electrons, but we cannot directly observe the required $\gamma \sim 1000$ electrons by their synchrotron emission. The powerlaw index of the inverse Compton emission will be the same as the radio emission, if a single population of electrons produces both. We do not know whether there is a single population. There could be a break in the powerlaw index, or a cut-off in the population.
Under the assumption of a single population, we previously estimated the magnetic field in the core of the cluster to be a 0.3-1$\mu$G [@SandersNonTherm05].
There are some potential issues with an inverse Compton explanation. If the source electron population creates a $\Gamma \sim 2$ powerlaw, we require significant amounts of additional absorption in the core of the cluster (Fig. \[fig:nhmaps\]), as the powerlaw is significant at low X-ray energies, unless it breaks in the X-ray band. Such absorption is not seen on the X-ray spectrum of the nucleus using *XMM* [@ChurazovPer03]. The excess absorption is not required with a flatter powerlaw index ($\Gamma \sim 1.5$). However a flat $\Gamma=1.5$ powerlaw will emit significant flux in the hard ($>20$ keV) band.
Hard X-rays were observed from Perseus using *HEAO 1* [@PriminiHeaoPer81] in the 20-50 keV band with a photon index of around 1.9. The flux translates to a total luminosity in the 2-10 keV band of around $1.6 \times 10^{44} \ergps$ above the thermal emission. This component was found to be variable over a four year timescale. [@NevalainenPds04] observed a flux around four times lower than this value using the high energy PDS detector on *BeppoSAX*. The variable component must be associated with the central nucleus rather than inverse Compton emission. The separation of the nuclear spectrum from any hard component is difficult, particularly if the nucleus varies on short timescales. [@NevalainenPds04] concluded that the central nucleus can account for all of the nonthermal emission observed from Perseus using *BeppoSAX*. More sensitive measurements of the hard flux from, for example, *Suzaku* are vital to resolve this issue. Observations with higher spatial resolution close in time would help remove uncertainty about the nuclear component.
The cooling time for electrons producing 10 keV electrons from inverse Compton scattering of CMB photons is $\sim 10^9$ yr. It is therefore likely that the spectrum could be broken at these energies. Such breaks could reduce the need for increased absorption for steeper powerlaw models here, or reduce the hard X-ray flux for those with flatter spectra.
![Inferred average nonthermal particle pressure calculated from the $\Gamma=1.5$ powerlaw plus multitemperature results in Fig. \[fig:hardmaps\], assuming inverse Compton emission. Also plotted is the average thermal gas electron pressure from [@SandersPer04].[]{data-label="fig:nonthermpress"}](fig24.eps){width="\columnwidth"}
If the emission is the result of inverse Compton emission, the pressure of the relativistic electrons, $P$, is related to the emissivity of the inverse Compton emission, $\mathcal{E}$, by (e.g. @Erlund06) $$P = \frac{1}{4} \frac{ \mathcal{E} \, m_\mathrm{e} \, c } { U \, \gamma
\, \sigma_\mathrm{T} },$$ where $U$ is the energy density of the photon field being scattered, $\gamma$ is the Lorentz factor of the electron scattering the photon to the observed waveband, $m_\mathrm{e}$ is the rest mass of the electron, $\sigma_\mathrm{T}$ is the Thomson cross-section and $c$ is the speed of light in a vacuum. If we assume that the depth of an emitting region in the hard X-ray maps is its radius, and that the X-ray emission is the result of scattering CMB and IR photons [@SandersNonTherm05] we can estimate the nonthermal pressure as a function of radius. We plot this in Fig. \[fig:nonthermpress\], showing that the nonthermal pressure is comparable to the thermal pressure near the centre of the cluster.
Thermal content of the radio bubbles {#sect:bubbles}
====================================
Earlier *Chandra* data have been used to limit the amount of thermal material material within the radio bubbles [@SchmidtPer02]. This analysis is made more difficult because the geometry of the core of the cluster is complex, and techniques accounting for projection appear not to work generally when examining the bubbles [@SandersPer04]. Here we place stringent limits on the volume filling factor of thermal gas using this 900-ks combined dataset using a comparative technique which depends far less on geometry.
We fit the projected spectrum from the inside of the bubble with a model made up of multiple temperature components at relatively low temperatures (fixed to 0.5, 1, 2, 3, 4 and 6 keV with normalisations varied and metallicities tied together) to account for the projected gas, plus a component fixed to a higher temperature to test for the existence of hot thermal gas within the bubble. We also do the fit again with an additional $\Gamma=2$ powerlaw to account for any possible non-thermal emission. We compare the normalisation per unit area of the component accounting for hot thermal gas in the bubble against that from a fit to a neighbouring region at the same radius in the cluster. The temperature of the gas in the bubble is stepped over a range of temperatures in the bubble and comparison regions.
The normalisation per unit area in the bubble and comparison region can be converted into an upper limit on the difference in normalisation per unit area between the two. We use any positive difference between the bubble region normalisation and background, plus twice the uncertainty on the difference (Using the positive uncertainty on the background and the negative uncertainty on the foreground. This is slightly pessimistic compared to symmetrising the errors), to make a 2$\sigma$ upper limit.
Assuming a volume for the bubble, an upper limit on the density of gas at that temperature can be calculated, assuming the gas is volume filling. If the gas is at pressure equilibrium with its surroundings and the pressure is known, then a limit of the volume filling fraction of gas at that temperature can instead be calculated.
![Regions used for examining the spectra inside and outside of the bubbles.[]{data-label="fig:bubbleregions"}](fig25.jpg.eps){width="\columnwidth"}
We examine the inner SW bubble and ghost NW bubbles using the regions shown in Fig. \[fig:bubbleregions\]. Also indicated are the regions used for background. The inner NE bubble is obscured by the High Velocity System, so we do not consider that. The ghost S bubble has somewhat uncertain geometry. We try to place the regions away from any contamination by low temperature gas (though we have tried alternative regions with little effect), and the background regions at a similar radius to the bubbles. We assume the bubble regions are cylindrical in shape, with depths of 9.4 kpc and 13.4 kpc for the inner SW and ghost NW bubbles, respectively. We take the deprojected electron pressure of the surrounding thermal gas from the mean value of the sectors surrounding the bubbles in figure 19 of [@SandersPer04]. This leads to values of 0.195 and $0.111 \keV \pcmcu$ for the inner SW and ghost NW bubbles, respectively.
![Hot thermal component normalisation per unit area (defined by the BACKSCAL header keyword) as a function of temperature inside the bubbles compared to outside. The results of including a $\Gamma=2$ powerlaw component are also shown.[]{data-label="fig:bubblenorms"}](fig26.eps){width="0.9\columnwidth"}
![2$\sigma$ upper limits to the volume filling fraction of the bubbles. Models including a $\Gamma=2$ powerlaw component are indicated.[]{data-label="fig:bubblevff"}](fig27.eps){width="0.9\columnwidth"}
For the two bubbles, including and not including the $\Gamma=2$ powerlaw component, we show the normalisation (emission measure) per unit area for the foreground and background regions in Fig. \[fig:bubblenorms\]. We convert these values to upper limits on the volume-filling fraction in Fig. \[fig:bubblevff\]. We find that if there is volume-filling thermal gas within the bubbles, it must have a temperature of less than $\sim 100 \keV$ for either bubble.
Discussion
==========
We have investigated several issues from the Chandra data of the Perseus cluster and now attempt to tie the interpretation of the various phenomena more closely together. In particular we try to understand the interplay between heating and cooling in the cluster core, and how energy is transported and distributed through the ICM.
Sound waves
-----------
Observations of Perseus and many other similar clusters show that jets feed relativistic plasma from around the massive central black hole into twin radio lobes. The power associated with this process is large and comparable to that required to balance radiative cooling within the regions of 50–100 kpc where the radiative cooling times is 3–5 Gyr [@Rafferty06; @DunnFabian06]. In the case of the Perseus cluster the jets are producing on average around $5\times
10^{44}-10^{45}\ergps$ in total, as estimated from the $P\mathrm{d}V$ work done over the 5-10 Myr age of the bubbles [@FabianCelottiPer02; @DunnFabian04]. These estimates depend on the filling factor of the bubbles by the relativistic plasma, which from the results given in Section \[sect:bubbles\] could well be unity. Therefore at least about 20–40 per cent of the bubble power goes into sound waves.
We expect the jets to provide power more or less continuously over the lifetime of the cluster core (several Gyr). This is supported by the high incidence of bubbles found in cluster cores that require heating [@DunnFabian06] and by the train of ghost bubbles seen in our Perseus cluster data [@FabianPer06]. Note however, that there is probably much fluctuation in the jet power over short timescales (the radio source has weakened in strength over the past 40 yr), but variations over a cooling time of $10^8\yr$ are probably less than an order of magnitude.
We now address the question of how the energy in the bubbling process, occurring on a timescale of 10 Myr, is fed into the bulk of the core and dissipated as heat which balances the cooling. The smooth cooling time profiles seen in cluster cores, and the peaked metal distributions argue for a relatively gentle, continuous (on timescales of $10^7-10^8\yr$ or more) and distributed heat source. The inflation of the bubbles does $P\mathrm{d}V$ work on the surroundings and thus creates pressure waves – sound waves – which carry the energy outward in a roughly isotropic manner. We discovered ripples in surface brightness in the first 200 ks of the Perseus cluster data [@FabianPer03] which we interpreted sound waves and have extended the analysis of them here.
Fig. \[fig:wavepowcuml\] shows that there is considerable power in these waves, around $2-3\times 10^{44}\ergps$ at radii of 30–70 kpc. It is similar to the level of heat required to offset radiative cooling within that region. The wave power drops with radius between 30 and 100 kpc indicating that the energy is being dissipated and confirming the idea that viscous dissipation of the sound waves is the distributed heat source. The dissipation length is comparable to that estimated on the basis of Spitzer-Braginski viscosity [@FabianPer03], although the process in the likely tangled magnetic, and cosmic-ray infused, plasma may be more complex and involve some form of bulk viscosity.
There is however a deficit of power shown by this analysis within 30 kpc. Although it could just be variability of the central power source (the jets), our results on the nonthermal component associated with the radio minihalo indicates that there is a significant nonthermal pressure from the cosmic rays there, comparable to the thermal pressure. This would quadruple (if the nonthermal wave pressure is the same as the thermal pressure in Equation \[eqn:wavepower\]) the predicted wave power in this region, bringing it into agreement with the region between 30 and 60 kpc and meaning that dissipation of sound waves can be the dominant heat source balancing radiative cooling.
The increase in power seen in Fig. \[fig:wavepowinst\] beyond 100 kpc can be explained as associated with the power $\sim 10^8\yr$ ago being 2–3 times larger than present. This may fit in with the presence of the long Northern optical filament [@Conselice01], which if drawn out from the centre by a bubble [@HatchPer06] must have been an exceptional bubble, possibly seen at 170 kpc north of the nucleus (Section \[sect:wavepower\]). Also the ghost bubbles seen to the South in the X-ray pressure maps [@FabianPer06] may be larger and so require more power than the present inner bubbles. If correct, the central source thus does vary by a factor of few on timescales of $10^8\yr$. Deeper X-ray observations at larger radii than the current ones are needed to test this interesting possibility further.
We conclude the discussion of sound waves by noting that typical sound waves will be difficult to see in current data on other clusters. The X-ray surface brightness of the central 50 kpc of the Perseus cluster is more than twice that of another cluster. Large ripples, interpreted as weak shocks have been seen in the Virgo cluster [@FormanM8706] and possible outer ripples may occur in A2199 and and 2A0335+096 [@SandersShock06]. These may just be the peaks in the distribution of sound waves in those clusters from the more exceptional power episodes of their central engines. Like tree rings or ice cores for the study of geological history, ripples in cluster cores offer the potential to track the past history of an AGN for more than $10^8\yr$. Observations of a yet wider region are required to see where the ripples eventually die out.
The nonthermal component
------------------------
We have reported on a hard X-ray component coincident with the radio minihalo (section \[sect:hxray\]). It is difficult to support a thermal origin for this emission in terms of a $\sim 16\keV$ gas associated with the HVC. A simple energy argument against the thermal hypothesis is to note that the crossing time of the 200 kpc diameter inner core of the cluster at $3000\kmps$ takes nearly $7\times
10^7\yr$. The radiative cooling time of 16 keV gas with a plausible density of $10^{-2}\pcmcu$ is $10^{10}\yr$, so the shocked gas would radiate only 0.7 per cent of its energy. With the luminosity of this component at $5\times 10^{43}\ergps$ we obtain a total injected energy of $2\times 10^{61}\erg$. This is the total kinetic energy of $2\times
10^{11}\Msun$ moving at $3000\kmps$. So an implausible large mass needs to be stripped from an apparently small galaxy in order to explain the hard component as due to shocking by the HVC.
The hard component is therefore most readily interpreted as inverse Compton emission from the minihalo. As discussed by [@SandersNonTherm05] this means that it must be well out of equipartition for the electrons and magnetic field with the electrons dominating the nonthermal pressure (as also deduced for the radio lobes, @FabianCelottiPer02). It is possible that the radio bubbles leak a small[^3] fraction of their cosmic-ray electron content (and presumably protons or positrons for charge neutrality) into their surroundings. These accumulate, losing their energy principally through inverse Compton losses on the Cosmic Microwave Radiation. A half-power break in the power-law spectrum is then expected around 10 keV in the hard X-ray flux if this process has, as expected, been continuing for more than a Gyr. This matches what is inferred of the observed spectrum.
The presence of the nonthermal component increases the heating within the inner 30 kpc if $\delta P$ is raised proportionately, and also means that some direct collisional heating of gas is possible. [@RuszkowskiCosmicRay07] have since investigated this idea in more detail.
The distribution of metals
--------------------------
The enhanced metallicity in the core of the Perseus cluster shows a spatial distribution which extends to the S along the axis of old bubbles, as expected if the bubbles push and drag the gas around [@Roediger07]. What is particularly interesting here is evidence that the distribution is clumpy and also that the central drop is a real effect.
The clumpiness and especially the sharp edges of at least one clump (Fig. \[fig:blob\_profile\]) likely require a magnetic field configured to prevent dispersal. The central metallicity drop is not easily explained as just due to some outer gas falling in to replace inner gas dragged out, since it has a significantly lower entropy. It could be accounted for if the metals are highly inhomogeneous on a small scale, with the higher metallicity gas which has the shorter cooling time cooling out [@MorrisFabian03].
An interesting possibility raised by the high nonthermal pressure near the centre is that the gas is made buoyant by the cosmic rays [@Chandran04; @Chandran05]. In this picture the simple entropy inferred from just the gas temperature and density are insufficient to determine its behaviour. The gas cosmic-ray mixture becomes convectively unstable where the cosmic-ray pressure begins to drop steeply outward, leading to the gas overturning. The pressure drop is inferred to occur at about 30 kpc (Fig. \[fig:nonthermpress\]) and is about the radius where the metallicity peaks (Fig. \[fig:Zmap\]). Such a turnover of gas may happen sporadically when the cosmic-ray density has built up for some time.
The optical filaments
---------------------
Finally we briefly discuss the H$\alpha$ filaments. These radiate most of their energy at Ly$\alpha$ (see @FabianNulsen84 for an image) and are mostly composed of molecular gas at a few thousand down to 50 K [@Hatch05; @Johnstone07; @Salome06]. We have shown that they are surrounded and mixed with soft X-ray emitting plasma at 0.5–1 keV temperature. In section \[sect:proffilament\] we consider one filament in detail. The H$\alpha$ luminosity of its peak is about one per cent of the total measured by [@Heckman89], so scaling the X-ray inferred mass cooling rate there of $0.06\Msunpyr$ we obtain a total mass cooling rate into filaments of $5\Msunpyr$.
This value assumes however that the X-ray emitting gas loses its energy solely by radiating X-rays. If in addition it loses energy by conduction, mixing or other means, with the cold gas in the filament and thereby powers the Ly$\alpha$ emission then we need to scale the above rate by a factor of 20 (the expected Ly$\alpha$/H$\alpha$ ratio for recombination) to obtain a total mass cooling rate of about $100\Msunpyr$. This value agrees of course with that obtained by just taking the total H$\alpha$ luminosity and assuming it is obtained from the thermal energy of the hot gas. It is about one third of the total inferred mass cooling rate obtained if there is no heating. We note that radiation at other wavelengths such as O <span style="font-variant:small-caps;">vi</span> emission [@Bregman06] can increase this fraction.
This implies that a high non-radiative mass cooling rate is possible in the Perseus cluster. By extension, it suggest that this happens in most cool-core clusters (which generally also have optical filaments; see @CrawfordBCS99). It can account for *part* of the lack of cool X-ray emitting gas in such cluster cores [@FabianCFlow02].
A cooling rate of $100\Msunpyr$ for 5 Gyr gives a total of $5\times
10^{11}\Msun$ of cold gas. This is about 10 times more than is inferred from CO measurements [@Salome06]. Star formation is another possible sink. NGC1275 has long been known to have an A-star spectrum and excess blue light. UV imaging of NGC1275 by [@Smith92] shows a lack of stars above $5\Msun$, which means that any continuous star formation (which would have been only $20\Msunpyr$) must have ended about 50 Myr ago. Burst models of star formation at rates up to many $100\Msunpyr$ 100 Myr or so ago are consistent with the UV data [@Smith92]. A picture in which gas accumulates through filaments and is then converted into stars sporadically on a 100 Myr timescale appears possible.
If however much of the power in the filaments is due to sources other than the hot gas, cosmic rays or kinetic motions for example, then the above star formation estimate is an upper limit.
Summary
-------
We have quantified the properties of the X-ray surface brightness ripples found in the core of the Perseus cluster and, assuming that they are due to sound waves, have determined the power propagated as sound waves. The power found in this way is sufficient to balance radiative cooling within the inner 70 kpc, provided that it is dissipated as heat over this lengthscale. This provides the quasi-isotropic, relatively gentle, heating mechanism required to prevent a full cooling flow developing. Ultimately the power is derived from the jets emitted by the central black hole.
A hard X-ray component is confirmed and argued to be plausibly due to inverse Compton scattering by cosmic-ray electrons in the radio minihalo. The electrons may have leaked out of the radio lobes of 3C84 and now have a pressure comparable to the thermal pressure of the hot gas in the innermost 30 kpc. The lobes themselves appear to be devoid of any thermal gas unless its temperature is very high (50–100 keV). The cosmic-ray electrons are important in enhancing the heating and possibly also in changing the convective stability of the central 30 kpc.
The X-ray data provide insight on the history of the past 100 Myr of activity of the nucleus of NGC1275. There are hints from the large ripples beyond 100 kpc, and from the the large Northern filament and the presence of many A stars, of a higher level of activity before that, a few 100 Myr ago. This can be tested by deep, high spatial resolution, observations of a wider region than covered out so far.
We infer that a close balance between heating and cooling is established in the core of the Perseus cluster over the past few 100 Myr. The average heating rate is, and has been, close to the radiative cooling rate, although there can be variations by a factor of a few on longer timescales. The primary energy source is the central black hole and jets; the energy is distributed by the sound waves generated by the inflation of the lobes. We suspect that this process is common to most cool core clusters and groups and is the mechanism by which heating of the cool core occurs. It will however be difficult to verify observationally in those other objects since the X-ray surface brightness is so much lower. Only the extreme peaks in the distribution of activity will generally be detectable.
Acknowledgements {#acknowledgements .unnumbered}
================
ACF acknowledges The Royal Society for support. We thank the *Chandra* team for enabling the superb images of the Perseus cluster to be obtained.
A Direct Spectral Deprojection Method {#appendix:deproj}
=====================================
Problems with existing spectral deprojection methods
----------------------------------------------------
Results from fitting cluster data using the <span style="font-variant:small-caps;">projct</span> model in <span style="font-variant:small-caps;">xspec</span> can be misleading. <span style="font-variant:small-caps;">projct</span> is a model to fit spectra from several annuli simultaneously, to account for projection. There are one or more components per deprojected annulus, each with parameters (e.g. temperature, metallicity). The projected sum of the components along line of sights (with appropriate geometric factors) are fitted against each of the spectra simultaneously. Often the resulting deprojected profiles (e.g. temperature) oscillate between values separated by several times the uncertainties on the values. This oscillation can disappear if different sized annuli are used. Sometimes halving the annulus width can halve the oscillation period, indicating they are not physical changes on the sky.
The oscillation can be alleviated by fitting the shells sequentially from the outside, freezing the parameters of components in outer shells before fitting spectra from shells inside them (see e.g. @SandersPer04). This helps to solve the issue where poorly modelled spectra near the centre can affect the results in outer annuli (The standard way to fit the data is simultaneous. All the spectra are used to calculate each point, even though interior shells and not projected in front of outer shells.) The difficulty with this method is that uncertainties calculated on parameters to the model are underestimated. They do not include the uncertainties on outer shells.
The outside-first fitting procedure does not fix every oscillating profile. Numerical experiments, when clusters are simulated and fit with <span style="font-variant:small-caps;">projct</span> (R. Johnstone, private communication), show that assuming the incorrect geometry when trying to account for projection does not produce oscillating profiles. Something which does create oscillating profiles are shells which contain more than one spectral component (e.g. several temperatures). It appears that <span style="font-variant:small-caps;">projct</span> tends to account for one of the components in one of the annulus fit results, and another in a different shell. By assuming that a spectral model is a good fit to the data, a very misleading result is produced. Other deprojection methods which assume a spectral model to do the deprojection will have similar issues.
Direct spectral deprojection
----------------------------
We describe here a method to create ‘deprojected spectra’, which appears to alleviate some of the issues found using <span style="font-variant:small-caps;">projct</span>. It is a model independent approach, assuming only spherical geometry (at present).
The routine takes spectra extracted from annuli in a sector, and their blank-sky background equivalents. From each of these foreground count rate spectra we subtract the respective blank sky background spectrum. Taking the outer spectrum, we assume that it was emitted from part of a spherical shell, and calculate the (count rate) spectrum per unit volume. This is then scaled by the volume projected onto the next innermost shell, and subtracted from the (count rate) spectrum from that annulus. After subtraction we calculate a spectrum per unit volume for the next innermost shell. We move inwards shell-by-shell, subtracting from each the calculated contributions from outer shells. This yields a set of deprojected spectra which are then directly fitted by spectral models.
To calculate the uncertainties in the count rates in the spectral channels in each spectrum we used a Monte Carlo technique. Firstly each of the input foreground and background spectra are binned using the same spectral binning, using a large number of counts per spectra channel (we used 100 in this work) so that Gaussian errors can be assumed in each spectral bin. We repeat the deprojection process 5000 times, creating new input foreground and background spectra by simulating spectra drawn from Gaussian distributions based on the initial spectra and their uncertainties. The output spectra are the median output spectra from this process, and the 15.85 and 84.15 percentile spectra were used to calculate the 1$\sigma$ errors on the count rates in each channel.
This technique assumes that the response of the detector does not change significantly over the detector. This is the case for the ACIS-S3 detector on *Chandra* used here. It also assumes that the effective area does not change significantly, as we do not account for the variation of the ancillary response. This could be incorporated into this method, but we have not implemented this yet.
[^1]: E-mail: jss@ast.cam.ac.uk
[^2]: <span style="font-variant:small-caps;">xspec</span> <span style="font-variant:small-caps;">mekal</span> and <span style="font-variant:small-caps;">apec</span> normalisations are defined as $\{10^{-14} \int n_\mathrm{e} n_\mathrm{H} \mathrm{d}V\} / \{4 \pi
[D_A (1+z)]^2\}$, where the source is at redshift $z$ and angular diameter distance $D_A$, and the electron number density $n_\mathrm{e}$ and Hydrogen number density $n_\mathrm{H}$ are integrated over volume $V$.
[^3]: Studies of bubbles in nearby clusters indicate they are not magnetic pressure dominated [@DunnFabian04; @DunnFabian06], so this limits the number of particles which can escape.
|
---
abstract: 'This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.'
address:
- 'Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002, USA'
- 'Department of Mathematics and Informatics, Vilnius University, Naugarduko 14, 2600 Vilnius, Lithuania'
- 'Department of Mathematics, Harvard University, Cambridge, MA 02138, USA'
author:
- 'David Cox, Rimvydas Krasauskas and Mircea Mustaţǎ[^1]'
title: Universal Rational Parametrizations and Toric Varieties
---
Introduction {#intro}
============
In geometric modeling rational curves and surfaces are widely used in the form of Bézier curves and surfaces or simple low-degree surfaces, e.g., various quadrics, torus surfaces, Dupin cyclides etc. Construction of curve arcs and patches on a given surface with the lowest possible parametrization degree is an important task. For instance this may help to solve data conversion problems which arise when translating from traditional solid modeling systems that deal with such simple surfaces to NURBS-based systems.
It follows that there is a need to understand *all* possible parametrizations of a given curve or surface. Is it somehow possible to find a “best” parametrization? In the case of toric surfaces (and, more generally, projective toric varieties of any dimension), this paper will offer one answer to this question, which we call a *universal rational parametrization*.
To illustrate what we mean by this, we give two examples of surfaces with universal rational parametrizations.
\[firstex\]
Consider a quadric surface $Q$ given by the homogeneous equation $u_0
u_3 = u_1 u_2$ in projective space $\PP^3$. Any rational parametrization of $Q$ can be represented by a collection of polynomials $H = (h_0,h_{2},h_{2},h_3)$ such that $$\label{fcond}
h_0 h_3 = h_1 h_2\quad \text{and} \quad
\gcd(h_0,h_{2},h_{2},h_3) = 1.$$ One obvious rational parametrization is given by $$\label{firstuniv}
P(x_{1}, x_{2},x_{3},x_{4}) = (x_2 x_3, x_1 x_3, x_2 x_4, x_1 x_4)$$ since $(x_2 x_3)(x_1 x_4) = (x_1 x_3)(x_2 x_4)$.
Now suppose that we have a collection of polynomials $$\label{hcond}
F = (f_1,f_{2},f_{3},f_4),\quad \gcd(f_1,f_{2}) = \gcd(f_{3},f_4) = 1.$$ Then let $H = P\circ F$, i.e., $$H = (h_0,h_{1},h_{2},h_3) = (f_2 f_3, f_1 f_3, f_2 f_4, f_1 f_4).$$ It is straightforward to show that $H$ satisfies . In other words, from the parametrization $P$ of , we get infinitely many others by composing with any $F$ satisfying .
But even more is true: Theorem \[upthm\] implies that *all* $H$’s satisfying arise in this way. In other words, such an $H$ is of the form $H = P\circ F$ for some $F$ as in . Furthermore, although $F$ is not unique, Theorem \[upthm\] describes the non-uniqueness precisely: given one $F$ with $H = P\circ F$, then all others are of the form $$(\lambda f_1,\lambda f_2,\lambda^{-1} f_3,\lambda^{-1} f_4).$$ for some nonzero scalar $\lambda$.
In the language of Theorem \[upthm\], we say that $P$ from is a *universal rational parametrization* of the quadric $Q$. The key property of the quadric $Q$ is that it comes from $\PP^{1} \times \PP^{1}$. If $x_{1},x_{2}$ are homogeneous coordinates on the first $\PP^{1}$ and $x_{3},x_{4}$ are homogeneous coordinates on the second, then $P$ induces an embedding $$\PP^{1} \times \PP^{1} \longrightarrow \PP^{3}$$ whose image is $Q$.
Here is the second example of a universal rational parametrization.
\[firststeiner\]
Consider the Steiner surface $S$ in $\PP^3$, which is defined in homogeneous coordinates by the equation $$u_1^2u_2^2+u_2^2u_3^2+u_3^2u_1^2 = u_0u_1u_2u_3.$$ Note that $S$ is not a smooth surface—its singular locus consists of the three lines $$\label{lines}
u_1 = u_2 = 0,\quad u_2 = u_3 = 0,\quad u_3 = u_1 = 0.$$ A rational parametrization of $S$ consists of polynomials $H
= (h_0,h_1,h_2,h_3)$ such that $$\label{stcond1}
h_1^2h_2^2+h_2^2h_3^2+h_3^2h_1^2 = h_0h_1h_2h_3\quad \text{and}\quad
\mathrm{gcd}(h_0,h_1,h_2,h_3) = 1.$$ One can easily show that $$\label{steinerp}
P(x_1,x_2,x_3) = (x_1^2+x_2^2+x_3^2, x_1x_2 , x_2x_3, x_3x_1)$$ is a parametrization of $S$. Furthermore, given polynomials $$\label{stcond}
F = (f_1,f_2,f_3),\quad \gcd(f_1,f_2,f_3) = 1,$$ we see that $$H = P \circ F = (f_1^2+f_2^2+f_3^2, f_1f_2 , f_2f_3, f_3f_1)$$ satisfies and hence is a rational parametrization of $S$.
In this situation, Theorem \[upthm\] tells us that *all* $H$’s satisfying are of the form $H = P \circ F$ for some $F$ satisfying , *provided* the image of $H$ does not lie in the lines . Furthermore, Theorem \[upthm\] implies that $F = (f_1,f_2,f_3)$ is unique up to $\pm 1$.
By Theorem \[upthm\], is the universal rational parametrization of the Steiner surface $S$. In this case, the key property of $S$ is that it came from $\PP^2$ via the map $\PP^2 \to S$ induced by $\eqref{steinerp}$. This map is not an embedding but is birational (i.e., is generically one-to-one). Furthermore, the three lines are where the map fails to have an inverse.
Both $\PP^1\times\PP^1$ and $\PP^2$ are examples of *smooth toric varieties*, and the coordinates $x_1,x_2,x_3,x_4$ for $\PP^1\times\PP^1$ and $x_1,x_2,x_3$ for $\PP^2$ are examples of *homogeneous coordinates* of toric varieties. Hence it makes sense that there should be a toric generalization of these examples. For instance, we will see that the gcd conditions and are dictated by the data which determines the toric variety.
The paper is organized into six sections as follows:
Section \[intro\]: Introduction
Section \[backgrd\]: Background and Related Work
Section \[projective\]: Universal Rational Parametrizations (Smooth Case)
Section \[projective\]: Universal Rational Parametrizations (Singular Case)
Section \[rational\]: Rational Maps to Smooth Toric Varieties
Section \[theory\]: Theoretical Justifications
In Section \[backgrd\] we will describe toric varieties and homogeneous coordinates along with a summary of related work. In Section \[projective\], we give a careful definition of rational parametrization and state Theorems \[upthm\], our main result about universal rational parametrizations when the toric variety involved is smooth. We also give numerous examples. Then, in Section \[singular\], we discuss Theorem \[singupthm\], which describes what happens when the toric variety is singular. However, in order to prove these results, we need to understand rational maps to smooth toric varieties. This is the subject of Section \[rational\], where the main result is Theorem \[ratmap\]. Finally, Section \[theory\] includes proofs of the results stated in Sections \[projective\], \[singular\] and \[rational\].
In this paper, we will work over the complex numbers $\CC$ so that we can apply the tools of algebraic geometry. Let $\CC^* =
\CC\setminus\{0\} = \{ z \in \CC \mid z \ne 0\}$.
Geometric modeling is mostly concerned with real varieties. In practice, many important real surfaces are real parts of complex toric surfaces with possibly non-standard real structures. The results of this paper hold over $\RR$, provided we use the standard real structure on the toric varieties involved. Our results can also be applied, with some straightforward modifications, to the case of non-standard real structures. The details about this situation and the practical issues of using universal rational parametrizations in geometric modeling will be presented elsewhere.
We would like to thank the referee for pointing out a problem in our original version of Theorem \[upthm\] and for suggesting the current form of Example \[hirzebruch\].
Background Material and Related Work {#backgrd}
====================================
The concept of universal rational parametrization was introduced at first for nonsingular quadric surfaces under the name of “generalized stereographic projection” in [@DHJ]. It was extended to more general rational surfaces in [@up] and [@M] (see also the recent paper [@Mu]).
Around the same time, homogeneous coordinates for toric varieties where defined by numerous people—see [@hc] for a complete list. Also important were maps into toric varieties, which were first explored in [@Guest] and [@Jac]. This led the first author to the description of maps to smooth toric varieties given in [@functor].
The relation between universal rational parametrizations and toric varieties was first realized when the second author defined the toric surface patches in [@rimas]. An account of this may also be found in [@zube].
Toric Varieties
---------------
In this paper, we will assume that the reader is familiar with the elementary theory of toric varieties, as explained in [@what]. A toric variety $X_\Sigma$ is determined by a fan $\Sigma$, which is a collection of cones $\sigma \subset \RR^n$ satisfying certain properties. We will assume that the union of the cones in $\Sigma$ is all of $\RR^n$. This means that $X_\Sigma$ is a compact toric variety.
Among the cones of $\Sigma$, the edges (= one-dimensional cones) play a special role. Suppose that the edges of $\Sigma$ are $\rho_1,\dots,\rho_r$. Then each $\rho_i$ corresponds to $x_i$, $n_i$ and $D_i$, where:
- The variable $x_i$ is in the homogeneous coordinate ring of $X_\Sigma$.
- The vector $n_i \in \ZZ^n$ is the first nonzero integer vector in $\rho_i$.
- The subvariety $D_i \subset X_\Sigma$ is defined by $x_i = 0$.
We think of $x_1,\dots,x_r$ as coordinates on $\CC^r$. We can use the $x_i$ to construct the toric variety $X_\Sigma$ as follows. Let $$\label{gdef}
G = \{(\mu_1,\dots,\mu_r) \in (\CC^*)^r \mid {\textstyle\prod_{i=1}^r}
\mu_i^{\langle m, n_i\rangle} = 1\ \text{for all}\ m \in \ZZ^n\},$$ where $\langle \, , \, \rangle$ is dot product on $\ZZ^n$. This is a subgroup of $(\CC^*)^r$ and hence acts on $\CC^r$ in the usual way. Also, for each cone $\sigma \in \Sigma$, let $$x^{\hat\sigma} = \prod_{n_i \notin \sigma} x_i$$ be the product of all variables corresponding to edges *not* lying in $\sigma$. Finally, let the *exceptional set* $Z \subset
\CC^r$ be defined by the equations $x^{\hat\sigma} = 0$ for all $\sigma \in \Sigma$. Then we get the quotient representation $$\label{quotient}
X_\Sigma = (\CC^r \setminus Z)/G.$$ As explained in [@what], this generalizes the quotient construction $$\PP^n = (\CC^{n+1} \setminus \{0\})/\CC^*.$$
One consequence of is that we have a natural map $\CC^r \setminus Z \to X_\Sigma$. We can think of this as a *rational* map from $\CC^r$ to $X_\Sigma$ which is not defined on the exceptional set $Z$. We will write this as $$\label{crxs}
\pi : \CC^r -\! -\!\hskip-1.5pt \to X_\Sigma,$$ where the broken arrow means that we have a rational map. The map will play an important role in what follows.
Polytopes
---------
In Sections \[projective\] and \[singular\], we will consider the projective toric variety $X_\Delta$ determined by an $n$-dimensional lattice polytope $\Delta \subset \RR^n$. The idea is that for each face of $\Delta$, we get the cone generated by the inward-pointing normals of the facets of $\Delta$ containing the face. This gives the *normal fan* $\Sigma_\Delta$ of $\Delta$, and the corresponding toric variety $X_{\Sigma_\Delta}$ is denoted $X_\Delta$.
Observe that edges of the normal fan correspond to facets of $\Delta$. Hence the homogeneous coordinates $x_1,\dots,x_r$ correspond to the facets of $\Delta$. For this reason, we call $x_1,\dots,x_r$ the *facet variables* of the polytope $\Delta$.
We can use $\Delta$ to obtain some interesting monomials in the facet variables. Represent $\Delta$ as the intersection $$\label{deltadesc}
\Delta = \bigcap_{i=1}^{r} \{m \in \RR^n \mid
\langle m, n_{i}\rangle \ge -a_{i}\}$$ of closed half-spaces. This gives the following monomials and polynomials.
\[dmsd\] For each lattice point $m \in \Delta\cap
\ZZ^n$, we define the [**$\Delta$-monomial**]{} $x^{m}$ to be $$\label{deltam}
x^{m} = \prod_{i=1}^{r} x_{i}^{\langle m, n_{i}\rangle+a_{i}}.$$ We also define $S_\Delta$ to be the linear span of the set of $\Delta$-monomials. Thus $$S_\Delta = \mathrm{Span}(x^{m} \mid m \in \Delta\cap \ZZ^n).$$
Since the $i$th facet is defined by $\langle m, n_{i}\rangle+a_{i} =
0$ and $\langle m, n_{i}\rangle+a_{i} \ge 0$ on $\Delta$ ($n_i$ points inward), we see that the exponent of $x_i$ measures the “distance” (in the lattice sense) from $m$ to the $i$th facet.
Here is an example of facet variables and $\Delta$-monomials.
\[p1p1b2first\] [Consider the polytope $\Delta$ $$\begin{matrix}
\begin{picture}(120,120)
\put(0,60){\line(1,0){120}}
\put(60,0){\line(0,1){120}}
\thicklines
\put(30,60){\line(0,1){30}}
\put(30,90){\line(1,0){60}}
\put(90,90){\line(0,-1){60}}
\put(90,30){\line(-1,0){30}}
\put(60,30){\line(-1,1){30}}
\put(18,75){$\scriptstyle{x_1}$}
\put(68,94){$\scriptstyle{x_5}$}
\put(32,40){$\scriptstyle{x_2}$}
\put(92,75){$\scriptstyle{x_4}$}
\put(68,23){$\scriptstyle{x_3}$}
\end{picture}
\end{matrix}$$ with vertices $(1,1), (-1,1), (-1,0), (0, -1), (1,-1)$. In terms of , we have $a_1 = \dots = a_5 = 1$. This gives a toric surface $X_\Delta$ with variables $x_1,\dots,x_5$ as indicated in the picture. The 8 points $m \in \Delta\cap\ZZ^2$ give the following 8 $\Delta$-monomials $x^m$: $$\begin{array}{cccccl}
x_2 x_3^2 x_4^2, & x_1 x_2^2 x_3^2 x_4, & x_1^2 x_2^3 x_3^2, \\
x_3 x_4^2 x_5, & x_1 x_2 x_3 x_4 x_5, & x_1^2 x_2^2 x_3 x_5, \\
& x_1 x_4 x_5^2, & x_1^2 x_2 x_5^2 .
\end{array}$$ We will say more about this example in Sections \[projective\] and \[rational\].]{}
We should also mention that polynomials $q \in S_\Delta$ have the following important property: given $\mu = (\mu_1,\dots,\mu_r)$ in the group $G$ defined in , one easily sees that $$\label{pmx}
q(\mu_1 x_1,\dots,\mu_r x_r) = \mu_\Delta\, q(x_1,\dots,x_r),$$ where $$\label{mud}
\mu_\Delta = \prod_{i=1}^r \mu_i^{a_i}.$$
Rational Maps to Projective Space
---------------------------------
Now pick a collection $P = (p_0,\dots,p_s)$ of $s+1$ polynomials in $S_\Delta$. This gives a rational map $$p: \CC^r -\! -\!\hskip-1.5pt \to \PP^s.$$ If $X$ denotes the Zariski closure of the image, then we write $p$ as $$p: \CC^r -\! -\!\hskip-1.5pt \to X \subset \PP^{s}.$$ We can relate $p$ to the toric variety $X_{\Delta}$ as follows.
\[pfactors\] In the above situation, the map $p$ factors $p = \Pi\circ \pi$, where $\pi : \CC^{r} -\! -\!\hskip-1.5pt \to X_{\Delta}$ is from and $$\Pi: X_\Delta -\! -\!\hskip-1.5pt \to X$$ is a rational map.
Given $a = (a_1,\dots,a_r) \in \CC^r\setminus
Z$ and $\mu = (\mu_1,\dots,\mu_r) \in G$, then we get $\mu\cdot a = (\mu_1a_1,\dots,\mu_ra_r)$. By , we have $$(p_0(\mu\cdot a),\dots,p_s(\mu\cdot a)) = \mu_\Delta
(p_0(a),\dots,p_s(a)).$$ This shows that $p$ induces $\Pi: (\CC^r\setminus Z)/G -\!
-\!\hskip-1.5pt \to \PP^s$. By , we can identify the quotient with $X_{\Sigma}$, and the proposition follows.
When $X_\Delta$ is smooth and $\Pi : X_\Delta -\! -\!\hskip-1.5pt \to
X$ is sufficiently nice, Theorem \[upthm\] asserts that $p$ is a universal rational parametrization. In Section \[projective\], we will use this theorem to explain Examples \[firstex\] and \[firststeiner\].
We will also see in Section \[singular\] that this doesn’t quite work when $X_\Delta$ is singular. In this case, Theorem \[singupthm\] will show that we get a universal rational parametrization by considering a suitable resolution of singularities.
Universal Rational Parametrizations (Smooth Case) {#projective}
=================================================
In this section, we will prove the existence of universal rational parametrizations for certain projective varieties which arise naturally from smooth toric varieties associated to polytopes.
Rational Parametrizations
-------------------------
We first give a definition of rational parametrization which is useful in geometric modeling. Given a projective variety $Y \subset
\PP^{k}$, we define its *affine cone* $C_{Y} \subset \CC^{k+1}$ to be $$C_{Y} = \pi^{-1}(Y) \cup \{0\} \subset \CC^{k+1},$$ where $\pi : \CC^{k+1} \setminus \{0\} \to \PP^{k}$ is the usual map. Using this, we can make the following definition.
\[rpdef\] Let $R = \CC[y_1,\dots,y_d]$ be the coordinate ring of $\CC^d$. A [**rational parametrization**]{} of a projective variety $Y
\subset \PP^{s}$ consists of $H = (h_{0},\dots,h_{s}) \in R^{s+1}$ such that $\mathrm{gcd}(h_{0},\dots,h_{s}) = 1$ and $H(\CC^{d})
\subset C_{Y}$.
In this paper, we use the convention that $\mathrm{gcd}(0,\dots,0) =
0$. Hence the gcd condition implies that the polynomials in a rational parametrization are not all zero. Then $H(\CC^{d}) \subset
C_{Y}$ implies that $H : \CC^d \to \CC^{s+1}$ induces a rational map $$h : \CC^d -\! -\!\hskip-1.5pt \to \PP^{s}$$ whose image lies in $Y$. It is important to note that in Definition \[rpdef\], we do not require that $h : \CC^d -\!
-\!\hskip-1.5pt \to Y$ be surjective or have dense image. Thus a rational parametrization might only parametrize a proper subvariety of $Y$. Also note that the gcd condition of Definition \[rpdef\] implies that two rational parametrizations $H$ and $H'$ give the same rational map to $\PP^{s}$ if and only if $H = cH'$ for $c \ne 0$ in $\CC$.
One Particular Parametrization
------------------------------
Let $\Delta$ be an $n$-dimensional lattice polytope in $\RR^n$. This gives the toric variety $X_\Delta$ determined by the normal fan $\Sigma_\Delta$ of $\Delta$. We will assume that $X_\Delta$ is smooth.
By Definition \[dmsd\], the facet variables $x_{1},\dots,x_{r}$ and the lattice points $\Delta\cap
\ZZ^{n}$ give rise to the vector space of polynomials $$S_{\Delta} = \mathrm{Span}(x^{m} \mid m \in \Delta \cap \ZZ^{n}),$$ where $x^{m}$ is the $\Delta$-monomial. As in Section \[backgrd\], a collection of $s+1$ polynomials $$\label{firstp}
P = (p_{0},\dots,p_{s}) \in S_{\Delta}^{s+1}$$ gives a rational map $$p : \CC^{r} -\!-\!\hskip-1.5pt \to X \subset \PP^{s}$$ where $X$ is the Zariski closure of the image. Then Proposition \[pfactors\] shows that $p = \Pi\circ\pi$, where $\pi :
\CC^{r} -\!-\!\hskip-1.5pt \to X_{\Delta}$ is from and $$\Pi : X_{\Delta} -\!-\!\hskip-1.5pt \to X \subset \PP^s$$ is a rational map. As already mentioned, the basic idea is that $P$ is a universal rational parametrization when $\Pi$ is sufficiently nice. However, we need to make some further definitions before we can state our main result.
Sufficiently Nice
-----------------
We can finally explain when $\Pi : X_\Delta -\!-\!\hskip-1.5pt \to X$ is sufficiently nice. Using the above notation, this means the following two things:
- First, $\Pi$ is *strictly defined*, which means for every $a \in \CC^r \setminus Z$, we have $p_i(a) \ne 0$ for some $0 \le i
\le s$. Using $X_\Delta = (\CC^d \setminus Z)/G$ and Proposition \[pfactors\], one can show that this condition ensures that $\Pi$ is defined everywhere. Thus we write $\Pi : X_\Delta \to
X$ when $\Pi$ is strictly defined.
- Second, $\Pi$ is *birational*, which means that $\Pi$ induces an isomorphism between dense open subsets of $X_\Delta$ and $X$.
When we discuss projections later in the section, we will give several conditions which are equivalent to being strictly defined. However, we will see in Example \[hirzebruch\] that being strictly defined is in general *stronger* than just assuming that the rational map $\Pi$ is defined everywhere on $X_\Delta$.
An important observation is that the $p_{i}$ in are relatively prime when $\Pi$ is strictly defined. To prove this, suppose that some nonconstant polynomial $q$ divides the $p_{i}$. Since the exceptional set $Z \subset \CC^{r}$ has codimension at least $2$, we can find $a \in \CC^{r} \setminus Z$ such that $q(a) = 0$. Hence $p_{i}(a) = 0$ for all $i$, which can’t happen when $\Pi$ is strictly defined. This proves that the $p_{i}$ are relatively prime. By Definition \[rpdef\], it follows that $P$ is a rational parametrization of $X$.
We also note that being strictly defined implies that $\Pi$ and hence $p$ are onto, i.e., $X$ is the image of $p : \CC^{r}
-\!-\!\hskip-1.5pt \to \PP^{s}$. This follows because $X_\Delta$ is compact.
Finally, note that when $\Pi$ is birational, there is a nonempty Zariski open subset $$U \subset X$$ on which $\Pi^{-1}$ is defined. We may assume that $U$ is the maximal such open set.
$\Sigma_\Delta$-Irreducible Polynomials
---------------------------------------
The rough idea of a universal rational parametrization is that any rational parametrization $H$ should arise from $P$ by composition with a polynomial map $\CC^d \to \CC^r$. But if the image of $\CC^d \to
\CC^r$ lies in the exceptional set $Z$, then the composition doesn’t make sense since $p$ is not defined on $Z \subset \CC^r$. It follows that we need to exclude certain polynomial maps. The precise definition is as follows. Let $R = \CC[y_1,\dots,y_d]$.
\[irreddef\] We say that $F = (f_1,\dots,f_r) \in R^r$ is [**$\Sigma_\Delta$-irreducible**]{} if $\mathrm{gcd}(f_{i_1},\dots,f_{i_k}) =
1$ whenever no cone of $\Sigma_\Delta$ contains $\rho_{i_1},\dots,\rho_{i_k}$.
Because of our gcd convention, Definition \[irreddef\] implies in particular that the edges $\rho_i$ such that $f_i = 0$ all lie in some cone of $\Sigma_\Delta$. In the discussion which follows, we will identify $F$ with the polynomial function $\CC^d \to \CC^r$ it induces.
Here are two examples of this definition.
\[p2irred\][Consider the toric variety $X_\Delta = \PP^2$ coming from the polytope $\Delta$ with vertices $(0,0), (2,0), (0,2)$. $$\begin{matrix}
\begin{picture}(260,80)
\thinlines
\put(20,20){\line(1,0){80}}
\put(40,0){\line(0,1){80}}
\thicklines
\put(40,20){\line(1,0){40}}
\put(40,20){\line(0,1){40}}
\put(40,60){\line(1,-1){40}}
\put(69,39){$\scriptstyle{x_3}$}
\put(24,39){$\scriptstyle{x_1}$}
\put(56,11){$\scriptstyle{x_2}$}
\put(200,40){\line(1,0){40}}
\put(200,40){\line(0,1){40}}
\put(200,40){\line(-1,-1){30}}
\put(228,44){$\scriptstyle{\rho_1}$}
\put(204,74){$\scriptstyle{\rho_2}$}
\put(188,11){$\scriptstyle{\rho_3}$}
\end{picture}
\end{matrix}$$ The polytope $\Delta$ is on the left with facet variables $x_1,x_2,x_3$ and the normal fan is on the right with edges $\rho_1,\rho_2,\rho_3$. The only choice for $\rho_{i_1},\dots,\rho_{i_k}$ in Definition \[irreddef\] is $\rho_1,\rho_2,\rho_3$, so that $F = (f_1,f_2,f_3)$ is $\Sigma$-irreducible if $\gcd(f_1,f_2,f_3) = 1$. This is the gcd condition in Example \[firststeiner\]. ]{}
\[p1p1irred\][Let $\Delta$ be the unit square in the plane with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. This gives $X_\Delta = \PP^1\times\PP^1$. $$\begin{matrix}
\begin{picture}(260,80)
\thinlines
\put(20,20){\line(1,0){80}}
\put(40,0){\line(0,1){80}}
\thicklines
\put(40,20){\line(1,0){40}}
\put(40,20){\line(0,1){40}}
\put(40,60){\line(1,0){40}}
\put(80,20){\line(0,1){40}}
\put(84,39){$\scriptstyle{x_2}$}
\put(56,67){$\scriptstyle{x_4}$}
\put(29,39){$\scriptstyle{x_1}$}
\put(56,11){$\scriptstyle{x_3}$}
\put(160,40){\line(1,0){80}}
\put(200,0){\line(0,1){80}}
\put(228,44){$\scriptstyle{\rho_1}$}
\put(204,74){$\scriptstyle{\rho_3}$}
\put(170,44){$\scriptstyle{\rho_2}$}
\put(204,6){$\scriptstyle{\rho_4}$}
\end{picture}
\end{matrix}$$ As in the previous example, $\Delta$ and the facet variables $x_1,x_2,x_3,x_4$ are on the left and $\Sigma_\Delta$ and the edges are $\rho_1,\rho_2,\rho_3,\rho_4$ are on the right. The minimal choices for $\rho_{i_1},\dots,\rho_{i_k}$ in Definition \[irreddef\] are $\rho_1,\rho_2$ and $\rho_3,\rho_4$ (you should check this). Thus $F = (f_1,f_2,f_3,f_4)$ is $\Sigma$-irreducible if $\gcd(f_1,f_2) =
\gcd(f_3,f_4) = 1$. This is the gcd condition in Example \[firstex\]. ]{}
The Main Result
---------------
Before stating our main result, we need to introduce some notation. As above, let $R = \CC[y_{1},\dots,y_{d}]$ be the coordinate ring of $\CC^{d}$. Also, given polynomials $F = (f_{1},\dots,f_{r}) \in R^r$ and $m \in \Delta\cap \ZZ^n$, we set $$f^{m} = \prod_{i=1}^{r} f_{i}^{\langle m,n_{i}\rangle+a_{i}}.$$ Recall the group $G$ from and that $\mu \in G$ gives $\mu_\Delta \in \CC^*$ defined in . The map $\mu \mapsto
\mu_\Delta$ is a group homomorphism $G \to \CC^*$. Let $G_\Delta$ be the kernel of this map. This group will measure the lack of uniqueness in Theorem \[upthm\]. Finally, let $\sum_m$ denote summation over all $m \in \Delta\cap \ZZ^n$.
Here is our precise result.
\[upthm\] Let ${P} = (p_0,\dots,p_s) = (\sum_{m} a_{0m} x^{m}, \dots,\sum_{m}
a_{sm}x^{m}) \in S_\Delta^{s+1}$, where $X_\Delta$ is smooth and $\Pi
: X_\Delta \to X$ is strictly defined and birational, with an inverse defined on $U \subset X$ which we assume to be maximal. Then $P$ is a [**universal rational parametrization**]{} of $X \subset \PP^s$ in the following sense:
1. If $F = (f_{1},\dots,f_{r}) \in R^{r}$ is $\Sigma_{\Delta}$-irreducible, then $${P}\circ F = ({\textstyle \sum_{m} a_{0m} f^{m}, \dots,\sum_{m}
a_{sm}f^{m}}) \in R^{s+1}$$ is a rational parametrization of $X \subset \PP^{s}$.
2. Conversely, given any rational parametrization $H \in R^{s+1}$ of $X$ whose image meets $U \subset X$, there is a $\Sigma_{\Delta}$-irreducible $F = (f_{1},\dots,f_{r}) \in R^{r}$ such that $H = {P}\circ F$.
3. If $F$ and $F'$ are $\Sigma_{\Delta}$-irreducible, then $P\circ
F = P\circ F'$ as rational parametrizations if and only if $F' =
\mu\cdot F$ for some $\mu \in G_\Delta$.
The proof will be given in Section \[theory\]. Here is a corollary of Theorem \[upthm\].
\[univcor\] Assume the same hypothesis as Theorem \[upthm\] and suppose that $H' = (h_0',\dots,h_s') \in R^{s+1}$ gives a rational map $\CC^d -\!
-\!\hskip-1.5pt \to \PP^{s}$ whose image lies in $X$ and meets $U$. Then there is a polynomial $q \in R$ and a $\Sigma_\Delta$-irreducible $F = (f_1,\dots,f_r) \in R^r$ such that $$H' = q\, P \circ F.$$
Write $H' = q\, H$, where the entries of $H$ are relatively prime. Since $H$ is a rational parametrization, we are done by Theorem \[upthm\].
Embeddings
----------
In order for Theorem \[upthm\] to be useful, we need to have a good supply of parametrizations $P = (p_0,\dots,p_s) \in S_\Delta^{s+1}$ which satisfy the hypotheses of the theorem. The first crucial observation is that since $X_\Delta$ is a smooth toric variety, it is a standard result that the collection of *all* $\Delta$-monomials gives a projective embedding (see [@Fulton Sec. 3.4]).
This means the following. Suppose that $\Delta\cap\ZZ^n =
\{m_0,\dots,m_\ell\}$ and let $$\label{pddef}
P_\Delta = (x^{m_0},\dots,x^{m_\ell})$$ Then $P_\Delta$ induces $p_\Delta : \CC^r -\!-\!\hskip-1.5pt \to
\PP^{\ell}$, and in the factorization $p_\Delta = \pi\circ \Pi_\Delta$ of Proposition \[pfactors\], the map $\Pi : X_\Delta \to \PP^{\ell}$ is an embedding. Hence we can write $X_\Delta \subset \PP^\ell$.
All of the hypotheses of Theorem \[upthm\] are satisfied in this situation, and the open set $U$ is all of $X = X_\Delta$. Thus $P_\Delta$ is a universal rational parametrization in the strong sense that *every* rational parametrization is of the form $P_\Delta\circ F$ for some $\Sigma_\Delta$-irreducible $F$.
Here are two examples of this result.
\[p1p1second\]
Let $\Delta$ be the unit square from Example \[p1p1irred\]. $$\begin{matrix}
\begin{picture}(120,80)
\put(20,20){\line(1,0){80}}
\put(40,0){\line(0,1){80}}
\thicklines
\put(40,20){\line(1,0){40}}
\put(40,20){\line(0,1){40}}
\put(40,60){\line(1,0){40}}
\put(80,20){\line(0,1){40}}
\put(84,39){$\scriptstyle{x_2}$}
\put(56,67){$\scriptstyle{x_4}$}
\put(29,39){$\scriptstyle{x_1}$}
\put(56,11){$\scriptstyle{x_3}$}
\end{picture}
\end{matrix}$$ The labeling of $x_1,x_2,x_3,x_4$ is consistent with Example \[p1p1irred\]. In terms of , $a_1 = a_3 =
0$ and $a_2 = a_4 = 1$. Then $$\begin{array}{ccccl}
P_\Delta &=&
(x_2 x_3, & x_1 x_3, & \\
&&\, x_2 x_4, & x_1 x_4)& \in S_\Delta^4
\end{array}$$ gives a universal rational parametrization of its image in $\PP^3$, which is the quadric $Q$ of Example \[firstex\]. This means that any parametrization of $Q$ is of the form $P_\Delta\circ F$, where $F
= (f_1,f_2,f_3,f_4)$ satisfies the gcd condition worked out in Example \[p1p1irred\].
To study uniqueness, we need to compute $$G = \{(\mu_1,\mu_2,\mu_3,\mu_4) \in (\CC^*)^4 \mid
{\textstyle\prod_{i=1}^4}
\mu_i^{\langle m, n_{i}\rangle} = 1\ \text{for all}\ m \in \ZZ^4\}.$$ Since it suffices to use $m = (1,0)$ and $(0,1)$, we see that $$G = \{(\mu_1,\mu_2,\mu_3,\mu_4) \in (\CC^*)^4 \mid
\mu_{1}\mu_{2}^{-1} = \mu_{3}\mu_{4}^{-1} =1\},$$ and it follows that $$G =\{(\mu_1,\mu_1,\mu_2,\mu_2) \mid \mu_1, \mu_2 \in \CC^*\}.$$ Then $a_1 = a_3 = 0$ and $a_2 = a_4 = 1$ imply that if $\mu =
(\mu_1,\mu_1,\mu_2,\mu_2) \in G$, then $\mu_\Delta = \mu_1\mu_2$. Hence $$G_\Delta = \{(\lambda,\lambda,\lambda^{-1},\lambda^{-1}) \mid \lambda
\in \CC^*\}.$$ Thus, when we write a rational parametrization of $Q$ as $P_\Delta\circ F$, we see that $F = (f_1,f_2,f_3,f_4)$ is unique up to $$(\lambda f_1,\lambda f_2,\lambda^{-1} f_3,\lambda^{-1} f_4).$$ for some nonzero scalar $\lambda$. It follows that we obtain precisely the description given in Example \[firstex\].
Here is how Theorem \[upthm\] applies to one of our earlier examples.
\[p1p1b2\]
Consider the toric variety $X_\Delta$ of the polytope $\Delta$ with vertices $(1,1), (-1,1), (-1,0), (0, -1), (1,-1)$ from Example \[p1p1b2first\]. $$\begin{matrix}
\begin{picture}(260,120)
\put(0,60){\line(1,0){120}}
\put(60,0){\line(0,1){120}}
\thicklines
\put(30,60){\line(0,1){30}}
\put(30,90){\line(1,0){60}}
\put(90,90){\line(0,-1){60}}
\put(90,30){\line(-1,0){30}}
\put(60,30){\line(-1,1){30}}
\put(18,75){$\scriptstyle{x_1}$}
\put(68,94){$\scriptstyle{x_5}$}
\put(32,40){$\scriptstyle{x_2}$}
\put(92,75){$\scriptstyle{x_4}$}
\put(68,23){$\scriptstyle{x_3}$}
\put(140,60){\line(1,0){120}}
\put(200,0){\line(0,1){120}}
\put(200,60){\line(1,1){60}}
\put(228,64){$\scriptstyle{\rho_1}$}
\put(204,94){$\scriptstyle{\rho_3}$}
\put(228,81){$\scriptstyle{\rho_2}$}
\put(158,64){$\scriptstyle{\rho_4}$}
\put(204,30){$\scriptstyle{\rho_5}$}
\end{picture}
\end{matrix}$$ As usual, the polytope is on the left and the normal fan of $\Delta$ is on the right. One can show that $X_\Delta$ is the blowup of $\PP^1
\times \PP^1$ at one point.
In terms of and the above labeling, we have $a_1 =
\dots = a_5 = 1$. The 8 points of $\Delta\cap \ZZ^2$ give an embedding of $X_\Delta$ into $\PP^7$. It follows that $$\begin{array}{cccccl}
P_\Delta &=&
(x_2 x_3^2 x_4^2, & x_1 x_2^2 x_3^2 x_4, & x_1^2 x_2^3 x_3^2,& \\
&& x_3 x_4^2 x_5, & x_1 x_2 x_3 x_4 x_5, & x_1^2 x_2^2 x_3 x_5,& \\
&&& x_1 x_4 x_5^2, & x_1^2 x_2 x_5^2)& \in S_\Delta^8.
\end{array}$$ is the universal rational parametrization of $X_\Delta$ by Theorem \[upthm\].
Let’s work out what this means. According to Definition \[irreddef\], $F = (f_{1},\dots,f_{5})$ is $\Sigma_\Delta$-irreducible provided that $$\gcd(f_{1},f_{3}) = \gcd(f_{1},f_{4}) = \gcd(f_{2},f_{4}) =
\gcd(f_{2},f_{5}) = \gcd(f_{3},f_{5}) = 1.$$ Then any rational parametrization of $X_{\Sigma}$ is of the form $P_\Delta\circ F$ for some $F$ satisfying this gcd condition. To determine the lack of uniqueness, we need to compute the group $G$. Using the methods of Example \[p1p1second\], one obtains $$G = \{(\lambda,\mu,\nu,\lambda \mu, \mu\nu) \mid \lambda,\mu,\nu \in
\CC^{*}\},$$ and then $a_1 = \dots = a_5 = 1$ imply that $$G_\Delta = \{(\lambda,\mu,\nu,\lambda \mu, \mu\nu) \mid \lambda,\mu,\nu \in
\CC^{*}, \lambda^2 \mu^3 \nu^2 = 1\}.$$ Thus rational parametrizations of $X_{\Sigma}$ are all of the form $P_\Delta\circ F$, where $F$ is unique up to $(\lambda,\mu,\nu,\lambda \mu,
\mu\nu)\cdot F$ for $\lambda^2 \mu^3 \nu^2 = 1$.
Projections
-----------
Although $P_\Delta = (x^{m_0},\dots,x^{m_\ell})$ from always gives a universal rational parametrization, it is rarely useful in practice since it usually gives an embedding into a projective space of high dimension. An important observation is that we can think of the general case $$P = (p_0,\dots,p_s) = ({\textstyle\sum_{i=0}^\ell a_{0i} x^{m_i},
\dots,\sum_{i=0}^\ell a_{si}x^{m_i}}) \in S_\Delta^{s+1}$$ in terms of projections. Let $\PP^\ell$ be a projective space with homogeneous coordinates $z_0,\dots,z_\ell$. Then the $s+1$ linear forms $\sum_{i=0}^\ell a_{ji} z_i$ define a projection $\PP^{\ell}
-\!-\!\hskip-1.5pt \to \PP^{s}$ defined by $$(z_0,\dots,z_\ell) \mapsto ({\textstyle\sum_{i=0}^\ell a_{0i} {z_i},
\dots,\sum_{i=0}^\ell a_{si}{z_i}}).$$ The *center* $L \subset \PP^{\ell}$ of this projection is defined by $\sum_{i=0}^\ell a_{ji} z_i = 0$ for $j = 0,\dots,s$. This tells us where the projection is not defined.
If we compose this projection with $p_\Delta : \CC^r -\!
-\!\hskip-1.5pt \to \PP^\ell$, then we get the rational map $p : \CC^r
-\! -\!\hskip-1.5pt \to \PP^s$ induced by $P$. Furthermore, since the image of $p_\Delta$ is $X_\Delta \subset \PP^\ell$, it follows that the variety $X$ parametrized by $P$ is the image of $X_\Delta$ under the projection.
From this point of view, we can think of $\Pi$ as a projection. It is then straightforward to check that $\Pi$ is strictly defined if and only if $X_{\Delta}$ is disjoint from the center $L$ of the projection. (For more sophisticated readers, we point out that being strictly defined is equivalent to the assertion that the linear system on $X_\Delta$ spanned by the $p_i$ has no base points. One can also show that $X_\Delta$ is the normalization of $X$ when $\Pi$ is strictly defined and birational.)
Let’s give an example from geometric modeling which involves the projection of a toric variety.
Consider the toric variety $X_\Delta = \PP^2$, where $\Delta$ is the polytope from Example \[p2irred\]. In terms of , we have $a_1 = a_2 = 0$ and $a_3 = 2$. The 6 points of $\Delta \cap \ZZ^2$ define $$P_\Delta = (x_1^2, x_2^2, x_3^2, x_1 x_2, x_2 x_3, x_3 x_1).$$ This gives the usual Veronese embedding of $\PP^2$ into $\PP^5$.
The composition of this map with the projection $\PP^5
-\!-\!\hskip-1.5pt \to \PP^3$ defined by $$\label{stpr}
(z_0,z_1,z_2,z_3,z_4,z_5) \mapsto (z_0+z_1+z_2,z_3,z_4,z_5)$$ gives a rational parametrization $$\label{upsteiner}
P = (x_1^2+x_2^2+x_3^2, x_1x_2 , x_2x_3, x_3x_1)$$ of the Steiner surface $S \subset \PP^3$ defined by $$u_1^2u_2^2+u_2^2u_3^2+u_3^2u_1^2 = u_0u_1u_2u_3,$$ where $u_0,u_1,u_2,u_3$ are homogeneous coordinates on $\PP^3$. We saw this equation earlier in Example \[firststeiner\].
It is easy to check that the center of the projection is disjoint from $X_\Delta$. Thus the map $\Pi : X_\Delta \to S$ is strictly defined. Furthermore, since $$x_1^2 = \frac{(x_1x_2)(x_3x_1)}{x_2x_3} = \frac{u_1u_3}{u_2},$$ one easily sees that $\Pi : X_\Delta \to S$ is birational with inverse $$\begin{aligned}
\Pi^{-1}(u_0,u_1,u_2,u_3) &= \Big(\frac{u_1u_3}{u_2},
\frac{u_1u_2}{u_3},\frac{u_2u_3}{u_1},u_1,u_2,u_3\Big)\\
&= (u_1^2 u_3^2, u_1^2 u_2^2, u_2^2u_3^2, u_1^2u_2u_3,
u_1u_2^2u_3, u_1u_2u_3^2).\end{aligned}$$ Also notice that $\Pi^{-1}$ is defined on the complement of the three lines $u_1 = u_2 = 0$, $u_2 = u_3 = 0$, $u_3 = u_1 = 0$.
By Theorem \[upthm\], is the universal rational parametrization of the Steiner surface $S$. It follows that if $H$ is a rational parametrization of $S$ whose image is not contained in any of the above three lines, then $H = P\circ F$, where $F =
(f_1,f_2,f_3)$. Furthermore, we know that $F$ is $\Sigma_\Delta$-irreducible, which by Example \[p2irred\] means $\gcd(f_1,f_2,f_3) = 1$.
Finally, we know that $G = \{(\lambda,\lambda,\lambda) \mid \lambda
\in \CC^*\} \simeq \CC^*$ in this case. Since $a_1 = a_2 = 0$ and $a_3 = 2$, we see that $\mu_\Delta = \lambda^2$ when $\mu =
(\lambda,\lambda,\lambda)$. It follows that the kernel of $\mu
\mapsto \mu_\Delta$ is $\pm(1,1,1)$, so that in $H = P\circ F$, $F$ is unique up to $\pm1$. Hence we recover the description of the rational parametrizations of the Steiner surface given in Example \[firststeiner\].
Observe that $F = (f_1,f_2,f_3)$ may fail to exist when the image of $H$ is contained in one of the three lines $z_1 = z_2 = 0$, $ z_2 =
z_3 = 0$, $z_3 = z_1 = 0$. For example, $H = (u,0,0,v)$ is a rational parametrization from $\CC^2$ to $S$ which is not of the form $P\circ
F$ for any $F \in \CC[u,v]^3$. Notice also that the union of these lines is the singular locus of $S$.
We next describe one important class of projections which always lead to universal rational parametrizations. Suppose that the smooth $n$-dimensional toric variety $X_\Delta$ is embedded into $\PP^\ell$ via $P_\Delta$. Then let $$P = (p_0,\dots,p_{n+1}) \in S_\Delta^{n+2}$$ be chosen generically. Then $X \subset \PP^{n+1}$ is the image of $X_\Delta$ under a generic projection. It is well-known that in this situation, $X_\Delta$ is disjoint from the center of the projection and the restriction of the projection to $X_\Delta$ is birational. Hence $P$ is a universal rational parametrization in this generic case. Notice that $X$ has dimension $n$ and hence is a hypersurface in $\PP^{n+1}$.
In particular, when $X_\Delta$ is a smooth toric surface, it follows that $$P = (p_0,\dots,p_3) \in S_\Delta^{4}$$ is a universal rational parametrization whenever the $p_i$ are chosen generically. Here, we parametrize a surface in $\PP^3$, which is the case of greatest interest in geometric modeling.
However, we should also mention that there are some non-generic projections which also work nicely. Here is another class of projections which are guaranteed to give universal rational parametrizations.
\[lattice\] Let $X_\Delta$ be the smooth toric variety of a polytope $\Delta$ and let $\mathcal{A} = \{\tilde{m}_0,\dots,\tilde{m}_k\} \subset
\Delta\cap \ZZ^n$. Assume that $\Delta$ is the convex hull of $\mathcal{A}$ and that $\mathcal{A}$ generates $\ZZ^n$ affinely over $\ZZ$ $($meaning that $\ZZ^n$ is the $\ZZ$-span of $\{m - m' \mid m,m'
\in \mathcal{A}\})$. Then $P_\mathcal{A} =
(x^{\tilde{m}_0},\dots,x^{\tilde{m}_k}) \in S_\Delta^{k+1}$ induces an everywhere defined birational map $$\Pi : X_\Delta \to X_\mathcal{A} \subset \PP^{k}$$ and $P_\mathcal{A}$ is the universal rational parametrization of $X_\mathcal{A}$.
Proposition 5.3 of [@sc] implies that the map $\Pi : X_\Delta \to
X_\mathcal{A}$ is the normalization map. (In [@sc], Proposition 5.3 does not assume that $\mathcal{A}$ generates $\ZZ^n$ affinely, but this is necessary since the proposition depends on Proposition 5.2, which does assume that $\mathcal{A}$ generates the lattice affinely.) It follows immediately that $\Pi$ is a birational morphism. One can also show that $\Pi$ is strictly defined in this case. Then the final assertion follows immediately from Theorem \[upthm\]. This completes the proof.
Here is an example of this proposition.
[ In the situation of Example \[p1p1b2\], let $\mathcal{A} \subset
\Delta\cap \ZZ^2$ be the five vertices of $\Delta$. Since $\mathcal{A}$ satisfies all of the conditions of Proposition \[lattice\], it follows that $$P_\mathcal{A} = (x_2 x_3^2 x_4^2, x_1^2 x_2^3 x_3^2, x_3 x_4^2 x_5,
x_1 x_4 x_5^2, x_1^2 x_2 x_5^2) \in S_\Delta^5$$ is the universal rational parametrization of $X_\mathcal{A} \subset
\PP^4$. Also note that $X_\mathcal{A}$ is the image of $X_\Delta
\subset \PP^7$ under the projection $\PP^7 -\!-\!\hskip-1.5pt \to
\PP^4$ determined by $\mathcal{A}$.]{}
One final comment about Propostion \[lattice\] is that $X_\mathcal{A}$ is itself a toric variety (possibly non-normal). In contrast, the image of $X_\Delta$ under a generic projection may fail to be a toric variety.
Further Examples
----------------
Here are two examples which show what happens when we violate one of the hypotheses of Theorem \[upthm\].
\[hirzebruch\]
Consider the quadrilateral $\Delta$ with vertices $(1,0)$, $(0,1)$, $(-1,1)$ and $(-1,0)$: $$\begin{matrix}
\begin{picture}(120,90)
\put(0,30){\line(1,0){120}}
\put(60,0){\line(0,1){90}}
\thicklines
\put(30,30){\line(0,1){30}}
\put(30,60){\line(1,0){30}}
\put(60,60){\line(1,-1){30}}
\put(30,30){\line(1,0){60}}
\put(48,23){$\scriptstyle{x_1}$}
\put(48,64){$\scriptstyle{x_3}$}
\put(18,45){$\scriptstyle{x_4}$}
\put(78,45){$\scriptstyle{x_2}$}
\end{picture}
\end{matrix}$$ The lattice points $\Delta\cap \ZZ^2$ give the five monomials $$\begin{array}{ccc}
x_1x_2 & x_1x_4 & \\
x_2^2x_3 & x_2x_3x_4 & x_3x_4^2
\end{array}$$ which in turn give an embedding $X_\Delta \subset \PP^4$.
Let $\mathcal{A} = \{(-1,1),(-1,0),(0,0)\} \subset \Delta\cap
\ZZ^2$. This gives $$P_\mathcal{A} = (x_1x_2, x_2^2x_3, x_2x_3x_4)$$ and the rational map $$\Pi : X_\Delta -\!-\!\hskip-1.5pt \to X_\mathcal{A} \subset \PP^2$$ defined by $$\label{projdef}
(x_1x_2, x_1x_4, x_2^2x_3, x_2x_3x_4, x_3x_4^2) \mapsto (x_1x_2,
x_2^2x_3, x_2x_3x_4).$$ The center of this projection is the line $L = \{(0,u,0,0,v) \mid (u,v)
\ne (0,0)\}$. One can check that $L$ is entirely contained in $X_\Delta$ and corresponds to those points where $x_2
=0$. Thus $\Pi$ is not strictly defined. The surprise is that $\Pi$ is nevertheless defined everywhere on $X_\Delta$. At first glance, this seems impossible, since $x_2 = 0$ corresponds to points $$(0,x_1x_4, 0,0, x_3x_4^2) \in X_\Delta$$ which project to $(0,0,0)$. We get around this difficulty by letting $x_2 = \varepsilon$, where $\varepsilon \in \CC$ is nonzero but close to zero. Then becomes $$(x_1\varepsilon, x_1x_4, \varepsilon^2x_3, \varepsilon x_3x_4, x_3x_4^2)
\mapsto
(x_1\varepsilon, \varepsilon^2x_3, \varepsilon x_3x_4) = (x_1, \varepsilon x_3,
x_3x_4) \in \PP^2$$ since $\varepsilon \ne 0$. Letting $\varepsilon \to 0$, this suggests that $$\Pi(0,x_1x_4, 0,0, x_3x_4^2) = (x_1, 0, x_3x_4) \in \PP^2.$$ In fact, one can prove rigorously that $\Pi$ is defined everywhere on $X_\Delta$.
We also note that $X_\mathcal{A} = \PP^2$ and that $\Pi$ is birational. This follows from $$\Pi^{-1}(u_0,u_1,u_2) = (u_0 u_1,u_0 u_2,u_1^2,u_1 u_2,u_2^2),$$ where $u_0,u_1,u_2$ are homogeneous coordinates on $\PP^2$. This is the inverse of $\Pi$ on the open subset of $\PP^2$ where $u_0 u_1 u_2
\ne 0$. So $\Pi$ is defined everywhere and is birational. However, Theorem \[upthm\] fails in this case. For example, $P_\mathcal{A}$ is not a rational parametrization since $x_2$ divides the polynomials of $P_\mathcal{A}$. Yet the definition of rational parametrization requires relatively prime polynomials. Hence $P_\mathcal{A}$ has no chance of being a universal rational parametrization.
Our final example concerns a singular polygon.
\[singex1\]
Consider the triangle $\Delta$ with vertices $(1,0)$, $(0,1)$, and $(-1,0)$: $$\begin{matrix}
\begin{picture}(120,90)
\put(0,30){\line(1,0){120}}
\put(60,0){\line(0,1){90}}
\thicklines
\put(30,30){\line(1,1){30}}
\put(60,60){\line(1,-1){30}}
\put(30,30){\line(1,0){60}}
\put(54,23){$\scriptstyle{z}$}
\put(37,45){$\scriptstyle{y}$}
\put(78,45){$\scriptstyle{x}$}
\end{picture}
\end{matrix}$$ The lattice points $\Delta\cap \ZZ^2$ give the four monomials $$\begin{array}{ccc}
& z & \\
x^2 & xy & y^2
\end{array}$$ which in turn give an embedding $X_\Delta \subset \PP^3$ as the singular quadric surface $u_2^2 = u_1 u_3$. One can show that $X_\Delta$ is the weighted projective space $\PP(1,1,2)$.
Even though $X_\Delta$ is singular, it is easy to see that $$P_\Delta = (z,x^2,xy,y^2)$$ satisfies the other hypotheses of Theorem \[upthm\]. So how close is $P_\Delta$ to being a universal rational parametrization?
For an example of what can go wrong, consider $H = (v,u,u,u)$. This is a rational parametrization of $X_\Delta$, yet $H$ is not of the form $P_\Delta\circ F$ for any $F = (f_1,f_2,f_3) \in \CC[u,v]^3$ since $u$ is not a square. So Theorem \[upthm\] fails in this case.
However, it is true that $H = P_\Delta\circ \widetilde{F}$, where $$\label{crazy2}
\widetilde{F} = (\sqrt{u},\sqrt{u},v).$$ So it may be that for singular toric varieties, square roots and other radicals appear naturally in considering what a universal parametrization means. But in the next section, we will learn a better method which uses resolution of singularities.
Universal Rational Parametrizations (Singular Case) {#singular}
===================================================
So far, we have always assumed that $X_\Delta$ is smooth, and we saw in Example \[singex1\] how things can go wrong when $X_\Delta$ has singularities. We will use a toric resolution of singularities to show that we still have universal rational parametrizations, where $\Delta$ is now allowed to be *any* $n$-dimensional lattice polytope in $\ZZ^n$.
As in Section \[projective\], assume that we have $$\label{pdelta}
P = (p_{0},\dots,p_{s}) \in S_\Delta^{s+1},$$ which induces a strictly defined birational map $$\Pi : X_{\Delta} \to X \subset \PP^s.$$ Our goal is to describe a universal rational parametrization of $X
\subset \PP^s$. Our main tool will be a toric resolution of singularities. As shown in [@Fulton], the normal fan $\Sigma_\Delta$ of $\Delta$ has a refinement $\Sigma$ such that $X_\Sigma$ is smooth. It follows that the induced toric morphism $$\varphi : X_\Sigma \to X_\Delta$$ is a resolution of singularities. We may assume that $\varphi^{-1}$ is defined on the smooth part of $X_\Delta$.
Let $x_1,\dots,x_{\tilde{r}}$ be the homogeneous coordinates of $X_\Sigma$ and let $\tilde{n}_i$ generate the edge of $\Sigma$ corresponding to $x_i$. Some of the $\tilde{n}_i$’s will be inner normals to facets of $\Delta$, while others will be new vectors added in the process of resolving singularities. We will regard the new $\tilde{n}_i$’s as inner normals to “virtual facet hyperplanes” of $\Delta$ in the following way.
Given $\tilde{n}_i$, we know that it lies in some cone $\sigma \in
\Sigma_\Delta$. We pick the smallest such cone. Its generators are facet normals of $\Delta$, and the intersection of the corresponding facets is a face $\Delta_\sigma$ of $\Delta$. Using the support functions defined in [@Fulton], one can prove that there is a unique integer $\tilde{a}_i$ such that $$\{m \in \RR^n \mid \langle m,\tilde{n}_i\rangle + \tilde{a}_i = 0\}
\cap \Delta = \Delta_\sigma.$$ We call $\{m \in \RR^n \mid \langle m,\tilde{n}_i\rangle + \tilde{a}_i
= 0\}$ the *virtual facet hyperplane* of $\Delta$ with $\tilde{n}_i$ as inner normal. When $\tilde{n}_i$ is the inner normal of a facet of $\Delta$, then one easily sees that the virtual facet hyperplane is the facet hyperplane $\langle m,\tilde{n}_i\rangle +
\tilde{a}_i = 0$ containing the corresponding facet of $\Delta$.
Let’s illustrate what this looks like in one of our previous examples.
\[singex2\][Consider the triangle $\Delta$ of Example \[singex1\] and let $\Sigma$ be the following refinement of its normal fan: $$\begin{matrix}
\begin{picture}(120,120)
\thinlines
\put(0,60){\line(1,0){120}}
\thicklines
\put(60,0){\line(0,1){120}}
\put(60,60){\line(-1,-1){60}}
\put(60,60){\line(1,-1){60}}
\put(94,30){$\scriptstyle{x_4}$}
\put(64,94){$\scriptstyle{x_1}$}
\put(16,30){$\scriptstyle{x_2}$}
\put(64,30){$\scriptstyle{x_3}$}
\end{picture}
\end{matrix}$$ (So the refinement is given by adding the edge corresponding to $x_3$.) Let $\tilde{n}_i$ generate the edge corresponding to $x_i$. Thus $\tilde{n}_1$, $\tilde{n}_2$ and $\tilde{n}_4$ are inner normals of facets of the triangle $\Delta$ of Example \[singex1\], while $\tilde{n}_3$ was added to make $X_\Sigma$ smooth. Then we can draw the virtual facet hyperplanes (= lines in this case) and their corresponding variables as follows: $$\begin{matrix}
\begin{picture}(120,90)
\put(0,30){\line(1,0){120}}
\put(15,15){\line(1,1){60}}
\put(105,15){\line(-1,1){60}}
\thicklines
\put(30,30){\line(1,1){30}}
\put(60,60){\line(1,-1){30}}
\put(30,30){\line(1,0){60}}
\put(56,23){$\scriptstyle{x_1}$}
\put(34,45){$\scriptstyle{x_4}$}
\put(78,45){$\scriptstyle{x_2}$}
\put(78,63){$\scriptstyle{x_3}$}
\multiput(15,60)(6,0){15}{\line(1,0){3}}
\end{picture}
\end{matrix}$$ The facet hyperplanes are solid lines, while the one virtual facet hyperplane is the dashed line corresponding to $x_3$. Note also that $$\tilde{a}_1 = 0\quad \text{and} \quad \tilde{a}_2 = \tilde{a}_3 =
\tilde{a}_4 = 1.$$ We will return to this example shortly.]{}
Given this setup, a lattice point $m \in \Delta \cap \ZZ^n$ gives the monomial $$\label{xds}
x^m = \prod_{i=1}^{\tilde{r}} x_i^{\langle m,\tilde{n}_i\rangle +
\tilde{a}_i}.$$ We call $x^m$ a $\Delta$-monomial of the toric variety $X_\Sigma$. Note that the exponent of $x_i$ in $x^m$ measures the lattice distance from $m$ to the corresponding virtual facet hyperplane. Here is an example.
\[singex3\][In the situation of Example \[singex2\], the lattice points of $\Delta\cap \ZZ^2$ give the $\Delta$-monomials $$\label{singlat}
\begin{array}{ccc}
& x_1 & \\
x_2^2x_3 & x_2x_3x_4 & x_3x_4^2
\end{array}$$ in the homogeneous coordinates of the toric variety $X_\Sigma$ which resolves the singularities of $X_\Delta$.]{}
One useful observation is that when dealing with lattice polygons, the only places we need to add virtual facet hyperplanes are at vertices whose adjacent inner normals do not form a basis of $\ZZ^2$ over $\ZZ$. Furthermore, in this situation, there is a unique minimal resolution of singularities which can be computed algorithmically—see [@Fulton Sec. 2.6]. Thus there is an algorithm for finding the virtual inner normals that need to be added at these vertices.
We are almost ready to state our main result. As above, $\Delta$ is an $n$-dimensional lattice polytope in $\RR^n$ and $\varphi: X_\Sigma
\to X_\Delta$ is a toric resolution. The lattice points in $\Delta\cap\ZZ^n$ determine $$S_\Delta = \mathrm{Span}(x^m \mid m \in \Delta\cap\ZZ^n)$$ where $x^m$ is now the $\Delta$-monomial in the homogeneous coordinates $x_1,\dots,x_{\tilde{r}}$ of $X_\Sigma$. Now let $$P = (p_0,\dots,p_s) = \Big({\textstyle \sum_m a_{0m} x^m, \dots,
\sum_m a_{sm} x^m}\Big) \in S_\Delta^{s+1}.$$ Then $P$ induces a rational map $p : \CC^{\tilde{r}} -\!-\!\skip1.5pt \to
\PP^s$, and similar to Proposition \[pfactors\], one can show that $p$ factors as $$\CC^{\tilde{r}} -\!-\!\skip1.5pt \to X_\Sigma \xrightarrow{\ \varphi\
} X_\Delta -\!-\!\skip1.5pt \to X \subset \PP^s,$$ where as usual, $X$ is the Zariski closure of the image of $p$. The map from $X_\Delta$ to $X$ will be denoted $\Pi$, and as in Theorem \[upthm\], we will assume that $\Pi$ is strictly defined and birational. Let $U \subset X$ be the maximal open set on which the inverse of $\Pi: X_\Delta \to X$ is defined, and then set $$\widetilde{U} = U \cap \big(X \setminus \Pi(\{ x \in X_\Delta \mid x\
\text{is a singular point of}\ X_\Delta\})\big).$$ Finally, we have $G \subset (\CC^*)^{\tilde{r}}$. Then $\mu =
(\mu_1,\dots,\mu_{\tilde{r}}) \in G$ gives $\mu_{\Delta,\Sigma} =
\prod_{i=1}^{\tilde{r}} \mu_i^{\tilde{a}_i}$. Let $G_{\Delta,\Sigma}$ be the kernel of the homomorphism $\mu \mapsto \mu_{\Delta,\Sigma}$. We use this notation because $\mu_{\Delta,\Sigma}$ and $G_{\Delta,\Sigma}$ depend not only on the polytope $\Delta$ but also on the fan $\Sigma$.
We now show that $P$ is a universal rational parametrization of $X$.
\[singupthm\] Let $\Delta$, $\Sigma$, ${P} = (p_0,\dots,p_s)$, $X$ and $\widetilde{U}$ be as above. Then $P$ is a [**universal rational parametrization**]{} of $X \subset \PP^s$ in the following sense:
1. If $F = (f_{1},\dots,f_{\tilde{r}}) \in R^{\tilde{r}} =
\CC[y_1,\dots,y_d]^{\tilde{r}}$ is $\Sigma$-irreducible, then $${P}\circ F = ({\textstyle \sum_{m} a_{0m} f^{m}, \dots,\sum_{m}
a_{sm}f^{m}}) \in R^{s+1}$$ is a rational parametrization of $X \subset \PP^{s}$.
2. Conversely, given any rational parametrization $H \in R^{s+1}$ of $X$ whose image meets the open set $\widetilde{U} \subset
X$, there is a $\Sigma$-irreducible $F = (f_{1},\dots,f_{\tilde{r}})
\in R^{\tilde{r}}$ such that $H = {P}\circ F$.
3. If $F$ and $F'$ are $\Sigma$-irreducible, then $P\circ
F = P\circ F'$ as rational parametrizations if and only if $F' =
\mu\cdot F$ for some $\mu \in G_{\Delta,\Sigma}$.
The proof will be given in Section \[theory\]. Note that the theorem uses the concept of $\Sigma$-irreducible. This uses the obvious modification of Definition \[irreddef\] which applies to any fan $\Sigma$.
Let’s apply Theorem \[singupthm\] to the singular example we’ve been studying.
\[singex4\]
Let $\Delta$ be the triangle of Examples \[singex1\], \[singex2\] and \[singex3\]. This gives the singular toric variety $X_\Delta$. The fan $\Sigma$ from Example \[singex2\] gives a resolution of singularites, and the $\Delta$-monomials $x^m$ for $m \in \Delta\cap \ZZ^2$ are given in . Let $$P = (x_1, x_2^2x_3, x_2x_3x_4, x_3x_4^2).$$ Since $\Pi : X_\Delta \to X \subset \PP^3$ is an isomorphism in this case, Theorem \[singupthm\] implies that $P$ is a universal rational parametrization of $X$.
This means the following. If $u_0,u_1,u_2,u_3$ are coordinates on $\PP^3$, then $X$ is defined by $u_2^2 = u_1u_3$. Hence, if $H =
(h_0,h_1,h_2,h_3)$ is a rational parametrization whose image is not the singular point $(1,0,0,0) \in X$, then there is $F =
(f_1,f_2,f_3,f_4)$ such that $$H = P\circ F = (f_1,f_2^2f_3, f_2f_3f_4, f_3f_4^2).$$ Furthermore, one can show that
- $F$ is $\Sigma$-irreducible if and only if $\gcd(f_1,f_3) =
\gcd(f_2,f_4) = 1$.
- $F = (f_1,f_2,f_3,f_4)$ is unique up to $(f_1,\lambda
f_2,\lambda^{-2} f_3,\lambda f_4)$ for $\lambda \in \CC^*$.
For $H = (v,u,u,u) \in \CC[u,v]^4$ as in Example \[singex1\], one easily sees that $H = P\circ F$ for $F = (v,1,u,1)$ in this case. So unlike Example \[singex1\], we don’t need square roots.
In the smooth case, we analyzed $P$ in terms of the embedding given by $P_\Delta$ followed by a projection. In the singular case, the analog of $P_\Delta$ need not give an embedding. However, when $\Delta$ is a toric surface, then it is. Hence the discussion of embeddings and projections given in Section \[projective\] applies to *any* toric surface.
Finally, we remark that while toric resolutions are in general not unique, in the surface case one can always find a minimal resolution which is unique up to isomorphism. It follows that we have a canonical choice of universal rational parametrization when $\Delta$ is a lattice polygon.
Rational Maps to Smooth Toric Varieties {#rational}
=======================================
In order to prove the results of Sections \[projective\] and \[singular\], we need to study rational maps to an abstract toric variety. So in this section we will assume that $X_\Sigma$ is a compact toric variety, possibly non-projective.
In algebraic geometry, there is a well-defined notion of a rational map between irreducible varieties, regardless of whether they are affine, projective or defined abstractly like $X_\Sigma$. Our goal here is to describe *all* rational maps $$\CC^d -\! -\!\hskip-1.5pt \to X_\Sigma$$ when $X_\Sigma$ is a smooth toric variety. Recall that this means that the generators of every $n$-dimension cone of $\Sigma$ are a $\ZZ$-basis of $\ZZ^n$.
The natural candidate for the universal rational map to $X_\Sigma$ is the rational map $$\pi: \CC^r -\! -\!\hskip-1.5pt \to X_\Sigma$$ of . So we need to explain what universal means in this context.
Given a polynomial map $F : \CC^d \to \CC^r$ such that $F$ is $\Sigma$-irreducible, we will show in Section \[theory\] that the composition $$\label{picircf}
\pi\circ F :\CC^d -\! -\!\hskip-1.5pt \to X_\Sigma.$$ is a well-defined rational map. One of the key assertions of Theorem \[ratmap\] below is that this gives *all* rational maps from $\CC^d$ to $X_\Sigma$.
However, the map $F$ in is not unique. Recall from that we have the subgroup $G \subset (\CC^*)^r$ which is used in the quotient representation of $X_\Sigma$. If $F =
(f_1,\dots,f_r) \in R^r$ is $\Sigma$-irreducible and $\mu =
(\mu_1,\dots,\mu_r)\in G$, then $$\mu\cdot F = (\mu_1 f_1,\dots,\mu_r f_r)$$ is also $\Sigma$-irreducible and gives the same rational map as $F$ when composed with $\pi$ (because of the quotient ). Another key assertion of Theorem \[ratmap\] is that this is the *only* way that two $\Sigma$-irreducible $F$’s can give the same $\pi\circ F$. Thus we have complete control of the lack of uniqueness.
We can now state the main result of this section. Let $R =
\CC[y_1,\dots,y_d]$.
\[ratmap\] Let $X_\Sigma$ be a smooth compact toric variety. Then:
1. If $F = (f_1,\dots,f_r) \in R^r$ is $\Sigma$-irreducible, then $\pi\circ F$ gives a well-defined rational map $\pi\circ F : \CC^d -\!
-\!\hskip-1.5pt \to X_\Sigma$.
2. If $F$ and $F'$ are $\Sigma$-irreducible, then $\pi\circ F =
\pi\circ F'$ as rational maps if and only if $F' = \mu\cdot F$ for some $\mu \in G$.
3. Finally, every rational map $f : \CC^d -\! -\!\hskip-1.5pt \to
X_\Sigma$ is of the form $\pi\circ F$ for some $\Sigma$-irreducible $F
\in R^r$.
Hence rational maps $f : \CC^d -\! -\!\hskip-1.5pt \to X_\Sigma$ correspond bijectively to $G$-equivalence classes of $\Sigma$-irreducible $(f_1,\dots,f_r) \in R^r$.
The proof will be given in Section \[theory\]. Here is an example of Theorem \[ratmap\].
\[p1p1b\]
Let $X_\Sigma$ be the toric variety of Example \[p1p1b2\]. There, we saw that $F = (f_{1},\dots,f_{5})$ is $\Sigma$-irreducible provided $$\gcd(f_{1},f_{3}) = \gcd(f_{1},f_{4}) = \gcd(f_{2},f_{4}) =
\gcd(f_{2},f_{5}) = \gcd(f_{3},f_{5}) = 1$$ and that $$G = \{(\lambda,\mu,\nu,\lambda \mu, \mu\nu) \mid \lambda,\mu,\nu \in
\CC^{*}\}.$$ By Theorem \[ratmap\], it follows that rational maps to $X_{\Sigma}$ are all of the form $\pi\circ F$, where $F$ is unique up to $(\lambda,\mu,\nu,\lambda \mu, \mu\nu)\cdot F$.
Let’s look at the specific example of the map $F' : \CC^{2} \to
\CC^{5}$ defined by $$\label{badmap}
F'(u,v) = (uv,1,u,v,1)$$ This induces a rational map $\pi\circ F' : \CC^2 -\! -\!\hskip-1.5pt
\to X_\Sigma$. However, $F'$ is not $\Sigma$-irreducible. To get a $\Sigma$-irreducible representation, observe that $$\label{bad2}
F' = (uv,1,u,v,1) = (uv,u^{-1},u,v,1)\cdot (1,u,1,1,1).$$ Since $(uv,u^{-1},u,v,1) \in G$ for $u,v \ne 0$, we see that $F'$ and $F = (1,u,1,1,1)$ give the same rational map. Since $F$ is $\Sigma$-irreducible, this is the representation given by Theorem \[ratmap\].
Notice that even though is given by polynomials, shows that it is not a polynomial multiple of the $\Sigma$-irreducible representation $F = (1,u,1,1,1)$.
We can also look at from the point of view of Theorem \[upthm\] and Corollary \[univcor\]. If we compose with $P_\Delta$ from Example \[p1p1b2\], we obtain $$H' = (u^2 v^2, u^3 v^2, u^4 v^2, u v^2, u^2 v^2,u^3 v^2, u v^2, u^2 v^2),$$ which satisfies the hypothesis of Corollary \[univcor\]. Factoring out the gcd $uv^2$, we can write this as $$H' = uv^2 (u, u^2, u^3, 1, u, u^2, 1, u) = uv^2 H.$$ Furthermore, one easily computes that $H = P_\Delta(1,u,1,1,1)$. Thus $$H' = uv^2 P_\Delta(1,u,1,1,1).$$ Notice also that unlike , the representation given by Corollary \[univcor\] involves only polynomials.
Finally, we observe that the smoothness assumption in Theorem \[ratmap\] is necessary, as shown by the following example.
\[crazyex\]
Consider the weighted projective plane $\PP(1,1,2)$. Here, represents this toric variety as the quotient of $\CC^3\setminus\{0\}$ under the action of $\CC^*$ given by $\lambda\cdot (x,y,z) = (\lambda x,\lambda y,\lambda^2 z)$. Then consider the rational map $$\CC^2 -\! -\!\hskip-1.5pt \rightarrow \PP(1,1,2)$$ defined by $$\label{crazy}
(u,v) \mapsto (\sqrt{u},\sqrt{u},v).$$ This looks crazy, but notice that $$(-1)\cdot (\sqrt{u},\sqrt{u},v) = ((-1)\sqrt{u},(-1)\sqrt{u},(-1)^2v)
= (-\sqrt{u},-\sqrt{u},v).$$ In fact, one can prove that gives a well-defined rational map whose image is the curve $x = y$ in $\PP(1,1,2)$. Since this map cannot be written in the form $\pi\circ F$ where $F$ consists of polynomials, we see that Theorem \[ratmap\] fails in this case.
We should also mention that this example is a version of Examples \[singex1\] and \[singex4\] in disguise. In fact, the “crazy” rational map is exactly the map we used in Example \[singex1\] to show that Theorem \[upthm\] fails for singular toric varieties. Recall also that we gave a purely polynomial version of this map by using the toric resolution described in Example \[singex4\].
Theoretical Justification {#theory}
=========================
The purpose of this section is to prove the three main results of this paper, Theorems \[singupthm\], \[upthm\] and \[ratmap\]. We begin with Theorem \[ratmap\] since it will be used to prove the other two theorems.
Suppose that $F = (f_1,\dots,f_r)$ is $\Sigma$-irreducible. The discussion following Definition \[irreddef\] implies that there is some $\sigma \in \Sigma$ such that $f_{i} \ne 0$ for all $\rho_{i}
\not\subset \sigma$. Thus $x^{\hat\sigma} = \Pi_{\rho_{i} \not\subset
\sigma} x_{i}$ does not vanish on the image of $F$, so that the image of $F$ is not contained in the exceptional set $Z =
\mathbf{V}(x^{\hat\sigma} \mid \sigma \in \Sigma)$. This shows that $V = F^{-1}(Z)$ is a proper subvariety of $\CC^{d}$. It follows that $F$ induces a map $$\CC^{d} \setminus V \xrightarrow{\ F\ } \CC^r \setminus Z
\xrightarrow{\ \pi\ } (\CC^r \setminus Z)/G = X_\Sigma.$$ Since $\CC^{d} \setminus V$ is a nonempty Zariski open subset of $\CC^{d}$, we get a well-defined rational map $\pi\circ F : \CC^d -\!
-\!\hskip-1.5pt \to X_\Sigma$. This proves assertion (1) of the theorem.
Before proving (2), we need to describe $Z$. Since $\mathbf{V}(x^{\hat\sigma})$ is a union of coordinate hyperplanes, their intersection $Z$ is a union of coordinate subspaces $W_1\cup
\dots \cup W_s$. Let one of these be $W_j =
\mathbf{V}(x_{i_1},\dots,x_{i_k})$. Suppose that $\rho_{i_1},\dots,\rho_{i_k} \subset \sigma$ for some $\sigma \in
\Sigma$. Then the point $a \in \CC^r$ with $a_{i_1} = \dots = a_{i_k}
= 0$ and $a_i = 1$ otherwise lies in $\mathbf{V}(x_{i_1},\dots,x_{i_k}) \subset Z$, yet $x^{\hat\sigma}$ is nonvanishing at $a$. This contradiction shows that no cone of $\Sigma$ contains $\rho_{i_1},\dots,\rho_{i_k}$.
Note that $V = F^{-1}(Z) = F^{-1}(W_1)\cup \dots \cup
F^{-1}(W_s)$, and if we write $W_j =
\mathbf{V}(x_{i_1},\dots,x_{i_k})$ as above, then $$F^{-1}(W_j) = \mathbf{V}(f_{i_1},\dots,f_{i_k}).$$ The above paragraph and Definition \[irreddef\] imply that $\mathrm{gcd}(f_{i_1},\dots,f_{i_k}) = 1$. This shows that each $F^{-1}(W_j)$ has codimension at least 2 in $\CC^{d}$, so that the same is true for their union $V$.
Now we can prove uniqueness. Suppose that another $\Sigma$-irreducible $F' = (f'_{1},\dots,f'_{r})$ gives the same rational map $f$. This means the following. Let $V' =
(F')^{-1}(Z)$. Then as above $V'$ has codimension at least 2 and the induced rational map $f'$ is defined on $\CC^{d}\setminus
V'$. Then $f = f'$ as rational maps implies that $f = f'$ as functions on $U = \CC^{d}\setminus(V\cup V')$. Since $X_\Sigma =
\big(\CC^r\setminus Z\big)/G$, this means that for each $y
\in U$, there is $\mu(y) \in G$ such that $\mu(y) \cdot F(y) = F'(y)$. Since $X_\Sigma$ is smooth, the quotient map $\CC^r\setminus Z \to X_\Sigma$ is a smooth fibration with fibers isomorphic to $G$. This implies that the map $y \mapsto
\mu(y)$ is an algebraic map $U \to G$.
Using $G \subset (\CC^*)^r$, $\mu$ gives maps $\mu_i : U \to \CC^*$ such that $f_i(y) = \mu_i(y) f'_i(y)$ for all $y \in U$. Now comes the key point: since $V \cup V'$ has codimension at least 2, $\mu_i$ must be constant. (To see this, write $\mu_i = A/B$, where $A,B$ are relatively prime polynomials. Then $A$ nonconstant $\Rightarrow$ the zeros of $\mu_i$ have codimension 1 and $B$ nonconstant $\Rightarrow$ the poles of $\mu_i$ have codimension 1. But $\mu_i$ is defined and nonzero outside a set of codimension at least 2.)
Hence the $\mu_i$ are constant. It follows $\mu \in G$ is also constant, and then $\mu$ gives the desired equivalence between $(f_{1},\dots,f_{r})$ and $(f'_{1},\dots,f'_{r})$. This completes the proof of (2).
It remains to show that all rational maps from $\CC^d$ to $X_\Sigma$ arise this way. So let $f : \CC^d -\! -\!\hskip-1.5pt \to X_\Sigma$ be a rational map. This is defined on some nonempty Zariski open subset of $\CC^{d}$, and the union $U$ of all such subsets is the maximal Zariski open subset on which $f$ is defined. The base points of $f$ are the complement $\CC^d \setminus U$, and since we are mapping into a compact space, the base points have codimension at least 2.
First consider the case when $f(U) \cap (\CC^*)^n \ne \emptyset$. Recall that $X$ has divisors $D_1,\dots,D_r$ corresponding to $x_1,\dots,x_r$. Since $X_\Sigma \setminus (\CC^*)^n = \bigcup_i
D_i$, it follows that $f(U) \not\subset D_i$ for all $i$. Thus $f^{-1}(D_i) \subset U$ is a proper subvariety (possibly empty) for each $i$. If $f^{-1}(D_i) = \emptyset$, set $f_i = 1$. Now suppose that $f^{-1}(D_i) \ne \emptyset$. Since $X_\Sigma$ is smooth, $D_i
\subset X_\Sigma$ is locally defined by a single equation, say $h =
0$, and then $f^{-1}(D_i) \subset U$ is defined locally by $h\circ f =
0$. It follows that every irreducible component of $f^{-1}(D_i)$ in $U$ has codimension 1, although the components may have multiplicities. Now, using $U \subset \CC^d$, we get the Zariski closure $Z_i \subset \CC^d$ of $f^{-1}(D_i)\subset U$. The irreducible components of $Z_i$ also have codimension 1, with the same multiplicities. It follows that there is $f_i \in R$, unique up to a constant, such that $\mathbf{V}(f_i) = Z_i$ with the same multiplicities.
We claim that $(f_1,\dots,f_r)$ is $\Sigma$-irreducible. Suppose that $\rho_{i_1},\dots,\rho_{i_k}$ are contained in no cone of $\Sigma$. Then the relation between cones and divisors implies that $D_{i_1}
\cap \dots \cap D_{i_k} = \emptyset$ in $X_\Sigma$. Thus, in $U$, we have $$f^{-1}(D_{i_1}) \cap \dots \cap f^{-1}(D_{i_k}) = \emptyset.$$ Since $\mathbf{V}(f_i) \cap U = f^{-1}(D_{i})$ for all $i$, it follows that $$\label{capu}
\mathbf{V}(f_{i_1},\dots,f_{i_k}) \cap U = \emptyset.$$ Hence $\mathbf{V}(f_{i_1},\dots,f_{i_k}) \subset \CC^d\setminus U$. Since $\CC^d\setminus U$ has codimension at least 2, this implies that $\mathrm{gcd}(f_{i_1},\dots,f_{i_k}) = 1$. Thus $(f_1,\dots,f_r)$ is $\Sigma$-irreducible. It follows that $(c_1 f_1,\dots,c_r f_r)$ is $\Sigma$-irreducible whenever $c_i \in \CC^*$. This will be useful below.
We next claim that there are $c_i \in \CC^*$ such that $(c_1
f_1,\dots,c_r f_r)$ gives the rational map $f$. Let $f'$ be the rational map determined by $(f_1,\dots,f_r)$. Using and our earlier description of $F^{-1}(Z)$, one easily shows that $f'$ is defined on $U$. Furthermore, the $f_i$ were defined so that in $U$, we have $$\label{ffp}
(f')^{-1}(D_i) = f^{-1}(D_i)$$ for all $i$. This equality also gives the correct multiplicities.
Now take a $n$-dimensional cone $\sigma \in \Sigma$. This gives the affine toric variety $U_\sigma \subset X_\Sigma$, and one easily sees that $U_\sigma = X_\Sigma \setminus \bigcup_{\rho_i \not\subset
\sigma} D_i$. Then implies that $(f')^{-1}(U_\sigma) =
f^{-1}(U_\sigma)$. Call this $U'_\sigma$ and note that $U'_\sigma \ne
\emptyset$ since $f(U) \cap (\CC^*)^n \ne \emptyset$. Thus $f$ and $f'$ give maps $U'_\sigma \to U_\sigma$. But since $X_\Sigma$ is smooth, we have $U_\sigma \simeq \CC^n$. We may assume that the edges of $\Sigma$ are labeled so that $\rho_1,\dots,\rho_n$ are the edges of $\sigma$. Then write $$\begin{aligned}
f{{\lower1pt\hbox{$|$}}_{\raise.5pt\hbox{${\scriptstyle U'_\sigma}$}}} &= (h_1,\dots,h_n) : U'_\sigma \to \CC^n\\
f'{{\lower1pt\hbox{$|$}}_{\raise.5pt\hbox{${\scriptstyle U'_\sigma}$}}} &= (h_1',\dots,h_n') : U'_\sigma \to \CC^n.\end{aligned}$$ We have set things up so that $D_i\cap U_\sigma$ is defined by the vanishing of the $i$th coordinate. Since respects multiplicities, we see that $h_i'/h_i = \beta_i$ is a nonvanishing function on $U'_\sigma$. Thus $$\beta_\sigma = (\beta_1,\dots,\beta_n) : U'_\sigma \longrightarrow
(\CC^*)^n$$ is an algebraic map which satisfies $$\beta_\sigma(y) \cdot f(y) = f'(y)$$ for all $y \in U'_\sigma$. If $\tau$ is another $n$-dimensional cone, then we get $\beta_\tau$ defined on $U'_\tau$ with similar properties. However, for any $y$ in the nonempty open subset $f^{-1}((\CC^*)^n)
\subset U'_\sigma \cap U'_\tau$, there is a unique element of $(\CC^*)^n$ which takes $f(y)$ to $f'(y)$. It follows easily that $\beta_\sigma = \beta_\tau$ on $U'_\sigma \cap U'_\tau$. Furthermore, the $U'_\sigma$ cover $U$ since the $U_\sigma$ cover $X$. It follows that we get an algebraic map $$\beta : U \longrightarrow (\CC^*)^n$$ with the property that $$\beta(y) \cdot f(y) = f'(y)$$ for all $y \in U$. But arguing as above, we see that $\beta$ must be constant since $\CC^d\setminus U$ has codimension at least 2. Thus there is $\beta \in (\CC^*)^n$ such that $\beta\cdot f(y) = f'(y)$ for all $y \in U$.
Since $(\CC^*)^r/G = (\CC^*)^n$, we can pick $(c_1,\dots,c_r) \in
(\CC^*)^r$ which maps to $\beta \in (\CC^*)^n$. We conclude that $(c_1 f_1,\dots,c_r f_r)$ is $\Sigma$-irreducible and gives $f$.
Finally, we need to discuss what happens when our rational map $f$ satisfies $f(U) \cap (\CC^*)^n = \emptyset$. Here, the idea is that there is a smallest torus orbit which meets $f(U)$. The Zariski closure of this orbit will be $D_{i_1}\cap \cdots\cap D_{i_\ell}$ where $\rho_{i_1},\dots,\rho_{i_\ell}$ are the edges of some $\sigma_0
\in \Sigma$. Let $\mathrm{orb}(\sigma_0)$ denote this orbit. Then make the following changes in the above proof:
1. First, let $f_{i_1} = \dots = f_{i_\ell} = 0$.
2. Second, replace $(\CC^*)^n$ with $\mathrm{orb}(\sigma_0)$
3. Third, for $\rho_i \not\subset \sigma_0$, pick $f_i$ so that $\mathbf{V}(f_i) \cap U = f^{-1}(D_{i})$ (with the same multiplicities).
4. Fourth, use $n$-dimensional cones $\sigma$ which contain $\sigma_0$ as a face.
With these changes, the above argument shows that $f$ comes from a $\Sigma$-irreducible element of $R^r$. We omit the details.
[In the existence part of the above proof, notice that the set $U$ was the maximal open subset of $\CC^d$ on which $f$ was defined. Yet the $f'$ we constructed was defined on a potentially bigger set, namely $\CC^d\setminus F^{-1}(Z)$. Once we prove $\beta\cdot f = f'$, it follows that $U = \CC^d\setminus
F^{-1}(Z)$. Using this, we obtain the following corollary.]{}
Let $f : \CC^d -\! -\!\hskip-1.5pt \to X_\Sigma$ be induced by a $\Sigma$-irreducible $F = (f_1,\dots,f_r) \in R^r$. Then the maximal open subset of $\CC^d$ on which $f$ is defined is given by $$U = \CC^d\setminus F^{-1}(Z).$$
We now turn to the proof of the existence of universal rational parametrizations for smooth toric projective toric varieties.
To prove (1), first assume that $F = (f_{1},\dots,f_{r})$ is $\Sigma_{\Delta}$-irreducible. We need to prove that the polynomials $\sum_{m} a_{im} f^{m}$ are relatively prime. So suppose that an irreducible polynomial $q \in R$ divides all of them.
We will use the interpretation of $\Pi : X_\Delta \to X$ as the composition of the embedding $X_\Delta \subset \PP^\ell$ given by $\Pi_\Delta$ followed by a projection. In particular, if $L$ is the center of the projection, then $X_\Delta \cap L = \emptyset$ since $\Pi$ is strictly defined on $X_\Delta$.
Suppose that we have $a \in \CC^{d}$ such that $q(a) = 0$. If $F(a)
\in \CC^{r} \setminus Z$, then $p(F(a))$ gives a point in $X_{\Delta}\cap L$, which is empty by assumption. It follows that $F(\mathbf{V}(q)) \subset Z$. Since $q$ is irreducible, $F(\mathbf{V}(q))$ must lie in some irreducible component of $Z$. By the proof of Theorem \[ratmap\], it follows that $F(\mathbf{V}(q))
\subset \mathbf{V}(x_{i_1},\dots,x_{i_k})$, where no cone of $\Sigma$ contains $\rho_{i_1},\dots,\rho_{i_k}$. Thus $f_{i_{j}}$ vanishes on $\mathbf{V}(q)$, so that $q$ divides $f_{i_{j}}$. But this is impossible since $F$ is $\Sigma_{\Delta}$-irreducible. This completes the proof of (1).
Next suppose that $H = (h_{0},\dots,h_{s})$ is a rational parametrization of $X$ whose image meets $U \subset X$. This gives a rational map denoted $h :\CC^d -\!-\!\hskip-1.5pt \to X$. Since $\Pi
: X_{\Delta} \to X$ is birational and $\Pi^{-1}$ is defined on $U$, we get a rational map $$f = \Pi^{-1}\circ h : \CC^{d} -\!-\!\hskip-1.5pt \to X_{\Delta}.$$ By Theorem \[ratmap\], $f$ is induced by a $\Sigma_\Delta$-irreducible $F = (f_{1},\dots,f_{r}) \in R^{r}$. It follows that $H$ and $P\circ F$ give the same rational map $\CC^{d}
-\! -\!\hskip-1.5pt \to \PP^{s}$. Since both satisfy the gcd condition of Definition \[rpdef\], we see that $H = c\,P\circ F$ for some constant $c \ne 0$.
We claim that there is $\mu \in G$ such that $H = P\circ (\mu\cdot
F)$. Recall from that if $\mu =
(\mu_{1},\dots,\mu_{r}) \in G$, then $$\label{pamud}
P(\mu\cdot (x_{1},\dots,x_{r})) = \mu_{\Delta}
P(x_{1},\dots,x_{r}),$$ where $$\mu_{\Delta} = \prod_{i=1}^{r} \mu_{i}^{a_{i}}.$$ Assume for the moment that the map $$\label{gdelta}
G \longrightarrow \CC^{*}$$ defined by $\mu \mapsto \mu_{\Delta}$ is surjective. Then we can find $\mu \in G$ such that $\mu_{\Delta} = c$. It follows that $$H = c\,P\circ F = \mu_{\Delta} P\circ F =
P\circ (\mu\cdot F),$$ as claimed. Since $G_\Delta$ is the kernel of , the uniqueness assertion of Theorem \[ratmap\] easily implies that $\mu\cdot F$ is unique up to $G_\Delta$-equivalence.
We still need to prove that is surjective. Since this map is a character, its image is either finite or all of $\CC^*$. Furthermore, it is well-known that $G$ is connected since $X_{\Delta}$ is smooth. Hence the image is either the identity or $\CC^*$. So all we need to prove is that is nonconstant.
If the map is constant, then $\mu_{\Delta} = 1$ for all $\mu \in G$. We claim this implies the existence of $m \in M$ such that $$\label{badm}
\langle m,n_i\rangle = a_i\quad\text{for all}\ i = 1,\dots,r.$$ We prove this as follows. As explained in [@hc], the inclusion $G \subset (\CC^{*})^{r}$ induces an exact sequence $$1 \longrightarrow G \longrightarrow (\CC^{*})^{r} \xrightarrow{\ \phi\ }
(\CC^{*})^{n} \longrightarrow 1.$$ The map $\mu \to \mu_{\Delta} = \prod_{i=1}^{r} \mu_{i}^{a_{i}}$ extends to the character $(\CC^{*})^{r} \to \CC^{*}$ corresponding to $(a_1,\dots,a_r) \in \ZZ^r$. If $\mu \to \mu_{\Delta}$ is constant on $G$, then above exact sequence shows that it induces a character $\chi^{m} : (\CC^{*})^{n} \to \CC^{*}$. Since the map $\phi$ is dual to the inclusion $\ZZ^{n} \to \ZZ^{r}$ which sends $m$ to $(\langle
m,n_{1}\rangle, \dots, \langle m,n_{r}\rangle)$, it follows that $(a_{1},\dots,a_r) = (\langle m,n_{1}\rangle,\dots,\langle
m,n_{r}\rangle)$, as claimed. Thus is proved.
However, if we compare to , we see that $-m$ lies in every facet of $\Delta$, which is clearly impossible. This contradiction shows that must be nonconstant, and we are done.
Finally, we prove the existence of universal rational parametrizations for arbitrary projective toric varieties.
Recall that $P$ induces a rational map $p : \CC^{\tilde{r}}
-\!-\!\skip1.5pt \to \PP^s$ which factors $$\CC^{\tilde{r}} -\!-\!\skip1.5pt \to X_\Sigma \xrightarrow{\ \varphi\
} X_\Delta \xrightarrow{\ \Pi\ } X \subset \PP^s.$$ Furthermore, the argument preceding the statement of Theorem \[upthm\] shows that $p$ is a rational parametrization of $X$. From here, the proof of (1) is identical to the proof of the first part of Theorem \[upthm\].
As for (2), observe that the composition $$X_\Sigma \xrightarrow{\ \varphi\ } X_\Delta \xrightarrow{\ \Pi\ } X$$ is birational. Furthermore, since $\Pi^{-1}$ is defined on $U$ and $\varphi^{-1}$ is defined on the smooth part of $X_\Delta$, it follows that $$\widetilde{\Pi} : X_\Sigma \to X$$ is a birational morphism whose inverse is defined on $\widetilde{U}$. Hence, if a rational parametrization $H$ induces a rational map $h :
\CC^{d} -\!-\!\skip1.5pt \to X \subset \PP^s$ whose image meets $\tilde{U}$, then $$f = \widetilde{\Pi}^{-1} \circ h : \CC^{d} -\!-\!\skip1.5pt \to
X_\Sigma$$ is a rational map. As in the proof of Theorem \[upthm\], Theorem \[ratmap\] implies that $f$ is given by a $\Sigma$-irreducible $F$. This means that $H$ and $P\circ F$ agree up to a constant.
For the third assertion of the theorem, the proof follows from what we did in the proof of Theorem \[upthm\]. This completes the proof of the theorem.
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[^1]: This research was partially done while the third author served as a Clay Mathematics Institute Long-Term Prize Fellow.
|
---
abstract: 'In this paper, motivated by a problem in stochastic impulse control theory, we aim to study solutions to a free boundary problem of obstacle-type. We obtain sharp estimates for the solution using nonlinear tools which are independent of the modulus of semi-convexity of the obstacle. This allows us to state a general estimate for solutions to free boundary problems of obstacle-type admitting obstacles with a general modulus of semi-convexity. We provide two applications of our result. We consider penalized fully nonlinear obstacle problems and provide sharp decay estimates for Hölder norms and we prove sharp estimates for the solution to a fully nonlinear stochastic impulse control problem.'
author:
- Rohit Jain
title: The Fully Nonlinear Stochastic Impulse Control Problem
---
Introduction
============
Stochastic impulse control problems ([@BL82], [@L73], [@M76], [@F79]) are control problems that fall between classical diffusion control and optimal stopping problems. In such problems the controller is allowed to instantaneously move the state process by a certain amount every time the state exits the non-intervention region. This allows for the controlled process to have sample paths with jumps. There is an enormous literature studying stochastic impulse control models and many of these models have found a wide range of applications in electrical engineering, mechanical engineering, quantum engineering, robotics, image processing, and mathematical finance. A key operator in stochastic impulse control problems is the intervention operator\
$$\textnormal{M}u(x) = \inf_{\xi \geq 0} (u(x+\xi) + 1).$$\
The operator represents the value of the strategy that consists of taking the best immediate action in state $x$ and behaving optimally afterward. Since it is not always optimal to intervene, this leads to the quasi-variational inequality\
$$u(x) \leq \textnormal{M}u(x) \; \; \forall x \in \mathbb{R}^{n}.$$\
From the analytic perspective one obtains an obstacle problem where the obstacle depends implicitly and nonlocally on the solution. More precisely we can consider the classical stochastic impulse control problem\
$$\begin{cases}
\Delta u(x) \geq f(x)& \forall x \in \Omega.\\
u(x) \leq Mu(x) & \forall x \in \Omega.\\
u = 0 & \forall x \in \partial \Omega.
\end{cases}$$\
Here we let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with a $C^{2,\alpha}$ boundary $\partial \Omega$, $u \in W_{0}^{1,2}(\Omega)$, $f \in L^{\infty}(\Omega)$, $ f \geq 0$, and $$\textnormal{M}u(x) = \inf_{\stackrel{\xi \geq 0}{x + \xi \in \bar{\Omega}}}(u(x + \xi) + 1).$$ The assumption $f \geq 0$ implies that the solution $\bar{u}$ to the boundary value problem $\Delta \bar{u} = f$ in $\Omega$ with $\bar{u} \in H_{0}^{1}(\Omega)$ satisfies $ \bar{u} \geq 0$. This implies in particular that the set of solutions to $v \leq M\bar{u}$ with $v \in H_{0}^{1}(\Omega)$ is nonempty. This allows for an iterative procedure to prove existence and uniqueness of the solution ([@BL82]). We also point out that the sharp $C^{1,1}_{loc}$ estimate in the classical stochastic impulse control problem has been previously obtained ([@CF79a], [@CF79b]). In this paper we consider a fully nonlinear problem. We let $F(D^{2}u)$ be a fully nonlinear uniformly elliptic operator i.e. $\lambda\|P\| \leq F(A+P) - F(P) \leq \Lambda\|P\| \; \forall P \geq 0$ and $A,P \in \mathbf{S}(n)$, where $\mathbf{S}(n)$ is the set of all $n \times n$ real symmetric matrices. We also assume that the operator is either convex or concave in the hessian variable. We define $\varphi_{u}(x)$ to be a semi-convex function with a general modulus of semi-convexity $\omega(r)$. We consider the following boundary value problem.\
$$\begin{cases}
F(D^{2}u) \leq 0 & \forall x \in \Omega. \\
u(x) \geq \varphi_{u}(x) & \forall x \in \Omega. \\
u = 0 & \forall x \in \partial \Omega. \\
\end{cases}$$\
In this work we are interested in proving sharp estimates for fully nonlinear obstacle problems admitting obstacles with a general modulus of semi-convexity. The following are our main results in this paper,
Consider the fully nonlinear obstacle problem with obstacle $\varphi_{u}$, admitting a modulus of semiconvexity, $\omega(r)$. Then the solution $u$ has a modulus of continuity $\omega(r)$ up to $C^{1,1}(\Omega)$.
As an application we apply our result to obtain a sharp estimate for the solution to the following fully nonlinear stochastic impulse control problems,
Let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with a $C^{2,\alpha}$ boundary $\partial \Omega$. Define\
$$\textnormal{M}u(x) = \varphi(x) + \inf_{\stackrel{\xi \geq 0}{x + \xi \in \bar{\Omega}}}(u(x + \xi)).$$\
Here $\varphi(x)$ is $\omega(r)$ semiconcave, strictly positive, bounded, and decreasing in the positive cone $\xi \geq 0$. Consider the solution to the following fully nonlinear stochastic impulse control problem,\
$$\begin{cases}
F(D^{2}u) \geq 0 & \forall x \in \Omega.\\
u(x) \leq \textnormal{M}u(x) & \forall x \in \Omega.\\
u = 0 & \forall x \in \partial \Omega.
\end{cases}$$\
Then, the solution $u$ has modulus of continuity $\omega(r)$ up to $C^{1,1}(\Omega)$.
We remark that as a corollary of this theorem we recover the sharp $C^{1,1}$ estimate for the classical stochastic impulse control problem.\
\
We proceed in stages to prove the stated theorems. The main point of interest in the first theorem is to improve the modulus of continuity for the obstacle $\varphi_{u}$ on the contact set $\{u = \varphi_{u}\}$. In particular the goal is to obtain a uniform modulus of continuity $\omega(r)$ for $\varphi_{u}$ which we can then propogate to the solution $u$. The second theorem follows from the first theorem once we establish semiconcavity estimates for the nonlocal obstacle $\textnormal{M}u(x)$. Moreover we can extend the free boundary regularity from the classical implicit constraint problem as considered in [@J15C] under the assumption that the data is analytic and $\varphi(x) = 1$. Finally as an application of the previous results we consider a singularly perturbed fully nonlinear obstacle problem and show optimal decay rates for Hölder norm estimates.\
\
*Acknowledgements* I would like to express my sincerest gratitude and deepest apprecation to my thesis advisors Professor Luis A. Caffarelli and Professor Alessio Figalli. It has been a truly rewarding experience learning from them and having their guidance.
Lipschitz Estimates for the Solution
====================================
To obtain the optimal estimate we first prove initial regularity estimates which we hope to extend. We fix $f =0$. All the proofs may be modified for a nonzero sufficiently regular $f$. We begin by first proving that solutions are indeed continuous.
Let $u$ be a solution to (4) with semiconvex obstacle $\varphi_{u}$. Then $u \in C(\Omega)$.
The lemma follows from a result due to G.C. Evans.
If $u$ is continuous in $\{ u = \varphi_{u} \}$, then $u$ is continuous in $\Omega$.
The possibility of a discontinuity is limited to a point on the free boundary, $\partial \{u > \varphi_{u} \}$. Consider $x_{0} \in \{u = \varphi_{u} \}$ and without loss of generality assume $u(x_{0}) = 0$. Suppose by contradiction, that there exists a sequence of points $\{x_{k}\}$ with the following properties:\
1. $\{x_{k}\} \to x_{0}$.\
2. $\forall k$, $x_{k} \in \{u > \varphi_{u} \}.$\
3. $\mu = \lim_{x_{k} \to x_{0}} u(x_{k}) > u(x_{0}) = 0.$\
\
By lower semicontinuity of $u$, we know that $\forall \delta > 0$ there exists a neighborhood of $x_{0}$ such that $u \geq -\delta$ for $\delta << \mu$. We consider $$r_{k} = \textnormal{dist} [ x_{k}, \{u = \varphi_{u} \} ].$$ For a large enough $k$ we can ensure that:\
1. $u(x) + \delta \geq 0 \; \; \textnormal{in} \; B_{r_{k}}(x_{k}).$\
2. $u(x_{k}) + \delta \geq \frac{\mu}{2}.$\
\
Moreover we know that $u(x) + \delta$ satisfies the equation in $B_{r_{k}}(x_{k})$. By the Harnack Inequality we obtain, $$\frac{\mu}{2} \leq u(x_{k}) + \delta \leq C \inf_{B_{\frac{r_{k}}{2}(x_{k})}} (u + \delta).$$ This implies for some $C_{0} > 0$ universal, $$\inf_{B_{\frac{r_{k}}{2}(x_{k})}} u \geq C_{0} \mu.$$
Since $u$ is also superharmonic, we know from the weak Harnack Inequality in $B_{4r_{k}}(y_{k})$ that, $$\begin{split}
u(y_{k}) & \geq c \left( \fint_{B_{2r_{k}}(y_{k})} u^{p} \right)^{\frac{1}{p}} \\
& = \frac{c}{|B_{2r_{k}}|^{1/p}} \left( \int_{B_{2r_{k}}(y_{k}) \setminus B_{\frac{r_{k}}{2}}(x_{k})} u^{p} + \int_{B_{\frac{r_{k}}{2}}(x_{k})} u^{p} \right)^{\frac{1}{p}} \\
& \geq \frac{c}{|B_{2r_{k}}|^{1/p}} \left(-(\delta)^{p} |B_{2r_{k}}| + (C_{0} \mu)^{p} |B_{\frac{r_{k}}{2}}| \right)^{\frac{1}{p}} \\
& \geq C_{1} \mu \; \; \; \textnormal{for} \; C_{1} > 0.
\end{split}$$
On the other hand $u(y_{k}) = \varphi_{u}(y_{k})$ and $y_{k} \to x_{0}$. This implies in particular that,\
1. $\varphi_{u}(y_{k}) \geq C_{1} \mu.$\
2. $\varphi(x_{0}) = u(x_{0}) = 0.$\
This is our desired contradiction.
We observe that the conditions on the obstacle may be relaxed in the proof of this lemma. In fact continuity of the obstacle is sufficient.
A generic semiconvex function with a general modulus of semiconvexity is known to be Lipschitz in the interior. We extend the previous result to show that solutions to a fully nonlinear obstacle problem admitting obstacles with a Lipschitz modulus of continuity grow from the free boundary with a comparable rate.
Let $u$ be a solution to (4) with semiconvex obstacle $\varphi_{u}$. Fix $0 \in \partial\{u > \varphi_{u}\}.$ Then $$\sup_{B_{r}(0)} u(x) \leq Cr.$$
Let $\gamma(r)$ denote the Lipschitz modulus of continuity for the obstacle $\varphi_{u}$ in $B_{r}(0)$. The obstacle condition $u \geq \varphi_{u}$ implies in $B_{r}(0)$ that $$u \geq \varphi_{u}(0) - \gamma(r).$$
Define $$v(x) = u - (\varphi_{u}(0) - \gamma(r)).$$
We note that $F(D^{2}v) = F(D^{2}u) \leq 0$ and $F(D^{2}v) = 0$ inside $\{u > \varphi_{u}\}.$ We consider $x \in B_{r/4}(0) \cap \{u > \varphi_{u}\}.$ Moreover we let $y$ be the closest free boundary point to $x$. Let $\rho$ be the distance of $x$ to its closest free boundary point $y$. From the Weak Harnack Inequality it follows, $$v(y) \geq C \left(\fint_{B_{2\rho(y)}} v^{p} \right)^{1/p}.$$
By the positivity of $v$ and Harnack Inequality in $B_{\rho(x)}$, it follows that the right hand side, $$\geq C \left( \frac{B_{\rho}(x)}{B_{2 \rho(y)}} \fint_{B_{\rho(x)}} v^{p} \right)^{1/p} \geq Cv(x).$$
Recall that $|\varphi_{u}(0) - \varphi_{u}(y)| \leq \gamma(r).$ Hence, $$0 \leq v(y) = \varphi_{u}(y) - \varphi_{u}(0) + \gamma(r) \leq 2\gamma(r).$$
Changing back to our solution $u$, we find, $$0 \leq u(x) - (u(0) - \gamma(r)) \leq Cv(y) \leq C \gamma(r).$$
In particular, $$u(x) - u(0) \leq C \gamma(r).$$
Optimal $C^{\omega(r)}$ Estimates for the Solution
==================================================
In the previous section we assumed that the obstacle had a uniform modulus of continutity. A priori for semi-concave functions you only know that the uniform modulus of continuity is Lipschitz. Our goal as in the classical case will be to study the interplay between the equation and the obstacle to improve regularity estimates for the obstacle on the contact set. We start this section by stating and proving a lemma in the particular case that our operator is the Laplacian. The motivating calculation will help us proceed to prove the desired estimate in the more general case. The content of the lemma says that for any given point $x_{1} \in \{u(x) > \varphi_{u}(x) \}$, $\exists x_{0} \in \{u(x) = \varphi_{u}(x)\}$ such that the solution grows at most by $\omega(2|x_{1} - x_{0}|)$ where $\omega(|x_{1} - x_{0}|)$ denotes the modulus of semiconvexity for the obstacle on the ball $B_{|x_{1} - x_{0}|}(x_{0})$. Since a lower estimate is available via the obstacle, what we aim to show is that around a fixed contact point the modulus of continuity of the solution is controlled by the modulus of semiconvexity of the obstacle.
Let $\varphi_{u}(x)$ be a semiconvex function with general modulus of semiconvexity $\omega(r)$. Consider the following obstacle problem: $$\begin{cases}
\Delta u \leq 0 & \forall x \in \Omega.\\
u(x) \geq \varphi_{u}(x) & \forall x \in \Omega.\\
u = 0 & \forall x \in \partial \Omega
\end{cases}$$
Fix $x\in \{u(x) > \varphi_{u}(x) \}$ and define $L_{x_{0}}(x) = \varphi_{u}(x_{0}) + \langle p, x-x_{0} \rangle$, the linear part of the obstacle at the point $x_{0}$. Then $\exists x_{0} \in \{u(x) = \varphi_{u}(x)\}$ and $C(n) >0$ such that $u(x) - L_{x_{0}}(x) \leq C(n)\omega(2|x - x_{0}|)$.
We fix $x_{1} \in \{u(x) > \varphi_{u}(x) \}$. Let $x_{0}$ denote the closest point to $x_{1}$ in $\{u = \varphi_{u} \}$. We denote this distance by $\rho = |x_{1} - x_{0}|$. Define $w(x) = u(x) - L_{x_{0}}(x)$. Using the mean value theorem for superharmonic functions in $B_{2\rho}(x_{0})$ we have, $$\begin{split}
0 & = w(x_{0}) \geq \frac{1}{\alpha(n)2^{n}\rho^{n}} \int_{B_{2\rho}(x_{0})} w(y) \;dy\\
& = K(n)\int_{B_{2\rho} (x_{0}) \smallsetminus B_{\rho} (x_{1})} w(y) \; dy \; + K(n) \int_{B_{\rho} (x_{1})} w(y) \; dy.
\end{split}$$
Semiconvexity of $w(x)$ in $B_{2\rho}(x_{0})$ and an application of the mean value theorem for harmonic functions in $B_{\rho}(x_{1})$ implies, $$\begin{split}
& \geq K(n) \int_{B_{2\rho} (x_{0}) \smallsetminus B_{\rho} (x_{1})} - \omega(2|y-x_{0}|) + C_{1}(n) w(x_{1})\\
& \geq -\tilde{C}(n)\omega(2\rho) + C_{1}(n) w(x_{1}).
\end{split}$$
In particular we obtain the desired bound, $$w(x_{1}) \leq C(n) \omega(2\rho).$$
We now look to generalize the previous argument in the fully nonlinear setting. In the preceding proof the lower bound on the obstacle was transferred to the solution at the contact point. Moreover we were able to renormalize the solution by subtracting off a linear part. We will also need a generalization of the mean value theorem that was used to connect pointwise information with information about the measure $\Delta u$.\
\
We consider again (4). For clarity we set $\omega(r) = \bar{C}r^{2}$ for some positive constant $\bar{C} > 0$. The arguments presented below can be trivially modified for the general semiconvex modulus by an appropriate rescaling. We make a remark in this direction towards the end of this section.
Let $x_{1} \in \{u > \varphi_{u} \}$. Then $\exists x_{0} \in \{u = \varphi_{u} \}$ such that for $w(x) = u(x) - L_{x_{0}}(x)$, where $L_{x_{0}}(x) = \varphi_{u}(x_{0}) + \langle p, x-x_{0} \rangle$ denotes the linear part of the obstacle at the point $x_{0}$, and a universal constant $K(n) > 0$, $$w(x_{1}) \leq K(n) |x_{1} - x_{0}|^{2}.$$
Fix $x_{1} \in \{u > \varphi_{u} \}$. Let $x_{0}$ be the closest point to $x_{1}$ in $\{u = \varphi_{u} \}$. We denote this distance by $\rho = |x_{1} - x_{0}|$. By the modulus of semiconvexity of the obstacle we know that $w(x) \geq -16\bar{C}\rho^{2}$ on $B_{4\rho}(x_{0})$. The idea of the proof is to zoom out to scale 1 and prove that the solution is bounded by a universal constant and then rescale back to obtain the desired bound. Consider the transformation $y = \frac{x-x_{0}}{\rho}$ and the scaled solution, $$v(y) = \frac{w(\rho y + x_{0})}{16\bar{C}\rho^{2}} + 1.$$
We note that $v(y)$ is a non-negative supersolution on $B_{4}(0)$ with $$\inf_{B_{4}(0)} v(y) \leq 1.$$
Moreover $v(y)$ is a solution in $B_{1}(y_{1})$, since $x_{0}$ is the closest point in the contact set to $x_{1}$. By the interior Harnack Inequality, $$v(y_{1}) \leq \sup_{B_{\frac{1}{2}}(y_{1})} v(y) \leq C \inf_{B_{\frac{1}{2}}(y_{1})} v(y).$$
We also know from the weak $L^{\epsilon}$ estimate for supersolutions that for universal constants $d$, $\epsilon$, $$|\{v \geq t\} \cap B_{2}(0)| \leq dt^{-\epsilon} \; \; \forall t >0.$$
We observe that $B_{\frac{1}{2}}(y_{1}) \subseteq B_{2}(0)$. Hence we can choose $t = t_{0}$ such that $$t_{0} = \left(\frac{\delta d}{|B_{\frac{1}{2}(y_{1})}|} \right)^{\frac{1}{\epsilon}},$$ for $\delta > 0$. It follows that, $$|\{v \leq t\} \cap B_{\frac{1}{2}}(y_{1})| > \delta |B_{\frac{1}{2}}(y_{1})| > 0.$$
Hence there exists a universal constant $C$ such that, $$v(y_{1}) \leq C.$$
This implies from (8), $$\frac{w(\rho y_{1} + x_{0})}{4\bar{C}\rho^{2}} \leq C.$$
Rescaling back we find, $$w(x_{1}) \leq K|x_{1} - x_{0}|^{2}.$$
We now prove a lemma that controls the oscillation of the solution between two arbitrary points on the contact set $\{u = \varphi_{u} \}$. As a corollary which we state after the proof, we improve the modulus of continuity for the obstacle $\varphi_{u}$ on the contact set $\{u = \varphi_{u}\}$.
Let $x_{1} \in \{u = \varphi_{u} \}$ and $x_{0} \in \{u = \varphi_{u} \}$. Then for $w(x) = u(x) - L_{x_{0}}(x)$, where $L_{x_{0}}(x) = \varphi_{u}(x_{0}) + \langle p, x-x_{0} \rangle$ denotes the linear part of the obstacle at the point $x_{0}$, and $K(n) > 0$ a universal constant, $$w(x_{1}) \leq K(n) |x_{1} - x_{0}|^{2}.$$
Assume by contradiction that for an arbitrary large constant $K > 0$ $$w > K|x_{1} - x_{0}|^{2}.$$
As before we denote the distance between the points by $\rho = |x_{1} - x_{0}|$. We begin with a claim.
$\exists \; \textnormal{Half Ball} \; HB_{\rho}(x_{1}) \; \textnormal{such that} \; \forall x \in HB_{\rho}(x_{1}), \; w(x) \geq \frac{K}{2} \rho^{2}.$
We define $\varphi_{w} = \varphi_{u} - L_{x_{0}}$, where as before $L_{x_{0}} = \varphi_{u}(x_{0}) + \langle p, x-x_{0} \rangle$ for $p \in D^{+} \varphi_{u}(x_{0})$ the superdifferential of $\varphi_{u}$ at the point $x_{0}$. We make the following observations:\
1. $w \geq \varphi_{w} \; \; \; \forall x \in B_{2\rho}(x_{0}).$\
2. $\varphi_{w}(x_{1}) = \varphi_{u}(x_{1}) - \varphi_{u}(x_{0}) + \langle p, x-x_{1} \rangle - \langle p, x-x_{0} \rangle \; \; \; \forall x \in B_{2\rho}(x_{0}).$\
\
In particular, $$w(x) \geq \varphi_{w}(x_{1}) + \varphi_{u}(x) - \varphi_{u}(x_{1}) - \langle p, x - x_{1} \rangle.$$
Now consider $d \in D^{+} \varphi_{u}(x_{1})$ and observe that $w(x_{1}) = \varphi_{w}(x_{1})$. This produces the following inequality, $$w(x) \geq w(x_{1}) + \varphi_{u}(x) - \varphi_{u}(x_{1}) - \langle d, x - x_{1} \rangle - \langle p-d, x - x_{1} \rangle.$$
By semiconvexity on $B_{\rho}(x_{1})$, (9), and fixing $x \in HB_{\rho}(x_{1}) = \{ x \in B_{\rho}(x_{1}) \; | \; \langle p-d, x - x_{1} \rangle \leq 0 \},$ we have, $$w(x) \geq K\rho^{2} - \bar{C} \rho^{2}.$$
We can choose $K$ large enough so that we obtain, $$w(x) \geq \frac{K}{2}\rho^{2}.$$
This is our desired half ball.
We now consider again the dilated solution $v(y)$ from the previous lemma (8). By the weak Harnack Inequality for supersolutions, $\exists C > 0$ universal and $\epsilon > 0$ such that, $$\int_{B_{4}(0)} |v(x)|^{\epsilon/2} \leq \left(C \inf_{B_{2}(0)}v(x)\right)^{\frac{2}{\epsilon}} \leq \left(Cv(0) \right)^{\frac{2}{\epsilon}} = C.$$
From our previous claim we obtain $$0 < |\{v(y) > \frac{K}{32\bar{C}} \} \cap B_{1}(y_{1})|.$$
Here $\bar{C}$ is our semiconvexity constant from before. We now have the following chain of inequalities, $$\begin{split}
0 & < |\{v(y) > \frac{K}{32\bar{C}} \} \cap B_{1}(y_{1})|\; (\frac{K}{32\bar{C}})^{\epsilon /2} \\
& = \int_{\{v(x) > \frac{K}{32\bar{C}} \} \cap B_{1}(y_{1})} (\frac{K}{32\bar{C}})^{\epsilon/2} \\
& \leq \int_{\{v(x) > \frac{K}{32\bar{C}} \} \cap B_{1}(y_{1})} |v(x)|^{\epsilon /2} \\
& \leq \int_{B_{4}(0)} |v(x)|^{\epsilon/2} \leq \left(C \inf_{B_{2}(0)}v(x) \right)^{\frac{2}{\epsilon}} \leq \left(Cv(0) \right)^{\frac{2}{\epsilon}} = C.
\end{split}$$
For $K$ large enough we obtain a contradiction. Hence for a universal constant $K(n) > 0$, $$w(x_{1}) \leq K(n) |x_{1} - x_{0}|^{2}.$$
Assume our obstacle is semiconvex on $B_{r}(x)$ with modulus of semiconvexity $\omega(r)$. We can translate our solution to the origin and scale by the modulus of semiconvexity of the obstacle. In particular, set $\rho$ to be the distance between our fixed points. $$v(y) =\frac{w(\rho y + x_{0})}{\omega(4\rho)}+1.$$
One can check that we get similar estimates in terms of the modulus of semiconvexity $\omega(\rho)$.
A corollary of the previous lemma is that on the contact set $\{u = \varphi_{u} \}$ the obstacle $\varphi_{u}$ has a modulus of continuity $\omega(r)$. In particular, $$\|\varphi_{u}\|_{C_{loc}^{\omega(r)}(\{u = \varphi_{u} \})} \leq C.$$
We can now state and prove a sharp estimate for our solutions.
Consider the boundary value problem (4) with semiconvex obstacle $\varphi_{u}$ admitting a modulus of semiconvexity, $\omega(r)$. Then the solution $u$ has modulus of continuity $\omega(r)$ up to $C^{1,1}(\Omega)$. In particuluar, $$\|u\|_{C^{\omega(r)}(\Omega)} \leq C.$$
To prove this theorem we consider three distinct cases.\
**Case 1**: $x_{1} \in \{ u > \varphi_{u}\}$, $x_{0} \in \{ u = \varphi_{u}\}$.\
Choose the closest point in the contact set to $x_{1}$ and call it $\bar{x}_{1}$. Then we apply Lemma 5 to obtain the correct oscillation estimate up to the free boundary. Then an application of Lemma 6 gives us the correct oscillation estimate between two contact points. Finally we use the triangle inequality to conclude.\
\
**Case 2**: $x_{1}$, $x_{0}$ $\in \{ u = \varphi_{u}\}$.\
This is the content of Lemma 6.\
\
**Case 3**: $x_{1}$, $x_{0}$ $\in \{ u > \varphi_{u}\}$.\
We distinguish two different subcases.\
*Case 3a*: $\max\{d(x_{1} , \{ u = \varphi_{u}\}), d(x_{0} , \{ u = \varphi_{u}\}) \} \geq 4|x_{1} - x_{2}|$.\
Suppose $\max\{d(x_{1} , \{ u = \varphi_{u}\}), d(x_{2} , \{ u = \varphi_{u}\}) \} = \rho$. Without loss of generality we assume that the maximum distance is realized at the point $x_{1}$. We observe that $B_{|x_{1}-x_{0}|}(x_{1}) \subseteq B_{\frac{\rho}{2}}(x_{1})$, and we consider $w = u - L_{x_{1}}$, where $L_{x_{1}}$ denotes the linear part of the solution at $x_{1}$. By an application of the Harnack Inequality we obtain, $$\sup_{B_{\rho}(x_{1})} w \leq C \inf_{B_{\rho / 2}(x_{1})} w \leq Cw(x_{2}) \leq C\omega(\rho).$$
Moreover we also appeal to the interior estimates for solutions to our fully nonlinear convex or concave operator, $F(D^{2}u) = 0$, $$\|w - w(x_{1}) \|_{C^{\omega(\rho)}(B_{\frac{\rho}{2}}(x_{1}))} \leq \frac{K}{\omega(\rho)}\|w - w(x_{1}) \|_{L^{\infty}(B_{\rho}(x_{1}))}.$$
Hence, $$\|w - w(x_{1}) \|_{C^{\omega(\rho)}(B_{\frac{\rho}{2}}(x_{1}))} \leq C.$$
*Case3b*: $\max\{d(x_{1} , \{ u = \varphi_{u}\}), d(x_{2} , \{ u = \varphi_{u}\}) \} < 4|x_{1}-x_{2}|$\
In this case one considers $\rho_{1} = d(x_{1} , \{ u = \varphi_{u}\})$ and $\rho_{0} = d(x_{0} , \{ u = \varphi_{u}\})$. Let $\bar{x}_{1}$ be the closest contact point to $x_{1}$ and $\bar{x}_{0}$ the closest contact point to $x_{0}$. We can apply Lemma 5 to obtain the desired oscillation estimate for each point up to the free boundary. We then apply Lemma 6 to control the oscillation between two contact points. Finally we apply the triangle inequality to conclude.
Application to Stochastic Impulse Control Theory
================================================
In the previous section we obtained a general estimate for fully nonlinear obstacle problems admitting an obstacle with a general modulus of semi-convexity. In this section we would like to apply the estimate to a particular fully nonlinear obstacle problem arising in stochastic impulse control theory. The idea is to prove that the given obstacle $Mu(x)$ is semi-concave with modulus of semiconcavity $ \omega(r)$. The strategy of the proof will follow the ideas presented in ([@CF79a]). We point out that the existence and uniqueness of a continuous viscosity solution to the fully nonlinear stochastic impulse control problem follows from introducing the Pucci Extremal Operators (see chapter 2 in [@CC95]) and adapting the arguments in ([@I95]). What one also needs to do is use Evans Lemma iteratively on a sequence of solutions to the fully nonlinear obstacle problem with continuous obstacle (See Remark 1).
Let $\varphi(x)$ be $\omega(r)$ semi-concave, strictly positive, bounded, and decreasing in the positive cone $\xi \geq 0$. Then the Obstacle $$Mu(x) = \varphi(x) + \inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi)$$ is semi-concave with modulus of semi-concavity $\omega(r)$.
We consider two distinct cases:\
1. $x_{0} \in \{u = Mu\}.$\
2. $x_{0} \in \{u < Mu\}.$\
\
**Case 1**: Fix $x_{0} \in \{u = Mu \}.$\
The proof in this case is based on characterizing the set where the infimum of $u$ occurs and establishing that this set is uniformly contained in the non-contact region $\{u < Mu \}$. This is the content of the following claims. We define the following sets:\
1. $\Sigma_{\geq x_{0}} = \{x_{0} + \xi \; : \; \xi \geq 0\}.$\
2. $\Sigma_{x_{0}} = \{\varphi(x_{0}) + u(x_{0} + \xi) = Mu(x_{0})\}.$\
\
The following claim characterizes $\Sigma_{x_{0}}$ as the set of points where $u$ realizes its infimum.
For every $y \in (\Sigma_{\geq x_{0}} \setminus \Sigma_{x_{0}})$ and for every $x \in \Sigma_{x_{0}}$, $u(x) \leq u(y).$
Fix $\bar{x} \in \Sigma_{x_{0}}$. Suppose by contradiction that $\exists x_{1} \in \Sigma_{\geq x_{0}} \setminus \Sigma_{x_{0}}$ such that $u(x_{1}) < u(\bar{x})$. This implies the following chain of inequalities,\
$$\begin{split}
\varphi(x_{0}) + u(x_{1}) & < \varphi(x_{0}) + u(\bar{x})\\
& = Mu(x_{0}) = \varphi(x_{0}) + \inf_{\stackrel{\xi \geq 0}{x_{0} + \xi \in \Omega}}u(x_{0} + \xi).
\end{split}$$\
In particular we obtain, $$u(x_{1}) < \inf_{\stackrel{\xi \geq 0}{x_{0} + \xi \in \Omega}}u(x_{0} + \xi).$$ This is our desired contradiction.
We now prove that pointwise the elements of $\Sigma_{x_{0}}$ are contained in the non-contact region, $\{u < Mu\}$.
Suppose the solution to the Boundary Value Problem $F(D^{2}\bar{u}) = 0$ satisfies $$\bar{u} < \inf_{\partial \Omega} \varphi.$$ Then $\forall x \in \Sigma_{x_{0}}$ it follows that $u(x) < Mu(x)$.\
Moreover in a neighborhood $N_{1}$ of $x$ we have $u \in C^{1,1}(N_{1})$
We observe that the first statement ensures that $\Sigma_{x_{0}} \cap (\partial \Omega) = \varnothing$. Suppose $x_{0} \in \Omega^{\circ}$, $x \in \partial \Omega$ and $x_{0} \leq x$. Then we observe,\
$$\begin{split}
Mu(x_{0}) & = u(x_{0}) \\
& \leq \bar{u}(x_{0}) < \inf_{\partial \Omega} \varphi \leq \varphi(x) + u(x) \leq \varphi(x_{0}) + u(x).
\end{split}$$\
The last inequality follows because $\varphi(x)$ is monotonically decreasing in the cone. Hence in particular $\Sigma_{x_{0}} \cap (\partial \Omega) = \varnothing$.\
\
Suppose now by contradiction that $\exists x \in \Sigma_{x_{0}}$ such that $u(x) = Mu(x).$ Then we have the following chain of inequalities,\
$$\begin{split}
u(x_{0}) & = Mu(x_{0}) \\
& = \varphi(x_{0}) + u(x)\\
& = \varphi(x_{0}) + Mu(x) \geq \varphi(x_{0}) + Mu(x_{0}) > Mu(x_{0})
\end{split}$$\
The last inequality follows from the strict positivity of the function $\varphi$. But we observe that the inequality contradicts the obstacle constraint $u(x_{0}) \leq Mu(x_{0}).$ Hence we have reached our desired contradiction.\
\
Finally the last statement of the claim follows from the continuity of $u$. The continuity of the solution implies that $\{u < Mu \}$ is an open set and thus in a small neighborhood $N_{1}$ of $x$, $u$ satisfies the equation, $F(D^{2}u) = 0.$ We can therefore apply interior regularity estimates to conclude.
We now strenghten the previous claim to obtain a uniform neighborhood of $\Sigma_{x_{0}}$ that is strictly contained in the non-contact region.
$\exists \delta_{0} >0$ such that $d \left \{\{u = Mu\}, \Sigma_{x_{0}}\right \} > \delta_{0}$.
Suppose by contradiction $\exists \{\delta_{k}\} \searrow 0$ and $\{x_{k}\} \subset \Sigma_{x_{0}}$, such that $$d(x_{k}, \{u = Mu\}) < \delta_{k}.$$ By definition, $x_{k} \in \Sigma_{x_{0}}$, implies $$\varphi(x_{0}) + u(x_{k}) = Mu(x_{0}) \; \; \forall k.$$ By the continuity of $u(x)$ this implies in particular that $\varphi(x_{0}) + u(\bar{x}) = Mu(x_{0})$ for some $\bar{x} \in \{u = Mu\}$. On the other hand, $\varphi(x_{0}) + u(\bar{x}) = Mu(x_{0})$ implies $\bar{x} \in \Sigma_{x_{0}}$. Hence from the previous claim we obtain, $$Mu(\bar{x}) = u(\bar{x}) < Mu(\bar{x}).$$ This is our desired contradiction.
We now state and prove a claim which allows us to redefine the obstacle in the neighborhood of a contact point.
For every $x, \bar{x} \in \Omega$, $\exists \delta >0$, such that if $|x -x_{0}| < \delta$, and $d(\bar{x}, \Sigma_{x_{0}}) >\delta$, then $u(x) < \varphi(x) + u(\bar{x})$. Moreover, if $x \in \{u = Mu \}$, then $\bar{x} \notin \Sigma_{x}$.
Suppose by contradiction that there exists a sequence of points $\{x_{k}\}$ and $\{\bar{x}_{k'}\}$ satisfying:\
1. $|x_{k}-x_{0}| = \delta_{k}.$\
2. $d(\bar{x}_{k'},\Sigma_{0}) > \delta_{k'} > 0$.\
3. $\{\delta_{k}\} \searrow 0$ and $\{\delta_{k'}\} \searrow 0$.\
4. $u(x_{k}) \geq \varphi(x_{k}) + u(\bar{x}_{k'}) \;$ $\forall k$ and $\forall k'$.\
\
We observe that from the previous claim $\exists k_{0}, k_{0}'$, such that $\forall k \geq k_{0}$ we have the following chain of inequalities,\
$$\begin{split}
Mu(x_{0} + \delta_{k}) & \leq Mu(\bar{x}_{k_{0}'}) \\
& \leq \varphi(\bar{x}_{k_{0}'}) + u(\bar{x}_{k_{0}'}) \\
& \leq \varphi(x_{0} + \delta_{k}) + u(\bar{x}_{k_{0}'}) \\
& \leq u(x_{0} + \delta_{k}) \leq Mu(x_{0} + \delta_{k}).
\end{split}$$\
Thus the above inequalities are all equalities. This implies $\forall k \geq k_{0}$, $$Mu(x_{0} + \delta_{k}) = \varphi(x_{0} + \delta_{k}) + u(\bar{x}_{k_{0}'}).$$ Letting $k \to \infty$ we obtain, $$Mu(x_{0}) = \varphi(x_{0}) + u(\bar{x}_{k_{0}'}).$$ Which implies in particular that $\bar{x}_{k_{0}'} \in \Sigma_{x_{0}}$. This is our desired contradiction.
From the last claim we can redefine the obstacle for $V_{\delta} = \{ |x-x_{0}| < \delta\}$. In particular by taking $\delta$ sufficiently small $\exists N_{2}$ neighborhood of $\Sigma_{x_{0}}$ such that, $$Mu(x) = \varphi(x) + \inf_{\stackrel{\xi \geq 0}{x + \xi \in N_{2}}}u(x + \xi).$$ For an even smaller $\delta$, $$Mu(x) = \varphi(x) + \inf_{\stackrel{\xi \geq 0}{x_{0} + \xi \in N_{3}}}u(x + \xi).$$ Where $N_{3}$ is such that, $$V_{\delta} + N_{3} - x_{0} \subseteq N_{1}.$$ Here $N_{1}$ is the neighborhood obtained in Claim 3. In particular for $x \in V_{\delta}$ and $\xi \in N_{3} - x_{0}$, we can bound the second incremental quotients. $$\delta^{2}u = u(x + h + \xi) + u(x - h + \xi) - 2u(x + \xi) \leq c|h|^{2}.$$ Moreover we know that for some $x + \bar{\xi}$ in $N_{1}$, we have, $$\inf_{\stackrel{\xi \geq 0}{x + \xi \in N_{1}}}u(x + \xi) = u(x + \bar{\xi}).$$ Now we consider the second incremental quotients of the obstacle $Mu(x).$ By the semi-concavity of $\varphi$ we obtain,\
$$\begin{split}
\delta^{2}Mu(x) & \leq \omega(h) + u(x + \bar{\xi} + h) + u(x + \bar{\xi} - h) - 2u(x + \bar{\xi})\\
& \leq \omega(h) + c|h|^{2} \\
&\leq C \omega(h).
\end{split}$$\
Thus, in a neighborhood of a contact point, $Mu(x)$ is semi-concave with semi-concavity modulus $ \omega(h)$.\
\
**Case 2**: Fix $x \in \{u < Mu\}$. We argue as before by considering the second incremental quotients of the obstacle, $\delta^{2}Mu(x)$. We observe that the infimum of $u$ in the positive cone, $\xi \geq 0$, must always be realized at a non-contact point. Suppose $ \exists \; x + \xi_{1} \in \{u = Mu \}$ satisfing, $$\inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi) = u(x + \xi_{1}).$$ Then from **Case 1** there exists $\xi_{2} \in \Sigma_{x + \xi_{1}} \subset \{u < Mu \}$ such that, $$\inf_{\stackrel{\xi \geq 0}{x + \xi_{1} + \xi \in \Omega}}u(x + \xi_{1} + \xi) = u(x + \xi_{1} + \xi_{2}).$$ Since $\xi_{1} + \xi_{2} \geq 0$, we have found a positive vector admissiable to $$\inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi).$$ Furthermore, $u(x + \xi_{1} + \xi_{2}) \leq u(x + \xi_{1})$. Hence we conclude that for a fixed $x \in \{u < Mu \}$ and for some $x + \bar{\xi}$ in $\{u < Mu\}$, $$\inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi) = u(x + \bar{\xi}).$$ Moreover from Claim 4 we know that $x + \bar{\xi}$ is a uniform positive distance away from the contact set $\{u = Mu \}$. Hence there exists a uniform neighborhood $N_{0}$ of points around $x + \bar{\xi}$ where $\{u < Mu\}$. In a smaller neighborhood $N_{1}$, $u \in C^{1, 1}(N_{1})$. In particular for $x + \xi \in N_{1}$, we can bound again the second incremental quotients, $$u(x + h + \xi) + u(x - h + \xi) - 2u(x + \xi) \leq c|h|^{2}$$ Using once more the semi-concavity estimate on $\varphi(x)$ and for some $x + \bar{\xi}$ in $N_{1}$ we find,\
$$\begin{split}
\delta^{2}Mu(x) & \leq \omega(h) + u(x + \bar{\xi} + h) + u(x + \bar{\xi} - h) - 2u(x + \bar{\xi})\\
& \leq \omega(h) + c|h|^{2} \\
&\leq C \omega(h).
\end{split}$$\
Thus, in a neighborhood of a non-contact point, $Mu(x)$ is semi-concave with semi-concavity modulus $\omega(h)$.
We are now in position to apply the general estimate obtained in the previous section.
Let u be the solution to the fully nonlinear stochastic impulse control problem\
$$\begin{cases}
F(D^{2}u) \geq 0 & \forall x \in \Omega. \\
u(x) \leq \textnormal{M}u(x) & \forall x \in \Omega.\\
u = 0 & \forall x \in \partial \Omega.
\end{cases}$$\
Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain with a $C^{2,\alpha}$ boundary $\partial \Omega$, and define the obstacle, $$Mu(x) = \varphi(x) + \inf_{\stackrel{\xi \geq 0}{x + \xi \in \bar{\Omega}}}(u(x + \xi)).$$ Where $\varphi(x)$ is $\omega(r)$ semi-concave, strictly positive, bounded, and decreasing in the positive cone $\xi \geq 0$. Then, the solution $u$ has a modulus of continuity $\omega(r)$ up to $C^{1,1}(\Omega)$.
The previous theorem shows that $Mu(x)$ is $\omega(r)$ semi-concave. Set $\varphi_{u}(x) = -Mu(x)$ and note that $\varphi_{u}$ is $\omega(r)$ semi-convex. We apply the estimates from the previous section to conclude.
Finally as in the classical case, assuming analytic data and $f(x) \leq f(x + \xi) \; \; \forall \xi \geq 0$, as well as concavity of $F(\cdot)$ in the hessian variable, it follows from an application of a nonlinear version of the Hopf Boundary Point Lemma [@BDl99] and the results of [@Lee98] that we obtain the following structural theorem for the free boundary,
Given the Fully Nonlinear Stochastic Impulse Control Problem $$\begin{cases}
F(D^{2}u) \geq f & \forall x \in \Omega.\\
u(x) \leq \textnormal{M}u(x) = 1 + \inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi). & \forall x \in \Omega.\\
u = 0 & \forall x \in \partial \Omega.
\end{cases}$$
It follows that, $\partial \{u < \textnormal{M}u \} = \Gamma^{1}(u) \cup \Gamma^{2}(u)$ where,\
1. $\forall x_{0} \in \Gamma^{1}(u)$ satisfying a uniform thickness condition on the coincidence set $\{u = \textnormal{M}u \}$, there exists some appropriate system of coordinates in which the coincidence set is a subgraph $\{x_{n} \leq g(x_{1}, \dots, x_{n-1}) \}$ in a neighborhood of $x_{0}$ and the function $g$ is analytic.\
2. $\Gamma^{2}(u) \subset \Sigma(u)$ where $\Sigma(u)$ is a finite collection of $C^{\infty}$ submanifolds.
We point out that the above theorem holds for the more general implicit constraint obstacle $$\textnormal{M}u = h(x) + \inf_{\stackrel{\xi \geq 0}{x + \xi \in \Omega}}u(x + \xi)$$ where the regularity of $\Gamma^{1}(u)$ corresponds to the regularity of $h(x)$.
Applications to a Penalized Problem
===================================
In this section we study a penalized fully nonlinear obstacle problem. The goal is to obtain optimal uniform estimates in the penalizing paramter $\epsilon.$ For this section we fix the modulus of semiconvexity to be linear, i.e. $\omega(r) = cr^{2}$. We point out that the followng can be suitably modified for a general modulus of semiconvexity. The idea to obtain the optimal estimate is to use the interplay between semiconvexity of the obstacle and the superharmonicity of the equation as before.
Consider the fully nonlinear penalized obstacle problem with obstacle $\varphi_{u}$, admitting a modulus of semiconvexity, $\omega(r) = Cr^{2}$ and a suitably defined class of penalizations $\beta_{\epsilon}$, $$\begin{cases}
F(D^{2}u) = \beta_{\epsilon}(u-\varphi_{u}) & \Omega,\\
u = 0 & \partial \Omega,\\
\varphi_{u} < 0 & \partial \Omega.
\end{cases}$$ Then the solution $u$ has a modulus of continuity $\omega(r)$ up to $C^{1,\alpha}(\Omega) \; \forall \alpha < 1$ independent of the penalizing parameter $\epsilon$.
Let $\rho(x)$ be a function in $C^{\infty}(\mathbb{R}^{n})$ with support in the unit ball, such that $\rho \geq 0$ and $\int_{\mathbb{R}^{n}} \rho = 1.$ Define for any $\delta > 0$, $$\rho_{\delta}(x) = \delta^{-n}\rho(\frac{x}{\delta}).$$
Consider the mollifier $$J_{\delta} [\varphi_{u}](x) = \int_{\Omega} \rho_{\delta}(x-y) \varphi_{u}(y) \; dy.$$
Recall that $\varphi_{u}$ semi-convex with a linear modulus implies that for any $\xi \in C^{\infty}_{0}(\Omega_{0})$, $\xi \geq 0$, where $\Omega_{0} \subset \Omega$ is an open set, it holds that for any directional derivative, $\frac{\partial}{\partial \eta}$ and some constant $C > 0$ independent of $\delta$, $$\int_{\Omega} \varphi_{u} \frac{\partial^{2} \xi}{\partial \eta^{2}} \geq -C.$$
Taking $\xi = \rho_{\delta}$, it follows that pointwise in $\Omega$, $$\frac{\partial^{2} J_{\delta} [\varphi_{u}]}{\partial \eta^{2}} \geq -C.$$
We consider, $\varphi_{u}^{\delta} = J_{\delta} [\varphi_{u} + \frac{C}{2} |x|^{2}] - \frac{C}{2} |x|^{2}.$ It follows that $$|D \varphi_{u}^{\delta}| \leq C.$$ $$\frac{\partial^{2} \varphi_{u}^{\delta}}{\partial \eta^{2}} \geq -C.$$ $$\varphi_{u}^{\delta} \to \varphi_{u} \; \; \textnormal{uniformly in} \; \Omega \; \textnormal{as} \; \delta \to 0.$$
Define $\beta_{\epsilon}(t) \in C^{\infty}$ for $0 < \epsilon < 1$ and $C$ a constant independent of $\epsilon$, such that,\
1. $\beta'_{\epsilon}(t) > 0.$\
2. $\beta_{\epsilon}(t) \to 0 \; \; \textnormal{if} \; t > 0, \epsilon \to 0$.\
3. $\beta_{\epsilon}(t) \to -\infty \; \; \textnormal{if} \; t < 0, \epsilon \to 0$.\
4. $\beta_{\epsilon}(t) \leq C$\
5. $\beta_{\epsilon}''(t) \leq 0$.\
Consider the penalized problem, $$\begin{cases}
F(D^{2}u) - \beta_{\epsilon}(u-\varphi_{u}^{\epsilon}) = 0 & \Omega,\\
u = 0 & \partial \Omega.
\end{cases}$$
Define for $N > 0,$ $$\beta_{\epsilon , N}(t) = \max \{ \min \{\beta_{\epsilon}, N \}, -N \}.$$
Consider the problem, $$\begin{cases}
F(D^{2}u) - \beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}) = 0 & \Omega,\\
u = 0 & \partial \Omega.
\end{cases}$$
It follows from $W^{2,p}$ theory for fully nonlinear equations that for each $v \in L^{p}(\Omega) \cap C^{0}(\bar{\Omega})$ $(1 < p < \infty)$, there exists a unique solution $w \in W^{2,p}(\Omega) \cap C^{0}(\bar{\Omega})$ solving, $$\begin{cases}
F(D^{2}w) - \beta_{\epsilon, N}(v-\varphi_{u}^{\epsilon}) = 0 & \Omega,\\
u = 0 & \partial \Omega,
\end{cases}$$ and for $\bar{C}$ independent of $v$, $$\|w\|_{W^{2,p}} \leq \bar{C}.$$
Define the solution map $T$ such that $Tv = w.$ Notice that $T$ maps $B_{\bar{C}}(0) \subset L^{p}(\Omega)$ into itself and is compact. Hence by Schauder’s fixed-point theorem, it follows, that there exists $u$ such that $Tu = u$. In particular, we have found a solution to $(16)$. Moreover $\beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}) \in C^{0,\alpha}$. Hence by Evans-Krylov $\|u\|_{C^{2,\alpha}} \leq C(\epsilon).$ We now estimate $\zeta = \beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}).$ By definition we know that $\beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}) \leq C$ for a constant $C$ independent of $N, \epsilon.$ Let $x_{0}$ be the minimum point of $\zeta.$ Without loss of generality we assume, $$\mu = \zeta(x_{0}), \; \; \mu \leq 0, \; \; \mu < \beta_{\epsilon}(0).$$
It follows that $x_{0} \notin \partial \Omega.$ If not, then, $$\mu = \zeta(x_{0}) = \beta_{\epsilon, N}(-\varphi_{u}^{\epsilon}) \geq \beta_{\epsilon, N}(0) \geq \beta_{\epsilon}(0).$$ A contradiction. On the other hand if $x_{0} \in \Omega$, then $\beta_{\epsilon}'(t) \geq 0$ implies that, $$\min_{\Omega}(u-\varphi_{u}^{\epsilon}) = u-\varphi_{u}^{\epsilon}(x_{0}) < 0.$$
Moreover it follows that $D^{2}(u-\varphi_{u}^{\epsilon})(x_{0}) \geq 0$. Hence $F(D^{2}(u-\varphi_{u}^{\epsilon})(x_{0})) \geq 0.$ By Ellipticity and the semiconvexity estimate it follows that, $$\begin{split}
\beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}) (x_{0}) & = F(D^{2}u_{\epsilon, N} - D^{2} \varphi_{u}^{\epsilon} + D^{2} \varphi_{u}^{\epsilon}) \\
& \geq F(D^{2}u_{\epsilon, N} - D^{2} \varphi_{u}^{\epsilon}) +\lambda \| D^{2} (\varphi_{u}^{\epsilon})^{+} \| - \Lambda \| D^{2} (\varphi_{u}^{\epsilon})^{-} \| \\
& \geq -C. \\
\end{split}$$
In particular, $|\beta_{\epsilon, N}(u-\varphi_{u}^{\epsilon}) | \leq C$ for a constant $C$ independent of $\epsilon$ and $N$. Furthermore $|F(D^{2}u)| \leq C$. It follows from elliptic estimates, $$\|u\|_{W^{2,p}} \leq C.$$ Hence for $N$ large enough $u$ is a solution for the penalized problem (15).
We now prove the optimal estimate as before
Consider the solution to the fully nonlinear penalized obstacle problem with obstacle $\varphi_{u}$, admitting a modulus of semiconvexity, $\omega(r) = Cr^{2}$ and a suitably defined class of penalizations $\beta_{\epsilon}$. Assume that $F(D^{2}u)$ is convex in the Hessian variable. Then the solution $u$ is $C^{1,1}$ independent of $\epsilon.$
Consider the penalization problem $$\begin{cases}
F(D^{2}u) = \beta_{\epsilon}(u-\varphi_{u}) & \Omega,\\
u = 0 & \partial \Omega.\\
\varphi_{u} < 0 & \partial \Omega.
\end{cases}$$ We aim to bound $\inf u_{\tau \tau}$ from below. The following computation continues to hold for viscosity solutions by using incremental quotients and recalling that second order incremental quotients are supersolutions of a convex equation. We fix a directional derivative $\tau$ and differentiate the penalization identity to obtain, $$F_{ij,kl}(D^{2}u)(D_{ij}u_{\tau})(D_{kl}u_{\tau}) + F_{ij}(D^{2}u)(D_{ij}u_{\tau \tau}) =$$ $$\beta_{\epsilon}''(u - \varphi_{u})(u - \varphi_{u})_{\tau}^{2} + \beta_{\epsilon}'(u - \varphi_{u})(u - \varphi_{u})_{\tau \tau}.$$
By convexity of the operator and the structural conditions on the penalization family $\beta_{\epsilon}(t)$ it follows that $$F_{ij}(D^{2}u))(D_{ij}u_{\tau \tau}) \leq \beta_{\epsilon}'(u - \varphi_{u})(u - \varphi_{u})_{\tau \tau}.$$
Suppose the minimum point of $u_{\tau \tau}$ is in the interior of the domain then, since $\beta'(t) > 0$, we find $(u - \varphi_{u})_{\tau \tau} \geq 0$. In particular, $u_{\tau \tau} \geq -C.$ Suppose now that the minimum point of $u_{\tau \tau}$ is realized on the boundary of the domain. We differentiate the equation with respect to $x_{\tau}$ for $\tau \in \{1, \dots, n-1\}$ and obtain, $$F_{ij}(D^{2}u)D_{ij}u_{\tau} = \beta_{\epsilon}'(u - \varphi_{u})(u - \varphi_{u})_{\tau}.$$ Recall $\varphi_{u} < 0$ on $\partial \Omega$. Hence for a fixed $\epsilon_{0} > 0$ it follows that $\varphi_{u} \leq u + \epsilon_{0}$ in $\{x \in \bar{\Omega} \; | \; d(x, \partial \Omega) \leq \frac{\epsilon_{0}}{2} \}.$ Morevover by the uniform continuity of $u^{\epsilon} \to u$ on $\bar{\Omega}$, there exists a small $\epsilon_{1}$, such that $\varphi_{u} \leq u + \frac{\epsilon_{0}}{2}$, $|\beta'| < \epsilon_{0}$, and $|\beta''| < \epsilon_{0}$ in $\{x \in \bar{\Omega} \; | \; d(x, \partial \Omega) \leq \frac{\epsilon_{0}}{2} \}$ for $0 < \epsilon < \epsilon_{1}.$ Hence it follows from the boundary Hölder estimates for linear non-divergence form equations, $$\|u_{\tau n} \|_{L^{\infty}(\partial B^{+}( \frac{\epsilon_{0}}{4}))} \leq C.$$
Moreover by uniform ellipticity we can use the equation to solve for $u_{nn}$ in terms of $\beta$ and $u_{kl}$ for $k \in (1, \dots, n-1)$ and $l \in (1, \dots , n).$ Hence we obtain after straightening the boundary, $$\|D^{2} u\|_{L^{\infty}(\partial \Omega)} \leq C.$$
Hence it follows that the solution is semiconvex with a linear modulus. Moreover $F(D^{2}u) = \beta_{\epsilon}(u - \varphi_{u}) \leq 0$ after choosing a penalization satisfying $\beta_{\epsilon}(t) \leq 0$ for $t \geq 0$. Finally an application of Lemma 6 proves that $u$ has a uniform $C^{1,1}$ estimate.
We point out that the above arguments give us a straightforward proof for $C^{1,1}$ estimates when the operator is convex. The previous section was based on $C^{1,\alpha}$ estimates for Fully Nonlinear equations hence did not have a restriction on the sign of the operator.
Previous computation and estimates can be generalized to Viscosity Solutions of convex operators (see [@CC95]).
Finally, as an application of the uniform estimates, we prove how the $C^{2,\alpha}$ estimate for the penalized problem decays in the penalizing parameter.
Consider the fully nonlinear penalized obstacle problem with obstacle $\varphi_{u}$, admitting a modulus of semi-convexity, $\omega(r) = Cr^{2}$ and a suitably defined class of penalizations $\beta_{\epsilon}$, $$\begin{cases}
F(D^{2}u) = \beta_{\epsilon}(u-\varphi_{u}) & \Omega,\\
u = 0 & \partial \Omega.\\
\varphi_{u} < 0 & \partial \Omega.
\end{cases}$$ Moreover assume that $F( \cdot)$ is convex in the hessian variable. Then for a constant $C$ independent of $\epsilon$, $$\|u\|_{C^{2,\alpha}} \leq C\epsilon^{-\alpha}.$$
It is well known that the penalization problem converges to the obstacle problem independent of the choice of penalizing family. Hence we fix a penalizing family, $$\beta_{\epsilon}(t) = \left\{
\begin{array}{lr}
\frac{t}{\epsilon^{2}} & \; \; t < 0.\\
0 & t \geq 0.
\end{array}
\right.$$
Without loss of generality we fix $\varphi_{u} = 0$. The following argument can be modified for non-zero obstacle $\varphi_{u}$. We consider the scaled function, $$v^{\epsilon}(x) = \frac{1}{\epsilon^{2}}u^{\epsilon}(\epsilon x).$$ We note that $$F(D^{2}v^{\epsilon}) = F(D^{2} (\frac{1}{\epsilon^{2}}u^{\epsilon}(\epsilon x))) = F(D^{2} u^{\epsilon}(\epsilon x)) = \frac{1}{\epsilon^{2}}u^{\epsilon}(\epsilon x) = v^{\epsilon}(x).$$
Hence we obtain for a constant $C$ independent of $\epsilon$, $$\|v^{\epsilon}\|_{C^{2, \alpha}} \leq C.$$
It follows, $$\begin{split}
|D^{2}u^{\epsilon}(x) - D^{2}u^{\epsilon}(y)| & = |\epsilon^{2}D^{2}v^{\epsilon}(\frac{x}{\epsilon}) - \epsilon^{2}D^{2}v^{\epsilon}(\frac{y}{\epsilon})| \\
& = |D^{2}v^{\epsilon}(\frac{x}{\epsilon}) - D^{2}v^{\epsilon}(\frac{y}{\epsilon})| \\
& \leq C | \frac{x}{\epsilon} - \frac{y}{\epsilon}|^{\alpha} \\
& \leq C\epsilon^{-\alpha} |x-y|^{\alpha}.
\end{split}$$
[\*\*\*\*\*]{}
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|
---
abstract: 'Numerous recent works have proposed pretraining generic visio-linguistic representations and then finetuning them for downstream vision and language tasks. While architecture and objective function design choices have received attention, the choice of pretraining datasets has received little attention. In this work, we question some of the default choices made in literature. For instance, we systematically study how varying similarity between the pretraining dataset domain (textual and visual) and the downstream domain affects performance. Surprisingly, we show that automatically generated data in a domain closer to the downstream task (e.g., VQA v2) is a better choice for pretraining than “natural” data but of a slightly different domain (e.g., Conceptual Captions). On the other hand, some seemingly reasonable choices of pretraining datasets were found to be entirely ineffective for some downstream tasks. This suggests that despite the numerous recent efforts, vision & language pretraining does not quite work “out of the box” yet. Overall, as a by-product of our study, we find that simple design choices in pretraining can help us achieve close to state-of-art results on downstream tasks without any architectural changes.'
author:
- 'Amanpreet Singh[^1]'
- 'Vedanuj Goswami$^\star$'
- Devi Parikh
bibliography:
- 'egbib.bib'
title: |
Are we pretraining it right?\
Digging deeper into visio-linguistic pretraining
---
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Marcus Rohrbach for helpful discussions and feedback.
[^1]: Equal contribution
|
---
abstract: 'We consider hydrodynamics with non conserved number of particles and show that it can be modeled with effective fluid Lagrangians which explicitly depend on the velocity potentials. For such theories, the “shift symmetry” $\phi\rightarrow\phi+$const. leading to the conserved number of fluid particles in conventional hydrodynamics is globally broken and, as a result, the non conservation of particle number appears as a source term in the continuity equation. The particle number non-conservation is balanced by the entropy change, with both the entropy and the source term expressed in terms of the fluid velocity potential. Equations of hydrodynamics are derived using a modified version of Schutz’s variational principle method. Examples of fluids described by such Lagrangians (tachyon condensate, k-essence) in spatially flat isotropic universe are briefly discussed.'
address: 'Dpto. de Física Teórica, Universidad del País Vasco, Apdo. 644, 48080, Bilbao, Spain.'
author:
- 'Alberto Díez-Tejedor[^1] and Alexander Feinstein[^2]'
title: 'Relativistic hydrodynamics with sources for cosmological K-fluids'
---
INTRODUCTION
============
It is well known that complex physical phenomena can be often modeled with good accuracy by an effective theory. One such effective macroscopic model, for example, is the hydrodynamical model of Landau [@landau], which has had a considerable success in explaining certain features of the collisions of highly relativistic nuclei [@bjorken; @cooper]. The universe, the most complex of all the physical systems, is in general successfully modeled by an isentropic perfect fluid. Hydrodynamic language, back in high regard, is now invoked to describe non-trivial field theories [@jackiw].
In cosmology, as mentioned above, the perfect fluid description, despite the generic complexity of the system, works fine. One of the usual assumptions in the conventional hydrodynamical description of the universe is that the universe expands adiabatically. Closely related to it is the assertion that the so-called mass, or particle number conservation, holds. Yet, one can imagine a universe where creation or destruction of particles takes place. This may happen due to the time variation and inhomogeneities of the gravitational field itself, not to discard a more speculative possibility of a universe filled with white and black holes where particles suddenly appear or disappear. What kind of an effective hydrodynamics would then describe such a universe?
There are several ways to approach the problem of the universe where particles are created or annihilated. If this happens due to quantum processes, then presumably the most direct approach would be to consider the quantization of the matter fields on a curved background using the machinery of the quantum field theory [@Birrel], and then evaluating the back-reaction of the created fields on the classical geometry. The promising direction within this approach is the study of stochastic gravity [@hu].
It is possible, though, that for some reason, one is not interested in the detailed description of the particle creation (destruction) mechanism. Then one would be trying to model the effects of the microscopic processes by a kind of an effective macroscopic model. In hadron-hadron collision theory [@bjorken; @cooper], such an effective macroscopic model is the Landau’s hydrodynamical description.
In the framework of cosmology with non conserved number of particles, a possible macroscopic description was put forward by Prigogine et al [@prigogine], and later generalised by Calvao et al [@calvao] some years ago. In this approach, the creation of particles is considered in the context of thermodynamics of open systems. What follows then, roughly speaking, is that an extra negative “viscous” pressure term appears in the energy-momentum tensor to account for the created particles.
Yet, there exists a more “economic” and elegant way to describe particle creation (annihilation) without a change in the form of the energy-momentum tensor, and without introducing an extra pressure term . To introduce a source term into the particle number conservation equation it is sufficient to allow entropy flow. The change in the particle number allowed by the continuity equation will then come at the expense of the entropy change.
In this paper we are interested in exploring a Lagrangian formulation of relativistic hydrodynamics with non conserved number of particles. In the conventional variational approach to relativistic hydrodynamics developed by Schutz [@schutz], the action does not depend on the velocity potential, but rather is a functional of its derivatives. This, in turn, maintains the symmetry $\phi\rightarrow\phi+$const. which allows particle number conservation. Here, we propose a Lagrangian formulation for the equations of hydrodynamics, where the Lagrangian, to break globally the symmetry leading to the particle number conservation, depends not only on the derivatives, but on the velocity potential itself.
We propose to modify Schutz’s original Lagrangian [@schutz; @schutz2; @brown], by introducing sources and sinks in the continuity equation, modeled by a velocity potential dependent function. Our formulation is matemathically selfconsistent, in that it gives the right set of hydrodynamical equations. Physically, the fluid Lagrangians we consider have connection to matter described by the rolling tachyon condensate [@sen] or by the K-essence [@armendariz; @armentesis].
THE HYDRODYNAMICS WITH PARTICLE NUMBER VARIATION\[sec:2\]
=========================================================
We start by assuming that we deal with simple thermodynamical systems (fluids), which are characterised by a fundamental equation of the form $U=U\left[S,V,N\right]$, where all the variables have their usual meanings, and we use $k_{B}=c=8\pi G=1$. Assuming the standard thermodynamic relations, one may show that such a system is completely specified by the energy density function $\rho(n,s)$ and the system’s size $V$: $$U\left[S,V,N\right]=V\rho(n,s),$$ where $n$ is the particle number density and $s$ is the entropy per particle. We can write the first law of thermodynamics as $$d\rho=hdn+nTds,\label{eq:priley}$$ where $h$ is the enthalpy per particle. Assuming further that the particle number in the system is not conserved, the equations of hydrodynamics take the following form: $$T_{\quad;\mu}^{\mu\nu}=0,\label{eq:consener}$$ $$\left(nu^{\mu}\right)_{;\mu}=\psi,\label{eq:varpati}$$ where $T^{\mu\nu}$ stands for the usual stress-energy tensor of a perfect fluid: $$T^{\mu\nu}=\left(\rho+p\right)u^{\mu}u^{\nu}+pg^{\mu\nu}.\label{eq:energia}$$ Here $p$ and $\rho$ are the pressure and the energy density of the fluid respectively, and $u^{\mu}$ is the four-velocity field ($u_{\mu}u^{\mu}=-1$). The equation (\[eq:consener\]) represents the conservation of the energy-momentum tensor, whereas (\[eq:varpati\]) is the continuity equation with the source ($\psi>0$) or the sink ($\psi<0$) term for the particles. We must further specify the equation of state $\rho=\rho(n,s)$, along with the source term $\psi$, which we take to have the form $\psi=\psi(n,s)$. To close this system of equations we add the first law of thermodynamics (\[eq:priley\]). The equations (\[eq:consener\]), (\[eq:varpati\]) and (\[eq:priley\]) form a self consistent field theory describing a fluid with particle number variation in terms of five macroscopic (or Eulerian) variables ($n,s,u^{\mu}$).
To obtain a more intuitive form of these equations it is convenient to project the energy conservation equation (\[eq:consener\]) along and, in the direction perpendicular, to the four-velocity. The parallel projection ($u_{\mu}T_{\quad;\nu}^{\mu\nu}=0$), after the balance equation (\[eq:varpati\]) and the thermodynamical relations have been substituted, gives the following continuity equation: $$s_{,\mu}u^{\mu}=-\frac{h\psi}{nT}.\label{eq:vars}$$ This equation was first given, in a somewhat different form, by Schutz and Sorkin [@schutz2]. One can appreciate how the change in the number of particles ($\psi$) is accompanied by a change in the entropy per particle ($u^{\mu}s_{,\mu}\neq0$). The fluid flow no longer follows lines of constant $s$, as it happens in the conventional hydrodynamics when no source is present ($\psi=0$). The projection perpendicular to the four-velocity gives ($P_{\mu\alpha}T_{\quad;\nu}^{\mu\nu}=0$, with $P_{\mu}^{\nu}\equiv u^{\nu}u_{\mu}+\delta_{\mu}^{\nu}$): $$\left(\rho+p\right)u_{\alpha;\nu}u^{\nu}=-p_{,\nu}P_{\alpha}^{\nu},$$ which is the relativistic Euler equation. The last two equations are completely equivalent to the eqs. (\[eq:consener\]) and (\[eq:varpati\]).
The variation rate of the number of particles $N$ and the total entropy $S$ of the fluid may still be written in a more suggestive way: $$\frac{dN}{d\tau}=V\psi,\quad\frac{dS}{d\tau}=-\frac{\mu}{T}\frac{dN}{d\tau},\label{eq:NS}$$ where $\mu=h-sT$ is the chemical potential. From the first of these two equations we see that the sign of the source term determines as to whether the particles are created or annihilated. The other equation describes the variation of the entropy, whose change is determined by both, the sign of the chemical potential and the source term.
THE ACTION PRINCIPLE\[sec:AN-ACTION\]
=====================================
The relativistic perfect fluid action functionals were developed by Taub [@taub1] and Schutz [@schutz]. Here we follow closely Schutz’s velocity potential formalism [@schutz]. In the case of the conventional hydrodynamics, where no particle creation takes place ($\psi=0$), one starts with the following action [@schutz; @schutz2; @brown]: $$S=\int d^{4}x\left\{ -\sqrt{-g}\rho(n,s)+J^{\mu}\left(\phi_{,\mu}+s\theta_{,\mu}+\beta_{A}\alpha_{\:,\mu}^{A}\right)\right\} ,$$ with $A$ taking the values $1,2$ and $3$. Here, $\phi$ and $\theta$ are Lagrange multipliers introduced to satisfy the particle number and the entropy conservation constraints respectively. One further assumes the existence of Lagrangian coordinates $\alpha^{A}$ which label the flow lines, and consequently introduces the $\beta_{A}$ potentials in form of Lagrange multipliers. $J^{\mu}$ is the particle number current-density, defined as $J^{\mu}\equiv\sqrt{-g}nu^{\mu}$. The expression for the current permits to write the particle number density as $n=\left|J\right|/\sqrt{-g}$. To include the gravity as a dynamical field into the picture, one adds, as usual, the Einstein-Hilbert term to the above action. The variables in which the action is formulated are, therefore: $g^{\mu\nu}$, $J^{\mu}$, $\phi$, $s$, $\theta$, $\beta_{A}$ and $\alpha^{A}$. Starting with this action, one derives both the hydrodynamical equations of motion and the energy-momentum tensor for the fluid [@brown].
We now consider the action principle for the hydrodynamics described in the previous section. For this purpose, we put forward the following action [@futuro]:$$S=\int d^{4}x\left\{ -\sqrt{-g}\rho(n,s)+J^{\mu}\left(\phi_{,\mu}+\beta_{A}\alpha_{\:,\mu}^{A}\right)\right\} ,$$ where now the entropy per particle $s$ is not an independent variable any more. We assume $s=s(\phi)$, so that the “shift symmetry” $\phi\rightarrow\phi+$const. present in the Schutz’s original action is globally broken. We have also suppressed the term $J^{\mu}s\theta_{,\mu}$ in the action, since now we do not expect the entropy per particle $s$ to conserve. To justify physically the functional dependence of the entropy on the velocity potential (apart from the fact that such an action leads to the equations of motion we expect) we suggest that since the non-conservation of the particles via the equation (\[eq:vars\]) leads to the entropy change, on one hand, and that we model the particle non-conservation by the symmetry breaking with the $\phi$-dependent term in the action on the other, it looks reasonable to introduce this dependence in the entropy term. One must bear in mind, however, that due to the particular parametrisation $s(\phi)$, the particle variation rate is neither arbitrary, nor generic, yet we find it sufficiently general for the purposes of the physics we are interested in. With this in mind, one may show [@futuro] that the equations of hydrodynamics as well as the form of the energy momentum tensor of the section \[sec:2\] can be recovered from the above action.
Introducing the equations of motion back into the action we obtain the on-shell expression $$S_{on-shell}=\int d^{4}x\sqrt{-g}p,$$ which coincides with the on-shell expression of Schutz for conventional hydrodynamics. Thus, we have that the Lagrangian for the hydrodynamics with particle non-conservation may still be given by the pressure of the fluid.
THE IRROTATIONAL FLOW \[sec:IRROTATIONAL\]
==========================================
We now assume the fluid flow to be irrotational. We also note, that although until now we have expressed the action in terms of $\rho(n,s)$, it is often convenient to use, with the help of the usual thermodynamic relations, a different parametrisation of the action [@brown]. In the context of the irrotational flow it will be more convenient to work with the equation of state $p=p(h,s)$.
First, let us see what happens in the conventional case when the particle number is conserved. In this case we have an isentropic fluid ($s=$const.) and the action may be expressed as $$S=\int d^{4}x\sqrt{-g}\left\{ p\left(\left|V\right|\right)-\left(\frac{\partial p}{\partial h}\right)_{s}\left[\left|V\right|-\frac{V^{\mu}\varphi_{,\mu}}{\left|V\right|}\right]\right\} ,$$ where we have defined the current $V^{\mu}\equiv hu^{\mu}$, and the subindex $s$ refers to the fact that the partial derivative $\left(\partial p/\partial h\right)$ is evaluated at constant $s$. The variables are $g^{\mu\nu}$, $V^{\mu}$ and $\varphi$, and the following equations of motion result: $$u_{\mu}=-h^{-1}\varphi_{,\mu},\label{eq:din1}$$ $$\left(nu^{\mu}\right)_{;\mu}=0,\label{eq:din2}$$ with the energy-momentum tensor given by $$T^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}}=\left(\frac{\partial p}{\partial h}\right)_{s}hu^{\mu}u^{\nu}+pg^{\mu\nu}.\label{eq:momento}$$ Comparing the last equation with the equation (\[eq:energia\]) allows to define the pressure and the energy density of the fluid as: $p=p$ and $\rho=nh-p$ (with $n=\left(\partial p/\partial h\right)_{s}$). The pressure and the density defined via the stress-energy tensor coincide with their usual thermodynamical definitions. The equation (\[eq:din1\]) is the expression of the fact that the fluid flow is irrotational, whereas the equation (\[eq:din2\]) is the particle number conservation equation.
The identity $u_{\mu}u^{\mu}=-1$ and the equation (\[eq:din1\]), lead to the following expression for the enthalpy: $$h=\sqrt{-\varphi_{,\mu}\varphi^{,\mu}}.\label{eq:entalpia}$$ To make contact with the now popular K-essence cosmology [@armendariz; @armentesis], we write the action on-shell as $$S_{on-shell}=\int d^{4}x\sqrt{-g}F(X),\label{eq:action}$$ where we define: $$X\equiv-\frac{1}{2}\varphi_{,\mu}\varphi^{,\mu}=\frac{h^{2}}{2},\label{eq:def1}$$ $$p(h)=p\left(\sqrt{2X}\right)\equiv F(X),\label{eq:def2}$$
Therefore, if one has an irrotational fluid where the number of particles is conserved, it is described by the hydrodynamics derived from the Lagrangian (\[eq:action\]), which depends only on the derivatives of the velocity potential defined by the equation (\[eq:din1\]). Moreover, the conservation equation (\[eq:din2\]), is just the Euler-Lagrange equation derived from the action (\[eq:action\]): $$-\left(nu^{\mu}\right)_{;\mu}=\left[F'(X)\varphi^{,\mu}\right]_{;\mu}=0,$$ where we have used $n=\left(\partial p/\partial h\right)_{s}=hF'(X)$, and the prime stands for the derivative of the function with respect to its argument. For completeness, we give the expression for the density and the pressure of the fluid in terms of the variable $X$: $$p=F(X),\quad\rho=2XF'(X)-F(X).\label{eq:presion}$$ These expressions are known in the context of K-field as purely kinetic K-field [@armendariz; @armentesis; @Scherrer].
Typically, in cosmology, one uses an equation of state $p=f(\rho)$ to describe an isentropic fluid. To obtain an action in the form (\[eq:action\]) describing such a fluid, one only has to express the energy density as $\rho=f^{-1}(p)$, insert the latter expression into the equation of energy density (\[eq:presion\]), and obtain the differential equation for $F,$ $f^{-1}(F)=2XF'-F$. This gives then for $F(X)$:
$$\int^{F}\frac{dF^{*}}{f^{-1}(F^{*})+F^{*}}=\ln\left[CX^{1/2}\right],\label{eq:ecF}$$
where $C$ is an arbitrary integration constant. The equation (\[eq:ecF\]) establishes how to pass from the standard hydrodynamical description of an isentropic irrotational perfect fluid ($p=f(\rho))$, to the language of an action principle (\[eq:action\]). Put differently, the purely kinetic K-field, is interpretable in terms of an isentropic perfect fluid with an equation of state which can be easily put into the form $p=p(\rho)$. Thus, *any solution to the Einstein’s field equations with the energy momentum tensor of the irrotational perfect fluid with the equation of state $p=p(\rho)$ is by default interpretable as a solution for the purely kinetic K-fluid*.
We now consider the irrotational flow where the number of particles is not conserved. In this case, the action can be expressed as [@futuro]:$$S=\int d^{4}x\sqrt{-g}\left\{ p\left(\left|V\right|,s\right)-\left(\frac{\partial p}{\partial h}\right)_{n}\left[\left|V\right|-\frac{V^{\mu}\phi_{,\mu}}{\left|V\right|}\right]\right\}$$ along with $s=s(\phi)$. The variables still are $g^{\mu\nu}$, $V^{\mu}$ and $\phi$, and the equations of motion that follow are$$u_{\mu}=-h^{-1}\phi_{,\mu},\label{eq:din3}$$ $$\left(nu^{\mu}\right)_{;\mu}=-nT\frac{ds}{d\phi}.\label{eq:din4}$$ In the last equation we have used $-nT=\left(\partial p/\partial s\right)_{h}$. The form of the energy-momentum tensor is left unchanged and is given by the equation (\[eq:momento\]). The equation (\[eq:din3\]) expresses again the fact that the flow is irrotational, and the continuity equation (\[eq:din4\]), if we define the particle creation rate as $$\psi\equiv-nT\frac{ds}{d\phi},\label{eq:crea}$$ is the balance equation (\[eq:varpati\]). We hence model the creation of particles through the function $s=s(\phi)$. Using the property $u_{\mu}u^{\mu}=-1$ and the equation (\[eq:din3\]), we obtain the equation (\[eq:entalpia\]), and now the on-shell action becomes: $$S_{on-shell}=\int d^{4}x\sqrt{-g}L(\phi,X),\label{eq:action2}$$ where we have used the equation (\[eq:def1\]) and have defined $$p(h,s)=p\left(\sqrt{2X},s(\phi)\right)\equiv L(\phi,X).\label{eq:ladensity}$$ We thus have succeeded in giving the action for the irrotational fluid flow where number of particles is not conserved in terms of the scalar velocity potential and its derivatives. Moreover, the continuity equation of the fluid (\[eq:din4\]), using $n=\left(\partial p/\partial h\right)_{s}=h\partial L/\partial X$, $u^{\mu}=-h^{-1}\phi^{,\mu}$ and $\psi=-nTds/d\phi=\partial L/\partial\phi$ becomes the Euler-Lagrange equation for the action (\[eq:action2\]): $$\left[\frac{\partial L}{\partial X}\phi^{,\mu}\right]_{;\mu}+\frac{\partial L}{\partial\phi}=0.\label{eq:ecdin}$$ We finally express the pressure and the density of the fluid in terms of the scalar field: $$p=L(\phi,X),\quad\rho=2X\frac{\partial L(\phi,X)}{\partial X}-L(\phi,X).\label{eq:denpres}$$
K-FLUID
=======
A special case arises when the fluid has a separable equation of state $p(h,s)=f(s)g(h)$. In this case, the action takes the form$$S=\int d^{4}x\sqrt{-g}K(\phi)F(X),\label{eq:landensity1}$$ with definitions $F(X)\equiv g(\sqrt{2X})$ and $K(\phi)\equiv f\left(s(\phi)\right)$. The entropy per particle $s$ can be then expressed as a function of the potential term $K(\phi)$:$$s=f^{-1}\left[K(\phi)\right],\label{eq:s}$$ and the equation (\[eq:denpres\]) permits to express the pressure and energy as: $$p=K(\phi)F(X),\quad\rho=K(\phi)\left[2XF'(X)-F(X)\right].\label{eq:pres1}$$ The above expressions are analogous to factorisable K-field theories [@armendariz; @armentesis], and we therefore refer to these fluids as K-fluids. The case in which there is no particle creation (purely kinetic K-fluid) is obtained with $K(\phi)=$const.
One usually assumes without lost of generality that $K(\phi)>0$ ($f(s)>0$), while $F(X)$ may be either positive or negative, allowing for tensions instead of pressure. Yet, we want to have a positive energy density, we therefore must have$$2XF'(X)-F(X)\geq0.\label{eq:con1}$$ In addition, the particle number density $n$ must also be positive, so we need $$F'(X)\geq0\label{eq:con2}$$ and $$\textnormal{sgn}\left[f'(s)\right]=-\textnormal{sgn}\left[p\right]\label{eq:con3}$$ to have positive temperature. One may further define the sound speed in a usual way: $$c_{s}^{2}=\left(\frac{\partial p}{\partial\rho}\right)_{s}=\frac{F'(X)}{2XF''(X)+F'(X)}$$ (cf [@Mukhanov]). For ordinary fluids one usually also imposes $0\leq c_{s}^{2}\leq1$, and therefore using (\[eq:con2\]), we have: $$F''(X)\geq0.\label{eq:con4}$$
We therefore refer to K-hydrodynamics, or K-fluid for short, as to an irrotational fluid with an equation of state $p(h,s)=f(s)g(h)$ and a particle variation rate given by $\psi(h,s)=k(s)g(h)$. The particle variation rate is further parametrised by $s=s(\phi)$, where $\phi$ is the velocity potential of this irrotational fluid, and the peculiar functional form of the particle production rate $\psi(h,s)$ is a consequence of this parametrisation choice. The action for the K-fluid is given by the equation (\[eq:landensity1\]). We impose positivity of the energy density (\[eq:con1\]), the particle number density (\[eq:con2\]) and the temperature (\[eq:con3\]), but are less stringent with the pressure, though one may always impose the positivity of the pressure as well. The fluid flow is stable as long as (\[eq:con4\]) holds.
We can now express the fluid parameters in terms of the scalar field. The particle number density and particle production rate take the form:$$n=K(\phi)\sqrt{2X}F'(X),\label{eq:particle1}$$ $$\psi=F(X)K'(\phi),\label{eq:rate}$$ and consequently the sign of the derivative of $K$ defines as to whether the creation or annihilation of particles takes place. When $K(\phi)=$const., the expression $N=Vn$ is the Noether’s charge associated with the “shift symetry” $\phi\rightarrow\phi+$const. of the action. These expressions above can be written in terms of the action without the explicit knowledge of the function $f(s)$. However, to evaluate the entropy per particle (\[eq:s\]), total entropy $S=Vns$ and temperature $$T=\frac{-f'(s)}{n}F(X),\label{eq:tem}$$ one must know the form of $f(s)$. Some examples will be given in the following section.
With the above hydrodynamical interpretations, let us look for a moment at the K-fluids where the number of particles is conserved. The entropy per particle is then a constant, say $s_{0}$, and therefore the equation of state has the form $p=p(h)$. This is an isentropic fluid characterised by the function $F(X)$ and the constant $f(s_{0})$. The action for the fluid becomes: $$S=\int d^{4}x\sqrt{-g}f(s_{0})F(X),\label{eq:ac}$$ where the function $F(X)$ is given by (\[eq:ecF\]) subject to the conditions (\[eq:con1\]), (\[eq:con2\]) and (\[eq:con4\]), while $f(s)$ must verify (\[eq:con3\]). The Lagrangian (\[eq:ac\]), up to a non-essential multiplicative constant, is the Lagrangian for the purely kinetic K-field [@Scherrer], for which we have defined the pressure, the energy density (\[eq:pres1\]), the entropy per particle (\[eq:s\]) $s=s_{0}$, the particle number density (\[eq:particle1\]) and the temperature (\[eq:tem\]).
To close this section we give the dynamical equation (\[eq:ecdin\]) in the case of the factorisable K-fluid theory: $$\nabla_{\mu}\left[K(\phi)F'(X)\phi^{,\mu}\right]+K'(\phi)F(X)=0.\label{eq:dyngen}$$
PARTICULAR EXAMPLES
===================
Let us consider some particular examples. We start by specifying the following equation of state: $$p(h,s)=e^{\mp s}g(h),\label{eq:ecstate}$$ where we have $-$ for $p>0$ and $+$ for $p<0$, in accordance with (\[eq:con3\]). We see that this equation of state is of the form described in the previous section. One must further specify the function $s(\phi)$ in terms of the particle creation rate, so that the entropy per particle (\[eq:s\]) can be expressed as a function of the potential: $$s=\mp\ln\left[K(\phi)\right].\label{eq:s1}$$ For the entropy per particle to be positive, one should impose $0<K(\phi)<1$ ($K(\phi)>1$) for $p>0$ ($p<0$). With the equation of state (\[eq:ecstate\]), the temperature of the fluid becomes$$T=\frac{-1}{n}\frac{\partial p}{\partial s}=\frac{\left|p\right|}{n}.$$ Note, that this expression for the temperature coincides with the expression one would have for a typical fluid composed of non-interacting physical particles (generalized to negative pressures), and is a consequence of the choice we made for the equation of state (\[eq:ecstate\]). In terms of the field we have the particle number density (\[eq:particle1\]), particle rate production (\[eq:rate\]), and with this choice of $f(s)$ we can compute the entropy per particle (\[eq:s1\]) and the temperature (\[eq:tem\]) $$T=\frac{\left|F(X)\right|}{\sqrt{2X}F'(X)}.\label{eq:temperature1}$$ If we consider the case where the particle number remains constant, the action for the fluid becomes $$S=\int d^{4}x\sqrt{-g}e^{\mp s_{0}}F(X),\label{eq:acsin}$$ where the function $F(X)$ is evaluated from (\[eq:ecF\]).
**Example 1**: Fluid with constant adiabatic index $p=w\rho$ ($w=\textnormal{const.}$). We have $\rho=f^{-1}(F)=F/w$. >From (\[eq:ecF\]) we obtain: $$F(X)=\pm X^{\frac{1+w}{2w}},\label{eq:F(X)}$$ where the sign $+$ corresponds to $w>0$, while the sign $-$ corresponds to the case $-1\leq w<0$, after the constraints (\[eq:con1\]) and (\[eq:con2\]) have been applied [^3]. In the case of the stable flow, the constraint (\[eq:con4\]) imposes the positivity of the pressure together with $0<w\leq1$. Such a fluid is then described by the action $$S=\int d^{4}x\sqrt{-g}e^{-s_{0}}X^{\frac{1+w}{2w}},$$ with $0<w\leq1$, and where $$p=e^{-s_{0}}X^{\frac{1+w}{2w}},\quad\rho=\frac{e^{-s_{0}}}{w}X^{\frac{1+w}{2w}},$$ $$n=e^{-s_{0}}\frac{(1+w)}{\sqrt{2}w}X^{\frac{1}{2w}},\quad T=\frac{\sqrt{2}w}{1+w}X^{\frac{1}{2}}.$$ In a spatially flat FRW universe ($ds^{2}=-dt^{2}+a^{2}(t)\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)\right]$), solving the dynamical (\[eq:dyngen\]) and Friedmann ($3H^{2}=\rho$) equations , we easily recover $X(a)\propto a^{-6w}$ and $a(t)\propto t^{2/3(1+w)}$.
Now, let us turn to the theories where the number of particles is not conserved. The kind of action we have is (\[eq:landensity1\]), sticking to the factorisable theories. A typical potential to use would be the well studied $K(\phi)\propto1/\phi^{2}$ [@armendariz; @armentesis; @Feinstein], due to the fact that it leaves one with solutions with constant enthalpy per particle in spatially flat isotropic universes. Yet, if we would use the equation of state (\[eq:ecstate\]) we would certainly run into trouble because of the restrictions on the function $K(\phi)$. There are two ways to circumvent this problem: either consider a different equation of state, or a different potential. If we stick to the above equation of state, then for example the following potentials $K(\phi)=A\cosh\phi$ with $A\geq1$ and $K(\phi)=A\exp(-\phi^{2})+B$, with $A,B>0$ and $A+B<1$, will do. Now, the only advantage of using the equation of state (\[eq:ecstate\]) is that the temperature is given as the ratio of the pressure to the particle number density. The next simplest choice for an equation of state would be the one for which the entropy function $f(s)$ is a power-law. We thus take $$p(h,s)=s^{b}g(h),\label{eq:eqstate2}$$ where $b$ is an arbitrary constant such that sgn$(b)=-$sgn$(p)$ to satisfy (\[eq:con3\]). The entropy then, according to (\[eq:s\]), is$$s=\left[K(\phi)\right]^{\frac{1}{b}},\label{eq:s2}$$ and is compatible with the potentials of the form $K(\phi)\propto1/\phi^{2}$. The particle number density and the particle rate production in terms of the field are given by (\[eq:particle1\]) and (\[eq:rate\]), whereas the temperature now takes the form $$T=\frac{\left|bF(X)\right|}{\sqrt{2X}F'(X)}\left[K(\phi)\right]^{-\frac{1}{b}}.\label{eq:temperature2}$$
We will now look at two “similar” fluids in a spatially flat FRW universe, but with different properties with respect to the particle number conservation. In the first case, the fluid is isentropic with a conserved particle number and provokes interest in both field theory [@jackiw] and cosmology [@Chap]. The second case represents the same fluid where the number of particles is not conserved and has a Lagrangian of the form of Sen’s tachyon condensate [@sen], which has recently become of considerable interest in cosmology [@Feinstein; @tachyon].
**Example 2**: Fluid with equation of state $p=-A/\rho$, $A=$const.$>0$ (Chaplygin gas)*.* In this case we have $\rho=f^{-1}(F)=-1/F$. Inserting this in (\[eq:ecF\]), we obtain, up to some unessential constants: $$F(X)=\pm\sqrt{1\pm X}.$$ We assume the constraints (\[eq:con1\]) and (\[eq:con2\]) hold, and we are left therefore with negative pressure: $$F(X)=-\sqrt{1-X}\label{eq:F(X)1}$$ with $0\leq X\leq1$, and there is no problem with the constraint (\[eq:con4\]), indicating that the flow is stable. We can think of such a fluid as a fluid with the equation of state $$p(h,s)=-s^{b}\sqrt{1-\frac{h^{2}}{2}}\label{eq:eqstate3}$$ in which the number of particles is conserved ($b=\textnormal{const.>0}$). The action is then $$S=-\int d^{4}x\sqrt{-g}\left(s_{0}\right)^{b}\sqrt{1-X},$$ and therefore $$p=-\left(s_{0}\right)^{b}\sqrt{1-X},\quad\rho=\frac{\left(s_{0}\right)^{b}}{\sqrt{1-X}},$$ $$n=\left(s_{0}\right)^{b}\sqrt{\frac{X}{2\left(1-X\right)}},\quad T=\frac{b}{s_{0}}\sqrt{\frac{2}{X}}\left(1-X\right).$$ Solving the field equation (\[eq:dyngen\]) in a spatially flat FRW model, one obtains $$X(a)=\frac{1}{1+Ba^{6}},$$ where $B$ is an integration constant. From here one may evaluate all the hydrodynamical parameters in terms of the scale factor, arriving to the unusual result that the temperature of the Chaplygin gas rises with the expansion. This basically happens due to the negative pressure of the fluid [@Lima]. One can further solve, as well, the Friedmann equation to find the behaviour of the scale factor as a function of time [@Chap].
**Example 3**: Tachyon condensate. The possibility of fluid description of tachyon condensate in bosonic and supersymmetric string theories discovered by Sen [@sen] has motivated a considerable amount of work studying the consequences of the rolling tachyon in cosmology [@Feinstein; @tachyon]. Here we are interested to look at tachyon condensate action in the light of the formalism developed above as a fluid where the number of particles is not conserved. The action for the K-fluid with the form of the tachyon condensate is [@Feinstein; @sen]$$S=-\int d^{4}x\sqrt{-g}K(\phi)\sqrt{1-X}.$$ We can think of the above action as one describing a fluid with equation of state (\[eq:eqstate3\]) in which the particle rate production is modeled by (\[eq:s2\]). From this action we can reed off: $$p=-K(\phi)\sqrt{1-X},\quad\rho=\frac{K(\phi)}{\sqrt{1-X}},$$ $$n=K(\phi)\sqrt{\frac{X}{2(1-X)}},\quad T=\frac{b}{\left[K(\phi)\right]^{\frac{1}{b}}}\sqrt{\frac{2}{X}}\left(1-X\right).$$ It is simple to obtain particular cosmological solutions for such a fluid if one assumes a spatially flat isotropic cosmology and a potential of the form $K(\phi)=\beta/\phi^{2}$ with $\beta>0$ [@Feinstein]. Solving Einstein’s equations one finds for the velocity potential $\phi(t)$ and the scale factor of the universe $a(t)$: $$\phi(t)=\sqrt{\frac{4}{3n}}t,\quad a(t)=t^{n},\label{eq:scale}$$ where $n$ is a constant given in terms of the parameter of the potential $n=n(\beta)$. Therefore the parameter $\beta$, the slope of the potential, defines the particle creation rate as well as different expansion rate. For these particular solutions the enthalpy per particle of the fluid (\[eq:entalpia\]) remains constant. This is in contrast with the case of Chaplygin gas, where the entropy per particle was constant.
We can use the expressions (\[eq:NS\]) to evaluate the increase of the number of particles and entropy of the system in a time interval $\Delta t$. For the toy models with equation of state of the form (\[eq:eqstate2\]), and a particle creation rate modeled by $K(\phi)=\beta/\phi^{2}$, we obtain the following expressions in a spatially flat FRW universe: $$\Delta N(t_{1},t_{2})=-\frac{8\pi\beta}{3}F(X)\int_{t_{1}}^{t_{2}}\left(\frac{a}{\phi}\right)^{3}dt,$$ $$\Delta S(t_{1},t_{2})=-\frac{8\pi\beta^{\frac{1+b}{b}}}{3}F(X)\left[1+\frac{2XF'(X)}{bF(X)}\right]\int_{t_{2}}^{t_{1}}\frac{a^{3}}{\phi^{\frac{2+3b}{b}}}dt.$$ For the tachyon-like model, taking into consideration (\[eq:scale\]), we have $$\Delta N\propto t^{3n-2},\quad\Delta S\propto t^{\frac{3nb-2\left(b+1\right)}{b}}.$$ Since we must impose $X<1$ for the action to be well-defined, one has $n>2/3$, and, interestingly enough, this implies that the particles are created in such a universe. The creation rate is best visualised by the expression
$$\frac{1}{N}\frac{dN}{dt}=\frac{1}{\sqrt{2X}}\frac{\frac{d}{d\phi}\left[\ln K(\phi)\right]}{\frac{d}{dX}\left[\ln F(X)\right]}.$$ For the above tachyon example we readily find that the creation rate fades with time as $t^{-1}$.
We see that the fluid we have is the same as in Chaplygin gas (has the same equation of state), but the production of particles changes the evolution of the universe. Changing the particle creation rate one changes the expansion rate of the model.
CONCLUSIONS
===========
In this paper we have considered a Lagrangian approach to a Relativistic Hydrodynamics in which the number of particles is not conserved. The particle number non-conservation is modeled by introducing an explicit velocity potential dependent term into the fluid Lagrangian. In doing so, the usual shift symmetry of the action is broken, resulting in the appearance of a source term in the continuity equation. The conservation equation derived from the stress-energy tensor indicates that the particle number non-conservation must be balanced by an entropy flow. Both the entropy flow and the change in the particle number are expressed as function of the velocity potential. Although such a description is valid for a general flow, we concentrate on the purely potential fluid motion without vorticity, to make contact with some modern theories used for the description of matter in the universe.
By identifying the K-essence field variable $2X$ with the *square* of the enthalpy per particle $h$ we identify the K-field theory and the hydrodynamical Lagrangians we look at. In the case of purely kinetic K-essence, we observe that this theory is identical to the isentropic perfect fluid, and give a ‘dictionary’ (\[eq:ecF\]) as to how to pass from the usual description in cosmology in terms of the equation os state $p=p(\rho)$ to the K-theory Lagrangians of the form $F(X)$. On a formal level, therefore, the purely kinetic K-essence is no ‘big news’, but rather a simple conventional hydrodynamics in a disguise.
The non-conventional hydrodynamics (K-hydrodynamics), the one analogous to the K-essence with the potential term, is rather more involved. First, one must interpret such a hydrodynamics as a flow where the number of particles is not conserved. This, in turn, leads to a change in the entropy per particle, as well as to a global entropy flow. The fluid now is not isentropic and to give an hydrodynamical description the two equations of state $p=f(s)g(h)$ and $\psi=\psi(s,h)$ must be specified. We have found [@futuro] that our parametrisation works for source terms of the form $\psi(s,h)=k(s)g(h)$, i.e. the source term must be separable in functions of entropy and enthalpy, and the enthalpy function must be the same as the one which appears in the pressure. This restricts the generality of the approach, nevertheless, it is of direct application to the K-essence-like cosmologies.
We have finally cosidered several examples of fluids with both conserved and non-conserved number of particles in the context of spatially flat isotropic universe. The telling example is the comparison of Chaplygin gas on one hand and a K-fluid with the form of tachyon condensate on the other. In the first case one deals with an isentropic perfect fluid where the number of particles is conserved. The peculiarity of this example is that the temperature of the gas rises up with the expansion. The second example represents a fluid with the same equation of state, but with the number of particles (entropy per particle) not conserved. It is interesting, however, that for special creation rates, those with the potential $K=\beta/\phi^{2}$ with $\beta>0$, the enthalpy per particle rather than the entropy remains constant in the course of the expansion. We also observe that in such a universe creation rather than destruction of particles takes place.
From a technical point of view it appears that the velocity potential/$X$-variable formalism is quite useful to study the dynamics of the cosmological models. It would be interesting in the future to obtain and study some of the K-fluid type Lagrangians obtained as effective theories derived from fundamental interactions. The work in this direction is in progress and will be presented elsewhere.
We are grateful to Manuel Valle for enlighting discussions. A.D.T. is grateful to Ruth Lazkoz and Sanjay Jhingan for encouragement. A.D.T. work is supported by the Basque Government predoctoral fellowship BFI03.134. This work is supported by the Spanish Science Ministry Grant 1/MCYT 00172.310-15787/2004.
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[^1]: wtbditea@lg.ehu.es
[^2]: wtpfexxa@lg.ehu.es
[^3]: The case of pressureless dust ($w=0$) should be treated separately. In this case it is convenient to work with the equation of state $\rho=\rho(n)$.
|
---
abstract: 'We present a theoretical study of the motion of the antihydrogen atom ($\bar{H}$) in the Earth’s gravitational field above a material surface. We predict that $\bar{H}$ atom, falling in the Earth’s gravitational field above a material surface, would settle in long-living quantum states. We point out a method of measuring the difference in energy of $\bar{H}$ in such states that allow us to apply spectroscopy of gravitational levels based on atom-interferometric principles. We analyze a general feasibility to perform experiments of this kind. We point out that such experiments provide a method of measuring the gravitational force ($Mg$) acting on $\bar{H}$ and they might be of interest in a context of testing the Weak Equivalence Principle for antimatter.'
author:
- 'A. Yu. Voronin, P. Froelich, V.V. Nesvizhevsky'
bibliography:
- 'hbarclock.bib'
title: Antihydrogen Gravitational Quantum States
---
= 10000
Introduction
============
Galileo, Newton and Einstein recognized that all bodies, regardless of their mass and composition, fall towards the Earth with an equal gravitational acceleration. Is that conclusion valid for antimatter? This has never been tested.
In the context of the general relativity theory, the universality of free fall is often referred to as the Weak Equivalence Principle (WEP). Violations of WEP could occur in ordinary matter-matter interactions e.g. as a result of the difference in the gravitational coupling to the rest mass and that to the binding energy. WEP is being tested with increasing sensitivity for macroscopic bodies. The best test so far confirms WEP to the accuracy of $2\cdot 10^{-13}$ (using a rotating torsion balance [@schl08]). Ongoing projects aim at the accuracy of 1 part in $10^{16}$ (laser tracking of a pair of test bodies in a freely falling rocket [@reas10]), or even to 1 part in $10^{18}$ (in an Earth orbiting satellite [@over09]). However, in view of difficulties in unifying the quantum mechanics with the theory of gravity, it is of great interest to investigate the gravitational properties of [*quantum mechanical objects*]{}, such as elementary particles or atoms. Such experiments have been already performed, e.g. using interferometric methods to measure the gravitational acceleration of neutrons [@NeutInter1; @NeutInter2] and atoms [@kase92; @pete99; @fray04; @clad05]. However, the experiments with [*anti*]{}atoms ( see [@ATRAP; @ALPHA] and references therein) are even more interesting in view of testing WEP, because the theories striving to unify gravity and quantum mechanics (such as supersymmetric string theories) tend to suggest violation of the gravitational equivalence of particles and antiparticles [@Sherk]. Experiments testing gravitational properties of antiatoms are on the agenda of all experimental groups working with antihydrogen (see e.g. ATHENA-ALPHA [@cesa05], ATRAP [@gabr10] and AEGIS [@Aegis1]). One of the challenging aspects in experiments of this kind is to control the initial parameters of antiatoms, such as their temperature and position, with sufficient accuracy [@walz04].
In the present paper we investigate a possibility to explore gravitational properties of antiatoms in the ultimate quantum limit. We study antihydrogen atoms levitating in the lowest gravitational states above a material surface. The existence of such gravitational states for *neutrons* was proven experimentally [@nesv02; @nesv03; @nesv05]. The existence of analogous states for antiatoms seems, at a first glance, impossible because of annihilation of antiatoms in the material walls. However, we have shown that ultracold antihydrogen atoms are efficiently reflected from material surface [@voro05l; @voro05] due to so-called quantum reflection from the Casimir-Polder atom-surface interaction potential. We have shown that antihydrogen atoms, confined by the quantum reflection via Casimir forces from below, and by the gravitational force from above, would form metastable gravitational quantum states. They would bounce on a surface for a finite life-time (of the order of 0.1 s) [@voro05l]. This simple system can be considered as a microscopic laboratory for testing the gravitational interaction under extremely well specified (in fact, quantized!) conditions.
The annihilation of ultraslow antiatoms in a wall occurs with a small but finite (few percent) probability. It provides a clear and easy-to detect signal, which might be used to measure continuously the antiatom density in the gravitational states as a function of time. If antiatoms are settled in a superposition of gravitational states, the antiatom density evolves with beatings, determined by the [*energy difference*]{} between the gravitational levels. The transition frequencies between the gravitational levels are related to the strength of the gravitational force $Mg$, acting on antiatoms; here $M$ is the gravitational mass of $\bar{H}$, and $g$ is the Earth’s local gravitational field strength. Also we show that a measurement of [*differences*]{} between the energy levels would allow us to disentangle $Mg$ in a way independent on effects of the antiatom-surface interaction. The plan of the paper is the following. In section \[gravstates\] we study the main properties of the quasi-stationary gravitational states; in section \[BounceHbar\] we present the time evolution of the $\bar{H}$ gravitational states superposition; in section \[QBE\] we discuss a concept of a quantum ballistic experiment, namely the spatial-temporal evolution of the $\bar{H}$ gravitational states superposition, in section \[Feas\] we analyze the feasibility of measuring $\bar{H}$ atom properties in gravitational quantum states. In the Appendix we derive useful analytical expressions for the quasi-stationary gravitational states scalar product.
$\bar{H}$ gravitational states {#gravstates}
==============================
In this section we discuss the properties of the $\bar{H}$ gravitational states above a material surface.
We consider an $\bar{H}$ atom bouncing above a material surface in the Earth’s gravitational field. $\bar{H}$ is confined due to the quantum scattering from the Casimir-Polder potential below, and the gravitational field above. The Schrödinger equation for the $\bar{H}$ wave-function $\Psi(z)$ in such a superposition of atom-surface and gravitational potentials is: $$\label{Schr}
\left[ -\frac{\hbar^2 \partial ^{2}}{2m\partial z^{2}}+V(z)+Mgz-E\right] \Psi
(z)=0$$ Here $z$ is the distance between the surface and the $\bar{H}$ atom, and $V(z)$ is the atom-surface interaction potential with a long-range asymptotic form $V(z)\sim-C_4/z^4$. We distinguish between the gravitational mass, that we refer to as $M$ and the inertial mass, denoted by $m$ hereafter. The wave-function $\Psi(z)$ satisfies the full absorption boundary condition at the surface ($z=0$) [@voro05], which stands for the annihilation of antiatoms in the material wall.
The characteristic length and energy scales are $$\begin{aligned}
\label{scaleL}
l_0 &=&\sqrt[3]{\frac{\hbar^{2}}{2mMg}},\\
l_{CP}&=&\sqrt{2mC_4}, \label{scaleCP}\\
\varepsilon_0 &=&\sqrt[3]{\frac{\hbar^2M^2g^2}{2m}},\label{scaleE}\\
\varepsilon_{CP}&=&\frac{\hbar^2}{4 m^2 C_4}.\label{scaleEcp}\end{aligned}$$
Here $l_0 =5.871$ $\mu m$ is the characteristic gravitational length scale, $l_{CP}=0.027$ $\mu m$ is the characteristic Casimir-Polder interaction length scale, $\varepsilon_0=2.211$ $10^{-14}$ a.u. is the characteristic gravitational energy scale, and $\varepsilon_{CP}=1.007$ $10^{-9}$ a.u. is the Casimir-Polder energy scale. As one can see, the gravitational length scale is much larger than the Casimir-Polder length scale $l_0\gg l_{CP}$, while the gravitational energy scale is much smaller than the Casimir-Polder energy scale $\varepsilon_0\ll \varepsilon_{CP}$. It is useful to introduce the gravitational time scale $\tau_0$: $$\label{tau0}
\tau_0=\hbar/\varepsilon_0\simeq 0.001s$$
For large atom-surface separation distances $z\gg l_{CP}$ the solution of eq.(\[Schr\]) has a form: $$\label{gravfree}
\Psi(z)\sim \mathop{\rm Ai}( \frac{z}{l_0}-\frac{E}{\varepsilon_0})+K(E)\mathop{\rm
Bi}(\frac{z}{l_0}-\frac{E}{\varepsilon_0})$$ where $\mathop{\rm Ai}(x)$ and $\mathop{\rm Bi}(x)$ are the Airy functions [@abra72]. The requirement of square integrability of the wave-function $\Psi(z\rightarrow \infty)\rightarrow 0$ results in the following equation for the energy levels of the gravitational states in the presence of the Casimir-Polder interaction: $$\label{eigenexact}
K(E_n)=0$$
The hierarchy of the Casimir-Polder and gravitational scales $l_{CP}\ll l_0$ suggests that the quantum reflection from the Casimir-Polder potential can be accounted for by modifying the boundary condition for the quantum bouncer (a particle bouncing in the gravitational field above a surface, the interaction of the latter with a particle is modeled by infinite reflecting wall). The quantum bouncer wave-function satisfies the following equation system:
$$\label{gravEq}
\left\{\begin{array}{cll}\left[ -\frac{\hbar^2 \partial ^{2}}{2m\partial z^{2}}+Mgz-E_n\right] \Phi_n
(z)=0\\
\Phi_n(z= 0)= 0
\end{array}
\right.$$
The quantum bouncer energy levels are known to be equal to [@nesv02]: $$\begin{aligned}
\label{En0}
E_n^0&=&\varepsilon_0 \lambda_n^0, \\ \label{lambda0}
\mathop{\rm Ai}(-\lambda_n^0)&=&0.\end{aligned}$$
Table \[Table1\] summarizes the eigenvalues and classical turning points $z_n^0=E_n^0/(Mg)$ for the first seven gravitational states of a quantum bouncer (with the mass of antihydrogen).
$n$ $\lambda_n^0$ $E_n^0$, peV $z_n^0$, $\mu m$
----- --------------- -------------- ------------------
1 2.338 1.407 13.726
2 4.088 2.461 24.001
3 5.521 3.324 32.414
4 6.787 4.086 39.846
5 7.944 4.782 46.639
6 9.023 5.431 52.974
7 10.040 6.044 58.945
: The eigenvalues, gravitational energies and classical turning points of a quantum bouncer with a mass of (anti)hydrogen in the Earth’s gravitational field. []{data-label="Table1"}
For the distances $l_{CP}\ll z \ll l_0$ one could neglect the gravitational potential in eq.(\[Schr\]). In this approximation, the solution of eq.(\[Schr\]) has the following asymptotic form: $$\label{free}
\Psi(z)\sim \sin(k z +\delta(E)).$$ Here $k$ is the wave vector $k=\sqrt{2mE}$, and $\delta(E)$ is the phase-shift of $\bar{H}$ reflected from the Casimir-Polder potential *in absence of the gravitational field* [@voro05]. Matching asymptotics in eq.(\[free\]) and eq.(\[gravfree\]) we get a relation between the phase-shift $\delta(E)$ and the $K-$function introduced in Eq.(\[gravfree\]): $$\label{Kapprox}
K(E)=-\frac{\tan(\delta(E))\mathop{\rm Ai'}(-E/\varepsilon_0)-k l_0 \mathop{\rm
Ai}(-E/\varepsilon_0)}{\tan(\delta(E))\mathop{\rm Bi'}(-E/\varepsilon_0)-k l_0 \mathop{\rm Bi}(-E/\varepsilon_0)}.$$ In deriving the above expression we took into account that the relation between $K(E)$ and $\delta(E)$ should not depend on the matching point $z_m$ and thus can be formally attributed to $z_m=0$. An equation for the distorted gravitational levels could be obtained by substitution of eq.(\[Kapprox\]) into eq.(\[eigenexact\]): $$\label{BC}
\frac{\tan(\delta(E_n))}{k l_0}=\frac{\mathop{\rm Ai}(-E_n/\varepsilon_0)}{\mathop{\rm Ai'}(-E_n/\varepsilon_0)}.$$ This equation is equivalent to the following boundary condition: $$\label{BCmodified}
\frac{\Phi( 0)}{\Phi'( 0)}=\frac{\tan(\delta(E_n))}{k }.$$ Thus the following equation system describes an $\bar{H}$ atom, confined by the Earth’s gravitational field and the quantum reflection from the Casimir-Polder potential: $$\label{gravEqmod}
\left\{\begin{array}{cll}\left[ -\frac{\hbar^2 \partial ^{2}}{2m\partial z^{2}}+Mgz-E_n\right] \Phi_n
(z)=0\\
\frac{\Phi( 0)}{\Phi'( 0)}=\frac{\tan(\delta(E_n))}{k }
\end{array}
\right.$$
For the lowest gravitational states the condition $k l_{CP} \ll 1$ is valid. Thus the scattering length approximation for the phase-shift $\delta(E)\approx -k a_{CP}$ is well justified. The *complex-value* quantity [@voro05]: $$\begin{aligned}
a_{CP}&=&-(0.10+i1.05)l_{CP},\\
a_{CP}&=&-0.0027-i0.0287 \mu m\end{aligned}$$ is the scattering length on the Casimir-Polder potential provided full absorbtion in the material wall.
Thus the equation for the lowest eigenvalues (\[BC\]) has a form: $$\label{BClow}
\frac{a_{CP}}{l_0}=-\frac{\mathop{\rm Ai}(-E_n/\varepsilon_0)}{\mathop{\rm Ai'}(-E_n/\varepsilon_0)}.$$
The above equation is equivalent to the following boundary condition for the wave-function $\Phi(z)$ of a particle in the gravitational potential eq.(\[Schr\]): $$\label{BClow1}
\Phi(z\rightarrow 0)\rightarrow z-a_{CP}$$
Because of the imaginary part of the scattering length $a_{CP}$, the gravitational states of $\bar{H}$ above a material surface are *quasi-stationary decaying* states. For low quantum numbers $n$, it is easy to relate the lowest quasi-stationary energy levels $E_n$ to the unperturbed gravitational energy levels $E_n^0$ of a quantum bouncer. Indeed the variable substitution $z=\widetilde{z}+a_{CP}$ transforms Eq.(\[gravEq\],\[BClow1\]) to the equation system for the quantum bouncer: $$\begin{aligned}
\label{gravEq1}
\left[ -\frac{\hbar^2\partial ^{2}}{2m\partial \widetilde{z}^{2}}+Mg\widetilde{z}-(E_n-Mg a_{CP})\right] \Phi_n
(\widetilde{z})=0\\
\Phi_n(\widetilde{z}\rightarrow 0)\rightarrow 0 \label{BC1}\end{aligned}$$
The eigenvalues $E_n$ and eigenfunctions $\Phi_n$ are : $$\begin{aligned}
\label{energy}
E_n=E_n^0+Mga_{CP},\\
\Phi_n(z)=\frac{1}{N_i}\mathop{\rm Ai}((z-a_{CP})/l_0-\lambda_n^0),\label{Phi}\end{aligned}$$ where $N_i$ is the normalization coefficient (see Eqs.(\[normex\],\[norm\]) in the Appendix). In the following, we will use the dimensionless eigenvalues $\lambda_n=E_n/\varepsilon_0$:
$$\label{lambdan}
\lambda_n=\lambda_n^0+a_{CP}/l_0$$
An important message from the above expression is that the complex shift $ Mg a_{CP}$ (due to the account of quantum reflection on the Casimir-Polder potential) is *the same* for all low-lying quasi-stationary gravitational levels. It means that the transition frequencies between the gravitational states are not affected by the Casimir-Polder interaction, provided the latter can be described by the complex scattering length $a_{CP}$. The scattering approximation is valid in the limit $k_n a_{CP}\rightarrow 0$, where $k_n=\sqrt{2mE_n}$ (let us note that for the first gravitational state $|k_1 a_{CP}|=0.0071$). However, accounting for the higher order $k$-dependent terms in Eq.(\[BC\]) would result in the state dependent shift of the gravitational states due to the Casimir-Polder interaction. We use a known low energy expansion of the $s$-wave phase-shift $\delta(E)$ in a homogeneous $1/z^4$ potential [@EffRad], in which we keep the two leading $k$-dependent terms: $$\tilde{a}_{CP}(k) \cot\left(\delta(k)\right)\simeq
-1+\frac{\pi}{3}\frac{l_{CP}}{a_{CP}}(l_{CP}k)+\frac{4}{3}(l_{CP}k)^2\ln\left(\frac{l_{CP}k}{4}\right)+...$$ We introduce a k-dependent modified “scattering length” $\tilde{a}_{CP}(k)\equiv-\delta(k)/k$ and get the following expression for $\tilde{a}_{CP}(k)$:
$$\label{ak}
\tilde{a}_{CP}(k)\simeq a_{CP}+\frac{\pi}{3}l_{CP} (l_{CP}k)+\frac{4}{3}a_{CP}(l_{CP}k)^2\ln\frac{l_{CP}k}{4}$$
The leading k-dependent term in the above expression $\frac{\pi}{3}l_{CP} (l_{CP}k)$ is real and independent on properties of the inner part of the Casimir-Polder interaction. It is determined by the asymptotic form of the potential, thus it depends on the Casimir-Polder length scale $l_{CP}$ only. Then the modified equation for the gravitational state energies is: $$E_n=E_n^0+Mg \tilde{a}_{CP}(E_n)$$ Taking into account the smallness of the k-dependent terms (for the lowest gravitational states) in expression (\[ak\]), we get: $$E_n\simeq E_n^0+Mg \tilde{a}_{CP}(k_n^0)= \varepsilon\left(\lambda_n^0+ a_{CP}/l_0+\frac{\pi l_{CP}}{3 l_0}
(l_{CP}k_n^0)+\frac{4 a_{CP}}{3l_0}(l_{CP}k_n^0)^2\ln\frac{l_{CP}k_n^0}{4}\right).$$ Here $k_n^0=\sqrt{2 m E_n^0}$.
The account for k-dependent terms in Eq.(\[ak\]) modifies the transition frequencies between the gravitational states in a way, dependent on the Casimir-Polder interaction. However, such modification is very weak. Indeed, taking into account, that $l_{CP} k_n^0\sim l_{CP}/l_0$ for the lowest gravitational states , the leading k-dependent term corrections to the gravitational energy are of the second order in a small parameter $l_{CP}/l_0$. The transition frequency between the first and second gravitational states equals $\omega_{12}=\omega_{12}^0+\Delta_{12}$, where $\omega_{12}^0=(E_2^0-E_1^0)/(2\pi\hbar)=254.54$ Hz, and $\Delta_{12}=Mg(\tilde{a}_{CP}(k_2^0)-\tilde{a}_{CP}(k_1^0))=
0.0017$ Hz.
An account of the first two terms in Eq.(\[ak\]) provides equal decay width for the lowest gravitational states. This width is determined by the probability of antihydrogen penetrating to the surface and annihilating. $$\Gamma_n= \varepsilon \frac{b}{2l_0}.\label{Wgrav}$$ Here we use a standard notation $b=4\mathop{\rm Im}a_{CP}$: $$b=0.115 \mbox{ } \mu m.$$ The widths of the gravitational states (\[Wgrav\]) are proportional to the ratio $\varepsilon/l_0$. Using Eqs. (\[scaleL\]) and (\[scaleE\]) we could find that this ratio is equal to the gravitational force $\varepsilon/l_0=Mg$ so that $$\Gamma_n = \frac{b}{2} Mg.$$
The corresponding life-time (calculated for an ideal conducting surface) is $$\tau=\frac{2\hbar}{Mgb} \simeq 0.1 \mbox{ s}. \label{time}$$ We note factorization of the gravitational effect (appearing in the above formula via a factor $Mg$) and the quantum reflection effect, manifestating through the constant $b$. Such a factorization is a consequence of the smallness of the ratio of the characteristic scales $b/(2l_0)\simeq 0.01$.
Comparing the $\bar{H}$ lifetime in the lowest gravitational states with the classical period $T=2\sqrt{\frac{2l_0
\lambda_1}{g}}\simeq 0.0033$ s of $\bar{H}$ with the ground state energy bouncing in the gravitational field, we see that $\bar{H}$ bounces in average about $30$ times before annihilating. This shows that the lowest gravitational states are well resolved quasi-stationary states.
It is interesting to estimate the maximum gravitational quantum number $N$, below which the gravitational states are still resolved, i.e: $$\label{resolve}
\frac{\tau_{N}}{T_{N}}=\frac{2\pi \hbar}{\Gamma(N)\frac{dE(n)}{dn}}> 1.$$ Here $\tau_{N}$ is the lifetime of the $N-$th gravitational state, $T_N$ is a classical period, corresponding to the $N-$th state via $T_N=2\hbar\pi/(dE(n)/dn)$. For such an estimation we transform eq.(\[BC\]) using the asymptotic form of the Airy function for a large negative argument and get: $$\label{Esemiclass}
\lambda_n=\left( \frac{3}{2}(\pi [n-\frac{1}{4}]-\delta(E_n))\right)^{2/3}.$$ The accuracy of the above equation increases with increasing $n$; it gives the energy value within a few percent even for $n=1$. In the energy domain of interest $|\delta(E)|\ll \pi [n-\frac{1}{4}]$ [@voro05], so: $$\label{Esemiclass1}
\lambda_n\simeq \lambda_n^0-\frac{\delta(E_n^0)}{\sqrt{\lambda_n^0}}.$$ Here we used the semiclassical approximation for $\lambda_n^0$ [@NVP]: $$\lambda_n^0\simeq \left( \frac{3}{2}\pi [n-\frac{1}{4}]\right)^{2/3}$$ One could verify that in the case of small $n$ the above equation reduces to eq.(\[lambdan\]). Substitution of eq.(\[Esemiclass1\]) into eq.(\[resolve\]) results to: $$\label{resolve1}
\frac{\tau_{n}}{T_{n}}\simeq \frac{1}{4\mathop{\rm Im}\delta(E_n^0)}.$$ The ratio $\tau(n)/T(n)$ expresses the number of $\bar{H}$ classical bounces during the lifetime of the $n$-th state. This dependence is shown in Fig.\[resol\].
![The number of $\bar{H}$ bounces during the lifetime of $n-$th gravitational state[]{data-label="resol"}](bouncen.eps){width="120mm"}
Using numerical values $\delta(E)$, calculated in [@voro05], we find that inequality (\[resolve\]) holds for $$n<N=30000.$$ The corresponding energy $E_{N}=6$ $10^{-11}$ a.u., and the characteristic size of such states is as large as $H_{N}=1.6$ $cm$. This means that the concept of the quasi-bound gravitational states is justified not only for the lowest states, but it also might be applied for highly excited states.
The quasi-stationary character of the antiatom gravitational states above a material surface manifests itself in a nonzero current through the bottom surface ($z=0$). Indeed, the expression for the current is: $$\label{j}
j(z,t)=\frac{i\hbar}{2M}\left(\Phi(z,t) \frac{d \Phi^*(z,t)}{dz}-\Phi^*(z,t) \frac{d \Phi(z,t)}{dz}\right).$$ taken at $z=0$ for a given gravitational state (\[Phi\]) turns out to be equal to: $$\label{j0}
j(0,t)=\varepsilon \exp(-\frac{\Gamma}{\hbar} t) \frac{\mathop{\rm Ai^*}(-\lambda_n)\mathop{\rm
Ai'}(-\lambda_n)-\mathop{\rm Ai}(-\lambda_n) \mathop{\rm Ai'^*}(-\lambda_n) }{N_iN_i^*}.$$ Here $\lambda_n$ is given by Eq. (\[lambdan\]), $N_i$ is the normalization factor. We take into account the smallness of the ratio $a_{CP}/l_0$ and Eq.(\[lambda0\]), and get :
$$\mathop{\rm Ai}(-\lambda_n)\approx-\frac{a_{CP}}{l_0}\mathop{\rm Ai'}(-\lambda_n^0),$$
which is exact up to the second order in the ratio $a_{CP}/l_0$. Now taking into account an explicit form of the normalization coefficients (Eq.(\[norm\]) in Appendix) $N_i=\mathop{\rm Ai'}((-\lambda_n^0)$, we get finally: $$\label{jex}
j(0,t)=-\varepsilon \frac{b}{2\hbar l_0}\exp(-\frac{\Gamma}{\hbar} t)=-\frac{\Gamma}{\hbar} \exp(-\frac{\Gamma}{\hbar}
t).$$ This result is in full agreement with Eq.(\[Wgrav\]) as far as: $$\frac{d}{dt}\int_0^{\infty}|\Phi(z,t)|^2dz=j(0,t)=-\frac{\Gamma}{\hbar}\exp(-\frac{\Gamma}{\hbar} t).$$
Bouncing antihydrogen {#BounceHbar}
=====================
In this section, we are interested in the evolution of an initially prepared arbitrary superposition of several lowest gravitational states of $\bar{H}$. In the following, we will limit our treatment to the scattering length approximation for describing the Casimir-Polder interaction, and will neglect all, except the first, term in the expression (\[ak\]), so that $\tilde{a}_{CP}(k)\approx a_{CP}$. The corresponding $\bar{H}$ wave-function is: $$\label{superpos}
\Phi(z,t)=\sum_{i=1}^n \frac{C_i}{N_i}\mathop{\rm Ai}(z/l_0-\lambda_i)\exp(-i \lambda_i \frac{t}{\tau_0}).$$ Here $\tau_0$ is the characteristic $\bar{H}$ bouncer time scale, $C_i$ are expansion coefficients and $N_i=\mathop{\rm Ai'}(-\lambda_i)$ are the normalization factors of the gravitational states (see the Appendix). We are interested in the evolution of the number of antihydrogen atoms as a function of time: $$\label{Flux}
F(t)=\int_0^\infty |\Phi(z,t)|^2dz= \sum_{i,j=1}^n \int_0^\infty \frac{C_{j}^*C_i}{N_j^*N_i}\mathop{\rm
Ai^*}(z/l_0-\lambda_{j})\mathop{\rm Ai}(z/l_0-\lambda_i)\exp(-i\varepsilon (\lambda_i-\lambda_{j}^*)t)dz$$
First, let us note that the above expression for the total number of particles is no longer constant because of the decay of the quasi-stationary gravitation states. Second, the quasi-stationary gravitational states corresponding to different energies are non-orthogonal: $$\frac{1}{N_iN_j}\int_0^\infty\mathop{\rm Ai^*}(z/l_0-\lambda_{j})\mathop{\rm Ai}(z/l_0-\lambda_i) dz\equiv
\alpha_{ij}\neq\delta_{ij}.$$ In the Appendix we will derive the following expression for the cross-terms $\alpha_{ij}$, exact up to the second order of the small ratio $a_{CP}/l_0$: $$\label{crossTA}
\alpha_{i\neq j}=i\frac{b/(2l_0)}{\lambda_j^0-\lambda_i^0+i b/(2l_0)}$$ As one can see, such cross-terms vanish if there is no decay, i.e. if $b=4\mathop{\rm Im}a_{CP}\rightarrow 0$.
Now we can calculate an expression for the number of antihydrogen atoms as a function of time (\[Flux\]): $$\label{Ft}
F(t)=\exp(-\frac{\Gamma}{\hbar} t) \left(\sum_{i}^n |C_i|^2+ 2\mathop{\rm Re}\sum_{i>j}^n\sum_{j}^n C_{j}^*C_{i}\frac{i
b/(2l_0)}{\lambda_j^0-\lambda_i^0+i b/(2l_0)} \exp(-i(\lambda_i^0-\lambda_j^0)\frac{t}{\tau_0})\right).$$
From Eqs.(\[Ft\]) and (\[Wgrav\]) we get the following expression for the disappearance (annihilation) rate $-\frac{dF(t)}{dt}$, keeping the terms up to the second order in the ratio $a_{CP}/l_0$: $$\label{Nt}
\frac{dF(t)}{dt}=-\frac{\Gamma}{\hbar} \exp(-\frac{\Gamma}{\hbar} t) \left(\sum_{i}^n |C_i|^2+2\mathop{\rm
Re}\sum_{i>j}^n\sum_{j}^n C_{j}^*C_{i}\exp(-i(\lambda_i^0-\lambda_j^0)\frac{t}{\tau_0})\right).$$
For a superposition of the two gravitational states with the equal coefficients $C_{1,2}$ (say $C_1=C_2=1$), the above expression gets a simple form: $$\label{2st}
\frac{dF_{12}(t)}{dt}=-\frac{\Gamma}{\hbar} \exp(-\frac{\Gamma}{\hbar} t)\left (1+\cos(\omega_{12}t)\right ).$$ Here $\omega_{12}=(\lambda_2^0-\lambda_1^0)/\tau_0$. The same result could be obtained by calculating the flux $j(0,t)$ Eq.(\[j\]) for a superposition of states (\[superpos\]).
One can see, that the disappearance rate decays as a function of time according to the exponential law with the width $\Gamma$ (the same for the lowest states), also it oscillates with the transition frequency between the first and second gravitational states (equal to $254.54$ Hz). We plot in Fig.\[twost\] the time evolution of $\bar{H}$ disappearance rate in a superposition of two lowest states. Curiously, the oscillation of disappearance rate is the direct consequence of decaying character of gravitational states. Indeed, such an oscillation is observable due to the nonvanishing contribution of the interference term in the expression for the total probability to find antihydrogen atoms, given by Eq.(\[Ft\]). As one can see from Eq.(\[crossTA\]) this contribution is proportional to the imaginary part of the scattering length, it would vanish in case there were no decay of gravitational states due to annihilation in the material wall, described by parameter $b/l_0$.
![Evolution of the annihilation rate of $\bar{H}$ atom in a superposition of the first and the second gravitational states.[]{data-label="twost"}](plot2st.eps){width="100mm"}
![Evolution of the annihilation rate of $\bar{H}$ atom in a superposition of first, second and third gravitational states.[]{data-label="threest"}](plot3st.eps){width="100mm"}
So far the oscillation frequency of the disappearance rate $N(t)$ corresponds to the energy difference between the unperturbed gravitational levels. Expression (\[Nt\]) does not include the shift of gravitational state energies $\mathop{\rm Re}a_{CP}/l_0$ as it is equal for all the gravitational states, thus it is canceled out in the energy difference. The account for higher order k-dependent terms in (\[ak\]) would result in a small (second order of the ratio $(a_{CP}/l_0)$) correction to the transition frequency. A measurement of the oscillation frequency $\omega_{12}$ given by Eq. (\[2st\]) would allow us to extract the following combination of the gravitational and the inertial masses from eq.(\[scaleE\]): $$\label{Mm}
\frac{M^2}{m}=\frac{2 \hbar \omega_{12}^3}{g^2(\lambda_2^0-\lambda_1^0)^3}.$$ Under the additional assumption of the equality of the known inertial mass of the *hydrogen* atom $m_H$ and that of antihydrogen, imposed by CPT, we get: $$M=\sqrt{\frac{2 m_H \hbar \omega_{12}^3}{g^2(\lambda_2^0-\lambda_1^0)^3}}.$$
The evolution of *three* gravitational state superposition provides information not only about the characteristic energy scale $\varepsilon_0$ but also about the level spacing as a function of quantum number $n$, characterized by the value $d^2E(n)/dn^2$. Such a study might be interesting for testing additional (to Newtonian gravitation) interactions (see [@Axion; @ShortRange] and references there in) between $\bar{H}$ and a material surface with the characteristic spatial scale of the order of micrometers. Such interactions would manifest as nonlinear additions to the gravitational potential, which would modify the spectrum character. In the case of three state superposition, the disappearance rate (\[Nt\]) has the form: $$\label{3st}
\frac{dF_{123}(t)}{dt}=-\frac{2}{3}\frac{\Gamma}{\hbar} \exp(-\frac{\Gamma}{\hbar}
t)\left(\frac{3}{2}+\cos(\omega_{12}t)
+\cos(\omega_{23}t)+\cos((\omega_{12}+\omega_{23})t)\right).$$ Here $\omega_{ij}=(\lambda_j^0-\lambda_i^0)/\tau_0$. One could verify that the period of coherence of $\cos(\omega_{12}t)$ and $\cos(\omega_{23}t)$ terms is: $$\label{Trev}
T_{r}=\frac{2\pi}{\omega_{12}-\omega_{23}}\simeq 0.02 s.$$ A semiclassical expression for $T_r$ is: $$T_r\approx \frac{2\pi}{|d^2E/dn^2|}.$$ One can see that the period $T_r$ is a quantum limit analog of a half revival period $T_{rev}=4\pi/|d^2E/dn^2|$ ($T_{rev}$ characterizes the time period after which the evolution of the wave-packet returns to the semiclassical behavior, see [@Reviv] for details and reference therein). In Fig.\[threest\] we plot the annihilation events as a function of time (\[3st\]) for a superposition of three lowest gravitational states. The period $T_r$ is clearly seen as a period of modulation of a rapidly oscillating function. The ratio $$T_r/\tau_0=\frac{2\pi}{\lambda_3-2\lambda_2+\lambda_1}$$ is sensitive to any nonlinear addition to the gravitational potential. Indeed while linear corrections to gravitational potential can only change $\varepsilon_0$, nonlinear additions change the derivative of levels density $|d^2E/dn^2|$.
Quantum ballistic experiment {#QBE}
============================
Two independent experiments are needed in order to determine the gravitational mass $M$ and the inertial mass $m$ of antihydrogen. In the previous section, we showed that a combination of gravitational and inertial masses $M$ and $m$, given by Eq.(\[Mm\]), can be extracted from the frequency measurement eq.(\[2st\], \[3st\]). An independent information could be obtained from measurement of the spatial density distribution of $\bar{H}$ in a superposition of the gravitation states, for instance, in the flow-throw experiment (a kind of a beam scattering experiment), in which $\bar{H}$-atoms with a wide horizontal velocity distribution move along the mirror surface. The time of flight along the mirror should be measured simultaneously with the spatial density distribution in a position-sensitive detector, placed at the mirror exit. Such a detector would be able to measure the density distribution along the vertical axis at a given time instant. The horizontal component of $\bar{H}$ motion could be treated classically. Due to a broad distribution of horizontal velocities in the beam, atoms would be detected within a wide range of time intervals between their entrance to the mirror and their detection at the exit. In such an approach, we could study the time evolution of $\bar{H}$ probability density at a given position $z$: $$\begin{aligned}
\label{Pzt}
|\Phi_{(12)}(z,t)|^2&=&\exp(-\frac{\Gamma}{\hbar} t)\left( |\Phi_{(12)}^{av}(z)|^2
+2\mathop{\rm Re}\Phi_{(12)}^{int}(z)\exp(-i\omega_{12} t)\right) \\
|\Phi_{(12)}^{av}(z)|^2&=&\left|\frac{\mathop{\rm Ai}(z/l_0-\lambda_1)}{\mathop{\rm Ai'}(-\lambda_1)}\right|^2+
\left|\frac{\mathop{\rm Ai}(z/l_0-\lambda_2)}{\mathop{\rm Ai'}(-\lambda_2)}\right|^2 \\
\Phi_{(12)}^{int}(z)&=& \frac{\mathop{\rm Ai}(z/l_0-\lambda_1)\mathop{\rm Ai}(z/l_0-\lambda_2)}{\mathop{\rm Ai'}(-\lambda_1)\mathop{\rm Ai'}(-\lambda_2)}\end{aligned}$$
The transition $\omega_{12}=254.54 Hz$ could be extracted from the probability density time evolution at a given $z$. The length scale $l_0$ could be extracted from the position of the zero $z_1^{(2)}$ of the wave-function. Here superscript stands for the state number, and a subscript corresponds to the number of zero for a given state. Thus such a position is determined by the condition $\mathop{\rm Ai}(z_1^{(2)}/l_0-\lambda_2)=0$; is equal to the following expression: $$z_1^{(2)}=(\lambda_2-\lambda_1)l_0=10.27\mu m.$$ The probability density in Eq.(\[Pzt\]) of a two states superposition at $z=z_1^{(2)}$ behaves like the probability density of the ground state alone: $$|\Phi_{12}(z_1^{(2)},t)|^2=\exp(-\frac{\Gamma}{\hbar} t)\left|\frac{\mathop{\rm
Ai}(z_1^{(2)}/l_0-\lambda_1)}{\mathop{\rm Ai'}(-\lambda_1)}\right|^2.$$ The probability density at a height $z_1^{(2)}$ does not exhibit any time-dependent oscillations. We show the probability density as a function of the height $z$ above a mirror (y-axis) and the time $t$ (x-axis) in a superposition of the first and second gravitational states in Fig.\[Fig2zt\].
![Color.The probability density of $\bar{H}$ in a superposition of the first and second gravitational states, as a function of the height $z$ above a mirror (vertical-axis) and the time $t$ (horizontal-axis). Dark shade: low probability density. Light shade: high probability density. The dashed line indicates the position of the node in the wave-function of the second state[]{data-label="Fig2zt"}](bounce.eps){width="100mm"}
The position of the node in the wave-function of the second state is shown in Fig.\[Fig2zt\] as a horizontal line, separating lower and upper rows of periodic maxima and minima in the probability density plot. The position $z_1^{(2)}$ does not depend on initial populations of the gravitational states, which makes it beneficial for extracting the spatial scale $l_0$.
From knowing the length $l_0$ and the energy $\varepsilon_0$ scales, one could get the following expressions for the inertial $m$ and gravitational $M$ masses of $\bar{H}$: $$\begin{aligned}
\label{mM}
m&=&\frac{\hbar^2}{2\varepsilon_0 l_0^2},\\
M&=&\frac{\varepsilon_0}{gl_0}.\end{aligned}$$ The equality $m=M$ postulated by EP relates $\varepsilon_0$ and $l_0$ as follows: $$\varepsilon_0=\hbar \sqrt{\frac{g}{2l_0}},$$ or, using the gravitational time scale Eq.(\[tau0\]): $$\tau_0= \sqrt{\frac{2l_0}{g}}.$$ One can easily recognize in the above expression a classical time of fall from the height $\l_0$ in the Earth’s gravitational field.
Thus a measurement of the temporal-spatial probability density dependence of $\bar{H}$ in a superposition of the two lowest gravitation states would provide a full information on the gravitational properties of antimatter. The superposition of three (and more) gravitational states could be useful to search for additional (to gravity) interactions with a spatial scale of the order of $l_0$. For such a purpose, it is useful to study the probability density at zeros of each Airy function in the state superposition. In particular, the positions corresponding to zeros of second and third gravitational states are the following: $$\begin{aligned}
z_1^{(2)}&=&(\lambda_2-\lambda_1)l_0=10.27 \mu m,\\ \label{z12}
z_1^{(3)}&=&(\lambda_3-\lambda_2)l_0=8.41 \mu m, \\ \label{z13}
z_2^{(3)}&=&z_1^{(2)}+z_1^{(3)}=18.68 \mu m, \label{z23}
\end{aligned}$$ The three state $(ijk)$ superposition probability density at position of zero $z_i^k$ is equal to the two state superposition $(jk)$ probability: $$|\Phi_{(ijk)}(z_n^{(k)},t)|^2=|\Phi_{(ij)}(z_n^{(k)},t)|^2.$$ This means that the three state probability density exhibits harmonic time-dependent oscillation with a frequency $\omega_{ij}$ at height of zero $z_n^k$. Let us mention that the time dependence of $|\Phi_{ijk}(z,t)|^2$ in any position $z$, except for the mentioned zeros is not harmonic; it is given by a superposition of three cosine functions with different frequencies, analogous to Eq(\[3st\]). This property allows us to extract the zeros positions using the probability density. One can see that the knowledge of zeros (Eq.(\[z12\]-\[z23\])) is analogous to the knowledge of the transition frequencies. A measurement of one zero position allows us to extract the spatial scale $l_0$, a measurement of positions of two or more zeros allows us to constrain hypothetical nonlinear additions to the gravitational potential. We show the probability density as a function of height $z$ above the mirror (y-axis) and the time $t$ (x-axis) in a superposition of first, second and third gravitational state in Fig.\[Fig3zt\].
![Color. The probability density of $\bar{H}$ in a superposition of the first, second and third gravitational states, as a function of the height $z$ above a mirror (vertical-axis) and the time $t$ (horizontal-axis). Dark shade: low probability density. Light shade: high probability density. The dashed lines indicate the positions of the nodes in the wave-functions of the second and third state.[]{data-label="Fig3zt"}](bounce2.eps){width="100mm"}
Feasibility of antihydrogen gravitational states experiment {#Feas}
===========================================================
In this section we study the feasibility of an experiment on the gravitationally bound quantum states of anti-hydrogen atoms. For such an estimation, we compare it with the already performed gravitational states experiments using ultra-cold neutrons (UCN) [@nesv02; @nesv03; @nesv05]. The UCN gravitational experiments can be used as a benchmark for such a comparison because 1) the neutron mass is nearly equal to the anti-hydrogen mass, 2) any modification to the $\bar{H}$ quantum state energies and wave-functions following from the precise shape of the Casimir-Polder potential are small compared to those of the quantum bouncer; UCN in the Earth’s gravitational field above a perfect mirror is well described by the quantum bouncer model, 3) our estimation of $\bar{H}$ lifetimes in the quantum states of $0.1$ s are compatible to or even longer than the time of UCN passage through the mirror-absorber installation, also 4) UCN velocities are comparable to velocities of ultra-cold $\bar{H}$ atoms produced in traps [@cesa05; @gabr10]. We will discuss here mainly the statistical limitations arising from an estimate of the spectra from sources of $\bar{H}$ atoms that are projected in the near future.
In the simplest configuration, the experimental method for observation of the neutron gravitational states consisted in measuring the UCN flux through a slit between the horizontal mirror and the flat absorber (scatterer) placed above it at a variable height as a function of the slit height (the integral measuring method), or analyzing the spatial UCN density distribution behind the horizontal bottom mirror exit (the differential measuring method) using position-sensitive neutron detectors. The slit height can be changed and precisely measured. The absorber acts selectively on the gravitational states, namely the states with a spatial size $H_n=\lambda_n^0 l_0$ smaller than the absorber height $H$ are weakly affected, while the states with $H_n>H$ are intensively absorbed [@NVP; @Adh; @West]. A detailed description of the experimental method, the experimental setup, the results of various applications of this phenomenon could be found, for instance in [@nesv02; @nesv03; @nesv05; @UFN].
Leaving aside numerous methodical difficulties in the experiments of this kind (as they have been already overcome in the neutron experiments) and a real challenge to get high phase-space densities of trapped anti-hydrogen atoms (they are aimed at anyway in the existing anti-hydrogen projects [@ATRAP; @ALPHA]), let us compare relevant phase-space densities in the two problems, keeping in mind that it is the principle parameter, which defines population of quantum states in accordance with the Liouville theorem. If the phase-space densities of anti-hydrogen atoms would be equal to those of UCN, we would just propose to use an existing UCN gravitational spectrometer [@nesv00; @kreuz] for anti-hydrogen experiments with minor modifications. UCN is an extremely narrow initial fraction in a much broader, and hotter neutron velocity distribution. Maximum UCN fluxes available today for experiments in a flow-though mode are equal to $4$ $10^3$ $UCN/cm^2/s$; such UCN populate uniformly the phase-space up to the so-called critical velocity of about $6$ $m/s$ (UCN with smaller velocity are totally reflected from surface under any incidence angle; thus they could be stored in closed traps and transported using UCN guides). If one uses pulsed mode with a duty cycle, say $10^{-3}$, the average flux would drop to $4$ UCN$/cm^2/s$. The pulse method provides more precise measurements, it is used in current experiments with the GRANIT spectrometer, and it will be used in gravitational interference measurements analogous to those performed with the centrifugal quantum states of neutrons [@Nature10; @CentrNJP]. Taking into account the phase-space volume available for UCN in the gravitationally bound quantum states in the GRANIT spectrometer [@kreuz], we estimate the total count rate of about $10^2$ events/day when the relative accuracy for the gravitational mass is $10^{-3}$ (we note that the accuracy in the mentioned experiment is defined by a width of a quantum transition, and a few events might be sufficient to observe the corresponding resonance).
The average flux of $\bar{H}$ atoms projected by AEGIS collaboration [@Aegis1; @Aegis2] is a few atoms per second; let’s take it equal $3$ $\bar{H}/s$ to have it defined. The cloud length is $\sim 8$ mm, its radius is $\sim1.5$ mm, thus the cloud volume is $\sim 5\times10^{-2}$ $cm^3$. For comparison, estimate of the average UCN flux, which would be emitted from a small UCN source with a volume of $\sim 5\times10^{-2}$ $cm^3$, the maximum UCN density available of $30$ $UCN/cm^3$, with a duty cycle of $\sim 10^{-3}$ gives $0.15$ $UCN/s$. It is $20$ times lower than the $\bar{H}$ flux estimated above. One should not forget, however, that the projected $\bar{H}$ temperature is $\sim 100$ $m$K, i.e. $100$ times larger than the effective UCN temperature. Thus we lose a factor of $\sqrt{100}=10$ because of larger spread of $\bar{H}$ vertical velocities. No geometrical factors are taken into account here as well as no constraints following from final solid angles allowed, final sizes of $\bar{H}$ detectors, mirrors etc. However, their account would decrease our estimation by only a few times provided proper experiment design (note that equal acceleration of all anti-hydrogen atoms would not decrease their phase-space density). Thus we could provide statistical power of $\bar{H}$ experiment compatible to that with UCN. As the projected temperature of anti-hydrogen atoms in another proposal [@Yam] is significantly lower ($\sim 1$ $m$K that is just equal to the effective UCN temperature), the phase-space density of anti-hydrogen atoms could be even higher. Another significant advantage of a lower temperature consists in a more compact setup design. Note that a gravitational spectrometer analogous to [@nesv00; @kreuz] selects just a very small fraction of UCN( $\bar{H}$) available (those with extremely small vertical velocity components) thus the count rate of “useful” events is extremely low in both cases.
Thus, we conclude that measurements of the gravitationally bound quantum states of anti-hydrogen atoms look realistic if they would profit from methodical developments available in neutron experiments plus high phase-space densities of anti-hydrogen atoms aimed at in future. Based on extensive analysis of the mentioned neutron experiments, we could conclude that measurements of the gravitational mass of anti-hydrogen atoms with an accuracy of at least $10^{-3}$ is realistic, provided that projected high $\bar{H}$ phase-space density is achieved.
Conclusions
===========
We argue on existence of long-living quasi-stationary states of $\bar{H}$ above a material surface in the gravitational field of Earth. A typical lifetime of such states above an ideally conducting plane surface is $\tau\simeq 0.1$ s. Quasi-stationary character of such states is due to the quantum reflection of ultra-cold (anti)atoms from the Casimir-Polder (anti)atom-surface potential. The relatively long life-time is due to the smallness of the ratio of the characteristic spatial antiatom-surface interaction scale $l_{CP}$ and the spatial gravitational scale $l_0$. We show that the spectrum of $\bar{H}$ decaying gravitational levels is quasi-discrete even for the highly excited states as long as their quantum number $n\ll 30 000$. We argue that low lying gravitational states provide an interesting tool for studying the gravitational properties of antimatter, in particular for testing the equivalence between the gravitational and inertial masses of $\bar{H}$. We show that, by counting the number of $\bar{H}$ annihilation events on the surface, both the transition frequencies between the gravitational energy levels, as well as the spatial density distribution of gravitational states superposition can be measured. An important observation in this context is that a modification of the above mentioned properties of gravitational states due to the interaction with a surface disappears in the first order of the small ratio $l_{CP}/l_0$. Finally, we show that actual measurements of quantum properties of $\bar{H}$ atoms, levitating above a material mirror in gravitational states are feasible, provided that the projected high phase-space density $\bar{H}$ is achieved.
Acknowledgments
===============
We would like to acknowledge the support from the Swedish Research Council, from the Wenner-Gren Foundations and the Royal Swedish Academy of Sciences.
Appendix
========
Here we derive an expression for the scalar product of two complex energy gravitational states eigenfunctions: $$\alpha_{ij}=\frac{1}{N_iN_j}\int_0^\infty\mathop{\rm Ai^*}(z/l_0-\lambda_{j})\mathop{\rm Ai}(z/l_0-\lambda_i)dz$$ with $$\label{normdef}
N_i^2=\int_0^\infty \mathop{\rm Ai^2}(z/l_0-\lambda_{j})dz.$$ We start with the equations for the eigenfunction $\mathop{\rm Ai}(z/l_0-\lambda_i)$ and the complex eigenvalues $\lambda_i=\lambda_{i}^0+a_{CP}/l_0$:
$$\label{Lambda1Eq}
-\mathop{\rm Ai''}(z/l_0-\lambda_i)+z\mathop{\rm Ai}(z/l_0-\lambda_i)=\lambda_i \mathop{\rm Ai}(z/l_0-\lambda_i).$$
The equations for the complex conjugated eigenfunction and the eigenvalue are: $$\label{Lambda2Eq}
-\mathop{\rm Ai^{*''}}(z/l_0-\lambda_j)+z\mathop{\rm Ai^*}(z/l_0-\lambda_j)=\lambda_{j}^{*} \mathop{\rm
Ai^*}(z/l_0-\lambda_j).$$ We multiply both sides of equation Eq.(\[Lambda1Eq\]) by $\mathop{\rm Ai^*}(z/l_0-\lambda_{j})$ and integrate them over $z$. Then we multiply both sides of Eq.(\[Lambda2Eq\]) by $\mathop{\rm Ai}(z/l_0-\lambda_{i})$ and integrate them over $z$. After substraction of the results of these operations we get: $$\mathop{\rm Ai^{*'}}(-\lambda_{j})\mathop{\rm Ai}(-\lambda_{i})-\mathop{\rm Ai^{*}}(-\lambda_{j})\mathop{\rm
Ai'}(-\lambda_{i})=(\lambda_j^*-\lambda_i)\int_0^\infty\mathop{\rm Ai^*}(z/l_0-\lambda_{j})
\mathop{\rm Ai}(z/l_0-\lambda_i)dz.$$ To get the above result, we integrated by parts the integrals with second derivatives and took into account that Airy functions vanish at infinity. Now we take into account the equality $\mathop{\rm Ai}(-\lambda_i^0)=0$ and smallness of the ratio $a_{CP}/l_0$, to get the following expressions, exact up to the second order in $a_{CP}/l_0$: $$\begin{aligned}
\mathop{\rm Ai}(-\lambda_i)=-\frac{a_{CP}}{l_0}\mathop{\rm Ai'}(-\lambda_i^0),\label{Lambda1Bc}\\
\mathop{\rm Ai^*}(-\lambda_j)=-\frac{a_{CP}^*}{l_0}\mathop{\rm Ai^{*'}}(-\lambda_j^0).\label{Lambda2Bc}
\end{aligned}$$ Up to the second order in $a_{CP}/l_0$ we get: $$\mathop{\rm Ai^{*'}}(-\lambda_{j})\mathop{\rm Ai}(-\lambda_{i})-\mathop{\rm Ai^{*}}(-\lambda_{j})\mathop{\rm
Ai'}(-\lambda_{i})=i\frac{b}{2l_0}\mathop{\rm Ai^{*'}}(-\lambda_{j}^0)
\mathop{\rm Ai'}(-\lambda_{i}^0).$$ The above result should be combined with the known expression for the normalization coefficient: $$\label{normex}
N_i^2=\mathop{\rm Ai'^2}(-\lambda_{i})+\lambda_i\mathop{\rm Ai^2}(-\lambda_{i}),$$ which, up to the second order in $a_{CP}/l_0$ turns to be: $$\label{norm}
N_i=\mathop{\rm Ai'}(-\lambda_{i}^0)(1+\lambda_i^0\frac{a_{CP}^2}{2l_0^2}).$$ Keeping first order terms, we finally get: $$\label{crossT}
\alpha_{i\neq j}=i\frac{b/(2l_0)}{\lambda_j^0-\lambda_i^0+i b/(2l_0)}.$$
|
---
abstract: 'A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin’s Theorem over division rings of characteristic zero implies that of Kaplansky’s Theorem over such division rings. Also, we give a generalization of Kaplansky’s Theorem over general fields. We show that this extension of Kaplansky’s Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky’s Theorem holds over $\Delta$.'
address: 'Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Department of Mathematics, Faculty of Sciences, University of Golestan, Gorgan 19395-5746, Iran'
author:
- 'Heydar Radjavi and Bamdad R. Yahaghi'
title: An extension of a theorem of Kaplansky
---
[**Introduction**]{}
A theorem of Kaplansky (see [@K Theorem H on p. 137] or [@RR1 Corollary 4.1.7]) unifies two previous results: that of Levitzki, stating that a semigroup of nilpotent matrices is triangularizable (see [@K Thoerem 35 on p. 135] or [@L], or [@Y2 Theorem 1.3] for a simple proof), and that of Kolchin deducing the same conclusion for a semigroup of unipotent matrices, i.e., those of the form $I+N$, where $I$ is the identity matrix and $N$ is nilpotent (see [@Ko] or [@K Theorem C on p. 100]). First, we present a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that this proof can be adjusted to show that the counterpart of Kolchin’s Theorem over division rings of characteristic zero implies that of Kaplansky’s Theorem over such division rings. Next, we give a generalization of Kaplansky’s Theorem. To be more precise, we prove that any semigroup of matrices with entries from a field of the form $T + N$, where $T$ comes from a triangularizable family $\mathcal{T}$ of matrices and $N$ is a nilpotent matrix coming from the commutant of $\mathcal{T}$ is triagularizable. This answers a question asked in [@RY] in the affirmative. Finally, we show that our extension of Kaplansky’s Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky’s Theorem holds over $\Delta$.
Let us begin by fixing some standard notation. Let $\Delta$ be a division ring and $M_n(\Delta)$ the algebra of all $n \times n$ matrices over $\Delta$. The division ring $\Delta$ could in particular be a field. By a semigroup $\mathcal{S} \subseteq M_n(\Delta)$, we mean a set of matrices closed under multiplication. An ideal $\mathcal{J}$ of $\mathcal{S}$ is defined to be a subset of $\mathcal{S}$ with the property that $SJ \in \mathcal{J}$ and $JS \in \mathcal{J}$ for all $ S \in \mathcal{S}$ and $ J \in \mathcal{J}$. We view the members of $M_n(\Delta)$ as linear transformations acting on the left of $\Delta^n$, where $\Delta^n$ is the right vector space of all $n\times 1$ column vectors. A semigroup $\mathcal{S}$ is called irreducible if the orbit of any nonzero $ x \in D^n$ under $ \mathcal{S}$ spans $\Delta^n$. When $n > 1$, this is equivalent to the members of $\mathcal{S}$, viewed as linear transformations on $\Delta^n$, having no common invariant subspace other than the trivial subspaces, namely, $\{0\}$ and $\Delta^n$. On the opposite of irreducibility is triangularizability, when the common invariant subspaces of the members of $\mathcal{S}$ include a maximal subspace chain (of length $n$) in $\Delta^n$, i.e., there are subspaces $$\{0\} = \mathcal{V}_0 \subseteq \mathcal{V}_1 \subseteq \cdots \subseteq \mathcal{V}_n = \Delta^n,$$ where $ \mathcal{V}_j$ is a $j$-dimensional subspace invariant under every $S \in \mathcal{S}$. For a collection $\mathcal{C}$ in $M_n(D)$, by the commutant of $\mathcal{C}$, denoted by $\mathcal{C}'$, we mean $$\mathcal{C}' := \{A \in M_n(D) : AB = BA \ \forall \ B \in \mathcal{C}\}.$$
We quote the following result from [@Y3 Theorem 2.2.10] for reader’s convenience. In fact the following theorem is a finite-dimensional version of [@Y Theorem 5] over general fields.
\[1.1\] [*Let ${\EuScript{V}}$ with $\dim {\EuScript{V}}> 1$ be a finite-dimensional vector space over a field $F$ and ${\EuScript{F}}$ a nonscalar triangularizable family of linear transformations on ${\EuScript{V}}$. Then ${\EuScript{F}}$ has a nontrivial hyperinvariant subspace.*]{}
[**Proof.**]{} We note that for every family ${\EuScript{F}}$ of linear transformations $${{\EuScript{F}}}^{'}=({\rm Alg}({{\EuScript{F}}}))^{'}.$$ Thus ${\EuScript{F}}$ has a nontrivial hyperinvariant subspace iff ${\rm Alg}({\EuScript{F}})$ does. Thus it suffices to prove the assertion for any nonscalar triangularizable algebra, say ${\EuScript{A}}$, of linear transformations.
Now either the algebra ${\EuScript{A}}$ is commutative or not. If it is a commutative algebra, note that by hypothesis there exists $ A \in {\EuScript{A}}$ that is not scalar. Let $\lambda$ be any eigenvalue of $A$, and ${\EuScript{M}}$ the corresponding eigenspace of $A$. Since ${\EuScript{A}}$ is commutative, for all $B \in {{\EuScript{A}}\cup {\EuScript{A}}}^{'}$ and $x \in {\EuScript{M}}$ we have $$ABx=BAx={\lambda}Bx,$$ i.e., $Bx \in {\EuScript{M}}$. So ${\EuScript{M}}$ is invariant under ${{\EuScript{A}}\cup {\EuScript{A}}}^{'}$. Now if the algebra ${\EuScript{A}}$ is not commutative, then there exist $A, B \in {\EuScript{A}}$ such that $AB-BA \neq 0$. Set $K_0=AB-BA$. Then $K_0$ is a nonzero nilpotent transformation in ${\EuScript{A}}$, for ${\EuScript{A}}$ is triangularizable. Define ${{\EuScript{A}}}_1:={{\EuScript{A}}}^{'}+ {{\EuScript{A}}}*{{\EuScript{A}}}^{'}$, where $${{\EuScript{A}}}*{{\EuScript{A}}}^{'}:= \{ \sum_{i=1}^{k} A_iA_i' :k \in {\Bbb N} ,\ \ A_i \in {{\EuScript{A}}},\ \ A_i' \in {{\EuScript{A}}}^{'}, \ (1 \leq i \leq k) \}.$$ Clearly, in view of the fact that $ {{\EuScript{A}}}^{'}$ is a unital subalgebra of ${\EuScript{L}}({\EuScript{V}})$, we see that ${\EuScript{A}}_1$ is a subalgebra of $ {\EuScript{L}}({\EuScript{V}})$ containing both $ {\EuScript{A}}$ and $ {{\EuScript{A}}}^{'}$. It thus suffices to prove that ${\EuScript{A}}_1$ has a nontrivial invariant subspace.
We claim that ${{\EuScript{A}}}_1K_0$, and hence ${{\EuScript{A}}}_1K_0{{\EuScript{A}}}_1$, the semigroup ideal generated by $K_0$ in $ {{\EuScript{A}}}_1$, consists of nilpotents. To this end, let $ A_0= A' + \sum_{i=1}^{k} A_iA_i' \in {{\EuScript{A}}}_1$ with $ A_i \in {{\EuScript{A}}},$ where $ A', A_i' \in {{\EuScript{A}}}^{'}, \ (1 \leq i \leq k, k \in {\Bbb N})$ be arbitrary. We prove that $A_0K_0$ is nilpotent: first of all we notice that $ A_0K_0= A'K_0 + \sum_{i=1}^{k} A_{i_0}A_i'$, where $ A_{i_0}=A_iK_0 \in {{\EuScript{A}}}$. Let $n= \dim {\EuScript{V}}$. Set $${{\EuScript{S}}}:= \{A\in {{\EuScript{A}}}: A^n=0\}.$$ Since ${\EuScript{A}}$ is triangularizable, it follows that ${\EuScript{S}}$ is a nonzero semigroup ideal of ${\EuScript{A}}$ consisting of nilpotent transformations (note that $0 \not= K_0 \in {\EuScript{S}}$). The set ${{\EuScript{S}}}{{\EuScript{A}}}^{'}$ is indeed a semigroup consisting of nilpotents because for all $ A \in {{\EuScript{A}}}, A' \in {{\EuScript{A}}}^{'}$ we have $ AA'=A'A$ and that ${\EuScript{S}}$ is a semigroup of nilpotents. Thus Levitzki’s Theorem shows that ${{\EuScript{S}}}{{\EuScript{A}}}^{'}$ is triangularizable. Therefore ${\rm Alg}({{\EuScript{S}}}{{\EuScript{A}}}^{'})$, the algebra generated by ${{\EuScript{S}}}{{\EuScript{A}}}^{'}$, consists of nilpotents. We have $$A_0K_0= K_0A' + \sum_{i=1}^{k} A_{i_0}A_i',$$ where $ A_{i_0}=A_iK_0 \in {{\EuScript{A}}}$. In fact $ A_{i_0}=A_iK_0 \in {{\EuScript{S}}}$, for $K_0 \in {\EuScript{S}}$ and ${\EuScript{A}}$ is triangularizable. Now clearly $A'K_0=K_0A' \in {{\EuScript{S}}}{{\EuScript{A}}}^{'}$ and $ A_{i_0}A_i'\in {{\EuScript{S}}}{{\EuScript{A}}}^{'}$. Therefore $ A_0K_0 \in {\rm Alg}({{\EuScript{S}}}{{\EuScript{A}}}^{'})$, and hence $A_0K_0$ is a nilpotent transformation. Thus ${{\EuScript{A}}}_1K_0{{\EuScript{A}}}_1$ is a nonzero semigroup ideal of ${{\EuScript{A}}}_1$ consisting of nilpotents which must be triangularizable, and hence reducible, by Levitzki’s Theorem. Now reducibility of the nonzero ideal ${{\EuScript{A}}}_1K_0{{\EuScript{A}}}_1$ implies that of ${{\EuScript{A}}}_1$, completing the proof.
The following is the counterpart of the preceding theorem over division rings.
\[1.2\] [*Let ${\EuScript{V}}$ with $\dim {\EuScript{V}}> 1$ be a finite-dimensional vector space over a division ring $D$ with center $F$ and ${\EuScript{F}}$ a triangularizable family of linear transformations on ${\EuScript{V}}$ such that the $F$-algebra generated by ${\EuScript{F}}$ contains a nonzero nilpotent linear transformation. Then ${\EuScript{F}}$ has a nontrivial hyperinvariant subspace.*]{}
[**Proof.**]{} The proof is an imitation of that of the preceding theorem, which is omitted for the sake of brevity. We refer the reader to [@Y3 Theorem 4.2.4] for a detailed proof.
Let ${\EuScript{V}}$ be a finite-dimensional vector space over a division ring $D$. For a triangularizable linear transformation $T \in {\EuScript{L}}({\EuScript{V}})$, we say that $\lambda \in D$ is an [*inner eigenvalue of $T$ relative to a triangularizing basis*]{} ${\EuScript{B}}$ for $T$ if $\lambda$ appears on the main diagonal of the matrix of $T$ with respect to the basis ${\EuScript{B}}$. It is easy to verify that if $\{S, T\} \subset {\EuScript{L}}({\EuScript{V}})$ is triangularizable and $T$ and $S$ have inner-eigenvalues in the center of $D$, then $ST-TS$ is nilpotent.
\[1.3\] [*Let ${\EuScript{V}}$ with $\dim {\EuScript{V}}> 1$ be a finite-dimensional vector space over a division ring $D$ with center $F$ and ${\EuScript{F}}$ a nonscalar triangularizable family of linear transformations on ${\EuScript{V}}$ with inner-eigenvalues in $F$. Then ${\EuScript{F}}$ has a nontrivial hyperinvariant subspace.*]{}
[**Proof.**]{} The assertion is easy if the family is commutative. If not, there exist $A, B \in {\EuScript{F}}$ such that $C:= AB - BA \not= 0$. Then, clearly, $C $ is a nilpotent linear transformation and belongs to the $F$-algebra generated by ${\EuScript{F}}$. The assertion now follows from Theorem \[1.2\].
A standard result in simultaneous triangularization over general fields is that the notion of triangularizability is preserved by passing to quotients. This result over division rings perhaps first appeared in [@S1], but it is implicit in [@RR1 Lemma 1.5.2] over fields and in [@S2 Lemma 2.4] over division rings. The proof given below is an imitation of the proof given over fields in [@Mom Lemma 1.2.4]. Recall that if ${\EuScript{V}}$ is a left (right) vector space over $D$ and ${\EuScript{M}}$ is a subspace of ${\EuScript{V}}$, then ${\EuScript{V}}/{\EuScript{M}}:= \{ \ x + {\EuScript{M}}: \ x \in {\EuScript{V}}\}$ is called the [*quotient space*]{}. If $A$ is a linear transformation on ${\EuScript{V}}$ and ${\EuScript{M}}\subset {\EuScript{N}}$ are invariant subspaces for $A$, then the quotient transformation $A_{{\EuScript{N}}/{\EuScript{M}}}$ on ${\EuScript{N}}/{\EuScript{M}}$ is defined by $A_{{\EuScript{N}}/{\EuScript{M}}} (x+ {\EuScript{M}}) = Ax +{\EuScript{M}}$ for each $x \in {\EuScript{N}}$; the invariance of ${\EuScript{M}}$ and ${\EuScript{N}}$ under $A$ guarantees that $A_{{\EuScript{N}}/{\EuScript{M}}}$ is well-defined. If ${\EuScript{C}}$ is a collection of linear transformations on ${\EuScript{V}}$, and if ${\EuScript{M}}\subset {\EuScript{N}}$ are two invariant subspaces for ${\EuScript{C}}$, then the [*collection of quotients of ${\EuScript{C}}$ with respect to*]{} $\{ {\EuScript{M}},{\EuScript{N}}\}$, denoted by ${\EuScript{C}}_{{\EuScript{N}}/{\EuScript{M}}}$, is the set of all quotient transformations $A_{{\EuScript{N}}/{\EuScript{M}}}$ on ${\EuScript{N}}/{\EuScript{M}}$, where $ A \in {\EuScript{C}}$. We say that a property ${\EuScript{P}}$ is [*inherited by quotients*]{} if every collection of quotients of a collection satisfying ${\EuScript{P}}$ also satisfies ${\EuScript{P}}$, e.g., the properties nilpotency, commutativity, having rank $\leq
1$, etc are inherited by quotients. The following asserts that the property of triagularizability is inherited by quotients.
\[1.4\] [*Let ${\EuScript{V}}$ be a finite-dimensional vector space over a division ring $D$, ${\EuScript{C}}$ a triangularizable collection of linear transformations on ${\EuScript{V}}$, and ${\EuScript{M}}$ and ${\EuScript{N}}$ with ${\EuScript{M}}\subseteq {\EuScript{N}}$ two invariant subspaces for the collection ${\EuScript{C}}$. Then ${\EuScript{C}}_{{\EuScript{N}}/{\EuScript{M}}}$ is triangularizable.*]{}
[**Proof.**]{} Without loss of generality assume that $\dim {\EuScript{V}}> 1$. Suppose ${\EuScript{M}}$ and ${\EuScript{N}}$ with ${\EuScript{M}}\subseteq {\EuScript{N}}$ and $ \dim \frac{{\EuScript{N}}}{{\EuScript{M}}} > 1$ are two invariant subspaces for the collection ${\EuScript{C}}$. We need to show that there exists an invariant subspace ${\EuScript{R}}$ of ${\EuScript{C}}$ such that $ {\EuScript{M}}\subsetneq {\EuScript{R}}\subsetneq {\EuScript{N}}$. To this end, let $$0 = {\EuScript{V}}_0 \subsetneq {\EuScript{V}}_1 \subsetneq \cdots \subsetneq {\EuScript{V}}_{n-1} \subsetneq {\EuScript{V}}_n = {\EuScript{V}},$$ where $ n = \dim {\EuScript{V}}$, be a triangularizing chain for ${\EuScript{C}}$. Set ${\EuScript{W}}_i := {\EuScript{N}}\cap ({\EuScript{M}}+ {\EuScript{V}}_i)$, where $ 0 \leq i \leq n$. It is plain that each ${\EuScript{W}}_i$ is an invariant subspace of ${\EuScript{C}}$ and that $${\EuScript{M}}= {\EuScript{W}}_0 \subseteq {\EuScript{W}}_1 \subseteq \cdots \subseteq {\EuScript{W}}_{n-1} \subseteq {\EuScript{W}}_n = {\EuScript{N}}.$$ Clearly, $ \dim \frac{{\EuScript{W}}_i}{{\EuScript{W}}_{i-1}} \leq \dim \frac{{\EuScript{V}}_i}{{\EuScript{V}}_{i-1}} = 1$. This implies that there is an $ 1 < i < n$ such that ${\EuScript{M}}\subsetneq {\EuScript{R}}:= {\EuScript{W}}_i \subsetneq {\EuScript{N}}$. This completes the proof.
We need the following useful lemma, which is a quick consequence of the preceding lemma, in the proof of one of our main results.
\[1.5\] [*Let $n \in \mathbb{N}$ and ${\EuScript{F}}$ a family of block upper triangular matrices in $M_n(D)$. Then ${\EuScript{F}}$ is triangularizable iff its diagonal blocks are triangularizable.*]{}
[**Proof.**]{} The proof, which is an quick consequence of Lemma \[1.4\], is omitted for the sake of brevity.
[**Main Results**]{}
We start off with a simple proof of Kaplnasky’s Theorem over fields of zero characteristic. We recall that if a semigroup $\mathcal{S}$ of matrices is irreducible, then so is every nonzero ideal $ \mathcal{J}$ of $\mathcal{S}$ (see [@RR1 Lemma 2.1.10]).
\[2.1\] [**(Kaplansky)**]{} Let $ n > 1$ and let $F$ be a field with characteristic zero and $ \mathcal S$ a semigroup in $ M_n(F)$ consisting of matrices with singleton spectra. Then the semigroup $ \mathcal S$ is triangularizable.
[**Proof.**]{} By passing to $F^*\mathcal{S}$, where $F^* = F \setminus \{0\}$, we may assume that $\mathcal{S}$ is closed under scalar multiplications by the nonzero elements of $F$. We only need to show that $\mathcal S$ is reducible. If the semigroup $\mathcal{S}$ contains a nilpotent element, then reducibility of $\mathcal{S}$ follows from that of the nonzero semigroup ideal of all nilpotent elements of $\mathcal{S}$. So it remains to prove the assertion when $\mathcal{S}$ contains no nonzero nilpotent element. It is then plain that $\mathcal{S}$ is reducible iff the set of all unipotent elements of $\mathcal{S}$ is reducible. Thus, in view of Kolchin’s Theorem, we will be done as soon as we prove that the set of all unipotent elements of $\mathcal{S}$ forms a semigroup. To this end, let $ I + N_1 , I + N_2 \in \mathcal{S}$ be arbitrary unipotent elements. We can write $ (I+ N_1)(I + N_2) = cI + N' \in \mathcal{S}$, where $c \in F^*$ and $N' $ is a nilpotent matrix. We need to show that $c=1$. If $N_2 = 0$, we have nothing to prove. Suppose $N_2 \not= 0$ so that $N_2^k \not= 0$ but $N_2^{k+1} = 0$ for some $k \in \mathbb{N}$ with $k < n$. Thus $$(I + N_2)^m = I + {m \choose 1} N_2 + \cdots + {m \choose k} N_2^k,$$ for all $ m \in \mathbb{N}$. Recall that ${m \choose k} := 0$ whenever $ m <k$. Clearly, we have $ (I+ N_1)(I + N_2)^m = c_mI + N_m' \in \mathcal{S}$ for all $ m \in \mathbb{N}$ with $c_m \in F$ an $n$th root of unity, i.e., $c_m^n = 1$ and $N'_m$ a nilpotent matrix. Since the set of the $n$th roots of unity in $F$ has at most $n$ elements, we see that there exists a subsequence $(m_i)_{i=1}^\infty$ and an $l \in \mathbb{N}$ such that $ (I+ N_1)(I + N_2)^{m_i} = c_{m_l}I + N'_{m_l}$ for all $ i \in \mathbb{N}$. Therefore, $$\Big( (I+ N_1)(I + N_2)^{m} - c_{m_l}I\Big)^n = 0,$$ for infinitely many $ m \in \mathbb{N}$. Now, fix $ 1 \leq i , j \leq n$ and note that the $(i, j)$ entry of the matrix $ \Big( (I+ N_1)(I + N_2)^{m} - c_{m_l}I\Big)^n$ is a polynomial of degree $kn$ in $m$ having infinitely many roots, namely, $m_j$’s ($j \in \mathbb{N}$). This implies that the $(i, j)$ entry of the matrix $ \Big( (I+ N_1)(I + N_2)^{m} - c_{m_l}I\Big)^n$ is zero for all $m \in \mathbb{N} \cup \{0\}$. Consequently, $$\Big( (I+ N_1)(I + N_2)^{m} - c_{m_l}I\Big)^n = 0,$$ for all $ m \in \mathbb{N} \cup \{0\}$. Setting $m=0 ,1$ in the above, we obtain $ c = c_{m_l}=1$, which is what we want.
\[2.2\] The counterpart of Kolchin’s Theorem over division rings of characteristic zero implies that of Kaplansky’s Theorem over such division rings. In other words, if every semigroup of unipotent matrices over a division ring $\Delta$ of characteristic zero is triangularizable, then so is every semigroup of matrices of the form $ cI + N$, where $c$ comes from the center of $\Delta$ and $ N$ is a nilpotent matrix with entries from $\Delta$.
[**Proof.**]{} Let $F$ denote the center of $\Delta$. Then, the proof is identical to that of the preceding theorem except that one should use the Gordon-Motzkin Theorem ([@La Theorem 16.4]) to get that $$\Big( (I+ N_1)(I + N_2)^{m} - c_{m_l}I\Big)^n = 0,$$ for all $ m \in \mathbb{N} \cup \{0\}$. Also one must use the Dieudonn[é]{} Determinant (see [@D Corollary 20.1]) to conclude that $c_m \in F$ is an $n$th root of unity whenever $ (I+ N_1)(I + N_2)^m = c_mI + N' \in \mathcal{S}$ with $m \in \mathbb{N}$, $c_m \in F$, and $N_1, N_2$ and $N'$ nilpotent.
Here is our extension of Kaplansky’s Theorem. This theorem affirmatively answers a question raised in [@RY].
\[2.3\] Let $ n \in \mathbb{N}$ and let $F$ be a field and $\mathcal{T}$ a triangularizable set of matrices in $ M_n(F)$, $\mathcal{N}$ the set of all nilpotents in $ M_n(F)$, and $ \mathcal S$ a semigroup in $ M_n(F)$ consisting of matrices of the form $T + N$, where $ T \in \mathcal{T}$ and $ N \in \mathcal{T}' \cap \mathcal{N}$. Then the semigroup $ \mathcal S$ is triangularizable.
[**Proof.**]{} We view the elements of $M_n(F)$ as linear transformations on $F^n$ and proceed by induction on $n$, the dimension of the underlying space. The assertion trivially holds for $n=1$. Assume $ n > 1$ and that the assertion holds for such semigroups of linear transformations acting on spaces of dimension less than $n$. If $\mathcal{T}$ consists of scalar matrices, then $\mathcal{S}$ is triangularizable by Kaplansky’s Theorem. If not, then by Theorem \[1.1\], $\mathcal{T}$ has a nontrivial hyperinvariant subspace. Therefore, there exists a nontrivial direct sum decomposition $F^n = \mathcal{V}_1 \oplus \mathcal{V}_2$ with respect to which $$T = \left(\begin{array}{cc} T_{11} & T_{12}\\ 0 & T_{22} \end{array} \right) , \ N = \left(\begin{array}{cc} N_{11} & N_{12}\\ 0 & N_{22} \end{array} \right)
,$$ for all $ T \in \mathcal{T}$ and $ N \in \mathcal{T}' \cap \mathcal{N}$. Thus, for each $S \in \mathcal{S}$, with respect to the decomposition $F^n = \mathcal{V}_1 \oplus \mathcal{V}_2$, we can write $$\begin{aligned}
S &= & T_S + N_S\\
& =& \left(\begin{array}{cc} T_{11} & T_{12} \\ 0 & T_{22} \end{array} \right) + \left(\begin{array}{cc} N_{11} & N_{12}\\ 0 & N_{22} \end{array} \right) \\
& =& \left(\begin{array}{cc} T_{11} + N_{11}& T_{12} + N_{12} \\ 0 & T_{22} + N_{22} \end{array} \right),\end{aligned}$$ where $$T_S = \left(\begin{array}{cc} T_{11} & T_{12}\\ 0 & T_{22} \end{array} \right) , \ N_S = \left(\begin{array}{cc} N_{11} & N_{12}\\ 0 & N_{22} \end{array} \right).$$ For $ 1 \leq i \leq 2$, use $\mathcal{S}_{ii}$, $\mathcal{T}_{ii}$, and $\mathcal{N}_{ii}$ to, respectively, denote the set of all $(i, i) $ block entries of $S \in \mathcal{S}$, $T \in \mathcal{T}$, and $ N \in \mathcal{T}' \cap \mathcal{N}$ with $S = T+ N$. For $i = 1, 2$, let $ n_i = \dim \mathcal{V}_i$ so that $n = n_1 + n_2$ and $\mathcal{N}_{i} $ denote the set of all nilpotent linear transformations on $\mathcal{V}_i$. Note that for each $i= 1, 2$, $\mathcal{T}_{ii}$ is triangularizable and $\mathcal{S}_{ii}$ is a semigroup of matrices of the form $ T_{ii} + N_{ii}$, where $T_{ii} \in \mathcal{T}_{ii}$, $N_{ii} \in \mathcal{T}_{ii}' \cap \mathcal{N}_{i}$. By Lemma \[1.4\], $ \mathcal{S}$ is triangularizable iff both $\mathcal{S}_{11}$ and $\mathcal{S}_{22}$ are triangularizable. But $\mathcal{S}_{ii}$ ($i=1, 2$) is triangularizable by the induction hypothesis because it consists of elements of the form $ T_{ii} + N_{ii}$, where $T_{ii} \in \mathcal{T}_{ii}$, $N_{ii} \in \mathcal{T}_{ii}' \cap \mathcal{N}_{i}$, $\mathcal{T}_{ii}$ is triangularizable, and $n_i < n$. This completes the proof.
Here is what we can say over general division rings.
\[2.4\] Let $ n \in \mathbb{N}$ and let $D$ be a division ring over which Kaplansky’s Therem holds, $\mathcal{T}$ a triangularizable set of matrices in $ M_n(D)$ with inner-eigenvalues in $F$, the center of $D$, $\mathcal{N}$ the set of all nilpotents in $ M_n(D)$, and $ \mathcal S$ a semigroup in $ M_n(D)$ consisting of matrices of the form $T + N$, where $ T \in \mathcal{T}$ and $ N \in \mathcal{T}' \cap \mathcal{N}$. Then the semigroup $ \mathcal S$ is triangularizable.
[**Proof.**]{} The proof, which we omit for the sake of brevity, is identical to that of the preceding theorem except that one has to make use of Corollary \[1.3\] as opposed to Theorem \[1.1\].
By a [*Kaplansky semigroup*]{} of matrices over a division ring $D$, we mean a semigroup of matrices of the form $cI + N$, where $c$ is in the center of $D$ and $N $ is a nilpotent matrix with entries from $D$. By [@RY Theorem 2.2], every finite Kaplansky semigroup of matrices over a general division ring is triangularizable. This result together with the proof of the preceding theorem implies the following.
\[2.5\] Let $ n \in \mathbb{N}$ and let $D$ be a division ring and $\mathcal{T}$ a triangularizable set of matrices in $ M_n(D)$ with inner-eigenvalues in $F$, the center of $D$, $\mathcal{N}$ the set of all nilpotents in $ M_n(D)$, and $ \mathcal S$ a finite semigroup in $ M_n(D)$ consisting of matrices of the form $T + N$, where $ T \in \mathcal{T}$ and $ N \in \mathcal{T}' \cap \mathcal{N}$. Then the semigroup $ \mathcal S$ is triangularizable.
[**Proof.**]{} In view of [@RY Theorem 2.2], the assertion is a quick consequence of the proof of the preceding theorem.
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abstract: 'We focus on emergence of the power-law cross-correlations from processes with both short and long term memory properties. In the case of correlated error-terms, the power-law decay of the cross-correlation function comes automatically with the characteristics of separate processes. Bivariate Hurst exponent is then equal to an average of separate Hurst exponents of the analyzed processes. Strength of short term memory has no effect on these asymptotic properties. Implications of these findings for the power-law cross-correlations concept are further discussed.'
address: |
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou Vezi 4, 182 08, Prague 8, Czech Republic\
Institute of Economic Studies, Faculty of Social Sciences, Charles University, Opletalova 26, 110 00, Prague 1, Czech Republic
author:
- Ladislav Kristoufek
bibliography:
- 'Bibliography.bib'
title: 'On the interplay between short and long term memory in the power-law cross-correlations setting'
---
power-law cross-correlations, long term memory, short term memory
*PACS codes: 05.10.-a, 05.40.-a, 05.45.-a*\
Introduction
============
Existence of statistically significant power-law cross-correlations between various series is a fascinating phenomenon important for modeling and forecasting time series. Several processes that possess such long-term correlations have been proposed in the literature. The most frequently discussed and applied ones are multivariate generalizations of the well-established fractionally integrated ARMA processes (usually labeled as FARIMA and ARFIMA) – VARFIMA or MVARFIMA processes – and fractional Gaussian noise processes or fractional Brownian motions, which are their integrated version (these are labeled as fGn and fBm in the literature, respectively) [@Lobato1997; @Ravishanker1997; @Martin1999; @Nielsen2004; @Shimotsu2007; @Sela2009; @Tsay2010; @Nielsen2011]. Construction of the multivariate ARFIMA process implies that the bivariate Hurst exponent is the average of the separate Hurst exponents [@Nielsen2011] and the same property holds for the fractional Brownian motion [@Amblard2011]. The long-range cross-correlations thus simply arise from the specification of these processes. However, most of the studies focus on a specific case when both studied series are themselves power-law correlated leaving aside a possibility that one of the processes is indeed not power-law correlated. Here, we focus on two specific cases – a pair of power-law correlated processes, and a combination of a power-law correlated and an exponentially correlated processes – and compare their properties in the power-law cross-correlations framework. For a sake of simplicity and straightforward results, we stick to the ARFIMA setting usually followed in the multidisciplinary physics literature [@Podobnik2008; @Podobnik2008a; @Zhou2008; @Jiang2011; @Kristoufek2011; @Kristoufek2013].
Power-law cross-correlated processes
====================================
Power-law cross-correlated processes are usually defined via a power-law decay of a cross-correlation function. Specifically, if the cross-correlation function $\rho_{xy}(k)$ between processes $\{x_t\}$ and $\{y_t\}$ decays as $\rho_{xy}(k) \propto k^{2H_{xy}-2} \equiv k^{-\gamma_{xy}}$ with lag $k \rightarrow +\infty$, we say that the processes are power-law cross-correlated. The characteristic bivariate Hurst exponent $H_{xy}$ has a similar interpretation as the univariate one so that if $H_{xy}>0.5$, the processes are cross-persistent, and if $H_{xy}<0.5$, the processes are cross-antipersistent. Both $H_{xy}$ and $\gamma_{xy}$ (in addition to other parameters) are used in the literature interchangeably, depending on an initial setting of the correlation structure.
Correlated ARFIMA processes
---------------------------
We start with ARFIMA processes with correlated error-terms simply structured as two ARFIMA(0,$d$,0) processes with parameters $d_1$, $d_2$, $a_n(d)=\frac{\Gamma(n+d)}{\Gamma(n+1)\Gamma(d)}$ and a specific correlation structure: $$\begin{gathered}
\label{eq1}
x_t=\sum_{n=0}^{\infty}{a_n(d_1)\varepsilon_{t-n}} \\
y_t=\sum_{n=0}^{\infty}{a_n(d_2)\nu_{t-n}} \nonumber \\
\langle \varepsilon_t \rangle = \langle \nu_t \rangle = 0 \nonumber\\
\langle \varepsilon_t^2 \rangle = \sigma_{\varepsilon}^2 < +\infty \nonumber\\
\langle \nu_t^2 \rangle = \sigma_{\nu}^2 <+\infty \nonumber\\
\langle \varepsilon_t\varepsilon_{t-n} \rangle = \langle \nu_t\nu_{t-n} \rangle = \langle \varepsilon_t\nu_{t-n} \rangle = 0\text{ for }n \ne 0 \nonumber\\
\langle \varepsilon_t\nu_t \rangle = \sigma_{\varepsilon\nu} <+\infty \nonumber.
\label{eq:ARFIMA_varcovar}\end{gathered}$$ Note that both processes are stationary [@Sowell1992; @Bertelli2002; @Samorodnitsky2006]. Cross-power spectrum $f_{xy}(\lambda)$ with frequency $0<\lambda \le \pi$ of the two processes can be written as $$\begin{gathered}
\label{eq:spectrum_ARFIMA_text}
f_{xy}(\lambda)=\frac{\sigma_{\varepsilon\nu}}{2\pi}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_k(d_1)a_l(d_2)\exp(i(k-l)\lambda)}=
\frac{\sigma_{\varepsilon\nu}}{2\pi}\left(1-\exp(i\lambda)\right)^{-d_1}\left(1-\exp(-i\lambda)\right)^{-d_2}. \nonumber\end{gathered}$$ To show whether the processes are power-law cross-correlated, we need to inspect an asymptotic behavior of the cross-correlation function $\rho_{xy}(n)$. Using the inverse Fourier transform of the cross-power spectrum, we can write the $n$th cross-correlation as $$\rho_{xy}(n)=\frac{\sigma_{\varepsilon\nu}}{2\pi\sigma_x\sigma_y}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_k(d_1)a_l(d_2)\int_{-\pi}^{\pi}\exp(i(n+k-l)\lambda)d\lambda}.
\label{eq:rhon}$$ Now, using the definition and properties of the Dirac delta function [@Dirac1958], we can rewrite the cross-correlation function in Eq. \[eq:rhon\] as $$\label{eqDirac}
\rho_{xy}(n)=\frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_k(d_1)a_l(d_2)\delta(n+k-l)}=\frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y}\sum_{k=0}^{\infty}{a_k(d_1)a_{n+k}(d_2)} \nonumber$$ and follow with $$\sum_{l=0}^{\infty}{a_k(d_1)a_l(d_2)\delta(n+k-l)}=\sum_{l=0}^{\infty}{a_k(d_1)a_l(d_2)\delta(l-n-k)}=\sum_{l=0}^{\infty}{a_k(d_1)a_{n+k}(d_2)} \nonumber$$ where $t=l$ and $a=n+k$. We can now rewrite $a_j(d)$ with a use of the Beta function so that $$a_j(d)=\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}=\frac{1}{kB(k,d)}. \nonumber
\label{eq:aj}$$ Using Stirling’s approximation of the Beta function $B(\bullet,\bullet)$ for fixed $d$ and $j \rightarrow +\infty$, we get $$\label{eq:Stirling1}
a_j(d) \approx \frac{1}{j}\frac{1}{\Gamma(d)j^{-d}}=\frac{j^{d-1}}{\Gamma(d)}.$$ Since we are interested in the asymptotic behavior, we can use Eq. \[eq:Stirling1\] and follow with $$\rho_{xy}(n) \approx \frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y\Gamma(d_1)\Gamma(d_2)}\sum_{k=0}^{\infty}{k^{d_1-1}(n+k)^{d_2-1}} \nonumber$$ given that $d_1,d_2,k,n+k>0$. Approximating the infinite sum with a definite integral, we can write $$\begin{gathered}
\label{eq:ARFIMA1}
\rho_{xy}(n) \approx \frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y\Gamma(d_1)\Gamma(d_2)}\int_{0}^{\infty}{k^{d_1-1}(n+k)^{d_2-1}dk}=\\
=\frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y\Gamma(d_1)\Gamma(d_2)}n^{d_1+d_2-1}\frac{\Gamma(d_1)\Gamma(1-d_1-d_2)}{\Gamma(1-d_2)}=\\
= \frac{\sigma_{\varepsilon\nu}\Gamma(1-d_1-d_2)}{\sigma_x\sigma_y\Gamma(1-d_2)\Gamma(d_2)}n^{d_1+d_2-1} \propto n^{d_1+d_2-1}=n^{-(1-d_1-d_2)} \nonumber\end{gathered}$$ given that $d_1+d_2<1$ and $n>0$. Therefore, given that $\sigma_{\varepsilon\nu}\ne 0$, the power-law cross-correlations emerge regardless of the level of correlation between error-terms $\{\varepsilon_t\}$ and $\{\nu_t\}$ as long as it is non-zero. Using the relationship between fractional differencing parameter and Hurst exponent $d=H-0.5$, we have $$H_{xy}=1-\frac{\gamma_{xy}}{2}=1-\frac{1-d_1-d_2}{2}=1-\frac{-(H_x+H_y)+2}{2}=\frac{H_x+H_y}{2}.
\label{eq:ARFIMA_H}$$ The bivariate Hurst exponent $H_{xy}$ is thus an average of the separate Hurst exponents $H_x$ and $H_y$ regardless the correlation between error-terms as long as it remains non-zero. This also covers the case showed in Ref. [@Podobnik2009] for two ARFIMA processes with the identical error-terms. We now turn to the combination of short and long term memory.
Combination of AR and ARFIMA processes
--------------------------------------
In the univariate case, distinguishing between short and long term memory is evident from the properties of the auto-correlation function. To see how these two types of memories interact in the bivariate setting, we investigate the case when one of the processes is long-range dependent, the other is short-range dependent and their error-terms are pairwise correlated. Let’s have ARFIMA process $\{x_t\}$ and AR(1) process $\{y_t\}$ defined as $$\begin{gathered}
\label{eq2}
x_t=\sum_{n=0}^{\infty}{a_n(d_1)\varepsilon_{t-n}} \\
y_t=\theta y_{t-1}+\nu_t \nonumber
\label{eq:ARFIMAAR}\end{gathered}$$ with $|\theta|<1$. Moments of the error-terms are specified as for the previous case and the processes are thus stationary [@Wei2006]. The cross-power spectrum has the following form $$\begin{gathered}
f_{xy}(\lambda)=\frac{\sigma_{\varepsilon\nu}}{2\pi}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_l(d_1)\theta^k\exp(i(k-l)\lambda)}=
\frac{\sigma_{\varepsilon\nu}}{2\pi}\left(1-\exp(-i\lambda)\right)^{-d_1}\left(1-\theta\exp(i\lambda)\right)^{-1}. \nonumber
\label{eq:spectrum_ARFIMA_AR1_text}\end{gathered}$$ Using the inverse Fourier transform, we get $$\rho_{xy}(n)=\frac{\sigma_{\varepsilon\nu}}{2\pi\sigma_x\sigma_y}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_l(d_1)\theta^k\int_{-\pi}^{\pi}\exp(i(n+k-l)\lambda)d\lambda}. \nonumber$$ Again, we use the definition of Dirac’s delta function and its properties to get $$\rho_{xy}(n)=\frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}{a_l(d_1)\theta^k\delta(n+k-l)}=\frac{\sigma_{\varepsilon\nu}}{\sigma_x\sigma_y}\sum_{k=0}^{\infty}{a_{n+k}(d_1)\theta^k}. \nonumber$$ Using the Stirling’s approximation and approximating the infinite sum by the definite integral, we get $$\rho_{xy}(n)\propto \int_{0}^{\infty}{(n+k)^{d_1-1}\theta^kdk}=\theta^{-n}\Gamma(d_1,-n\log\theta)(-\log\theta)^{-d_1} \nonumber$$ where $\Gamma(\bullet,\bullet)$ is the incomplete upper Gamma function [@Wall1948]. Using the approximation of the incomplete upper Gamma function [@Blahak2010], we can write $$\begin{gathered}
\rho_{xy}(n)\propto \theta^{-n}(-\log\theta)^{-d_1}(-n\log\theta)^{d_1-1}\exp(n\log\theta)\\
=\theta^{-n}\theta^{n}(-\log\theta)^{-d_1}(-\log\theta)^{d_1+1}n^{d_1-1}\propto n^{d_1-1} \nonumber\end{gathered}$$ Therefore, we have $$H_{xy}=1-\frac{\gamma_{xy}}{2}=1-\frac{1-d_1}{2}=1-\frac{-H_x+1.5}{2}=\frac{H_x+0.5}{2}$$ which is perfectly in hand with Eq. \[eq:ARFIMA\_H\] for $H_y=0.5$, i.e. the process $\{y_t\}$ is not long-range dependent with $d_2=0$. Note that the asymptotic relationship is again independent of $\sigma_{\varepsilon\nu}$ as long as $\sigma_{\varepsilon\nu}\ne0$.
Discussion and conclusions
==========================
Two types of processes generating power-law cross-correlations have been studied in detail. Importantly, such cross-correlations very easily arise from a very simple specification of the separate processes. As long as the error-terms are correlated, the power-law decay of the cross-correlation function emerges from the correlation structure of the separate processes. Moreover, we have presented that even if one of the analyzed processes is only short-range (exponentially) correlated, the long term memory of the other processes dominates and the processes together form a power-law cross-correlated pair. This is true regardless of a strength of the short term memory component.
These theoretical results are extremely important for empirical studies of power-law cross-correlations across various disciplines. They imply that the usually reported result of $H_{xy}>0.5$ connected with $H_{xy}\approx \frac{1}{2}(H_x+H_y)$ is not necessarily a sign of complex dependence between the analyzed series but it might simply emerge from the fact that at least one of the series is power-law correlated and the error-terms of the processes are at least somehow correlated. As the former case – power-law correlated separate processes – is quite frequent, we focus on the latter one – correlated error-terms. This brings us to a very understanding and interpretation of the error-term in statistical analysis. Even though there are many approaches, we stick to the two most common ones. Error-term can be understood primarily as a measurement error which brings uncertainty into a well defined model. Possible correlation between measurement errors of two series is thus usually not expected. Note that such definition is prevalent in experimental studies. However, error-terms in time series analysis are more frequently taken as unexpected innovations to the system or sometimes also as an unpredictable information flow. This is nicely illustrated in definitions of ARFIMA and AR processes in Eqs. \[eq1\] and \[eq2\]. The unpredictable innovations flow into the system and they get translated into final processes $\{x_t\}$ and $\{y_t\}$ based on their memory characteristics. Such innovations can either have a long-term or a short-term effect on the overall dynamics of a given process which is characterized by a specification as either AR or ARFIMA process (or generally many other possible specifications).
For the time series analysis, the latter understanding of error-terms prevails. In many systems, it is meaningful to expect correlated error-terms. As the power-law correlations and cross-correlations are heavily examined in economic and financial series, let us illustrate the concept on a simple example from econophysics. There, examination of cross-correlations between returns, volatility (riskiness of an asset) and traded volume is very popular. Now assume that an unexpected (not necessarily extreme) negative event occurs, e.g. during a quarterly profit announcement. Such event is negative information coming into the system and it is thus an innovation, an error-term (or a part of it) for the examined processes. The information in turn affects all three studied variables – price and thus also returns react negatively, volatility increases due to a magnified uncertainty and traders become more active as they try to rebalance their positions according to a new market situation. Therefore, all three variables react to the same impulse and their error-terms are thus correlated. Moreover, volatility and traded volume are power-law correlated which means that emerging power-law cross-correlations between returns, volatility and traded volume are present automatically as shown in this paper.
However, this should not discard the power-law cross-correlations analysis as a concept. Our findings stress that the analysis needs to be complete, without only partial results of the bivariate Hurst exponent. Empirical analyses should not be reported in a purely technical manner but they should be put into a correct context of the examined series. Another direction of research also further opens for processes with $H_{xy} \ne \frac{1}{2}(H_x+H_y)$ which have been studied only marginally [@Podobnik2008a; @Kristoufek2013; @Sela2012]. Branch of the power-law cross-correlations thus still remains an open field with numerous possibilities for future research.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank the anonymous referees for valuable comments and suggestions which helped to improve the paper significantly. Support from the Czech Science Foundation under project No. 14-11402P and Grant Agency of the Charles University in Prague under project No. 1110213 is gratefully acknowledged.
References {#references .unnumbered}
==========
|
---
abstract: '[PG1115+080]{} is a quadruply lensed quasar at $z=1.72$ whose image positions are well fit by simple models of the lens galaxy (at $z=0.31$). At optical wavelengths, the bright close pair of images exhibits a modest flux ratio anomaly (factors of $\sim$1.2–1.4 over the past 22 years) with respect to these same models. We show here that as observed in X-rays with [*Chandra*]{}, the flux ratio anomaly is far more extreme, roughly a factor of 6. The contrasting flux ratio anomalies in the optical and X-ray band confirm the microlensing hypothesis and set a lower limit on the size of the optical continuum emission region that is $\sim$10–100 times larger than expected from a thin accretion disk model.'
author:
- 'David Pooley, Jeffrey A. Blackburne, Saul Rappaport, Paul L. Schechter, & Wen-fai Fong'
title: 'A Strong X-Ray Flux Ratio Anomaly in the Quadruply Lensed Quasar [PG1115+080]{}$^1$'
---
Introduction {#sec:intro}
============
[PG1115+080]{} was the second gravitationally lensed quasar to be discovered, and the first found to be quadruple [@1980Natur.285..641W]. It has been the subject of numerous studies at wavelengths ranging from radio to mid-infrared to optical to UV to X-ray. The lensing galaxy is a member of a small group of galaxies, the tide from which produces the quadrupole moment needed to produce four images [@1997AJ....114..507K]. It was the first gravitationally lensed system to yield multiple time delays [@1997ApJ...475L..85S]. The optical images show uncorrelated flux variations on a timescale of order one year, presumably the result of microlensing by stars in the lensing galaxy .
[PG1115+080]{} is an example of what @2003AJ....125.2769S call an “inclined quad,” a system with a close, bright pair of images that results when a lensed source lies just inside a “fold” caustic [@2005ApJ...635...35K]. Several very similar systems have subsequently been discovered . In each case, one of the two close images is a minimum of the light travel time surface, and the other is a saddlepoint. From quite general considerations, if the gravitational potential is smooth, one expects the close, bright pair to be mirror images of each other and therefore very nearly equal in brightness [@2002ApJ...567L...5M]. [*All*]{} of the known inclined quads violate this prediction, despite the fact that such models fit the observed image positions to within a few percent. This phenomenon has come to be known as the “flux ratio anomaly” problem.
In this regard [PG1115+080]{} is the [*least*]{} anomalous among the inclined quads. In the earliest images that resolved the close pair, the ratio of the flux of the saddlepoint ($A_2$) to that of the minimum ($A_1$) was very nearly unity . By the mid-1980s the ratio had decreased to $\sim$$2/3$. Recent optical observations (see §\[sec:opt-obs\]) give a ratio closer to $\sim$$5/6$. By contrast the corresponding optical ratios for inclined quads WFIJ2026$-$4536, HS0810+2554, MG0414+0534, and SDSSJ0924+0219 are approximately $3/4$, $1/2$, $1/3$, and $1/10$, respectively . For this last case, @2006ApJ...639....1K argue that microlensing by stars (rather than millilensing by dark matter subcondensations) is responsible for the anomaly. @2004AAS...205.2806P and @2006astro.ph..1523M predict a substantial brightening of the faint saddlepoint in SDSSJ0924+0219 on a timescale of roughly one decade if the microlensing hypothesis is correct.
@2004AAS...205.2806P argue that the saddlepoint in [PG1115+080]{} would likewise be expected to get substantially (a factor of 2 or more) fainter on a similar timescale. But over the course of a quarter century [PG1115+080]{} has declined to cooperate, at least at optical wavelengths.
In the present paper we report that [PG1115+080]{} has indeed been exhibiting microlensing of the expected amplitude, but at X-ray wavelengths rather than at optical wavelengths. In §\[sec:obs\] we describe the X-ray and optical observations and our analysis. In §\[sec:discuss\] we discuss implications for the lensing galaxy and for the relative sizes of the quasar’s optical and X-ray emitting regions. We summarize our conclusions in §\[sec:conclusions\]. Throughout, we assume a “concordance” cosmology with $\Omega_M=0.3$, $\Omega_{\Lambda}=0.7$, and $h=0.72$.
0.4cm 0.3cm 0.4cm
Observations and Analysis {#sec:obs}
=========================
X-ray observations {#sec:xray-obs}
------------------
[PG1115+080]{} was observed for 26.5 ks on 2000 Jun 02 (ObsID 363) and for 9.8 ks on 2000 Nov 03 (ObsID 1630) with the *Chandra X-ray Observatory’s* Advanced CCD Imaging Spectrometer (ACIS). These observations were used by @2004ApJ...610..686G to study the X-ray properties of the lensing group of galaxies. The data were taken in timed-exposure mode with an integration time of 3.24 s per frame, and the telescope aimpoint was on the back-side illuminated S3 chip. The data were telemetered to the ground in faint mode.
The data were downloaded from the [*Chandra*]{} archive, and reduction was performed using the CIAO3.3 software provided by the [*Chandra*]{} X-ray Center[^1]. The data were reprocessed using the CALDB3.2.1 set of calibration files (gain maps, quantum efficiency, quantum efficiency uniformity, effective area) including a new bad pixel list made with the [acis\_run\_hotpix]{} tool. The reprocessing was done without including the pixel randomization that is added during standard processing. This omission slightly improves the point spread function. The data were filtered using the standard [*ASCA*]{} grades and excluding both bad pixels and software-flagged cosmic ray events. Intervals of strong background flaring were searched for, but none were found.
For each observation, an image was produced in the 0.5–8 keV band with a resolution of 00246 per pixel (see Figure \[fig:images\]). To determine the intensities of each lensed quasar image, a two-dimensional model consisting of four Gaussian components plus a constant background was fit to the data. The background component was fixed to a value determined from a source-free region near the lens. The relative positions of the Gaussian components were fixed to the separations determined from [*Hubble Space Telescope*]{} observations [@1993AJ....106.1330K], but the absolute position was allowed to vary. Each Gaussian was constrained to have the same full-width at half-maximum, but this value was allowed to float. The fits were performed in Sherpa [@2001SPIE.4477...76F] using Cash statistics [@1979ApJ...228..939C] and the Powell minimization method. The intensity ratios (relative to image C) are listed in Table \[tab:xrayfits\]. The best-fit full-width at half maximum (fwhm) was $0\farcs83 \pm 0\farcs01$ for ObsID 363 and $0\farcs80 \pm 0\farcs02$ for ObsID 1630; both consistent with the overall width of the instrumental point spread function (PSF) as found in the [*Chandra*]{} PSF Library [@2001ASPC..238..435K] supplied by the [*Chandra*]{} X-ray Center. In addition to the Gaussians, models of the form $f(r)=A[1+(r/r_0)^2]^{-\alpha}$ were also tried; these gave similar results to the values in Table \[tab:xrayfits\].
Based on the best fit Gaussian shape and the relative intensities, we constructed a “pseudo” maximum entropy method (MEM) representation of the data. Here we have simply plotted Gaussians of a common width (fwhm = $0.22''$), with the fitted intensities and at the fitted locations (see Table \[tab:xrayfits\]). We used the largest source width consistent (at 3$\sigma$ confidence) with no blurring of the intrinsic [*Chandra*]{} PSF. A maximum likelihood deconvolution of the image is presented by @2004mmu..sympE...1C and appears consistent with our “pseudo” MEM image.
[rccl]{}\
$A_1/C$ & $3.9\pm0.3$ & $4.3\pm0.5$ & 3.91\
$A_2/C$ & $0.6\pm0.1$ & $1.2\pm0.3$ & 3.73\
$B/C$ & $1.0\pm0.1$ & $0.9\pm0.1$ & 0.67\
$A_2/A_1$& $0.16\pm0.03$ & $0.29\pm0.08$ & 0.96
Spectra of the quasar images were extracted using the [ACIS Extract]{} package v3.94 [@Broos02]. A single spectrum of $A_1$ and $A_2$ was extracted because of the significant overlap, but $B$ and $C$ were extracted separately. Both the [*Chandra*]{} effective area and PSF are functions of energy, and [ACIS Extract]{} corrected the effective area response for each spectrum based on the fraction of the PSF enclosed by the extraction region (at 1.5 keV, these fractions were 0.9 for $A_1+A_2$, 0.8 for $B$, and 0.9 for $C$). The spectra were grouped to contain at least ten counts per bin, and $\chi^2$ fitting was performed in Sherpa using a simple absorbed power law model. The column density was fixed at the Galactic value of $3.56\times10^{20}$ cm$^{-2}$ . The individual fits were all acceptable and yielded consistent results, so joint fits were performed with the power law indices tied to each other and the normalizations allowed to float. The best fit photon index for ObsID 363 is $1.57\pm0.04$ and for ObsID 1630 is $1.54\pm0.07$, which compares well with the values found from the fits of image C alone ($1.55\pm0.09$ and $1.46\pm0.08$, respectively). Based on the individually fitted power laws, the unabsorbed 0.5–8 keV flux of image C is $(6.2\pm0.4)\times10^{-14}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{} in ObsID 363 and $(6.9\pm0.9)\times10^{-14}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{} in ObsID 1630. These serve as useful reference fluxes since image C is fairly uncontaminated by flux from the other images and is also a minimum image and therefore less susceptible to fluctuations.
[ACIS Extract]{} was also used to obtain light curves from the above extraction regions for each observation. No significant signs of short-term variability were found within either observation; Kolmogorov-Smirnov tests showed that each light curve had a greater than 10% chance of being consistent with a constant count rate. The light curve for the $A_1+A_2$ region is plotted in Figure \[fig:lc363\].
Given the time delays among the lensed images, it is fair to ask if intrinsic short-term quasar variability combined with a time delay could masquerade as a genuine X-ray flux ratio anomaly. We can rule this out in the X-ray band for ObsID 363. The time delay between $A_1$ and $A_2$ from our lens model (see §\[sec:model\]) is $14.5\pm2$ ks (with $A_1$ leading). The 26.5 ks observation therefore covers 1.8 time delay cycles. If we split the observation into two equal parts, we obtain the same $A_1/A_2$ ratio as in Table \[tab:xrayfits\]. To produce this ratio as well as the constant $A_1+A_2$ lightcurve in Figure \[fig:lc363\] purely by variability is highly implausible.
![Light curve of the 0.5–8 keV count rate of $A_1+A_2$ in ObsID 363 showing a rather constant flux. Horizontal bars indicate the 2 ks time bins, and vertical bars show 1-$\sigma$ errors.[]{data-label="fig:lc363"}](f2.eps){width="47.00000%"}
Optical observations {#sec:opt-obs}
--------------------
[PG1115+080]{} has been observed repeatedly with the Magellan 6.5-meter Baade and Clay telescopes at Las Campanas Observatory between 2001 March and 2006 February using the Raymond and Beverly Sackler Magellan Instant Camera (MagIC). The instrument has a scale of 00691 per pixel and a 2.36 arcminute field. We present here results from three epochs for which the seeing was especially good, making the decomposition of $A_1$ and $A_2$ easier and less uncertain, and reducing the contamination from the lensing galaxy. Three 60-second exposures were obtained with a Johnson V filter on UT 2001 March 26. Two 60-second exposures each were obtained obtained with a Sloan $i'$ filter on UT 2004 Feb 22 and 2005 June 07.
The data were flattened using standard procedures. [ClumpFit]{}, an empirical PSF-fitting photometry program based on [DoPHOT]{}, was used to measure fluxes and positions for the four quasar images and for the lensing galaxy. The profile for the galaxy was taken to be an elliptical pseudo-Gaussian. As we presently concern ourselves only with flux ratios, we have not put our photometry onto a standard system. The fluxes for the $A_1$, $A_2$ and $B$ images are given relative to the $C$ image, for which the microlensing fluctuations are expected to be smallest. It should be remembered that variations of 0.1 mag have been seen on a timescale of weeks and that image $C$ leads the $A$ images and the $B$ image by 10 and 25 days, respectively [@1997ApJ...475L..85S; @1997ApJ...489...21B]. The results of our photometry are given in Table \[tab:opthistory\], along with selected results (typically those obtained in the best seeing) from prior epochs.
We note that the flux ratios for contemporaneous observations appear to be consistent to within a few percent over the optical wavelength region. We therefore make no attempt to account for bandpass in presenting the present and past optical results.
[lcccccc]{} 1984 Mar 26 & B & 075 & $-1.26$ & $-1.21$ & 0.41 & $0.95 \pm 0.07$\
1985 Mar 19 & V & 062 & $-1.18$ & $-0.83$ & 0.49 & $0.73 \pm 0.04$\
1986 Feb 19 & V & 06 & $-1.27$ & $-0.99$ & 0.48 & $0.77 \pm 0.03$\
1986 Feb 19 & B & 06 & $-1.23$ & $-0.97$ & 0.48 & $0.79 \pm 0.03$\
1991 Mar 03 & F785LP & [*HST*]{}& $-1.46$ & $-1.07$ & 0.50 & $0.70 \pm 0.01$\
1991 Mar 03 & F555W & [*HST*]{}& $-1.47$ & $-1.02$ & 0.50 & $0.66 \pm 0.01$\
1995 Dec 20 & V & 085 & $-1.50$ & $-1.04$ & 0.47 & $0.66 \pm 0.01$\
2001 Mar 26 & V & 056 & $-1.48$ & $-1.04$ & 0.42 & $0.68 \pm 0.01$\
2004 Feb 22 & $i'$ & 048 & $-1.40$ & $-1.18$ & 0.42 & $0.81 \pm 0.01$\
2005 Jun 07 & $i'$ & 043 & $-1.40$ & $-1.19$ & 0.42 & $0.81 \pm 0.01$\
\
Lens Model & & & $-1.48$ & $-1.43$ & 0.44 & 0.96\
Discussion {#sec:discuss}
==========
Modeling the lens {#sec:model}
-----------------
Using Keeton’s (2001) [Lensmodel]{} software, we modeled the lensing potential as a singular isothermal sphere accompanied by a second, offset singular isothermal sphere, which provides a quadrupole moment. This choice of model was motivated by the presence of a group of galaxies to the southwest of the lensing galaxy. We used the image positions provided by the CASTLES Lens Survey[^2], and did not constrain the fluxes. Our best-fit model predicts an Einstein radius of 10 for the primary lensing galaxy, with a second mass having an Einstein radius of 26 located 125 away at a position angle $116^{\mathrm o}$ west of north. This places it close to the observed location of the associated group of galaxies. The model yields a total reduced $\chi^2$ of 3, with the greatest contribution coming from the position of the primary lensing galaxy. The flux ratios predicted by this model are listed in Tables \[tab:xrayfits\] and \[tab:opthistory\], and may be expected to vary between different plausible models of the lens at the 10% level.
Anomalous flux ratios and microlensing {#sec:microlensing}
--------------------------------------
Simple smooth analytic models [@2002ApJ...567L...5M] predict that the $A_2/A_1$ flux ratio should be very nearly equal to unity. For our lens model, the ratio is 0.96. @2005ApJ...627...53C observe a mid-infrared flux ratio of $0.93 \pm 0.06$, consistent with this prediction. In 1984, measured a flux ratio of $0.95 \pm 0.07$, but since then, as seen in Table \[tab:opthistory\], the optical flux ratio has varied on a timescale of years between 0.66 and 0.81. As noted in §\[sec:xray-obs\], the contemporaneous X-ray flux ratio is less than 0.2, inconsistent not only with the predictions of the smooth models, but with the optical observations as well.
Microlensing by stars in the lensing galaxy could in principle account for such flux ratios, but only if the source is small compared to the Einstein radii of the microlensing stars. Our simple model has convergence, $\kappa$, and shear, $\gamma$, roughly equal at the image positions, with magnifications $\mu$ of 19.9 for the $A_1$ image and $-$19.0 for the $A_2$ image. Examples of point source magnification histograms for pairs of images very much like those in [PG1115+080]{} are presented by Schechter and Wambsganss, with magnifications for $A_1$ and $A_2$ of 10 and 16, respectively [@2002ApJ...580..685S]. They present histograms both for the case when 100% of the convergence is due to stars and for the case when only 20% of the convergence is due to stars and the rest is due to a smooth dark component. The X-ray flux ratio rules out neither hypothesis but is considerably more likely if dark matter is present.
Until now, it was a bit of a puzzle why the optical flux anomalies had failed to deviate from unity as much as was predicted by these histograms. Now it appears that it was because the optical region is too large to be strongly microlensed (see §\[sec:sizes\]). As Schechter & Wambsganss note, the determination of the dark matter fraction of lensing galaxies using the statistics of flux ratio anomalies is made considerably more difficult if the source size is comparable to that of a stellar Einstein ring. It seems now that the X-ray flux ratio anomalies offer a cleaner determination of the dark matter fraction than the optical anomalies.
Long-term X-ray variability {#sec:xrayvar}
---------------------------
According to the microlensing model for flux-ratio anomalies, discussed below, $A_2$ is expected to brighten in X-rays on a timescale of $\sim$10 years, and follow-up [*Chandra*]{} observations will be able to directly test this. As $A_2$ brightens, the unresolved flux will also increase. To look for past signs of this effect, we searched the High Energy Astrophysics Science Archive Research Center, provided by NASA’s Goddard Space Flight Center, for other X-ray observations of [PG1115+080]{} and found two [*ROSAT*]{} observations and three relevant [*XMM-Newton*]{} observations. The [*ROSAT*]{} observations and an earlier [*Einstein*]{} observation are analyzed in @2000ApJ...531...81C.
The [*ROSAT*]{} count rates were converted to unabsorbed 0.5–2 keV fluxes using WebPIMMS [@1993Legac...3...21M] with the assumptions of an absorbed power law of photon index 1.65 and a column density of $3.56\times10^{20}$ cm$^{-2}$. For the [*XMM*]{} observations, we extracted spectra of [PG1115+080]{} from the EPIC-PN and both EPIC-MOS detectors. We performed joint spectral fits (on all quasar images added together) in the 0.5–10 keV band for each observation with simple absorbed power laws with the column density fixed at the Galactic value. These gave acceptable fits, from which we computed the unabsorbed 0.5–2 keV fluxes. We also used our previous [*Chandra*]{} joint fits to compute the total 0.5–2 keV fluxes (from all quasar images added together) from the [*Chandra*]{} observations. The long-term X-ray light curve is shown in Figure \[fig:ltlc\].
From the seven measurements of the lensed flux from [PG1115+080]{} over the course of 12.5 years, the mean is $1.75\times10^{-13}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{}, and the sample standard deviation is $6.7\times10^{-14}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{}, or $\sim$40%. There is no evidence for strong short term variability from the individual lensed images in the [*Chandra*]{} data, nor is there evidence for strong short term variability within the three [*XMM*]{} observations (in which the individual images are unresolved).
As discussed above, if the demagnification of $A_2$ is due to microlensing, the unresolved flux will rise as $A_2$ becomes less demagnified. The observed relative X-ray fluxes of the four images $A_1:A_2:B:C$ are $1:0.16:0.25:0.25$ (based on ObsID 363; see Table \[tab:xrayfits\]). If $A_2$ were to rise in flux to match $A_1$, the overall change in flux would be $\sim$50%. The recent [*XMM*]{} observations show that the X-ray flux has risen $\sim$30% since the [*Chandra*]{} observations from six years ago (Figure \[fig:ltlc\]). However, there is an obvious degeneracy between a rise in the flux of $A_2$ and typical quasar variability over the course of many years.
![Long-term X-ray light curve of [PG1115+080]{} showing the combined flux of all four images. For most observations, the plotted error bars are smaller than the plotting symbols.[]{data-label="fig:ltlc"}](f3.eps){width="47.00000%"}
Sizes of quasar emission regions {#sec:sizes}
--------------------------------
The size scales of the emission regions in quasars are difficult to probe directly since they are on the microarcsecond scale or smaller. The use of temporal variability for inferring sizes is indirect and becomes impractical for distant quasars. By contrast, microlensing directly explores angular scales of (by definition) microarcseconds. Of the emission features of the quasar, only those which subtend smaller angles on the sky than the Einstein radius of the microlenses will exhibit strong variations in flux.
![Source sizes at X-ray and optical wavelengths (see text).[]{data-label="fig:fxi"}](f4.eps){width="47.00000%"}
Figure \[fig:fxi\] displays the results from our study of [PG1115+080]{}. Here we have plotted the ratio of angular scale for different regions of the quasar, $\theta_s$, to the Einstein radius of a solar-mass microlens, $\theta_E$. For ratios greater than unity, microlensing should be strongly suppressed [for a detailed analysis see @2005ApJ...628..594M]. The ratio $\theta_s/\theta_E$ is plotted against the assumed mass of the central black hole, $M_{\rm BH}$. For every value of $M_{\rm BH}$ there is a corresponding Eddington luminosity which can be compared to the observed values of [$L_\mathrm{x}$]{} ($2.4\times 10^{44}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{}; 0.5–8 keV; this work) and $L_{\rm opt}$ ($1.2\times 10^{45}$ [$\mathrm{erg~cm}^{-2}~\mathrm{s}^{-1}$]{}, from a sum of the V-, I-, and H-band data provided by the CASTLES Lens Survey) for [PG1115+080]{} (see the top axis label). Within the $\theta_s/\theta_E$ vs. $M_{\rm BH}$ plane we plot contours of constant size in units of $R_g$, the gravitational radius of the black hole ($GM_{\rm BH}/c^2$). As is evident from the plot, the X-rays, which should arise deep in the gravitational potential well of the black hole, should be microlensable for any $M_{\rm BH} \lesssim 10^{10}\,M_\odot$. This is in clear agreement with the large X-ray flux ratio anomalies observed for [PG1115+080]{} and for two other quad lenses: RXJ0911+0554 and RXJ1131$-$1231 [@2001ApJ...555....1M; @2006ApJ...640..569B]. By contrast, the broad-line emission region should not be microlensable, except for a lower mass black hole (i.e., $M_{\rm BH} \lesssim 3 \times 10^7\,M_\odot$). Finally, the dotted and dashed curves mark the radii within which 50% of the power in the $I$ and $V$ bands emerge, respectively, for a simple thin accretion disk model . According to these curves, the optical continuum ought to be microlensed by approximately the same amount as in the X-ray band, in agreement with @2005ApJ...628..594M. But clearly it is not!
Using [*HST*]{} spectra, @2005MNRAS.357..135P found that the $A_2/A_1$ ratio in the ultraviolet continuum is $\sim$0.5 and decreases to shorter wavelengths, indicating that the UV is more severely microlensed than the optical but less microlensed than the X-rays.
Therefore, within the microlensing scenario, we can conclude that the continuum optical emission from [PG1115+080]{} comes from much further out than the UV, which in turn comes from further out than the X-rays. In particular, we find that the optical emission comes from a region $\sim$10–100 times larger than expected for a thin accretion disk model (for $M_{\rm BH}$ in the range $3 \times 10^9\,\rightarrow 10^8\,M_\odot$). Since $L_{\rm opt}$ dominates [$L_\mathrm{x}$]{} in [PG1115+080]{} (and for many other luminous quasars), this is difficult to understand from an energetics point of view, since the energy released goes as $r^{-1}$. Of course the optical light could be scattered by a large-scale plasma region; however, in that case one would expect the X-rays to be scattered as well, and hence share a similar effective emission region. Thus, while the X-ray images clearly appear to be microlensed, the bulk of the optical emission must be coming from $\sim$100–3000$R_g$ from the central black hole (for $M_{\rm BH}$ in the range $3 \times 10^9\,\rightarrow 10^8\,M_\odot$).
In coming to these conclusions, we have neglected special- and general-relativistic effects in the emissions from the accretion disk, except for cosmological redshift. In addition, we have followed @2005ApJ...628..594M in assuming a Kerr black hole with a large spin parameter ($a=0.88$). This is consistent with estimates for a typical quasar [@2006astro.ph..3813W], and implies an innermost disk radius of $2.5 R_g$ and a binding energy per mass $\eta=0.146$. We have also set the bolometric luminosity to 33% of the Eddington luminosity, as advocated by @2005astro.ph..8657K. Neither of these parameter assumptions has a strong effect on the size of the predicted optical emission region for a thin accretion disk model.
Conclusions {#sec:conclusions}
===========
We have made use of optical data collected over the past 22 years to demonstrate that the bright, close pair of lensed images of [PG1115+080]{} has a consistent flux ratio ($A_2/A_1$) of $\sim$0.7–0.8. X-ray observations with [*Chandra*]{}, covering two epochs separated by 5 months, indicate a much more extreme flux ratio of $\sim$0.2. Both the optical and X-ray ratios are anomalous with respect to smooth lensing models, which predict a flux ratio of 0.96. We used a comparison of the optical and X-ray flux ratio anomalies to argue in favor of the microlensing origin of the anomalies, and to show that the optical emission region is much larger (i.e., $\sim$$10-100$) than predicted by a simple thin accretion disk model.
We thank Joachim Wambsganss, George Chartas, and the anonymous referee for useful comments. DP gratefully acknowledges support provided by NASA through Chandra Postdoctoral Fellowship grant number PF4-50035 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. SR received some support from Chandra Grant TM5-6003X. JAB and PLS acknowledge support from NSF Grant AST-0206010. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center.
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[^1]: <http://asc.harvard.edu>
[^2]: http://www.cfa.harvard.edu/castles/
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---
abstract: 'We construct a model of spontaneous CP violation in E$_{\text 6}$ supersymmetric grand unified theory. In the model, we employ an SU(2)$_\text{F}$ flavor symmetry and an anomalous U(1)$_{\text A}$ symmetry. The SU(2)$_\text{F}$ flavor symmetry is introduced to provide the origin of hierarchical structures of Yukawa coupling and to ensure the universality of sfermion soft masses. The anomalous U(1)$_{\text A}$ symmetry is introduced to realize the doublet-triplet mass splitting, to provide the origin of hierarchical structures of Yukawa couplings, and to solve the $\mu$ problem. In the model, CP is spontaneously broken by the SU(2)$_\text{F}$ breaking in order to provide a Kobayashi-Maskawa phase and to evade the supersymmetric CP problem. However, a naive construction of the model generally leads to unwanted outcome, Arg$[\mu b^*]=\cal O$(1), when CP violating effects in the flavor sector are taken into account. We cure this difficulty by imposing a discrete symmetry and find that this prescription can play additional roles. It ensures that realistic up-quark mass and Cabibbo angle are simultaneously realized without cancellation between $ \mathcal O(1)$ coefficients. Also, severe constraints from the chromo electric dipole moment of the quark can be satisfied without destabilizing the weak scale. The discrete symmetry reduces the number of free parameters, but the model is capable of reproducing quark and lepton mass spectra, mixing angles, and a Jarlskog invariant. We obtain characteristic predictions $V_{ub} \sim \cal O$($\lambda^4$) ($\lambda=0.22$) and $| V_{cb} Y_b | = |Y_c|$ at the grand unified theory scale.'
---
IUHET-534\
Cavendish-HEP-09/18\
DAMTP-2009-66\
[**Spontaneous CP violation in E$_{\text 6}$ SUSY GUT\
with SU(2) flavor and anomalous U(1) symmetries**]{}
M. Ishiduki$^1$, S.-G. Kim$^{23}$[^1], N. Maekawa$^1$[^2], and K. Sakurai$^4$[^3]
$^1$[*Department of Physics, Nagoya University,\
Nagoya 464-8602, Japan*]{}\
$^2$[*Department of Physics, Tohoku University,\
Sendai 980-8578, Japan*]{}\
$^3$[*Department of Physics, Indiana University,\
Bloomington IN 47405, USA*]{}\
$^4$[*DAMTP, Wilberforce Road, Cambridge, CB3 0WA, UK\
Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, UK* ]{}\
Introduction
============
Low-energy supersymmetry (SUSY) is one of the leading candidates for physics beyond the standard model (SM) [@Nilles:1983ge]. In particular, the minimal supersymmetric extensions of the SM (MSSM) exhibit remarkable coincidence of three SM gauge coupling constants near $10^{16}$GeV, and the supersymmetric grand unified theory (SUSY GUT) appears to be plausible model behind the MSSM. In these models, however, if generic soft SUSY breaking terms are introduced, these terms can induce flavor changing neutral current (FCNC) processes and CP violating observables like electric dipole moments (EDMs) that are too large to be consistent with the present experiments [@Gabbiani:1996hi]. These issues are called the SUSY flavor problem and the SUSY CP problem, respectively. In this paper, we address the later problem in the context of SUSY GUT. We choose E$_{\text 6}$ as a unification group and introduce an SU(2) flavor symmetry. Based on this framework, we construct a model of spontaneous CP violation (SCPV) in the flavor sector to provide the origin of the Kobayashi-Maskawa (KM) phase and to evade the SUSY CP problem.
Among several possible grand unification groups [@Georgi:1974sy; @Fritzsch:1974nn; @Gursey:1975ki], the E$_{\text 6}$ group with SU(2)$_{\text F}$ flavor symmetry exhibits a numbers of attractive features [@Gursey:1975ki; @Bando:1999km; @Bando:2001bj].
1. Three fundamental representations of E$_6$, $\bold{ 27}_1$, $\bold{ 27}_2$, and $\bold{ 27}_3$, contain all of the SM fermions and the right-handed neutrinos.
2. Various realistic hierarchical structures of masses and mixings of quarks and leptons can be naturally derived from the one basic hierarchy of E$_{\text 6}$ invariant Yukawa couplings.
3. This basic hierarchy results from the SU(2)$_{\text F}$ flavor symmetry and its breaking effects.[^4]
4. The SUSY flavor problem can be avoided by the SU(2)$_{\text F}$ flavor symmetry without destabilizing the weak scale.[^5] To be more concrete, at the leading order, the model has the following sfermion soft mass matrices at the GUT scale, as a consequence of SU(2)$_{\text F}$ symmetry [@Maekawa:2002eh]. $$m_{\bold {10}}^2 =
\left(
\begin{array}{ccc}
m_0^2 & 0 & 0 \cr
0 & m_0^2 & 0 \cr
0 & 0 & m_3^2
\end{array}
\right), \quad
m_{\bar{ \bold {5}}}^2=
\left(
\begin{array}{ccc}
m_0^2 & 0 & 0 \cr
0 & m_0^2 & 0 \cr
0 & 0 & m_0^2
\end{array}
\right)
\label{eq:mod-uni}$$ (Here, $m_{\bf 10}$ and $m_{\bf\bar 5}$ are the mass matrices of the ${\bf 10}$ and ${\bf\bar 5}$ fields of SU(5), where the SM matter fields are embedded in these multiplets in a standard way [@Georgi:1974sy].) In this paper, we call this form of sfermion soft mass matrix “$\it{modified}$ universality"[@Maekawa:2002eh; @Kim:2006ab; @Kim:2008yta; @Ishiduki:2009gr; @Kim:2009nq]. In this set up, if the gluino mass ($M_3$), the up-type Higgs mass ($m_{H_U}$), and the supersymmetric Higgs mass ($\mu $) as well as $m_3$ are the weak scale, then the weak scale can be stabilized.[^6] Also, even if there are some deviations from the “$\it{modified}$ sfermion universality" in flavor space, the FCNC constraints can be satisfied by raising $m_0$, and this does not destabilize the weak scale as long as $m_3$ is maintained in the weak scale.[^7] Note that this prescription is more natural compared to the case for the deviation from the “universality" that is assumed in minimal supergravity where $m_3$ is set equal to $m_0$.
Now, let us move on to the SUSY CP problem. In order to avoid the SUSY CP problem, it is desirable that the dimensionful soft SUSY breaking parameters and the $\mu$ parameter are all real. On the other hand, the Yukawa couplings should be complex parameters so as to provide a KM phase [@Cabibbo:1963yz]. Is it possible to realize these observations naturally? To answer this question, in this paper we examine the idea of spontaneous CP violation (SCPV) [@Lee:1973iz; @Barr:1988wk] within the context of E$_{\text 6}\times $SU(2)$_{\text F}$ SUSY GUT. First of all, by imposing CP symmetry, all the coupling constants of the original Lagrangian, including soft SUSY breaking terms, can be taken to be real. Then we make a model in which CP is spontaneously broken by complex vacuum expectation values (VEV) of flavon fields; i.e., the SU(2)$_\text{F}$ flavor symmetry breaking is also responsible for the CP violation. Here, once renormalizability is assumed, it seems difficult to generate complex Yukawa couplings. Therefore we allow higher dimensional operators so that complex dimensionless (Yukawa) couplings can be induced via nonrenormalizable interaction terms. Since CP is broken in the SU(2)$_\text{F}$ sector, we can expect that the SU(2)$_\text{F}$ singlet SUSY breaking parameters can remain real after the SCPV, contrary to the Yukawa couplings. Also, even if flavor nonsinglet SUSY breaking parameters acquire CP phases, their effects on the CP violating observables can be suppressed in the current framework by taking $m_0$ much larger than the weak scale.[^8] Therefore, SCPV provides a way to solve the SUSY CP problem.
In the previous papers of Ref.[@Maekawa:2001uk], an anomalous U(1)$_{\text A}$ gauge symmetry [@Witten:1984dg; @Froggatt:1978nt] was introduced to solve the doublet-triplet splitting (DTS) problem and to provide the origin of the hierarchal structures of Yukawa couplings. The U(1)$_{\text A}$ symmetry was also applied to solve the $\mu$ problem [@Maekawa:2001yh]. However, as we will see in the following section, if we naively apply the U(1)$_\text{A}$ symmetry to the model of SCPV, it generally leads to an unwanted outcome, Arg$[\mu b^*] =\cal O$(1), and gives rise to the SUSY CP problem. We cure this difficulty by imposing discrete symmetry. Interestingly, this discrete symmetry can play additional roles in the model. It ensures that realistic up-quark mass and Cabibbo angle are simultaneously realized without cancellation between $ \mathcal O(1)$ coefficients. Also, the severe constraints from the chromo-EDMs (CEDM) can be satisfied without destabilizing the weak scale. As a result of the discrete symmetry, the number of free parameters of the model is reduced. However, the model is capable of reproducing quark and lepton mass spectra, mixing angles, and a Jarlskog invariant. We also obtain characteristic predictions $V_{ub} \sim \cal O$($\lambda^4$) ($\lambda=0.22$) and $|Y_b V_{cb}| = |Y_c|$ at the GUT scale.
This paper is organized as follows. In the next section, we give a brief review of the model of E$_{\text 6}$ SUSY GUT and SU(2)$_{\text F}$ flavor symmetry. In Sec.3, an anomalous U(1)$_{\text A}$ gauge symmetry and its specific vacua are summarized. Based on these two sections, we show a simple model of the SCPV that is caused by the VEVs of flavon fields, in Sec.4. In the same section, we also discuss CP violation effects on $\mu$- and $b$-terms and indicate that they lead to Arg$[\mu b^*] =\cal O$(1). Then, we introduce discrete symmetry in order to solve this difficulty. In Sec.5, we summarize a model of SCPV in E$_{\text 6}$ SUSY GUT with SU(2)$_\text{F} $, U(1)$_\text{A}$, and discrete symmetries. Then, we examine the consequent Yukawa couplings and derive some predictions of the model. The last section is devoted to a summary and discussion.
E$_{\text 6}\times$SU(2)$_{\text F}$ SUSY GUT {#sec:e6*su2 susy gut}
=============================================
In this section we briefly summarize the model of E$_{\text 6}\times$SU(2)$_{\text F}$ SUSY GUT discussed in Refs.[@Bando:2001bj; @Maekawa:2002eh]. We mainly focus on the Yukawa sector and sfermion soft masses of the model.
E$_6$ GUT {#subsec:e6 gut}
---------
In this subsection, we discuss the Yukawa sector of E$_{\text 6}$ SUSY GUT. Here the twisting mechanism among SU(5) ${\bar {\bold{ 5 }}}$ fields plays important roles [@Bando:1999km; @Bando:2001bj; @Maekawa:2002eh].
For the E$_{\text 6}$ group, $\textbf{27}$ is the fundamental representation, and in terms of E$_{ 6}\supset $SO(10)$\times $U(1)$_{V'}$ (and \[SO(10)$\supset$SU(5)$\times$U(1)$_{V}$\]) it is decomposed as[^9] $$\begin{aligned}
\textbf{27} = \textbf{16}_1 [ \textbf{10}_{-1} + \bar {\textbf{5}}_3 + \textbf{1}_{-5}]
+ \textbf{10}_{-2} [ \textbf{5}_2 + \bar{\textbf{5}}_{-2}^\prime ] + \textbf{1}_4 [ \textbf{1}_0^\prime]
\, .
\label{eq:e6 decomposition}\end{aligned}$$ As shown in , each $\textbf{27}$ incorporates two $\bar {\textbf{5}} $’s(and ${\textbf{1}} $’s) of SU(5). This nature allows a model to simply produce different and realistic Yukawa structures of quark and lepton from a single hierarchical structure of an E$_6$ invariant Yukawa coupling [@Bando:1999km; @Bando:2001bj].
Let us introduce the following superpotential: $$\begin{aligned}
W_{\textnormal E_{ 6}} \supset
Y_{ij}^H \Psi_i^{\textbf{27}} \Psi_j^{\textbf{27}} H^{\textbf{27}} + Y_{ij}^C \Psi_i^{\textbf{27}} \Psi_j^{\textbf{27}} C^{\textbf{27}}
\label{eq:E_6 Yukawa}\end{aligned}$$ and assume that the original Yukawa hierarchies are $$\begin{aligned}
Y_{ij}^{H} \sim Y_{ij}^{C} \sim \left(
\begin{array}{c c c}
\lambda ^6 & \lambda ^5 & \lambda ^3 \\
\lambda ^5 & \lambda ^4 & \lambda ^2 \\
\lambda ^3 & \lambda ^2 & 1
\end{array}
\right) .
\label{eq:original Yukawa}\end{aligned}$$ Here $\Psi_i^{\textbf{27}} \, (i=1-3) $ are matter fields, and $H ^{\textbf{27}} $ and $C ^{\textbf{27}}$ are Higgs fields that break E$_{ 6}$ into SO(10) and SO(10) into SU(5), respectively.[^10] We assume that the MSSM Higgs doublets are included in $H ^{\textbf{27}} $ and $C ^{\textbf{27}}$. Hereafter, we parametrize the several hierarchical structures of couplings and VEVs of various fields in terms of the Cabibbo angle $\lambda$. (We set $\lambda \equiv 0.22$.) In Eq., we only parametrize the hierarchical structures, and the $\cal O$(1) coefficients are suppressed for the moment.
In Eq., once $H^{\textbf{27}} $ and $C^{\textbf{27}}$ acquire VEVs in the components of $\textbf{1}_4 ( \textbf{1}^\prime_0 )$ and $\textbf{16}_1 (\textbf{1}_{-5})$, respectively, a superheavy mass matrix of rank 3 among $\bold{ 5}_i, \bar {\bold{ 5 }}^\prime_i$, and $\bar {\bold{ 5 }}_i$ of $\Psi_i^{\textbf{27}}$ is induced, through the Yukawa coupling of Eq.. Therefore the 3 degrees of freedom among $\bar {\bold{ 5 }}^\prime_i$ and $\bar {\bold{ 5 }}_i$ decouple at the GUT scale. In consequence, the up-type quark Yukawa coupling ($Y_U$) remains in the original form of Eq. but the down-type quark ($Y_D$) and charged lepton Yukawa couplings ($Y_E$) differ from it. Importantly, the three massless modes among $\bar {\bold{ 5 }}^\prime_i$ and $\bar {\bold{ 5 }}_i$ mainly originate from the first two generations of $\Psi_{i}^{27} $ because the third generation $\bar {\bold{ 5 }}^\prime_3$ and $\bar {\bold{ 5 }}_3$ from $\Psi_{3}^{27} $ have large Yukawa couplings, i.e., large GUT scale masses. This feature is also important for making sfermion universality for the ${\bar {\bold{ 5 }}}_{}$ sector, as discussed in the following subsection. As an example, when $\langle C \rangle/ \langle H \rangle \sim \lambda^{0.5}$, we end up with the following milder hierarchies for $Y_D$ and $Y_E$ [@Bando:2001bj]: $$\begin{aligned}
Y_D \sim Y_E^T \sim \left(
\begin{array}{c c c}
\lambda ^6 & \lambda ^{5.5} &\lambda ^5 \\
\lambda ^5 & \lambda ^{4.5} &\lambda ^4 \\
\lambda ^3 & \lambda ^{2.5} &\lambda ^2
\end{array}
\right).
\label{eq:yd and ye}\end{aligned}$$ Note that the hierarchies of Eqs. and are adequate to reproduce several mass spectra and flavor mixing angles of quarks and leptons.[^11] However, an exception is the up-quark Yukawa coupling $Y_u$. The hierarchy of Eq. leads to $Y_u \sim \lambda^6$ but the realistic value is $Y_u \sim \lambda^8 $.[^12] We will get back to this issue in Sec.\[subsec:susy cp problem and discrete symmetry\]
SU(2) flavor symmetry and modified universality {#subsec:su(2)f and modified universality}
-----------------------------------------------
In this subsection, we briefly discuss the effect of SU(2)$_{\text F}$ flavor symmetry on the sfermion soft masses and see the emergence of the $\it{modified}$ universal form of Eq. [@Maekawa:2002eh].
First of all, we assume that the original hierarchy of Eq. (partly) originates from the flavor symmetry breaking effects. In order to naturally incorporate the $\cal O$(1) top quark Yukawa coupling, here, we employ SU(2)$_{\text F}$ flavor symmetry and treat the first two generations of matter fields as the doublet $\Psi^{\textbf{27}}_a$ whereas the third generation field $\Psi_3^{\textbf{27}}$ and all the Higgs fields are treated as singlets under SU(2)$_{\text F}$. We also introduce flavon fields $F_a$ and $\bar F^a$ that are singlets under E$_{\text 6}$ and a doublet and an anti-doublet under SU(2)$_{\text F}$, respectively. Then, as shown later, $F_a$ and $\bar F^a$ acquire VEVs and break the the SU(2)$_{\text F}$. These effects generate hierarchical structures of the Yukawa coupling of Eq..
Suppose that the soft SUSY breaking terms are mediated to the visible sector above the scale where E$_{\text 6}$ and SU(2)$_{\text F}$ symmetries are respected, such as in gravity mediation. Then, $\Psi^{\textbf{27}}_a$ acquires a soft mass ($m_0$) that is different from the mass of $\Psi^{\textbf{27}}_3$ ($m_3$), in general, in an E$_{\text 6}\times$SU(2)$_{\text F}$ symmetric way. As a result, this guarantees whole sfermion soft mass degeneracy at the GUT scale, except for $\textbf{10}_3$, as shown in Eq., because ${\bold{ 10 }_1}$, ${\bold{ 10 }_2}$, and all the MSSM ${\bar {\bold{ 5 }}}$ fields originate from an identical field $\Psi^{\textbf{27}}_a$.[^13]
Anomalous U(1) gauge symmetry {#sec:u1a}
=============================
In the previous papers of Ref.[@Maekawa:2001uk], an anomalous U(1)$_{\text A}$ gauge symmetry [@Witten:1984dg; @Froggatt:1978nt] was introduced to solve the doublet-triplet mass splitting and to provide the origin of the hierarchical structures of Yukawa couplings. In this section, we summarize essential points and comment on what kinds of fields acquire VEVs, and how the magnitudes of these VEVs are determined in the framework of U(1)$_{\text A}$.
We introduce an anomalous U(1)$_{\text A}$ gauge symmetry whose anomalies are canceled by the Green-Schwarz mechanism [@Green:1984sg]. The theory possesses the Fayet-Iliopoulos term ($\xi^2$), and we assume its magnitude as $\xi = \lambda \Lambda$. (Here $\Lambda$ is the cutoff scale of the theory and we set $\Lambda=1$.) Let us denote the symmetries of the theory, except for U(1)$_{\text A}$, as $G_a $. Then it is shown in Ref.[@Maekawa:2001uk] that a system consisting of all the $G_a$ and U(1)$_{\text A}$ invariant terms[^14] has a supersymmetric vacuum where all the fields that are negatively charged under U(1)$_{\text A}$ get VEVs in the following way.[^15] $$\begin{aligned}
\left \{
\begin{array}{lc}
\langle Z_i^+ \rangle = 0 \nonumber & (z_i^+>0) \\
\langle Z^-_i \rangle \simeq \lambda^{-z^-_i} & (z_i^-<0)
\end{array}
\right. \quad \eqref{eq:u1a vev}
\label{eq:u1a vev}\end{aligned}$$ Here $Z_i$ denotes the $G_a$ singlet field, but the argument can be extended to the case where $Z_i $ are composite operators that are made by $G_a $ nonsinglet fields. For example, in case the where $Z^-= \bar X X $ (here $X$ denotes the $G_a $ nonsinglet field and $\bar X$ is the antirepresentation of $X$), $\langle \bar X X \rangle = \lambda^{-(x + \bar x)} $ leads to $\langle X \rangle = \langle \bar X \rangle = \lambda^{-(x + \bar x)/2} $, once the D-flatness condition of $G_a $ is taken into account.
A U(1)$_{\text A}$ symmetry and its specific SUSY vacuum are applied in several aspects of phenomenological model building. For example, an appropriate U(1)$_{\text A}$ charge assignment for the Higgs sector can ensure the DTS via the DW mechanism [@Maekawa:2001uk; @Dimopoulos:1981xm]. Here note that since a positively charged field does not acquire a VEV in this U(1)$_{\text A}$ vacuum, only the terms that are linear in a positively charged field are relevant to the determination of the symmetry breaking structure of the model.
It is also important to mention that a term whose total U(1)$_{\text A}$ charges are negative does not appear at the U(1)$_{\text A}$ breaking vacuum. The reason is that this type of term should originally be accompanied by at least one positively charged field but its VEV is always vanishing according to Eq. (SUSY-zero mechanism) [@Nir:1993mx; @Maekawa:2001uk]. Importantly, this property can be applied to solve the $\mu$ problem [@Maekawa:2001yh]. To be more precise, when the U(1)$_{\text A}$ charges of the Higgs fields are set to be negative, the supersymmetric $\mu$-term is forbidden by the SUSY-zero mechanism. However, this SUSY-zero mechanism will be broken by the amount of the SUSY breaking scale, once soft SUSY breaking effects are taken into account in the total Lagrangian. So the appropriate scale of the $\mu$-term can be induced. Employing these properties of U(1)$_{\text A}$, in the following sections we construct and examine a model of SCPV in the E$_{\text 6}$ SUSY GUT.
Spontaneous CP violation {#sec:spontaneous cp violation}
========================
In this section, we first discuss the mechanism of SCPV that is caused by the SU(2)$_{\text F}$ breaking sector. After specifying a source for the CP violating phase, we examine how this phase will appear in the parameters of the MSSM. Particularly, we refer to a solution for the $\mu$ problem in the context of U(1)$_{\text A}$ and indicate that it leads to an unwanted outcome, Arg$[\mu b^*] = \cal O$(1), once CP violating effects are taken into account. Next, we discuss discrete symmetry in order to solve this difficulty.
Spontaneous CP violation {#subsec:spontaneous CP violation}
------------------------
In this subsection, adopting a U(1)$_{\text A}$ symmetry and its characteristic vacuum, we discuss the mechanism of SCPV that is caused by the SU(2)$_{\text F} $ breaking sector.
Let us introduce the E$_{\text 6} \times $SU(2)$_{\text F}$ singlet field $S (s > 0)$. Then, in general, we obtain the following superpotential made of $F_a (f<-1) $, $\bar F^a ( \bar f <-1)$, and $S$: $$\begin{aligned}
W_S = \lambda^{s}S [ \sum_{n=0}^{n_f } c_{n} \lambda^{ (f+\bar f) n } (\bar F F)^{n} ]
\label{eq:general Ws}\end{aligned}$$ We assume that $\cal O$(1) coefficients $c_{n}$ are all real and that the theory is originally CP invariant.[^16] $n$ are natural numbers which are truncated at $n_f$, where $s+ (f+\bar f) (n_f + 1) $ becomes negative (SUSY-zero mechanism). The appearance of $\lambda $ in Eq. can be understood as originating from the VEV of an operator whose U(1)$_{\text A}$ charge is $-1$. From Eq., the $F$-flatness condition with respect to $S$ leads to $$\lambda ^s[c_0 + c_1 \lambda ^{(f+\bar f)} \langle \bar FF \rangle +\dots +c_{n_f} \lambda ^{n_f (f+\bar f)} \langle \bar FF \rangle ^{n_f}]=0 \, .
\label{F=0}$$ Thus, when $n_f \geq 2$, it generally forces $\langle \bar F F \rangle $ to acquire an imaginary phase. Therefore, at this stage the CP invariance is spontaneously broken. Using SU(2)$_{\text F}$ gauge symmetry and taking its $D$-flatness condition into account, hereafter we adopt the following basis where only $\langle F_2 \rangle $ acquires an imaginary phase: $$\begin{aligned}
\langle F_a \rangle
\sim
\left(
\begin{array}{c}
0 \\
e^{i\rho }\lambda ^{-\frac{(f+\bar f)}{2}}
\end{array}
\right),
\indent
\langle \bar F^a \rangle
\sim
\left(
\begin{array}{c}
0 \\
\lambda ^{-\frac{(f+\bar f)}{2}}
\end{array}
\right)
\label{<F_a>}\end{aligned}$$
Spontaneous CP violation and $\mu$-term generation {#subsec:spontaneous CP violation and mu-term generation}
--------------------------------------------------
In this subsection, we discuss the effect of Eq on the generation of the $\mu$- and $b$-terms. Here an appropriate $\mu$-term is induced by the SUSY breaking effect on the SUSY-zero mechanism.[^17]
Let us assume that the MSSM Higgs doublets are contained in $\bold{ 5 }$ and ${\bar {\bold{ 5 }}'}$ of $H^\bold{27}$. Then if the U(1)$_{\text A}$ charge of $H^\bold{27}$ is negative (h$<$0), there is no supersymmetric mass term for the MSSM Higgs doublets. This is because the positively charged field $S$ does not acquire VEVs in the SUSY limit (SUSY-zero mechanism). Here, a possible term is $\lambda^{s+3h} S H^{\bold{ 27 }}H^{\bold{ 27 }}H^{\bold{ 27 }} \rightarrow \lambda^{s+3h} \langle S \rangle \langle H^{\bold{ 1 }} \rangle H^{\bold{ 10 }}H^{\bold{ 10 }}$, but $\langle S \rangle $ is vanishing in the SUSY limit.
However, the situation changes once we take SUSY breaking effects into account in the total Lagrangian [@Hall:1983iz; @Maekawa:2001yh]. For a schematic explanation, we first discuss a simplified version of the system which consists of E$_{\text 6}\times$SU(2)$_{\text F}$ singlet fields $S$($s>0$) and $Z$($z<-1$). The superpotential is given as $$\begin{aligned}
W \supset \lambda^{s}S + \lambda^{s+z}SZ + \dots
\label{eq:superpotentials for s}\end{aligned}$$ and we also introduce the corresponding general soft SUSY breaking terms[^18] $$\begin{aligned}
V_{soft} = \textstyle \sum \limits_{X=S,Z} \, \tilde m_X^2 |X|^2 + A_{S} \lambda^{s}S + A_{Z} \lambda^{s+z}SZ + \dots .
\label{eq:soft-terms-sz}\end{aligned}$$ Here we assume that all the soft parameters $\tilde m_S$, $\tilde m_Z$, $A_S $ and $A_Z $ are the weak scale. As usual, the whole potential is composed of $V= \textstyle \sum \limits _{X=S,Z} \, |F_X|^2 + V_{soft} \, $,[^19] where $$\begin{aligned}
|F_S|^2 = |\frac{\delta W}{\delta S}|^2 = | \lambda^{s} + \lambda^{s+z} Z |^2, \quad
|F_Z|^2 =|\frac{\delta W}{\delta Z}|^2 = |\lambda^{s+z}S |^2.
\label{eq:fz}\end{aligned}$$ Here note that, in the SUSY limit, $\langle Z \rangle$ is determined so that $|F_S|$ of Eq. vanishes. Also, $\langle S \rangle =0$ satisfies the vanishing $|F_Z|$. However, once SUSY breaking effects are taken into account, there is no need for $F$-flatness conditions to be satisfied. Actually, the total potential $V$ which includes SUSY breaking effects makes $\langle S \rangle $ and $\langle F_S \rangle $ nonvanishing. First of all, the extremum condition of $V$ with respect to $S$ determines $\langle S \rangle$ as$$\begin{aligned}
\langle S \rangle \sim
\lambda^{-(s+2z)} (A_S+A_Z).
\label{eq:<S>}\end{aligned}$$ Here the shift of $\langle S \rangle $ from zero can be understood as the balance between the SUSY mass term and the tadpole term that is induced by the SUSY breaking effect.[^20]
![ SUSY breaking effect on the potential of $S$ []{data-label="SUSY breaking effect"}](SUSYB_effect.eps){width=".4\textwidth"}
Also, $F_S$ is fixed by the extremum condition of V with respect to $\Delta Z$, where $\Delta Z$ is defined as the deviation of $\langle Z \rangle$ from its SUSY VEV as $\langle Z \rangle =\lambda^{-z}+\Delta Z$. $$\begin{aligned}
\frac{\delta V}{\delta \Delta Z} &=& F_S^* \lambda^{s+z} + \tilde m_Z^2 \lambda^{-z} + A_Z \lambda^{s+z} S=0 \nonumber \\
&\rightarrow &
\langle F_S^* \rangle= \lambda^{-(s+2z)} \tilde m_Z^2 +A_Z \langle S \rangle
\label{eq:<F_s>}\end{aligned}$$ Combining Eqs. and , the $\mu$- and b-terms are induced as follows: $$\begin{aligned}
{\cal L} &\supset& \lambda^{s+3h} S H H \langle H \rangle |_{\theta ^2}+
\lambda^{s+3h} A_{SH}S H H \langle H\rangle \nonumber \\
&\rightarrow& \lambda^{s+ 2h + \frac{h - \bar h}{2}} ( \langle S\rangle HH |_{\theta ^2} + (\langle F_S \rangle+\langle S\rangle A_{SH}) HH )
\label{eq:myu1}\end{aligned}$$ Therefore, when $z$ is appropriately chosen, the $\mu$ and $b$ parameters appear as the weak scale.
Now, let us examine the effects of the CP violation of Eq. on the generation of the $\mu$ and $b$ parameters. So far we have assumed that all the soft SUSY breaking terms and $\cal O$(1) coefficients were real. Therefore, the $\mu$ and $b$ parameters are real. However, this result can be changed, in general, once flavon fields acquire complex VEVs. In order to see this, we extend Eq. to include Eq.. Also, we take the corresponding SUSY breaking terms $$\begin{aligned}
A \lambda^{s} S [ \sum_{n=0}^{n_f } c'_{n} \lambda^{ (f+\bar f) n } (\bar F F)^{n} ]
\label{eq:complex A}\end{aligned}$$ into account. Here note that each $\cal O$(1) coefficient $c_n'$ appearing in Eq. is different from that of Eq., in general.[^21] Therefore, $$\begin{aligned}
A \lambda^{s} S \{ c_0' + c_1' \lambda^{ f + \bar f} \langle \bar F F \rangle + \dots + \Sigma c_{n_f}' \lambda^{ (f + \bar f) n_f} \langle \bar F F \rangle ^{n_f} \}
\label{eq:non-vanishing A}\end{aligned}$$ is generically nonvanishing, even though the left-hand side of vanishes. Note that, from Eq., each higher dimensional operator in Eq. gives a contribution that is of the same order of the original tadpole term, $c'_0A \lambda^s S$, but the effective coefficient is complex. Therefore, Eq. turns into $A_S \lambda^s S$ of Eq., where $A_S$ is complex and Arg$[A_S] \sim \mathcal O(1)$. Then, we conclude from Eqs., and that the $\mu$ and $b$ parameters turn out to be complex parameters and Arg$[ \mu b^*] = \cal O$(1), in general.
SUSY CP problem and discrete symmetry {#subsec:susy cp problem and discrete symmetry}
-------------------------------------
As we saw in the previous subsection, CP violating VEVs of flavon fields generically lead to complex $\mu$ and $b$ parameters and Arg$[ \mu b^*] = \cal O$(1). This rephasing invariant complex phase induces (C)EDMs of quarks and leptons, and gives rise to the SUSY CP problem. Though the decoupling procedure can reduce the SUSY contribution to the (C)EDMs, we discuss another way to avoid this issue in order not to destabilize the weak scale.
As we discussed below Eq., the couplings among $S$ and flavon fields led to complex $\mu$ and $b$ parameters and Arg$[ \mu b^*] = \cal O$(1). Therefore, we introduce a discrete symmetry so that coupling among $S$ and $\bar F F$ is forbidden.[^22] Specifically, we exploit the discrete symmetry in order to discriminate $F$ from the others assigning a nontrivial discrete charge for $F$, since only $F$ acquires a complex VEV in the basis defined in Eq.. One may think that the introduction of an additional symmetry seems ad hoc, but we can find the utility of this discrete symmetry from several different point of view, as follows.
As we mentioned at the end of Sec.2.1, a naive assumption for the Yukawa hierarchy of Eq. leads to a relatively large up-quark Yukawa coupling. Therefore, there should be cancellation between $\cal O$(1) coefficients of Eq., in order to reproduce realistic up-quark Yukawa coupling ($Y_u \sim \lambda^8$) [@Fusaoka:1998vc; @Ross:2007az]. However, the situation can be improved if the following types of interactions are only responsible for the generation of $Y_H$: $$\begin{aligned}
\left(
\begin{array}{ccc}
0 & \lambda ^{2 \psi_a + a} \Psi^{a \bold{27}} A^\bold{78} \Psi_a^\bold{27} & 0 \\
\lambda ^{2 \psi_a + a} \Psi^{a \bold{27}} A^\bold{78} \Psi_a^\bold{27} & \lambda ^{2( \psi_a + \bar f)} \bar F^a \Psi_a^\bold{27} \bar F^b \Psi_b^\bold{27} & \lambda^{\psi_a + \psi_3 + \bar f} \bar F^a \Psi_a \Psi_3 \\
0 & \lambda^{\psi_a + \psi_3 + \bar f} \Psi_3 \bar F^a \Psi_a & \lambda^{2 \psi_3} \Psi_3 \Psi_3
\end{array}
\right) \lambda ^h H^\bold{27} \nonumber \\
\rightarrow
\left(
\begin{array}{ccc}
0 & Q_{B-L} \lambda^5 \Psi_1 \Psi_2 & 0\\
Q_{B-L} \lambda^5 \Psi_2 \Psi_1 & \lambda^4 \Psi_2 \Psi_2 & \lambda^2 \Psi_2 \Psi_3 \\
0 & \lambda^2 \Psi_3 \Psi_2 & \Psi_3 \Psi_3
\end{array}
\right) H \label{eq:yh110}\end{aligned}$$ Here $A^{\bold{78}}$ is assumed to acquire $B-L$ conserving VEVs [@Maekawa:2001uk] and $Q_{B-L}$ represents the $B-L$ charge of the corresponding component of $\Psi^{\bold{27}}$. Then, Eq. leads to $Y_u \sim Q_{B-L}^2 \lambda^6 $, which can be an order of magnitude smaller than the original expectation, $Y_u \sim \lambda^6 $.[^23] Simply, Eq. is ensured if there is a discrete symmetry which discriminates $F$ from the others and forbids $ F^a \Psi_a^\bold{27} F^b \Psi_b^\bold{27} H^\bold{27} $ and $ F^a \Psi_a^\bold{27} \bar F^b \Psi_b^\bold{27} H^\bold{27} $.[^24]
Also, another utility of discrete symmetry arise from the examination of the SUSY CP problem that is caused even if the $\mu$- and $b$-terms are real. In the case where the $\it{modified}$ sfermion universality of Eq. is adopted, the nonuniversality of the up-type squark sector ($m_{\bold{10}}^2$) induces quark a CEDM [@Hisano:2004tf; @Ishiduki:2009gr]. This CEDM exceeds experimental bound if $m_3$ is placed in the weak scale. Here, the CP violating phase is provided by the up-type quark Yukawa coupling, even if $\mu$, $b$, $M_3$, and $A$ parameters are real. Therefore, when we respect the weak scale stability, the up-type quark Yukawa coupling should also be real parameters other than $M_3$, $\mu$, and $b$.[^25] This requirement can also be ensured if there is a discrete symmetry which discriminates $F$ from the others and $F$ does not contribute to the generation of $Y_H$, since only $F$ acquires complex VEV.[^26]
In connection with this type of SUSY CP problem that is caused by the flavor off-diagonal complex soft term, the SU(2)$_\text{F}$ breaking effect can also induce similar flavor off-diagonal complex soft term. For example, a higher dimensional term in the K$\ddot {\text a}$hler potential, $\tilde m^2 (\Psi_3^\bold{27})^\dagger \epsilon^{ab} F_a \Psi_b^\bold{27}$, induces a complex $(m_{\bf 10}^2)_{13}$ component of the order of $e^{i \rho} \lambda^{3} \tilde m^2 $, and it contributes to the up-quark (C)EDM. However, this effect can also be forbidden if there is a discrete symmetry which discriminates $F$ from the others.
Following these observations, we introduce a discrete symmetry to our model so that all the above requirements can be fulfilled. When we adopt a cyclic symmetry, $Z_3$ is the smallest group that can allow $ \bar F^a \Psi_a^\bold{27} \bar F^b \Psi_b^\bold{27} H^\bold{27} $ but forbid $ F^a \Psi_a^\bold{27} F^b \Psi_b^\bold{27} H^\bold{27} $. Then, e.g., $F$ is assigned $Z_3=1$ and all the other fields are assigned $Z_3=0$, in order to discriminate $F$ from the others. Here, an exception is $C^\bold{27} $. As explained in the next section, the model can not be realistic without $F^a \Psi_a^\bold{27} \bar F^b \Psi_b^\bold{27} C^\bold{27} $ and $F^a \Psi_a^\bold{27} \Psi_3^\bold{27} C^\bold{27} $, since it supplies the KM phase through the down-type quark Yukawa coupling. Therefore, $C^\bold{27}$ should be assigned $Z_3=2$. Once a Higgs field is charged under this new $Z_3$ symmetry, some modifications are necessary in the Higgs sector that is originally composed to ensure the DTS [@Maekawa:2001uk]. In the following discussion, therefore, we extend $Z_3$ into $Z_6$ where $Z_6$ incorporates the $Z_2$ symmetry that was used to ensure that the adjoint Higgs field acquired B-L conserving VEV [@Barr:1997hq; @Maekawa:2001uk]. Here, a simple choice of $Z_6$ charges is $Z_6[F]=2$ and $Z_6[C]=4$. However, we set $Z_6[F]=1$ and $Z_6[C]=5$ in order to utilize the degrees of freedom of the enlarged discrete symmetry.
A model of SCPV in E$_{\text 6}$ SUSY GUT {#sec:a modle of scpv in e6 susy gut}
=========================================
In this section, we summarize a model of SCPV in the E$_6$ SUSY GUT with SU(2)$_{\text F}$, U(1)$_{\text A}$, and Z$_{\text 6}$ symmetries. After giving brief explanations for the field contents, we focus on the effects of the discrete symmetry on the Yukawa couplings of the model. We examine consequent Yukawa couplings and derive some predictions from them.
Field content {#subsec:field contens}
-------------
Here we summarize the field content of the model and its representations under the E$_{\text 6}$, SU(2)$_{\text F} $, U(1)$_{\text A}$ and $Z_6$ symmetries.
$\Psi_a^\bold{27}$ $\Psi_3^\bold{27}$ $F_a$ $\bar F^a$ $H^\bold{27}$ $\bar H^{\overline {\bold{27}}} $ $C^{\bold{27}}$ $\bar C^{\overline {\bold{27}}}$ $C'^{\bold{27}}$ $\bar C'^{\overline {\bold{27}}}$ $A^{\bold{78}}$ $A'^{\bold{78}}$
----------- -------------------- -------------------- ---------------- ------------------ --------------- ----------------------------------- ----------------- ---------------------------------- ------------------ ----------------------------------- ----------------- ------------------ -- -- --
$E_6$ $\bold{27}$ $\bold{27}$ $\bold{1}$ $\bold{1}$ $\bold{27}$ ${\overline {\bold{27}}}$ $\bold{27}$ ${\overline {\bold{27}}}$ $\bold{27}$ ${\overline {\bold{27}}}$ $\bold{78}$ $\bold{78}$
$SU(2)_F$ $\bold{2}$ $\bold{1}$ $\bold{2}$ $\bar{\bold{2}}$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$
$U(1)_A$ 4 $\frac{3}{2}$ -$\frac{3}{2}$ -$\frac{5}{2}$ -3 2 -4 0 7 9 -1 4
$Z_6$ 0 0 1 0 0 0 5 0 3 3 3 3
: Field contents and charge assignment under E$_{\text 6}\times$SU(2)$_{\text F} \times$U(1)$_{\text A}\times$Z$_{\text 6}$[]{data-label="tb:z6complete"}
$Z_0$ $Z_3$ $Z_4$ $S$
----------- ------------ ------------ ------------ ------------
$E_6$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$
$SU(2)_F$ $\bold{1}$ $\bold{1}$ $\bold{1}$ $\bold{1}$
$U(1)_A$ -1 -2 -5 9
$Z_6$ 0 3 4 0
: Field contents and charge assignment under E$_{\text 6}\times$SU(2)$_{\text F} \times$U(1)$_{\text A}\times$Z$_{\text 6}$[]{data-label="tb:z6complete"}
We introduce the following fields, which are listed in Table \[tb:z6complete\]. First of all, $\Psi^{ {\bold{27}}}$ is a matter field, and we make its first two generations as a doublet ($\Psi_a^{ {\bold{27}}}$) and the third generation as a singlet ($\Psi_3^{{\bold{27}}}$) of the SU(2)$_{\text F}$, respectively. The SU(2)$_{\text F}$ and CP are simultaneously broken by the VEVs of flavon fields $F_a$ and $\bar F^a$ as in Eq.. All the other fields are singlets under SU(2)$_{\text F}$. $H^{{\bold{27}}}$ is the field whose VEV breaks E$_{\text 6}$ into SO(10), and $\bar H^{\overline {\bold{27}}}$ is introduced to maintain the D-flatness condition. $C^{ {\bold{27}}}$ is the field whose VEV breaks SO(10) into SU(5), and $\bar C^{\overline {\bold{27}}}$ is introduced to maintain the corresponding D-flatness condition. $A^\bold{78} $ and ${A'}^\bold{78} $ are the adjoint fields. Here, the F-flatness conditions with respect to ${A'}^\bold{78} $ make $A^\bold{78} $ acquire DW-type VEVs for the SO(10) adjoint component to solve the DTS problem [@Dimopoulos:1981xm]. $C'^{ {\bold{27}}}$, $\bar C'^{\overline {\bold{27}}}$, and $Z$ are introduced to give masses for the NG modes and to maintain DW-type VEVs [@Barr:1997hq].
In Table \[tb:z6complete\], U(1)$_\text{A}$ charges are assigned so that the DTS and appropriate Yukawa hierarchies are ensured. Also, $Z_6$ charges are determined so that the requirement discussed in Sec.\[subsec:susy cp problem and discrete symmetry\] can be fulfilled. A more detailed discussion for the symmetry breaking in this model can be found in [@Maekawa:2001uk]. As a result of the SUSY-zero mechanism and $Z_6$ discrete symmetry, possible forms of Yukawa couplings are characteristically restricted. In the following subsections, we examine consequent Yukawa couplings and derive some predictions of the model.
Yukawa couplings {#subsec:yukawa couplings}
----------------
In this subsection, we examine Yukawa couplings of the model restoring $ \mathcal O(1)$ coefficients.
Under the charge assignment of Table \[tb:z6complete\], the following interactions between matter and Higgs fields are allowed. $$\begin{aligned}
Y_H &:&
\left(
\begin{array}{ccc}
0 & d \Psi^a (A,Z_3) \Psi_a & 0 \\
d \Psi^a (A,Z_3) \Psi_a & c \lambda^{2 (\psi_a + \bar f)} \bar F^a \Psi_a \bar F^b \Psi_b & b \lambda^{\psi_a + \psi_3 + \bar f} \bar F^a \Psi_a \Psi_3 \\
0 & b \lambda^{\psi_a + \psi_3 + \bar f} \Psi_3 \bar F^a \Psi_a & a \lambda^{2 \psi_3 } \Psi_3 \Psi_3
\end{array}
\right) \lambda^h H
\label{eq:yhz6}\end{aligned}$$ $$\begin{aligned}
Y_C &:&
\left(
\begin{array}{ccc}
0 & f \lambda^{2 \psi_a + f+ \bar f} F^a \Psi_a \bar F^b \Psi_b & g \lambda^{\psi_a + \psi_3 + f} F^a \Psi_a \Psi_3 \\
f \lambda^{2 \psi_a + f+ \bar f} \bar F^a \Psi_a F^b \Psi_b & 0 & 0 \\
g \lambda^{\psi_a + \psi_3 + f } \Psi_3 F^a \Psi_a & 0 & 0
\end{array}
\right) \lambda^c C \nonumber \\
\label{eq:ycz6}\end{aligned}$$ These interactions are responsible for the generation of $Y_H$ and $Y_C$. Here we explicitly write in $\cal O$(1) coefficients, $a,b,c,d,g$ and $f$, for later discussion. In Eq.$, \Psi^a (A,Z_3) \Psi_a H$ includes $\lambda^{2 \psi_a + a+z_3+h } \Psi^a Z_3 A \Psi_aH$, $\lambda^{2 (\psi_a +a) +h } \Psi^a A^2 \Psi_a H $, etc. Since $A$ acquires $B$-$L$ conserving VEVs ($\langle A \rangle \sim Q_{B-L} \lambda$), the effects of $d \Psi^a \langle (A,Z_3) \rangle \Psi_a H$ differ for the different components of $\Psi_a^\bold{27}$, as we reparametrize below. Note that all the $\cal O$(1) coefficients are assumed to be real numbers because of the original CP symmetry.
When the Higgs fields and flavon fields acquire VEVs, Eqs. and induce the following mass matrix for ${\bold 5}_{i}$, $\bar {\bold 5}_{i}'$ and $\bar {\bold 5}_{i}$: $$\begin{aligned}
\bordermatrix{
& \bar {\bold{ 5 }}^\prime_1 &\bar {\bold{ 5 }}^\prime_2 & \bar {\bold{ 5 }}^\prime_3 & {\bar {\bold{ 5 }}}_1 &{\bar {\bold{ 5 }}}_2 &{\bar {\bold{ 5 }}}_3\cr
\bold{ 5 }_1& 0 & \alpha d_5 \lambda^5 & 0 & 0 & f e^{i\delta} \lambda^{5.5} & g e^{i\delta} \lambda^{3.5} \cr
\bold{ 5 }_2& -\alpha d_5 \lambda^5 & c \lambda^4 & b \lambda^2 & f e^{i\delta} \lambda^{5.5} & 0 & 0 \cr
\bold{ 5 }_3& 0 & b \lambda^2 & a & g e^{i\delta} \lambda^{3.5} & 0 & 0 \cr}
\langle H \rangle.
\label{eq:z655bp5b}\end{aligned}$$ Here each power of $\lambda$ is determined by the corresponding U(1)$_{\text A}$ charge. It is important to note that $\alpha=1$ for the colored Higgs components ($H^C, \bar H^{\bar C}$) of $\bold{ 5 }$ and $ \bar {\bold{ 5 }}^\prime_{}$, and $\alpha=0$ for the doublet Higgs components ($H_u, H_d$) of $\bold{ 5 }$ and $ \bar {\bold{ 5 }}^\prime_{}$, since the (1,2) and (2,1) elements of Eq. originate from the B-L conserving VEV of $A$.[^27] Then, we take the following conventions $d \Psi \langle (Z, A) \rangle \Psi H \supset $ $d_5 \lambda^5 \epsilon ^{a b} H^C[\Psi_a] \bar H^{\bar C}[\Psi_b] {\bold{1}'}[H]$, $\frac{-1}{2} d_q \lambda^5 \epsilon^{ab} Q[\Psi_a] q_R^C[\Psi_b] H_u[H]$(or $H_d[H]$), $\frac{-3}{2} d_l \lambda^5 \epsilon^{ab} e_R^C[\Psi_a] L[\Psi_b] H_d[H]$, where $\epsilon^{12}=-\epsilon^{21}=1$, $\epsilon^{11}=\epsilon^{22}=0$ and $d_5, d_q, d_l$ are the $ \mathcal O(1)$ coefficients that are different from each other in general. In Eq., since $\alpha =0$ for $H_u$ and $ H_d$ components of $\bold{ 5 }$ and $ \bar {\bold{ 5 }}^\prime_{}$, the $H_d[\bar {\bold 5}'_1$\] component becomes a purely massless mode. In other words, $L[\bar {\bold 5}_3$\] does not contain a massless mode whose main mode is $H_d[\bar {\bold 5}'_1$\]. This fact requires the down-type Higgs doublet to be composed of not only $H_d[H^{ \bar {\bold{5}}^\prime }$\] but also $ L $\[$C^{{ \bar {\bold{ 5}}}}$\]; otherwise the determinant of $Y_E$ vanishes, as we will see later.
Now we derive the Yukawa couplings $Y_U$, $Y_D$, and $Y_E$ in a semianalytical way, assuming that each $\cal O$(1) coefficient does not largely alter the hierarchical structures that originate in Eqs., and . First of all, $Y_U$ is simply derived from Eq. by extracting the corresponding components as[^28]$$\begin{aligned}
Y_U =
\bordermatrix
{
& {U_{R}^C}_1 & {U_{R}^C}_2 & {U_{R}^C}_3 \cr
Q_{1} & 0 & -\frac{1}{2} d_q \lambda^{5} & 0 \cr
Q_{2} & \frac{1}{2} d_q \lambda^5 & c \lambda^{4} & b \lambda^{2} \cr
Q_{3} & 0 & b \lambda^{2} & a \cr
}.
\label{eq:z6Yu}\end{aligned}$$ For $Y_D$ and $Y_E$, we first derive the relation between the gauge eigen modes and the mass eigen modes of ${\bar{\bold{5}}}$ from Eq.. Then we replace $\bar {\bold 5}_{i}$ and $\bar {\bold 5}_{i}'$ of $\Psi_a$ and $\Psi_3$, whic appeared in Eqs. and , in terms of massless modes. In the same step, we also replace $H_d[\bar {\bold 5}_{H}'$\] and $L[\bar {\bold 5}_{C}$\] by $H_D$ and $\beta_H \lambda^{0.5} H_D$, respectively. Here $\beta_H$ denotes another $\cal O$(1) coefficient which enters in this step. When we leave only the leading order contributions, this procedure leads to the following Yukawa couplings: $$\begin{aligned}
&Y_D = \nonumber \\
&\bordermatrix
{
& {D_{R}^C}_1 & {D_{R}^C}_2 & {D_{R}^C}_3 \cr
Q_{1} & -\{(\frac{bg-af}{g})^2 \frac{1}{ac-b^2} +1 \}\frac{g}{a} g \beta_H e^{2 i\delta} \lambda^{6} & -\frac{bg-af}{g} \frac{d_5}{ac-b^2} g \beta_H e^{i\delta} \lambda^{5.5}& -\frac{1}{2} d_q \lambda^5 \cr
Q_{2} & (\frac{d_q}{2}- \frac{d_5}{ac-b^2} \frac{bg-af}{g} b) \lambda^5 & (-\frac{ad_5^2}{ac-b^2} \frac{b}{g} e^{-i\delta} + \beta_H f e^{i\delta}) \lambda^{4.5} & (\frac{ac-b^2}{a} + \frac{bg-af}{g} \frac{b}{a}) \lambda^{4} \cr
Q_{3} & -\frac{ad_5}{ac-b^2} \frac{bg-af}{g} \lambda^3 & (-\frac{ad_5^2}{ac-b^2} \frac{a}{g} e^{-i\delta} + \beta_H g e^{i\delta}) \lambda^{2.5} & \frac{bg-af}{g} \lambda^{2} \cr
},
\nonumber \\
\label{eq:z6Yd}\end{aligned}$$ $$\begin{aligned}
Y_E^T =
\bordermatrix
{
& L_1 & L_2 & L_3 \cr
{E_R^C}_{1} & -\{(\frac{bg-af}{g})^2 \frac{1}{ac-b^2} +1 \}\frac{g}{a} g \beta_H e^{2i\delta} \lambda^{6} & 0 & -\frac{3}{2} d_l \lambda^5 \cr
{E_R^C}_{2} & \frac{3}{2} d_l \lambda^5 & \beta_H f e^{i\delta} \lambda^{4.5} & (\frac{ac-b^2}{a} + \frac{bg-af}{g} \frac{b}{a}) \lambda^{4} \cr
{E_R^C}_{3} & 0 & \beta_H g e^{i\delta} \lambda^{2.5} & \frac{bg-af}{g} \lambda^{2} \cr
}.
\nonumber \\
\label{eq:z6Ye}\end{aligned}$$
As a result of discrete symmetry, Eq. leads to $Y_u \sim (\frac{d_q}{2})^2 \lambda^6$, where $d_q $ is proportional to a $ Q_B$ charge.[^29] Also, $Y_U$ turns out to be a real parameters, contrary to $Y_D$ which possesses nonremovable phases that are converted to the KM phase. In Eq.(\[eq:z6Ye\]), one can see that the determinant vanishes when $\beta_H$ is set to zero, i.e., $H_D \sim H_d[H^{ \bar {\bold{5}}^\prime }$\]. Also note that nonremovable phase in Eq. vanishes when $\beta_H$ is set to zero. This is the reason why we assigned U(1)$_\text{A}$ and $Z_6$ charges so that the down-type Higgs doublet is composed as $H_D \sim H_d[H^{ \bar {\bold{5}}^\prime }$\]+$\lambda^{0.5} L $\[$C^{{ \bar {\bold{ 5}}}}$\].
Model analysis {#subsec:model analysis}
--------------
The resultant Yukawa couplings of Eqs., , have rather restricted forms and small numbers of $ \mathcal O(1)$ coefficients. In this subsection, we examine these Yukawa couplings and derive some predictions. Also, we discuss the compatibility with observables.
First of all, in a semianalytical derivation, where we collect leading contributions in each element, Eqs. and lead to the following Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. $$\begin{aligned}
V_{CKM}^\text{leading} =
\begin{pmatrix}
1 & \frac{ a (bg-af)^2 }{2(ac-b^2) } \frac{ (d_q + 2 d_5) \beta_H }{ \{ (bg-af)^2+(ac-b^2)g^2 \} \beta_H -a^2 d_5^2 e^{2 i \delta} } \lambda & 0 \\
- V_{12}^* & 1 & \frac{g}{a} \frac{ ac-b^2 }{ bg-af } \lambda^2 \\
V_{12}^* V_{23} & - V_{23} & 1
\end{pmatrix}
\label{eq:e6 ckm}\end{aligned}$$ Here, $V_{ub}$ vanishes at the order of $ \mathcal O(\lambda^3)$ and appers at the next order in $\lambda$ in this model. It is interesting to note that the current global fit for the CKM matrix elements at low-energy [@Amsler:2008zz] or the high-energy extrapolation [@Ross:2007az] indeed suggest this type of hierarchical structure.
Also, in the leading order analysis, we found that Eqs. and lead to the following relation, $$\begin{aligned}
{\left| V_{cb} Y_b \right|}= {\left| Y_c \right|}
\label{eq: vcb yb = yc}\end{aligned}$$ where $Y_b$ and $Y_c$ denote bottom and charm quark Yukawa couplings. We can guess the origin of this relation as follows. The (23) element of $V_{CKM}$ is given as $V_{23} \sim \frac{Y_{D23}}{Y_{D33}} - \frac{Y_{U23}}{Y_{U33}}$, but now, $Y_{D23} \sim Y_{U22}$ and $Y_{D33} \sim Y_{U32}$ because ${\bar{\bold{5}}}_2$ turns into ${\bar{\bold{5}}}_3$, after ${\bar{\bold{5}}}$ mixing. Then $V_{23} Y_b \sim Y_{D23} - \frac{ Y_{U23} }{ Y_{U33} } Y_{D33} \sim \frac{Y_{U22} Y_{U33} - Y_{U23} Y_{U32} }{ Y_{U33} }$ and this is roughly $Y_c$. For comparison with the observables, we quote the results of Ref.[@Ross:2007az], $$\begin{aligned}
| V_{cb} Y_b | \sim \frac{10}{6} | Y_c |
\label{eq:Ross}\end{aligned}$$ where low-energy inputs of $V_{cb}$, $Y_b$ and $Y_c$ are extrapolated to the GUT scale. Since Eq. is derived in the case of $\tan\beta$=10, Eq. fixes $\tan\beta\sim 6$ in the current model[^30]
Finally, as an example, we numerically calculate the eigenvalues of quark and lepton Yukawa couplings, a mixing matrix, and a Jarlskog invariant ($\text{J}_\text{CP}$), substituting a set of trial values for the $ \mathcal O(1)$ coefficients that appeared in Sec.5.2. We set $a=0.6,\, b=-0.5,\, c=-.7,\, d_5 =-0.9, \,d_q=0.4,\, d_l=-0.5,\, f=1.5,\, g=-0.9,\, \beta_H= 0.9,\, \delta=1.4$, and repeat the processes described in Sec.5.2. We obtain the following eigenvalues of the quark and lepton Yukawa couplings, a mixing matrix, and a Jarlskog invariant. $$\begin{aligned}
\begin{array}{c}
Y_t = 6(5) \times 10^{\text{-1}} \\
Y_c = 3(1) \times 10^{\text{-3}} \\
Y_u = 4(3) \times 10^{\text{-6}}
\end{array}
\quad
\begin{array}{c}
Y_b = 2(3) \times 10^{\text{-2}} \\
Y_s = 5(6) \times 10^{\text{-4}} \\
Y_d = 8(3) \times 10^{\text{-5}}
\end{array}
\quad
\begin{array}{ccc}
Y_\tau = 3(4) \times 10^{\text{-2}} \\
Y_\mu = 1(3) \times 10^{\text{-3}} \\
Y_e = 3(1) \times 10^{\text{-5}}
\end{array}
\label{eq:z6massspectra}\end{aligned}$$ $$\begin{aligned}
|V_{CKM}| = \left(
\begin{array}{ccc}
1 & 2(2) \times 10^{-1} & 2(4) \times 10^{\text{-3}} \\
2(2) \times 10^{-1} & 1 & 10(4) \times 10^{\text{-2}} \\
20(7) \times 10^{-3} & 10(4) \times 10^{-2} & 1
\end{array}
\right)
\label{eq:z6ckm}\end{aligned}$$ $$\begin{aligned}
\text{J}_\text{CP}= 1(3) \times 10^{\text{-5}}
\label{eq:z6jinv}\end{aligned}$$ Here the parenthetic digits are the corresponding values of Ref.[@Ross:2007az] that are extrapolated to the GUT scale from the low-energy inputs using the MSSM two-loop RGEs[^31]. From Eqs.-, we see that the current model has the capability to reproduce the quark and lepton mass spectra, mixing angles, and a Jarlskog invariant.
Summary and discussion {#sec:summary and discussion}
======================
In this paper we discussed a model of spontaneous CP violation in the E$_{\text 6}$ SUSY GUT with SU(2)$_{\text F}$ flavor and anomalous U(1)$_{\text A}$ symmetries.
We made a model where CP symmetry is spontaneously broken in the flavor sector in order to provide the origin of the KM phase and to evade the SUSY CP problem. However, as we saw in Sec.\[subsec:spontaneous CP violation and mu-term generation\], a naive construction of the model generally leads to an unwanted outcome, Arg$[\mu b^*]=\cal O$(1), when the CP violating effect is taken into account. Then in Sec.\[subsec:susy cp problem and discrete symmetry\], we introduced a discrete symmetry in order to cure this difficulty. Interestingly, this discrete symmetry plays additional roles. It ensures that realistic up-quark mass and Cabibbo angle are simultaneously realized without cancellation between $ \mathcal O(1)$ coefficients. Also, severe constraints from the chromo-electric dipole moment of quark can be satisfied without destabilizing the weak scale.
The discrete symmetry reduces the number of free parameters, but the model is capable of reproducing quark and lepton mass spectra, mixing angles, and a Jarlskog invariant, as we see in Eqs.-. We also obtain the characteristic predictions $V_{ub} \sim \cal O$($\lambda^4$) and $| V_{cb} Y_b | = |Y_c|$ at the GUT scale. Note that, for the trial set of $\cal O$(1) coefficients that is used to calculate Eqs.-, Arg\[Det\[$Y_U Y_D $\]\] is $\mathcal O(1)$ and it leads to the strong CP phase. However, this phase can be set to zero by the VEV of the axion that originates from the U(1)$_{\text A}$ breaking.
A more precise fitting of the $ \mathcal O(1)$ coefficients is beyond the scope of the current study, since it requires detailed study of RGEs and threshold corrections. Note that, in the model, there are many superheavy fields which affect the RGEs. In particular, among the Higgs fields listed in Table \[tb:z6complete\], positively charged fields $C'$, $\bar C'$, and $A'$ provide relatively light fields and these fields start to influence the flow of the gauge couplings from the intermediate scales, as depicted in Fig.1 of the third paper of Ref.[@Bando:2001bj]. Then, this also results in a change of the flows of the Yukawa couplings.
In Sec.\[sec:a modle of scpv in e6 susy gut\], we discussed the Yukawa couplings of quarks and charged leptons. As for the neutrino sector, the left-handed neutrino masses are induced via the see-saw mechanism [@seesaw]. Here, the right-handed neutrino mass terms are obtained by the following interactions, $\Psi_{i}^{\bold {27}} \Psi_{j}^{\bold {27}} \bar H^{\overline {\bold {27}}} \bar H^{\overline {\bold {27}}}$, $\Psi_{i}^{{\bold {27}}} \Psi_{j}^{ {\bold {27}}} \bar H^{\overline {\bold {27}}} \bar C^{\overline {\bold {27}}}$, $\Psi_{i}^{ {\bold {27}}} \Psi_{j}^{ {\bold {27}}} \bar C^{\overline {\bold {27}}} \bar C^{\overline {\bold {27}}}$, adding appropriate fields like $F_a, \bar F^a, C^{ {\bold {27}}}$ and $\bar C^{\overline {\bold {27}}}$ to make these terms singlets under the symmetries listed in Table \[tb:z6complete\]. Then, when we follow the procedure of Ref.[@Bando:2001bj], the solar and atmospheric neutrino mass squared differences are roughly derived as $\Delta m^2_{\odot }=\cal O$($10^{\text{-6}}) \text{eV}^2$ and $\Delta m^2_{\text{atm}}=\cal O$($10^{\text{-3}}) \text{eV}^2$, respectively. Note that actual values can change by an order of magnitude, since several new $ \mathcal O(1)$ coefficients appear in the neutrino sector. Therefore, $\Delta m^2_{\odot }$ and $\Delta m^2_{\text{atm}}$ can be consistent with the current experimental constraints [@Amsler:2008zz]. This is the same for the neutrino mixing angles. In the current model, the Maki-Nakagawa-Sakata (MNS) matrix elements [@Maki:1962mu] are roughly given as $$\begin{aligned}
|V_{MNS}| \sim \left(
\begin{array}{ccc}
\lambda^{0.5} & 1 & \lambda \\
1 & \lambda^{0.5} & \lambda^{0.5} \\
\lambda^{0.5} & \lambda & 1
\end{array}
\right),
\label{eq:z6ckm}\end{aligned}$$ and these values may vary by an order of magnitude when we replace $ \mathcal O(1)$ coefficients with real numbers. Also, for example, when we change the U(1)$_\text{A}$ charges as $\frac{1}{2} \{c-\bar c -(h-\bar h) \}=-0.5$, $|(V_{MNS})_{13}|$ changes to a more realistic value of $ \mathcal O(\lambda^{1.5})$. Finally, it is interesting to note that, in the model described in Sec.\[sec:a modle of scpv in e6 susy gut\], one of the left-handed neutrino becomes massless.
Acknowledgement {#acknowledgement .unnumbered}
===============
S.K is supported in part by Grants-in-Aid for JSPS fellows. N.M is supported in part by Grants-in-Aid for Scientific Research from the MEXT of Japan. K.S is supported by the U.K. Science and Technology Facilities Council. This work was partially supported by the Grant-in-Aid for Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos," from the MEXT of Japan.
Higgs sector {#sec:higgs sector}
============
In this section, we briefly outline the Higgs sector of the model that is made to leave $H_u[\bold{10}_H]$ and $H_d[\bold{10}_H] + \lambda^{0.5} L[\bold{16}_C]$ massless. These components are identified as the MSSM Higgs doublets $H_U$ and $H_D $, respectively. As stressed in Sec.5.2, $H_D \sim H_d[\bold{10}_H] + \lambda^{0.5} L[\bold{16}_C]$ is crucial for the nonvanishing KM phase and electron mass.
In order to distinguish the pair of low-energy massless modes that originate from the Higgs sector, here we compose the operator matrix $O$ which induces mass terms among the SU(2)$_\text{L}$ doublet components of $H,\bar H,C,\bar C,C',\bar C',A$ and $A'$[^32]. $O$ consists of 16 rows and 16 columns. In the following, we take a base where indices are in the order $\bold{10}_{H}$, $\bold{10}_{C} $, $\bold{16}_{C}$, ${\bold{16}}_{A}$, ${\bold{10}}_{C'}$, ${\bold{10}}_{\bar C'}$, ${\bold{16}}_{C'}$, ${\bold{16}}_{A'}$, ${\bold{10}}_{\bar H}$, ${\bold{10}}_{\bar C}$, ${\bold{16}}_{H}$ for rows and $\bold{10}_{H}, \bold{10}_{C} $, $\overline{\bold{16}}_{\bar C}$, $\overline{\bold{16}}_{A}$, ${\bold{10}}_{C'}$, ${\bold{10}}_{\bar C'}$, $\overline{\bold{16}}_{\bar C'}$, $\overline{\bold{16}}_{A'}$, ${\bold{10}}_{\bar H}$, ${\bold{10}}_{\bar C}$, $\overline{\bold{16}}_{\bar H}$ for columns, respectively. Here we divide $O$ into the following 9 blocks: $$\begin{aligned}
O =
\bordermatrix{
& \text{\small{$\bold{10}_{H}$\dots}} & \dots & \text{\small{\dots $\overline{\bold{16}}_{\bar H}$}}\cr
\text{\quad \rotatebox[origin=c]{-90}{\small{\rotatebox[origin=c]{90}{$\bold{10}_{H}$} $\dots$}}} & \bold{0}_{4 \times 4} & A_{4 \times 4} & \bold{0}_{4 \times 3} \cr
\quad \,\,\, \text{\rotatebox[origin=c]{-90}{\small{\quad $\dots$ \quad}}} & B_{4 \times 4} & C_{4 \times 4} & D_{4 \times 3} \cr
\text{\quad \rotatebox[origin=c]{-90}{\small{$\dots$ \rotatebox[origin=c]{90}{$\bold{16}_{H}$}}}} & E_{3 \times 4} & F_{3 \times 4} & G_{3 \times 3} \cr
}
\label{eq:operator matrix}\end{aligned}$$ where subscripts indicate the number of rows and columns of the partial matrices. Here the $\bold{0}$’s have no entries and $E$ has a few entries since the corresponding fields are mostly negatively charged under U(1)$_\text{A}$ (SUSY-zero mechanism). On the contrary, the elements of $C, D, F$ and $G$ are almost filled since the corresponding fields are mostly positively charged. Therefore, in the following, we concentrate on $A$ and $ B$ which determine the composition of the massless modes. Here $B$ is responsible for the composition of $H_U$ and $A$ is responsible for the composition of $H_D$.
Under the charge assignment given in Table.\[tb:z6complete\], $A$ and $B$ are filled with the following terms. $$\begin{aligned}
A =
\bordermatrix{
&\bold{10}_{C'}&\bold{10}_{\bar C'}&\overline{\bold{16}}_{\bar C'} & \overline{\bold{16}}_{A'} \cr
\bold{10}_H & H^2 A C' & 0 & HH \bar C'\bar C & 0 \cr
\bold{10}_C & 0 & \bar C'(AF\bar F +Z_4)C & 0 & 0 \cr
\bold{16}_C & 0 & 0 & \bar C'(AF\bar F+Z_4)C & 0 \cr
\bold{16}_A & 0 & \bar C'AC & \bar C'AH &A'A \cr
} \label{eq:aigen}\end{aligned}$$ $$\begin{aligned}
B =
\bordermatrix{
&\bold{10}_{H}&\bold{10}_{C}&\overline{\bold{16}}_{\bar C} & \overline{\bold{16}}_{A} \cr
\bold{10}_{C'} & H^2 A C'& 0 &C'H (A+Z_3)\bar C\bar C& \bar CAC'Z_3 \cr
\bold{10}_{\bar C'} &0& \bar C'(AF\bar F+Z_4)C& \bar C'(A+Z_3) \bar C^2 & \bar C'A\bar C \bar H Z_3 \cr
\bold{16}_{C'} & 0 & 0 & 0 & \bar H A C'\cr
\bold{16}_{A'} & 0 & 0 & 0 & A'A \cr
}
\label{eq:bigen}\end{aligned}$$ Then, after symmetry breaking, Eqs. and induce the following mass terms among SU(2)$_\text{L}$ components: $$\begin{aligned}
A =
\bordermatrix{
&\bold{10}_{C'}&\bold{10}_{\bar C'}&\overline{\bold{16}}_{\bar C'} & \overline{\bold{16}}_{A'} \cr
\bold{10}_H & 0 & 0 & \lambda^{5.5} & 0 \cr
\bold{10}_C & 0 & \lambda^{5.5} & 0 & 0 \cr
\bold{16}_C & 0 & 0 & \lambda^{5} & 0 \cr
\bold{16}_A & 0 & \lambda^{6.5} & \lambda^{5.5} & \lambda^{3} \cr
}
\label{eq:al-hhbar-1r1/2}\end{aligned}$$ $$\begin{aligned}
B_L =
\bordermatrix{
&\bold{10}_{H}&\bold{10}_{C}&\overline{\bold{16}}_{\bar C} & \overline{\bold{16}}_{A} \cr
\bold{10}_{C'} & 0 & 0 & \lambda^{8.5} & \lambda^9 \cr
\bold{10}_{\bar C'} &0& \lambda^5& \lambda^{13} & \lambda^{12.5} \cr
\bold{16}_{C'} & 0 & 0 & 0 & \lambda^{8.5} \cr
\bold{16}_{A'} & 0 & 0 & 0 & \lambda^3 \cr
}
\label{eq:bl-hhbar-1r1/2}\end{aligned}$$ Note that each (1,1) entry vanishes since $A$ acquires $B-L$ conserving VEV. Therefore, Eq. and appropriately realize $H_D \sim H_d[\bold{10}_H] + \lambda^{0.5} L[\bold{16}_C]$ and $H_U \sim H_u[\bold{10}_H]$.
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[^1]: kimsg@indiana.edu
[^2]: maekawa@eken.phys.nagoya-u.ac.jp
[^3]: sakurai@hep.phy.cam.ac.uk
[^4]: An anomalous U(1)$_\text{A}$ symmetry which we discuss in Sec. 3 also plays a role to generate hierarchical structures of Yukawa couplings.
[^5]: Here we assume that the $D$-term contributions are small. A possible way to justify this assumption is by adopting non-Abelian discrete symmetry instead of the gauged symmetry which we use in this paper.
[^6]: In order to satisfy the lightest CP-even Higgs boson mass bound set by LEPI$\!$I, one may employ $\it{maximal\, mixing}$ of the stop sector [@Carena:1999xa]. Another interesting option is $\it{light\, Higgs\, scenario}$ or $\it{inverted\, hierarchy}$ where one assumes that the heaviest CP-even Higgs boson behaves as the SM Higgs boson [@Kane:2004tk].
[^7]: However, the muon $g-2$ anomaly is not explained if $m_0 \gtrsim $1TeV, since SUSY contributions decouple [@Hagiwara:2006jt].
[^8]: If $m_3$ is placed in the weak scale in view of the naturalness, there are nondecoupling SUSY contributions to the CP violating observables. These effects can be suppressed if CP is not effectively broken in the up-type (s)quark sector [@Ishiduki:2009gr]. We will comment this point in Secs.4 and 5.
[^9]: Here acutes are used to distinguish different $\bar {\textbf{5}}$($\textbf{1}$)s. Each subscript represents corresponding U(1) charge.
[^10]: In the following discussion, we also introduce antifundamental fields, ${\bar H}^{\bar {\bold{ 27 }}}$ and ${\bar C}^{\bar {\bold{ 27 }}}$, to maintain corresponding D-flatness conditions. Also, the adjoint field $A ^{\textbf{78}} $ is introduced to realize the DTS, employing the Dimopoulos-Wilczek (DW) mechanism [@Dimopoulos:1981xm]. A complete list of field contents is given in Sec.\[sec:a modle of scpv in e6 susy gut\].
[^11]: See Ref.[@Bando:2001bj] for details.
[^12]: See, e.g., Ref.[@Fusaoka:1998vc; @Ross:2007az].
[^13]: Since possible deviations from Eq., which are induced by the SU(2)$_{\text F}$ and E$_6$ breaking effects, are argued in Ref.[@Maekawa:2002eh], we do not repeat the detailed argument here but the FCNC constraints can be satisfied in the current model by employing the argument given in the Introduction.
[^14]: Here we include all the nonrenormalizable operators. Also, each term is assumed to have an $\cal O$(1) coefficient.
[^15]: From now on, each superfield is denoted by an uppercase letter, whereas the corresponding lower case letter indicates an associated U(1)$_{\text A}$ charge. The consistency of Eq. requires the number of positively charged fields to be larger than that of negatively charged fields by one.
[^16]: In general, following terms can also appear in the bracket of Eq.. $$\begin{aligned}
\Sigma \lambda^{(h+ \bar h) n_h} (H^
\bold{ 27 } {\bar H}^{\overline {\bold{ 27 }}})^{n_h} + \dots
\label{eq:possible-contribution-to-Ws}\end{aligned}$$ However, these terms are not important for the determination of $\langle \bar F F \rangle$. From Eq., it is understood that their signatures are summarized in $c_0$ of Eq..
[^17]: Note that, within the context of the U(1)$_{\text A}$ summarized in the previous section, the Giudice-Masiero mechanism [@Giudice:1988yz] does not work well since it requires ${\left| 2 h \right|} \leq 1$ [@Nir:1995bu].
[^18]: In Eq., abbreviation denotes $\lambda^{2s+z}S^2Z$, $\lambda^{s+2z}SZ^2$, etc. Since SUSY breaking effects does not largely change Eq(3.1), these terms are not relevant and omitted in the following arguments. This is the same for Eq..
[^19]: Note that the U(1)$_\text{A}$ D-flatness is satisfied by the VEV of an operator whose U(1)$_{\text A}$ charge is $-1$.
[^20]: We plot the potential respect to $S$ in Fig.\[SUSY breaking effect\]. Here the dotted parabola shows the SUSY mass term ($|\lambda^{s+z}S|^2$) and the dashed-dotted line indicates the SUSY breaking tadpole contribution ($A_S \lambda^{s} S + A_Z \lambda^{s+z} S\langle Z \rangle$). The solid line is the sum of these contributions.
[^21]: When each $ \mathcal O(1)$ coefficient appearing in a superpotential term equals to the corresponding coefficient of a soft SUSY breaking term, the above mechanism for the generation of $\mu$-term does not work.
[^22]: In actual model building we should introduce an additional positive singlet field $S'$ which plays a role of $S$ appeared in Eq.. Then this $S'$ may acquire complex VEV in its scalar and F components. However, the coupling between $S'$ and $H^3$ can be forbidden by an appropriate symmetry. Or, since the magnitude of $\langle S' \rangle $ is dictated by the smallest negative charge of the relevant field, another $Z'$ field can make $\langle S' \rangle $ sufficiently small, and ensures $\mu$- and $b$-terms to be almost real.
[^23]: In general $Y_U$ and $Y_D$ are the combinations of $Y_H$ and $Y_C$. However, we make Higgs sector so that the MSSM up-type Higgs doublet $H_U$ originates solely from $H_u [H^\bold{ 5 }$\] but the down-type Higgs doublet $H_D$ originates as $H_D= H_d[H^{{\bar {\bold{ 5 }}'}}$\]+$\lambda^{0.5} L $\[$C^{{\bar {\bold{ 5 }}}}$\]. (See Appendix \[sec:higgs sector\].) In this case, $Y_D$ is combinations of $Y_H$ and $Y_C$ but $Y_U$ is directly given by $Y_H$. \[footnote:Higgs mixing\]
[^24]: Note that, in this case, $F^a \Psi_a \Psi_3 \rightarrow \lambda^3 \Psi_{1} \Psi_{3}$ is also forbidden.
[^25]: We assume that each Yukawa coupling has a corresponding interaction with the SUSY breaking spurion field ($X=\theta^2 m_{SUSY}$), like $c_{ij}^{H(C)} X {Y}_{ij} \Psi_i \Psi_j H(C)$, where $c_{ij}^{H(C)}$ are the real $ \mathcal O(1)$ coefficient. Then if $Y_f$ is real (complex), the corresponding $A_f$ term is also real (complex).
[^26]: See \[footnote:Higgs mixing\].
[^27]: This is true even if higher dimensional term, like $Z_3 \Psi^a A^3 \Psi_a H $, is taken into account. In terms of SU(4)$\times$SU(2)$_{\text L}\times$SU(2)$_{\text R}$, $H$ and $A$ acquire VEVs in components of $H(\bold{1},\bold{1},\bold{1})$ and $A(\bold{15},\bold{1},\bold{1})$, respectively. Therefore two $\Psi_a(\bold{1},\bold{2},\bold{2})$ need to make singlet by themselves in $Z_3 \Psi^a A^3 \Psi_a H $, but anti-symmetric contractions respect to SU(2)$_{\text F}$, SU(2)$_{\text L}$ and SU(2)$_{\text R}$ indices end up zero.
[^28]: See \[footnote:Higgs mixing\] and Appendix \[sec:higgs sector\].
[^29]: We assume that this nature is maintained even if higher dimensional terms, like $Z_3 \Psi^a A^3 \Psi_a H $, is taken into account. In terms of SU(4)$\times$SU(2)$_{\text L}\times$SU(2)$_{\text R}$, SU(4) indices of $\Psi_a(\bold{4},\bold{2},\bold{2})$ can not be contracted with that of $\Psi_a(\bar {\bold{4}},\bold{2},\bold{2})$, because of the SU(2)$_{\text F}$ symmetry. Then, it is expected that quark components among $\Psi_a(\bold{4},\bold{2},\bold{2})$ and $\Psi_a(\bar {\bold{4}},\bold{2},\bold{2})$ pick up at least two $Q_B$ charges from the contraction with the adjoint fields $A(\bold{15},\bold{1},\bold{1})$. Strictly speaking, VEVs of $\bold{15}$-plets that are made by the direct product of adjoint fields can give contributions which are not proportional to $Q_B$ for quark components of $(\Psi_a(\bold{4},\bold{2},\bold{2}) \times \Psi_a(\bar {\bold{4}},\bold{2},\bold{2}))_\bold{15}$. However, these contributions generally possess another suppression factors that are determined so that irreducible representations do not contain a singlet component.
[^30]: Since [@Ross:2007az] uses the MSSM renormalization group equations (RGE), the discussion can be slightly changed once heavy fields contributions, that are relevant near the GUT scale, are taken into account.
[^31]: For the eigenvalues of down-quark and charged lepton Yukawa couplings, that are originally calculated in $\tan \beta =$10, we naively multiply $6/10$ to assess the corresponding values in case of $\tan \beta$=6.
[^32]: See [@Maekawa:2001uk] for more detailed discussion.
|
---
abstract: |
We present a method for the direct evaluation of the difference between the free energies of two crystalline structures, of different symmetry. The method rests on a Monte Carlo procedure which allows one to sample along a path, through atomic-displacement-space, leading from one structure to the other by way of an intervening transformation that switches one set of lattice vectors for another. The configurations of both structures can thus be sampled within a single Monte Carlo process, and the difference between their free energies evaluated directly from the ratio of the measured probabilities of each. The method is used to determine the difference between the free energies of the [*fcc*]{} and [*hcp*]{} crystalline phases of a system of hard spheres.
PACS numbers: 64.70Kb, 02.70.Lq, 71.20Ad\
address: |
Department of Physics and Astronomy, The University of Edinburgh\
Edinburgh, EH9 3JZ, Scotland, United Kingdom
author:
- 'A.D. Bruce, N.B. Wilding & G.J. Ackland'
title: 'Free energies of crystalline solids: a lattice-switch Monte Carlo method'
---
pnumwidth[2em]{}
specialpagefalse oddhead[ 3.0Released November 10, 1992]{} evenheadoddhead oddfoot[@font@page=1 ]{} evenfootoddfoot
One of the fundamental tasks of theoretical condensed matter physics is to understand the observed structures of crystalline materials in terms of microscopic models of the atomic interactions. The principles involved are well known: one needs to evaluate which of the candidate structures has the lowest free energy for given (model and thermodynamic) parameters. In practice the task is rather less straightforward. Conventional Boltzmann importance sampling Monte Carlo (MC) methods do not yield the free energy [@binder]. It is therefore customary to resort to integration methods (IM) which determine free energies by integrating free-energy [*derivatives*]{} measured at intervals along a parameter-space-path connecting the system of interest to a reference system whose free energy is already known. This procedure has been used widely, and with ingenuity [@frenkelladd]. Nevertheless it leaves much to be desired. In particular, to determine the [*difference*]{} between the free energies of two phases one has to relate [*each*]{} of them separately to some reference system, with uncertainties which are not always transparent, and which can be significant on the scale of the free energy difference of interest. Clearly, one would prefer a method which focuses more directly on this difference. The elements of such a strategy are to be found in the extended-sampling techniques pioneered by Torrie and Valleau [@torr;vall], and recently revitalized in the multicanonical method of Berg and Neuhaus [@bn]. The key concept underlying this method is that of a configuration-space-path comprising the macrostates of some chosen macroscopic property ${\cal M}$. The method utilizes a sampling distribution customized to even out the probabilities of different ${\cal M}$-macrostates. In principle, it allows one to sample along a path (whose canonical probability is generally extremely small) chosen to connect the distinct regions of configuration space associated with two phases; the difference between the free energies of the two phases can then be obtained directly from the ratio of the probabilities with which the system is found in each of the two regions. This idea has been applied in the investigation of the phase behavior of ferromagnets [@bn], fluids [@nbw], and lattice gauge theories [@lgt]. However its application to [*structural*]{} phase behavior faces a distinctive problem: finding a path that links the regions of configuration space associated with two different crystal structures [@grsadb], [*without*]{} traversing regions of non-crystalline order, which present problems [@blocks] for even multicanonical Monte-Carlo studies. We show here how one may construct such a path, and use it for direct high-precision measurement of free-energy differences of crystal structures.
The idea is simple; we describe it first in general and qualitative terms. The atomic position coordinates are written, in the traditions of lattice dynamics, as the sum of a lattice vector [@useoflattice], and a displacement vector. The configurations associated with a particular structure are explored by MC sampling of the displacements. Given any configuration of one structure one may identify a configuration of the other, by [*switching*]{} one set of lattice vectors for the other, while keeping the displacement vectors [*fixed*]{}. Such lattice switches can be incorporated into the MC procedure by regarding the lattice type as a stochastic variable. Lattice switches have an intrinsically low acceptance probability, since typically they entail a large energy cost. But the multicanonical method can be used to draw the system along a path comprising the macrostates of this ‘energy cost’, and thence into a region of displacement-space in which the ‘energy cost’ is low, and the lattice switch can be implemented. The net result is a MC procedure which visits both structures in the course of a single simulation, while never moving out of the space of crystalline configurations. The method is, we believe, potentially very general. We illustrate it here by using it to determine the difference between the free energies of the two close-packed structures ([*fcc*]{} and [*hcp*]{}) of a system of hard spheres. This problem has a long history [@aldercarteryoung]. The difference between the free energies (effectively, the [*entropies*]{}) is extremely small, and
recent IM studies have disagreed on its value [@woodcock; @bolfrenk]. It thus provides an exacting and topical testing ground for the method proposed here [@widersignificance].
We consider a system of $N$ particles with spatial coordinates $\{\vec{r}\}$. The particles are confined in a fixed volume $V$, with periodic boundary conditions [@novolumechange]. We make the decomposition $$\vec{r}_i=\vec{R}_i+\vec{u}_i
\label{eq:decomposition}$$ where the vectors $\vec{R}_i, i=1\ldots N
\equiv\{\vec{R}\}_\alpha$ define the sites of a lattice of type $\alpha$ (here, either [*fcc*]{} or [*hcp*]{}). Clearly there are many transformations that will map one set of vectors into the other; the mapping we have chosen is explained in Fig. 1(a),(b): it exploits the fact that the two structures differ only in the stacking pattern of the close-packed planes.
We define a partition function (and free energy) associated with the structure $\alpha$ by [@generalnotation] $$\begin{aligned}
Z(N,V,T,\alpha)& =&
\int_{\{\vec{u}\} \in \alpha} \prod_i [ d\vec u _i ]
\exp\left[- \Phi(\{\vec{u}\},\alpha) \right] \nonumber \\
&\equiv& \exp\left[- F_{\alpha} (N,V,T)/kT\right]
\label{eq:canpartdefa}\end{aligned}$$ where $\Phi$ represents the dimensionless configurational energy. In the present context $$\Phi(\{\vec{u}\},\alpha) \equiv \Phi(\{\vec{r}\})= \left\{
\begin{array}{ll}
0 & \mid \vec{r}_i -\vec{r}_j\mid >\sigma\,\, \forall i,j \\
\infty & \mbox{otherwise}
\end{array}
\right .
\label{eq:Phidef}$$ where $\sigma$ is the hard sphere diameter. The $\alpha$-label attached to the integral in Eq. (\[eq:canpartdefa\]) signifies that it must include only contributions from configurations within the subspace associated with the structure $\alpha$[@associated].
Consider now the canonical ensemble with probability distribution
$$P(\{\vec{u}\},\alpha\mid N,V,T) = \frac
{\exp\left[- \Phi(\{\vec{u}\},\alpha) \right]}
{Z(N,V,T)}
\label{eq:rawcanconfdist}$$
where $Z(N,V,T) \equiv\sum_{\alpha}Z(N,V,T,\alpha)$. The probability that the system will be found to have structure $\alpha$ provides a measure of the associated partition function: $$\begin{aligned}
P(\alpha\mid N,V,T)& \equiv&
\int_{\{\vec{u}\} \in \alpha} \prod_i [ d\vec u _i ]
P(\{\vec{u}\},\alpha\mid N,V,T) \nonumber\\
&=& \frac{Z(N,V,T,\alpha)}{Z(N,V,T)}
\label{eq:phaseprobs}\end{aligned}$$
The difference between the free-energies of the two structures may be thus be expressed as $$F_{\mbox{\it hcp}}(N,V,T)-F_{\mbox{\it fcc}}(N,V,T) \equiv NkT \Delta f =
kT \ln{{\cal R}}
\label{eq:deltaf}$$ where, $${\cal R} = \frac{Z(N,V,T,\mbox{\it fcc})} {Z(N,V,T,\mbox{\it hcp})}
= \frac{P(\mbox{\it fcc}\mid N,V,T)} {P(\mbox{\it hcp}\mid N,V,T)}
\label{eq:Rdef}$$ This identification is useful [*only*]{} if one can devise a MC procedure that will actually visit the configurations $\{\vec{u}\},\alpha$ with the probabilities prescribed by Eq. (\[eq:rawcanconfdist\]). To do so one must deal with the ergodic block against lattice switches (‘updates’ of the lattice label, $\alpha$): almost invariably such a switch maps an accessible configuration of one structure onto an inaccessible configuration of the other (one which violates the hard-sphere constraint implied by Eq. (\[eq:Phidef\])). Fig. 1(b) provides an example. The resolution is to [*bias*]{} the sampling procedure so as to favor the occurrence of configurations which transform [*without*]{} violating this constraint. To do so we define an [*overlap order parameter*]{}
$${\cal M}(\{\vec{u}\})
\equiv
M(\{\vec{u}\},{\mbox{\it hcp}})
-M(\{\vec{u}\},{\mbox{\it fcc}})
\label{eq:overlapdef}$$
where $M(\{\vec{u}\},\alpha)$ counts the number of pairs of overlapping spheres associated with the configuration $\{\vec{u}\},\alpha$ (again, see Fig. 1(b)). Since $M(\{\vec{u}\},\alpha)$ will necessarily be zero for any set of displacements $\{\vec{u}\}$ [*actually visited*]{} when the system has lattice $\alpha$, the order parameter ${\cal M}$ is necessarily $\ge 0$ ($\le 0$) for realizable configurations of the [*fcc*]{} ([*hcp*]{}) structure. The displacement configurations with ${\cal M}=0$ are accessible in [*both*]{} structures and thus offer no barrier against lattice switches. Accordingly the set of ${\cal
M}$-macrostates provides us with the required ‘path’ connecting the two phases, through a lattice-switch at ${\cal M}=0$. To pick out this path we must sample from the biased configuration distribution $$P(\{\vec{u}\},\alpha \mid N,V,T, \{\eta \})\propto
P(\{\vec{u}\},\alpha \mid N,V,T)e^{ \eta ({\cal M}(\{\vec{u}\}) )}
\label{eq:multicanconfdist}$$ where $\{\eta\} \equiv \eta ({\cal M}), {\cal M} =0,\pm 1,\pm 2 \ldots$ define a set of multicanonical weights [@bn], which have to be determined such that configurations of all relevant ${\cal M}$-values are sampled. Once this is done, one can measure the weighted distribution of ${\cal M}$-values, and reweight (unfold the bias) to determine the true canonical form of this distribution: $$P({\cal M}\mid N,V,T) \propto
P({\cal M} \mid N,V,T, \{\eta\})e^{-\eta ({\cal M})}
\label{eq:candist}$$ Finally, the difference between the free energies of the two structures may be read off from this distribution through the identification (cf Eqs. (\[eq:deltaf\]),(\[eq:Rdef\])) $\Delta f = N^{-1} \ln {\cal R}$, with $${\cal R}= \frac{
\sum_{{\cal M} > 0} P({\cal M}\mid N,V,T)
}{
\sum_{
{\cal M} < 0} P({\cal M}\mid N,V,T)
}
\label{eq:Rvalue}$$
We have implemented this procedure to study systems of N=216, 1728 and 5832 hard spheres (forming, respectively 6, 12 or 18 close-packed layers). The volume $V$ was chosen such that the fraction of space filled, $\rho$, satisfies $\rho/\rho_{cp} = 0.7778$ [@whythisdensity], where $\rho_{cp}\equiv 0.7404$ is the space filling fraction in the closest packing limit. The MC procedure entails sampling the displacement variables $\{\vec{u}\}$ and the lattice label $\alpha$. The variables $\{\vec{u}\}$ were updated by drawing new values from a top-hat distribution [@useoftophat] and accepting them provided they satisfy the hard sphere constraint; the lattice switches were attempted (and accepted with probability $1/2$) only when the system is in the ${\cal M}=0$ macrostate. The weights (which enable the system to reach this special macrostate) were obtained using methods explained elsewhere [@grsadb]. We allowed typically $2\times10^4$ sweeps for equilibration and up to $5\times10^7$ sweeps for final sampling runs on the largest system size. The simulations were conducted on DEC ALPHA workstations using overall some 800 hours CPU time.
Fig. 2 shows the measured overlap distribution for the system of $N=1728$ spheres; the inset shows the probability on a logarithmic scale, exposing the enormity of the entropic ‘barrier’ (probability ‘trough’) that the multicanonical weighting enables us to negotiate. The difference between the free energies of the two structures is identifiable immediately and transparently from the ratio of the integrated weights of the two essentially gaussian peaks. Our results for this system and other system sizes are gathered together in Table 1, along with the results of other authors.
From Table 1 it is apparent that the present work greatly refines the largely inconclusive results of the original IM study [@frenkelladd]. Our results are consistent with –though substantially more precise than– very recent IM studies of Bolhuis and Frenkel [@bolfrenk]. They are inconsistent with the result reported by Woodcock [@woodcock], given that $\Delta f$ is believed to [*decrease*]{} as the density is reduced, towards melting [@pressurecalc]. While we have not attempted an explicit analysis of the finite-size behavior, the close agreement between our results for $N=1728$ and $N=5832$ indicates that the latter should provide an extremely good estimate of the thermodynamic limit, confirming the stability of the [*fcc*]{} structure at this density.
Our principal concern here, however, is with the [*general*]{} lessons that can be learned about the method introduced in this work. The precision we have achieved with this method is self-evidently a significant advance on that of IM studies. Admittedly, this level of precision has entailed substantial processing [*time*]{}, principally because of the relative slowness of the diffusive exploration of the multicanonically weighted configuration space. But the point is that the procedure is [*practicable*]{} [@notaeons], with a computational strategy that is, we suggest, less complex and more transparent than that of IM. Thus, for example, the method described in [@frenkelladd] involves integration (of a mean-square displacement) along a parameter-space-path connecting each structure to a reference system, comprising an Einstein-model of the same structure; the MC integral then has to be combined with the known free energy of the Einstein model, a virial correction, and a correction to the virial correction, before taking the difference between the results for the two structures. The uncertainties in all the contributions have to be assessed separately. By contrast, the present method focuses directly on the quantity of interest (the relative weights of the peaks in Fig. 2); and the precision with which it is prescribed is defined by standard MC sampling theory.
Finally we comment briefly on the more general applicability of the method. For systems other than hard spheres, the role of the overlap order parameter is played by the [*energy barrier*]{} encountered in the lattice switch; the generalization of the weighting procedure should be straightforward. It seems unlikely that many problems will require the level of precision needed here, where the two phases are so finely balanced. However some circumstances will not generally prove as favorable. In the present case one can readily identify a lattice-to-lattice mapping which [*guarantees*]{} no overlaps (high-energy-cost interactions) amongst subsets of the atoms (those lying within the [*same*]{} close-packed plane). The optimal form of mapping may not always be so evident. It may also prove advantageous to relax the constraint imposed in Eq. (\[eq:decomposition\]) that the coordinates $\{\vec{u}\}$ represent [*identical*]{} displacement patterns in the two structures. At the cost of only little extra computational complexity the true displacements may be represented as structure-dependent (even site-dependent) [*functions*]{} of a (still common) set of coordinates $\{\vec{u}\}$. It should then be possible to ensure that a typical displacement pattern in one structure maps onto a typical pattern in the other. These matters are the subject of ongoing study.
K. Binder, [*J. Comp. Phys.*]{} [**59**]{} 1 (1985) D. Frenkel & A.J.C. Ladd, [*J. Chem. Phys*]{} [**81**]{}, 3188 (1984). G.M. Torrie & J.P. Valleau, [*Chem. Phys. Lett.*]{} [**28**]{}, 578 (1974). B.A. Berg & T. Neuhaus, [*Phys. Lett. B*]{} [**267**]{}, 249 (1991); [*Phys. Rev. Lett.*]{} [**68**]{}, 9 (1992). N.B. Wilding, [*Phys. Rev. E*]{} [**52**]{},602 (1995). B. Grossmann, M.L. Laursen, T. Trappenberg & U.J. Wiese, [*Phys. Lett. B*]{} [**293**]{}, 175 (1992). G.R. Smith & A.D. Bruce, [*Phys. Rev. E*]{} [**53**]{}, 6530 (1996) apply multicanonical methods to a structural phase transition which involves [*no change of symmetry*]{} where an appropriate path is identified simply by ${\cal M} =\rho$. Eg: the ergodic block associated with re-crystallization. We use the term ‘lattice vector’ a little loosely: we mean the set of vectors identified by the orthodox crystallographic lattice, convolved with the orthodox basis. B.J.Alder, B.P. Carter & D.A. Young, [*Phys Rev*]{} [**183**]{}, 831 (1969). L.V. Woodcock, [*Nature*]{} [**384**]{}, 141 (1997). P.G.Bolhuis & D. Frenkel (unpublished). The hard sphere system has a wider significance. Y.Choi [*et al*]{},[*J. Chem. Phys.*]{} [**99**]{} 9917 (1993) show that predictions for the phase diagram of a Lennard Jones solid depend extremely sensitively on the hard-sphere free-energy difference $\Delta
f$ computed here. Colloids provide experimental realizations of near-hard-sphere systems:
P.N. Pusey [*et al*]{}, [*J. Phys: Condens. Matter*]{} [**6**]{} A29 (1994). In common with previous studies we work at constant [*volume*]{}, and constrain the $c/a$ ratio in the [*hcp*]{} structure to its closest-packing value. Generalization to the constant [*pressure*]{} ensemble is straightforward in principle. We use a general notation; formally the properties of the hard-sphere system are independent of $T$. In the MC context the configurations [*associated*]{} with a given structure are identified as the set which are actually [*accessed*]{} in a simulation initialized within the set. This value of $\rho$ was chosen to coincide with one of those studied in Ref [@frenkelladd]. This choice of sampling procedure ensures that the center of mass is effectively fixed. For consistency the width of the top-hat distribution must be large compared to the range of displacements actually [*accepted*]{}. This is the implication of studies of the pressure in the two structures: B.J.Alder, D.A. Young, M.R. Mansigh & Z.W. Salsburg, [*J. Comp. Phys*]{} [**7**]{}, 361 (1971). The time required is measured on a scale of hours rather than the eons required if one were to attempt such a ‘direct’ method [*without*]{} the multicanonical strategy provided here: recall the scale on the inset of Fig. 2.
------------------ ------- ------ ------- ----------------
$\rho/\rho_{cp}$ N Ref.
0.7360 216 90 (135) [@frenkelladd]
0.7360 12000 500 (100) [@woodcock]
0.7360 12906 90 (20) [@bolfrenk]
0.7778 1152 -120 (180) [@frenkelladd]
0.7778 216 101 (4) PW
0.7778 1728 83 (3) PW
0.7778 5832 86 (3) PW
------------------ ------- ------ ------- ----------------
: Results for the difference between the free energy of [*hcp*]{} and [*fcc*]{} structures, as defined in Eq. (\[eq:deltaf\]) with associated uncertainties in parentheses. Results attributed to Ref. [@frenkelladd] were deduced by combining the separate results for [*fcc*]{} and [*hcp*]{} given there. PW signifies the present work. The PW error bounds were computed from the statistical uncertainties in the weights of the peaks in $P({\cal M})$.
\[tab:coexden\]
|
---
abstract: 'We present a method for Baxterizing solutions of the constant Yang-Baxter equation associated with $\mathbb{Z}$-graded Hopf algebras. To demonstrate the approach, we provide examples for the Taft algebras and the quantum group $U_q\left[sl(2)\right]$.'
---
0.5cm
[**Universal Baxterization for $\mathbb{Z}$-graded Hopf algebras**]{}
0.8cm
K.A. Dancer[^1], P.E. Finch[^2] and P.S. Isaac[^3]
0.9cm
*Centre for Mathematical Physics, School of Physical Sciences,*
*The University of Queensland, Brisbane 4072, Australia.*
0.9cm
\[section\] \[section\] \[section\] \[section\] \[section\]
ł
Introduction
============
The word “Baxterization” was originally coined by V.F.R. Jones [@Jones; @Jones2] to refer to the insertion of a parameter into a solution of the constant Yang-Baxter equation so that it becomes a solution of the parameter dependent Yang-Baxter equation. This is done in such a way that the resultant parametric solution reduces to the original constant one in some suitable limit.
There exist well-studied methods of Baxterization, especially those associated with quantum groups. Both universal (i.e. representation independent) [@Jones2; @KRS; @KulishResh; @ZhangGould] and representation dependent [@Jones; @Jones2; @Jimbo; @Jimbo3; @ZGB; @DGZ; @ChangGeXue; @ZhangKauffmanGe; @WellyMartins; @DIL] approaches have been developed.
In this paper, we introduce a new method of Baxterizing universal R-matrices arising from $\mathbb{Z}$-graded associative algebras. In particular we focus on $\mathbb{Z}$-graded Hopf algebras. The prime examples with which we demonstrate our results are the Taft algebras [@Taft; @Chen].
The (multiplicative) parameter dependent Yang-Baxter equation (YBE) is $$R_{12}(x)R_{13}(xy)R_{23}(y) = R_{23}(y)R_{13}(xy)R_{12}(x).$$ Here $R$, known as an $R$-matrix, is an operator on $V \otimes V$ for some vector space $V$. We use the standard notation that $R_{13}\in \End(V\otimes V\otimes V)$ represents $R$ operating on the $1$st and $3$rd components of $V\otimes V\otimes V$, and similarly for $R_{12}, R_{23}$. This equation has a variety of applications, particularly in exactly solvable models in statistical mechanics [@Bax] and quantum field theory [@Faddeev]. Consequently, it is always of interest to develop new methods of solving this equation. Certainly there already exists a body of elegant works dedicated to solving this equation, some noteworthy articles being [@Drinfeld; @Drinfeld2; @Jimbo]. For a good overview of the parameter dependent Yang-Baxter equation and its solutions, see for example [@Jimbo2] or [@Lambe].
By contrast, the constant Yang-Baxter equation $$R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$$ has no parameter dependence and hence is easier to solve. Solutions are known in many different contexts, most significantly Drinfeld’s universal solution arising from the quantum double construction for Hopf algebras [@Drinfeld]. From a solution of the parameter dependent YBE, one can easily obtain a solution of the constant YBE (by taking some suitable limit), but the converse is not true. As mentioned above, there are well established Baxterization techniques for quantum groups, however these methods do not extend to Hopf algebras in general.
In this paper we present a straightforward method of obtaining a universal parameter dependent solution from a constant solution in the context of $\mathbb{Z}$-graded Hopf algebras. To demonstrate the method, we provide specific examples for the finite dimensional Taft algebras and the quantum group $U_q\left[sl(2)\right]$.
Universal Baxterization
=======================
Let $H$ be an associative algebra with unit with multiplication $m$. Let $A$ be a subalgebra of $H$. If we can find $\{ A^p| p \in \mathbb{Z}\}$ such that
- $A = \bigoplus_{p} A^p$
and
- $m: A^p \otimes A^q \rightarrow A^{p+q},$
then we say $A$ is $\mathbb{Z}$-graded, and call $A = \bigoplus_p A^p$ the $\mathbb{Z}$-grading of $A$. If there exists some $p \neq 0$ such that $A^p \neq \{ 0\}$ we say the $\mathbb{Z}$-grading is nontrivial.
\[Baxterization\] Let $H$ be an associative algebra with unit. Suppose $H$ has subalgebras $A,B$ with $\mathbb{Z}$-gradings $A = \bigoplus_p A^p$ and $B= \bigoplus_q B^q$ respectively. If $H$ contains a solution of the constant Yang–Baxter equation of the form
$$R = \sum_{i,\a} a_\a^i \otimes b_\a^i$$ where $a_\a^i \in A^i$ and $b_\a^i \in B^i$, then
$$R(\mu) = \sum_i \mu^i \sum_\a a_\a^i \otimes b_\a^i$$ is a solution of the multiplicative parametric Yang–Baxter equation.\
[**Proof:**]{} It is given that $R = \sum_{i,\a} a_\a^i \otimes b_\a^i$ satisfies the constant Yang-Baxter equation
$$R_{12}R_{13} R_{23} = R_{23} R_{13} R_{12}.$$ Substituting in, this is equivalent to stating
$$\sum_{i,j,k,\a,\b,\g} a^i_\a a^j_\b \otimes b^i_\a a^k_\g \otimes b^j_\b b^k_\g =
%
\sum_{p,q,r,\d,\e,\k} a^q_\e a^r_\k \otimes a^p_\d b^r_\k \otimes b^p_\d b^q_\e.$$ In particular, we can equate the entries belonging to $A^s \otimes H \otimes B^q$, giving
$$\sum_{j,\a,\b,\g} a^{s-j}_\a a^j_\b \otimes b^{s-j}_\a a^{t-j}_\g \otimes b^j_\b b^{t-j}_\g =
%
\sum_{q,\d,\e,\k} a^q_\e a^{s-q}_\k \otimes a^{t-q}_\d b^{s-q}_\k \otimes b^{t-q}_\d b^q_\e.$$ Now we substitute the parametrized $R$-matrix $R(\mu)$ into the parametric Yang-Baxter equation:
$$\begin{aligned}
R_{12}(\mu) R_{13}(\mu \nu) R_{23}(\nu)
&= \sum_{i,j,k,\a,\b,\g} \mu^{i+j}\nu^{j+k} \, a^i_\a a^j_\b \otimes b^i_\a a^k_\g \otimes b^j_\b b^k_\g \\
&= \sum_{s,t} \mu^s \nu^t \sum_{j,\a\b\g} a^{s-j}_\a a^j_\b \otimes b^{s-j}_\a a^{t-j}_\g \otimes b^j_\b b^{t-j}_\g \\
&= \sum_{s,t} \mu^s \nu^t \sum_{q,\d,\e,\k} a^q_\e a^{s-q}_\k \otimes a^{t-q}_\d b^{s-q}_\k \otimes b^{t-q}_\d b^q_\e \\
&= \sum_{p,q,r,\d,\e,\k} \mu^{q+r} \nu^{p+q} \, a^q_\e a^r_\k \otimes a^p_\d b^r_\k \otimes b^p_\d b^q_\e \\
&= R_{23}(\nu) R_{13}(\mu \nu) R_{12}(\mu) \end{aligned}$$
as required.
It is possible to generalize the result of Proposition \[Baxterization\] to include $\mathbb{Z}^n$-graded algebras. The result is that if $R$ is an element of $\bigoplus_{p\in\mathbb{Z}^n}A^p\otimes B^p$, then a universal Baxterization exists. Explicitly, let $$R = \sum_{i\in\mathbb{Z},\ p\in\mathbb{Z}^n} a_i^p\otimes b_i^p,$$ where $a_i^p\in A^p,$ $b_i^p\in B^p$. The Baxterized solution will then be $$R(\mu) =
\sum_{p\in\mathbb{Z}^n}\mu^{\tau(p)}\sum_{i\in\mathbb{Z}}a_i^p\otimes b_i^p,$$ where $\tau:\mathbb{Z}^n\rightarrow \mathbb{Z}$ (or some other appropriate codomain) is a group homomorphism under addition. The proof of this result is essentially the same as the proof of Proposition \[Baxterization\]. We will not, however, make use of this generalization in the current article.
One algebraic structure where a nontrivial $\mathbb{Z}$-grading may arise is the Drinfeld double of a Hopf algebra. To understand the Drinfeld double, we first introduce the dual of a finite Hopf algebra $H$, which we denote $H^*$.
The vector space underlying $H^*$ is the set of linear maps $f:H \rightarrow \C$. We choose the bilinear form
$$\langle f, x \rangle = f(x), \quad \forall x \in H.$$ If $H$ has basis $\{ a_i \}$, then we choose $\{a_i^*\}$ as a basis for $H^*$, where
$$\langle a_i^*, a_j \rangle = \delta_{ij}.$$ The structure of $H^*$ is induced by that of $H$. Specifically, if $H$ has multiplication $m$, unit $u$, coproduct $\Delta$ and counit $\e$ then $H^*$ becomes a Hopf algebra with multiplication $m^*$, unit $u^*$, coproduct $\Delta^*$ and counit $\e^*$ defined by:
$$\begin{aligned}
{2}
&\langle m^*(a_{i}^{*} \otimes a_{j}^{*}), a_{k} \rangle = \langle a_{i}^{*} \otimes a_{j}^{*}, \Delta(a_{k}) \rangle , & \qquad &
\langle u^*(\textit{k}), a_{i} \rangle = k\epsilon(a_{i}),\; \forall k \in \C, \\
&\langle \Delta^*(a_{i}^{*}), a_{j} \otimes a_{k} \rangle = \langle a_{i}^{*}, m(a_{j} \otimes a_{k}) \rangle, & &
\epsilon^*(a_{i}^{*}) = \langle a_{i}^{*},e \rangle.\end{aligned}$$
The Drinfeld double of a finite Hopf algebra $H$, which we denote $D(H)$, is a quasitriangular Hopf algebra spanned by elements of the form $\{gh^*|g \in H, h^* \in H^*\}$. Details of the algebraic structure and costructure of $D(H)$ can be found in [@Drinfeld]. Of particular relevance here is the property that $D(H)$ contains a canonical solution of the Yang–Baxter equation of the form
$$R = \sum_{i} a_i \otimes a_i^*,$$ where $\{a_i\}$ is a basis for $H$. Here we identify $a_i$ with $a_i \epsilon$ and $(a_i)^*$ with $u(a_i)^*$ where $\epsilon$ and $u$ are the counit and unit of $H$ respectively.
Using this universal $R$-matrix, we have the following result:
\[double\] Let $H$ be a finite-dimensional $\mathbb{Z}$-graded Hopf algebra with nontrivial $\mathbb{Z}$-grading $H = \bigoplus_p A^p$. If the coproduct of $H$ satisfies $$\Delta: A^p \rightarrow \bigoplus_q A^q \otimes A^{p-q}, \qquad \forall p \in \mathbb{Z},$$ then $D(H)$ nontrivially satisfies the conditions for Proposition \[Baxterization\].
[**Proof:**]{} Let $A^{p}$ have the basis $\{ a^{p}_{i} \}$ and $B^{p}$ have the basis $\{ (a^{p}_{i})^*|a^p_i \in A^p \}$. Clearly the dual of $H$ can be written as $H^* = \bigoplus_p B^p.$ Moreover an $R$-matrix for $D(H)$ is $\sum_{j} a_j \otimes (a_j)^* \; \in \bigoplus_p A^p \otimes B^p$. Thus it remains only to show $m: B^p \otimes B^q \rightarrow B^{p+q}$ where $m$ represents multiplication within $D(H)$.
But $$\begin{aligned}
\langle m \bigl( (a^p_i)^* \otimes (a^q_j)^* \bigr), a^r_k \rangle
&= \langle (a^p_i)^* \otimes (a^q_j)^*, \Delta(a^r_k) \rangle \\
&= 0 \hspace{4cm} \mbox{if $r \neq p + q$}.\end{aligned}$$ Thus $m: B^{p} \otimes B^{q} \rightarrow B^{p+q}$ for all $p, q \in \Z$. Hence $D(H)$ satisfies the conditions for Proposition \[Baxterization\].
Example: $U_q[sl(2)]$
=====================
The $q$-deformed Lie algebra $U_q[sl(2)]$ has generators $e,f,h$ satisfying
$$[e,f] = \frac{q^h - q^{-h}}{q-q^{-1}}, \quad [h,e] = 2e, \quad [h,f] = -2f,$$ where $q$ is the deformation parameter. Define $[n]_q$ and $[n]_q!$ as follows:
$$\begin{aligned}
[n]_q &= \frac{q^n - q^{-n}}{q-q^{-1}}, \\
[n]_q! &= [n]_q [n-1]_q ... [1]_q.\end{aligned}$$
Then $U_q[sl(2)]$ contains the following universal $R$-matrix [@CP]:
$$\mathfrak{R} = \sum_{n=0}^\infty \frac{q^{\frac 12 n(n+1)}(1-q^{-2})^n}{[n]_q!} q^{\frac 12 (h \otimes h)} e^n \otimes f^n.$$ Now set $H = \langle h \rangle$ to be the subalgebra generated by $h$, $A = \langle h,e \rangle$ to be the subalgebra generated by $e$ and $h$, and $B = \langle h,f \rangle$ to be the subalgebra generated by $f$ and $h$. Then $A$ has the natural $\mathbb{Z}$-grading $A = \bigoplus_{k \in \mathbb{N}} A^k$ where $A^k = H e^k, \, k \geq 0$. Similarly, $B$ has the $\mathbb{Z}$-grading $B = \bigoplus_{k \in \mathbb{N}} B^k$ where $B^k = H f^k, \, k \geq 0$. Note that with these $\mathbb{Z}$-gradings $R \in \bigoplus_{k \in \mathbb{N}} A^k \otimes B^k$, so we can apply Proposition \[Baxterization\]. We find that
$$\mathfrak{R}(\mu) = \sum_{n=0}^\infty \mu^n \frac{q^{\frac 12 n(n+1)}(1-q^{-2})^n}{[n]_q!} q^{\frac 12 (h \otimes h)} e^n \otimes f^n$$ is a solution of the mulitplicative parametric Yang-Baxter equation.
Applying the spin-$\frac 12$ representation, which is given by
$$e = \begin{pmatrix} 0 & 1 \\ 0 & 0
\end{pmatrix}, \quad
f = \begin{pmatrix} 0 & 0 \\ 1 & 0
\end{pmatrix}, \quad
h = \begin{pmatrix} 1 & 0 \\ 0 & -1
\end{pmatrix},$$
this becomes
$$R(\mu) = \begin{pmatrix}
q^{\frac 12} & 0 & 0 & 0 \\
0 & q^{-\frac 12} & \mu q^{-\frac 12} (q - q^{-1}) & 0 \\
0 & 0 & q^{-\frac 12} & 0 \\
0 & 0 & 0 & q^{\frac 12}
\end{pmatrix}.$$ Similarly, applying the spin-$1$ representation, which is given by
$$e = \sqrt{q + q^{-1}} \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1 \\0 & 0 & 0
\end{pmatrix}, \quad
f = \sqrt{q + q^{-1}} \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0
\end{pmatrix}, \quad
h = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -2
\end{pmatrix},$$
the parametric $R$-matrix becomes [$$R(\mu) = \begin{pmatrix}
q^2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & \mu(q^2 - q^{-2}) & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & q^{-2} & 0 & \mu q^{-2}(q^2 - q^{-2}) & 0 & \mu^2 q^{-1}(q-q^{-1})^2(q+q^{-1})& 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & \mu(q^2 - q^{-2}) & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & \mu(q^2 - q^{-2}) & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & q^{-2} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & q^2
\end{pmatrix}.$$ ]{}
Example: Taft algebras
======================
The Taft algebra $T_{N,q}$ [@Taft] over a field $\mathbb{F}$ is an $N^2$-dimensional algebra with unit $e$ generated by $\langle a,x|a^N = e, \, x^N = 0, xa = qax \rangle$. Here $q$ is a primitive $N^{\rm th}$ root of unity in $\mathbb{F}$. We choose $\{ a^i x^j| 0 \leq i,j < n\}$ as a basis for $T_{N,q}$, and note that multiplication of two basis elements is given by $(a^i x^j)( a^k x^l) = q^{jk} a^{i+k} x^{l+j}$.
The Taft algebra $T_{N,q}$ becomes a Hopf algebra when endowed with a costructure and antipode defined on the generators $a,x$ by:
$$\begin{aligned}
{3}
&\Delta(a) = a \otimes a, & & \epsilon(a) = 1, & & \gamma(a) = a^{-1}, \\
&\Delta(x) = x \otimes e + a \otimes x, \qquad && \epsilon(x) = 0, \qquad && \gamma(x) = -a^{-1} x.\end{aligned}$$
Here $\Delta, \, \epsilon$ and $\gamma$ represent the coproduct, counit and antipode respectively.
The coproduct and counit extend as homomorphisms to all of $T_{N,q}$. Following the notation of [@Chen], we define $(n)_q = 1 + q + ... + q^{n-1}$ and $(n)_q! = (n)_q (n-1)_q ... (1)_1.$ Set
$${n \choose m}_{q} = \frac{(n)_{q}!}{(m)_{q}!(n-m)_{q}!}.$$
Then for all elements $a^i x^j \in T_{N,q}$, we find the coproduct is given by
$$\Delta(a^{i}x^{j}) = \sum_{k=0}^{j} {j \choose k}_{q} a^{j-k+i}x^{k} \otimes a^{i}x^{j-k}.$$
The Drinfeld double $D(T_{n,q})$ contains a universal $R$-matrix given by
$$R = \sum_{i,j=0}^{N-1} a^i x^j \otimes (a^i x^j)^*.$$ But $T_{N,q}$ has the $\Z$-grading $T_{N,q} = \bigoplus_p A^p$ where $A^p$ has basis $\{a^i x^p| 0 \leq i < N \}$. Under this grading, the coproduct satisfies $$\Delta: A^p \rightarrow \bigoplus_q A^q \otimes A^{p-q} \quad \forall p \in \Z.$$
Thus we note from Propositions \[Baxterization\] and \[double\] that the Drinfeld double $D(T_{N,q})$ contains an algebraic solution of the parametric Yang-Baxter equation given by:
$$R(\mu) = \sum_{i,j=0}^{N-1} \mu^j a^i x^j \otimes (a^i x^j)^*.$$
This can in turn give rise to several matrix solutions of the parametric Yang-Baxter equation, as the representation theory of the Taft algebras has been developed by Chen [@Chen]. Explicitly, the $N^2$ irreducible representations of $T_{N,q}$ are given by
$$\pi_{n,l}(a^{i}x^{j}) = \sum_{k = 1}^{n-j} q^{(k-l-n)i} \frac{(k+j-1)_{q}!}{(k-1)_{q}!} \Pi_{p=0}^{j-1}(1-q^{p+k-n}) e_{k,k+j}$$ and $$\pi_{n,l}((a^{i+l-1}x^{j})^{*}) = \left\{
\begin{array}{lcl}
\frac{e_{i+j,i}}{(j)_{q}!}, & & 1 \leq i \leq n - j \mbox{ mod(N)},\\
0 &\quad & \mbox{otherwise},
\end{array} \right.$$ where $1 \leq n,l \leq N.$ Here $e_{i,j}$ is the $n \times n$-dimensional elementary matrix whose only non-zero entry is a 1 in the $(i,j)$ position.
There are also $N$-dimensional indecomposable representations of $T_{N,q}$, which can be found in [@Chen]. They are given by:
$$\pi_\a (a^{i}x^{j}) = \alpha q^{-i(j+l)} \frac{(N-2)_{q}!}{(N-j-1)_{q}!} \prod_{p=1}^{j-1} (1 - q^{-p}) e_{N+1-j,1} + \sum_{k=2}^{N-j}q^{i(k-1-l)}\frac{(k+j-1)_{q}!}{(k-2)_{q}!} \prod_{p=0}^{j-1} (1 - q^{k+p}) e_{k,k+j}$$ and, as before,
$$\pi_\a((a^{i+l-1}x^{j})^{*}) = \left\{
\begin{array}{lcl}
\frac{e_{i+j,i}}{(j)_{q}!}, & & 1 \leq i \leq n - j \mbox{ mod(N)},\\
0 &\quad & \mbox{otherwise}.
\end{array} \right.$$
When the representation [@Chen] arising from $\tilde{V}_{3,l} \otimes \tilde{V}_{3,l}$ is applied to the universal R-matrix of $D(T_{N,q})$, $N\geq 3$, it gives the Baxterized $R$-matrix [$$R(\mu) =
\left(
\begin{array}{ccc ccc ccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & q^{-l-2} & 0 & (1-q^{-2}) \mu & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & q^{-2(l+2)} & 0 & q^{-l-4}(q^{2}-1)\mu & 0 & (1-q^{-1})(1-q^{-2})\mu^{2} & 0 & 0 \\
0 & 0 & 0 & q^{l} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & q^{-1} & 0 & q^{l+1}(1-q^{-2})\mu & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & q^{-l-2} & 0 & (1-q^{-2})\mu & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & q^{2l} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & q^{l} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$$]{} It is worth noting that for $l=N-1$, this matrix is a special case of the $9\times 9$ matrix associated with $U_q[sl(2)]$ given earlier.
Although the examples we have given are all upper triangular, this is not true in general. For a resultant matrix which is not triangular, we require a $\mathbb{Z}$-grading which is not an $\mathbb{N}$-grading. For example, the quantum double of $U_q[sl(2)]$ contains such a grading.
[999]{} V.F.R. Jones, [*On a Certain Value of the Kauffman Polynomial*]{}, Commun. Math. Phys. [**125**]{} (1989), 459–467. V.F.R. Jones, [*Baxterization*]{}, Int. J. Mod. Phys. B [**4**]{} (1990), 701–713. P.P. Kulish, N.Yu. Reshetikhin, E.K. Sklyanin, [*Yang-Baxter Equation and Representation Theory : I*]{}, Lett. Math. Phys. [**5**]{} (1981), 393–403. P.P. Kulish, N.Yu. Reshetikhin, [*Quantum linear problem for the Sine-Gordon equation and higher representations*]{}, J. Soviet Math. 23 (1983), 2435–2446. Y.-Z. Zhang, M.D. Gould, [Quantum Affine Algebras and Universal $R$-Matrix with Spectral Parameter]{}, Lett. Math. Phys. [**31**]{} (1994), 101–110. M. Jimbo, [*A $q$-Difference Analogue of $U(g)$ and the Yang-Baxter Equation*]{}, Lett. Math. Phys. [**10**]{} (1985), 63–69. M. Jimbo, [*Quantum $R$ Matrix for the Generalized Toda System*]{}, Commun. Math. Phys. [**102**]{} (1986), 537–547. R.B. Zhang, M.D. Gould, A.J. Bracken, [*From representations of the braid group to solutions of the Yang-Baxter equation*]{}, Nucl. Phys. B [**354**]{} (1991), 625–652. G.W. Delius, M.D. Gould, Y-.Z. Zhang, [*On the Construction of Trigonometric Solutions of the Yang-Baxter Equation*]{}, Nucl. Phys. B [**432**]{} (1994), 377–403. Y. Chang, M.L. Ge, K. Xue, [*Yang-Baxterization of Braid Group Representations*]{}, Commun. Math. Phys. [**136**]{} (1991), 195–208. Y. Zhang, L.H. Kauffman, M.L. Ge, [*Yang-Baxterizations, Universal Quantum Gates and Hamiltonians*]{}, Quantum Inf. Process. [**4**]{} (2005), 159–197. W. Galleas, M.J. Martins, [*New $R$-matrices from representations of braid-monoid algebras based on superalgebras*]{}, Nucl. Phys. B [**732**]{} (2006), 444–462. K.A. Dancer, P.S. Isaac, J. Links, [*Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equation*]{}, J. Math. Phys. 47, 103511 (2006). E.J. Taft, [*The Order of the Antipode of Finite-dimensional Hopf Algebra*]{}, Proc. Nat. Acad. Sci. USA [**68**]{} (1971), 2631–2633. H.-X. Chen, [*Irreducible Representations of a Class of Quantum Doubles*]{}, J. Algebra [**225**]{} (2000), 391–409. R.J. Baxter, [*Exactly Solvable Models in Statistical Mechanics*]{}, Academic Press, London (1982). L.D. Faddeev, [*Two dimensional Integrable Models in Quantum Field Theory*]{}, Physica Scripta 24 (1981), 832–835. V.G. Drinfeld, [*Quantum Groups*]{}, Proc. Internat. Cong. Math. Berkeley (1986), 789–820. V.G. Drinfeld, [*Hopf Algebras and the Quantum Yang-Baxter Equation*]{}, Soviet Math. Dokl. [**32**]{} (1985), 254–258. M. Jimbo, [*Introduction to the Yang-Baxter Equation*]{}, Int. J. Mod. Phys. A [**4**]{} (1989), 3759–3777. L. Lambe, [*On the Quantum Yang-Baxter Equation with Spectral Parameter I*]{}, Proc. London Math. Soc. (3), [**68**]{} (1994), 31–50. V. Chari and A. Pressley, [*A Guide to Quantum Groups*]{}, Cambridge University Press (1994).
[^1]: dancer@maths.uq.edu.au
[^2]: pfinch@maths.uq.edu.au
[^3]: psi@maths.uq.edu.au
|
---
author:
- |
Bin Chen\
Department of Physics,\
and State Key Laboratory of Nuclear Physics and Technology,\
and Center for High Energy Physics,\
Peking University,\
Beijing 100871, P.R. China\
- |
Jiang Long\
Department of Physics,\
Peking University,\
Beijing 100871, P.R. China\
title: Hidden Conformal Symmetry and Quasinormal Modes
---
Introduction
============
Very recently, motivated by the work in [@Andy2010], it has been found that in many black holes which have holographic 2D CFT descriptions there exists a hidden conformal symmetry. The hidden conformal symmetry is realized by two sets of locally defined vector fields $\{V_i, \bar{V}_i\}$ satisfying $SL(2,R)$ Lie algebra. This symmetry is not globally defined, and is broken by the periodic identification on angular variable. It could not be used to generate new solutions. Nevertheless, it determines the scattering amplitudes by acting on the solution space. More explicitly, the scalar Laplacian could be written as the $SL(2,R)$ quadratic Casimir in some limit region. The hidden conformal symmetry was considered to be essential to implement a holographic description of a black hole. It was widely studied in various kinds of black holes, including the 4D Kerr-Newman[@Chen:2010xu], 4D Kerr-Newman-AdS-dS[@Chen:2010bh], 3D black holes, extremal black holes[@Chen:2010fr] and others[@71].
In retrospect, the appearance of hidden conformal symmetry is not a surprise, considering the fact that the black hole is dual to a 2D CFT. On the CFT side, the conformal symmetry restricts the form of the correlation functions of the operators. Correspondingly, the conformal symmetry acts on the solution space and determines the scattering amplitudes. In the 4D Kerr case, the hidden conformal symmetry only become manifest in the low frequency limit and in the near region, but it determines not only the low frequency scattering amplitudes but also the super-radiant scattering ones. In 3D cases, the hidden conformal symmetry generically manifests itself more clearly, in all regions. Such features in 3D black holes are due to the fact that the black holes are always locally isomorphic to their covering spaces.
On the other hand, from AdS/CFT correspondence the quasinormal modes, which determine the relaxation time of the perturbations about the black hole, are related to the poles of the retarded correlation function in the momentum space in the dual conformal field theory. In a 2D CFT, the retarded Green’s functions for an operator with fixed conformal weights, fixed charges with respect to chemical potentials are determined by the conformal symmetry[@Cardy:1984rp]. The poles in the retarded Green’s function could be read easily. On the gravity side, the quasinormal modes could be read from the eigenfunctions satisfying the purely ingoing boundary condition at the black hole horizon and appropriate boundary condition at the asymptotical infinity[@quasinomral1999]. One has to solve the equations of motion explicitly in order to get the eigenfunctions, whose analytic forms are often out of reach. It is thus interesting to see that the equation of motion with the hidden conformal symmetry acting on could always be solved in terms of hypergeometric functions, due to mathematical fact that the hypergeometric functions could form the representation of the $SL(2,R)$ group. As a result, the quasinormal modes could be read exactly. Actually one aim of this paper is to show that even without solving the equations of motion explicitly, we can determine the quasinormal modes in an elegant algebraic way.
Another issue on hidden conformal symmetry is how it acts on the vector and tensor fields. In all the studies in the literature, the hidden conformal symmetry has kept being discussed in the scalar equation of motion. As it is an intrinsic property of the black hole, it should also act on the other kinds of perturbations. In this paper, we address this issue. For the locally defined vector fields, they act on the vector and the tensor fields via Lie-derivatives. We show that for 3D black holes, the hidden conformal symmetry acts on the vector and tensor fields in a subtle way. We find that only after an appropriate combination, the equations of motion of the vector and tensor fields could be written as the Lie-induced quadratic Casimir: (\^2+m\_t\^2)T\_+=0 where \^2- \_[V\_0]{}\_[V\_0]{}+(\_[V\_1]{}\_[V\_[-1]{}]{}+\_[V\_[-1]{}]{}\_[V\_1]{}) is the Casimir commuting with the Lie-derivatives $\mathcal{L}_{V_i}$, and $T_+$ is an appropriate superposition of tensor components. Actually the scalar equation of motion could also be cast into the same form (1.2).
The fact that the equations of motion of all perturbations could be written as (1.2) allows us to construct the quasinormal modes in an uniform way. We start from the highest-weight mode, which not only satisfies the equations of motion but also obeys the condition \_[V\_1]{}\^[(0)]{}=0 , \_[V\_0]{}\^[(0)]{}=h\_R\^[(0)]{}, then construct the descendent modes \^n=(\_[V\_[-1]{}]{})\^n\^[(0)]{}. It is nice to find that the descendent modes constitute an infinite tower of quasinormal modes. We will show that all the information of quasinormal modes is encoded in the hidden conformal symmetry. The frequencies of the quasinormal modes take the following form: \_1\_R\^[(n)]{}=\_2 k+i(h\_R+n\_R), |\_1\_L\^[(n)]{}=|\_2 k+i(h\_L+n\_L)where $n_{L,R}$ are non-negative integers and $\l_i,\bar{\l}_i$ are parameters in the hidden conformal symmetry. The spectrum of all kinds of quasinormal modes share the same structure, with the difference being from the conformal weights which are decided by the $m^2_t$ term.
The way we approach the quasinormal modes is partly motivated by the work in [@Sachs08]. In this paper, Ivo Sachs and Sergey N. Solodukhin showed that quasinormal modes of the BTZ black hole in topologically massive gravity may be derived from the Killing vector fields. The essential aspect is that the Killing vectors form a $SL(2,R)$ Lie algebra locally so that they can build an infinite tower of quasinormal modes. Our treatment is in spirit similar to theirs, but differs in detail. In particular, our investigation on the vector and tensor fields has not been presented anywhere else before, to our knowledge. Moreover our discussion includes the warped AdS$_3$ black hole and self-dual warped AdS$_3$ black hole of topological massive gravity, whose hidden conformal symmetry is nontrivial, in contrast to the BTZ black hole, which is locally isomorphic to AdS$_3$ so that the hidden conformal symmetry is not a real surprise. Actually the equations of motion in the warped spacetime are of the form (\^2+b|\_[|[V\_0]{}]{}\^2+m\_t\^2)T\_+=0, which is slightly different from (1.2) but still allows us to construct the quasinormal modes in the similar way. But now the conformal weight depends not only on the mass but also on the extra quantum numbers.
In the next section, we briefly review the realization of hidden conformal symmetry. In Sec. III, we study the scalar perturbation and determine the quasinormal modes as a warm-up. In Sec. IV, we investigate the action of the hidden conformal symmetry on the vector and gravitational perturbations. In Sec. V, we discuss the quasinormal modes of the BTZ black hole, and reproduce the well-known results. In Sec. VI, we try to generalize the method to the warped $AdS_3$ and self-dual warped $AdS_3$ black holes, which need a minor modification of our construction. We will end with discussions in Sec. VII. Some technical details are put into two appendixes.
Hidden Conformal Symmetry
=========================
In this paper,we will restrict to generic nonextremal black holes which have the hidden conformal symmetry. Generically we may introduce the vector fields V\_0&=&\_1\_t+\_2\_,\
V\_1&=&e\^[\_1 t+\_2 ]{}\[(A +B ) \_t+(C +D )\_+\_r\],\[conformal1\]\
V\_[-1]{}&=&e\^[-\_1 t-\_2 ]{}\[(A+B) \_t+(C +D )\_- \_r\],where $\lambda_1,\lambda_2,\mu_1,\mu_2,A,B,C,D$ are all constants satisfying \_1\_1+\_2\_2=-1,\
\_1=2A,\
\_2=2C,\
\_1 B+\_2 D=0,and $\Delta=(r-r_+)(r-r_-)$, $\Delta'=\frac{d\Delta}{dr}$. The above vector fields form a $SL(2,R)$ algebra. =V\_[1]{}, \[V\_[+1]{},V\_[-1]{}\]=2V\_0 And similarly we can define the left sector $\bar{V}_0,\bar{V}_{\pm1}$ with parameters $\bar{\mu}_i,
\bar{\l}_i,\bar{A},\bar{B},\bar{C},\bar{D}$.
The essential aspect is that the scalar Laplacian can be written as the $SL(2,R)$ quadratic Casimir. More explicitly, the radial scalar field equation in a black hole with holographic description is of the form (V\^2+m\_s\^2)(r)=0, \[vscalar\] where $V^2=-V_0^2+\frac{1}{2}(V_1V_{-1}+V_{-1}V_1)$ is the $SL(2,R)$ quadratic Casimir operator and $m_s^2$ is a constant. This is true for the 4D Kerr(-Newman) black hole in the low frequency and the near region, and is always true for 3D black holes in the whole region. Actually, one can give the explicit form of the Casimir. But we would not give it here, instead we will derive it in the next section in the general framework of Lie-derivative operation.
Scalar Modes
============
In this section we will derive the scalar equation using the Lie derivatives. This seems useless since we have known the results in Sec. II. But we will see that it is valuable to reproduce it in another way, which could be generalized to discuss the vector and tensor modes.
First we define Lie-induced quadratic Casimir \^2- \_[V\_0]{}\_[V\_0]{}+(\_[V\_1]{}\_[V\_[-1]{}]{}+\_[V\_[-1]{}]{}\_[V\_1]{})\[lieCasimir\] where $\mathcal{L}_{V_i},i=0,\pm1$ are the Lie derivatives with respect to the vector fields $V_i$. $\mathcal{L}^2$ is analogue to the $SL(2,R)$ quadratic Casimir $V^2$.
Let $\Phi$ be a scalar field and we immediately have \^2= \^\_\_+\^\_where we have defined \^&&(V\_1\^V\_[-1]{}\^+V\_1\^V\_[-1]{}\^)-V\_0\^V\_0\^, \[Pi\]\
\^&&(V\_1\^\_V\_[-1]{}\^+V\_[-1]{}\^\_V\_1\^)-V\_0\^\_V\_0\^.\[Sigma\] The explicit expressions of $\Pi$’s and $\Sigma$’s can be found in Appendix A. We use them to find \^2=-\_r\_r +\[\_+\^2-\_-\^2\]where $\sigma_\pm=(\pm A(r_+-r_-)+B)\partial_t+(\pm
C(r_+-r_-)+D)\partial_\phi$. Since we focus on the black holes which have a hidden conformal symmetry, the scalar equation can be written formally as (\^2+m\_s\^2)=0 \[lscalar\] where $m_s$ is a constant which is related to the conformal weight of the scalar. It varies for different black holes. Certainly for the scalar, (\[lscalar\]) is exactly the same as (\[vscalar\]).
To construct the tower of scalar quasinormal modes, we first impose the condition: \_[V\_1]{}\^[(0)]{}=0 , \_[V\_0]{}\^[(0)]{}=h\_R\^[(0)]{}\[highestweight\] to define the “highest-weight” mode. Since =\_[\[X,Y\]]{},\_[aX]{}=a\_X \[lie\] where $X,Y$ are arbitrary vectors and $a$ is an arbitrary constant, we get the following relation from the scalar Eq. (\[lscalar\]): h\_R\^2-h\_R-m\_s\^2=0. This determines the conformal weight h\_R=(1+). We have chosen the “$+$" root to simplify our discussion. But the other choice can also be considered easily.
From the mode $\Phi^{(0)}$, we construct an infinite tower of quasinormal scalar modes $\Phi^{(n)}$ as \^[(n)]{}=(\_[V\_[-1]{}]{})\^n\^[(0)]{}, nN. All the $\Phi^{(n)}$ are descendents of the mode $\Phi^{(0)}$. Since the Casimir $\mathcal{L}^2$ commutes with $\mathcal{L}_{V_i}$, $
i=0,\pm1$, $\Phi^{(n)}$ satisfy the scalar equation as well. To compute the frequency of the quasinormal modes, we may expand the scalar as =e\^[-iøt+i k ]{}, as $\p_t$ and $\p_\phi$ are always the Killing vectors of the black holes. For the highest-weight mode $\Phi^{(0)}$, we have $$\lambda_1\omega_0-\lambda_2 k_0=ih_R,$$ where $\omega_0$ and $k_0$ are its frequency and angular momentum. In principle, $k_0$ could be complex in the solution. Taking the highest mode as quasinormal modes require $k_0$ be real. For the descendent mode $\Phi^{(n)}$, we have \_[V\_0]{}\^[(n)]{}=(-i\_1\_R\^[(n)]{}+i\_2 k\^[(n)]{})\^[(n)]{}, where its frequency $\omega_R^{(n)}$ and angular momentum $k_R^{(n)}$ are related to $\o_0$ and $k_0$ via the relation $$\omega_R^{(n)}=\omega_0-in\mu_1,\hs{5ex}
k_R^{(n)}=k_0+in\mu_2.\label{relationRk}$$ To be a well-defined quasinormal mode, the angular momentum $k_R^{(n)}$ should be real, which requires a choice of complex $k_0$. Note that the real part of the $k_0$ and $k_R^{(n)}$ are always the same, taken as $k$. From the relation (\[relationRk\]) and the first relation in (2.2), we obtain \_1\_R\^[(n)]{}=\_2 k+i(h\_R+n). \[qnscalar\] Alternatively it is more convenient to use just the algebraic relation (\[lie\]) to get this relation.
The relation (\[qnscalar\]) gives the frequencies of the scalar quasinormal modes. We find that the frequencies of the modes only depend on the parameters which appear in the hidden conformal symmetry. Our construction relates the hidden conformal symmetry to the structure of quasinormal modes directly.
Note that we can also determine the left sector modes from the other set of vector fields $\{\bar{V}_i\}$ according to the following rules:\
(i) $R\to L$\
(ii)$\lambda_i\to \bar{\lambda}_i,{\mu}_i\to \bar{\mu}_i$, where $i=1,2$.
In the next section, we will see that the similar construction could be applied to the vector and gravitational modes, with subtle modifications.
One can solve the highest-weight condition (\[highestweight\]) explicitly. The solution is just \^[(0)]{}=C\_0(r-r\_+)\^[-a-]{}(r-r\_-)\^[-a+]{}, where $C_0$ is a integration constant and a&=&-iAø+iCk,\
b&=&-iBø+iD k. To satisfy the ingoing boundary condition at the horizon $r=r_+$, we need A+ <0. We will see that this is indeed the case for the black holes studied in this paper. Asymptotically, the solution behaves as \^[(0)]{} \~r\^[-h\_R]{}. So we see that the solution has the right behavior as the quasinormal mode. It is easy to find that the other quasinormal modes have the same asymptotical behavior.
Vector and Tensor Modes
=======================
Let us first consider the vector modes. Motivated by the impressive result on scalar modes, we try to compute $\mathcal{L}^2A_\mu$ and expect a similar structure. However it turns out to be more complicated: \^2A\_=\^\_\_A\_+\^\_A\_+\_\^ A\_+\_\^\_A\_, \[lvector\] where $A_\mu$ is a vector field and $\Pi^{\rho\sigma},\Sigma^\sigma$ are defined in (\[Pi\]) and (\[Sigma\]). $\Upsilon_\mu^{\rho\sigma}$ is defined as \_\^V\_1\^\_V\_[-1]{}\^+V\_[-1]{}\^\_V\_1\^-2V\_0\^\_V\_0\^. At first glance, (\[lvector\]) looks quite different from the scalar Eq. (\[lscalar\]). Especially the fact that the different components are mixed together make things untractable. Nevertheless, we will show that for 3D black holes, the relation (\[lvector\]) could be simplified. The detailed discussion on the vector and the tensor perturbations in 3D black holes could be found in Appendix B.
Notice that the first and the second terms on the right-hand side of (\[lvector\]) are similar to the terms that appeared in the scalar modes. The third term vanishes if we only consider the $A_t$ and $A_\phi$ components since $\Sigma^\sigma$ is only a function of $r$ and independent of $t$ and $\phi$. To focus only on the $A_t$ and $A_\phi$ components is plausible since the $A_r$ component can be determined by the other components in 3D. The only trouble comes from the fourth term, which cannot vanish automatically. The trick is that we should consider the superposition of $A_t$ and $A_\phi$. Let us define: A\_+=\_1A\_t+\_2A\_, where $\k_{1,2}$ are the constants to be determined. We find that a suitable choice of $\kappa_1$ and $\kappa_2$ can make all the components of $\Upsilon_\mu^{\rho\sigma}$ vanish. Actually, if (\_1\_t+\_2\_)V\_i\^=0 where $i=0,\pm1$ and $\sigma=t,\phi,r$, then $\Upsilon_\mu^{\rho\sigma}=0$. The above condition can be satisfied if \_1:\_2=-\_2:\_1. Thus, we get \^2A\_+=\^\_\_A\_++\^\_A\_+. This shows that $A_+$ transform like a scalar. Now the question is if the equation of $A_+$ could be written like a scalar: (\^2+m\_v\^2)A\_+=0. \[lvector2\] Certainly $m_v^2$ may be different from the scalar case, depending on the backgrounds as well. We will show for the 3D black holes in this paper, (\[lvector2\]) is always true.
Next we turn to the tensor fields. For the tensor field $T_{\mu\nu}$, we have \^2 T\_=\^\_\_T\_+\^\_T\_+\_\^T\_+\_\^T\_\
+\_\^T\_+\_\^\_T\_+\_\^\_T\_where we have defined \_\^\_V\_1\^\_V\_[-1]{}\^+\_V\_1\^\_V\_[-1]{}\^-2\_V\_0\^\_V\_0\^. By introducing T\_[+]{}=\_1T\_[tt]{}+\_2T\_[t]{}+\_3T\_[t]{}+\_4T\_, we find that when (\_1\_t+\_2\_)V\_i\^=0,\
(\_1\_t+\_3\_)V\_i\^=0,\
(\_2\_t+\_4\_)V\_i\^=0,\[tensor\]\
(\_3\_t+\_4\_)V\_i\^=0,all the redundant terms vanish and \^2T\_[+]{}=\^\_\_T\_[+]{}+\^\_T\_[+]{}. The condition (\[tensor\]) can be obeyed if the parameters $\mu_i,\k_i$ satisfy the relations \_1:\_2=-\_2:\_1=\_3:\_4,\_2=\_3.\[kappa\] As the vector case, we expect that the equations of motion of the tensor is (\^2+m\_t\^2)T\_[+]{}=0, \[ltensor\] for some constant $m_t$. We will show that for 3D black holes this is the case in the next section.
The above construction may be generalized to the higher-rank tensor fields. In general, for a rank $n$ tensor, we have \_[V]{}T\_[l\_1l\_2l\_n]{}=V\^\_T\_[l\_1l\_2l\_n]{}+\_[l\_1]{}V\^T\_[l\_2l\_n]{}++\_[l\_n]{}V\^T\_[l\_1 l\_2l\_[n-1]{}]{}. We can define a tensor as T\_+=, where the summation is over all $\sigma_i=t,\phi$. Then we can choose the $2^n$ coefficients $\kappa_{\cdots}$ such that \_[V\_i]{}T\_+=V\_i\^\_T\_+ with $i=0,\pm1$. Note that this means that $T_+$ transform as a scalar under $SL(2,R)$. This could be satisfied if (\_[\_1\_[j]{}t\_[j+2]{}\_[n]{}]{}\_t+\_[\_1\_[j]{}\_[j+2]{}\_[n]{}]{}\_)V\^\_i=0. There are $n\cdot 2^{n-1}$ constraints while there are only $2^n$ degrees of freedom. But the above equations are not independent and we can still determine the $2^n$ coefficients. One can begin with $\kappa_{tt\cdots t}$ and end with $\kappa_{\phi\phi\cdots\phi}$ step by step. Then one finds that \^2T\_+=(\^\_\_+\^\_) T\_+ and we wish that the equation of motion of $T_+$ could be written as (\^2+m\_[hs]{}\^2)T\_+=0 with $m_{hs}$ being a constant. In this paper, we just focus on the vector and rank $2$ tensor and leave the general case for a future study.
Before we go into the concrete examples, we would like to discuss the physical implications of (\[lvector2\]) and (\[ltensor\]) on the quasinormal modes. It is not hard to see that if we have the relations (\[lvector2\]) and (\[ltensor\]), all the treatment on the scalar modes could be applied to the vector and tensor modes. That is to say, we can define the “highest-weight” modes $\Psi^{(0)}$, where $\Psi^{(0)}$ can be either $A_+$ or $T_{+}$, as \_[V\_1]{}\^[(0)]{}=0 , \_[V\_0]{}\^[(0)]{}=h\_R\^[(0)]{}. Moreover, since (\[lie\]) holds for arbitrary tensor fields, we can determine the conformal weight to be $h_R=\frac{1}{2}(1+\sqrt{1+4m_i^2})$ with $m_i^2=m_v^2$ or $m_t^2$. Similarly we can construct a tower of quasinormal modes $\Psi^{(n)}$ as \^n=(\_[V\_[-1]{}]{})\^n\^[(0)]{}. The frequency of the quasinormal vector and tensor modes share the same structure as the scalar modes (\[qnscalar\]), with the difference coming from the conformal weights. Certainly we can construct the left sector modes in a similar way.
quasinormal Modes in BTZ Black Hole
===================================
In this section, we take the BTZ black hole as a typical example to illustrate the above constructions of quasinormal modes. The scalar, vector and spinor quasinormal modes of the BTZ black hole were discussed in , while the massive gravitational one in TMG theory was studied in [@Sachs08](see also [@Afshar:2010ii]). The metric of a BTZ black hole is[@BTZ] ds\^2=-dt\^2+dr\^2+r\^2(d-dt)\^2 The left and right moving temperature are T\_L=, T\_R=
From the scalar equation we find the hidden conformal symmetry in the BTZ black hole. In the BTZ case, we should replacing $r$ to $r^2$ and $\partial_r$ to $\partial_{r^2}$ in the conformal coordinates and the vector fields defined in (\[conformal1\]). It turns out the parameters in (\[conformal1\]) should be \_1=-\_2=-,\_1=-\_2=2T\_R,\
A=-C=-,B=D=-\
|[\_1]{}=|[\_2]{}=-,|[\_1]{}=|[\_2]{}=2T\_L,\
|[A]{}=|[C]{}=-,|[B]{}=-|[D]{}=-and we can also find that m\_s\^2=m\^2 where $m$ is the scalar mass. By substituting $\lambda_i$ into (\[qnscalar\]), we find \_R\^[(n)]{}=-k-i4T\_R(n\_R+h\_R),\_L\^[(n)]{}=k-i4T\_L(n\_L+h\_L),n\_L,n\_R\[BTZscalar\] where $h_L=h_R=\frac{1}{2}(1+\sqrt{1+m^2})$. This is in complete agreement with [@Briminghan01].
To check the vector modes, we begin with the vector field equation in three-dimensional spacetime: \_\^\_A\_=-mA\_. We can show that for the BTZ black hole, the above equation can be written as A\_t=m\^2A\_t+2mA\_,\
A\_=m\^2A\_+2mA\_t.where we have defined the operator $\tilde{\Delta}=\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}$, which is an analogue to the Laplacian operator acting on the scalar field. See Appendix B.1 for more details. We immediately get A\_=(m\^22m)A\_where $A_\pm=A_t\pm
A_{\phi}$. Note that this is just what we want. Using the language in the above section, as $-\mu_2:\mu_1=1$ and $-\bar{\mu}_2:\bar{\mu}_1=-1$, we may choose $\kappa_1=\kappa_2=1$ and $\bar{\kappa}_1=-\bar{\kappa}_2=1$. Hence $A_\pm$ transform like a scalar mode and m\_v\^2=(m\^2+2m) for the right-moving sector and |[m]{}\_v\^2=(m\^2-2m) for the left-moving sector. Then we get h\_R=+1, h\_L= which is in agreement with the general result that $|h_L-h_R|=s$. The frequencies of the quasinormal vector modes are still given by (\[BTZscalar\]).
Next, we turn to the gravitational modes. For the standard 3D gravity, there is no propagating gravitational mode. However, for the topological massive gravity, there is a massive graviton, whose equation of motion could be written as a linear equation[@Sachs08] \_\^\_h\_+mh\_=0. Analogue to the vector mode, we can show that for the BTZ black hole, h\_[tt]{}&=&m\^2h\_[tt]{}+2mh\_[t]{}+2mh\_[t]{}+h\_[tt]{}+2h\_,\
h\_[t]{}&=&m\^2h\_[t]{}+2mh\_+2mh\_[tt]{}+h\_[t]{}+2h\_[t]{},\
h\_&=&m\^2h\_+2mh\_[t]{}+2mh\_[t]{}+h\_+2h\_[tt]{}.See Appendix B for more detail. After defining $h_{\pm}=h_{tt}\pm h_{t\phi}\pm h_{\phi
t}+h_{\phi\phi}$, we get h\_=(m\^24m+3)h\_ The above equations are precisely what we expect. Since in this case, we should choose $\kappa_1=\kappa_2=\kappa_3=\kappa_4=1$ and $\bar{\kappa}_1=-\bar{\kappa}_2=-\bar{\kappa}_3=\bar{\kappa}_4=1$. Then $h_\pm$ are just the $T_{\pm}$ we have defined in the previous section. Consequently, we find that m\_t\^2=(m\^2+4m+3) for the right-moving sector and |[m]{}\_t\^2=(m\^2-4m+3) for the left-moving sector. The right and left conformal weight are respectively h\_R=,h\_L=, \[tensorweight\]which again is consistent with the fact that $|h_L-h_R|=s$. The frequencies of the gravitational quasinormal modes take the same form as (\[BTZscalar\]).
quasinormal Modes in Warped $AdS_3$ and Self-dual Warped $AdS_3$ Black Hole
===========================================================================
In this section, we will generalize the algebraic method to the warped $AdS_3$ and the self-dual warped $Ad S_3$ black holes. For the warped black holes, the scalar equations could not be simply written as (\[lscalar\]). Actually they take the following form: (\^2+b|\_[|[V]{}\_0]{}\^2+m\_s\^2)=0 with $b$ and $m_s^2$ being constants. This is a little different from the previous discussion due to the presence of the $b\bar{\mathcal{L}}_{\bar{V_0}}^2$ term. Nevertheless, we can still construct a tower of right-moving modes by imposing the conditions \_[V\_1]{}\^[(0)]{}=0,\_[V\_0]{}\^[(0)]{}=h\_R\^[(0)]{},\^[(n)]{}=(\_[V\_[-1]{}]{})\^n\^[(0)]{}. The first two conditions just define the “highest-weight” mode. And the last equation construct the descendent modes. Because of the commutative relation $[\mathcal{L}_{V_{-1}},\bar{\mathcal{L}}_{\bar{V_0}}]=0$, all the modes $\Phi^{(n)}$ satisfy the scalar equation as well. The following discussion is similar to the one in Sec. III. Here we only give the results: h\_R=(1+), \_1\_R\^[(n)]{}=\_2 k+i(h\_R+n) where $q$ is defined by $\bar{\mathcal{L}}_{\bar{V_0}}\Phi^{(0)}=q\Phi^{(0)}$. However,we can not construct the left-moving modes due to the noncommutative relation of $\bar{\mathcal{L}}_{\bar{V_0}}$ and $\bar{\mathcal{L}}_{\bar{V}_{-1}}$. Because of the presence of $b\bar{\mathcal{L}}_{\bar{V_0}}^2$ term in the scalar equation, the conformal weight depends on the quantum number $q$. This fact is in consistency with the known result.
Next, we try to generalize the above discussion to the vector modes. In this case, we find that for any vector $A_{\mu}$ |\_[|[V\_0]{}]{}\^2A\_=|[V]{}\_0\^\_|[V]{}\_0\^\_A\_. All the redundant term of $\partial_{\mu}\bar{V}_0^{\lambda}$ vanish since $\bar{V}_0^{\lambda}$ are constant numbers. This implies that we can still define $A_+=\kappa_1A_t+\kappa_2A_{\phi}$ with $\kappa_1:\kappa_2=-\mu_2:\mu_1$. We still expect that it transforms as a scalar. More explicitly, we wish (\^2+b|\_[|[V\_0]{}]{}\^2+m\_v\^2)A\_+=0. If this is true, we can discuss the vector modes parallel to the treatment on the scalar modes. We will check this point in the warped $AdS_3$ and self-dual warped $AdS_3$ black hole backgrounds in the next two subsections.
For the warped spacetime, the equation of motion of the gravitational mode could not be written as a linear equation[@Anninos:2009zi] and is much more involved. Here we just assume that there is a massive rank 2 symmetric tensor mode in the warped spacetime. In 3D dimension, its equation of motion is \_\^\_h\_+mh\_=0. In this case, we can still define $T_+=\kappa_1T_{tt}+\kappa_2T_{t\phi}+\kappa_3T_{\phi
t}+\kappa_4T_{\phi\phi}$ with $\k_i$’s satisfying (\[kappa\]) and wish it to satisfy the equation of the form (\^2+b|\_[|[V\_0]{}]{}\^2+m\_t\^2)T\_+=0. If this is true, it allows us to construct the tensor quasinormal modes in the similar way.
Warped $AdS_3$ black hole
--------------------------
The metric of the spacelike stretched warped $AdS_3$ black hole is[@Andy08] ds\^2=dt\^2+2M(r)dtd+N(r)d\^2+Q(r)dr\^2 where M&=&r-,\
N&=&\[3(\^2-1)r+(\^2+3)(r\_++r\_-)-4\],\[warpedmetric\]\
Q&=&.From warped AdS/CFT correspondence, the right- and left-moving temperatures in the dual 2D CFT are T\_L=(r\_++r\_–),T\_R=. The hidden conformal symmetry of the warped AdS$_3$ black hole has been discussed in [@Fareghbal:2010yd]. From the scalar equation, we find that \_1=-,\_2=,\_1=0,\_2=-2T\_R\
A=-,B=-,C=,D=0\
|\_1=-,|\_2=0,|\_1=,|\_2=2T\_L\
|[A]{}=-,|[B]{}=,|[C]{}=0,|[D]{}=and $b,q,m_s^2$ are b=,q=i,m\_s\^2= where $m$ is the scalar mass. Hence the scalar conformal weight is h\_R=(1+) As emphasized in [@ChenXu2], to compare with the poles of the correlation functions in the dual CFT, we should take the following identification on quantum numbers into account[@ChenXu2] =,=, where $\tilde{\o},\tilde{k}$ are the quantum numbers of global warped AdS$_3$ spacetime. Then we find the scalar quasinormal modes with the frequencies\_R\^[(n)]{}=(-4T\_L-i4T\_R(n+h\_R)). \[QNwarped\]This is in agreement with the result in .
Next, we check the vector modes. Since in this case, $\mu_1=0$ indicates $\kappa_1=1, \kappa_2=0$, we should choose $A_+=A_t$. In Appendix B.1 we show that $A_t$ satisfy A\_t=(m\^2+2m)A\_t, \[warpedvector\] which could be rewritten as (\^2+b|\_[|[V\_0]{}]{}\^2+m\_v\^2)A\_+=0 with $m_v^2=\frac{m^2+2m\nu}{\nu^2+3}$ and $b$ has been given in the scalar case. This is in agreement with our expectation. Hence, the vector conformal weight is h\_R=(1+) where we have used the identification $\tilde{k}=\frac{2\nu\omega}{\nu^2+3}$. The result is in perfect match with the result in [@ChenXu2]. The spectrum of the vector quasinormal modes takes the same form as (\[QNwarped\]).
For the tensor mode, as $\mu_1=0, \mu_2\neq 0$, we may choose \_1=1, \_2=\_3=\_4=0 and have T\_+=h\_[tt]{}.From the equation of motion, we learn that h\_[tt]{}=(m\^2+4m+3\^2)h\_[tt]{} which could be rewritten as (\^2+b|\_[|[V\_0]{}]{}\^2+m\_t\^2)T\_+=0 with $m_t^2=\frac{m^2+4m\nu+3\nu^2}{\nu^2+3}$. Thus we find the conformal weight of the tensor mode h\_R=(1+). When $\nu=1$, the warped black hole reduces to the BTZ black hole and the tensor conformal weight reduces to $h_R$ in (\[tensorweight\]). The spectrum of tensor quasinormal modes takes the same form as (\[QNwarped\]).
Self-dual warped $AdS_3$ black hole
-----------------------------------
The self-dual warped AdS$_3$ black hole is a vacuum solution of 3D topological massive gravity. It could be described by the metric \[selfBH\] ds\^2 = (-(r-r\_+)(r-r\_-)dt\^2 + dr\^2.\
.+ ( d+ (r-) dt)\^2), where the coordinates range as $t\in[-\infty,\infty]$, ${r}\in[-\infty,\infty]$ and $\phi \sim \phi + 2\pi$.
The hidden conformal symmetry of this black hole has been discussed in [@Li:2010zr][^1]. It turns out that the parameters in the vector fields take the following values \_1=-,\_2=0,\_1=2T\_R,\_2=0,\
A=-,B=0,C=0,D=,\
|\_1=0,|\_2=-,|\_1=0,|\_2=2T\_L,\
|[A]{}=0,|[B]{}=1,|[C]{}=-,|[D]{}=0,where the right- and left- moving temperature are T\_L=, T\_R=. The $b,q,m_s^2$ are b=,q=-,m\_s\^2=. The conformal weight for the scalar is just h\_R=(1+).
For the vector field, as $\mu_2=0$, we may choose $\l_2=1$ so that $A_+=A_{\phi}$. In Appendix B.1, we see that $A_{\phi}$ satisfy A\_=(m\^2-2m)A\_, which could be cast into the form (\^2+b|\_[|[V\_0]{}]{}\^2+m\_v\^2)A\_+=0 with $m^2_v=\frac{m^2-2m\nu}{\nu^2+3}$. The conformal weight for the vector is then h\_R=(1+).
For the tensor field, as $\mu_2=0, \mu_1\neq 0$, we may choose \_4=1, \_1=\_2=\_3=0 and have T\_+=h\_.From the equation of motion, we learn that h\_=(m\^2-4m+3\^2)h\_[tt]{} which could be rewritten as (\^2+b|\_[|[V\_0]{}]{}\^2+m\_t\^2)T\_+=0 with $m_t^2=\frac{m^2-4m\nu+3\nu^2}{\nu^2+3}$. The conformal weight of the tensor field is just h\_R=(1+), which could reduce to $h_L$ in (\[tensorweight\]) at $\nu=1$.
In all cases, the quasinormal modes can be written as \_R\^[(n)]{}=-i2T\_R(n+h\_R) where $h_R$ can be the scalar, the vector or the tensor conformal weight. The results all in good agreement with .
Discussions
===========
In this paper, we have studied the relation between the hidden conformal symmetry and the quasinormal modes. We found that the spectrum of the quasinormal modes may be directly read out from the action of the hidden conformal symmetry on various perturbations. Our construction provides a direct rule to find the spectrum. The rule is simple and show clearly that the quasinormal modes are determined completely by the hidden conformal symmetry. We found that in the spectrums, ø-i2T(h+n), which is in accordance with the structure of the poles of the correlation functions of the dual operators in CFT.
Our construction is based on the relation (\[lie\]) on the Lie-derivatives and the fact that the Lie-induced Casimir $\mathcal{L}^2$ defined in (\[lieCasimir\]) commutes with the Lie-derivatives. Starting from the highest-weight mode, we can construct its infinite tower of descendent modes. For the scalar, the construction is straightforward, as shown in Sec. III. However, the action of the hidden conformal symmetry on the vector and tensor field is highly nontrivial. We observed that only after some suitable composition the vector and the gravitation modes behaved like the scalar modes. This allowed us to treat the scalar, vector and gravitational modes in a uniform way. From our construction, the spectrum of various kinds of quasinormal modes are in agreement with the CFT prediction and previous study. Moreover, our discussion in Sec. IV suggested that our treatment could be applied to higher-rank tensor fields. It would be nice to have a detailed study on this question. Another interesting issue is to study if the hidden conformal symmetry can determine the fermionic quasinormal modes.
We applied our method to the case of the BTZ black hole and find perfect agreement with the known results. For the warped AdS$_3$ and self-dual warped AdS$_3$ black holes, the discussion is subtler. Even for the scalar mode, the scalar equation could not be simply written as the $SL(2,R)$ quadratic Casimir for all quantum numbers. Nevertheless, we can still apply our treatment with a minor modification. For all the scalar, vector and tensor modes, we managed to construct towers of the quasinormal modes, in agreement with the ones found in the literature. Strictly speaking, we only succeeded in finding one set of the quasinormal modes, corresponding to the poles of the correlation functions of the right-moving sector in the dual CFT. It would be nice to find the other set, corresponding to the left-moving ones.
In this paper, we studied the quasinormal modes of the nonextremal black holes. Since the coordinates that are used to implement the hidden conformal symmetry are different in the extremal case[@Chen:2010fr], it is interesting to see if the same construction works for the extreme black holes. We expect an similar conclusion.
We discussed the action of the hidden conformal symmetry on the vector and tensor fields in three-dimensional spacetime. It would be interesting to investigate this issue in the Kerr/CFT correspondence in higher dimensions. However in this case, the problem is much more complicated because we have to apply the Newman-Penrose formalism to obtain the Teukolsky master equation of high spin perturbations. It is not clear how the hidden conformal symmetry is realized in this framework.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work was in part supported by NSFC Grant No. 10775002, 10975005. We would like to thank for KITPC for hospitality, where this project was initiated.
Appendix A {#appendix-a .unnumbered}
==========
The explicit forms of $\Pi^{\rho\sigma}$ and $\Sigma^\sigma$ are the following: \^[rr]{}&=&-\
\^[tt]{}&=&(A+B)\^2-\_1\^2\
\^&=&(C+D)\^2-\_2\^2\
\^[t]{}&=&\^[t]{}=(AB)(C+D)-\_1\_2\
\^[rt]{}&=&\^[rt]{}=\^[r]{}=\^[r]{}=0and \^t=\^=0, \^r=-’.
Appendix B:Vector and Tensor Perturbation In (2+1)-dim. Black Holes {#appendix-bvector-and-tensor-perturbation-in-21-dim.-black-holes .unnumbered}
===================================================================
In this section, we give a discussion of the vector and the tensor perturbations in (2+1)-dim. black holes. The discussion is not restricted to the black holes studied in this paper. In fact,we only require the following conditions on the metric: \_tg\_=0,\_g\_=0,g\_[rt]{}=g\_[r]{}=0.\[metriccon\]
### B.1 Vector perturbation {#b.1-vector-perturbation .unnumbered}
We begin with the vector equation in 3D spacetime \_\^\_A\_=-mA\_, which could be written in components A\_r&=&-\_r\^[t]{}(\_rA\_-\_A\_t),\
\_rA\_t&=&\_tA\_r-m,\
\_rA\_&=&\_A\_r-m.Obviously the $A_r$ component could be decided by $A_t$ and $A_\phi$. Our goal is to find an equation which is analogue to the scalar equation \_g\^\_=. This motivates us to compute $\tilde{\Delta}A_i=\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}A_i$, with $i=t,\phi$. The results are A\_t&=&m\^2A\_t+m\^[tr]{}(g\_[t]{}’A\_t-g\_[tt]{}’A\_),\
A\_&=&m\^2A\_+m\^[rt]{}(g\_[t]{}’A\_-g\_’A\_t),where $\epsilon_{\lambda}^{\mu\nu}$ is the Levi-Civita tensor and $\tilde{\epsilon}^{tr\phi}$ is the Levi-Civita symbol with $\tilde{\epsilon}^{tr\phi}=1$.
For the BTZ black hole, all the $r$ coordinates should be replaced by $r^2$, and then g\_[t]{}’=0,g\_[tt]{}’=-1,g\_’=1,=. Note that the derivative should be taken with respect to $r^2$. Then we get A\_t=m\^2A\_t+2mA\_,\
A\_=m\^2A\_+2mA\_t.
For the spacelike stretched warped AdS$_3$ black hole, since $g_{tt}'=0, g_{t\phi}'=\nu, \sqrt{-g}=\frac{1}{2}$, we find that A\_t=(m\^2+2m)A\_t. For the self-dual warped AdS$_3$ black hole, $g_{t\phi}'=\frac{4\nu^2\alpha}{(\nu^2+3)^2},g_{\phi\phi}'=0,\sqrt{-g}=\frac{2\nu \alpha}{(\nu^2+3)^2}$, then $A_{\phi}$ satisfy A\_=(m\^2-2m)A\_.
### B.2 Tensor perturbation {#b.2-tensor-perturbation .unnumbered}
In 3D spacetime, the rank 2 symmetric tensor perturbation obeys the equation \_\^\_h\_+mh\_=0. For the BTZ black hole in 3D TMG theory, this is the equation for a massive graviton. However for the warped AdS spacetime, the gravitational perturbation could not be put into such a simple form[@Anninos:2009zi]. Nevertheless we can still assume a massive tensor perturbation in the backgrounds, satisfying this equation.
From this equation, we can easily find h\_’&=&\_h\_[r]{}+\_[()]{}+m\_[()]{},\
h\_[t]{}’&=&\_t h\_[r]{}+\_[(t)]{}+m\_[(t)]{},\
h\_[t]{}’&=&\_h\_[rt]{}+\_[(t)]{}+m\_[(t)]{},\
h\_[tt]{}’&=&\_[t]{}h\_[rt]{}+\_[(tt)]{}+m\_[(tt)]{},and h\_[rt]{}&=&(a\_[rt]{}+m\_[(rt)]{}),\
h\_[r]{}&=&(a\_[r]{}+m\_[(r)]{}),\
h\_[rr]{}&=&g\_[rr]{}\^2(\^[rt]{})\^2(g\_[tt]{}h\_-2g\_[t]{}h\_[t]{}+g\_h\_[tt]{}),where we have defined \_[()]{}&=&\^\_[r]{}h\_-\^\_h\_[r]{}\
\_[(t)]{}&=&\^\_[r]{}h\_[t]{}-\^\_[t]{}h\_[r]{}\
\_[(t)]{}&=&\^\_[rt]{}h\_-\^\_[t]{}h\_[r]{}\
\_[(tt)]{}&=&\^\_[rt]{}h\_[t]{}-\^\_[tt]{}h\_[r]{}\
m\_[()]{}&=&mg\_[rr]{}\^[rt]{}(g\_[t]{}h\_-g\_h\_[t]{})\
m\_[(t)]{}&=&mg\_[rr]{}\^[tr]{}(g\_[t]{}h\_[t]{}-g\_[tt]{}h\_)\
m\_[(t)]{}&=&mg\_[rr]{}\^[rt]{}(g\_[t]{}h\_[t]{}-g\_h\_[tt]{})\
m\_[(tt)]{}&=&mg\_[rr]{}\^[tr]{}(g\_[t]{}h\_[tt]{}-g\_[tt]{}h\_[t]{})\
m\_[(rt)]{}&=&-mg\_[rr]{}\^[rt]{}(\_th\_[t]{}-\_h\_[tt]{})\
m\_[(r)]{}&=&-mg\_[rr]{}\^[rt]{}(\_th\_-\_h\_[t]{})\
W&=&m\^2-(\^[rt]{})\^2(g\_[t]{}’g\_[t]{}’-g\_[tt]{}’g\_’) and a\_[rt]{}&=&-g\_[rr]{}(\^[rt]{})\^2\[-g\_[tt]{}’(\_t h\_-\_h\_[t]{})+g\_[t]{}’(\_th\_[t]{}-\_h\_[tt]{})\]\
a\_[r]{}&=&-g\_[rr]{}(\^[rt]{})\^2\[-g\_’(\_ h\_[tt]{}-\_[t]{}h\_[t]{})+g\_[t]{}’(\_h\_[t]{}-\_[t]{}h\_)\].Similar to the vector case, we find the following equations: h\_&=&m\^2h\_+\_+h\_+\_+(I)\_+(II)\_,\
h\_[t]{}&=&m\^2h\_[t]{}+\_[t]{}+h\_[t]{}+\_[t]{}+(I)\_[t]{}+(II)\_[t]{},\
h\_[t]{}&=&m\^2h\_[t]{}+\_[t]{}+h\_[t]{}+\_[t]{}+(I)\_[t]{}+(II)\_[t]{},\
h\_[tt]{}&=&m\^2h\_[tt]{}+\_[tt]{}+h\_[tt]{}+\_[tt]{}+(I)\_[tt]{}+(II)\_[tt]{},where \_&=&2g\_[t]{}’h\_-2g\_’h\_[t]{}\
\_[t]{}&=&-g\_’h\_[tt]{}+g\_[tt]{}’h\_\
\_[t]{}&=&-g\_[tt]{}’h\_+g\_’h\_[tt]{}\
\_&=&2g\_[t]{}’h\_[tt]{}-2g\_[tt]{}’h\_[t]{}\_&=&g\_[t]{}’g\_[t]{}’h\_-g\_’g\_[t]{}’h\_[t]{}-g\_’g\_[t]{}’h\_[t]{}+g\_’g\_’h\_[tt]{}\
\_[t]{}&=&g\_[t]{}’g\_[tt]{}’h\_-g\_’g\_[tt]{}’h\_[t]{}-g\_[t]{}’g\_[t]{}’h\_[t]{}+g\_’g\_[t]{}’h\_[tt]{}\
\_[t]{}&=&g\_[t]{}’g\_[tt]{}’h\_-g\_’g\_[tt]{}’h\_[t]{}-g\_[t]{}’g\_[t]{}’h\_[t]{}+g\_[t]{}’g\_’h\_[tt]{}\
\_[tt]{}&=&g\_[tt]{}’g\_[tt]{}’h\_-g\_[tt]{}’g\_[t]{}’h\_[t]{}-g\_[tt]{}’g\_[t]{}’h\_[t]{}+g\_[t]{}’g\_[t]{}’h\_[tt]{}(I)\_&=&\[-(\_r)\_(\_th\_[t]{}-\_h\_[tt]{})+(\_r)\_(\_th\_-\_h\_[t]{})\]\
(I)\_[t]{}&=&\[-(\_r)\_t(\_th\_[t]{}-\_h\_[tt]{})+(\_r)\_t(\_th\_-\_h\_[t]{})\]\
(I)\_[t]{}&=&\[-(\_r)\_(\_h\_[t]{}-\_[t]{}h\_)+(\_r)\_(\_h\_[tt]{}-\_[t]{}h\_[t]{})\]\
(I)\_[tt]{}&=&\[-(\_r)\_[t]{}(\_h\_[t]{}-\_[t]{}h\_)+(\_r)\_[t]{}(\_h\_[tt]{}-\_[t]{}h\_[t]{})\](II)\_&=&(\_r)(-g\_h\_[t]{}+g\_[t]{}h\_)-(\_r)(g\_[t]{}h\_[t]{}-g\_h\_[tt]{})\
(II)\_[t]{}&=&(\_r)(-g\_[t]{}h\_[t t]{}+g\_[tt]{}h\_[t]{})-(\_r)(g\_[t]{}h\_[t]{}-g\_[tt]{}h\_)\
(II)\_[t]{}&=&(\_r)(-g\_[t]{}h\_+g\_h\_[t]{})-(\_r)(g\_[t]{}h\_[t]{}-g\_h\_[tt]{})\
(II)\_[tt]{}&=&(\_r)(-g\_[tt]{}h\_[t]{}+g\_[t]{}h\_[tt]{})-(\_r)(g\_[t]{}h\_[t]{}-g\_[tt]{}h\_)and $\beta$ is defined as =g\_[t]{}’g\_[t]{}’-g\_[tt]{}’g\_’. The above equations are our main results for the rank 2 tensor perturbations in three-dimensional black hole backgrounds satisfying the conditions (\[metriccon\]).
For the BTZ black hole, we replace $r$ to $r^2$ and $\partial_r$ to $\partial_{r^2}$, then we find (I)\_[ij]{}=(II)\_[ij]{}=0,=1,=\
\_=h\_[tt]{},\_[t]{}=h\_[t]{},\_[t]{}=h\_[t]{},\_[rr]{}=h\_\
\_=-2h\_[t]{},\_[t]{}=-h\_-h\_[tt]{},\_[t]{}=h\_[tt]{}+h\_,\_[tt]{}=2h\_[t]{}such that h\_[tt]{}&=&m\^2h\_[tt]{}+2mh\_[t]{}+2mh\_[t]{}+h\_[tt]{}+2h\_,\
h\_[t]{}&=&m\^2h\_[t]{}+2mh\_+2mh\_[tt]{}+h\_[t]{}+2h\_[t]{},\
h\_[t]{}&=&m\^2h\_[t]{}+2mh\_+2mh\_[tt]{}+h\_[t]{}+2h\_[t]{},\
h\_&=&m\^2h\_+2mh\_[t]{}+2mh\_[t]{}+h\_+2h\_[tt]{}.
For the warped spacetimes, the discussion is similar but more tedious. It turns out that for the spacelike stretched AdS$_3$ black hole, the component $h_{tt}$ obeys the equation h\_[tt]{}=(m\^2+4m+3\^2)h\_[tt]{}, while for the self-dual AdS$_3$ black hole, the component $h_{\phi\phi}$ obeys h\_=(m\^2-4m+3\^2)h\_.
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[^1]: The scalar quasinormal mode of self-dual black hole has been discussed in [@Li:2010zr] in the similar way as the one in [@Sachs08].
|
---
abstract: 'The process of heralded noiseless amplification, and the inverse process of heralded noiseless attenuation, have potential applications in the context of quantum communications. Although several different physical implementations of heralded noiseless amplifiers have now been demonstrated, the research on heralded noiseless attenuators has been largely confined to a beam-splitter based approach. Here we show that an optical parametric amplifier (OPA), combined with appropriate heralding, can also serve as a heralded noiseless attenuator. The counterintuitive use of an optical amplifier as an attenuator is only possible due to the probabilistic nature of the device.'
author:
- 'R.A. Brewster'
- 'I.C. Nodurft'
- 'T.B. Pittman'
- 'J.D. Franson'
title: Noiseless attenuation using an optical parametric amplifier
---
Although quantum optical signals cannot be deterministically amplified without adding noise [@caves1982], it has recently been shown that non-deterministic noiseless amplification is possible [@ralph2009]. Broadly speaking, the idea is related to the linear optics quantum computing paradigm [@knill2001], where heralding signals are used to indicate the successful operation of a probabilistic device. This type of heralded noiseless amplifier has led to a large number of implementations and applications (see, for example, [@xiang2010; @ferreyrol2010; @zavatta2011; @gisin2010; @curty2011; @pitkanen2011; @minar2012; @scott2013; @bruno2013; @chrzanowski2014; @ulanov2015; @donaldson2015]). Somewhat surprisingly, the inverse process of heralded noiseless attenuation is also significant [@micuda2012; @gagatsos2014; @zhao2017]. In fact, Micuda [*et.al.*]{} have shown that noiseless amplification and noiseless attenuation can be combined to conditionally suppress the effects of loss in a quantum communication channel [@micuda2012].
At the present time, heralded noiseless attenuators have only been studied using a beam-splitter based approach [@micuda2012; @gagatsos2014]. This opens the question of whether other systems may be useful for this application. In this brief paper, we show that a conventional optical parametric amplifier (OPA) –when combined with appropriate heralding– can be used as a heralded noiseless attenuator. This unconventional use of an amplifier as an attenuator is somewhat related to the idea that the annihilation operator $\hat{a}$ can actually increase the average number of photons for certain states [@mizrahi2002]. It can also be viewed as an example of a more general equivalence between beam-splitters and OPA’s in conditional measurements [@ban1997; @barnett1999].
![(Color online) Two implementations of a heralded noiseless attenuator for coherent states. (a) shows a conventional beam-splitter based approach [@micuda2012; @gagatsos2014]. (b) shows the heralded noiseless parametric attenuator (NPA) formed by seeding a parametric down-converter (PDC) with the input state and vacuum. In both cases, the successful operation of the attenuator is heralded by detecting exactly zero photons in an auxiliary detector. The similar topologies of (a) and (b) highlights the similarities between multiphoton interference at a beam-splitter and stimulated emission [@sun2007].[]{data-label="fig:fig1"}](fig1.pdf){width="3.25in"}
An overview of the basic idea is shown in Figure \[fig:fig1\]. First, recall that the goal of the noiseless amplifer is to increase the amplitude of a coherent state by the transformation $|\alpha\rangle \rightarrow |g\alpha\rangle$, where $g\geq1$ is the gain and certain limits apply [@ralph2009; @pandey2013]. In analogy, the goal of the noiseless attenuator is to reduce the amplitude as $|\alpha\rangle \rightarrow |\nu\alpha\rangle$, where $\nu\leq1$ is the attenuation parameter. In the beam-splitter based approach shown in Figure \[fig:fig1\](a), the input state and a vacuum state are mixed at a beam-splitter with amplitude transmittance $\nu < 1$, and the output state is only accepted when an auxiliary single-photon detector registers exactly zero photons [@micuda2012; @gagatsos2014]. In this case, the heralded coherent state attenuation occurs because the post-selected probability amplitude of the Fock states $|n\rangle$ in the number-state expansion is reduced by a factor of $\nu^{n}$ [@micuda2012].
In contrast, Figure \[fig:fig1\](b) shows the design of the noiseless attenuator considered in this paper. This basic arrangement of an OPA with heralding on one of the output modes has been extensively studied within the context of general conditional quantum state engineering by multiphoton addition [@sperling2014]. Here we specifically focus on the case of heralding on zero photons to realize a noiseless attenuator, which we refer to as a heralded noiseless parametric attenuator (NPA).
As shown in Figure \[fig:fig1\](b), the input state and a vacuum state are used to seed the signal and idler inputs of a parametric down-converter (PDC), and the output state is only accepted when the auxiliary detector registers exactly zero photons in the idler mode. In this case, the desired attenuation essentially occurs because higher number Fock states $|n\rangle$ are increasingly more likely to stimulate the emission of a signal-idler photon pair in the PDC process [@sun2007]. Consequently as $|n\rangle$ increases, the more likely it is to produce an idler photon, which renders it less likely to “survive” the heralding process. This results in a corresponding decrease in the relative amplitudes of these terms in the coherent state expansion, and a corresponding overall attenuation of the coherent state amplitude.
In order to explicitly calculate this attenuation, we use a relatively simple method based on the time evolution (two-mode squeezing) operator for an OPA given by $$\hat{S}(r)={{e}^{r(\hat{a}\hat{b}-{{{\hat{a}}}^{\dagger }}{{{\hat{b}}}^{\dagger }})}},
\label{eq:squeezeOp}$$ where $r=\kappa t$, $\kappa$ is the coupling between the pump and the signal and idler modes, $t$ is the time, and $\hat{a}$ and $\hat{b}$ are the annihilation operators for the signal and idler modes respectively [@agarwalBook]. It has been shown by Schumaker and Caves that equation (\[eq:squeezeOp\]) can be rewritten in a factored form in the following way [@schumaker1985; @caves2012] $$\hat{S}(r)=\frac{1}{g}{{e}^{-\sqrt{{{g}^{2}}-1}{{{\hat{a}}}^{\dagger }}{{{\hat{b}}}^{\dagger }}/g}}{{g}^{-({{{\hat{a}}}^{\dagger }}\hat{a}+{{{\hat{b}}}^{\dagger }}\hat{b})}}{{e}^{\sqrt{{{g}^{2}}-1}\hat{a}\hat{b}/g}},
\label{eq:factoredOp}$$ where $g=\cosh(r)$ is the gain of the OPA.
The idler mode will be assumed to initially be in its vacuum state ${{\left| 0 \right\rangle }_{i}}$. The unitary transformation $\hat{S}(r)$ followed by the heralding process gives the projection ${{\left| 0 \right\rangle }_{i}}\left\langle {{0}_{i}} \right|\hat{S}(r){{\left| \alpha \right\rangle }_{s}}{{\left| 0 \right\rangle }_{i}}$. In evaluating this expression, the last exponential on the right-hand side of equation (\[eq:factoredOp\]) reduces to $${{e}^{\sqrt{{{g}^{2}}-1}\hat{a}\hat{b}/g}}{{\left| \alpha \right\rangle }_{s}}{{\left| 0 \right\rangle }_{i}}={{\left| \alpha \right\rangle }_{s}}{{\left| 0 \right\rangle }_{i}},
\label{eq:identityExample}$$ since $\hat{b}|0\rangle_i=0$. In the same way, the first exponential on the right-hand side of equation (\[eq:factoredOp\]) reduces to the identity operator when acting to the left, since $\langle0|_i\hat{b}^\dagger=0$.
Combining equations (\[eq:factoredOp\]) and (\[eq:identityExample\]) with the usual expression for a coherent state gives the heralded state of the output mode as $$\begin{split}
& {{\left\langle 0 \right|}_{i}}\hat{S}(r){{\left| \alpha \right\rangle }_{s}}{{\left| 0 \right\rangle }_{i}} \\
& =\frac{1}{g}{{e}^{-{{\left| \alpha \right|}^{2}}/2}}\sum\limits_{n=0}^{\infty }{\frac{{{\alpha }^{n}}}{\sqrt{n!}}{{g}^{-{{{\hat{a}}}^{\dagger }}\hat{a}}}}{{\left| n \right\rangle }_{s}} \\
& =\frac{1}{g}{{e}^{-{{\left| \alpha \right|}^{2}}/2}}\sum\limits_{n=0}^{\infty }{\frac{1}{\sqrt{n!}}{{\left( \frac{\alpha }{g} \right)}^{n}}}{{\left| n \right\rangle }_{s}}. \\
\end{split}
\label{eq:calculation}$$ The unnormalized state $|\psi\rangle_\text{out}$ of the output mode can be rewritten as $${{\left| \psi \right\rangle }_{\text{out}}}=\frac{1}{g}{{e}^{-({{g}^{2}}-1){{\left| \alpha \right|}^{2}}/2{{g}^{2}}}}{{\left| \alpha /g \right\rangle }_{s}}.
\label{eq:cohOut}$$ If we define the attenuation parameter by $\nu =1/g$, then it can be seen from equation (\[eq:cohOut\]) that the NPA results in the transformation $|\alpha \rangle \to |\nu \alpha \rangle $. The probability ${{P}_{s}}$ of success is just the square of the coefficient in front of the state in Eq. (5), which gives $${{P}_{s}}=\frac{1}{{{g}^{2}}}{{e}^{-({{g}^{2}}-1){{\left| \alpha \right|}^{2}}/{{g}^{2}}}}.
\label{eq:cohProb}$$ Equation (\[eq:cohOut\]) is somewhat remarkable in that the heralding process converts a coherent state into an attenuated coherent state rather than a more complicated result [@footnote1].
Simply passing a coherent state through a beam-splitter (or absorbing filter) without heralding would also give an attenuated coherent state. That is not the case, however, for more complicated superposition states such as a Schrödinger cat state $|\psi\rangle=(1/\sqrt{2})(|\alpha\rangle+|-\alpha\rangle)$, for example. Although each component would have a reduced amplitude, the coherence between the two states would be eliminated by which-path information left in the environment [@kirby2013; @franson2017]. The heralded noiseless attenuator described here or in references [@micuda2012; @gagatsos2014] avoids decoherence of that kind by post-selecting on a single state of the “environment", namely the idler mode in this case.
This property also allows the NPA to coherently attenuate the most basic superposition state: a single-rail qubit [@footnote2]. It can be shown that applying the operator $\hat{S}(r)$ to a Fock state $|n\rangle $ after proper heralding on a zero-photon detection event in the idler mode gives $${{\left\langle 0 \right|}_{i}}\hat{S}(r){{\left| n \right\rangle }_{s}}{{\left| 0 \right\rangle }_{i}}=\frac{1}{{{g}^{n+1}}}{{\left| n \right\rangle }_{s}}.
\label{eq:numOut}$$ Applying equation (\[eq:numOut\]) to, for example, the single rail qubit $|\psi\rangle_\text{in}=(1/\sqrt{2})(|0\rangle+|1\rangle)$ would give an output state of the form $${{\left| \psi \right\rangle }_{\text{out}}}=\frac{1}{g\sqrt{2}}\left( {{\left| 0 \right\rangle }_{s}}+\frac{1}{g}{{\left| 1 \right\rangle }_{s}} \right).
\label{eq:qubitOut}$$ It can be seen from equation (\[eq:qubitOut\]) that the single rail qubit is also attenuated by a factor $\nu=1/g$, while the probability of success is $${{P}_{s}}=\frac{{{g}^{2}}+1}{2{{g}^{4}}}.
\label{eq:numProb}$$ In summary, we have introduced the heralded noiseless parametric attenuator (NPA) as an alternative to the beam-splitter based approach to noiseless attenuation [@micuda2012; @gagatsos2014]. A heralding signal is essentially used to convert the gain $g$ of a conventional OPA into an attenuation factor $\nu=1/g$. Larger gain values result in higher attenuation, but a rapidly decreasing probability of success (cf. equations (\[eq:cohProb\]) and (\[eq:numProb\])). From a practical point of view, the experimental realization of an NPA appears to be feasible with existing technology. The effective gain values for state-of-the-art single-pass single-mode waveguide-based PDC sources can be remarkably large [@harder2013], and the use of currently available single-photon detectors with low dark count rates and high detection efficiencies would help overcome the difficulties associated with heralding on “zero photons" [@eisaman2011].
As is the case with a heralded noiseless amplifier [@ralph2009], we note that the NPA studied here does not add or subtract photons, but simply reweights the coefficients in the Fock state expansion of the input state. It is exactly this reweighting that allows the combination of a noiseless attenuator and a noiseless amplifier to suppress loss in quantum communication channels [@micuda2012]. In addition, we emphasize that this unconventional use of an amplifier as an attenuator is only possible due to the probabilistic (non-unitary) nature of the heralding process.
In this brief paper, we have specifically considered the action of the NPA on coherent states and single-rail qubit inputs. The transformation of more exotic states is also interesting, and may have further applications in robust quantum communications [@brewster2017].
[**Acknowldegments:**]{} This work was supported in part by the National Science Foundation under grant No. 1402708. We acknowledge fruitful conversations with G.T. Hickman and H. Lamsal.
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¶[[**P**]{}]{} 1[[**CP**]{}\^1]{} 3[(\_3 -1 )]{}\
[0.4cm]{}
On exotic algebraic structures on affine spaces
\
**M. Zaidenberg**
\
We review the recent developement on the subject, emphasizing its analytic aspects and pointing out some open problems. The study of exotic $\C^n$-s is at the very beginning, and hopefully this survey would be useful to learn more about these unusual and beautiful objects.
All algebraic varieties considered below are usually assumed being smooth, reduced, irreducible, and defined over $\C$. [*Isomorphism*]{} means biregular isomorphism of algebraic varieties and is denoted by $\simeq$.\
It is my pleasure to thank Sh. Kaliman and P. Russell. Without their friendly help and advice this survey would not be written.\
Contents
\
[**1. Product structures**]{}\
[**2. Kaliman’s modification**]{}\
[**3. Hyperbolic modification**]{}\
[**4. Dimca’s and Kaliman’s examples of exotic hypersurfaces**]{}\
[**5. Russell’s $\C^*$–threefolds and the Makar–Limanov invariant**]{}\
[**6. APPENDIX: Simply connectedness of $\C^*$–equivariant cyclic coverings**]{}\
[**7. Concluding remarks**]{}\
[**1. Product structures**]{}
\
[**1.1. Definition.**]{} Let $X$ be a smooth affine algebraic variety. It is called [*an exotic $\C^n$*]{} if $X$ is diffeomorphic to $\R^{2n}$ but not isomorphic to $\C^n$.
The following characterization of $\C^2$, due to C. P. Ramanujam \[Ram\] shows that there is no exotic $\C^2$.\
[**1.2. Ramanujam’s Theorem.**]{} [*A smooth affine algebraic surface $X$ is isomorphic to $\C^2$ iff the groups ${\rm H}_2 (X; \Z),\,\pi_1 (X)$, and $\pi_1^{\infty} (X)$ all are trivial. In particular, if $X$ is homeomorphic to $\C^2$, then $X$ is isomorphic to $\C^2$.* ]{}\
Here $\pi_1^{\infty} (X)$ denotes the fundamental group at infinity of $X$. If $X$ is the interior of a compact real manifiold with the boundary $\partial X$, then $\pi_1^{\infty} (X) = \pi_1 (\partial X)$.
To recognize an exotic $\C^n$ one has to verify two properties from Definition 1.1. As for the first one, the next criterion could be useful.\
[**1.3. Ramanujam–Dimca’s Theorem**]{} (\[Ram; Di\], see also \[Ka 1, Lemma 1; tD 1, Theorem 3.14\]). [*Let $X$ be a smooth affine algebraic variety of complex dimension $n \ge 3$. Then $X$ is diffeomorphic to $\R^{2n}$ iff $X$ is contractible, or, equivalently, iff $\pi_1 (X) = {\bf 1}$ and ${\rm H}_i (X;
\Z) = 0$ for $i = 1,\dots,n$.* ]{}\
The proof is based on the h-cobordism theorem. The main point is to show that being contractible, $X$ possesses a smooth simply connected boundary $\partial X$. The latter follows from the Lefschetz Hyperplane Section Theorem. In the case when $X$ as below is a product of two contractible varieties, instead of Lefschetz–type arguments one can apply the Van Kampen Theorem.\
As follows from \[Ram\] the statement of Theorem 1.3 does not hold for $n =
2$. Indeed, Ramanujam constructed an example of a smooth contractible affine surface $S_0$ which is not homeomorphic to $\R^4$ and, moreover, has an infinite group $\pi_1^{\infty} (S_0) $. See e.g. \[GuMi, FlZa, PtD 3\] for further information on such surfaces.\
[**1.4.**]{} We recall [**Zariski’s Cancellation Problem**]{}:\
[*Given an isomorphism $X \times \C^k \simeq \C^{n + k}$, does it follows that $X \simeq \C^n$?*]{}\
For $n \ge 3$ the problem is still open. For n = 2 the positive answer was obtained by the Miyanishi-Sugie and Fujita Theorem \[MiSu, Fu 1\]. This theorem provides us, in certain cases, with a tool to distinguish exotic $\C^n$–s. Despite of the fact that it was proven later on, the first examples of exotic $\C^n$–s for any $n \ge 3$ had been alluded to already in \[Ram\]: take $X^n = S_0 \times \C^{n - 2}$, where $S_0$ is the Ramanujam surface. See \[Za 1,2,4\] for some further examples of exotic structures of product type.\
[**1.5. Exotic $\C^n$–s of log-general type.**]{} More generally, let $X_i, \,i=1,\dots,m$, be smooth contractible affine varieties. Then by Ramanujam–Dimca’s Theorem 1.3 $X := (\prod_{i = 1}^m X_i ) \times \C^r$ is diffeomorphic to $\R^{2n}$ as soon as $n :={\rm dim}_{\C}\,X $ is at least $3$. To distinguish such a product structure $X$ from the standard $\C^n$, the logarithmic Kodaira dimension $\k$ could be available (see \[Ii 1\]). But, if $r
> 0$, then $\k (X) = \k (\C^n) = - \infty$. So, assume for the moment that $r =
0$. Then $\k (X) = \sum_{i=1}^m \k(X_i) \ge 0$ as far as $\k(X_i) \ge 0$ for each $i = 1,\dots,m$. Therefore, in this case $X$ is an exotic $\C^n$.
For instance, put $X = (S_0)^m ,\,m > 1$, where $S_0$ is the Ramanujam surface. Then $X$ is an exotic $\C^n$ of log-general type, i.e. $\k(X) = n$, where $n =
2m ={\rm dim}_{\C}\,X $ (indeed, by \[Ii 2\] we have $\k(S_0) = 2$). Using other contractible surfaces of log-general type as factors, one may construct more such examples [^1], and up to now these are the only known ones. Note that all of them are of even dimensions.\
[**1.6.**]{} [**Remark.**]{} Let $S$ be a contractible, or at least acyclic smooth surface of non–negative Kodaira dimension. Then $\k(S) = 1$ or $2$ by the Fujita classification of open surfaces with $\k = 0$ \[Fu 2\][^2]. Thus, for the product structure $X = \prod_{i = 1}^m S_i$ obtained by means of surfaces $S_i$ as above its log-Kodaira dimension takes the values in the interval $m \le
\k(X) \le 2m = {\rm dim}_{\C}\,X $. Later on we will see that there exist exotic $\C^n$–s $X$ with $1 \le \k(X) \le [n/2]$.\
[**1.7. Examples.**]{} A source of examples provide contractible affine surfaces of log-Kodaira dimension one. The complete list of them was obtained by R. V. Gurjar and M. Miyanishi \[GuMi\]. T. tom Dieck and T. Petrie \[PtD 1\] realized some of them as hypersurfaces in $\C^3$. Namely, put $p_{k,\,l}
(x,\,y,\,z) = ((xz + 1)^k - (yz + 1)^l)/z \in \C [x,\,y,\,z]$, where $(k, l) =
1,\, k, \,l \ge 2$. Then all the fibres of the polynomial $p_{k,\,l}$ are contractible surfaces in $\C^3$. All its non-zero fibres are smooth surfaces of log-Kodaira dimension one, isomorphic to $S_{k,\,l} = p_{k,\,l}^{-1} (1)$ (the zero fibre has non–isolated singularities).
Due to Theorem 1.8 below, the product $S_{k,\,l} \times \C^{n - 2}$ is a hypersurface in $\C^{n + 1}$ which is an exotic $\C^n$. Moreover, all the fibres of $p_{k,\,l}$ regarded as polynomial on $\C^{n + 1}$ are contractible hypersurfaces, and all but the zero one are exotic $\C^n$–s.
Recently Sh. Kaliman and L. Makar–Limanov \[KaML 3\] have shown that all the log-Kodaira dimension one contractible surfaces from the Gurjar–Miyanishi list admit embeddings into $\C^3$, thus providing deformation families of contractible hypersurfaces (see (1.9) below).\
[**Proposition (Sh. Kaliman, L. Makar–Limanov).**]{} [*Any smooth contractible surface $S$ of log-Kodaira dimension one can be given as a surface in $\C^3$ with the equation $p_{k,\,l,\,m,\,f} (x,\,y,\,z) = 0$, where $$p_{k,\,l,\,m,\,f} (x,\,y,\,z) = [(z^m x + f(z))^k - (z^m y + g(z))^l -
z]/z^m\,\,,$$ $(k,\,l) = 1, \,k,\,l \ge 2, \,m \ge 1,\,f,\,g \in \C[z],
\,\,{\rm deg}\,f,\,{\rm deg}\,g < m,\,f(0) = g(0) = 1$ and where $g$ is uniquely defined by $f$ (which can be taken arbitrary) in view of the assumption that $p_{k,\,l,\,m,\,f}$ is a polynomial.* ]{}\
[**1.8.**]{} Besides the log-Kodaira dimension, one may equally use other invariants to distinguish exotic product–structures, for instance the logarithmic plurigenera ${\bar P}_{m_1,\dots,m_l}$ \[Ii 1,2\]. Indeed, if $X :=
(\prod_{i = 1}^m X_i )$ and at least one of the factors $X_i$ has a non–zero log–plurigenus, then by Künneth’s Formula the corresponding log–plurigenus of $X$ does not vanish. Moreover, we have the following theorem.\
[**Iitaka–Fujita’s Strong Cancellation Theorem**]{} \[IiFu\].
*Let $X,\,Y,\,A_1,\,A_2$ be algebraic varieties such that ${\rm dim}_{\C}\,X = {\rm
dim}_{\C}\,Y,\,\k(X) \ge 0$, and the log–plurigenera ${\bar P}_{m_1,\dots,m_l}
(A_i)$ all vanish, $i = 1,2$. Then, given an isomorphism $\Phi \,:\,X \times
A_1 \to Y \times A_2$, there exists an isomorphism $\varphi\,:\,X \to Y$ making the following diagram commutative:\
*
(200,80) 0.2em (62,25)[$X \times A_1$]{} (105,25)[$Y \times A_2$]{} (84,27)[$\vector(1,0){15}$]{} (66,5)[$X$]{} (84,6)[$\vector(1,0){15}$]{} (109,5)[$Y$]{} (69,22)[$\vector(0,-1){11}$]{} (111,22)[$\vector(0,-1){11}$]{} (91,10)[$\varphi$]{} (91,30)[$\Phi$]{} (160,11)[$(1)$]{}
\
(the vertical arrows are the canonical projections).
\
Applying this theorem to the case when $A_1 = A_2 = \C^r$ we see that, as soon as $\k(X) \ge 0$, the factor $\C^r$ cancells. Therefore, if $X$ and $Y$ are contractible varieties such that $n :=\d X + r \ge 3$ and $0 \le \k(X) \neq
\k(Y)$, then $X \times \C^r$ is an exotic $\C^n$ non–isomorphic to $Y \times
\C^r$. This allows us to distinguish some exotic product structures of negative log–Kodaira dimension.\
[**1.9. Deformable exotic product structures**]{}\
[**Theorem (Flenner–Zaidenberg)**]{} \[FlZa\]. [*For any $n \ge 3$ and $m
\ge 1$ there exists a family of versal deformations ${\tilde f}\,:\,{\tilde
X}^{n + m} \to B^m$ of exotic product structures on $\C^n$ over a smooth quasiprojective base $B^m$ of dimension $m$.*]{}\
Here $\tilde f$ is a smooth morphism such that for any $b \in B^m$ the fibre $X_b = {\tilde f}^{-1} (b)$ is an exotic product structure on $\C^n$ and $X_b \not\simeq X_{b'}$ if $b \neq b'$ are generic points of $B^m$.
The construction uses families of versal deformations of contractible surfaces of log–Kodaira dimension one. If $ f\,:\,{\tilde S}^{2 + m} \to B^m$ is such a family, then by the Iitaka–Fujita Theorem 1.8 we can take ${\tilde X}^{n + m}
= {\tilde S}^{2 + m} \times \C^{n - 2}$. Due to \[KaML 3\] (see 1.7) this leads to families of exotic hypersurfaces in $\C^{n+1}$.
In the next section we will give another construction of deformable exotic $\C^n$–s due to Sh. Kaliman. But still we do not know the answer to the following\
[**Question.**]{} [*Does there exist a deformable exotic $\C^n$ of log–general type?*]{}[^3]\
[**1.10. Analytically exotic $\C^n$–s.**]{} We say that an exotic $\C^n$ is [*analytically exotic*]{} if it is not even biholomorphic to $\C^n$. The following result shows that among the product structures constructed above there are analytically exotic ones.\
[**Strong Analytic Cancellation Theorem**]{} \[Za 2\]. [*Let $X,\,Y$ be smooth quasi–projective varieties of log–general type. If $\Phi\,:\, X \times
\C^r \to Y \times \C^m$ is a biholomorphism, then $m = r$ and there exists an isomorphism $\varphi\,:\,X \to Y$ which makes diagram $(1)$ commutative, where $A_i, \, i=1,2$ are replaced by $\C^r$.* ]{}\
Thus, if $S$ is a contractible surface of log–general type and $n \ge
3$, then $X = S \times \C^{n - 2}$ is an analytically exotic $\C^n$. The same is true for the product structures $X = S \times M^{n - 2}$, where $S$ is as above and $M^{n - 2}$ is any contractible affine variety of dimension $n - 2$. Indeed, for such an $X$ its Eisenman–Kobayashi intrinsic 2–measure form $E^{(2)}_X$ does not vanish identically (this useful remark is due to Sh. Kaliman \[Ka 2\]). More generally, one can consider the maximal value of $k$ for which $E^{(k)}_X$ does not vanish identically. This yields a coarse analytic invariant which replaces the log–Kodaira dimension as it has been used in 1.8 above, and so it permits to distinguish certain analytically exotic $\C^n$-s up to a biholomorphism.
Another remarkable property of the above exotic product structures on $\C^n$ is that they contain no copy of $\C^{n-1}$.\
[**1.11. Theorem**]{} \[Za 2; Ka 2\]. [*Let $X = S \times \C^{n - 2}$ be an exotic product structure on $\C^n$, where $S$ is a contractible surface of log-general type. Then there is no regular injection $\C^{n - 1} \to X$; in particular, there is no algebraic hypersurface in $X$ isomorphic to $\C^{n -
1}$. Moreover, there is no proper holomorphic injection $\C^{n - 1} \to X$, and therefore there is no closed analytic hypersurface in $X$ biholomorphic to $\C^{n - 1}$.* ]{}\
Theorem 1.10 shows that the following is likely to be true.\
[**1.12. Conjecture.**]{} [*Any exotic $\C^n$ is analytically exotic.*]{}\
[**1.13. Problem.**]{} [*Does there exist a pair of biholomorphic but not isomorphic exotic $\C^n$-s? Does there exist a non-trivial deformation family of exotic $\C^n$-s with the same underlying analytic structure?*]{}\
We even do not know whether the deformation families of exotic product–structures constructed in the proof of Theorem 1.9 are versal in the analytic sense. To this point, the knowledge of the collection of entire curves (i.e. holomorphic images of $\C$) in the contractible surfaces of log–Kodaira dimension one could be useful. What is the set of tangent directions of such curves in the tangent bundle? Does the degeneration locus of the Kobayashi pseudo–distance provide a non-trivial analytic invariant of such surfaces?\
**2. Kaliman’s modification**
\
[**2.1. Definition**]{} (cf. \[Ka 2\]). Consider a triple $(X,\,H,\,C)$ consisting of an algebraic variety $X$, an irreducible hypersurface $H$ in $X$ and a closed subvariety $C$ of $H$ with ${\rm codim}_X C \ge 2$. Let $\sigma_C\,:\,{\tilde X} \to X$ be the blow-up of $X$ at the ideal sheaf of $C$. Let $E \subset {\tilde X}$ be the exceptional divisor of $\sigma_C$ and ${\tilde H} \subset {\tilde X}$ be the proper transform of $H$. [*The Kaliman modification*]{}[^4] consists in replacing the triple $(X,\,H,\,C)$ by the pair $(X',\,E')$, where $X' = {\tilde X}
\setminus {\tilde H}$ and $E' = E \setminus {\tilde H}$. We will also say that [*$X'$ is the Kaliman modification of $X$ along $H$ with center $C$*]{}.
A triple $(X,\,H,\,C)$ resp. a pair $(X',\,E')$ as above will be called [*a smooth contractible affine triple*]{} resp. [*a smooth contractible affine pair*]{} if all its members are smooth contractible affine varieties.\
[**2.2. Theorem**]{} \[Ka 2, Theorem 3.5\]. [*The Kaliman modification of a smooth contractible affine triple is a smooth contractible affine pair.*]{}\
The statement of the theorem remains valid under the assumption that the hypersurface $H$ (not necessarily smooth any more) is a toplogical cell and $C
\subset {\rm reg}\,H$, while all other conditions on $X$ and $C$ being preserved (see \[Ka 2, Theorem 3.5\]). However, we do not know whether the smoothness of $C$ is essential[^5].\
[**2.3. Examples.**]{} The Kaliman modification produces new examples of analytically exotic $\C^n$–s and of their versal deformations. Let $(X,\,H,\,C)$ be a smooth contractible affine triple, where $X$ is an exotic $\C^n$ such that certain Eisenman–Kobayashi form $E^{(k)}_X$ is different from zero at the points of an open subset $U \subset X$. Performing the Kaliman modification we arrive again at an exotic $\C^n$, call it $X'$, which has a non-zero form $E^{(k)}_{X'}$. Indeed, by Theorems 1.3 and 2.2 $X'$ is diffeomorphic to $\R^{2n}$. Furthermore, the restriction $
\sigma_C\,|\,X'\,:\,X' \to X$ is a dominant holomorphic mapping which is a contraction with respect to the Eisenman–Kobayashi forms, i.e. $\sigma_C^* \,
E^{(k)}_X \le E^{(k)}_{X'}$, and whence $E^{(k)}_{X'} \not\equiv 0$. Thus, $X'$ is an analytically exotic $\C^n$.
For instance, fix a point $s_0$ in a smooth contractible affine surface $S$ of log–general type and put $X = S \times \C^{n-2} ,\,H = S \times \C^{n-3}$ and $C = \{s_0\} \times \C^k$, where $0 \le k \le n-3$. By Sakai’s Theorem \[Sak\] $S$ is measure hyperbolic, i.e. $E^{(2)}_S$ is positive on a subset of $S$ whose complement has measure zero. Therefore, $E^{(2)}_X = {\rm pr}^*_S\,
E^{(2)}_S$ is different from zero at the points of a massive subset of $X$ (where ${\rm pr}_S\,:\,X \to S$ is the natural projection). Performing the Kaliman modification, by Theorem 2.2, we obtain a smooth contractible pair $(X',\,E')$, where $X'$ is an exotic $\C^n$ which have a non-zero form $E^{(2)}_{X'}$. It is easily seen that $E' \simeq \C^{n-1}$. Hence, $X'$ is not biholomorphic to any ${\tilde X} = {\tilde S} \times \C^{n-2}$ as above. Indeed, by Theorem 1.11 such an ${\tilde X}$ does not contain any biholomorphic image of $\C^{n-1}$. And also it is not biholomorphic to any exotic ${\tilde
X} = {\tilde S} \times \C^{n-2
}$, where $\k ({\tilde S}) = 1$, because for the latter product the form $E^{(2)}_{\tilde X}$ vanishes at a Zariski open subset. [^6] This proves the following\
[**2.4. Proposition.**]{} [*For $n = 3\,$ $X'$ as above is an exotic $\C^3\,$ which is not biholomorphic to any exotic product–structure on $\C^3$.*]{}\
[**2.5. Deformable analytically exotic $\C^n$**]{}\
By iterating the construction used in the preceeding example, Sh. Kaliman obtained the following result (cf. Theorem 1.9 above).\
[**Theorem**]{} \[Ka 2, sect.4\]. [*For any $n \ge 3$ there exist versal deformation families of analytically exotic $\C^n$–s with any given number of moduli*]{}.\
The proof proceeds as follows. Start with an exotic $\C^n\,$, $X = S
\times \C^{n-2}$, as above. Fix $m$ distinct points $s_1,\dots, s_m \in S$ and $m$ disjoint affine hyperplanes $A_1,\dots, A_m$ in $\C^{n-2}$. Put $H_i = S
\times A_i,\,C = \{s_i\} \times A_i$, and perform the Kaliman modifications along $H_i$ with centers $C_i$ for $i=1,\dots,m$. Then we result with a family $X' = X'\,(s_1,\dots, s_m ,\,A_1,\dots, A_m)$ of analytically exotic $\C^n$–s endowed each one with $m$ disjoint hypersurfaces $E'_1,\dots,E'_m$ isomorphic to $\C^{n-1}$. And they are the only biholomorphic images of $\C^{n-1}$ in $X'$ for fixed data \[Ka 2, Lemma 4.1\]. Now it is not difficult to check that the positions of the points $s_1,\dots, s_m$ in $S$ and of the hyperplanes $A_1,\dots, A_m$ in $\C^{n-2}$ provide the moduli of these exotic structures considered up to a biholomorphism.\
**EXOTIC AFFINE HYPERSURFACES**
\
In Sections 3–5 below we review some explicit constructions of contractible hypersurfaces in $\C^{n+1}$. We discuss different approaches to the recognition problem for exotic hypersurfaces.
By the Abhyankar–Moh and Suzuki Theorem \[AM, Su\] the only smooth irreducible simply connected curves in $\C^2$ are those obtained from the affine line $x =
0$ by means of polynomial coordinate changes. By the Lin–Zaidenberg Theorem \[LiZa\] the only simply connected affine plane curves, up to the action of the group of biregular automorphisms ${\rm Aut}\,\C^2$, are the quasihomogeneous ones. In particular, each irreducible singular such curve is equivalent to one and the only one from the sequence $\Gamma_{k,\,l} = \{x^k - y^l =
0\},\,(k,\,l) = 1, \,k>l\ge 2$. Thus, in the case $n = 2$ this classifies completely the contractible hypersurfaces in $\C^n$.
In contrast, we have already seen in Section $1$ above that there are even deformation families of smooth contractible surfaces of log–Kodaira dimension one in $\C^3$, and so for any $n \ge 3$ there are deformation families of hypersurfaces in $\C^{n+1}$ which are exotic $\C^n$–s. They are far from being classified in any sense. In particular, [*no exotic hypersurface in $\C^{n+1}$ of log–general type is known.*]{}\
**3. Hyperbolic modification**
\
In \[tD 1\] T. tom Dieck introduced a general construction which, under certain conditions, represents a given topological resp. complex manifold $Z$ with possible singularities as an algebraic quotient of an action of a real resp. complex Lie group $G$ on another such manifold $X$, canonically defined by Z and a given representation of $G$. This representation should have the unique fixed point, which should be of hyperbolic type, and so the correspondence $Z
\longmapsto X$ was called [*the hyperbolic modification*]{}.
When $Z \subset \C^n$ is an affine algebraic variety and $G = \C^*$, the hyperbolic modification $X$ of $Z$ is an affine algebraic variety in $\C^{n+1}$ endowed with a $\C^*$–action, and $Z \is X//\C^*$. The $n$–th iterate $X_n$ of the hyperbolic modification of $Z$ is endowed with an action of the $n$–torus $T_n = (\C^* )^n$, and $Z \is X_n//T_n$.
The main advantage of this transform is that it leads, in the case of hypersurfaces, to new amazing examples of exotic families.\
[**Proposition 3.1**]{} \[tD 1, KaML 3\]. [*For any $n \ge 3$ there exist effectively defined polynomials $p_{k,\,l}^{(n)}$ on $\C^{n+1}$, where $(k,\,l) = 1,\,k > l \ge 2$, such that all the fibres $(p_{k,\,l}^{(n)})^{-1}
(c),\,c \in \C$, are exotic $\C^n$–s.*]{}\
Since we are interested mainly in the hypersurface case, and it is simpler, we give the precise definition only in this case.\
[**3.2. Definition**]{} \[tD 1\]. Let $p \in \C [x_1,\dots,x_n] \setminus
\{0\}$ be a polynomial on $\C^n$ such that $p({\bar 0}) = 0$. The polynomial $\tp (x, z) = p(xz)/z \in \C[x,\,z]$ on $\C^{n+1}$ is called [*the hyperbolic modification of $p$*]{}, and its zero fibre $X_0 = \tp^{-1} (0) \subset
\C^{n+1}$ is called [*the hyperbolic modification of the zero fibre $Z_0 =
p^{-1} (0) \subset \C^n$ of $p$*]{}.\
In the case of the simply connected curves $\Gamma_{k,\,l} = \{x^k - y^l = 0\}$ the hyperbolic modification was already used in \[PtD 1\]. The origin having been placed at the smooth point $(1,\,1) \in \Gamma_{k,\,l}$, the hyperbolic modification gives rise to the Petrie–tom Dieck polynomials $p_{k,\,l} = ((xz
+ 1)^k - (yz + 1)^l)/z$ (see 1.6; see also \[PtD 2\] for some related constructions).\
[**3.3. Some properties of the hyperbolic modification.**]{} Let $\tp (x,
z)$ be the hyperbolic modification of a polynomial $p(x)$ on $\C^n$. Consider the $\C^*$–action $G(\lambda,\, x, \,z) = (\lambda x,\,\lambda^{-1} z)$ on $\C^{n+1}$. It is easily seen that $\tp$ is a quasi–invariant of $G$ of weight $1$, i.e. $\tp \circ G_{\lambda} = \lambda \tp ,\,\lambda \in \C^*$. Denote $X_1 = \tp^{-1} (1) \subset \C^{n+1}$. Then the restriction $G \,|\, (\C^*
\times X_1)$ yields an isomorphism $\C^* \times X_1 \simeq \C^{n+1} \setminus
X_0$. In particular, $X_1$ is a smooth hypersurface, and all the fibres $X_c =
\tp^{-1} (c),\,c \in \C \setminus \{0\}$, are isomorphic to $X_1$.
Thus, being applied to a hypersurface $Z_0$ in $\C^n$, the hyperbolic modification produces actually a pair of distinct hypersurfaces $X_0$ and $X_1$ in $\C^{n+1}$. The zero fibre $X_0$ inherits the $\C^*$–action $G\,|\,X_0$. The ring of $G$–invariants coincides with the subring $\C[zx_1
,\dots, zx_n] \subset \C[x, \,z]$, and $\pi\,:\,\C^{n+1} \ni (x, \,z)
\longmapsto xz \in \C^n$ is the canonical quotient morphism, as well as the restriction $\pi \,|\,X_0 \,:\,X_0 \to Z_0$. Thus, $Z_0 = X_0 //\C^*$ is the algebraic quotient. Note that $X_0$ is smooth iff $Z_0$ is so.\
For the morphism $\pi\,:\,X_0 \to Z_0$ the Kawamata Addition Theorem \[Kaw\] and the Iitaka inequality \[Ii 1\] imply that $\k (Z_0) \le \k (X_0) \le \d Z_0$. The same holds for any iterated hyperbolic modification $X_0^{(n)}$ of $Z_0$.\
The restriction $\pi \,|\,X_1 \,:\,X_1 \to \C^n$ is a birational morphism. From Proposition 3.6 in \[tD 1\] it follows that the generic fibre $X_1$ is the Kaliman modification of $\C^n$ along $Z_0$ with center at the origin (see 2.1). Combining several statements from \[tD 1, (1.1), (1.3), (2.1), (3.1); Ka 2, (3.5)\] we obtain the following [^7]\
[**3.4. Theorem.**]{}
*Let $Z_0 = p^* (0)$, where $p \in
\C[x_1,\dots,x_n]$. Assume that $Z_0$ is an irreducible reduced divisor on $\C^n$, which contains the origin and is smooth at the origin. Let $X_0$ resp. $X_1 \subset \C^{n+1}$ be the zero fibre resp. the generic fibre of the hyperbolic modification $\tp$ of the polynomial $p$.*
a\) Let $Z_0$ be smooth. Then both the hyperbolic modification $X_0$ and the Kaliman modification $X_1$ of $Z_0$ are acyclic resp. contractible as soon as $Z_0$ is acyclic resp. contractible.
b\) Let $Z_0$ be a topological manifold and has at most isolated singularities. Then $X_1$ is acyclic resp. contractible if $Z_0$ is acyclic resp. a topological cell.
\
This leads to the following result \[tD 1, Theorem 3.12\].\
[**3.5. Theorem (T. tom Dieck).**]{} [*If $Z_0 = p^{-1}(0)$ is a smooth contractible hypersurface in $\C^n,\, n\ge 3$, then all the fibres of the hyperbolic modification $\tp$ of $p$ are smooth hypersurfaces in $\C^{n+1}$ diffeomorphic to $\R^{2n}$. If, furthermore, $\k (Z_0 ) \ge 0$, then the zero fibre $X_0$ of $\tp$ is an exotic $\C^n$ of non–negative log–Kodaira dimension.*]{}\
Starting with the Petrie–tom Dieck surface $S_{k,\,l}$ in $\C^3$ of log–Kodaira dimension one (see 1.7) and iterating the hyperbolic modification, for any given $n \ge 3$ one can effectively find a polynomial $p^{(n)}_{k,\,l}$ on $\C^{n+1}$ such that all its fibres are smooth and diffeomorphic to $\R^{2n}$, and the zero fibre is an exotic $\C^n$. In fact, in this particular case all of them are exotic $\C^n$–s, as has been recently shown in \[KaML 3\]. More precisely, it was shown that none of these fibres is dominated by $\C^n$. This proves Proposition 3.1 above. The Brieskorn–Pham polynomials provide another examples of this type (in this case one has to apply (b) of Theorem 3.4; see \[tD 1, Section 4\]).
To manage the general case, it would be useful to prove the following conjecture, which seems to be interesting by itself. It is easily checked for $n = 2$.\
[**3.6. Conjecture.** ]{} [*Let $X_0$ be a special fibre and $X_1$ be a generic fibre of a primitive polynomial $p \in \C[x_1,\dots,x_n]$. Let $X'_0$ be the desingularization of an irreducible component of $X_0$. Then $\k (X'_0)
\le \k (X_1)$. In other words, the log–Kodaira dimension is lower semi–continuous on the fibres of a polynomial in $\C^n$.* ]{}\
Note that for $n \ge 5$ the exotic $\C^n$ which is the zero fibre $X_0 =
(p^{(n)}_{k,\,l})^{-1} (0)$ is different from the exotic product structures on $\C^n$ considered in 1.5 above, since here $1 \le \k (X_0) \le 2$.\
[**4. Dimca’s and Kaliman’s examples of exotic hypersurfaces**]{}
As we have seen in the preceeding section, the hyperbolic modification $\tp$ of a polynomial $p$ on $\C^n$ is a quasi–invariant of weight $1$ of a linear $\C^*$–action $G$ on $\C^{n+1}$ of mixed type (the latter means that the $\C^*$–action $G$ has one fixed point only and the linear part of $G$ in the fixed point has weights of different signs). In the examples considered below the defining polynomials of exotic hypersurfaces in $\C^{n+1}$ will be quasi–invariants of weights $> 1$ of regular $\C^*$–actions of mixed type. The generic fibre $X_1$ of such a polynomial does not need to be contractible any more. Its zero fibre $X_0$ will be presented as a cyclic branched covering of $\C^n$ ramified along a hypersurface $Z_0$ with certain properties, which ensure that $X_0$ is contractible.\
[**4.1. Dimca’s list**]{} \[Di, ChoDi\]. A. Libgober \[Lib\] discovered that the (singular) projective hypersurface $${\bar H}_{n,\,d} = \{x_0^{d-1} x_1 +
x_1^{d-1} x_2 +\dots + x_{n-1}^{d-1} x_n + x_{n+1}^d = 0\} \subset \P^{n+1}$$ has the same homology groups as $\P^n, \,n$ odd. In fact, ${\bar H}_{n,\,d}$ is a completion of $\C^n$; namely, $H_{n,\,d} \simeq \C^n$, where $H_{n,\,d} =
{\bar H}_{n,\,d} \setminus \{x_0 = 0\}$.
Generalizing this example, G. Barthel and A. Dimca \[BaDi\] found some others homology $\P^n$–s with isolated singularities [^8]. These are the projective closures in $\P^{n+1}$ of the affine hypersurfaces $H_{n,\,d,\,a} \subset \C^{n+1}$ with the equations $$p_{n,\,d,\,a} (x) = x_1 + x_1^{d-1}x_2 + \dots + x_{n-2}^{d-1} x_{n-1} +
x^{d-a}_{n-1}x_n^a + x^d_{n+1} = 0$$ where $n$ is odd and $1 \le a <
d,\,(a,\,d) = (a,\,d-1) = 1$.\
[**4.2. Proposition**]{} \[Di, Propositions 5, 7; ChoDi, Theorem 5, Proposition 6\].
**
a\) $H_{n,\,d,\,a} \subset \C^{n+1}$ is diffeomorphic to $\R^{2n}$.
b\) For $a = 1$ the map $p_{n,\,d,\,a}\,:\,\C^{n+1} \to \C$ is a smooth fibre bundle with the fibre diffeomorphic to $\R^{2n}$.
c\) For $a > 1$ the generic fibre $X_1 = p_{n,\,d,\,a}^{-1} (1)$ is not contractible (in fact, its Euler characteristic is different from $1$).
d\) $H_{3,\,d,\,1} \simeq \C^3$ and the fibration $p_{3,\,d,\,1}\,:\,\C^4
\to \C$ is algebraically trivial.
[^9]\
A. Dimca posed the question:
[*Is it true that for $a > 1$ all the hypersurfaces $H_{3,\,d,\,a} =
\{x + x^{d-1}y + y^{d-a}z^a + t^d = 0\} \subset \C^4$ as above are exotic $\C^3$–s?*]{}
The positive answer was done by \[KaML 2\], see Theorem 5.10 below. It is still unknown whether the same is true in higher dimensions.\
The following criterion of contractibility of cyclic coverings, proposed in \[Ka 1, Theorem A\] (see also \[tD 2\]), allows one to establish that the hypersurfaces like those in the previous examples and more general ones are contractible.\
[**4.3. Theorem (Kaliman)**]{}.
*Let a polynomial $q \in \C[x_1,\dots,
x_n]$ be a quasi–invariant of a positive weight $l$ of a regular $\C^*$–action $G$ on $\C^n$. Suppose that the zero fibre $Z_0 = q^* (0)$ is a smooth, reduced, and irreducible divisor in $\C^n$ such that*
i\) $\pi_1 (\C^n \setminus Z_0 ) \approx \Z$ [^10];
ii\) for some prime $k$ coprime with $l$, $\,{\rm H}_i (Z_0;\,\Z / k\Z ) =
0,\,i=1,\dots,n$, i.e. the fibre $Z_0$ is $\Z / k\Z$–acyclic.
Then the zero fibre $X_0 = p^{-1} (0) \subset \C^{n+1}$ of the polynomial $p(x,\,z) = q(x) + z^k$ is diffeomorphic to $\R^{2n}$.
\
Note that the polynomial $p$ is a quasi–invariant of weight $kl$ of the $\C^*$–action ${\tilde G}(\lambda,\,x,\,z) = (G(\lambda^k,\,x),\,\lambda^l z)$ on $\C^{n+1}$. The morphism $\pi\,:\,X_0 \ni(x,\,z) \longmapsto x \in \C^n$ represents $X_0$ as a $k$–fold branched cyclic covering over $\C^n$ ramified along $Z_0$. This covering is equivariant with respect to the actions $G$ on $\C^n$ and ${\tilde G}\,|\,X_0$ on $X_0$.
The generic fibre $X_1 = p^{-1} (1)$ topologically is the joint $Z_1 \ast \Z /
k\Z$, where $Z_1 = q^{-1} (0) \subset \C^n$ is the generic fibre of $q$ \[Ne\]. Therefore, $X_1$ is not contractible as soon as $Z_1$ is not contractible \[Ka 1, Lemma 9\].
The assumption (i) is always fulfilled for a generic fibre of a primitive polynomial \[Ka 1, Lemma 8\]. Though the zero fibre of a $\C^*$–quasi–invariant is usually non–generic, there exist non–trivial examples where this and all the other conditions of Theorem 4.3 are satisfied. [^11]\
[**4.4. Proposition**]{} \[Ka 1, Theorem 10\]. [*Put $$q(x,\,y,\,z) = x +
x^ay^b + y^cz^d\,,$$ $l = bd$ and $G(\lambda,\,x,\,y,\,z) = (\lambda^l
x,\,\lambda^{-r} y, \,\lambda^s z)$, where $r = d(a-1),\, s = c(a-1) + b$. If $(s,\, d) = 1$, then the polinomial $q$ and the $\C^*$–action $G$ verify all the assumptions of Theorem 4.3. Furthermore, the Euler characteristic of the generic fibre $Z_1 = q^{-1} (1)$ is different from $1$ as soon as $b,\,d \ge
2$.* ]{}\
Let $p(x,\,y,\,z,\,t) = q(x,\,y,\,z) + t^k$, where $q$ is as above and $k$ is a prime such that $(bd,\,k) = 1$. Then, by Theorem 4.3, $X_0 =
X_{a,\,b,\,c,\,d,\,k} = p^{-1} (0)$ is a smooth contractible hypersurface in $\C^4$. Later on we will see that most of these threefolds are exotic $\C^3$–s.
The polynomial $p$ being $\tilde G$–quasi–invariant, the threefold $X_0$ carries a $\C^*$–action ${\tilde G}\,|\,X_0$. In general, a polynomial $f$ on $\C^n$ is a quasi–invariant of a linear diagonalized $\C^*$–action iff its Newton diagram is linearly degenerate, i.e. if it lies in an affine hyperplane. This is always the case when $f$ consists of $n$ monomials only, like in the preceeding examples. Since all the hypersurfaces discussed in this section carry $\C^*$–actions, none of them is of log–general type. However, as we will see, for some of them the log–Kodaira dimension is non–negative.
The next result provides an estimate from below of the log-Kodaira dimensions of ramified coverings, and so makes it possible, in certain cases, to recognize exotic $\C^n$–s among hypersurfaces as in Theorem 4.3.\
[**4.5. Proposition**]{} \[Ka 1, Corollaries 12, 13\]. [*Let $f\,:\,W \to
V$ be a morphism of smooth quasiprojective varieties, which is a branched covering ramified over a divisor $R \subset V$ of simple normal crossing type. Assume that the Sakai analytic dimension $k_c$ of the complement $V \setminus
R$ is non–negative (see \[Sa\]) [^12]. If the ramification order of $f$ on each of the irreducible components of $f^{-1} (R) \subset W$ is high enough, then $\k (W) \ge \k (V \setminus R)$. Consequently, in this case $W$ is of log-general type if $V \setminus R$ is so.*]{}\
More carefull analysis with the same type of arguments \[Ka 1, Theorem B\] leads to the conclusion that $\k (X_{a,\,b,\,c,\,d,\,k}) = 2$ if $k > a \ge 4, \,d=
a-1, \,(b, \,d) = (bd, \,k) = 1$ and $ (b,\,c) > d^2 k$. For instance, $\k(X) =
2$ for $X = X_{4,\,46,\,92,\,3,\,5} = \{x + x^4 y^{46} + y^{92}z^3 + t^5 = 0\}
\subset \C^4$. See \[Ru 1, Ka 3\] for some other cases when the log–Kodaira dimension of these threefolds is at least non–negative or more. Note that an exotic threefold of non–negative log–Kodaira dimension cannot be an exotic product structure on $\C^3$, which is always of log–Kodaira dimension $-\infty$.\
**5. Russell’s $\C^*$–threefolds and the Makar–Limanov invariant**
\
By [*a $\C^*$–variety*]{} we mean a smooth irreducible algebraic variety endowed with a regular effective $\C^*$–action. Most of the exotic $\C^n$–s which are known, except some Kaliman modifications, are $\C^*$–varieties. Thus, we come to the following\
[**5.1. Problem.**]{} [*Classify contractible $\C^*$–varieties up to equivariant isomorphism.*]{}\
This includs the famous linearization problem for $\C^*$–actions on the affine space $\C^3$, which is still open (see e.g. \[KoRu, Kr 2\]).\
[**5.2. Koras–Russell bicyclic covering construction.**]{} Analysing Dimca’s examples 4.2 from the point of view of the previous work with M. Koras \[KoRu\], P. Russell \[Ru 1\] came to a remarkable general method of constructing contractible $\C^*$–threefolds. In particular, it yields all of them of a certain restricted type (namely, [*tame of mixed type*]{}; see (5.4) and Theorem 5.5 below), including those of 4.2 and 4.4 above. We discuss here some principal points of this construction.
Denote by $\omega_r$ the cyclic group of the complex $r$–roots of unity. Let $B$ be a smooth contractible algebraic variety and $Z_1,\, Z_2 \subset B$ be two smooth divisors which meet normally. For a pair $(\alpha_1,\,\alpha_2)$ of coprime positive integers consider the bicyclic covering over $B$ branched to order $ \alpha_i$ over $Z_i, \,i=1,2$. We get a commutative diagram\
(200,100) 0.2em (75,39)[$X$]{} (51,18)[$X_1$]{} (97,18)[$X_2$]{} (75,-2)[$B$]{}
(72,37)[$\vector(-1,-1){15}$]{} (80,37)[$\vector(1,-1){15}$]{} (58,16)[$\vector(1,-1){15}$]{} (96,16)[$\vector(-1,-1){15}$]{} (77,34)[$\vector(0,-1){28}$]{}
(57,32)[$\omega_{\alpha_2}$]{} (89,32)[$\omega_{\alpha_1}$]{} (57,6)[$\omega_{\alpha_1}$]{} (90,6)[$\omega_{\alpha_2}$]{} (79,20)[$\omega_{\alpha_1 \alpha_2}$]{}
(160,22)[$(2)$]{}
\
[**Question.**]{} [*Under which assumptions on $Z_1,\,Z_2$ the resulting variety $X$ is contractible?*]{}\
Here $X$ is acyclic if both $\omega_{\alpha_1}, \, \omega_{\alpha_2}$ act trivially on the homologies $H_* (X; \Z)$. Indeed, in this case the homologies would be $\alpha_i$–torsions, $i = 1,2$, and hence trivial \[Ru 1\]. This is so if $B$ is a $\C^*$–variety and the divisors $Z_1,\, Z_2$ are invariant under the $\C^*$–action and, furthermore, they are given as $Z_i = q_i^{-1}
(0),\,i=1,2$, where $q_i \in \C[B]$ is a regular $\C^*$–quasi–invariant of weight which is relatively prime with $\alpha_i, \,i=1,2$. Indeed, under these assumptions (2) is a diagram of equivariant morphisms of $\C^*$–varieties, and $\omega_{\alpha_i}, i=1,2$, acts on $X$ via the $\C^*$–action. Therefore, it induces the trivial action in the homologies. We also need, of course, $X$ to be simply connected (cf. Appendix).
From now on we restrict the consideration by $\C^*$–threefolds, and namely by those of mixed (or hyperbolic) type. We say that a $\C^*$–threefold $X$ with a $\C^*$–action $G$ is [*of mixed type*]{} if $G$ has the unique fixed point $X^G = \{x_0\}$ and the weights of the diagonalized linear action $dG\,|\,T_{x_0} X$ are of different signs. One may assume that $$dG
(\lambda,\,x,\,y,\,z) = (\lambda^{-a} x,\, \lambda^b y,\,\lambda^c
z),\,\,\lambda \in \C^* ,$$ where $a,\,b,\,c > 0$. Since $G$ is effective, $a,\,b,\,c$ are relatively prime: $(a,\,b,\,c) = 1$. The triple $(a,\,b,\,c)$ is called [*reduced*]{} if, moreover, $(a,\,b) = (b,\,c) = (a,\,c) = 1$.
We will say that a mixed $\C^*$–threefold $X$ is [*tame*]{} if the algebraic quotient $X//G$ is isomorphic to $\C^3 //dG$, or, what is the same, to one of the surfaces $\C^2 // \omega_r$, where $\omega_r$ acts diagonally in $\C^2$. $X$ is said to be [*linearizable*]{} if it is equivariantly isomorphic to $\C^3$ with a linear $\C^*$–action. The following statement is unexplicitly contained in \[KoRu, Ko\].\
[**5.3. Proposition.**]{} [*If $X$ is a tame contractible affine $\C^*$–threefold of mixed type with a reduced triple of weights at the fixed point, then $X$ is linearizable.*]{}\
The next example due to P. Russell shows the importance of the assumption on $X$ being tame.\
[**Example.**]{} Let $S$ be a contractible surface of non–negative log–Kodaira dimension. Put $Y = S \times \C$, and let $X$ be the Kaliman modification of $Y$ along $S \times \{0\}$ with center at a point $\{s_0\}
\times \{0\}$. The tautological $\C^*$–action on $\C$ lifts to a $\C^*$–action on $Y$, which in turn induces a $\C^*$–action of mixed type on $X$ with the reduced triple of weights $(-1,\,-1,\,1)$. Here $X// \C^* \simeq S
\not\simeq \C^2 // \omega_r$, and being an exotic $\C^3$, $\,X$ is not linearizable.\
Now the idea of \[KoRu, Ru 1\] can be described as follows. Starting with a tame contractible $\C^*$–threefold $X$ with a non–reduced triple of weights $(-a,\,b,\,c)$, we may factorise it by a cyclic subgroup of $\C^*$ to make the triple of weights being reduced. By Proposition 5.3, we arrive in this way to $\C^3$ with a linear $\C^*$–action. Namely, put $$\alpha = (b,\,c),\,\beta =
(a,\,c),\,\gamma = (a,\,b)\,\,\,\,{\rm and}\,\,\,\,a' = a/\beta\gamma,\,b' =
b/\alpha\gamma,\,c' = c/\alpha\beta\,.$$
Then both $(\alpha,\,\beta,\,\gamma)$ and $(a',\,b'\,c')$ are reduced triples and $(a',\,\alpha) = (b',\,\beta) = (c',\,\gamma) = 1$. Furthermore, $B
= X / \omega_{\alpha\beta\gamma}$ is a tame $\C^*$–threefold of mixed type with the reduced triple of weights $(-a',\,b',\,c')$, which is, due to Proposition 5.3, isomorphic to $\C^3$ with a linear $\C^*$–action of the same type.\
[**5.4. Russell’s threefolds.**]{} P. Russell \[Ru 1\] realized the converse procedure. Starting now with $\C^3$ with a linear diagonalized reduced mixed $\C^*$–action and passing to the corresponding tricyclic coverings, he reconstructed all possible tame contractible affine $\C^*$–threefolds of mixed type. In what follows we call them [*Russell’s threefolds*]{}.
The principal point of Russell’s construction is the choice of branching divisors $Z_0,\,Z_1,\,Z_2 \subset \C^3$ of the tricyclic covering. The first of them $Z_0$ appears naturally. Indeed, let $X$ be a contractible $\C^*$–threefold of the mixed type $(-a,\,b,\,c)$ with the fixed point $x_0
\in X$. Put $$X^+ = \{x \in X\,|\,\lim_{\lambda \to 0} G_{\lambda} x = x_0
\}\,,$$ $$X^- = \{x \in X\,|\,\lim_{\lambda \to 0} G_{\lambda^{-1}} x = x_0
\}\,.$$
Then $X^+ ,\,X^-$ are isomorphic to $\C^2$ and $\C$ respectively and meet transversally. We call them [*the positive plane*]{} resp. [*the negative axis*]{}. If $\sigma\,:\,X \to B,\,B = X / \omega_{\alpha\beta\gamma} \simeq
\C^3$, is the quotient morphism, then $\sigma (X^{\pm}) = B^{\pm}$ are, respectively, the coordinate plane $x = 0$ and the coordinate axis $y = z = 0$, and $\sigma$ is branched to order $\alpha$ along $Z_0 = B^+$.
The two other divisors $Z_1,\,Z_2$ are chosen as follows. In order to get $X_i
\simeq \C^3$ in diagram (2) we take $Z_i \simeq \C^2,\,i=1,2$. Moreover, after passing to the first covering we would like to have in $X_i \simeq
\C^3\,(i=1,2)$ the situation described in Theorem 4.3. To this point these three embedded planes $Z_0,\,Z_1,\,Z_2 \subset \C^3$ should satisfy the following conditions:\
**
i\) $Z_i$ is equivalent to a coordinate plane under a tame automorphism of $\C^3,\, i=1,2$;
ii\) $Z_0 \cup Z_1 \cup Z_2$ is a normal crossing divisor;
iii\) $Z_i$ are invariant under the $\C^*$–action $(\lambda,\,x,\,y,\,z)
\longmapsto (\lambda^{-a'} x,\,\lambda^{b'} y,\,\lambda^{c'} z)$ on $\,\C^3,\,
i=0,1,2$;
iv\) locally at the origin the triple $Z_0,\,Z_1,\,Z_2$ determines a quasi–homogeneous coordinate system [^13]
v\) the intersection $Z_1 \cap Z_2$ consists of the negative axis $B^-$ and of $r-1$ closed $\C^*$–orbits, where $r \ge 1$.
\
If $Z_1,\,Z_2$ can be linearized simulteneously (this corresponds to $r =
1$), then clearly $X \simeq \C^3$. Therefore, in interesting cases $r > 1$.
Put $C_i = Z_i \cap P,\,i=1,2$, where $P = \{x = 1\} \subset \C^3$. Note that the surface $Z_i \subset \C^3$ is the closure of the orbit of the curve $C_i$ under the $\C^*$–action. The affine plane $P$ is invariant with respect to the induced $\omega_{a'}$–action, and by (iii) the curves $C_1, \,C_2 \subset P$ should be $\omega_{a'}$–invariant, too. Furthermore, $C_1,\,C_2$ are isomorphic to $\C$ and meet normally at the origin $P \cap B^-$ and in $r-1$ other points. If $C_i$ are given in the plane $P$ by the equations $p_i(y,\,z)
= 0, \,p_i \in \C[y,\,z],\,i=1,2$, then the equations of $Z_i$ are $\p_i
(x,\,y,\,z) = 0$, where $\p_i \in \C [x,\,y,\,z], \,i=1,2$, are defined as follows: $$\p_1 (x^{a'},\,y,\,z) = x^{-b'} p_1 (x^{b'} y,\,x^{c'} z)\,\,\,{\rm
and}\,\,\,\p_2 (x^{a'},\,y,\,z) = x^{-c'} p_2 (x^{b'} y,\,x^{c'}
z)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$$ As soon as a pair of plane curves $C_1,\,C_2$ as above is chosen in such a way that $\p_i$ are polynomials, $i =
1,2$, the corresponding triple $Z_0,\,Z_
1,\,Z_2$ satisfies all the conditions (i)-(v).\
[**5.5. Theorem.**]{} a) \[KoRu, Ru 1\] [*Fix two reduced triples $(a',\,b',\,c')$ and $(\alpha,\,\beta,\,\gamma)$ of positive integers such that $(a',\,\alpha) = (b',\,\beta) = (c',\,\gamma) = 1$. Let $C_i = p_i^{-1} (0),
\,Z_i = \p_i^{-1} (0),\,i=1,2,$ and $Z_0 = \{x = 0\}$ be as above. Let $X \to
\C^3$ be the tricyclic covering ramified to order $\alpha$ over $Z_0$, to order $\beta$ over $Z_1$ and to order $\gamma$ over $Z_2$[^14]. Then $X$ is a Russell threefold. Conversely, any Russell threefold is obtained by the above construction.*]{}
b\) \[ML, KaML 1,2\] [*X as above is an exotic $\C^3$ except in the cases when $$(r-1)(\beta - 1)(\gamma - 1) = 0\,.$$*]{} [**5.6. Remark.**]{} Putting $p_2
(y,\,z) = z$, which is possible, we may present a Russell threefold $X$ as the hypersurface in $\C^4$ with the equation $$\p_1(x^{\alpha},\,y,\,z^{\gamma}) +
t^{\beta} = 0\,.$$
[**5.7. Examples.**]{} Put $(-a', \,b',\,c') = (-r + 1,\,1,\,1)$, where $r \ge 2$, and $p_1 (y,\,z) = y + y^r + z$. Then $\p_1 (x,\,y,\,z) = y + x y^r
+ z$ and $$X = \{y + x^{\alpha} y^r + z^{\gamma} + t^{\beta} = 0\} \subset
\C^4\,.$$ In the simplest non–trivial case $r=2,\,\alpha = 1,\,\beta = 3,
\gamma = 2$ we get the affine cubic $X_0 \subset \C^4$ with the equation $$y +
xy^2 + z^2 + t^3 = 0\,,$$ which is an exotic $\C^3$ \[ML\].\
The following result makes precise the statement of Theorem 5.5, b). It was obtained by a rather elementary method of analyzing the defining equations of Russell’s threefolds \[KaML 1,3\],\
[**5.8. Theorem**]{} \[KaML 1\]. [*Let $X$ be a Russell threefold constructed by the data $(r,\,\alpha,\,\beta,\,\gamma)$, where $r > 1,\,\alpha
\ge 2,\, \beta,\,\gamma \ge 4$. Then there is no dominant morphism $\C^3 \to
X$.* ]{}\
It was previously known \[Ru 1\] that $\k (X) = 2$ if $ \alpha \ge a'\beta\gamma$ and $\beta,\,\gamma >> 1$; furthermore, $\k (X) \ge 0$ if $a' = 1,\, \alpha
\ge 2, \, \beta,\,\gamma \ge 4$. But $\k (X) = -\infty$ for the Russell’s threefolds $X$ as in 5.7 with $\alpha = 1$, since the complement $X \setminus
\{y = 0\}$ is isomorphic to $\C^* \times \C^2$. Furthermore, the cubic $X_0$ from 5.7 is dominated by $\C^3$.\
[**5.9. The Makar–Limanov invariant**]{}\
[**Definition.**]{} Recall that a derivation $\partial$ of a ring $A$ is called [*locally nilpotent*]{} if each element $a \in A$ is vanished by an appropriate positive power $n = n(a)$ of $\partial$, i.e. $\partial^n (a) = 0$. Denote $A^{\partial} = {\rm Ker}\,\partial$; $A^{\partial}$ is called [*the ring of constants of*]{} $\partial$. Let $LN(A)$ denote the set of all locally nilpotent derivations of $A$. Put $A_0 = \bigcap_{\partial \in LN(A)}
A^{\partial}$. We call $A_0$ [*the ring of absolute constants of $A$*]{}. Note that $A_0 = \{\C\}$ if $A = \C[x_1,\,\dots,\,x_n]$ is a polynomial ring.\
The subring $A_0 \subset A$ of the absolute constants is invariant under ring isomorphisms. It was introduced in \[ML\], where it was shown that it is non–trivial in the case of the algebra $A=\C[X_0]$ of the regular functions on the Russell cubic $X_0$ (see 5.7). We call $A_0$ [*the Makar–Limanov invariant of $A$*]{}.\
[**5.10. Theorem**]{} \[KaML 2, Theorem 8.3\].
*Let $A = \C[X]$, where $X$ is a Russell threefold. Then $A_0 = A$ (i.e. there is no locally nilpotent derivations on $A$) except in the following two cases:*
i\) if $X = \{x + x^ry + z^{\beta} + t^{\gamma} = 0\}$, then $A_0 =
\C[x]$;
ii\) if $X = \{x + (x^r + z^{\beta})^l y + t^{\gamma} = 0\}$, then $A_0 =
\C[x,\,z]$.
\
This proves (b) of Theorem 5.5.\
**6. APPENDIX: Simply connectedness of $\C^*$–equivariant cyclic coverings**
\
The results of this section are due to Sh. Kaliman [^15]. The original presentation has been modified by P. Russell by picking out the group theoretic component (see Proposition 6.2). In particular, he used the following\
[**6.1. Definition**]{}. Let $G$ be a group. We say that a subgroup $H
\subset G$ is [*normally generated by elements*]{} $a_1,\dots, a_n \in H$ if it is generated by the set of all elements conjugate to $a_1,\dots, a_n$. Thus, $H$ is the minimal normal subgroup of $G$ that contains $a_1,\dots, a_n$. We denote it by $\,<<a_1,\dots, a_n>>$.
We say that $G$ is [*normally one–generated*]{} if $G = \,<<a>>\,$ for some element $a \in G$. If $A, \,B \subset G$, then $[A, \,B]$ denotes the subgroup generated by all the commutators $[a,\,b] = aba^{-1}b^{-1}$, where $a \in A,\,b
\in B$.\
[**6.2. Proposition.**]{} [*Put $K = [G,\,G]$. Assume that $G =
\,<<a>>\,$ is normally one–generated and that $G_{\rm ab} \simeq \Z$ where $G_{\rm ab} = G/K$ is the abelianization of $G$. Then the following statements hold.\
a) $K = \,<<[a,\,K]>>$.\
b) Suppose that $[a^l,\,K] = {\bf 1}$ for some $l \neq 0$. Fix any $k \in
\Z$ with $(k,\,l) = 1$. Let $G_k = \rho^{-1} (H_k)$, where $\rho\,:\,G \to
G_{\rm ab} = G/K$ is the canonical epimorphism and $H_k \subset G_{\rm ab}$ is the subgroup generated by $a^k ({\rm mod}\,K)$. Then $G_k = \,<<a^k>>$.* ]{}\
[*Proof.*]{} a) Denote by $A$ the cyclic subgroup generated by $a$: $A =
\,<a>\, \subset G$. Set $M_a = \,<< \,[A,\,G]\, >>\, = \,<<\,[a^k,\,G]\,|\,k
\in \Z\,>>$.\
[**Claim 1.**]{} $K = M_a$.\
[*Proof.*]{} The abelianization $G_{\rm ab}$ of $G$ is a free cyclic group generated by the class $a \,({\rm mod}\,K) \neq 0$. From the short exact sequence $${\bf 1} \to K \to G \to G_{\rm ab} \simeq \Z \to {\bf 0}$$ it follows that the commutator subgroup $K = [G,\,G]$ consists of the elements $$g
= \prod\limits_{i=1}^r c_ia^{m_i} c_i^{-1} \in G$$ such that $\sum\limits_{i=1}^r m_i = 0$.
Set $k_0 = 0$ and $k_i = \sum\limits_{j=1}^i m_j$, so that $m_i = -k_{i-1} +
k_i$. Denote by $\sim b$ any element conjugate to $b$. Since $ca^{k+l}c^{-1} =
(ca^kc^{-1})(ca^lc^{-1})$, with the above notation every element $g \in G$ can be written as [^16] $$g =
\prod\limits_{i=1}^r (\sim a^{m_i} ) = \prod\limits_{i=1}^r (\sim a^{-k_{i-1} +
k_i}) = (\prod\limits_{i=1}^{r-1} (\sim a^{k_i})(\sim a^{-k_i}))\,a^{k_r}\,,$$ where $k_r = 0$ iff $g \in K$. Furthermore, note that $$(\sim a^k) ( \sim
a^{-k}) = ca^kc^{-1}da^{-k}d^{-1} = c(a^kc^{-1}da^{-k}d^{-1}c)c^{-1} =
c[a^k,\,b]c^{-1} = \,\sim [a^k,\,b]\,,$$ where $b = c^{-1}d$. Thus, we have $$g
= (\prod\limits_{i=1}^{r-1} (\sim[a^{k_i} ,\,b_i]))\,a^{k_r}\,.$$ If $g \in K$, then $k_r = 0$ and, therefore, $g \in M_a$. This proves the inclusion $K
\subset M_a$. Since, evi
dently, $M_a \subset K$, we have $K = M_a$.\
Put $$N_a = \,<<[a^{\epsilon},\,G]\,|\,\epsilon = \pm1>>\, \,\subset M_a\,.$$
[**Claim 2.**]{} $M_a = N_a$.\
[*Proof.*]{} From the identity $$[a^k,\,b] =
(a^{k-1}[a,\,b]a^{-k+1})[a^{k-1},\,b]\,,\,\,\,k\in\Z_{>0}\,,$$ we obtain by induction $$[a^k,\,b] =
(a^{k-1}[a,\,b]a^{-k+1})(a^{k-2}[a,\,b]a^{-k+2})\dots(a[a,\,b]a^{-1})[a,\,b]
\in N_a\,.$$ To show that $[a^{-k} ,\,b] \in N_a,\,k\in\Z_{>0}$, it is enough to replace $a$ by $a^{-1}$ in the above identities. Thus, $M_a \subset N_a$.\
[**Claim 3.**]{} $N_a = <<[a,\,K]>>$.\
[*Proof.*]{} Since $G_{\rm ab} = <a ({\rm mod}\,K) >$ is a cyclic group, any element $b \in G$ can be written as $b = a^md$, where $d \in K$. Therefore, $$[a^{\epsilon},\,b] = a^{\epsilon}(a^m d)a^{-\epsilon}(a^m d)^{-1} = a^m
(a^{\epsilon} d a^{-\epsilon} d^{-1} )a^{-m} =\, \sim [a^{\epsilon} ,\,d] \in
\,<<[a^{\epsilon} ,\,K]>>\,.$$ Finally, $$[a^{-1},\,b]^{-1} = a^{-1}[a,\,b] a
\in \,<<[a,\,K]>>\,,$$ and hence $[a^{-1},\,b] \in \,<<[a,\,K]>>$.\
Now (a) follows from Claims 1–3.\
To prove (b), we start with the following\
[**Claim 4.**]{} [*Under assumptions of (b), $K = \,<<[a^k,\,K]>>$.*]{}\
[*Proof.*]{} Represent $1 = \mu l + \nu k$, where $\mu, \nu \in \Z$. Then for $d \in K$ we have $$[a^{\epsilon} ,\,d] = a^{\epsilon} d a^{-\epsilon}
d^{-1} = (a^k)^{\epsilon\nu} d (a^k)^{-\epsilon\nu} d^{-1} \in
\,<<[a^k,\,K]>>\,.$$ Since $\,<<[a^k,\,K]>>\, \subset K$ and by (a) $K =
\,<<[a,\,K]>>\,$, the Claim follows.\
Furthermore, note that for any $c \in G_k$ we have $c = a^{mk} d$, where $d \in
K$. Therefore, to prove (b) it suffices to show that $K \subset \,<<a^k>>$.
Take $[g,\,h] \in K$ arbitrary. Due to Claim 4 we have the presentation $$[g,\,h] = \prod\limits_{i=1}^N d_i[a^{\nu_i k},\,c_i] d_i^{-1} =
\prod\limits_{i=1}^N (d_i a^{\nu_i k}d_i^{-1})((d_ic_i)a^{-\nu_i
k}(d_ic_i)^{-1})$$ $$= \prod\limits_{i=1}^N (\sim a^{\nu_i k}) (\sim a^{-\nu_i
k}) \in \,\, <<a^k>>\,.$$ This proves (b).\
[**6.3. Notation.**]{} Let $X$ be a smooth irreducible algebraic variety, $q \in \C[X],\,F_0 = q^{*}(0)$ and $F_1 = q^{-1}(1)$. Assume that $F_0$ is a reduced and irreducible divisor. Fix a smooth complex disc $\Delta\subset X$ which meets $F_0$ normally at a smooth point of $F_0$, and a small positive simple loop $\delta \subset \Delta$ around $F_0$. It defines uniquely up to conjugacy an element $\alpha \in \pi_1(X \setminus F_0)$ [^17]. Following Fujita \[Fu 2, (4.17)\] we call such an $\alpha$ [*the vanishing loop of the divisor $F_0$*]{}, and the group $\,<<\alpha>>\, \subset
\pi_1(X \setminus F_0)$ [*the vanishing subgroup of $F_0$*]{}, keeping in mind that $\,<<\alpha>>\,$ is contained in the kernel of the natural surjection $i_*\,:\,\pi_1(X \setminus F_0) \to \pi_1(X)$.\
The following statement should be well known. However, in view of the lack of references we sketch the proof.\
[**6.4. Lemma.**]{} ${\rm Ker}\,i_* = \,<<\alpha>>\,$.\
[*Proof.*]{} Let a loop $\gamma\,:\,S^1 \to X \setminus F_0$ represents the class $[\gamma] \in {\rm Ker}\,i_*$, i.e. $\gamma$ is contractible in $X$. Fix a stratification of $F_0$ which satisfies the Whitney condition A and contains the regular part ${\rm reg}\,F_0$ of $F_0$ as an open stratum. By the Thom Transversality Theorem the homotopy $S^1 \times [0,\,1] \to X$ of $\gamma
= \gamma_0$ to the constant loop $\gamma_1 \equiv {\rm const}$ can be chosen being transversal to the stratification, and therefore such that its image meets the divisor $F_0$ in a finite number of its regular points only. We may also assume that these intersection points $p_1,\,\dots,\,p_n \in {\rm
reg}\,F_0\,, \,\,\,p_i \in \gamma_{t_i} \cap F_0$, correspond to different values $0 < t_1 < \dots < t_n < 1$ of the parameter of homotopy $t \in [0, 1]$. If $s_i \in [0,\,1],\,\,s_i < t_i < s_{i+1},$ and ${\bar \gamma}_i =
\gamma_{s_i}\,:\,S^1 \to X \setminus F_0 , \, i=1,\dots,n+1$, then clearly ${\bar \gamma}_{i+1}^{-1}\cdot {\b
ar \gamma}_i \approx \delta_i^{\epsilon_i}$, i.e. ${\bar \gamma}_i \approx
{\bar \gamma}_{i+1}\cdot\delta_i^{\epsilon_i}$, where $\delta_i$ is a vanishing loop of $F_0$ at the point $p_i$ and $\epsilon_i = \pm 1$, and ${\bar
\gamma}_{n+1} \approx {\rm const}$. Thus, $[\gamma] = [{\bar \gamma}_1] =
[\delta_n]^{\epsilon_n}\cdot \dots \cdot [\delta_1]^{\epsilon_1} \in
\,<<\alpha>>\,$, and we are done.\
[**6.5. Corollary.**]{} [*If $\pi_1 (X) = {\bf 1}$, then the group $G =
\pi_1 (X \setminus F_0)$ is normally one–generated by the vanishing loop $\alpha$ of $F_0$.*]{}\
[**6.6.**]{} Assume further that the restriction $q\,|\,(X \setminus
F_0)\,:\,X \setminus F_0 \to \C^*$ is a smooth fibration. Then we have the exact sequence $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf 1} \to \pi_1(F_1) {\stackrel{i_*}{\longrightarrow}} \pi_1 (X \setminus F_0) {\stackrel{q_*}{\longrightarrow}} \Z \to {\bf 0} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)$$ such that $q_*(\alpha) = 1 \in \Z$.\
[**6.7. Lemma.**]{} [*In the assumptions as above suppose additionally that $X$ is simply connected. Then $i_*\pi_1(F_1) = K = [G,\,G]$, and $G_{\rm
ab} = H^1 (X \setminus F_0) \simeq \Z$.*]{}\
[*Proof.*]{} By Corollary 6.5 we have $G = \,<<\alpha>>\,$. From the exact sequence (4) it follows that $$i_*\pi_1(F_1) = {\rm Ker}\,q_* = \{g =
\prod\limits_{i=1}^r (\sim\alpha^{k_i})\,\,|\,\,\sum\limits_{i=1}^r k_i = 0\} =
K$$ (see the proof of Claim 1 in Proposition 6.2). This proves the first assertion. The second one follows by applying (4) once again.\
[**6.8. Lemma.**]{} [*Let, in the notation as above, $q$ be a quasi–invariant of a positive weight $l$ of a regular $\C^*$–action $t$ on $X
\setminus F_0$. Then $[\alpha^l,\,K] = {\bf 1}$.*]{}\
[*Proof.*]{} Let $\varphi_l\,:\,Y_l \to X \setminus F_0$ be the cyclic covering of order $l$: $$Y_l = \{(x,\,z) \in (X \setminus F_0) \times
\C\,|\,z^l = q(x)\}\,.$$ Put $q_l = q \circ \varphi_l\,:\,Y_l \to \C^*$. Define a morphism $\theta\,:\,F_1 \times \C^* \to Y_l$ as follows: $$\theta
(x,\,\lambda) = (t(\lambda,\,x),\,\lambda)\,, \,x\in F_1,\,\lambda \in \C^*\,.$$ It is easily seen that $\theta\,:\,F_1 \times \C^* \to Y_l$ is an isomorphism. We have $$\varphi_* \pi_1(Y_l) = \varphi_* (\pi_1(F_1) \times \Z)
= <i_*\pi_1(F_1),\,\alpha^l>\,.$$ This implies that $\alpha^l$ commutes with $K
= i_*\pi_1(F_1)$.\
The next theorem is the main result of this section.\
[**6.9. Theorem (Sh. Kaliman).**]{} [*Let $X$ be a simply connected smooth irreducible algebraic variety, $q \in \C[X]$ be a regular function on $X$ such that\
i) $F_0 = q^*(0)$ is a smooth reduced irreducible divisor, and\
ii) $q\,|\,(X \setminus F_0)$ is a quasi–invariant of weight $l > 0$ of a regular $\C^*$–action $t$ on $X \setminus F_0$.\
Let $\sigma_k \,:\,X_k \to X$ be the cyclic covering branched to order $k$ over $F_0$: $$X_k = \{(x,\,z) \in X \times \C\,|\,q(x) =
z^k\}\,,\,\,\,\,\,\,\,\,\sigma_k (x,\,z) = x\,.$$ If $(k,\,l) = 1$, then $X_k$ is simply connected.* ]{}\
[*Proof.*]{} Put $q_k = q\circ \sigma_k \in \C[X_k]$ and $F_{k,0} =
q_k^{-1}(0) \subset X_k$. Since $X_k \setminus F_{k,0} \to X \setminus F_0$ is a non–ramified $k$-sheeted cyclic covering, the induced homomorphism $$(\sigma_k)_*\,:\,\pi_1(X_k \setminus F_{k,0}) \to \pi_1(X \setminus F_0) =
G$$ is an injection onto a normal subgroup $G_k$ of $G$ of index $k$, and $G/G_k \simeq \Z/k\Z$. Clearly, $\alpha^k \in G_k$ is covered by a vanishing loop $\beta \in \pi_1(X_k \setminus F_{k,0})$ of the smooth divisor $F_{k,0}
\subset X_k$, i.e. $(\sigma_k)_* (\beta) = \alpha^k$. Therefore, $\,<<\alpha^k>> \,\subset G_k$.
In fact, $G_k$ has the same meaning that in Proposition 6.2(b), i.e. $G_k =
q_*^{-1} (H_k)$, where $H_k = k\Z \subset \Z \simeq G_{\rm ab}$. Indeed, by the universal property of the commutator subgroup, under the homomorphism $\tau\,:\, G \to G/G_k \simeq \Z/k\Z$ we have $K \subset {\rm Ker}\,\tau =
G_k$, and hence $G_k = q_*^{-1} (q_* (G_k))$. Furthermore, $q_* (G_k) \supset
k\Z = H_k$, because $\alpha^k \in G_k$ and $q_* (\alpha_k)=1 \in \Z$. Actually, $q_* (G_k) = H_k$, since $[G\,:\,G_k] = k$. It follows that $G_k = q_*^{-1}
(H_k)$.
By Lemma 6.8, we have $[\alpha^l,\,K] = {\bf 1}$, so that Proposition 6.2(b) can be applied. Due to this Proposition, $G_k = \,<<\alpha^k>>\,$. Or, what is the same, $\pi_1(X_k \setminus F_{k,0}) = \,<<\beta>>\,$. The inclusion $i\,:\,X_k \setminus F_{k,0} \hookrightarrow X_k$ induces an epimorphism $i_*\,:\,\pi_1(X_k \setminus F_{k,0}) \to \pi_1(X_k)$ with the kernel $\,<<\beta>>\,$ (see Lemma 6.4). Thus, $\pi_1(X_k) = {\bf 1}$, as desired.
**7. Concluding remarks**
\
Of course, in such a short survey it is impossible to touch all the interesting related topics. Let us make just a few remarks.\
[**7.1.**]{} Due to a lemma of T. Fujita \[Fu 2, (2.4)\] any smooth acyclic algebraic surface is affine. In general, this does not hold in higher dimensions. Indeed, J. Winkelmann \[Wi\] constructed a free regular $\C_+$–action on $\C^5$ with the quotient $\C^5 // \C_+ = Q \setminus Z$, where $Q$ is a smooth affine quadric of complex dimension four and $Z \subset
Q$ is a smooth codimension two subvariety. This quotient is diffeomorphic to $\R^8$, but it is not Stein.\
[**7.2.**]{} By the Gurjar–Shastri Theorem \[GuSha\] any smooth acyclic surface is rational. All the exotic $\C^n$–s in the present paper are rational as well. For instance, Russell’s threefolds are rational being $\C^*$–varieties with a rational quotient. The general problem (posed by Van de Ven \[VdV\]) [*whether a smooth contractible quasiprojective variety is rational*]{}, is still open (for this and the related Hirzebruch problem on compactifications of $\C^n$ see e.g. \[MS, Fur\]).\
[**7.3.**]{} At last, we remind the well known Abhyankar–Sathaye problem on equivalence of embeddings $\C^k \hookrightarrow \C^n$, where $k < n \le
2k+1$. Note that for $n \ge 2k+2$ all such embeddings are equivalent (Jalonek–Kaliman–Nori–Srinivas; see e.g. \[Ka 4, Sr\] and references therein), while this is unknown already for the embeddings $\C \hookrightarrow \C^3$ and $\C^2 \hookrightarrow \C^3$.\
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Mikhail Zaidenberg
Université Grenoble I
Laboratoire de Mathématiques
associé au CNRS
BP 74
38402 St. Martin d’Hères–cédex
France
e-mail: ZAIDENBE@FOURIER.GRENET.FR
[^1]: this remark is due to T. tom Dieck \[tD 1\]
[^2]: by the Miyanishi-Sugie and Fujita Theorem \[MiSu, Fu 1\] (see also \[Ru 2\]) $\C^2$ is the only contractible, or even acyclic, surface of $\k = -\infty$.
[^3]: cf. the Rigidity Conjecture for acyclic surfaces of log–general type in \[FlZa\].
[^4]: in the surface case a similar transform was called a half–point attachement (detachment) in \[Fu 2\].
[^5]: See \[Ka 2\] for an example which shows that the theorem does not work without the condition $C \subset {\rm reg}\,H$, even with a smooth $C$.
[^6]: we are grateful to Sh. Kaliman for this remark.
[^7]: Another approach to the proof of (a certain part of) this theorem, based on Kempf–Ness and Neeman results on algebraic group actions (see e.g. \[Kr 1, §4\] and references therein) was proposed in the lectures of T. Petrie at the Workshop on Open Algebraic Varieties, CRM, Montréal, December 5–9, 1994; see also \[Pe\].
[^8]: and in particular, all homology $\P^2$–s which are normal surfaces in $\P^3$ endowed with a $\C^*$–action.
[^9]: Sh. Kaliman \[Ka 1\] noted that for $d = 3$ it is trivialized by the Nagata automorphism.
[^10]: see footnote 11 below.
[^11]: Sh. Kaliman has informed me that in fact the condition (i) of Theorem 4.3 is superfluous; see Appendix.
[^12]: being non–negative $k_c$ must coincide with the log–Kodaira dimension $\k$ \[Sa\]
[^13]: this follows, of course, from ii) and iii).
[^14]: that is, $X = {\rm
spec}\,A$, where $A$ is the extension of $\C[x,\,y,\,z]$ by the corresponding roots of $x,\,\p_1$ and $\p_2$.
[^15]: Letter to the author from 12.03.1995. We get them placed here with the kind permission of Sh. Kaliman and P. Russell.
[^16]: Although the equality $\sim a^{k + l} = \,(\sim a^k) (
\sim a^l)\,\,\,$ is not symmetric any more, this does not cause problems, as well as the use of the non–commutative product symbol $\prod$.
[^17]: indeed, this easily follows from the connectedness of the smooth part ${\rm reg}\,F_0$ of $F_0$.
|
---
abstract: 'We performed a series of hydrodynamical calculations of an ultra-relativistic jet propagating through a massive star and the circumstellar matter to investigate the interaction between the ejecta and the circumstellar matter. We succeed in distinguishing two qualitatively different cases in which the ejecta are shocked and adiabatically cool. To examine whether the cocoon expanding at subrelativistic speeds emits any observable signal, we calculate expected photospheric emission from the cocoon. It is found that the emission can explain early thermal X-ray emission recently found in some long gamma-ray bursts. The result implies that the difference of the circumstellar environment of long gamma-ray bursts can be probed by observing their early thermal X-ray emission.'
author:
- AKIHIRO SUZUKI and TOSHIKAZU SHIGEYAMA
title: 'EARLY THERMAL X-RAY EMISSION FROM LONG GAMMA-RAY BURSTS AND THEIR CIRCUMSTELLAR ENVIRONMENTS'
---
INTRODUCTION\[intro\]
=====================
Since the discovery of gamma-ray bursts (GRBs), numerous studies have been done to understand their progenitors, the mechanism to produce their highly energetic emission, and the central engine [see, e.g., @1999PhR...314..575P; @2006RPPh...69.2259M for review]. It is currently known that long GRBs are triggered by the gravitational collapse of massive stars. The spatial and temporal coincidence of GRB 980425 and SN 1998bw[@1998Natur.395..670G] has revealed the connection between long GRBs and a special class of type Ic supernovae (broad lined type Ic SNe), i.e., the firmly established SN-GRB connection . For example, well-known GRBs associated with SNe are GRB030329/SN 2003dh [@2003Natur.423..847H; @2003ApJ...591L..17S], GRB 060218/SN 2006aj [@2006Natur.442.1008C; @2006Natur.442.1011P; @2006Natur.442.1018M], GRB 100316D/SN 2010bh . The increasing number of detected samples of GRB-associated SNe has enabled us to investigate their circumstellar environments.
Especially, whether the circumstellar matter (CSM) of the progenitor is dilute or dense is of particular interest, because it is expected that the CSM interacts with the ejecta and results in producing high-energy emission. The CSM may originate from the stellar material ejected prior to the explosion as a wind or the common envelope if the progenitor of the GRB was in a binary system [@2004ApJ...607L..17P].
Recently, it is reported that thermal components are found in X-ray spectra of some long GRBs, which are taken by [*Swift*]{} satellite 100-1000 seconds after the trigger [@2006Natur.442.1008C; @2011MNRAS.411.2792S; @2011MNRAS.416.2078P; @2012MNRAS.427.2950S; @2012MNRAS.427.2965S]. The component is seen as an excess superposed on a power-law non-thermal component that is usually attributed to synchrotron emission from the forward shock, i.e., the afterglow emission. Spectral analyses reveal that the component can be fitted by a single blackbody spectrum with temperature of $k_\mathrm{B}T=0.1$-$0.9$ keV [see, @2012MNRAS.427.2950S]. The luminosity ranges from $10^{45}$ to $10^{49}$ erg s$^{-1}$. The contribution of the thermal emission to the total X-ray flux is typically a few % up to several 10 %. The emitting radii inferred from the fitting results are $10^{12\mathrm{-}13}$ cm, which are much larger than the typical radius of the progenitor star $\la 10^{11}$ cm. Their durations are several 100 seconds, up to 1000 seconds for the longest case, GRB 060218, which is classified as a low luminosity GRB associated with a supernova SN 2006aj. The number of GRBs whose spectra exhibit the thermal component now reaches several dozens [see, @2012MNRAS.427.2950S; @2012MNRAS.427.2965S].
Several models to explain this emission component have been presented. As an example, it is proposed that the supernova shock breakout can be responsible for the emission of some GRBs [e.g., @2007ApJ...667..351W; @2007MNRAS.375..240L]. On the other hand, for GRB 060218/SN 2006aj, it is pointed out that the radiated energy and the inferred emitting radius are too large to ascribe the emission to the supernova shock breakout from the progenitor surface [e.g., @2007MNRAS.382L..77G]. Therefore, some authors ascribe the large emitting radius to the presence of a stellar wind with a high mass-loss rate. In this model, the shock emerges from the photosphere located in the wind. Another proposed model is the cocoon emission. The cocoon is a hot plasma resulting from the interaction between the jet and the stellar material. It emerges from the star at the same time the collimated jet penetrates the stellar surface and then expands spherically at mildly relativistic speeds. [@2006ApJ...652..482P] investigated emission from the cocoon by combining a numerical radiative transfer calculation with an analytical treatment of the dynamical evolution of the cocoon. While their model is easy to treat, it is necessary to check whether some parameters used there, such as, the total energy of the cocoon, are realized in actual situations by using hydrodynamical calculations. In particular, by performing hydrodynamical calculations, one can estimate the amount of energy deposited into the cocoon out of the total injected energy in a self-consistent way. Furthermore, the large emitting radii inferred from spectral analyses indicate that the emission comes from the region where the CSM is expected to be present. If so, the ejecta-CSM interaction must give rise to thermal X-ray emission. This effect should also be investigated by hydrodynamical calculations. In addition, the cocoon emission might be important as a source of seed photons for inverse Compton to produce high-energy photons with energies of $\sim$ 100 MeV, as pointed out by [@2009ApJ...707.1404T].
In this Letter, to investigate the interaction between the ejected matter and the CSM, we perform special relativistic hydrodynamical calculations of the propagation of a relativistic jet emanating from a massive star in the CSM. In Section 2, we describe our method to calculate the evolution of the jet and the interaction with the CSM. Results of the hydrodynamical simulations and the expected light curves of the emission from the cocoon are presented in Section 3. Finally, in Section 4, we discuss implications from the results and conclude this Letter.
METHOD\[method\]
================
In this section, we briefly explain setups of the hydrodynamical calculations performed in this study. The detailed code description is found in [@thesis].
Hydrodynamics
-------------
We perform hydrodynamical calculations of the propagation of an ultra-relativistic jet in a massive star and the subsequent interaction with the CSM by using the special relativistic hydrodynamics code in 2D spherical coordinates $(r,\theta)$ developed by one of the authors. In this code, we adopt a mapping procedure, in which the width of the radial zones is doubled as the jet head reaches a fraction ($\sim 0.9$) of the maximum of the radial coordinate, in order to calculate the propagation of the jet till $t\sim 1800$ s. Thus, the radial resolution becomes coarser as the time elapses. At $t=0$, the radial coordinate ranges from $r=10^9$ cm to $r=10^{11}$ cm. At the end of the calculations, the maximum of the radial coordinate reaches $r\sim 6\times 10^{13}$ cm. The radial zone is divided into $N_r$ uniform cells and the number $N_r=1024$ is fixed. The angular coordinate $\theta$ ranges from $\theta=0$ to $\theta=\pi/2$ and is composed of $N_\theta=256$ uniform cells.
Simulation setup
----------------
As a presupernova model, we adopt 16TI model in [@2006ApJ...637..914W], which is commonly used in calculations of collapsar jets. In this study, we consider several models to clarify the effect of the ejecta-CSM interaction. Since the spatial distribution of the CSM is highly uncertain, we adopt the simplest steady wind model whose density profile is given by, $$\rho_\mathrm{w}(r)=\frac{\dot{M}}{4\pi r^2v_\mathrm{w}}.$$ The density profile is uniquely determined for a given ratio of the mass-loss rate $\dot{M}$ and the wind velocity $v_\mathrm{w}$. In this study, the wind velocity $v_\mathrm{w}$ is fixed to be $1000$ km s$^{-1}$. We performed calculations with the mass-loss rates of $\dot{M}=10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4},$ and $10^{-3}\ M_\odot\ \mathrm{yr}^{-1}$. In the following, we especially focus on the two extreme cases, the models with $\dot{M}=10^{-7}$ and $10^{-3}\ M_\odot\ \mathrm{yr}^{-1}$(hereafter they are referred to as the dense and dilute CSM models).
The jet is injected from the inner boundary $r=10^9$ cm from $t=0$ to $t=60$ s at a constant energy injection rate by using the same method as the previous works [e.g., @2003ApJ...586..356Z; @2007ApJ...665..569M; @2011ApJ...732...26M]. The parameters specifying the jet injection condition are as follows: the total energy $E_\mathrm{tot}=3\times 10^{52}$ erg, the energy injection rate $\dot{E}=5\times 10^{50}$ erg/s, the opening angle $\theta_\mathrm{j}=10^\circ$, the initial Lorentz factor $\Gamma_\mathrm{0}=5$, and the specific internal energy $\epsilon_0/c^2=20$.
![Color-coded Lorentz factor and density distributions at $t\sim 200$ s for the model with $\dot{M}=10^{-3}\ M_\odot$ yr$^{-1}$ (lower panel) and $\dot{M}=10^{-7}\ M_\odot$ yr$^{-1}$ (upper panel).[]{data-label="figure_compare"}](f2.eps)
RESULT\[result\]
================
Jet dynamics
------------
A lot of previous works on an ultra-relativistic jet emanating from the progenitor star have been carried out and unveiled the dynamical evolution of the jet, such as, the formation of the recollimation shock and the realization of the well-known fireball solution. [e.g., @2003ApJ...586..356Z; @2007ApJ...665..569M; @2011ApJ...732...26M]. Our calculations successfully reproduce and confirm their findings. Some snapshots of the spatial distributions of the Lorentz factor and the density of the dilute CSM model are shown in Figure \[figure\_snapshots\]. The jet propagates in the interior of the progenitor star and then breaks out, and ejects stellar materials into the circumstellar space. As seen in the top right panel, the emergence of a hot material from the jet cavity follows the breakout of the collimated jet. The ejecta rapidly expand to form a spherical cocoon as seen in the bottom left panel of Figure \[figure\_snapshots\]. It is noteworthy that the cocoon expands at mildly relativistic speeds. The appearance and the subsequent expansion of the cocoon have also been reported and investigated by several previous works [see, e.g., @2000ApJ...531L.119A; @2002MNRAS.337.1349R; @2003ApJ...586..356Z; @2005ApJ...629..903L].
Effect of CSM interaction
-------------------------
Results of the dense and dilute CSM models are compared in Figures \[figure\_compare\] and \[figure\_radial\]. Figure \[figure\_compare\] represents the spatial distribution of the Lorentz factor (left) and the pressure (right) at $t\sim 200$ s for the dense CSM model (lower panel) and the dilute CSM model (upper panel).
Near the jet axis ($\theta<10-20^\circ$), no difference between the two models is recognized. On the other hand, in the region with large inclination angles ($\theta>20^\circ$), we can see differences between the models. Denser CSM reduces the size of the cocoon in comparison with dilute CSM. In addition, the pressure distribution shows shell-like structure in the dense CSM.
This difference can also be seen in the radial profiles of some physical variables of the cocoon, as illustrated in Figure \[figure\_radial\]. In both cases, the expansion velocities are mildly relativistic as seen in the top panel. From the bottom panel showing the pressure profiles, one can see that the reverse shock forms as a result of the cocoon-dense CSM interaction. On the other hand, in dilute CSM, the rarefaction wave propagates toward the center in the cocoon.
![Radial profiles along $\theta=45^\circ$ at $t\sim 200$ for the dense CSM model (solid line) and the dilute CSM model (dashed line). Each panel represents radial velocity normalized by the speed of light, the density, and the pressure from top to bottom.[]{data-label="figure_radial"}](f3.eps)
This is due to the aspherical distribution of the energy deposited into the ejecta. Near the jet axis, the energy carried by the jet is too enormous for the CSM to affect the jet propagation. On the other hand, the energy deposited into the cocoon component is much smaller than that of the jet. The kinetic energy and mass of the cocoon component, which are now defined as those confined in the region outside the star and $\theta>10^\circ$, can be obtained from results of the simulation. They are found to be $3\times 10^{50}$ erg and $2\times 10^{-3}$ M$_\odot$ at $t=10$ s and $10^{51}$ erg and $2\times 10^{-2}$ M$_\odot$ at $t=20$ s. These values are almost independent of the mass-loss rate, because the dissipation of the kinetic energy of the jet to form the cocoon takes place in the star. The kinetic energy and mass of the cocoon increase due to the continuous energy and mass injection by the jet. In our calculations, the injection of the jet is terminated at $t=60$ s , which means that the injection of the mass and kinetic energy into the cocoon lasts even after the cocoon begins to expand. The kinetic energy of the cocoon is up to a few $\%$ of the total injected energy and thus has a potential for producing thermal X-ray photons with the observed luminosity $\sim 10^{45-48}$ erg s$^{-1}$ for several hundreds seconds. On the other hand, the mass of the ultra-relativistic jet component, which is defined as the material with the Lorentz factor larger than 100, is $3\times 10^{-6}$ M$_\odot$, while a substantial fraction ($\sim 10^{51}$ erg) of the injected energy is carried by this component.
As the radial profiles along $\theta=45^\circ$ at $t=200$ s in Figure \[figure\_radial\] shows, a reverse shock is formed in the dense CSM model. The other model with the mass-loss rates $10^{-4}$ $M_\odot\ \mathrm{yr}^{-1}$ also form a reverse shock. The energy of the matter ejected immediately after the breakout of the jet from the surface results from the dissipation of a part of the kinetic energy of the jet for the initial several seconds. Denoting the dissipated kinetic energy by $E_\mathrm{dis}$ and the fraction of the internal energy to the total by $\epsilon$, the pressure of the cocoon scales as $$P_\mathrm{c}\sim \frac{\epsilon E_\mathrm{dis}}{4\pi(v_\mathrm{exp}t)^3},$$ where we have assumed that the cocoon is spherically expanding at the velocity $v_\mathrm{exp}$. On the other hand, the ram pressure of the CSM behind the forward shock is given by, $$\rho_\mathrm{w}\Gamma^2c^2\sim \frac{\dot{M}\Gamma_\mathrm{exp}^2c^2}{4\pi v_\mathrm{w}(v_\mathrm{exp}t)^2},$$ where $\Gamma_\mathrm{exp}=(1-v_\mathrm{exp}^2/c^2)^{-1/2}$. The reverse shock forms when the pressure $P_\mathrm{c}$ of the cocoon becomes comparable to the ram pressure $\rho_\mathrm{w}\Gamma^2c^2$ of the shocked CSM. The balance between the pressure of the cocoon and the ram pressure yields the following expression for the time of the reverse shock formation, $$\begin{aligned}
t&\sim& \frac{\epsilon E_\mathrm{dis}v_\mathrm{w}}
{\dot{M}v_\mathrm{exp}\Gamma_\mathrm{exp}^2c^2}
\sim
10^2
\left(\frac{\epsilon}{5.0\times 10^{-4}}\right)
\left(\frac{E_\mathrm{dis}}{10^{51}\ \mathrm{erg}}\right)\\
&&\hspace{5em}\times
\left(\frac{v_\mathrm{w}}{10^8\ \mathrm{km\ s}^{-1}}\right)
\left(\frac{\dot{M}}{10^{-4}\ M_\odot \mathrm{yr}^{-1}}\right)^{-1}\ \mathrm{s},
\nonumber
%0.0005*1e51/((1-0.8^2)^(-1)*(1e-3*2e33/(365*24*60*60))*3e10^2)*(1e8/(3e10*0.9))=11.68\end{aligned}$$ where we have derived the final expression by assuming $v_\mathrm{exp}=0.9c$. The value of the fraction $\epsilon$ is found from the result of hydrodynamical calculations. This rough estimation is consistent with the fact that a reverse shock is observed for models with $\dot{M}=10^{-3}$ and $10^{-4}$ $M_\odot$ yr$^{-1}$ at $t=200$ s and no reverse shock for models with lower mass-loss rates.
![Light curves of the photospheric emission calculated for the models with $\dot{M}=10^{-3}\ M_\odot$ yr$^{-1}$ (solid line) and $\dot{M}=10^{-7}\ M_\odot$ yr$^{-1}$ (dashed line).[]{data-label="figure_emission"}](f4.eps)
Photospheric emission
---------------------
In the following, we investigate whether thermal X-ray emission from GRBs can probe their circumstellar environments. We derive the expected light curve and the spectra of thermal X-ray emission from our models by calculating the photospheric emission.
According to [@2012MNRAS.427.2950S], thermal emission with the isotropic luminosity of the order of $10^{47}$ erg s$^{-1}$ and the photon temperature $\sim 0.1-0.9$ keV is observed. Illuminated by the radiation, heavy atoms, such as oxygen and carbon, are rapidly photo-ionized. The recombination time scale much longer than the ionization time scale keeps those ions fully ionized and the dominant opacity source becomes electron scattering. Thus, we calculated the Thomson photosphere from a distant observer along the axis of the jet ($\theta=0$). In deriving light curves and spectra, we have assumed that the matter and radiation are strongly coupled and the internal energy density on the photosphere is dominated by that of radiation. At first, we briefly consider the properties of the emission. Since the ejecta move at mildly relativistic velocities with the Lorentz factor of a few, the relativistic beaming effect strengthen the emission, especially, in the early phase. From the top and bottom panels of Figure \[figure\_radial\], the radiation temperature of the shocked region for the dense CSM can be estimated to be, $$\Gamma k_\mathrm{B} T_\mathrm{ph}=\Gamma k_\mathrm{B}\left(\frac{3p}{a_\mathrm{r}}\right)^{1/4}\sim 0.1\mathrm{-}0.2\ \mathrm{keV},$$ in the observer frame. This is consistent with observed values.
The resultant light curves and time-integrated $\nu F_\nu$ spectra of the photospheric emission for both models are presented in Figures \[figure\_emission\] and \[figure\_spectra\]. At first, for both models, the photospheric emission is very bright for the first $\sim 200$ sec. This is because the cocoon is hot immediately after the emergence from the stellar or wind photosphere. After the early phase, the radial velocity at the photosphere decreases as the photosphere moves inward and the cocoon gradually cools. This corresponds to the decrease of the luminosity. The cocoon cools in different ways for the dense and the dilute CSM. As seen in Figure \[figure\_radial\], the cocoon component is shocked in the dense CSM model. The shock converts the kinetic energy of the cocoon into the thermal energy and keeps the shocked region hot. As a result, the photospheric emission remains luminous even at $t\sim 1000$ sec. In the dilute CSM model, on the other hand, the cocoon adiabatically cools.
Figure \[figure\_spectra\] shows the time-integrated $\nu F_\nu$ spectra for the dense (upper panel) and dilute (lower panel) CSM models. In each panel, $\nu F_\nu$ spectra integrated over $t=0$-$200$ s (solid line), $t=200$-$1500$ s (dashed line), and $t=0$-$1500$ s (dotted line) are plotted. If we fit a blackbody spectrum with a single temperature to each of these spectra, the temperature is found to be $k_\mathrm{B}T=0.16,$ $0.077$, and $0.13$ keV for the $0$-$200$ s, $200$-$1500$ s, and $0$-$1500$ s spectra of the dense CSM model and $k_\mathrm{B}T=0.061,$ $0.038,$ and $0.051$ keV for the dilute CSM model. In high-energy part, however, a deviation from the planck function is prominent. This shows that each spectrum is actually superposition of blackbody spectra with different temperatures.
![Time-integrated $\nu F_\nu$ spectra of the photospheric emission calculated for the models with $\dot{M}=10^{-3}\ M_\odot$ yr$^{-1}$ (upper panel) and $\dot{M}=10^{-7}\ M_\odot$ yr$^{-1}$ (lower panel). In each panel, $\nu F_\nu$ spectra integrated over $t=0$-$200$ s (solid line), $t=200$-$1500$ s (dashed line), and $t=0$-$1500$ s (dotted line) are plotted.[]{data-label="figure_spectra"}](f5.eps)
DISCUSSION AND CONCLUSIONS
==========================
In this study, we performed hydrodynamical simulations of a jet emerging from a massive star surrounded by the CSM. Especially, we focus on the effect of the interaction between the ejecta and the CSM. The CSM is assumed to be a steady wind with the wind velocity $v_\mathrm{w}=1000$ km s$^{-1}$ and the mass-loss rates ranging from $\dot{M}=10^{-7}\ M_\odot$ yr$^{-1}$ to $10^{-3}\ M_\odot$ yr$^{-1}$. We found that the dynamical behavior of the cocoon, which expands at sub-relativistic speeds, is significantly affected by the ejecta-CSM interaction, while the collimated jet is so energetic that the CSM can not decelerate it even for the dense CSM model. In the dense CSM, the cocoon is shocked and thus remains hot even at $\sim$1000 s after the jet injection. On the other hand, in the dilute CSM, the cocoon cools adiabatically. Furthermore, calculating the photospheric emission from the cocoon, we found that the difference can be detected by observing their early thermal X-ray emission.
Here we compare our results with the calculation by [@2006ApJ...652..482P]. They consider emission from a freely expanding spherical cocoon with the initial internal energy of $3\times 10^{51}$, $10^{52}$, and $3\times 10^{52}$ erg. On the other hand, the internal energy of the cocoon reproduced in the present calculations is less than these assumed values. As a result, for the dilute CSM model, where a freely expanding cocoon is realized, the photon temperature of the photospheric emission is lower than $0.1$ keV. For the dense CSM model, a fraction of the kinetic energy of the cocoon can be converted into the internal one by the reverse shock, which leads to a higher photon temperature and brighter photospheric emission. This effect is not taken into account by [@2006ApJ...652..482P].
From GRB 060218/SN 2006aj, a bright thermal X-ray emission with the temperature $k_\mathrm{B}T\simeq 0.1$-$0.2$ keV is observed even in a few thousand seconds after the trigger. [@2006Natur.442.1008C] ascribed the long-lived thermal X-ray emission to a high mass-loss rate of $\dot{M}\simeq 3\times 10^{-4}$ $M_\odot$ yr$^{-1}$. Our results show that the shocked cocoon is realized at this mass-loss rate. Interestingly, some fundamental features of the emission, such as the photon temperature and the long duration, are also reproduced by the photospheric emission from the shocked cocoon . Therefore, this event might occur in a dense circumstellar environment. However, we cannot reproduce the detailed temporal evolution of the observed photon temperature and the light curve by the present calculations in which the steady wind model is assumed. We attribute this discrepancy to the inhomogeneities of the CSM profile as described below.
We should note that there are some uncertainties in this study. In particular, the properties of the photospheric emission calculated in this study are expected to strongly depend on the spatial distribution of the CSM. Although we adopted a steady wind model in this study, it may not be realized in actual circumstellar environments of progenitor systems of long GRBs. Of course, the slope of the density profile of the wind depends on its mass-loss history prior to the gravitational collapse. If the progenitor had been rapidly rotating, the wind could have angular dependence. Furthermore, some authors point out that long GRB progenitors have evolved in binary systems [e.g., @2004ApJ...607L..17P], in which the structure of the material surrounding the system is expected to be much more complicated than single star progenitor cases. We regard the influence of the spatial distribution of the CSM on the properties of early thermal X-ray emission as one of future works.
We appreciate A. Heger for kindly providing us the presupernova model used in this study. This work has been partly supported by Grant-in-Aid for JSPS Fellows (21$\cdot$1726) of the Ministry of Education, Science, Culture, and Sports in Japan. Numerical computations were carried out in part on the Cray XT4 and the middle cluster at the Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan.
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abstract: 'Cache attacks exploit memory access patterns of cryptographic implementations. Constant-Time implementation techniques have become an indispensable tool in fighting cache timing attacks. These techniques engineer the memory accesses of cryptographic operations to follow a uniform key independent pattern. However, the constant-time behavior is dependent on the underlying architecture, which can be highly complex and often incorporates unpublished features. *CacheBleed* attack targets cache bank conflicts and thereby invalidates the assumption that microarchitectural side-channel adversaries can only observe memory with cache line granularity. In this work, we propose [*MemJam*]{}, a side-channel attack that exploits false dependency of memory read-after-write and provides a high quality intra cache level timing channel. As a proof of concept, we demonstrate the first key recovery attacks on a constant-time implementation of AES, and a SM4 implementation with cache protection in the current Intel Integrated Performance Primitives (Intel IPP) cryptographic library. Further, we demonstrate the first intra cache level timing attack on SGX by reproducing the AES key recovery results on an enclave that performs encryption using the aforementioned constant-time implementation of AES. Our results show that we can not only use this side channel to efficiently attack memory dependent cryptographic operations but also to bypass proposed protections. Compared to *CacheBleed*, which is limited to older processor generations, [*MemJam*]{} is the first intra cache level attack applicable to all major Intel processors including the latest generations that support the SGX extension.'
author:
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bibliography:
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title: '[*MemJam*]{}: A False Dependency Attack against Constant-Time Crypto Implementations'
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Introduction
============
In cryptographic implementations, timing channels can be introduced by key dependent operations, which can be exploited by local or remote adversaries [@brumley2005remote; @osvik2006cache]. Modern microarchitectures are complex and support various shared resources, and the operating system (OS) maximizes the resource sharing among concurrent tasks [@schimmel1994unix; @marr2002hyper]. From a security standpoint, concurrent tasks with different permissions share the same hardware resources, and these resources can expose exploitable timing channels. A typical model for exploiting microarchitectural timing channels is for a spy process to cause resource contention with a victim process and to measure the timing of its own or of the victim operations [@ristenpart2009hey; @aciiccmez2007new; @tromer2010efficient; @irazoqui2015s]. The observed timing behavior give adversaries strong evidence on the victim’s resource usage pattern, thus they leak critical runtime data. Among the shared resources, attacks on cache have received significant attention, and their practicality have been demonstrated in scenarios such as cloud computing [@ristenpart2009hey; @zhang2012cross; @irazoqui2015s; @inci2016cache; @yarom2014flush; @gruss2016flush]. A distinguishable feature of cache attacks is the ability to track memory accesses with high temporal and spatial resolution. Thus, they excel at exploiting cryptographic implementations with secret dependent memory accesses [@tsunoo2003cryptanalysis; @osvik2006cache; @benger2014ooh; @inci2015seriously]. Examples of such vulnerable implementations include using S-Box tables [@webster1986design], and efficient implementations of modular exponentiation [@kocc1995analysis].
The weakness of key dependent cache activities has motivated researchers and practitioners to protect cryptographic implementations against cache attacks [@brickell2006software; @tromer2010efficient]. The simplest approach is to minimize the memory footprint of lookup tables. Using a single 8-Bit S-Box in Advanced Encryption Standard (AES) rather than T-Tables makes cache attacks on AES inefficient in a noisy environment, since the adversary can only distinguish accesses between 4 different cache lines. Combining small tables with cache state normalization, i.e., loading all table entries into cache before each operation, defeats cache attacks in asynchronous mode, where the adversary is only able to perform one observation per operation. More advanced side channels such as exploitation of the thread scheduler [@gullasch2011cache], cache attack on interrupted execution of Intel Software Guard eXtension (SGX) [@moghimi2017cachezoom], performance degradation [@allan2016amplifying] and leakage of other microarchitectural resources [@aciiccmez2007predicting; @aciiccmez2010new] remind us the importance of constant-time software implementations. One way to achieve constant-time memory behavior, is the adoption of small tables in combination with accessing all cache lines on each lookup [@tromer2010efficient]. The overhead would be limited and is minimized by the parallelism we can achieve in modern processors. Another constant-time approach adopted by some public cryptographic schemes is interleaving the multipliers in memory known as scatter-gather technique [@brickell2006mitigating].
Constant-time implementations have effectively eliminated the first generation of timing attacks that exploit obvious key dependent leakages. The common view is that performance penalty is the only downside which, once paid, there is no need to be further worried. However, this is far from the reality and constant-time implementations may actually give a false sense of security. A commonly overlooked fact is that constant-time implementations and related protections are relative to the underlying hardware [@ge2016contemporary]. In fact, there are major obstacles preventing us from obtaining true constant-time behavior. Processors constantly evolve with new microarchitectural features rolled quietly with each new release and the variety of such subtle features makes comprehensive evaluation impossible. A great example is the cache bank conflicts attack on OpenSSL RSA scatter-gather implementation: it shows that adversaries with intra cache level resolution can successfully bypass constant-time techniques relied on cache-line granularity [@yarom2017cachebleed]. As a consequence, what might appear as a perfect constant-time implementation becomes insecure in the next processor release–or worse–an unrecognized behavior might be discovered, invalidating the earlier assumption.
Our Contribution
----------------
We propose an attack named [*MemJam*]{} by exploiting false dependency of memory read-after-write, and demonstrate key recovery against two different cryptographic implementations which are secure against cache attacks with experimental results on both regular and SGX environments. In summary:
- **False Dependency Attack:** A side-channel attack on the false dependency of memory read-after-write. We show how to dramatically slow down the victim’s accesses to specific memory blocks, and how this read latency can be exploited to recover low address bits of the victim’s memory accesses.
- **Attack on protected AES and SM4:** Attacks utilizing the intra cache level information on AES and SM4 implementations protected against cache attacks. The implementations are chosen from Intel Integrated Performance Primitives (Intel IPP), which is optimized for both security and speed.
- **Attack on SGX Enclave:** The first intra cache level attack against SGX Enclaves supported by key recovery results on the contant-time AES implementation. The aforementioned constant-time implementation of AES is part of the SGX SDK source code.
- **Protection Bypass:** Bypasses of remarkable protections such as proposals based on constant-time techniques [@brickell2006mitigating; @tromer2010efficient], static and runtime analysis [@zhang2016cloudradar; @irazoqui2016mascat] and cache architecture [@liu2016catalyst; @costan2016sanctum; @kayaalp2017ric; @xu2017vcat].
Experimental Setup and Generic Assumptions
------------------------------------------
Our experimental setup is a Dell XPS 8920 desktop machine with Intel(R) Core i7-7700 processor running Ubuntu 16.04. The Core i7-7700 has 4 hyper-threaded physical cores. Our only assumptions are that the attacker is able to co-locate on one of the logical processor pairs within the same physical core as the victim. In the cryptographic attacks, the attacker can measure the time of victim encryption. The attacker further knows which cryptographic implementation is used by the victim, but she does not need to have any knowledge of the victim’s binary or the offset of the S-Box tables. We will discuss assumptions that are specific to the attack on SGX at Section \[sec:sgx\].
Related Work
============
**Side channels** including power, electromagnetic and timing channels have been studied for a few decades [@kocher2011introduction; @brumley2005remote; @carluccio2005electromagnetic]. Timing side channels can be constructed through the processor cache to perform key recovery attacks against cryptographic operations such as RSA [@inci2015seriously], ECDSA [@benger2014ooh], ElGamal [@zhang2012cross], DES [@tsunoo2003cryptanalysis] and AES [@irazoqui2015s; @osvik2006cache]. On multiprocessor systems, attacks on the shared LLC—a shared resource among all the cores—perform well even when attacker and victim reside in different cores [@irazoqui2015s]. Flush+Reload, Prime+Probe, Evict+Reload, and Flush+Flush are some of the proposed attack methodologies with different adversarial scenarios [@yarom2014flush; @gruss2016flush; @osvik2006cache]. Performance degradation attacks can improve the channel resolution [@gullasch2011cache; @allan2016amplifying]. LLC attacks are highly practical in cloud, where an attacker can identify where a particular victim is located [@ristenpart2009hey; @zhang2012cross]. Despite the applicability of LLC attacks, attacks on core-private resources such as L1 cache are as important [@aciiccmez2010new; @bonneau2006cache]. Attacks on SGX in a system level adversarial scenario are notable examples [@lee2016inferring; @moghimi2017cachezoom]. There are other shared resources, which can be utilized to construct timing channels [@ge2016survey]. Exploitation of Branch Target Buffer (BTB) leaks if a branch has been taken by a victim process [@aciiccmez2007predicting; @aciiccmez2010new; @lee2016inferring]. Logical units within the processor can leak information about the arithmetic operations [@aciicmez2007cheap; @andrysco2015subnormal]. *CacheBleed* proposes cache bank conflicts and false dependency of memory write-after-read as side channels with intra-cache granularity [@yarom2017cachebleed]. However, cache bank conflicts leakage does not exist on current Intel processors, and we verify the authors’ claim that the proposed write-after-read false dependency side channel does not allow efficient attacks.
**Defense:** software and hardware strategies have been proposed such as alternative lookup tables, data-independent memory access pattern, static or disabled cache, and cache state normalization to defend against cache attacks [@tromer2010efficient]. Scatter-Gather techniques have been adopted by RSA and ECC implementations [@brickell2006mitigating]. In particular, introducing redundancy and randomness to the S-Box tables for AES has been proposed [@brickell2006software]. A custom memory manager [@zhou2016software], relaxed inclusion caches [@kayaalp2017ric] and solutions based on cache allocation technology (CAT) such as Catalyst [@liu2016catalyst] and vCat [@xu2017vcat] are proposed to defend against LLC contention. Sanctum [@costan2016sanctum] and Ozone [@aweke2017ozone] are new processor designs with respect to cache attacks. Detection-based countermeasures have also been proposed using performance counters, which can be used to detect cache attacks in cloud environments [@zhang2016cloudradar; @briongos2017cacheshield]. MASCAT [@irazoqui2016mascat] is proposed to block cache attacks with code analysis techniques. CachD [@CachD] detects potential cache leakage in the production software. Nonetheless, these proposals assume that the adversary cannot distinguish accesses within a cache line. That is, attacks with intra cache-line granularity are considered out-of-scope. Doychev et al. proposed the only software leakage detector that consider full address bits as its leakage model [@doychev2017rigorous].
Background
==========
Multitasking
------------
The memory management subsystem shares the dynamic random-access memory (DRAM) among all concurrent tasks, in which a virtual memory region is allocated for each task transparent to the physical memory. Each task is able to use its entire virtual address space without meddling of memory accesses from others. Memory allocations are performed in pages, which each virtual memory page can be stored in a DRAM page with a virtual-to-physical page mapping. The logical processors are also shared among these tasks and each logical processor executes instructions from one task at a time, and switches to another task. Memory write and read instructions work with virtual addresses, and the virtual address is translated to the corresponding physical address to perform the memory operation. The OS is responsible for page directory management and virtual page allocation. The OS assists the processor to perform virtual-to-physical address translation by performing an expensive page walk. The processor saves the address translation results in a memory known as Translation Look-aside Buffer (TLB) to avoid the software overhead introduced by the OS. Intel microarchitecture follows a multi-stage pipeline and adopts different optimization techniques to maximize the parallelism and multitasking during the pipeline stages [@inteloptimze]. Among these techniques, hyper-threading allows each core to run multiple concurrent threads, and each thread shares all the core-private resources. As a result, if one resource is busy by a thread, other threads can consume the remaining available resources. Hyper-threading is abstracted to the software stack: OS and applications interact with the logical processors.
Cache Memory
------------
DRAM memory is slow compared to the internal CPU components. Modern microarchitectures take advantage of a hierarchy of cache memories to fill the speed gap. Intel processors have two levels of core-private cache (L1, L2), and a Last Level Cache (LLC) shared among all cores. The closer the cache memory is to the processor, the faster, but also smaller it is compared to the next level cache. Cache memory is organized into different sets, and each set can store some number of cache lines. The cache line size, which is 64 byte, is the block size for all memory operations outside of the CPU. The higher bits of the physical address of each cache line is used to determine which set to store/load the cache line. When the processor tries to access a cache line, a cache hit or miss occurs respective of its existence in the relevant cache set. If a cache miss occurs, the memory line will be stored to all 3 levels of cache and to the determined sets. Reloads from the same address would be much faster when the memory line exists in cache. In a multicore system, the processor has to keep cache consistent among all levels. In Intel architecture, cache lines follow a write-back policy, i.e, if the data in L1 cache is overwritten, all other levels will be updated. The LLC is inclusive of L2 and L1 caches, which means that if a cache line in LLC is evicted, the corresponding L1 and L2 cache lines will also be evicted [@inteloptimze]. These policies help to avoid stale cached data where one processor reads invalid data mutated by another processor.
L1 Cache Bottlenecks
--------------------
L1 cache port has a limited bandwidth and simultaneous accesses will be block each other. This bottleneck is critical in super-scalar multiprocessor systems. Older processors’ generation adopted multiple banks as a workaround to this problem [@agneroptimize], in which each bank can operate independently and serve one request at a time. While this partially solved the bandwidth limit, it creates the cache bank conflicts phenomena which simultaneous accesses to the same bank will be blocked. Intel resolved the cache bank conflicts issue with the Haswell generation [@inteloptimze]. Another bottleneck mentioned in various resources is due to the false dependency of memory addresses with the same cache set and offset [@inteloptimze; @agneroptimize]. Simultaneous read and write with addresses that are multiples of 4kB is not possible, and they halt each other. The processor cannot determine the dependency from the virtual address, and addresses with the same last 12 bits have the chance to map to the same physical address. Such simultaneous access can happen between two logical processors and/or during the out-of-order execution, where there is a chance that a memory write/read might be dependent on a memory read/write with the same last 12 bits of address. Such dependencies cannot be determined on the fly, thus they cause latency.
Cache Attacks
-------------
Cache attacks can be exploited by adversaries where they share system cache memory with benign users. In scenarios where the adversary can colocate with a victim on the same core, she can attack core-private resources such as L1 cache, e.g., OS adversaries [@lee2016inferring; @moghimi2017cachezoom]. In cloud environment, virtualization platforms allow sharing of logical processors to different VMs; however, attacks on the shared LLC have a higher impact, since LLC is shared across all the cores. In cache timing attacks, the attacker either measure the timing of the victim operations, e.g, *Evict+Time* [@osvik2006cache] or the timing of his own memory accesses, e.g, *Prime+Probe* [@irazoqui2015s]. The attacker needs to have access to an accurate time resource such as the *RDTSC* instruction. In the basic form, attacks are performed by one observation per entire operation. In certain scenarios, these attacks can be improved by interrupting the victim and collecting information about the intermediate memory states. Side channel attacks exploiting cache bank conflicts rely on synchronous resource contention. *CacheBleed* methodology is somewhat similar to Prime+Probe, where the attacker performs repeated operations, and measures it’s own access time [@yarom2017cachebleed]. In a cache bank conflicts attack, the adversary repeatedly performs simultaneous reads to the same cache bank and measures their completion time. A victim on a colocated logical processor who access the same cache bank would cause latency to the attacker’s memory reads.
loop:
rdtscp;
mov %eax, (%r9);
movb 0x0000(%r10), %al;
movb 0x1000(%r10), %al;
movb 0x2000(%r10), %al;
movb 0x3000(%r10), %al;
movb 0x4000(%r10), %al;
movb 0x5000(%r10), %al;
movb 0x6000(%r10), %al;
movb 0x7000(%r10), %al;
add $4, %r9;
dec %r11;
jnz loop;
loop:
rdtscp
mov %eax, (%r9);
movb %al, 0x0000(%r10);
movb %al, 0x1000(%r10);
movb %al, 0x2000(%r10);
movb %al, 0x3000(%r10);
movb %al, 0x4000(%r10);
movb %al, 0x5000(%r10);
movb %al, 0x6000(%r10);
movb %al, 0x7000(%r10);
add $4, %r9
dec %r11
jnz loop
{width=".99\linewidth"}
[*MemJam*]{}: Read-After-Write Attack
=====================================
[*MemJam*]{} utilizes *false dependencies*. Data dependency occurs when an instruction refers to the data of a preceding instruction. In pipelined designs, hazards and pipeline stalls can occur from dependencies if the previous instruction has not finished. There are cases where false dependencies occur, i.e. the pipeline stalls even though there is no true dependency. Reasons for false dependencies are register reuse and limited address space for the Arithmetic Logic Unit (ALU). False dependencies degrade instruction level parallelism and cause overhead. The processor eliminates false dependencies arising from register reuse by a register renaming approach. However, there exist other false dependencies that need to be addressed during the software optimization [@inteldev; @inteloptimze].
In this work, we focus on a critical false dependency mentioned as *4K Aliasing* where data that is multiples of 4k apart in the address space is seen as dependent. 4k Aliasing happens due to virtual addressing of L1 cache, where data is accessed using virtual addresses, but tagged and stored using physical addresses. More than one virtual addresses can refer to the same data with the same physical address and the determination of dependency for concurrent memory accesses, requires virtual address translation. Physical and virtual address share the last 12 bits, and any data accesses whose addresses differ in the last 12 bits (i.e. the distance is not 4k) cannot have a dependency. For the fairly rare remaining cases, address translation needs to be done before resolving the dependency, which causes latency. Note that the *granularity* of the potential dependency, i.e. whether two addresses are considered “same”, depends also on the microarchitecture, as dependencies can occur at the *word* or cache *line* granularity (i.e. ignoring the last 2 or last 6 bits of the address, respectively). These rare false dependencies due to 4K aliasing can be exploited to attack memory, since the attacker can deliberately process falsely dependent data by matching the last 12 bits of his own address with a security critical data inside a victim process.
4K Aliasing has been mentioned in various places as an optimization problem existing on all major Intel processors [@agneroptimize; @inteloptimze]. We verify the results of Yarom et al. [@yarom2017cachebleed], the only security related work regarding false dependencies, which exploited *write-after-read* dependencies. The resulting timing leakage by write stall after read is not sufficient to be used in any cryptographic attack. [*MemJam*]{} exploits a different channel due to the false dependency of *read-after-write*, which causes a higher latency and is thus simply observable. Intel Optimization Manual highlights the *read-after-write* performance overhead in various sections [@inteloptimze]. As described in Section 11.8, this hazard occurs when a memory write is closely followed by a read, and it causes the read to be reissued with a potential 5 cycles penalty[^1]. In Section B.1.4 on memory bounds, write operations are treated under the store bound category. In contrast to load bounds, Top-down Microarchitecture Analysis Method (TMAM)[^2] reports store bounds as fraction of cycles with low execution port utilization and small performance impact. These descriptions in various sections highlight that *read-after-write* stall is considered more critical than *write-after-read* stall.
Memory Dependency Fuzz Testing
------------------------------
We performed a set of experiments to evaluate the memory dependency behavior between two logical processors. In these experiments, we have thread [$\mathcal A$]{} and [$\mathcal B$]{} running on the *same* physical core, but on *different* logical processors, as shown in Figure \[fig:hypert\]. Both threads perform memory operations; only thread [$\mathcal B$]{} measures its timing and hence the timing impact of introduced false dependencies.
**Read-after-read (RaR):** In the first experiment, the two logical threads [$\mathcal A$]{} and [$\mathcal B$]{} read from the same shared cache and can potentially block each other. This experiment can reveal cache bank conflicts, as used by *CacheBleed* [@yarom2017cachebleed]. [$\mathcal B$]{} uses Listing \[lst:probe\_read\] to perform read measurements and [$\mathcal A$]{} constantly reads from different memory offsets and tries to introduce conflicts. [$\mathcal A$]{} reads from three different type of offsets: **(1)** Different cache line than [$\mathcal B$]{}, **(2)** same cache line, but different offset than [$\mathcal B$]{}, and **(3)** same cache line and same offset as [$\mathcal B$]{}. As depicted, there is no obvious difference between the histograms for three cases in Figure \[fig:histogram\_1\] verifying the lack of cache bank conflicts on 7th generation CPUs.
**Write-after-read (WaR):** The histogram results for the second experiment on false dependency of write-after-read is shown in Figure \[fig:histogram\_2\], in which the cache line granularity is obvious. Thread [$\mathcal A$]{} constantly reads from different type of memory offsets, while thread [$\mathcal B$]{} uses Listing \[lst:probe\_write\] to perform write measurements. The standard deviation for conflicted cache line (blue) and conflicted offset (red) between thread [$\mathcal A$]{} and [$\mathcal B$]{} is distinguishable from the green bar where there is no cache line conflict. This shows a high capacity cache granular behavior, but the slight difference between conflicted line and offset verifies the previous results stating a weak offset dependency [@yarom2017cachebleed].
**Read-after-write (RaW):** Figure \[fig:histogram\_3\] shows an experiment on measuring false dependency of read-after-write, in which, thread [$\mathcal A$]{} constantly writes to different memory offsets. Thread [$\mathcal B$]{} uses Listing \[lst:probe\_read\] to perform read measurements. Accesses to three different types of offsets are clearly distinguishable. The conflicted cache line accesses (blue) are distinguishable from non-conflicted accesses (green). More importantly, conflicted accesses to the same offset (red) are also distinguishable from conflicted cache line accesses, resulting in a side channel with intra cache-line granularity. There is an average of 2 cycle penalty if the same cache line has been accessed, and a 10 cycle penalty if the same offset has been accessed. Note that the word offsets in our platform have 4 byte granularity. From an adversarial standpoint, this means that an adversary learns about bits 2-11 of the victim memory access, in which 4 bits (bits 2-5) are related to intra cache-line resolution, and thus goes beyond any other microarchitectural side channels known to exist on 6th and 7th generation Intel processors (Figure \[fig:address\]).
{width=".85\linewidth"} \[fig:slowwrite\]
{width=".85\linewidth"} \[fig:read\_cost\]
**Read-after-weak-Write (RawW):** In this experiment on the read-after-write conflicts, we followed a less greedy strategy on the conflicting thread. Rather than constantly writing to the same offset, [$\mathcal A$]{} executes write instructions to the same offset with some gaps filled with other memory accesses and instructions. As shown in Figure \[fig:slowwrite\], the channel dramatically became less effective. This tells us that causing read access penalty will be more effective with constant writes to the same offset without additional instruction. In this regard, we will use Listing \[lst:conflict\] in our attack to achieve the maximum conflicts.
![Intra Cache Level Leakage: [*MemJam*]{} latency is related to 10 address bits, in which 4 bits are intra cache level bits.[]{data-label="fig:address"}](images/address.pdf){width=".98\linewidth"}
mov %[target], %rax;
write_loop:
.rept 100;
movb $0, (%rax);
.endr;
jmp write_loop;
**Read-after-Write Latency:** In the last experiment, we tested the delay of execution over a varying number of conflicting reads. We created a code stub that includes 64 memory read instructions and a random combination of instructions between memory reads to create a more realistic computation. The combination is chosen in a way to avoid unexpected halts and to maintain the parallelism of all read operations. We measure the execution time of this computation on [$\mathcal B$]{}, while [$\mathcal A$]{} is writing to a conflicting offset. First, we measured the computation with 64 memory reads to addresses without conflicts. Our randomly generated code stub takes an average of 210 cycles to execute. On each step of the experiments, as shown in Figure \[fig:read\_cost\], we change some of the memory offsets to have the same last 12 bits of address as of [$\mathcal A$]{}ś conflicting write offset. We observe a growth on read accesses’ latency by increasing the number of conflicting reads. Figure \[fig:read\_cost\] shows the results for a number of experiments. In all of them, the overall execution time of [$\mathcal B$]{} is strongly dependent on the number of conflicting reads. Hence, we can use the RaW dependency to introduce strong timing behavior using bits 2-11 of a chosen target memory address.
[*MemJam*]{} Correlation Attack
===============================
[*MemJam*]{} uses read-after-write false dependencies to introduce timing behavior to otherwise constant-time implementations. The resulting latency is then exploited using a correlation attack. [*MemJam*]{} proceeds with the following steps:
1. Attacker launches a process constantly writing to an address using Listing \[lst:conflict\] where the last 12 bits match the virtual memory offset of a *critical* data that is read in the victim’s process.
2. While the attacker’s conflicting process is running, attacker queries the victim for encryption and records a ciphertext and execution time pair of the victim. Higher time infers more accesses to the *critical* offset.
3. Attacker repeats the previous step collecting ciphertext and time pairs.
The attack methodology resembles the *Evict+Time* strategy originally proposed by Tromer et al. [@tromer2010efficient], except that the attacker uses false dependencies rather than evictions to slow down the target *and* that the slowdown only applies to an 4-byte block of a cache line. Furthermore, *all* of the victim’s accesses addresses with the same last 12 bits are slowed down while an eviction only slows the first memory access(es).
Based on the intra cache level leakage in Figure \[fig:address\], we divide a 64 byte cache line into 4-byte blocks and hypothesize that the access counts to a block are correlated with the running time of victim, while the attacker jams memory reads to that block, i.e, the attacker expects to observe a higher time when there are more accesses by the victim to the targeted 4-byte block and lower time when there are lower number of accesses. Based on this hypothesis, we apply a classical correlation based side-channel approach [@kocher2011introduction] to attack implementations of two different block ciphers, namely AES and SM4, a standard cipher. SM4 among AES, Triple DES, and RC4 are the only available symmetric ciphers as part of Intel’s IPP crypto library [@intelIPP][^3]. Both implementations have optimizations to hinder cache attacks. In fact, the AES implementation features a constant cache profile and can thus be considered resistant to most microarchitectural attacks including cache attacks and high-resolution attacks as described in [@moghimi2017cachezoom]. [*MemJam*]{} can still extract the keys from both implementations due to the intra cache-line spatial resolution as depicted in Figure \[fig:address\]. We describe the targeted implementations next, as well as the correlation models we use to attack them.
Attack 1: IPP Constant-Time AES {#sec:aes}
-------------------------------
AES is a cipher based on substitution permutation network (SPN) with 10 rounds supporting 128-bit blocks and 128/192/256-bit keys [@daemen2013design]. The SubBytes is a security-critical operation and the straightforward way to implement AES SubBytes operation efficiently in software is to use lookup tables. SubBytes operates on each byte of cipher state, and it maps an 8-bit input to an 8-bit output using a non-linear function. A precomputed 256 byte lookup table known as S-Box can be used to avoid recomputation. There are efficient implementations using T-Tables that output 32-bit states and combine SubBytes and MixColumns operations. T-Table implementations are highly vulnerable to cache attacks. During AES rounds, a state table is initiated with the plaintext, and it holds the intermediate state of the cipher. Round keys are mixed with states, which are critical S-Box inputs and the main source of leakage. Hence, even an adversary who can partially determine which entry of the S-Box has been accessed is able to learn some information about the key.
Among the efforts to make AES implementations more secure against cache attacks, `Safe2Encrypt_RIJ128` function from Intel IPP cryptographic library is noteworthy. This implementation is the only production-level AES software implementation that features true cache constant-time behavior and does not utilize hardware extensions such as AES-NI or SSSE3 instruction sets. This implementation is also part of the Linux SGX SDK [@linuxsgx] and can be used for production code if the SDK is compiled from the scratch, i.e., it does not use prebuilt binaries. We verified the match between the implementation in Intel IPP binary and SGX SDK source code through reverse engineering. This implementation follows a very simple direction: **(1)** it implements AES using 256byte S-Box lookups without any optimization such as T-Tables, **(2)** instead of accessing a single byte of memory on each S-Box lookup, it fetches four values from the same vertical column of 4 different cache lines and saves them to a local cache aligned buffer, finally, **(3)** It performs the S-Box replacement by picking the correct S-Box entry from the local buffer. This implementation is depicted in Figure \[fig:aes\_sbox\]. This implementation protects AES against any kind of cache attacks, as the attacker sees a constant cache access pattern: The S-Box table only occupies 4 cache lines, and on each SubBytes operation, all of them will sequentially be accessed. This implementation can be executed in less than 2000 cycles on a recent laptop processor. This is fast enough for many cryptographic applications, and it provides full protection against cache attacks, even if the attacker can interrupt the execution pipeline.
{width=".97\linewidth"}
{width=".97\linewidth"}
Based on [*MemJam*]{} 4-byte granular leakage channel, and the design of AES, we can create a simple correlation model to attack this implementation. The accessed table index of the last round for a given ciphertext byte $c$ and key byte $k$ is given as $index = S^{-1}(c \oplus k)$. We define matrix $\mathbf A$ for the access profile where each row corresponds to a known ciphertext, and each column indicates the number of accesses when $index < 4$. While we assume that the attacker causes slow-downs to the first 4-byte block of S-Box, we define matrix $\mathbf L$ for leakage where each row corresponds to a known ciphertext and each column indicates the victim’s encryption time. Then our correlation attack is defined as the correlation between $\mathbf A$ and $\mathbf L$, in which the higher the number of accesses, the higher the running time. Our results will verify that correlation is high, even though the implementation has dummy accesses to the monitored block. These can be ignored as noise, slightly reducing our maximum achievable correlation.
**AES Key Recovery Results on Synthetic Data:** We first verified the correctness of our correlation model on synthetic data using a noise free leakage (generated by PIN [@pintool]). For each of the 16 key bytes using a vector that matches exactly to the number of accesses to the targeted block of S-Box for different ciphertexts, all the correct key bytes will have the highest correlation after 32,000 observations with the best and worst correlations of 0.046 and 0.029 respectively.
**AES Key Recovery Results using [*MemJam*]{}:** Relying on the verification of Synthetic Data, we plugged in the real attack data vector, which consists of pairs of ciphertext and time measured through repeated encryption of unknown data blocks. Results on AES show that we can effectively exploit the timing information, and break the so-called constant-time implementation. The victim execution of AES encryption function takes about 1700 and 2000 cycles without and with an active thread on the logical processor pair, respectively. The target AES implementation performs 640 memory accesses to the S-Box, including dummy accesses. If the spy thread constantly writes to any address that collides with a S-Box block offset, the time will increase to a range between 2000 and 2300 cycles. The observed variation in this range has a correlation with the number of accesses to that block. Figure \[aes\_key\_linearity\] shows the linear relationship between the correlation of synthetic data and real attack data for one key byte after 2 million observations. Most of the possible key candidates for a target key byte have a matching peak and hill between the two observations. The highest correlation points in both cases declare the correct key byte (0.038 red, 0.014 blue). The quantitative difference is due to the expected noise in the real measurements.
Figure \[fig:aes\_key\_cor\] shows the correlation of 4 different key bytes after 2 million observations with the correct key bytes having the highest correlations. Our repeated experiments with different keys and ciphertexts show that 15 correct key bytes have the highest correlation ranks, and there is only the key byte at index 15 that has a high rank but not necessarily the highest. Figure \[fig:aes\_key\_ranks\] shows the key ranks over the number of observations. Key byte ranks take values between 1 and 256, where 1 means that the correct key byte is the most likely one. As it is shown, after only 200,000 observations, the key space is reduced to a computationally insecure space and a key can be found with an efficient key enumeration method [@glowacz2015simpler]. After 2 million observations, all key bytes except one of them are recovered. The non-optimized implementation of this attack processes this amount of information in 5 minutes.
{width=".7\linewidth"}
Attack 2: IPP Cache Protected SM4
---------------------------------
SM4 is a block cipher[^4] that features an unbalanced Feistel structure and supports 128-bit blocks and keys [@diffie2008sms4]. SM4 design is known to be secure and no relevant cryptanalytic attacks exist for the cipher. Figure \[fig:sm4\] shows a schematic of one round of SM4. T1-T4 are $4\times32$-bit state variables of SM4. Within each round, the last three state variables and a 32-bit round key are mixed, and each byte of the output will be replaced by a non-linear S-Box value. After the non-linear layer, the combined 32-bit output of S-Boxes $x$ are diffused using the linear function L. The output of $L$ is then mixed with the first 32-bit state variable to generate a new random 32-bit state value. The same operation is repeated for 32 rounds, and each time a new 32-bit state is generated as the next round T4 state. The current T2, T3, T4 are treated as T1, T2, and T3 for the next round. The final 16 bytes of the entire state after the last round produce the ciphertext. SM4 Key schedule produces $32\times32$-bit round keys from a 128-bit key. Since the key schedule is reversible, recovering 4 repeated round keys provides enough entropy to reproduce the cipher key.
All the SM4 operations except the S-Box lookup are performed in 32-bit word sizes. Hence, SM4 implementation is both simple and efficient on modern architectures. We chose the function `cpSMS4_Cipher` from Intel IPP Cryptography library. Our target is based on the straightforward cipher algorithm with addition of S-Box cache state normalization. We recovered this implementation through reverse engineering of Intel IPP binaries. The implementation preloads four values from different cache lines of S-Box before the first round, and it mixes them with some dummy variables, forcing the processor to fill the relevant cache lines with S-Box table. This cache prefetching mechanism protects SM4 against asynchronous cache attacks. On our experimental setup, the implementation runs in about 700 cycles, which informs us that this implementation maintain a high speed while secure against asynchronous attacks. Interrupted attacks that leak intermediate states would not be as simple, since the interruption need to happen faster than 700 cycles. We will further discuss the difficulty of correlating any cache-granular information, even if we assume the adversary can interrupt the encryption and perform some intermediate observations.
**Single-round attack on SM4:** We define $c_{1}, c_{2}, c_{3}, c_{4}$ as the four 32-bit words of a ciphertext and $k_{r}$ as the secret round key for round $r$. We recursively follow the cipher structure from the last round with our ciphertext words as inputs, and write the last 5 rounds’ relations as Equation \[eq:sm4\]. In each round, $x^{i}_{r}$ is the S-Box index, and $i$ is the byte offset of the 32-bit word $x_{r}$. With a similar approach to the attack on AES, we define matrix $\mathbf A$ for the access profile, where each row corresponds to a known ciphertext, and each column indicates the number of accesses when $x^{i}_{r} < 4$. Then we define the matrix $\mathbf L$ for the observed timing leakage and the correlation between $\mathbf A$ and $\mathbf L$ similar to the AES attack. In contrast, S-Box indices in the AES attack are defined based on a non-linear inverse S-Box operation of key and ciphertext, which eventually maps to all possible key candidates. In SM4, the index $x^{i}_{r}$ is defined before any non-linear operation. As a result, an attack capable of distinguishing accesses of 4 out of 256 S-Box entries reveals only 6 bits per key byte. In the mentioned relations, performing the attack using this model on $x^i_{32}$, recovers the 6 most significant bits of each key byte $i$ for the last round key (Total of 24 out of the 32 bits).
$$\begin{aligned}
&x_{32} = c_{1} \oplus c_{2} \oplus c_{3} \oplus k_{32}\hspace{5ex} d_{2} = c_{1}, d_{3} = c_{2}, d_{4} = c_{3}\nonumber\\
&d_{1} = L(s(x^1_{32}), s(x^2_{32}), s(x^3_{32}), s(x^4_{32})) \oplus c_{4}\nonumber\\
&x_{31} = d_{1} \oplus d_{2} \oplus d_{3} \oplus k_{31}\hspace{5ex} e_{2} = d_{1}, e_{3} = d_{2}, e_{4} = d_{3}\nonumber\\
&e_{1} = L(s(x^1_{31}), s(x^2_{31}), s(x^3_{31}), s(x^4_{31})) \oplus d_{4}\nonumber\\
&x_{30} = e_{1} \oplus e_{2} \oplus e_{3} \oplus k_{30}\hspace{5ex} f_{2} = e_{1}, f_{3} = e_{2}, f_{4} = e_{3}\nonumber\\
&f_{1} = L(s(x^1_{30}), s(x^2_{30}), s(x^3_{30}), s(x^4_{30})) \oplus e_{4}\nonumber\\
&x_{29} = f_{1} \oplus f_{2} \oplus f_{3} \oplus k_{29}\hspace{5ex} g_{2} = f_{1}, g_{3} = f_{2}, g_{4} = f_{3}\nonumber\\
&g_{1} = L(s(x^1_{29}), s(x^2_{29}), s(x^3_{29}), s(x^4_{29})) \oplus f_{4}\nonumber\\
&x_{28} = g_{1} \oplus g_{2} \oplus g_{3} \oplus k_{28}\nonumber\\
\label{eq:sm4}\end{aligned}$$
**Multi-round attack on SM4:** The relationship for round $31$ can be used not only to recover 6-bit key candidates of round $31$, but also the remaining unknown 8 bits of entropy for round $32$. This is due to the linear property of function L and the recursive nature of newly created state variables. After the attack on round $32$, similar to the round key, we only have certainty about 24 bits of the new state variable $d_{1}$, but this information will be propagated as the input to round $31$. The next round of attack for key byte of round $31$ needs more computation to process an 8 bit of unknown key and 8 bit of unknown state (total of 16 bit), but this is computationally feasible, and the 8-bit key from round $32$ with highest correlation can be recovered by attacking the S-Box indices in round $31$. We recursively applied this model to each round resulting a correlation attack with the following steps, which gives us enough entropy to recover the key:
{width="\linewidth"}
{width="\linewidth"}
1. $x_{32}$ $\rightarrow$ 24 bits of $k_{32}$.
2. $x_{31}$ $\rightarrow$ 24 bits of $k_{31}$ + 8 bits of $k_{32}$
3. $x_{30}$ $\rightarrow$ 24 bits of $k_{30}$ + 8 bits of $k_{31}$
4. $x_{29}$ $\rightarrow$ 24 bits of $k_{29}$ + 8 bits of $k_{30}$
5. $x_{28}$ $\rightarrow$ 24 bits of $k_{28}$ + 8 bits of $k_{29}$
6. Recover the key from $k_{32}, k_{31}, k_{30}, k_{29}$
**SM4 Key Recovery Results on Synthetic Data:** Our noise-free synthetic data shows that 3000 observations are enough to find all correct 6-bit and 8-bit round key candidates with the highest correlations. Even in an interrupted cache attack or without cache protection, targeting this implementation using a cache-granular information would be much harder and inefficient due to the lack of intra cache-line resolution. If we only distinguish the 64-byte cache lines out of a 256-byte S-Box, we only learn $4\times2$-bit (total of 8 bits) out of 32-bit round keys, and on each round, we need to solve 8 bits + 24 bits of uncertainty. Although solving 32-bit of uncertainty sounds possible for a noise-free data, it is computationally much harder in a practical noisy setting. Our intra cache line leakage can exploit SM4 efficiently in a known-ciphertext scenario, while the best efficient cache attack on SM4 requires chosen plaintexts [@nguyen2012improved].
**SM4 Key Recovery Results using [*MemJam*]{}:** The results on SM4 show even more effective key recovery against this implementation compared to AES. Figure \[fig:sm4\_6bit\] shows the correlation for 6-bit round keys after 5 rounds of repeated attack, and the correlation for 12-bit key candidates can be seen in Figure \[fig:sm4\_8bit\]. The attack expects assurance on the correct key candidates for each round of attack before proceeding to the next round due to the recursive structure of SM4. In our experiment using real measurement data, we have noticed that 40,000 observations are sufficient to have assurance of correct key candidates with the highest correlations. Our implementation of the attack can recover the correct 6-bit and 8-bit keys, and it takes about 5 minutes to recover the cipher key. In Figure \[fig:sm4\_8bit\], we plotted the accumulated per byte correlations for all 8-bit candidates within each round of attack. During the computation of 6-bit candidates, the 8-bit candidates relate to 4 different state bytes. This accumulation greatly increases the result and the correct 8-bit key candidates have a very high aggregated correlation compared to the 6-bit candidates.
[*MemJam*]{}ing SGX Enclave {#sec:sgx}
===========================
Intel SGX is a trusted execution environment (TEE) extension released as part of Skylake processor generation [@linuxsgx]. The main goal of SGX is to protect runtime data and computation from system and physical adversaries. Having said that, SGX must remain secure in the presence of malicious OS, thus modification of OS resources for facilitation of side-channel attacks is relevant and within the considered threat model. Previous works demonstrate high resolution attacks with 4kB page [@xu2015controlled; @van2017telling] and 64B cache line granularity [@BrasserGrand; @moghimi2017cachezoom]. Intel declared microarchitectural leakages out of scope for SGX, thus pushing the burden of writing leakage free constant-time code onto enclave developers. Indeed, Intel follows this design paradigm and ensures constant cache-line accesses for its AES implementation, making it resistant to *all* previously known microarchitectural attacks in SGX.
In this section, we verify that [*MemJam*]{} is also applicable to SGX enclaves, as there is no fundamental microarchitectural changes to resist against memory false dependencies. We repeat the key recovery results against Intel’s constant-time AES implementation after moving it into an SGX enclave. The results verify the exploitability of intra cache level channel against SGX secure enclaves. In fact, the attack can be reproduced in a straightforward manner. The only difference is a slower key recovery due to the increased measurement noise resulting from the enclave context switch.
{width=".98\linewidth"}
{width=".95\linewidth"}
SGX Enclave Experimental Setup and Assumptions
----------------------------------------------
Following the threat model of *CacheZoom* [@lee2016inferring; @moghimi2017cachezoom], we assume that the system adversary has control over various OS resources. Please note that SGX was exactly designed to thwart the threat of such adversaries. The adversary uses its OS-level privileges to decrease the setup noise: We isolate one of the physical cores from the rest of the running tasks, and dedicate its logical processors to [*MemJam*]{} write conflict thread and the victim enclave. We further disable all the non-maskable interrupts on the target physical core and configure the CPU power and frequency scaling to maintain a constant frequency. We assume that the adversary can measure the execution time of an enclave interface that performs encryption, and the enclave interface only returns the ciphertext to the insecure environment. Both plaintexts and the secret encryption key are generated at runtime using *RDRAND* instruction, and they never leave the secure runtime environment of SGX enclave. The *RDTSC* instruction cannot be used inside an enclave. The attacker uses it right before the call to the enclave interface and again right after the enclave exit. As a result, the entire execution of the enclave interface, including the AES encryption, is measured. As before, an active thread causing read-after-write conflicts to the first 4-byte of AES S-Box is executed on the neighboring virtual processor of the SGX thread.
AES Key Recovery Results on SGX
-------------------------------
Execution of the same AES encryption function as Section \[sec:aes\] inside an SGX enclave interface takes an average of 14,600 cycles with an active thread causing read-after-write conflicts to the first 4-byte of AES S-Box. The additional overhead is caused by the enclave context switch, which significantly increases the noise of the timing channel due to the variable timing behavior. Having that, this experiment shows a more practical timing behavior where adversaries cannot time the exact encryption operation, and they have to measure the time for a batch of operations. This not only shows that SGX is vulnerable to [*MemJam*]{} attack, but it also demonstrates that [*MemJam*]{} is applicable in a realistic scenario. Figure \[fig:sgx\_aes\_key\_cor\] shows the key correlation results using 50 million timed encryptions in SGX, collected in 10 different time frames. We filtered outliers, i.e. measurements with high noise by only considering samples that are in the range of 2000 cycles of the mean. Among the 50 million samples, 93% pass the filtering, and we only calculated the correlations for the remaining traces. Figure \[fig:sgx\_aes\_key\_ranks\] shows that we can successfully recover 14 out of 16 key bytes, revealing sufficient information for key recovery after 20 million observations.
These results show that even cryptographic libraries designed by experts that are fully aware of current attacks and of the leakage behavior of the target device may fail at writing unexploitable code. Modern microarchitectures are so complex that assumptions such as *constant cache line profiles* result in unexploitable constant-time implementations are seemingly impossible to fulfill.
Discussion
==========
The `Safe2Encrypt_RIJ128` AES implementation has been designed to achieve a constant cache access profile by ensuring that the same cache lines are accessed every time regardless of the processed data. The 4-byte spatial resolution of [*MemJam*]{}, however, thwarts this countermeasure by providing intra cache-line resolution. One approach to restore security and protect against [*MemJam*]{}is to apply constant memory accesses with a 4-byte granularity. That would require accessing every fourth byte of the table for each memory lookup for the purpose of maintaining a uniform memory footprint. At that point, it might be easier to just do a *true* constant time implementation and access *all* entries each time, resting assured that there is no other effect somewhere hidden in the microarchitecture resulting in a leak with byte granularity. As we discussed in the related work, system-wide defense proposals that apply to cache attacks are not relevant and cannot detect or prevent [*MemJam*]{}. Also, an adversary performing the [*MemJam*]{} attack does not need to know about the offset of S-Box in the binary, since she can simply scan the 10-bits address entropy through introducing conflicts to different offsets and measuring the timing of victim. This is important when it comes to obfuscated binaries or scenarios, where the offset of S-Box is unknown.
Hardware based, e.g, AES-NI or hardware assisted, e.g, SIMD-based bit-sliced implementations of AES or SM4 should exclusively be used to protect the targeted implementation in an efficient manner. Intel IPP has different variants optimized for various generations of Intel instruction sets [@intelcpudispatch]. Intel IPP features different implementations of AES as well as SM4 in these variants. A list of these variants and implementations are given in Table \[tab:implementations\]. All of them have at least one vulnerable implementation. In cases where there is an implementation based on the AES-NI instruction set (or SSSE3 respectively), the library falls back to the basic version at runtime if the instruction set extensions are not available. The usability of this depends on the compilation and runtime configuration. Developers are allowed to statically link to a more risky variants [@intelIPPlinkage], and they need to assure not to use the vulnerable versions during linking. These ciphers should be avoided in cases where the hardware does not provide support, e.g, Core and Nehalem does not support AES-NI, e.g, AES-NI can be disabled in some BIOS. After all, the current hardware support for cryptographic primitives are restricted and if any other cipher is demanded, this limitation and vulnerability endangers the security of cryptographic systems. A temporary workaround to defend against the source of leakage on current Intel microarchitectures is to disable hyper-threading.
Prior to [*MemJam*]{} it might have seemed reasonable to design SGX enclaves under the paradigm that constant cache line accesses result in leakage-free code. However, the increased 4-byte intra cache-line granularity of [*MemJam*]{} shows that only code with true constant-time properties, i.e. constant execution flow and constant memory accesses can be expected to have no remaining leakage on modern microarchitectures.
**Responsible Disclosure** We have informed the Intel Product Security Incident Response Team of our findings on August 2nd, 2017. They have acknowledged the receipt and are currently reviewing our findings.
\[tab:implementations\]
**Implementation Technique** **Function Name** **l9 n0 y8 k0 e9** **m7 mx** **n8** **Linux SGX SDK**
------------------------------ ------------------- -------------------- -------------- -------------- -------------------------
$\checkmark$ $\times$ $\times$ $\checkmark$ (prebuilt)
$\checkmark$ $\times$ $\checkmark$ $\checkmark$ (prebuilt)
$\times$ $\checkmark$ $\times$ $\checkmark$ (source)
$\checkmark$ $\times$ $\times$ N/A
$\checkmark$ $\checkmark$ $\checkmark$ N/A
**Release** **Family** **Cache Bank Conflicts** **4K Aliasing**
------------- ------------ -------------------------- -----------------
$\checkmark$ $\checkmark$
$\times$ $\checkmark$
$\checkmark$ $\checkmark$
$\times$ $\checkmark$
$\times$ $\checkmark$
$\times$ $\checkmark$
: Intel processor families and availability of the leakage channels. Major Intel processors suffer from 4k aliasing, and are vulnerable to [*MemJam*]{} [@agneroptimize].[]{data-label="tab:cpus"}
Conclusion
==========
This work proposes [*MemJam*]{}, a new side-channel attack based on false dependencies. For the first time, we discovered new aspects of this side channel and its capabilities, and show how to extract secrets from modern cryptographic implementations. [*MemJam*]{} uses false read-after-write dependencies to slow down accesses of the victim to a particular 4-byte memory blocks *within* a cache line. The resulting latency of otherwise constant-time implementations was exploited with state-of-the art timing side-channel analysis techniques. We showed how to apply the attack to two recent implementations of AES and SM4. According to the available resources, the source of leakage for[*MemJam*]{} attack is present in all Intel CPU families released in the last 10 years [@agneroptimize; @inteloptimze]. Our results also verified that [*MemJam*]{} is the first intra cache level attack applicable to SGX enclaves. Table \[tab:cpus\] highlights the availability of the cache bank conflicts and 4k aliasing leakage source. [*MemJam*]{} is another piece of evidence that modern microarchitectures are too complex and constant-time implementations cannot simply be trusted with wrong assumptions about the underlying system. The remaining data-dependent addressing within a cache line is exploitable.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the National Science Foundation, under grant CNS-1618837.
[^1]: `LD_BLOCKS_PARTIAL.ADDRESS_ALIAS` Performance Monitoring Unit (PMU) event counts the number of times reads were blocked.
[^2]: Top-Down Characterization is a hierarchical organization of event-based metrics that identifies the dominant performance bottlenecks in an application. [@tmam]
[^3]: Patents investigated by Intel verify the importance of SM4 [@gueron2016sm4; @wolrich2016sms4; @yap2016sms4]
[^4]: Formerly SMS4, the standard cipher for Wireless LAN Wired Authentication and Privacy Infrastructure (WAPI)
|
---
abstract: 'In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^p\to L^p$ norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.'
address:
- 'Béchir Amri, Université de Tunis, Institut Préparatoire aux Etudes d’Ingénieurs de Tunis (IPEIT), Département de Mathématiques, 1089 Montfleury Tunis, Tunisie'
- 'Jean–Philippe Anker, Université d’Orléans & CNRS, Laboratoire MAPMO (UMR 6628), Fédération Denis Poisson (FR 2964), Bâtiment de Mathématiques, B.P. 6759, 45067 Orléans cedex 2, France'
- 'Mohamed Sifi, Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisie'
author:
- 'Béchir Amri, Jean–Philippe Anker & Mohamed Sifi'
title: Three results in Dunkl analysis
---
Introduction {#section_introduction}
============
Dunkl theory generalizes classical Fourier analysis on $\mathbb{R}^N$. It started twenty years ago with Dunkl’s seminal work [@D1] and was further developed by several mathematicians. See for instance the surveys [@R4; @G] and the references cited therein.
In this setting, the Paley–Wiener theorem is known to hold for balls centered at the origin. In [@J2], a Paley–Wiener theorem was conjectured for convex neighborhoods of the origin, which are invariant under the underlying reflection group, and was partially proved. Our first result in Section \[section\_PaleyWiener\] is a proof of this conjecture in the crystallographic case, following the third approach in [@J2].
Generalized translations were introduced in [@R2] and further studied in [@T2; @R5; @TX]. Apart from their abstract definition, we lack precise information, in particular about their integral representation $$(\tau_xf)(y)=\int_{\mathbb{R}^N}\hspace{-1mm}f(z)\,d\gamma_{x,y}(z)\,,$$ which was conjectured in [@R2] and established in few cases, for instance in dimension $N\!=\!1$ or when $f$ is radial. Our second result in Section \[section\_bounds\] is an optimal bound for the integral $$\int_{\,\mathbb{R}}|d\gamma_{x,y}(z)|$$ in dimension $N\!=\!1$, improving upon earlier results in [@R1; @TX]. Our bound depends on the multiplicity $k\!\ge\!0$ and tends from below to $\sqrt{2\,}$, as $k\to+\infty$. Our third result in Section \[section\_support\] deals with the support of the distribution $\gamma_{x,y}$ in higher dimension, that we determine rather precisely in the crystallographic case.
Background {#section_background}
==========
In this section, we recall some notations and results in Dunkl theory and we refer for more details to the articles [@D1; @J1] or to the surveys [@R4; @G].
Let $G\!\subset\!\text{O}(\mathbb{R}^N)$ be a finite reflection group associated to a reduced root system $R$ and $k:R\rightarrow[0,+\infty)$ a $G$–invariant function (called multiplicity function). Let $R^+$ be a positive root subsystem, $\Gamma_{\!+}$ the corresponding open positive chamber, $\overline{\Gamma_{\!+}}$ its closure, $\overline{\Gamma^+}\hspace{-1mm}
=\!\sum_{\,\alpha\in R^+}\mathbb{R}_+\alpha$ the dual cone, and let us denote by $x_+$ the intersection point of any orbit $G.x$ in $\mathbb{R}^N$ with $\overline{\Gamma_{\!+}}$.
The Dunkl operators $T_\xi$ on $\mathbb{R}^N$ are the following $k$–deformations of directional derivatives $\partial_\xi$ by difference operators: $$\textstyle
T_\xi f(x)=\partial_\xi f(x)
+\sum_{\,\alpha\in R^+}\!k(\alpha)\,\langle\alpha,\xi\rangle\,
\frac{f(x)-f(\sigma_\alpha.\,x)}{\langle\alpha,\,x\rangle}\,,$$ where $\sigma_\alpha.\,x=
x-\frac{\langle\alpha,\,x\rangle}{2\,|\alpha|^2}\,\alpha$ denotes the reflection with respect to the hyperplane orthogonal to $\alpha$. The Dunkl operators are antisymmetric with respect to the measure $w(x)\,dx$ with density $$\textstyle
w(x)=\,\prod_{\,\alpha\in R^+}|\,\langle\alpha,x\rangle\,|^{\,2\,k(\alpha)}\,.$$
The operators $\partial_\xi$ and $T_\xi$ are intertwined by a Laplace–type operator $$\begin{aligned}
\label{vk}
V\hspace{-.25mm}f(x)\,
=\int_{\mathbb{R}^N}\hspace{-1mm}f(y)\,d\mu_x(y)\end{aligned}$$ associated to a family of compactly supported probability measures $\{\,\mu_x\,|\,x\!\in\!\mathbb{R}^N\hspace{.25mm}\}$. Specifically, $\mu_x$ is supported in the the convex hull $$C^{\,x}=\,{\operatorname{co}}(G.x)\,.$$
For every $\lambda\!\in\!\mathbb{C}^N$, the simultaneous eigenfunction problem $$T_\xi f=\langle\lambda,\xi\rangle\,f
\qquad\forall\;\xi\!\in\!\mathbb{R}^N$$ has a unique solution $f(x)\!=\!E(\lambda,x)$ such that $E(\lambda,0)\!=\!1$, which is given by $$\label{EV}
E(\lambda,x)\,
=\,V(e^{\,\langle\lambda,\,.\,\rangle})(x)\,
=\int_{\mathbb{R}^N}\hspace{-1mm}e^{\,\langle\lambda,y\rangle}\,d\mu_x(y)
\qquad\forall\;x\!\in\!\mathbb{R}^N.$$ Furthermore $\lambda\mapsto E(\lambda,x)$ extends to a holomorphic function on $\mathbb{C}^N$ and the following estimate holds: $$|E(\lambda,x)|\leq e^{\,\langle({\operatorname{Re}}\lambda)_+,\,x_+\rangle}
\quad\forall\;\lambda\!\in\!\mathbb{C}^N,\;\forall\;x\!\in\!\mathbb{R}^N.$$
In dimension $N\!=\!1$, these functions can be expressed in terms of Bessel functions. Specifically, $$\textstyle
E(\lambda,x)=j_{k-\frac12}(\lambda\,x)
+\frac{\lambda\,x}{2\hspace{.25mm}k+1}\,j_{k+\frac12}(\lambda\,x)\,,$$ where $$\textstyle
j_\nu(z)\,=\;\Gamma(\nu\!+\!1)\;
{\displaystyle\sum\nolimits_{\,n=0}^{+\infty}}\;
\frac{(-1)^n}{n\hspace{.25mm}!\,\Gamma(\nu+n+1)}\;
\bigl(\frac z2)^{2n}$$ are normalized Bessel functions.
The Dunkl transform is defined on $L^1(\mathbb{R}^N\!,w(x)dx)$ by $$\mathcal{D}f(\xi)={\textstyle\frac1c}
\int_{\mathbb{R}^N}\!f(x)\,E(-i\,\xi,x)\,w(x)\,dx\,,$$ where $$c\,=\int_{\mathbb{R}^N}\!e^{-\frac{|x|^2}2}\,w(x)\,dx\,.$$ We list some known properties of this transform:
- The Dunkl transform is a topological automorphism of the Schwartz space $\mathcal{S}(\mathbb{R}^N)$.
- (*Plancherel Theorem*) The Dunkl transform extends to an isometric automorphism of $L^2(\mathbb{R}^N\!,w(x)dx)$.
- (*Inversion formula*) For every $f\!\in\!\mathcal{S}(\mathbb{R}^N)$, and more generally for every $f\!\in\!L^1(\mathbb{R}^N\!,w(x)dx)$ such that $\mathcal{D}f\!\in\!L^1(\mathbb{R}^N\!,w(\xi)d\xi)$, we have $$f(x)=\mathcal{D}^2\!f(-x)\qquad\forall\;x\!\in\!\mathbb{R}^N.$$
- (*Paley–Wiener theorem*) The Dunkl transform is a linear isomorphism between the space of smooth functions $f$ on $\mathbb{R}^N$ with ${\operatorname{supp}}f\!\subset\hspace{-.5mm}\overline{B(0,R)}$ and the space of entire functions $h$ on $\mathbb{C}^N$ such that $$\label{PaleyWiener0}\textstyle
\sup_{\,\xi\in\mathbb{C}^N}\,(1\!+\!|\xi|)^M\,e^{-R\,|{\operatorname{Im}}\xi\,|}\,|h(\xi)|<+\infty
\qquad\forall\;M\!\in\!\mathbb{N}\,.$$
A geometric Paley–Wiener theorem {#section_PaleyWiener}
================================
In this section, we prove a geometric version of the Paley–Wiener theorem, which was looked for in [@J2; @T2; @J3], under the assumption that $G$ is crystallographic. The proof consists merely in resuming the third approach in [@J2] and applying it to the convex sets considered in [@A1; @A2; @A3; @A4] instead of the convex sets considered in [@O1]. Recall that the second family consists of the convex hulls $$C^\Lambda={\operatorname{co}}(G.\Lambda)$$ of $G$–orbits $G.\Lambda$ in $\mathbb{R}^N$, while the first family consists of the polar sets $$C_\Lambda=\{\,x\!\in\!\mathbb{R}^N\mid
\langle x,g.\Lambda\rangle\!\le\!1\hspace{2mm}\forall\;g\!\in\!G\,\}\,.$$
\[c\][$\Lambda$]{} {width="55mm"} {width="40mm"}
\[c\][$\Lambda$]{} {width="50mm"} {width="50mm"}
Before stating the geometric Paley–Wiener theorem, let us make some remarks about the sets $C^\Lambda$ and $C_\Lambda$. Firstly, they are convex, closed, $G$–invariant and the following inclusion holds:
$
C^\Lambda\,\subset\,|\Lambda|^2\,C_\Lambda\,.
$
Secondly, we may always assume that $\Lambda\!=\!\Lambda_+$ belongs to the closed positive chamber $\overline{\Gamma_{\!+}}$ and, in this case, we have $$\begin{aligned}
C^\Lambda\cap\,\overline{\Gamma_{\!+}}\,
&=\;\textstyle\overline{\Gamma_{\!+}}\,\cap
\bigl(\,\Lambda-\overline{\Gamma^+}\bigr)\,,\\
C_\Lambda\cap\,\overline{\Gamma_{\!+}}\,
&=\,\{\,x\!\in\!\overline{\Gamma_{\!+}}\mid
\langle\Lambda,x\rangle\le1\,\}\,.\end{aligned}$$ Thirdly, on one hand, every $G$–invariant convex subset in $\mathbb{R}^N$ is a union of sets $C^\Lambda$ while, on the other hand, every $G$–invariant closed convex subset in $\mathbb{R}^N$ is an intersection of sets $C_\Lambda$. For instance, $$\overline{B(0,R)}\,
=\;\bigcup\nolimits_{\,|\Lambda|=R}C^{\Lambda}\,
=\;\bigcap\nolimits_{\,|\Lambda|=R^{-1}}\hspace{-.5mm}C_{\Lambda}\,.$$ Fourthly, we shall say that $\Lambda\!\in\!\overline{\Gamma_+}$ is [*admissible*]{} if the following equivalent conditions are satisfied:
- $\Lambda$ has nonzero projections in each irreducible component of $(\mathbb{R}^N\hspace{-.5mm},
\hspace{-.25mm}R)$,
- $C^\Lambda$ is a neighborhood of the origin,
- $C_\Lambda$ is bounded.
In this case, we may consider the gauge $$\textstyle
\chi_\Lambda(\xi)
=\max_{\,x\in C_\Lambda}\langle x,\xi\rangle
=\min\,\{\,r\!\in[\hspace{.5mm}0,+\infty)\mid\xi\!\in\!r\,C^\Lambda\,\}$$ on $\mathbb{R}^N$.
\[PaleyWiener1\] Assume that $\Lambda\!\in\!\overline{\Gamma_+}$ is admissible. Then the Dunkl transform is a linear isomorphism between the space of smooth functions $f$ on $\mathbb{R}^N$ with ${\operatorname{supp}}f\!\subset\hspace{-.5mm}C_\Lambda$ and the space of entire functions $h$ on $\mathbb{C}^N$ such that $$\label{PW1}\textstyle
\sup_{\,\xi\in\mathbb{C}^N}
(1\!+\!|\xi|)^M\,e^{-\chi_\Lambda({\operatorname{Im}}\xi)}\,|h(\xi)|
<+\infty
\qquad\forall\;M\!\in\!\mathbb{N}\,.$$
Following [@J2], this theorem is first proved in the trigonometric case, which explains the restriction to crystallographic groups, and next obtained in the rational case by passing to the limit. The proof of Theorem \[PaleyWiener1\] in the trigonometric case is similar to the proof of the Paley–Wiener Theorem in [@O1; @O2], and actually to the initial proof of Helgason for the spherical Fourier transform on symmetric spaces of the noncompact type. This was already observed in [@S] and will be developed below for the reader’s convenience. The limiting procedure, as far as it is concerned, is described thoroughly in [@J2] and needs no further explanation.
Thus assume that $h$ is an entire function on $\mathbb{C}^N$ satisfying and, by resuming the proof of [@O1 Theorem 8.6(2)], let us show that its inverse Cherednik transform $$\label{ICT}
f(x)\hspace{.25mm}=\hspace{.5mm}{\operatorname{const.}}\int_{\mathbb{R}^N}\!
h(\xi)\,\widetilde E(i\hspace{.25mm}\xi,x)\,\widetilde w(\xi)\,d\xi$$ vanishes outside $C_\Lambda$. Firstly, one may restrict by $G$–equivariance to $x\!=\!g_0.x_+$, where $x_+\hspace{-1mm}\in\!
\Gamma_+\hspace{-1mm}\smallsetminus \!C_\Lambda$ and $g_0$ denotes the longest element in $G$, which interchanges $\Gamma_+$ and $-\Gamma_+$. Secondly, by expanding $$\Bigl\{\,\prod\nolimits_{\,\alpha\in R^+}
(\hspace{.25mm}\langle\check\alpha,\xi\rangle\!
-\!k_\alpha\hspace{.25mm})\hspace{.25mm}\Bigr\}\;
\widetilde E(\xi,x)\,
=\,\sum\nolimits_{\,g\in G}\sum\nolimits_{\,q\in Q^+}\!
{\mathbf c}(-g.\xi)\,\widetilde E_q(g,g.\xi)\,
e^{\hspace{.25mm}\langle\hspace{.25mm}g.\xi
+\varrho\hspace{.2mm}+\hspace{.1mm}q,
\hspace{.5mm}x\hspace{.25mm}\rangle}$$ becomes $$f(x)\hspace{.25mm}=\hspace{.5mm}{\operatorname{const.}}\hspace{.5mm}
\sum\nolimits_{\,g\in G}\det g\,\sum\nolimits_{\,q\in Q^+}f_{g,q}(x)\,
e^{\hspace{.25mm}\langle\hspace{.25mm}\varrho\hspace{.2mm}+\hspace{.1mm}q,
\hspace{.5mm}x\hspace{.25mm}\rangle}\,,$$ where $$\label{coefficients}
f_{g,q}(x)\,=\int_{\mathbb{R}^N}\!h(g^{-1}\hspace{-.5mm}.\hspace{.25mm}\xi)\,
\widetilde E_q(g,i\hspace{.25mm}\xi)\,
e^{\hspace{.5mm}i\hspace{.25mm}\langle\hspace{.25mm}\xi,
\hspace{.5mm}x\hspace{.25mm}\rangle}\,
\Bigl\{\hspace{.5mm}\prod\nolimits_{\,\alpha\in R^+}\hspace{-1mm}\textstyle
\frac{\Gamma(\hspace{.25mm}i\hspace{.25mm}\langle\hspace{.25mm}\check\alpha,
\hspace{.5mm}\xi\hspace{.25mm}\rangle\hspace{.25mm}
+\hspace{.25mm}k_\alpha\hspace{.25mm})}
{\Gamma(\hspace{.25mm}i\hspace{.25mm}\langle\hspace{.25mm}\check\alpha,
\hspace{.5mm}\xi\hspace{.25mm}\rangle\hspace{.25mm}
+\hspace{.25mm}1\hspace{.25mm})}\hspace{.25mm}\Bigr\}\,d\xi\,.$$ Thirdly, one shows that all expressions vanish, by shifting the contour of integration from $\mathbb{R}^N$ to $\mathbb{R}^N\!+i\hspace{.5mm}t\hspace{.5mm}
g_0\hspace{.1mm}.\hspace{.2mm}\Lambda$ with $t\!>\!0$, which produces an exponential factor $e^{-c\hspace{.25mm}t}$ with $c\hspace{-.25mm}=\!\langle\Lambda,x_+\rangle\!-\!1\hspace{-.5mm}
>\hspace{-.5mm}0$, and by letting $t\!\to\!+\infty$.
Since every $G$–invariant convex compact neighborhood of the origin in $\mathbb{R}^N$ is the intersection of admissible sets $C_\Lambda$, Theorem \[PaleyWiener1\] generalizes as follows.
\[PaleyWiener2\] Let $C$ be a $G$–invariant convex compact neighborhood of the origin in $\mathbb{R}^N$ and $\chi(\xi)\!=\!\max_{\,x\in C}\langle x,\xi\rangle$ the dual gauge. Then the Dunkl transform is a linear isomorphism between the space $\mathcal{C}_C^\infty(\mathbb{R}^N)$ of smooth functions $f$ on $\mathbb{R}^N$ with ${\operatorname{supp}}f\!\subset\hspace{-.5mm}C$ and the space $\mathcal{H}_\chi(\mathbb{C}^N)$ of entire functions $h$ on $\mathbb{C}^N$ such that $$\textstyle
\sup_{\,\xi\in\mathbb{C}^N}
(1\!+\!|\xi|)^M\,e^{-\chi({\operatorname{Im}}\xi)}\,|h(\xi)|
<+\infty
\qquad\forall\;M\!\in\!\mathbb{N}\,.$$
Notice that the Dunkl transform $\mathcal{D}$ always maps $\mathcal{C}_C^\infty(\mathbb{R}^N)$ into $\mathcal{H}_\chi(\mathbb{C}^N)$ and that the assumption that $G$ is crystallographic is only used to prove that $\mathcal{D}$ is onto.
$L^p$ bounds for generalized translations in dimension $1$ {#section_bounds}
==========================================================
Dunkl translations are defined on $\mathcal{S}(\mathbb{R}^N)$ by $$(\tau_xf)(y)=
{\textstyle\frac1c}\int_{\mathbb{R}^N}\hspace{-1mm}
\mathcal{D}f(\xi)\,E(i\hspace{.25mm}\xi,x)\,E(i\hspace{.25mm}\xi,y)\,w(\xi)\,d\xi
\qquad\forall\;x,y\!\in\!\mathbb{R}^N.$$ They have an explicit integral representation [@R1] in dimension $N\!=\!1$: $$(\tau_xf)(y)=\int_{\mathbb{R}}f(z)\,d\gamma_{x,y}(z)\,,$$ where $$\label{gm}
d\gamma_{x,y}(z)\,=\,\begin{cases}
\,\gamma(x,y,z)\,|z|^{2k}\,dz
&\text{if \,}x,y\!\in\!\mathbb{R}^*\\
\,d\delta_y(z)
&\text{if \,}x\!=\!0\\
\,d\delta_x(z)
&\text{if \,}y\!=\!0
\end{cases}$$ is a signed measure such that $\displaystyle\int_{\mathbb{R}}d\gamma_{x,y}(z)=1$. Specifically, $$\textstyle
\gamma(x,y,z)\,=\,d\;\sigma(x,y,z)\;\rho(|x|,|y|,|z|)\;{1\hspace{-1mm}\text{l}}_{\,I_{|x|,|y|}}(|z|)
\qquad\forall\;x,y,z\in\mathbb{R}^*,$$ where $$\begin{aligned}
d&=\textstyle
\frac{\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)\vphantom{\frac12}}\,,\\
\sigma(x,y,z)
&=\textstyle
1-\frac{x^2+\,y^2\,-z^2}{2\,x\,y}
+\frac{z^2+\,y^2\,-x^2}{2\,z\,y}
+\frac{x^2+\,z^2\,-y^2}{2\,x\,z}\\
&=\textstyle
\frac{(z+x+y)\,(z+x-y)\,(z-x+y)}{2\,x\,y\,z}
\qquad\forall\;x,y,z\in\mathbb{R}^*,\\
\rho(a,b,c)&=\textstyle
\frac{\{\,c^2-\,(a-b)^2\}^{k-1}\,
\{\,(a+b)^2-\,c^2\}^{k-1}\vphantom{\frac12}}
{(\,2\,a\,b\,c\,)^{\,2k-1}\vphantom{\frac12}}\\
&=\textstyle
\frac{(\,2\,b^2c^2+\,2\,a^2c^2+\,2\,a^2b^2
-\,a^4-\,b^4-\,c^4\vphantom{\frac12})^{\,k-1}}
{(\,2\,a\,b\,c\,)^{\,2k-1}\vphantom{\frac12}}
\qquad\forall\;a,b,c>0\,,
\end{aligned}$$ and $I_{a,b}$ denotes the interval $[\,|\,a\!-\!b\,|,a\!+\!b\,]$. Notice the symmetries $$\label{gamma}
\gamma(x,y,z)\,=\,\begin{cases}
\,\gamma(y,x,z)\,,\\
\,\gamma(-x,-y,-z)\,,\\
\,\gamma(-z,y,-x)=\gamma(x,-z,-y)\,.
\end{cases}$$
\[measure\] The following inequality holds, for every $x,y\!\in\!\mathbb{R}$: $$\label{A}
\int_{\,\mathbb{R}}\,\bigl|d\gamma_{x,y}(z)\bigr|\leq A_k=
\textstyle\sqrt{2}\,
\frac{\{\,\Gamma(k+\frac12)\}^2}{\Gamma(k+\frac14)\,\Gamma(k+\frac34)}\,.$$ Actually there is equality if $x\!=\!y\!\in\!\mathbb R^*$. Moreover $A_k\!\overset<\longrightarrow\!\sqrt{2}$ as $k\!\to\!+\infty$.
This result improves earlier bounds obtained in [@R1] and [@TX], which were respectively $4$ and $3$.
Let $x,y\!\in\!\mathbb{R}^*$.
*Case 1*: Assume that $x\,y\!<\!0$. Then $|\,|x|\!-\!|y|\,|=|\,x\!+\!y\,|$ and $|x|\!+\!|y|=|\,x\!-\!y\,|$, hence $\sigma(x,y,z)\,{1\hspace{-1mm}\text{l}}_{\,I_{|x|,|y|}}(|z|)
=\frac{z\,+\,x\,+\,y}z\,\frac{(x-y)^2-\,z^2}{-\,2\,x\,y}\,{1\hspace{-1mm}\text{l}}_{\,I_{|x|,|y|}}(|z|)$ and $\gamma_{x,y}$ are positive. Thus $$\int_{\mathbb{R}}|d\gamma_{x,y} (z)|
=\int_{\mathbb{R}}d\gamma_{x,y} (z)
=1\,.$$ *Case 2*: Assume that $x\,y\!>\!0$. By symmetry, we may reduce to $0\!<\!x\!\leq\!y$. Then $$\begin{aligned}
\int_{\,\mathbb{R}}|d\gamma_{x,y} (z)|
&\,=\int_{-\infty}^{\,0}\hspace{-1mm}|d\gamma_{x,y}(z)|\,
+\int_{\,0}^{+\infty}\hspace{-1.5mm}|d\gamma_{x,y} (z)|\\
&\,=\,2\,d\int_{y-x}^{\,y+x}\hspace{-1.5mm}
\textstyle
\frac{x\,+\,y}{2\,x\,y\,z}
\bigl(\frac{z^2-\,x^2-\,y^2\,+\,2\,x\,y}{2\,x\,y\,z}\bigr)^k\,
\bigl(\frac{x^2+\,y^2\,+\,2\,x\,y\,-\,z^2}{2\,x\,y\,z}\bigr)^{k-1}\,
z^{2k}\,dz\,.
\end{aligned}$$ After performing the change of variables $z=\!\sqrt{\,x^2\!+y^2\!-2\,x\,y\cos\theta\,}$ and setting $y\!=\!s\,x$, we get $$\label{gam}
\int_{\,\mathbb{R}}|d\gamma_{x,y}(z)|\,
=\,{\textstyle\frac{\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)}}\,
(1\!+\!s)\int_{\,0}^{\,\pi}\!\textstyle
\frac{(1\,-\,\cos\theta)\,\sin^{2k-1}\!\theta}
{\sqrt{1\,+\,s^2-\,2\,s\,\cos\theta}}\,d\theta\,.$$ Denote by $F(s)$ the right hand side of . Since $$F'(s)=\,{\textstyle\frac{\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)}}\,
(1\!-\!s)\int_{\,0}^{\,\pi}\hspace{-1mm}\textstyle
\frac{\sin^{2k+1}\!\theta\vphantom{\big|}}
{(1\,+\,s^2-\,2\,s\,\cos\theta)^{\frac32}}\,d\theta$$ is nonpositive, $F(s)$ is a decreasing function on $[\,1,+\infty\,)$, which reaches its maximum at $s\!=\!1$. Let us compute it: $$\begin{aligned}
A_k=\,F(1)
&=\,{\textstyle\frac{\sqrt{2}\,\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)}}
\int_{\,0}^{\,\pi}\!(1\!-\!\cos\theta)^{k-\frac12}\,
(1\!+\!\cos\theta)^{k-1}\,\sin\theta\;d\theta\\
&=\,2^{2k}\,{\textstyle\frac{\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)}}
\int_{\,0}^{\,1}t^{k-\frac12}\,(1\!-\!t)^{k-1}\,dt\\
&=\textstyle
\,2^{2k}\,{\textstyle\frac{\Gamma(k+\frac12)}{\sqrt{\pi}\,\Gamma(k)}}\;
B(k\!+\!\frac12,k)=\textstyle
\,2^{2k}\,
\frac{\{\,\Gamma(k+\frac12)\}^2}
{\sqrt{\pi}\;\Gamma(2k+\frac12)}\,
=\,\sqrt{2}\;
\frac{\Gamma(k+\frac12)}{\Gamma(k+\frac14)}\,
\frac{\Gamma(k+\frac12)}{\Gamma(k+\frac34)}\;,
\end{aligned}$$ after performing the change of variables $t\!=\!\frac{1-\cos\theta}2$ and using standard properties of the beta and gamma functions.
Finally let us show that $A_k\!\overset<\longrightarrow\!\sqrt{2}$ as $k\!\to\!+\infty$. Write $$\textstyle
A_k=\sqrt{2}\;\frac{G(k+\frac14)}{G(k+\frac12)}\;,
\quad\text{where}\quad
G(u)=\frac{\Gamma(u+\frac14)}{\Gamma(u)}
\quad\forall\;u\!>\!0\,.$$ Since the logarithmic derivative $\frac{\Gamma'}\Gamma$ of the gamma function is a strictly increasing analytic function on $(0,+\infty)$, the logarithmic derivative $$\textstyle
\frac{G'(u)}{G(u)}
=\frac{\Gamma'(u+\frac14)}{\Gamma(u+\frac14)}
-\frac{\Gamma'(u)}{\Gamma(u)}$$ is positive. Hence $G$ is an strictly increasing function and $A_k\!<\!\sqrt{2}$. On the other hand, using Stirling’s formula $$\textstyle
\Gamma(u)\,\sim\,\sqrt{2\pi}\;u^{u-\frac12}\;e^{-u}
\quad\text{as}\quad u\to+\infty\,,$$ we get $G(k\!+\!\frac14)\sim G(k\!+\!\frac12)$ hence $A_k\to\sqrt{2}$, as $k\to+\infty$.
As a first consequence, we obtain the $L^1\!\to\!L^1$ operator norm of Dunkl translations in dimension $N\!=\!1$.
\[L1\] Let $x\!\in \mathbb{R}^*$. Then $\tau_x$ is a bounded operator on $L^1(\mathbb{R},|x|^{2k}dx)$, with $\|\tau_x\hspace{.25mm}\|_{L^1\to L^1}=A_k$.
The inequality $\|\tau_x\hspace{.25mm}\|_{L^1\to L^1}\!\le\hspace{-.25mm}A_k$ follows from , together with , and it remains for us to prove the converse inequality. By symmetry, we may assume that $x\!>\!0$. Since $$A_k=\,\lim\nolimits_{\,y\to x}\int_{\,\mathbb{R}}|\gamma(x,y,z)|\,|z|^{2k}\,dz\,,$$ for every $0\!<\!\varepsilon\!<\!A_k$, there exists $0\!<\!\eta\!<\!x$ such that, for every $y\!\in\hspace{-.25mm}
[\hspace{.25mm}x\!-\!\eta,x\!+\!±\eta\hspace{.25mm}]$, $$\label{crucial}
\int_{\,\mathbb{R}}|\gamma(x,y,z)|\,|z|^{2k}\,dz\hspace{.5mm}
>\hspace{.25mm}A_k\hspace{-.5mm}-\hspace{-.25mm}\varepsilon\,.$$ Let $f$ be a nonnegative measurable function on $\mathbb{R}$ such that $${\operatorname{supp}}f\!\subset\hspace{-.25mm}
[-\hspace{.25mm}x\!-\!\eta,-\hspace{.25mm}x\!+\!±\eta\hspace{.5mm}]
\quad\text{and}\quad
\bigl\|\hspace{.25mm}f\hspace{.25mm}\bigr\|_{L^1}\hspace{-.5mm}
=\hspace{-.5mm}\displaystyle\int_{\,\mathbb{R}}\!f(z)\,|z|^{2k}dz=1\,.$$ Since $$\begin{cases}
\,\gamma(x,y,z)\ge0
&\forall\;y\!<\!0\,,\;\forall\;z\!<\!0\,,\\
\,\gamma(x,y,z)\le0
&\forall\;y\!>\!0\,,\;\forall\;z\!<\!0\,,
\end{cases}$$ we have $$\bigl|(\tau_xf)(y)\bigr|=
\int_{-x-\eta}^{-x+\eta}\hspace{-1mm}
f(z)\,|\gamma(x,y,z)|\,|z|^{2k}\,dz\,.$$ Hence, using and , $$\bigl\|\hspace{.5mm}\tau_xf\hspace{.5mm}\bigr\|_{L^1}
=\int_{\,\mathbb{R}}\hspace{.5mm}\bigl|(\tau_xf)(y)\bigr|\,|y|^{2k}\,dy\,
=\int_{-x-\eta}^{-x+\eta}\Bigl\{\,
\int_{\,\mathbb{R}}|\gamma(x,-z,-y)|\,|y|^{2k}\,dy\,
\Bigr\}\,f(z)\,|z|^{2k}\,dz$$ is bounded from below by $A_k\hspace{-.5mm}-\hspace{-.25mm}\varepsilon$. Consequently $\|\tau_x\hspace{.25mm}\|_{L^1\to L^1}\!
\ge A_k\hspace{-.5mm}-\hspace{-.25mm}\varepsilon$ and we conclude by letting $\varepsilon\!\to\!0$.
Let us next compute the $L^2\!\to\!L^2$ operator norm of Dunkl translations.
\[L2\] Let $x\!\in \mathbb{R}$. Then $\tau_x$ is a bounded operator on $L^2(\mathbb{R},|x|^{2k}dx)$, with $\|\tau_x\hspace{.25mm}\|_{L^2\to L^2}=1$.
The proof is straightforward, via the Plancherel formula, and generalizes to higher dimensions. On one hand, the inequality $\|\tau_x\|_{L^2\to L^2}\!\le\!1$ follows from the estimate $|E(i\hspace{.25mm}\xi,x)|\!\le\!1$. On the other hand, let $$f_\varepsilon(x)=
\varepsilon^{\hspace{.25mm}k+\frac12}\,
f(\varepsilon\hspace{.25mm}x)$$ be a rescaled normalized function in $L^2(\mathbb{R},|x|^{2k}\hspace{.25mm}dx)$. Then $$\|\hspace{.5mm}f_\varepsilon\hspace{.5mm}\|_{L^2}
=\|\hspace{.5mm}f\hspace{.5mm}\|_{L^2}
=1$$ while $$\begin{aligned}
\|\hspace{.5mm}\tau_x\hspace{.5mm}f_\varepsilon\hspace{.5mm}\|_{L^2}^{\,2}
&=\int_{\,\mathbb{R}}|E(i\hspace{.25mm}\xi,x)|^2\,
\varepsilon^{-2k-1}\,|\mathcal{D}\hspace{-.25mm}f(\varepsilon^{-1}\xi)|^2\,
|\xi|^{2k}\,d\xi\\
&=\int_{\,\mathbb{R}}|E(i\hspace{.25mm}\varepsilon\hspace{.25mm}\xi,x)|^2\,
|\mathcal{D}\hspace{-.25mm}f(\xi)|^2\,|\xi|^{2k}\,d\xi
\end{aligned}$$ tends to $$\int_{\,\mathbb{R}}
|\hspace{.25mm}\mathcal{D}\hspace{-.25mm}f(\xi)|^2\,
|\xi|^{2k}\,d\xi\hspace{.5mm}
=\|f\|_{L^2}^{\,2}
=1$$ as $\varepsilon\to0$. This concludes the proof of the lemma.
Eventually, Corollary \[L1\] and Lemma \[L2\] imply the following result, by interpolation and duality.
\[Lp\] Let $x\!\in \mathbb{R}$ and $1\!\le\!p\!\le\!\infty$. Then $\tau_x$ is a bounded operator on $L^p(\mathbb{R},|x|^{2k}dx)$, with $\|\tau_x\hspace{.25mm}\|_{L^p\to L^p}\le
A_{\hspace{.25mm}k}^{2\,|1/p\hspace{.25mm}-1/2\hspace{.25mm}|}$.
In the product case, where $G\!=\!\mathbb{Z}_2^N$ acts on $\mathbb{R}^N$, we have $$\bigl\|\,\tau_x\,\bigr\|_{L^p\to L^p}\le\,
A_{\hspace{.25mm}k}^{2\,|\frac1p-\frac12|N}$$ for every $x\!\in\!\mathbb{R}^N$ and $1\!\le\!p\!\le\!\infty$.
A support theorem for generalized translations {#section_support}
==============================================
As mentioned in the introduction, we lack information about Dunkl translations in general. In this section, we locate more precisely the support of the distribution $$\langle\,\gamma_{x,y},f\,\rangle=(\tau_xf)(y)\,$$ which is known [@T2] to be contained in the closed ball of radius $|x|\!+\!|y|$.
\[support\] [(i)]{} The distribution $\gamma_{x,y}$ is supported in the spherical shell $$\bigl\{\,z\!\in\!\mathbb{R}^N\bigm|
\bigl||x|\!-\!|y|\bigr|\le|z|\le|x|\!+\!|y|\,\bigr\}\,.$$ [(ii)]{} If $G$ is crystallographic, then the support of $\gamma_{x,y}$ is more precisely contained in $$\bigl\{\,z\!\in\!\mathbb{R}^N\bigm|
z_+\hspace{-1mm}\preccurlyeq\hspace{-.5mm}
x_+\hspace{-1mm}+\hspace{-.5mm}y_+,\,
z_+\hspace{-.25mm}\succcurlyeq
y_+\hspace{-1mm}+\hspace{-.4mm}g_0.x_+
\hspace{1mm}\text{and }\hspace{1mm}
x_+\hspace{-1mm}+\hspace{-.4mm}g_0.y_+
\,\bigr\}\,.$$ Here $g_0$ denotes the longest element in $G$, which interchanges the chambers $\Gamma_{\!+}$ and $-\Gamma_{\!+}$, and $\preccurlyeq$ the partial order on $\mathbb{R}^N$ associated to the cone $\overline{\Gamma^+}$: $$a\preccurlyeq b
\quad\Longleftrightarrow\quad
b-a\in\overline{\Gamma^+}\,.$$
\[r\][$x+y$]{} \[l\][$x-y$]{} ![Support of $\gamma_{x,y}$ for the root system $A_1\!\times\!A_1$[]{data-label="fig:supportA1A1"}](supportA1A1.eps "fig:"){height="55mm"}
\[c\][$x+y$]{} \[c\][$x-y$]{} ![Support of $\gamma_{x,y}$ for the root system $B_2$[]{data-label="fig:supportB2"}](supportB2.eps "fig:"){height="75mm"}
Let $h\!\in\!\mathcal{C}_c^\infty(\mathbb{R}^N)$ be an auxiliary radial function such that $$\int_{\mathbb{R}^N}\hspace{-1mm}h(x)\,w(x)\,dx=1$$ and ${\operatorname{supp}}h\!\subset\!-{\operatorname{co}}(G.u)$, where $u\!\in\!\Gamma_{\!+}$ is a unit vector. For every $\varepsilon\!>0$ and $x,y,z\!\in\!\mathbb{R}^N$, set $$\gamma_\varepsilon(x,y,z)\,
=\,{\textstyle\frac1{c^2}}\int_{\mathbb{R}^N}\hspace{-1mm}
\mathcal{D}h(\varepsilon\,\xi)\,
E(i\,\xi,x)\,E(i\,\xi,y)\,E_k(-i\,\xi,z)\,w(\xi)\,d\xi\,.$$ Firstly, according to and , $$\xi\,\longmapsto\,\mathcal{D}h(\varepsilon\,\xi)\,E(i\,\xi,x)\,E(i\,\xi,y)$$ is an entire function on $\mathbb{C}^N$ satisfying $$\label{PW}
\textstyle
\bigl|\,\mathcal{D}h(\varepsilon\,\xi)\,E(i\,\xi,x)\,E(i\,\xi,y)\,\bigr|\;
\le\,C_M\,(1\!+\!|\xi|)^{-M}\,
e^{-\langle\,g_0.(x_+\hspace{-.25mm}+\hspace{.25mm}y_+\hspace{-.25mm}
+\hspace{.25mm}\varepsilon\hspace{.2mm}u),\,({\operatorname{Im}}\xi)_+\rangle}\,,$$ where $g_0$ is the longest element in $G$, which interchanges the chambers $\Gamma_{\!+}$ and $-\Gamma_{\!+}$. Secondly, $$\begin{aligned}
\langle\hspace{.5mm}\gamma_{x,y},f\hspace{.5mm}\rangle\hspace{.5mm}
&=\,{\textstyle\frac1c}
\int_{\mathbb{R}^N}\hspace{-1mm}\mathcal{D}f(\xi)\,
E(i\,\xi,x)\,E(i\,\xi,y)\,w(\xi)\,d\xi\\
&=\,{\textstyle\lim_{\,\varepsilon\to0}\,\frac1c}
\int_{\mathbb{R}^N}\hspace{-1mm}
\mathcal{D}h(\varepsilon\,\xi)\,\mathcal{D}f(\xi)\,
E(i\,\xi,x)\,E(i\,\xi,y)\,w(\xi)\,d\xi\\
&=\,{\textstyle\lim_{\,\varepsilon\to0}}
\int_{\mathbb{R}^N}\hspace{-1mm}f(z)\,\gamma_\varepsilon(x,y,z)\,w(z)\,dz
\end{aligned}$$ i.e. the distribution $\gamma_{x,y}$ is the weak limit of the measures $\gamma_\varepsilon(x,y,z)\,w(z)\,dz$. Thirdly, notice the symmetries $$\label{symmetry}
\gamma_\varepsilon(x,y,z)\,=\,\begin{cases}
\,\gamma_\varepsilon(y,x,z)\,,\\
\,\gamma_\varepsilon(g.x,g.y,g.z)
\quad\forall\;g\!\in\!G\cup\{-\text{Id}\}\,,\\
\,\gamma_\varepsilon(-z,y,-x)=\gamma_\varepsilon(x,-z,-y)\,.
\end{cases}$$
If $G$ is crystallogaphic, we use Corollary \[PaleyWiener2\] (actually the third version of the Paley–Wiener theorem in [@J2]), and deduce from that the function $z\,\longmapsto\gamma_\varepsilon(x,y,z)$ is supported in $${\operatorname{co}}\,\{G.(x_+\hspace{-.75mm}+\hspace{-.25mm}y_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon\,u)\}
=\hspace{.5mm}{\operatorname{co}}(G.x)+{\operatorname{co}}(G.y)+\varepsilon{\operatorname{co}}(G.u)\,.$$ Equivalently, $$\gamma_\varepsilon(x,y,z)\ne0
\quad\Longrightarrow\quad
z_+\!\prec
x_+\hspace{-.75mm}+\hspace{-.25mm}y_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon\hspace{.25mm}u\,.$$
Using the symmetries , we see that $\gamma_\varepsilon(x,y,z)\ne0$ implies also $$\begin{cases}
\,-\hspace{.25mm}g_0.x_+\!\prec
-\hspace{.25mm}g_0.z_+\hspace{-.75mm}+\hspace{-.25mm}y_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon u
&\text{i.e.}\quad
z_+\!\succ
x_+\hspace{-.75mm}+\hspace{-.25mm}g_0.y_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon\hspace{.25mm}g_0.u\,,\\
\,-\hspace{.25mm}g_0.y_+\!\prec
-\hspace{.25mm}g_0.z_+\hspace{-.75mm}+\hspace{-.25mm}x_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon u
&\text{i.e.}\quad
z_+\!\succ
g_0.x_+\hspace{-.75mm}+\hspace{-.25mm}y_+
\hspace{-.75mm}+\hspace{-.25mm}\varepsilon\hspace{.25mm}g_0.u\,.
\end{cases}$$ The conclusion of Theorem \[support\] in the crystallographic case is obtained by letting $\varepsilon\!\to\!0$.
If $G$ it not crystallographic, we can only use the spherical Paley–Wiener theorem and we obtain this way that $\gamma_\varepsilon(x,y,z)\ne0$ implies $$\begin{cases}
\,|z|\le|x|+|y|+\varepsilon\,,\\
\,|x|\le|z|+|y|+\varepsilon\,,\\
\,|y|\le|x|+|z|+\varepsilon\,,
\end{cases}$$ hence $$\bigl|\,|x|-|y|\,\bigr|-\varepsilon\le|z|\le|x|+|y|+\varepsilon\,.$$ We conclude again by letting $\varepsilon\!\to\!0$.
[XX]{}
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---
address: |
Service de Physique Théorique, CEA, CE-Saclay\
F-91191 Gif-sur-Yvette Cedex, France
author:
- 'R. PESCHANSKI'
title: 'HIGH ENERGY SCATTERING FROM THE $AdS/CFT$ CORRESPONDENCE[^1] '
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
Historically, the interpretation of strong interactions in terms of a string theory has raised much hope [@frampton] but was a deceiving adventure. Indeed, while the Veneziano (resp. Shapiro-Virasoro) amplitudes for Reggeon (resp. Pomeron) exchanges were very promising and at the root of the open (resp. closed) string theories, problems arise when looking for the internal consistency of the whole scheme in our 4-dimensional world: a quantum anomaly requires 26 or 10 dimensions, gravitons and zero-mass vectors unavoidably appear in the spectrum of strong interaction states. So a stringy description remains an open problem for $QCD_4.$
Recently, the proposal of an AdS/CFT correspondence [@adscft] seems to overcome some difficulties met during the last 30 years. In very brief terms (for a extended review, see [@review]) the idea is to unify a “microscopic” and a “microscopic” description of a configuration of $N \gg 1$ three-branes in the so-called Type II-B string theory in 10 dimensions.
In the “microscopic” description, the system gives rise to a 4-dimensional $SU(N)$ gauge theory (with $ {\cal N} = 4$ supersymmetries), while in the “macroscopic” one, it is the source of a gravitational background equipped with a $AdS_5 \otimes S_5$ metric with the physical Minkowski space lying at the boundary of $AdS_5 $. A duality property is conjectured between the 4-dimensional $SU(N)$ gauge theory at [*strong*]{} coupling and the gravitational background at [*weak*]{} coupling. Interestingly enough, the dynamical rôle of the fifth dimension in $AdS_5 $ is crucial for the validity of the correspondence.
The case of a gauge theory with $ {\cal N} = 4$ supersymmetries corresponds to a 4-dimensional, non-confining, conformal field theory. The conjecture could be enlarged to confining theories without supersymmetry (e.g. see [@witten]) by introducing a “horizon” scale in the 5-th dimension. The term “horizon” comes from the consideration of a black hole metric in the bulk of $AdS$ space in order to break supersymmetry. Even if the exact dual of $QCD_4$ has not yet been identified, these dualities give a laboratory framework for gauge observables at strong coupling. For instance, the Wilson area criterion for confinement can be explicitely verified [@Wilson].
High energy amplitudes: obervables and results
==============================================
Let us briefly outline the derivation of our papers [@janik1; @janik2]. Scattering amplitudes in the high energy limit (and small momentum transfer) can be conveniently expressed in terms of a correlator of Wilson loops [@Nachtr]. $$\label{e.ampinit}
A(s,q^2) = -2is \int d^2{x_\perp}e^{iq{x_\perp}}
{\left\langle{\frac{W_1W_2}{{\left\langle{W_1}\right\rangle}{\left\langle{W_2}\right\rangle}}-1}\right\rangle}$$ where the two [*tilted*]{} Wilson loops follow elongated trajectories along the light-cone direction with transverse separation $a$ and a tilting angle $\theta$ around the impact parameter axis. This corresponds to the scattering of colorless quark-antiquark pairs of mass $m \sim a^{-1}$. Indeed, the geometrical parameters of the configuration can be related to the energy scales by analytic continuation $\cos\theta \to \cosh \chi\equiv \frac{1}{\sqrt{1-v^2}}=\frac{s}{2m^2}-1$ where $\chi=\frac{1}{2}\log \frac{1+v}{1-v}$ is the Minkowski angle (rapidity) between the two lines, and $v$ is the relative velocity.
The results we obtain distinguish between the large and the small impact parameter kinematics. At large impact parameter, we could use [@janik1] the supergravity approximation of the appropriate type II-B string theory, since the fields, in particular gravity, are weak. We computed the exchange contribution of all zero-mode fields between the two separated $AdS_5$ surfaces whose geometry is fixed by area minimization with the two initial Wilson loops at the boundary. Looking to the contribution of the various fields (dilaton, Kaluza-Klein scalars, antisymmetric tensors mixed with Ramond-Ramond forms and the graviton) we find a hierarchy of real phase-shifts $\delta(b) \equiv \log
{\left\langle{\frac{W_1W_2}{{\left\langle{W_1}\right\rangle}{\left\langle{W_2}\right\rangle}}}\right\rangle}$ contributing to elastic scattering at large impact parameter only. Indeed, this hierarchy is different from the static Wilson loop correlator [@Wilson], since the graviton is dominant and not the Kaluza-Klein scalars. The potential problems with unitarity are avoided, since the weak field approximation appears to be valid only at very large impact parameter $L/a \gg s^{2/7}$ where the scattering amplitudes are purely elastic. Note, however, the retentivity of the gravitational interaction, which is still mysterious in the general context of the AdS/CFT correspondence where the decoupling from gravity is expected.
In a second paper [@janik2], we addressed the problem of small impact parameter and the origin of inelasticity, i.e. imaginary contributions to the phase shift. We concentrated on a situation where the difficulty with supergravity field exchanges does not arise, since there exists a single connected minimal surface which gives the dominant contribution to the scattering amplitude in the strong coupling regime. This allows us to extend our study to small impact parameters, where inelastic channels are expected to play an important rôle. Moreover, it is possible to investigate both cases of conformal (non-confining) and confining cases by considering the appropriate geometries in $AdS$ spaces. Our goal was to understand the rôle of confining geometries in the characteristic features of scattering amplitudes at high energy. The main expected feature is Reggeization, i.e. the determination of the amplitudes by singularities (poles and cuts) in the complex plane of the crossed channel partial waves, moving with $t \equiv q^2.$
Our main result [@janik2] is that high energy amplitudes are governed by the geometry of minimal surfaces, generalizing the [*helicoid*]{} in different $AdS$ geometries with the elongated tilted Wilson loops at the boundary. Indeed, the confining geometries have the remarkable properties to admit approximately flat configurations near the horizon scale in the fifth dimension and thus the tilting angle induces (approximate) helicoidal solutions for the minimal surface problem. For this solution and after analytic continuation, one finds a Regge singularity corresponding to a linear double Regge pole trajectory with intercept one $$\label{e.bhregge}
\alpha(t)= 1+\frac{R_0^2}{4 \sqrt{2g^2_{YM}N}}t\ ,$$ where $R_0$ is the horizon scale and $g_{YM},$ the gauge theory coupling.
The results in confining geometries for impact parameter larger or of the order of $R_0$ can be contrasted with the conformal (non-confining) $AdS_5 \otimes S_5$ case which, using an asymptotic evaluation (the mathematical knowledge on minimal surfaces embedded in $AdS$ spaces is yet limited!), leads to amplitudes with flat trajectories of the type $$\label{ampli}
A(s,t)\sim i s^{1+\frac{ 2\pi^4}{\Gamma(1/4)^4} \cdot
\frac{\sqrt{2g^2_{YM}N}}{2\pi}}
t^{-1-\frac{F(\pi/2)}{2\pi} \
\frac{\sqrt{2g^2_{YM}N}}{2\pi}},$$ where $F(\pi/2)\sim .3\pi$ comes from an anomalous dimension computed in [@drukker].
However, even in the confining cases, it may of course happen that the impact parameter distance between the two Wilson lines becomes much smaller than $R_0.$ In this case (see Fig. 1) the minimal surface problem becomes less affected by the black hole geometry and will just probe the small $z$ region of the geometry.
The precise behaviour at these shorter distances will depend on the type of gauge theory and, in particular, on the small $z$ limit of the appropriate metric. In fact this limit resembles the original $AdS_5\times S^5$ geometry. Note that the same behaviour can be equivalently obtained through rescaling, by keeping the impact parameter fixed and putting the scale $R_0 \to \infty$. The conformal behaviour of the amplitude (\[ampli\]) may thus give a hint of the small impact parameter limit also present in the physical confining case of $QCD_4,$ and thus a kind of hard-soft transition in impact parameter.
\[fig:fig1\]
Outlook
=======
Using the $AdS/CFT$ correspondence, we found a relation between high-energy amplitudes in gauge theories at strong coupling and minimal surfaces generalizing the helicoid in various $AdS$ backgrounds. We considered three cases: (i) flat metric approximation of an $AdS$ black hole metric giving rise to Regge amplitudes with linear trajectories, (ii) an approximate evaluation for the conformal $AdS_5 \times S^5$ geometry leading to flat Regge trajectories and (iii) evidence for a transition, in a confining theory, from behaviour of type (i) to (ii) when the impact parameter decreases below the interpolation scale set by the horizon radius. It would be quite useful to supplement the approximations made in our investigations by an evaluation of the string fluctuation pattern around the classical configurations we analyzed, in order to have a more precise determination of the predictions based on the $AdS/CFT$ correspondence. After that, we will be able to discuss the validity and usefulness of this stimulating conjecture in a deeper way.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was done in tight collaboration with R. Janik whose name should be associated to all the results mentioned in this contribution. This work was supported in part by the EU Fourth Framework Programme ‘Training and Mobility of Researchers’, Network ‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’, contract FMRX-CT98-0194 (DG 12 - MIHT).
References {#references .unnumbered}
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[^1]: Talk presented at the DIS00 workshop, Liverpool, April 2000.
|
---
abstract: 'We study the influence of atomic interactions on quantum simulations in momentum-space lattices (MSLs), where driven atomic transitions between discrete momentum states mimic transport between sites of a synthetic lattice. Low energy atomic collisions, which are short ranged in real space, relate to nearly infinite-ranged interactions in momentum space. However, the distinguishability of the discrete momentum states coupled in MSLs gives rise to an added exchange energy between condensate atoms in different momentum orders, relating to an effectively attractive, finite-ranged interaction in momentum space. We explore the types of phenomena that can result from this interaction, including the formation of chiral self-bound states in topological MSLs. We also discuss the prospects for creating squeezed states in momentum-space double wells.'
author:
- Bryce Gadway
- Fangzhao Alex An
- 'Eric J. Meier'
- 'Jackson Ang’ong’a'
title: 'Interacting atomic quantum fluids on momentum-space lattices'
---
Quantum simulation with ultracold atoms [@Bloch-RMP08; @AtomsRev-NatPhys-2012] has been a powerful tool in the study of many-body physics and nonequilibrium dynamics. There has been recent interest in extending quantum simulation studies from real-space potentials to synthetic lattice systems composed of discrete internal [@Boada-Synth; @Celi-ArtificialDim] or external [@NateGold-TrapShake] states. These synthetic dimensions enable many unique capabilities for quantum simulation, including new approaches to engineering nontrivial topology [@Celi-ArtificialDim; @Wall-SpinOrb], access to higher dimensions [@Boada-Synth], and potential insensitivity to finite motional temperature.
The recent development of momentum-space lattices (MSLs), based on the use of discrete momentum states as effective sites, has introduced a fully synthetic approach to simulating lattice dynamics [@Gadway-KSPACE; @Meier-AtomOptics; @Meier-SSH; @Alex-2Dchiral; @Alex-Annealed]. As compared to partially synthetic systems [@Fallani-chiral-2015; @Stuhl-Edge-2015], fully synthetic lattices offer complete microscopic control of system parameters. While this level of control is analogous to that found in photonic simulators [@SzameitReview-2010; @PhotRev-NatPhys-2012], matter waves of atoms can interact strongly with one another.
However, fully synthetic systems also present apparent challenges for studying nontrivial atomic interactions. Synthetic systems based purely on internal states suffer from limited state spaces, sensitivity to external noise for generic, field-sensitive states [@Sugawa-NonAb], and possible collisional relaxation [@Soding-relaxation] and three-body losses [@Weiner-Collisions]. Furthermore, for isotropic scattering lengths as in $^{87}$Rb [@Sugawa-NonAb] and alkaline earth atoms [@Pagano-AlkalineEarth], interactions in the synthetic dimension are nearly all-to-all. Similarly, $s$-wave contact interactions relate to nearly infinite-ranged momentum-space interactions at low energy, and should naively be decoupled from particle dynamics in MSLs.
Here, we investigate the role of atomic interactions in MSLs, showing that finite-ranged interactions in momentum space result from the exchange energy of bosonic condensate atoms in distinguishable momentum states. We explore potential interaction-driven phenomena that can be studied in topological MSLs, showing that chiral propagating bound states can emerge in the presence of an artificial magnetic flux. We additionally discuss the use of momentum-space double wells for the generation of squeezed many-particle states.
MSLs provide a bottom-up approach to engineering designer Hamiltonians with field-driven transitions. This technique is based on the coherent coupling of multiple atomic momentum states via two-photon Bragg transitions, synthesizing an effective lattice of coupled modes in momentum space. In the general case, the transition frequency associated with each Bragg transition is unique. For free non-interacting particles, this stems from the quadratic dispersion $E_p^0 = p^2/2m$, with momentum $p$ and atomic mass $m$. Considering atoms initially at rest and driven by laser fields of wavelength $\lambda$ and wavevector $k = 2\pi/\lambda$, a discrete set of momentum states $p_n = 2n\hbar k$ may be coupled, having energies $4 n^2 E_r$, with $E_r = \hbar^2 k^2 / 2m$ being the photon recoil energy. By individually addressing the unique Bragg transition resonances, one may realize MSLs with full “local” and temporal parameter control. Specifically, single-particle tight-binding models of the form $$H^{sp} \approx \sum_n t_n(e^{i \varphi_n} \hat{c}^\dag_{n+1} \hat{c}_n + \mathrm{h.c.}) + \sum_n \varepsilon_n \hat{c}^\dag_n \hat{c}_n \ ,
\label{EQ:e00}$$ may be realized by a single pair of Bragg lasers, where $\hat{c}_n$ ($\hat{c}^\dagger_n$) is the annihilation (creation) operator for the state with momentum $p_n$. Here, nearest-neighbor tunneling elements are controlled through the amplitude and phase of individual frequency components of the Bragg laser field, which drive first-order, two-photon Bragg transitions [@Kozuma-Bragg]. Similarly, an effective potential landscape of site energies $\varepsilon_n$ is controlled by small frequency detunings of the laser fields from Bragg resonances.
![\[FIG:fig1\] **Interaction shifts of Bragg tunneling resonances.** **(a)** Energy dispersion of non-interacting massive atoms $E_p^0$ (red solid line) in units of the recoil energy $E_r$, and the Bogoliubov dispersion $E_p$ of a homogeneous gas with weak repulsive interactions and a mean-field energy $\mu = 4 E_r$ (blue dashed line). **(b)** The effective interaction potential (normalized to $\mu$) experienced by weakly-coupled excitations with momentum $p$, shown for the cases $\mu / E_r = \ $0.1, 1, and 4. **(c)** Semi-log plot of the effective interaction potentials in (b), shown for a larger range of momenta, compared to the form $\mu - \mu^2/2 E_p^0$ (dotted lines) relevant in the free-particle limit ($E_p^0 \gg 2\mu$). **(d)** Cartoon depiction of effective site energies shifted by central population density, for $\mu/E_r = 0$ (no interactions) and $\mu/E_r = 4$. ](Fig1.pdf){width="\columnwidth"}
While there have been several demonstrations [@Meier-AtomOptics; @Meier-SSH; @Alex-2Dchiral; @Alex-Annealed] of the ability to engineer diverse single-particle Hamiltonians using MSLs, the prospects for studying interactions and correlated dynamics have not yet been examined. In typical real-space atomic quantum simulations, two-body contact interactions are the dominant mechanism leading to correlated behavior [@Greiner-SF-MI-2002]. The two-body contact potential $V(\textbf{r},\textbf{r'})$, being nearly zero ranged in real space, relates to a nearly infinite-ranged interaction potential $V(\textbf{k},\textbf{k'})$ in momentum space. At first glance, it appears as though only all-to-all interactions should result (considering only mode-preserving interactions), which unfortunately cannot give rise to correlated behavior for a fixed total density.
However, we find that finite-ranged, attractive interactions arise in the MSL due to atom statistics in the quantum fluid. As the landscape of MSL site energies is determined by synthesized detunings from the Bragg transition resonances, an interaction potential results from density-dependent modifications to the free-particle energy dispersion. We consider the case of small-amplitude momentum excitations of a homogeneous bosonic quantum gas at rest with uniform particle density $n$. The quadratic dispersion $E_p^0$ for a non-interacting gas is shown in Fig. , along with that of Bogoliubov quasiparticles of a weakly-interacting quantum gas [@Stenger-Bragg; @Ozeri-Bog-RMP; @Hadzibabic-Bragg-Strong], $E_p = \sqrt{E_p^0 (E_p^0 + 2\mu)}$ (ignoring a uniform energy shift of $\mu$ for all states). Here, $\mu = g n$ is the uniform condensate mean-field energy, with the interaction parameter $g$ related to the $s$-wave scattering length $a$ as $g = 4\pi \hbar^2 a/m$.
{width="\textwidth"}
The interactions mainly modify the shape of the free-particle dispersion at low momenta, with a characteristic linear dispersion near $p = 0$. At higher momenta ($E_p^0 \gg 2\mu$), the Bogoliubov quasiparticles are free-particle-like and have a roughly quadratic dispersion which is shifted in energy by the chemical potential ($E_p \approx E_p^0 + \mu - \mu^2 / 2 E_p^0$). This extra energy shift of order $\mu$ for high momentum states is a consequence of exchange interactions with the zero-momentum condensate. Indeed, the modification of the energy-momentum dispersion results not from momentum-dependent collisional interactions, but rather from the quantum statistics of the identical bosons in distinguishable motional states [@exch; @Kaufman-Entang; @Larson-MultibandBosons]. This is analogous to the effective magnetic interactions of electrons in condensed matter that result from spin-independent Coulomb interactions and exchange statistics.
The shifts of the state energies from their non-interacting values ($E_p - E_p^0$) are plotted in Fig. . The Bragg transition frequencies, which depend only on the energy differences between pairs of adjacent momentum states, are most strongly perturbed near the initially populated zero-momentum site. The density-dependent perturbations to the transition frequencies result in an effective interaction potential that is finite ranged and attractive, as depicted in Fig. . For $\mu \gtrsim E_r$, the momentum-space interaction potential has significant off-site contributions, the effective range of which increases with larger ratios $\mu / E_r$. There is, however, a natural limitation on the compatibility of long-ranged interactions with the scheme for engineering MSLs. This method breaks down when unique spectral addressing of the individual Bragg transitions is lost, occurring when multiple momentum orders populate the linear phonon branch (occurring roughly when $\mu$ exceeds $8 E_r$, the bare energy spacing of the Bragg resonances).
For concreteness, we now focus on the limit $\mu \ll 2 E_r$, where all coupled momentum orders are approximately distinguishable and the interaction potential is effectively local in momentum space. This allows us to move beyond a description of weakly-coupled condensate excitations, and describe the more general case where atomic population is arbitrarily distributed among many momentum orders. We explore new phenomena that may be opened up to investigation by combining this simple local interaction with the wide range of tunable lattice models enabled by MSLs.
The role of momentum-space interactions in the limit of purely distinguishable momentum orders can be simply described by a multimode nonlinear Schrödinger equation, where the self- and cross-phase modulation terms describing intra-mode and inter-mode interactions differ by the exchange energy. We make the simplifying assumption that all momentum orders share a common spatial wavefunction throughout the dynamics. This single-mode approximation is valid on only relatively short timescales, and does not capture the spatial separation of momentum wavepackets. To focus on the unique contributions of this exchange-driven momentum-space interaction, we additionally assume single-spin (internal state) bosonic atoms with effectively one-dimensional dynamics. In this restricted scenario, four-wave mixing processes [@Deng-FWM-1999; @Trippenbach-FWM-theory; @Rolston-NL-2002; @Pertot-10-PRL] are not allowed, and the individual state populations are conserved by the atomic interactions.
In this single-mode approximation, assuming a homogeneous condensate with fixed total atom number $N$ and thus fixed density, we may represent the condensate wavefunction simply with appropriately normalized (to unity) complex amplitudes $\phi_n$ of the various discrete plane-wave momentum orders with momenta $p_n = 2 n \hbar k$ [@Trippenbach-FWM-theory]. We furthermore remove a global energy term $2\mu$ by redefining the $\phi_n$, thus transforming the momentum-space interaction into an effectively attractive self-interaction term for atoms residing in the same order (valid for $\mu \ll 2E_R$). Taking into account the contributions of these interactions to the effective tight-binding models of Eq. \[EQ:e00\], the dynamical evolution of the atoms becomes governed by $$i \hbar \dot{\phi}_n = \sum_m H^{sp}_{mn} \phi_m - \mu |\phi_n|^2 \phi_n \ ,
\label{EQ:e0d}$$ a tunable lattice tight-binding model with local attractive interactions.
Even with only two coupled sites, interactions are expected to significantly alter the system dynamics. For the simple case of population initialized in one of two equal-energy sites, weak interactions ($\mu/t < 4$) lead to a slowdown of two-mode Rabi dynamics, giving way to critical slowing for $\mu / t \approx 4$. For stronger interactions ($\mu/t > 4$), nonlinear self trapping [@Smerzi-Joseph-Atoms-1997; @Raghavan-Jos-1999] occurs, preventing population from ever fully leaving the initially-populated site. Moreover, many analogs of behavior found in tunnel-coupled superconductors, including plasma oscillations, the ac and dc Josephson effects, hysteresis, and macroscopic quantum self trapping, can be expected to emerge in double wells of Bragg-coupled states [@Smerzi-Joseph-Atoms-1997; @Raghavan-Jos-1999].
While such interaction-driven phenomena are well studied in the two-mode case [@Albiez-Oberth-2005; @Levy-ACDC-2007; @Leblanc-Joseph; @Eckel-Hysteresis-2014; @Trenkwalder-ParSymmBreak-2016; @Chang-spinor-josephson; @Tomko-Oberth-2017], MSLs offer unique capabilities for engineering multiply-connected lattice geometries [@Ryu-SQUID]. In particular, we consider the effects that local, attractive interactions can have on particle dynamics in MSLs with closed tunneling pathways, where artificial magnetic fluxes play a nontrivial role [@Alex-2Dchiral; @An-prep]. In Fig. \[FIG:fig2\], we explore atom dynamics on lattices with triangular and rhomboidal geometries, attainable in one physical dimension with a combination of first- and second-order Bragg transitions (nearest- and next-nearest-neighbor tunnelings, respectively) [@An-prep]. We first consider the simplest such configuration, consisting of three sites with periodic boundary conditions, which is an analog of multiply-connected superconducting quantum interference devices or atomtronic circuits [@Ryu-SQUID].
Figure illustrates triple-well dynamics in the presence of a very weak applied flux ($0.001 \pi$) and uniform tunneling amplitude $t$. Without interactions (upper plot), an initially localized wavepacket spreads almost evenly to the neighboring sites. For sufficiently large interactions ($\mu/t = 6$, lower plot), however, the initial onset of a slightly asymmetric chiral current induces the formation of a fully chiral soliton-like mode [@Tromb-Breather]. For still larger values of the nonlinear interaction, self trapping occurs.
We extend this investigation to a many-site zig-zag ladder system, shown in Fig. , with a uniform distribution of effective magnetic fluxes $\varphi$ and tunnelings $t$. The role of nonlinear interactions in such a topological lattice model is of interest for its connection to emergent topological phenomena in kinetically frustrated systems. Here we again examine the case of population initialized to a single, central mode ($n=0$), exploring dynamics following a tunneling quench. The distributions of normalized site populations $P_n$ at various evolution times $\tau$ are shown in Fig. . With no interactions (black solid line), chiral currents are present, but with a rapid ballistic spreading of the atomic distribution. We find that moderate interactions stabilize the atomic distribution, leading to soliton- or breather-like states [@Tromb-Breather]. For non-zero flux values ($\pm \pi/6$ for the blue and orange solid lines), these self-stabilized states propagate in a chiral fashion along the zig-zag ladder. For much larger interaction strengths ($\mu/t \gtrsim 9$), the atoms remain localized at $n = 0$ for all flux values.
This interaction-driven stabilization of chiral wavepackets is further summarized in Fig. , through the average position $\langle n \rangle$ and largest site population $P_n^{max}$. Figure contrasts the dynamics of $\langle n \rangle$ for interaction values $\mu/t = 0, 7.2,$ and 12, all for a uniform magnetic flux $\varphi = \pi/6$. While some dynamics of $\langle n \rangle$ can be seen even for the self-trapped scenario at $\mu/t = 12$, the position of the most highly populated site (filled circles) never deviates from the initial location. The dynamics of this site population $P_n^{max}$ are shown in Fig. for these same cases. Without interactions, ballistic spreading leads to a continuous decrease of the maximum density. For a range of moderate interactions, we find that the distribution self stabilizes at short timescales ($\tau \approx \hbar/t$). Finally, for very large interactions, behavior analogous to macroscopic quantum self trapping inhibits particle transport and population remains largely localized at the central site.
![\[FIG:fig3\] **Squeezing in a momentum-space double well.** (**a**) Visualization of many-particle ($N = 100$) spin states $|\Psi\rangle$ through their overlap with different coherent spin states $|\langle \theta,\varphi|\Psi\rangle|^2$. Shown are the cases of an initial coherent spin state $|\pi/2,\pi\rangle$ (upper plot), and the transformed state after evolution under $H_{\mathrm{sq}}$ for a time $\kappa \tau_{\mathrm{sq}}/\hbar = 0.0173 \pi$ (lower plot). (**b**) Squeezing along the $\hat{z}$-axis, $\xi_z$, for different evolution times $\kappa \tau_{\mathrm{sq}}/\hbar = \{0.1,0.25,0.5,0.75,1\}\times 0.0173 \pi$ (solid lines, with colors varying from blue to red) and for different angles of rotation $\varphi_{\mathrm{rot}}$ of the final distribution about $\hat{J}_x$. ](Fig3.pdf){width="\columnwidth"}
Populating flat energy bands [@DiamondLattice-2016; @Vito-Flatband-14; @Ehud-FlatBand-10; @FlatBand-Nonlinear; @Ani-synthz-zigzag] of similar topological models with interacting atoms should lead to interesting, emergent many-body dynamics. Interacting gases, combined with engineered MSLs having arbitrary and time-fluctuating disorder [@Alex-Annealed], should also enable highly controllable explorations into the physics of many-body localization [@aleiner:finite_temperature_disorder_2010; @Deissler-DisorderWithInteractions-2010]. For both scenarios, the most interesting open questions relate to phenomena driven by quantum fluctuations, which are not captured by Eq. \[EQ:e0d\].
The simplest MSL to capture such physics is a single momentum-space double well. For fixed total particle number $N = N_1 + N_2$, one can define effective angular momentum operators relating to the coherences and macroscopic occupations $N_1$ and $N_2$ of two Bragg-coupled momentum orders (ignoring thermal and quantum depletion). These are given by $\hat{J}_x = (\hat{c}_1^\dag \hat{c}_2 + \hat{c}_2^\dag \hat{c}_1)/2$, $\hat{J}_y = i(\hat{c}_1^\dag \hat{c}_2 - \hat{c}_2^\dag \hat{c}_1)/2$, and $\hat{J}_z = (\hat{c}_2^\dag \hat{c}_2 - \hat{c}_1^\dag \hat{c}_1)/2$, where $\hat{c}_n$ ($\hat{c}^\dagger_n$) is the annihilation (creation) operator for mode $n$ [@RaghavanBigelow]. Here, Dicke states $|j,m\rangle$ with total spin $j = N/2$ and $\hat{z}$-projection $m = (N_2 - N_1) / 2$ describe collective two-mode number states. Coherent spin states (CSSs) of the form $|\theta , \varphi\rangle = \sum_{m = -j}^{j} f_{m}^j (\theta, \varphi) e^{-i(j+m)\varphi}|j,m\rangle$, for $f_{m}^j (\theta, \varphi) = \binom{2j}{j+m}^{1/2}\cos(\theta/2)^{j-m}\sin(\theta/2)^{j+m}$, result from a global rotation of spin-polarized states $|j,j\rangle$ about the spin vector $\hat{n}_\varphi = \cos(\varphi) \hat{J}_x + \sin(\varphi) \hat{J}_y$ by an amount $\theta$ [@CSS-RMP]. In this description, the momentum-space interaction relates to an effective nonlinear squeezing Hamiltonian $H_{\mathrm{sq}} = \kappa \hat{J}_z^2$, for $\kappa = -3\mu/2N$ [@RaghavanBigelow].
We examine the case of the CSS $|\pi/2,\pi\rangle$ initially aligned along $-\hat{J}_x$. For short evolution times, the nonlinear Hamiltonian $H_{\mathrm{sq}}$ leads to a “shearing” of such coherent states. This is depicted in Fig. , for the initial CSS (upper plot) and the sheared, non-classical squeezed state after a time $\kappa\tau_{\mathrm{sq}}/\hbar = 0.0173 \pi$ (lower plot), through the overlap of these states with CSSs of varying $\theta$ and $\varphi$ values. For ease of calculation, dynamics are shown for the case of only $N = 100$ atoms ($j = 50$).
Figure shows, for sheared distributions relating to various evolution times $\tau_{\mathrm{sq}}$, the $\hat{z}$-axis squeezing parameter $\xi_z = 2 j \frac{\langle \Delta\hat{J}_z^2\rangle}{j^2 - \langle\hat{J}_z\rangle^2}$ as a function of rotation angle $\varphi_{rot}$ about the $\hat{J}_x$ spin axis. For typical experimental parameter values ($N = 10^5$ atoms, $\mu / \hbar = 2 \pi \times 1$ kHz), an optimal squeezing of $\xi_z^{min} \approx (3j^2)^{-1/3}$, relating to $-33$ dBm (after rotation), would be expected after a total duration $\tau_{\mathrm{sq}} \approx 1.13 j^{1/3}(\hbar/\mu) \approx 6.6$ ms [@RaghavanBigelow]. The practical enhancement of atom interferometry from such squeezing will be challenged by decoherence and dephasing, such as from heating and laser phase noise. However, refocusing $\pi$ (echo) pulses or twist-and-turn squeezing schemes [@Muessel-TWIST] can be used to mitigate decoherence due to spatial separation of the momentum orders. Looking beyond this simple two-mode case, the straightforward extension to multiple momentum states should also allow for unique investigations into multi-mode squeezing and quantum phase transitions in multi-mode analogs of the Lipkin-Meshkov-Glick model [@Trenkwalder-ParSymmBreak-2016; @Tomko-Oberth-2017].
We have shown that local real-space interactions of atomic gases can give rise to finite-ranged effective interactions in momentum space. These momentum-space interactions can lead to correlated dynamics of atoms in highly-tunable MSLs, opening up new possibilities for exploring interacting topological and disordered fluids. To focus on this effectively attractive momentum-space interaction, we have restricted our investigations to the case of mode-preserving collisions. It will be interesting to explore the effect of mode-changing collisions on MSL studies, with four-wave mixing relevant in higher dimensions [@Deng-FWM-1999; @Trippenbach-FWM-theory; @Rolston-NL-2002] and for spinful condensates [@Pertot-10-PRL], as these give rise to effectively infinite-ranged, correlated hopping terms. While beyond the scope of this work, it will also be interesting to examine the roles of inhomogeneous density, finite temperature, elastic $s$-wave scattering, departures of the momentum excitations from the form of Bogoliubov quasiparticles [@Hadzibabic-Bragg-Strong], and energy-dependent modifications to the $s$-wave scattering length.
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abstract: |
A new type of integral representation is proposed for the propagators of the massive Klein-Gordon field minimally coupled to the gravity of the de Sitter expanding universe. This representation encapsulates the effects of the Heaviside step functions of the Feynman propagators making possible for the first time the calculation of Feynman diagrams involving scalar particles on this background. In order to convince that, a simple example is given outlining the amplitudes in the second order of perturbations of the Compton effect in the de Sitter scalar quantum electrodynamics..
Pacs: 04.62.+v
author:
- 'Ion I. Cotăescu'
- 'Ion Cotăescu Jr.'
date: 'Received: date / Revised version: date'
title: Integral representation of the scalar propagators on the de Sitter expanding universe
---
Introduction
============
The classical or quantum scalar fields are the principal pieces used in various models on curved spacetime. Of a special interest in cosmology is the de Sitter expanding universe carrying scalar fields variously coupled to gravity whose quantum modes can be analytically solved [@Nach; @CT; @BODU; @T; @Csc; @Csc1]. Nevertheless, despite of this opportunity, we have not yet a complete scalar quantum field theory (QFT) on de Sitter backgrounds based on perturbations and renormalization procedures able to describe all the processes involving scalar bosons. This is because of the technical difficulties in calculating Feynman diagrams affecting the fields of any spin on the de Sitter expanding universe.
The source of these difficulties is the fact that the causal propagators, expressed explicitly in terms of Heaviside step functions depending on time, lead to the fragmentation of the time integrals of the chronological products of free fields giving the transition amplitudes in different orders of perturbations. Under such circumstances, these integrals cannot be evaluated forcing one to restrict so far only to the first order amplitudes of the de Sitter QFT which do not involve propagators [@Lot1; @Lot2; @Lot3; @R1; @R2; @R3; @AuSp1; @AuSp2; @CQED; @Cr1; @Cr2]. Note that the processes in the first order of perturbations which are forbidden in special relativity by the energy-momentum conservation are allowed on the de Sitter spacetimes where the momentum and energy cannot be measured simultaneously [@CGRG]. However, the calculations in the first order of perturbations are only the first step to a complete QFT involving propagators for which we must get over the above mentioned difficulties.
In the traditional QFT on the Minkowskian spacetime this problem is solved by the Fourier representations of the causal propagators which encapsulate the effect of the Heaviside step functions [@BDR]. Unfortunately, in the de Sitter case such Fourier representations do not hold as we shall explain in what follows. Therefore, we must look for another type of integral representation able to take over the effects of the Heaviside functions. Recently we succeeded to find a new integral representation of the propagators of the Dirac field on the de Sitter [@Cint1] or any spatially flat FLRW [@Cint2] spacetimes which is different from the usual Fourier integrals allowed in special relativity. Here we would like to continue this study applying the same method to the massive and charged Klein-Gordon fields, minimally coupled to the gravity of the de Sitter expanding universe, writing down for the first time the new integral representation of their propagators.
Moreover, we show that this new integral representation plays the same role as the familiar Fourier one in special relativity, helping us to calculate the Feynman diagrams of the de Sitter scalar quantum electrodynamics (SQED) in a similar manner as in the flat case. In order to convince that we present as a premiere the amplitudes of the Compton effect on the de Sitter expanding universe, written in a closed form thanks to our integral representation.
We start in the second section presenting briefly the massive scalar field whose mode functions are written in the conformal local chart with Cartesian coordinates. The next section is devoted to our principal result reported here, demonstrating that the integral representation we propose gives just the Feynman propagator after applying the method of contour integrals [@BDR]. In the third section we derive the amplitudes of the Compton effect on the de Sitter expanding portion. Finally some concluding remarks are presented.
Massive scalar field
====================
Let us consider the de Sitter expanding universe defined as the expanding portion of the $(1+3)$-dimensional de Sitter manifold, equipped with the spatially flat FLRW chart whose coordinates, $x^{\mu}$ ($\alpha,...\mu,\nu,...=0,1,2,3$), are the proper time $x^0=t$ and the Cartesian coordinates $x^i$ ($i,j,k,...=1,2,3$) for which we use the vector notation, ${\bf x}=(x^1,x^2,x^3)$. For technical reasons we work here mainly in the conformal chart having the conformal time $$\label{tc}
t_c=-\frac{1}{\omega}e^{-\omega t}<0\,,$$ and the same space coordinates. In these charts the line element reads [@BD] $$\begin{aligned}
ds^2=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}&=&dt^2-a(t)^2(d{\bf x}\cdot d{\bf x})\nonumber\\
&=& a(t_c)^2(dt_c^2-d{\bf x}\cdot d{\bf x})\,,\label{mrw}\end{aligned}$$ where $$\label{atc}
a(t)=e^{\omega t} \to a(t_c)=-\frac{1}{\omega t_c}\,,$$ depend on the Hubble constant of the de Sitter spacetime denoted here by $\omega$.
In the conformal chart, the Klein-Gordon equation of a charged scalar field of mass $m$ takes the form $$\label{KG1}
\left( \partial_{t_c}^2-\Delta -\frac{2}{t_c}\,
\partial_{t_c}+\frac{m^2}{\omega^2 t_c^2}\right)\phi(t_c, {\bf x})=0\,,$$ allowing the well-known solutions that can be expanded in terms of plane waves of positive and negative frequencies as [@BD] $$\begin{aligned}
\phi(x)&=&\phi^{(+)}(x)+\phi^{(-)}(x)\nonumber\\
&=&\int d^3p \left[f_{\bf p}(x)a({\bf
p})+f_{\bf p}^*(x)b^*({\bf p})\right]\,, \label{field1}\end{aligned}$$ where the fundamental solutions have the general form $$f_{\bf p}(x)=f_{\bf p}(t_c,{\bf x})=\frac{1}{\sqrt{\pi\omega}}\frac{e^{i {\bf p}\cdot {\bf x}}}{[2\pi a(t_c)]^{\frac{3}{2}}} {\cal F}_{\nu}(t_c) \,.$$ The time modulation function ${\cal F}_{\nu}$ may be any arbitrary linear combination of Bessel functions. In what follows it is convenient to consider modified Bessel functions $K_{\nu}$ instead of the usual Hankel ones such that we can write $${\cal F}_{\nu}(t_c)=c_1 K_{i\nu}(ipt_c)+c_2 K_{i\nu}(-ipt_c)\,,$$ where $\nu=\sqrt{\frac{m^2}{\omega^2}-\frac{9}{4}}$ (in the minimal coupling) while $c_1$ and $c_2$ are arbitrary complex valued constants.
The fundamental solutions must satisfy the orthonormalization relations $$\begin{aligned}
\langle f_{\bf p},f_{{\bf p}'}\rangle=-\langle f_{\bf p}^*,f_{{\bf
p}'}^*\rangle&=&\delta^3({\bf p}-{\bf p}')\,,\\
\langle f_{\bf p},f_{{\bf p}'}^*\rangle&=&0\,,\end{aligned}$$ with respect to the relativistic scalar product [@BD] $$\label{SP2}
\langle \phi,\phi'\rangle=i\int \frac{d^3x}{(\omega t_c)^2}\, \phi^*(x)
\stackrel{\leftrightarrow}{\partial_{t_c}} \phi'(x)\,,$$ that hold only if we set $$|c_1|^2-|c_2|^2=1\,,$$ as it results from Eqs. (\[SP2\]) and (\[KuKu\]). These constants define the vacuum [@P1] as in the case of the Bunch-Davies vacuum [@BuD], where $c_1=1$ and $c_2=0$, which is a particular case of adiabatic vacuum [@GM; @Zel; @ZelS; @GMM; @allen; @bousso]. In what follows we use this vacuum for brevity but all our formulas we derive here can be rewritten easily in any other vacuum.
After fixing he constants $c_1$ and $c_2$ the quantization can be done in a canonical manner replacing the wave functions of the field (\[field1\]) by field operators, $a({\bf p}) \to {\frak a}({\bf p})$ and $b({\bf p}) \to {\frak b}({\bf p})$, such that $b^{*}\to {\frak b}^{\dagger}$ [@BDR]. Then we assume that the particle (${\frak a}$, ${\frak a}^{\dagger}$) and antiparticle (${\frak b}$, ${\frak b}^{\dagger}$) operators fulfill the standard commutation relations in the momentum representation, among which the non-vanishing ones are $$\label{com1}
[a({\bf p}), a^{\dagger}({\bf p}^{\,\prime})]=[b({\bf p}), b^{\dagger}({\bf
p}^{\,\prime})] = \delta^3 ({\bf p}-{\bf p}^{\,\prime})\,.$$
In the configuration representation the partial commutator functions of positive or negative frequencies, $$iD^{(\pm)}(x,x')=[\phi^{(\pm)}(x),\phi^{(\pm)\,\dagger}(x')]$$ give the total one, $D=D^{(+)}+D^{(-)}$. These functions are solutions of the Klein-Gordon equation in both the sets of variables and obey $[D^{(\pm)}(x,x')]^*=D^{(\mp)}(x,x')$ such that $D$ is a real valued function. These functions can be written as mode integrals as, $$\begin{aligned}
iD^{(+)}(x,x')&=&iD^{(+)}(t_c,t_c',{\bf x}-{\bf x}')=\int d^3 p \, f_{\bf p}(x)f_{\bf p}(x')^* \nonumber\\
&=&\frac{1}{\pi\omega}\frac{1}{[4\pi^2 a(t_c)a(t_c')]^{\frac{3}{2}}}\nonumber\\
&\times& \int d^3 p\,e^{i{\bf p}\cdot({\bf x}-{\bf x}')}K_{i\nu}(ipt_c)K_{i\nu}(-ipt_c')\,,\label{Dp}\\
iD^{(-)}(x,x')&=&iD^{(-)}(t_c,t_c',{\bf x}-{\bf x}')=-\int d^3 p \, f_{\bf p}(x)^*f_{\bf p}(x') \nonumber\\
&=&-\frac{1}{\pi\omega}\frac{1}{[4\pi^2 a(t_c)a(t_c')]^{\frac{3}{2}}}\nonumber\\
&\times& \int d^3 p\,e^{i{\bf p}\cdot({\bf x}-{\bf x}')}K_{i\nu}(-ipt_c)K_{i\nu}(ipt_c')\,,\label{Dm}\end{aligned}$$ taking similar forms after changing ${\bf p}\to -{\bf p}$ in the last integral. Note that these integrals can be solved in terms of hypergeometric functions obtaining well-known closed formulas [@BD].
Integral representations of the scalar propagators
==================================================
The commutator functions allow us to construct the propagators, i. e. the Green functions corresponding to initial conditions at $t=\pm\infty$, without solving the Green equation. As in the scalar theory on Minkowski spacetime, we may use the Heaviside step functions for defining the retarded, $D_R$, and advanced, $D_A$, propagators, $$\begin{aligned}
D_R(t_c,t_c',{\bf x}-{\bf x}')&=& \theta(t_c-t_c')D(t_c,t_c',{\bf x}-{\bf x}')\,,\label{DR}\\
D_A(t_c,t_c',{\bf x}-{\bf x}')&=& -\,\theta(t_c'-t_c)D(t_c,t_c',{\bf x}-{\bf x}')\,,\label{DA}\end{aligned}$$ while the causal Feynman propagator has the well-known form [@BDR], $$\begin{aligned}
\label{DF}
iD_F(t_c,t_c',{\bf x}-{\bf x}')&=& \langle 0|T[\phi(x)\phi^{\dagger}(x')]\,|0\rangle
\nonumber\\
&=& \theta (t_c-t_c') D^{(+)}(t_c,t_c',{\bf x}-{\bf x}')\nonumber\\
&-&\theta(t_c'-t_c)D^{(-)}(t_c,t_c',{\bf x}-{\bf x}')\,.\end{aligned}$$ Our main goal here is to find the suitable integral representation of these propagators which should encapsulate the effect of the Heaviside step functions.
As mentioned, the propagators (\[DR\]), (\[DA\]) and (\[DF\]) cannot be used in the concrete calculations of Feynman diagrams as they stay because of their explicite dependence on the Heaviside $\theta$-functions. In the case of the Minkowski spacetime this problem is solved by representing these propagators as $4$-dimensional Fourier integrals which take over the effects of the Heaviside functions according to the method of the contour integrals [@BDR]. In this manner one obtains a suitable integral representation of the Feynman propagators allowing one to work exclusively in momentum representation.
In de Sitter spacetimes we also have a momentum representation but we do not know how to exploit it since in this geometry the propagators are functions of two time variables, $t-t'$ and $tt'$, instead of the unique variable $t-t'$ of the Minkowski case. This situations generates new difficulties since apart from a Fourier transform in $t-t'\in {\Bbb R}$ a suplementary Mellin transform for the new variable $tt'\in {\Bbb R}^+$ [@GR] might be considered. Obviously, an integral with two more variables of integration is not a convenient solution for representing the Feynman propagators.
This means that we must look for an alternative integral representation based on the method of the contour integrals [@BDR] but avoiding the mentioned Fourier or Mellin transforms. As in the Dirac case [@Cint1; @Cint2], the explicit forms of the partial commutator functions (\[Dp\]) and (\[Dm\]) suggest us to postulate the following integral representation of the Feynman propagator $$\begin{aligned}
&&D_F(x,x')\equiv D_F(t_c,t_c',{\bf x}-{\bf x}')=\frac{1}{\pi^2\omega}\frac{1}{[4\pi^2 a(t_c)a(t_c')]^{\frac{3}{2}}} \nonumber\\
&&~~~~~~\times \int d^3p\, e^{i{\bf p}\cdot({\bf x}-{\bf x}')}\int_{-\infty}^{\infty} ds\, |s| \,\frac{K_{i\nu}(ist_c)K_{i\nu}(-ist_c')}{s^2-p^2-i\epsilon}\,. \label{DFI}\end{aligned}$$ It remains to prove that this integral representation gives just the Feynman propagator (\[DF\]) according to the well-known method of contour integrals [@BDR]. Focusing on the last integral of Eq. (\[DFI\]) denoted as $${\cal I}(t_c,t_c')=\int_{-\infty}^{\infty}ds\,M(s,t_c,t_c')\,,$$ we observe that for large values of $|s|$ the modified Bessel functions can be approximated as in Eqs. (\[Km0\]) obtaining the asymptotic behavior $$M(s,t_c,t_c')\sim \frac{e^{-is(t_c-t_c')}}{s}\,.$$ Now we can estimate the integrals on the semicircular parts, $c_{\pm}$, of the contours pictured in Fig. 1 taking $s\sim R e^{i\varphi}$ and using Eq. (3.338-6) of Ref. [@GR] which gives $$\int_{c_{\pm}}ds\,M(s,t_c,t_c')\sim I_0[\pm R(t_c-t_c')]\sim \frac{1}{\sqrt{R}}\,e^{\pm R(t_c-t_c')}\,,$$ since the modified Bessel function $I_0$ behaves as in the first of Eqs. (\[Km0\]). In the limit of $R\to \infty$ the contribution of the semicircle $c_+$ vanishes for $t_c'>t_c$ while those of the semicircle $c_-$ vanishes for $t_c>t_c'$. Therefore, the integration along the real $s$-axis is equivalent with the following contour integrals $${\cal I}(t_c,t_c')=\left\{
\begin{array}{lll}
\int_{\small C_+}ds\,M(s,t_c,t_c')={\cal I}_+(t_c,t_c')&{\rm for}& t_c<t_c'\\
\int_{\small C_-}ds\,M(s,t_c,t_c')={\cal I}_-(t_c,t_c')&{\rm for}&t_c>t_c'
\end{array}\right. \,,\nonumber$$ where the contours $C_{\pm}$ are the limits for $R\to \infty$ of those of Fig. 1. Then we may apply the Cauchy’s theorem [@Complex], $${\cal I}_{\pm}(t_c,t_c')=\pm 2\pi i \left.{\rm Res}\left[M(s,t_c,t_c')\right]\right|_{s=\mp p\pm i\epsilon}\,,$$ taking into account that in the simple poles at $s=\pm p\mp i\epsilon$ we have the residues $$\left.{\rm Res}\left[M(s,t,t')\right]\right|_{s=\pm p\mp i\epsilon}=\pm\frac{1}{2}\,K_{\nu}(\pm ip t_c){K}_{\nu}(\mp i pt'_c)\,.$$ Consequently, the integral ${\cal I}_-(t_c,t_c')$ gives the first term of the Feynman propagator (\[DF\]) with $D^{(+)}$ expanded as in Eq. (\[Dp\]) while the integral ${\cal I}_+(t_c,t_c')$ yields its second term with $D^{(-)}$ in the form (\[Dm\]), proving that the integral rep. (\[DFI\]) is correct.
The other propagators, $D_A$ and $D_R$, can be represented in a similar manner by changing the positions of the poles as in the flat case [@BDR], $$\begin{aligned}
&&{D}_{\substack{R\\
A}}(x,x')={D}_{\substack{R\\
A}}(t_c,t_c^{\prime},{\bf x}-{\bf x}')
=\frac{1}{\pi^2\omega} \frac{1}{[4\pi^2 a(t_c) a(t_c')]^{\frac{3}{2}}}\nonumber\\
&&~~~~~~\times\int {d^3p}\,{e^{i{\bf p}\cdot({\bf x}-{\bf x}')}}\int_{-\infty}^{\infty}ds\, |s|\,\frac{K_{i\nu}(ist_c)K_{i\nu}(-ist_c')}{(s\pm i\epsilon)^2-p^2}\,,\label{SRA}\end{aligned}$$ but in our integral representation instead of the Fourier one.
Finally we note that the above integral representations can be rewritten at any time in the FLRW chart, $\{t,{\bf x})\}$, substituting $t_c\to t$ and $a(t_c)\to a(t)$ according to Eqs. (\[tc\]) and (\[atc\]).
Example: Compton effect in SQED
===============================
We succeeded thus to derive the specific integral representations of the scalar propagators on the de Sitter expanding universe that can be used for calculating the Feynman diagrams of the physical effects involving the Klein-Gordon field. Here we give a simple example outlining how our approach works in the SQED on the de Sitter expanding universe, deriving the amplitudes of the Compton effect in the second order of perturbations.
We consider that our massive charged scalar field $\phi$ is coupled minimally to the electromagnetic field $A_{\mu}$ through the interaction Lagrangian $${\cal L}_{int}=-i e \sqrt{g(x)}\,g^{\mu\nu}(x)A_{\mu}(x)\left[\phi^{\dagger}(x)\stackrel{\leftrightarrow}{\partial_{\nu}}\phi(x)\right]\,,$$ where $e$ is the electrical charge. We know that in the chart $\{t_c,{\bf x}\}$ the electromagnetic potential can be expanded in terms of similar plane waves as in the Minkowski spacetime since the Maxwell equations are conformally invariant if we work exclusivelly in the Coulomb gauge where $A_0=0$ [@CMax; @CQED]. Therefore, we may write the expansion $$\label{Max}
{A_i}(x)=\int d^3k
\sum_{\lambda}\left[{\mu}_{{\bf k},\lambda;\,i}(x) \alpha({\vec
k},\lambda)+{\mu}_{{\bf k},\lambda;\,i}(x)^* \alpha^{\dagger}({\bf k},\lambda)\right]\,,$$ in terms of the mode functions, $$\label{fk}
{\mu}_{{\bf k},\lambda;\,i}(t_c,{\bf x}\,)=
\frac{1}{(2\pi)^{3/2}}\frac{1}{\sqrt{2k}}\,e^{-ikt_c+i{\bf
k}\cdot {\bf x}}\,{\varepsilon}_{i} ({\bf k},\lambda)\,,$$ depending on the components of the polarization vectors ${\varepsilon}_{i}({\bf k},\lambda)$ of momentum ${\bf k}$ ($k=|{\bf k}|$) and helicity $\lambda=\pm 1$ [@CQED]. Note that the polarization vector is orthogonal to momentum, $k^i\varepsilon_{i}({\bf k},\lambda)=0$.
With these preparations we can write the first Compton amplitude [@BDR] with a self-explanatory notation as $$\begin{aligned}
&&{\cal A}_{\lambda_1,\lambda_2}({\bf p}_1, {\bf k}_1, {\bf p}_2, {\bf k}_2)\equiv\langle out\,{\bf p}_2, ({\bf k}_2,\lambda_2)|in \,{\bf p}_1, ({\bf k}_1,\lambda_1\rangle \nonumber\\
&&~~~~~~~~= -i \,\frac{e^2}{2} \int d^4x\,d^4x' \sqrt{g(x)g(x')}\,g^{ij}(x)g^{kl}(x') \nonumber\\
&&\times\mu_{{\bf k}_2,\lambda_2, j}(x)^*\mu_{{\bf k}_1,\lambda_1, l}(x')\left[f_{{\bf p}_2}^*(x) \stackrel{\leftrightarrow}{\partial_{i}} D_F(x,x') \stackrel{\leftrightarrow}{\partial_{k}'}f_{{\bf p}_1}(x') \right]\,,\nonumber\\\end{aligned}$$ where $D_F$ is given by our integral representation (\[DFI\]). We perform first the space integrals generating Dirac $\delta$-functions which have to assure the momentum conservation after integrating over the internal momentum ${\bf p}$ of $D_F$. Thus, after a little calculation we may write $$\begin{aligned}
&&{\cal A}_{\lambda_1,\lambda_2}({\bf p}_1, {\bf k}_1, {\bf p}_2, {\bf k}_2)=-i\frac{e^2}{2}\,\delta^3({\bf p}_1+{\bf k}_1-{\bf p}_2-{\bf k}_2)\nonumber\\
&&~~~~~~~~~~\times p_1^i\varepsilon_i({\bf k}_1,\lambda_1)p_2^j\varepsilon_j({\bf k}_2,\lambda_2)^*\nonumber\\
&&~~~~~~~~~~\times \int_{-\infty}^{\infty} ds |s| \frac{{\cal V}(p_2,k_2,s)^*{\cal V}(p_1,k_1,s)}{s^2-|{\bf p}_1 +{\bf k}_1|^2-i\epsilon}\,,\label{A1}\end{aligned}$$ where we introduced the vertex functions defined up to a phase factor as $$\begin{aligned}
{\cal V}(p,k,s)&=&\frac{1}{2\omega^2 \pi^3 \sqrt{k}}\nonumber\\
&\times&\int_0^{\infty}d\tau\, K_{i\nu}\left(-i\frac{p}{\omega}\tau\right)K_{i\nu}\left(i\frac{s}{\omega}\tau\right)e^{i\frac{k}{\omega}\tau}\,,\label{vert}\end{aligned}$$ after changing the variable of integration $t_c\to \tau=-\omega t_c$. We obtained thus a closed form of the first Compton amplitude bearing in mind that the second one can be obtained directly by changing ${\bf k}_1 \leftrightarrow {\bf k}_2$ in Eq. (\[A1\]) [@BDR]. We must stress that this result is due to our integral representation since otherwise the Heaviside step functions of the original form (\[DF\]) would have mix up the time integrals.
The Compton amplitudes obtained here are complicated since, in general, the quantum effects in de Sitter spacetimes are described by formulas involving integrals of the form (\[vert\]). For example, in the de Sitter QED in Coulomb gauge [@CQED] the amplitudes in the first order of perturbations are given by integrals of this form whose analyze required an extended analytical and numerical study [@CQED; @Cr1; @Cr2]. A similar study can be performed in the case of the Compton effect calculated here but this exceeds the purposes of this paper where the principal objective was to define our new integral representation of the scalar propagators.
Concluding remarks
==================
The above example shows that the integral representation of the scalar propagators proposed here is crucial for calculating the Feynman diagrams in any order of the SQED in the presence of the gravity of the de Sitter background. Thus one could find new observable effects involving interacting fields, allowed by the local gravity, whose indirect influence could be better measured than its direct interaction with the quantum matter which is very weak.
It remains to study the renormalization observing that here it is not certain that the standard regularization procedures, as for example the Pauli-Villars method, will work as in the flat case. This is because of the structure of the propagators studied here which depend on mass only indirectly through the index of the $K$-functions. Thus a priority task is to find suitable methods of regularization and renormalization looking for alternative methods or adapting the well-known regularization procedures of the two-point functions [@Sch1; @Sch2; @Kel; @DW; @Br].
Concluding we can say with a moderate optimism that now we have all the tools we need for calculating at least the non-gravitational effects of the massive scalar field in the presence of the gravity of the de Sitter expanding universe.
Appendix A: Modified Bessel functions {#appendix-a-modified-bessel-functions .unnumbered}
-------------------------------------
The modified Bessel functions $I_{\nu}(z)$ and $K_{\nu}(z)=K_{-\nu}(z)$ are related to the Hankel ones such that their Wronskian [@NIST] gives the identity $$\label{KuKu}
K_{\nu}(i s)
\stackrel{\leftrightarrow}{\partial_{s}}K_{\nu}(-is)=W[K_{\nu}(is),K_{\nu}(-is)]=\frac{i\pi}{s}\,.$$ For $|z|\to \infty$ and any $\nu$ we have, $$\label{Km0}
I_{\nu}(z) \to \sqrt{\frac{\pi}{2z}}e^{z}\,, \quad K_{\nu}(z) \to K_{\frac{1}{2}}(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\,.$$
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|
---
author:
- |
\
Physik Department, Technische Universität München, D-85748 Garching, Germany\
E-mail:
title: '$K$ and $B$ Physics in the Custodially Protected Randall-Sundrum Model'
---
The custodially protected Randall-Sundrum model
===============================================
During the past ten years Randall-Sundrum (RS) models [@Randall:1999ee] have attracted a lot of attention. Apart from their original motivation to solve the gauge hierarchy problem, models with bulk fields [@Chang:1999nh] are also able to address the flavour problem [@Huber:2003tu; @Agashe:2004cp]. As the localisation of the fermionic zero modes depends exponentially on their bulk mass parameters, the observed hierarchies in the effective Yukawa couplings can naturally be generated from $\mathcal{O}(1)$ fundamental parameters.
The simplest RS model with only the SM gauge group in the bulk has severe problems with the electroweak precision parameters, so that the lowest-lying KK modes have to be at least $\sim10\,\text{TeV}$ and therefore beyond the LHC reach. However RS models with an enlarged custodial symmetry [@Agashe:2003zs] can be made consistent with electroweak precision data for KK scales even as low as $(2-3)\,\text{TeV}$ [@Carena:2007ua]. A detailed description, including a set of Feynman rules, of such a custodially protected RS model has been presented in [@Albrecht:2009xr], where further references can be found.
As the fermions’ couplings to the KK gauge bosons depend on the overlaps of the respective bulk profiles, flavour non-universalities and therefore flavour changing neutral currents (FCNCs) arise already at the tree level. These effects are transmitted also to the $Z$ couplings via electroweak symmetry breaking. Fortunately the protective RS-GIM mechanism [@Agashe:2004cp] is at work: FCNCs are strongly suppressed by the hierarchies responsible for the effective Yukawa couplings. Still interesting and potentially large new physics (NP) contributions to flavour violating observables arise.
Meson-antimeson mixing and fine-tuning [@Blanke:2008zb]
=======================================================
Meson-antimeson mixings in the $K$ and $B_{d,s}$ systems receive significant contributions from KK gluons and the new heavy $Z_H$ gauge boson. In particular new contributions to the scalar left-right operator $(\bar s P_L d)(\bar s P_Rd)$ are generated by the exchange of KK gluons which, due to the strong chiral and QCD enhancement, lead to large NP effects in $K-\bar K$ mixing. Assuming anarchic, i.e. structureless 5D Yukawa couplings, the constraint from the CP-violating parameter $\varepsilon_K$ leads to the generic lower bound $M_\text{KK}{
\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}20\,\text{TeV}$ [@Csaki:2008zd]. However, by partly abandoning the anarchic ansatz, it is possible to lower the KK scale down to $M_\text{KK}\simeq 2.5\,\text{TeV}$. As we can see in the left panel of Fig.\[fig1\] even for such low KK masses it is possible to obtain an agreement with the $\varepsilon_K$ data without the need for large fine-tuning in the fundamental 5D Yukawa couplings.
The data on other $\Delta F=2$ observables turn out to be less restrictive, so that a simultaneous agreement with all these observables can be obtained. Having imposed these constraints we show in the right panel of Fig.\[fig1\] the correlation between the mixing-induced CP-asymmetry $S_{\psi\phi}$ and the semi-leptonic CP-asymmetry $A^s_\text{SL}$, both measuring CP-violating effects in $B_s-\bar B_s$ mixing. Thanks to the new sources of flavour and CP-violation, the custodially protected RS model can easily account for the recently observed possible non-SM effects in that system [@Punzi].
Rare $K$ and $B$ decays [@Blanke:2008yr]
========================================
In contrast to the $\Delta F=2$ sector where the dominant NP contributions arise from KK gauge boson exchanges, $\Delta F=1$ processes are dominated by the new tree level contributions of the $Z$ boson. As the left-handed $Z d_L^i \bar d_L^j$ couplings are protected by the enlarged custodial symmetry [@Blanke:2008zb; @Buras:2009ka], rare $K$ and $B$ decays are dominated by the $Z$ boson couplings to right-handed down-type quarks. Consequently, as the hierarchy in the right-handed down sector can only partly compensate the CKM hierarchy, the NP effects in $K$ decays are much larger than in $B$ decays.
As seen from the left panel of Fig.\[fig2\], the $K\to\pi\nu\bar\nu$ branching ratios can be enhanced by as much as $(100-200)\%$ with respect to the SM. In addition, as a result of the new left-right operator contributions to $\varepsilon_K$ no visible correlation arises [@Blanke:2009pq]. Thus the $K\to\pi\nu\bar\nu$ decays can be used to distinguish this NP model from other frameworks such as the Littlest Higgs model with T-parity whose specific flavour structure results in a strict two-branch correlation in the $K\to\pi\nu\bar\nu$ plane [@LHT]. Also in the case of other rare $K$ decays large deviations from the SM can be found and specific correlations arise that can be used to test this model. Interestingly large effects in the $K$ decays can [*not*]{} appear simultaneously with a large non-SM $S_{\psi\phi}$.
On the other hand the NP effects in rare $B$ decays are generally small ($\mathcal{O}(10\%)$) in the model in question and therefore difficult to disentangle from the SM (see e.g. the $B_{s,d}\to\mu^+\mu^-$ modes in the right panel of Fig.\[fig2\]). Due to the presence of new flavour and CP-violating interactions the predictions from models with Minimal Flavour Violation can be strongly violated.
The analysis of correlations among various flavour violating observables thus provides a promising tool to understand the origin of NP, complementary to direct searches at collider experiments.
Acknowledgements {#acknowledgements .unnumbered}
================
Warmest thanks are given to my collaborators in the project summarised here: M.E.Albrecht, A.J.Buras, B.Duling, K.Gemmler, S.Gori and A.Weiler. This work was partially supported by the Max-Planck-Institut für Physik München, the Deutsche Forschungsgemeinschaft (DFG) under contract BU 706/2-1 and the DFG Cluster of Excellence ‘Origin and Structure of the Universe’.
[99]{}
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K. Agashe, A. Delgado, M. J. May, and R. Sundrum, [*[RS1, custodial isospin and precision tests]{}*]{}, [*JHEP*]{} [**08**]{} (2003) 050, \[[[hep-ph/0308036]{}](http://xxx.lanl.gov/abs/hep-ph/0308036)\]. C. Csaki, C. Grojean, L. Pilo, and J. Terning, [*[Towards a realistic model of Higgsless electroweak symmetry breaking]{}*]{}, [*Phys. Rev. Lett.*]{} [ **92**]{} (2004) 101802, \[[[ hep-ph/0308038]{}](http://xxx.lanl.gov/abs/hep-ph/0308038)\]. K. Agashe, R. Contino, L. Da Rold, and A. Pomarol, [*[A custodial symmetry for $Z b \bar b$]{}*]{}, [*Phys. Lett.*]{} [**B641**]{} (2006) 62–66, \[[[hep-ph/0605341]{}](http://xxx.lanl.gov/abs/hep-ph/0605341)\].
M. S. Carena, E. Ponton, J. Santiago, and C. E. M. Wagner, [*[Electroweak constraints on warped models with custodial symmetry]{}*]{}, [*Phys. Rev.*]{} [**D76**]{} (2007) 035006, \[[[ hep-ph/0701055]{}](http://xxx.lanl.gov/abs/hep-ph/0701055)\]. C. Bouchart and G. Moreau, [*[The precision electroweak data in warped extra-dimension models]{}*]{}, [*Nucl. Phys. *]{}[**B810**]{} (2009) 66 \[[[ 0807.4461]{}](http://xxx.lanl.gov/abs/0807.4461)\].
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M. Blanke, A. J. Buras, B. Duling, S. Gori, and A. Weiler, [*[$\Delta F=2$ Observables and Fine-Tuning in a Warped Extra Dimension with Custodial Protection]{}*]{}, [*JHEP*]{} [**0903**]{} (2009) 001, \[[[ 0809.1073]{}](http://xxx.lanl.gov/abs/0809.1073)\].
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G. Punzi, [*Flavour physics at the Tevatron*]{}, these proceedings.
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M. Blanke [*et. al.*]{}, [*[Rare and CP-violating $K$ and $B$ decays in the Littlest Higgs model with T-parity]{}*]{}, [*JHEP*]{} [**01**]{} (2007) 066, \[[[hep-ph/0610298]{}](http://xxx.lanl.gov/abs/hep-ph/0610298)\]. M. Blanke, A. J. Buras, B. Duling, S. Recksiegel, and C. Tarantino, [*[FCNC Processes in the Littlest Higgs Model with T-Parity: a 2009 Look]{}*]{}, [[ 0906.5454]{}](http://xxx.lanl.gov/abs/0906.5454).
|
---
abstract: 'This chapter is a short introduction to Sullivan models. In particular, we find the Sullivan model of a free loop space and use it to prove the Vigué-Poirrier-Sullivan theorem on the Betti numbers of a free loop space.'
address: |
Faculte des Sciences\
2 Boulevard Lavoisier, 49045 Angers Cedex 01, France\
email:
author:
- Luc Menichi
bibliography:
- 'Bibliographie.bib'
title: 'Rational homotopy – Sullivan models'
---
In the previous chapter, we have seen the following theorem due to Gromoll and Meyer.
Let $M$ be a compact simply connected manifold. If the sequence of Betti numbers of the free loop space on $M$, $M^{S^1}$, is unbounded then any Riemannian metric on $M$ carries infinitely many non trivial and geometrically distinct closed geodesics.
In this chapter, using Rational homotopy, we will see exactly when the sequence of Betti numbers of $M^{S^1}$ over a field of caracteristic $0$ is bounded (See Theorem \[nombres de betti lacets libres pas bornes\] and its converse Proposition \[monogene donne Betti bornes\]). This was one of the first major applications of rational homotopy.
Rational homotopy associates to any rational simply connected space, a commutative differential graded algebra. If we restrict to almost free commutative differential graded algebras, that is “Sullivan models”, this association is unique.
Graded differential algebra
===========================
Definition and elementary properties
------------------------------------
All the vector spaces are over ${{\mathbb{Q}}}$ (or more generally over a field ${{\mathbf{k}}}$ of characteric $0$). We will denote by $\mathbb{N}$ the set of non-negative integers.
A (non-negatively upper) *graded vector space* $V$ is a family $\{V^n\}_{n\in{{\mathbb{N}}}}$ of vector spaces. An element $v\in V_i$ is an element of $V$ of *degree* $i$. The degree of $v$ is denoted $\vert v\vert$. A [*differential*]{} $d$ in $V$ is a sequence of linear maps $d^n:V^n\rightarrow V^{n+1}$ such that $d^{n+1}\circ d^{n}=0$, for all $n\in{{\mathbb{N}}}$. A differential graded vector space or *complex* is a graded vector space equipped with a differential. A morphism of complexes $f:V\buildrel{\simeq}\over\rightarrow W$ is a *quasi-isomorphism* if the induced map in homology $H(f):H(V)\buildrel{\cong}\over\rightarrow H(W)$ is an isomorphism in all degrees.
A *graded algebra* is a graded vector space $A=\{A^n\}_{n\in{{\mathbb{N}}}}$, equipped with a multiplication $\mu:A^p\otimes A^q\rightarrow A^{p+q}$. The algebra $Â$ is *commutative* if $ab=(-1)^{\vert a\vert\vert b\vert}ba$ for all $a$ and $b\in A$.
A differential graded algebra or *dga* is a graded algebra equipped with a differential $d:A^n\rightarrow A^{n+1}$ which is also a *derivation*: this means that for $a$ and $b\in A$ $$d(ab)=(da)b+(-1)^{\vert a\vert}a(db).$$ A *cdga* is a commutative dga.
\[example cdga\] 1) Let $(B,d_B)$ and $(C,d_C)$ be two cdgas. Then the tensor product $B\otimes C$ equipped with the multiplication $$(b\otimes c)(b'\otimes c'):=(-1)^{\vert c\vert\vert b'\vert} bb'\otimes cc'$$ and the differential $$d(b\otimes c)=(db)\otimes c+(-1)^{\vert b\vert}b\otimes dc.$$ is a cdga. The [*tensor product of cdgas*]{} is the sum (or coproduct) in the category of cdgas.
2\) More generally, let $f:A\rightarrow B$ and $g:A\rightarrow C$ be two morphisms of cdgas. Let $B\otimes_A C$ be the quotient of $B\otimes C$ by the sub graded vector spanned by elements of the form $bf(a)\otimes c-b\otimes g(a)c$, $a\in A$, $b\in B$ and $c\in C$. Then $B\otimes_A C$ is a cgda such that the quotient map $B\otimes C\twoheadrightarrow B\otimes_A C$ is a morphism of cdgas. The cdga $B\otimes_A C$ is the pushout of $f$ and $g$ in the category of cdgas: $$\xymatrix{
A\ar[r]^f\ar[d]_g
& B\ar[d]\ar@/^/[ddr]\\
C\ar[r]\ar@/_/[drr]
&B\otimes_A C\ar@{.>}[dr]|-{\exists!}\\
&&D
}$$
3\) Let $V$ and $W$ be two graded vector spaces. We denote by $\Lambda V$ the free graded commutative algebra on $V$.
If $V={{\mathbb{Q}}}v$, i. e. is of dimension $1$ and generated by a single element $v$, then
-$\Lambda V$ is $E(v)={{\mathbb{Q}}}\oplus {{\mathbb{Q}}}v$, the exterior algebra on $v$ if the degree of $v$ is odd and
-$\Lambda V$ is ${{\mathbb{Q}}}[v]=\oplus_{n\in{{\mathbb{N}}}} {{\mathbb{Q}}}v^n$, the polynomial or symmetric algebra on $v$ if the degree of $v$ is even.
Since $\Lambda$ is left adjoint to the forgetful functor from the category of commutative graded algebras to the category of graded vector spaces, $\Lambda$ preserves sums: there is a natural isomorphism of commutative graded algebras $\Lambda (V\oplus W) \cong\Lambda V\otimes \Lambda W$.
Therefore $\Lambda V$ is the tensor product $E(V^{odd})\otimes S(V^{even})$ of the exterior algebra on the generators of odd degree and of the polynomial algebra on the generators of even degree.
Let $f:A\rightarrow B$ be a morphism of commutative graded algebras. Let $d:A\rightarrow B$ be a linear map of degree $k$. By definition, $d$ is a *$(f,f)$-derivation* if for $a$ and $b\in A$ $$d(ab)=(da)f(b)+(-1)^{k\vert a\vert}f(a)(db).$$
\[proprietes universelles\]
1\) Let $i_B:B\hookrightarrow B\otimes\Lambda V$, $b\mapsto b\otimes 1$ and $i_V:V\hookrightarrow B\otimes\Lambda V$, $v\mapsto 1\otimes v$ be the inclusion maps. Let $\varphi:B\rightarrow C$ be a morphism of commutative graded algebras. Let $f:V\rightarrow C$ be a morphism of graded vector spaces. Then $\varphi$ and $f$ extend uniquely to a morphism $B\otimes \Lambda V\rightarrow C$ of commutative graded algebras such that the following diagram commutes $$\xymatrix{
B\ar[r]^\varphi\ar[dr]_{i_B}
&C
&V\ar[l]_f\ar[dl]^{i_V}\\
& B\otimes \Lambda V\ar@{.>}[u]|-{\exists!}
}$$
2\) Let $d_B:B\rightarrow B$ be a derivation of degree $k$. Let $d_V:V\rightarrow B\otimes\Lambda V$ be a linear map of degree $k$. Then there is a unique derivation $d$ such that the following diagram commutes.
$$\xymatrix{
B\ar[r]^-{i_B}
&B\otimes \Lambda V
&V\ar[l]_{d_V}\ar[dl]^{i_V}\\
B\ar[r]^-{i_B}\ar[u]^{d_B}& B\otimes \Lambda V\ar@{.>}[u]|-{\exists!d}
}$$
3\) Let $f:\Lambda V\rightarrow B$ be a morphism of commutative graded algebras. Let $d_V:V\rightarrow B$ be a linear map of degree $k$. Then there exists a unique $(f,f)$-derivation $d$ extending $d_V$: $$\xymatrix{
V\ar[r]^{d_V}\ar[d]_{i_V}
&B\\
\Lambda V\ar@{.>}[ur]_{\exists!d}
}$$
1\) Since $\Lambda V$ is the free commutative graded algebra on $V$, $f$ can be extended to a morphism of graded algebras $\Lambda V\rightarrow C$. Since the tensor product of commutative graded algebras is the sum in the category of commutative graded algebras, we obtain a morphism of commutative graded algebras from $B\otimes \Lambda V$ to $C$.
2\) Since $b\otimes v_1\dots v_n$ is the product $(b\otimes 1)(1\otimes v_1)
\dots (1\otimes v_n)$, $d(b\otimes v_1\dots v_n)$ is given by $$d_B(b)\otimes v_1\dots v_n
+\sum_{i=1}^n (-1)^{k(\vert b\vert+\vert v_1\vert+\dots+\vert v_{i-1}\vert)} (b\otimes v_1\dots v_{i-1})(d_V v_i) (1\otimes v_{i+1}\dots
v_n)$$
3\) Similarly, $d(v_1\dots v_n)$ is given by $$\sum_{i=1}^n (-1)^{k(\vert v_1\vert+\dots+\vert v_{i-1}\vert)} f(v_1)\dots f(v_{i-1})d_V(v_i) f(v_{i+1})\dots
f(v_n)$$
Sullivan models of spheres {#modeles de Sullivan des spheres}
--------------------------
[**Sullivan models of odd spheres $S^{2n+1}$, $n\geq 0$.**]{}
Consider a cdga $A(S^{2n+1})$ whose cohomology is isomorphic as graded algebras to the cohomology of $S^{2n+1}$ with coefficients in ${{\mathbf{k}}}$: $$H^*(A(S^{2n+1}))\cong H^*(S^{2n+1}).$$ When ${{\mathbf{k}}}$ is ${{\mathbb{R}}}$, you can think of $A$ as the De Rham algebra of forms on $S^{2n+1}$. There exists a cycle $v$ of degree $2n+1$ in $A(S^{2n+1})$ such that $$H^*(A(S^{2n+1}))=\Lambda [v].$$ The inclusion of complexes $({{\mathbf{k}}}v,0)\hookrightarrow A(S^{2n+1})$ extends to a unique morphism of cdgas $m:(\Lambda v,0)\rightarrow
A(S^{2n+1})$(Property \[proprietes universelles\]):
$$\xymatrix{
({{\mathbf{k}}}v,0)\ar[r]\ar[d]
& A(S^{2n+1}) \\
(\Lambda v,0)\ar@{.>}[ur]_{\exists!m}
}$$ The induced morphism in homology $H(m)$ is an isomorphism. We say that $m:(\Lambda v,0)\buildrel{\simeq}\over\rightarrow
A(S^{2n+1})$ is a Sullivan model of $S^{2n+1}$
[**Sullivan models of even spheres $S^{2n}$, $n\geq 1$.**]{}
Exactly as above, we construct a morphism of cdga $m_1:(\Lambda v,0)\rightarrow A(S^{2n})$. But now, $H(m_1)$ is not an isomorphism:
$H(m_1)(v)=[v]$. Therefore $H(m_1)(v^2)=[v^2]=[v]^2=0$. Since $[v^2]=0$ in $H^*(A(S^{2n}))$, there exists an element $\psi\in A(S^{2n})$ of degree $4n-1$ such that $d\psi=v^2$.
Let $w$ denote another element of degree $4n-1$. The morphism of graded vector spaces ${{\mathbf{k}}}v\oplus {{\mathbf{k}}}w\hookrightarrow A(S^{2n})$, mapping $v$ to $v$ and $w$ to $\psi$ extends to a unique morphism of commutative graded algebras $m:\Lambda(v,w)\rightarrow A(S^{2n})$ (1) of Property \[proprietes universelles\]): $$\xymatrix{
{{\mathbf{k}}}v\oplus {{\mathbf{k}}}w
\ar[r]\ar[d]
& A(S^{2n}) \\
\Lambda(v,w)\ar@{.>}[ur]_{\exists!m}
}$$
The linear map of degree $+1$, $d_V:V:={{\mathbf{k}}}v\oplus {{\mathbf{k}}}w\rightarrow \Lambda(v,w)$ mapping $v$ to $0$ and $w$ to $v^2$ extends to a unique derivation $d:\Lambda(v,w)\rightarrow \Lambda(v,w)$ (2) of Property \[proprietes universelles\]). $$\xymatrix{
{{\mathbf{k}}}v\oplus {{\mathbf{k}}}w\ar[r]^{d_V}\ar[d]
&\Lambda(v,w)\\
\Lambda(v,w)\ar@{.>}[ur]_{\exists!d}
}$$ Since $d$ is a derivation of odd degree, $d\circ d$ (which is equal to $1/2[d,d]$) is again a derivation. The following diagram commutes $$\xymatrix{
V\ar[r]^{d_V}\ar[d]
&\Lambda V\ar[r]^{d}
&\Lambda V\\
\Lambda V\ar[urr]_{d\circ d}\ar[ur]^{d}
}$$ Since the composite $d\circ d_V$ is null, by unicity (2) of Property \[proprietes universelles\]), the derivation $d\circ d$ is also null. Therefore $(\Lambda V,d)$ is a cdga. This is the general method to check that $d\circ d=0$.
Denote by $d_A$ the differential on $A(S^{2n})$. Let’s check now that $d_A\circ m=m\circ d$. Since $d_A$ and $d$ are both $(id,id)$-derivations, $d_A\circ m$ and $m\circ d$ are both $(m,m)$-derivations.
Since $d_A(m(v))=d_A(v)=0=m(0)=m(d(v))$ and $d_A(m(w))=d_A(\psi)=v^2=m(v^2)=m(d(w))$, $d_A\circ m$ and $m\circ d$ coincide on $V$. Therefore by unicity (3) of Property \[proprietes universelles\]), $d_A\circ m=m\circ d$. Again, this method is general. So finally, we have proved that $m$ is a morphism of cdgas. Now we prove that $H(m)$ is an isomorphism, by checking that $H(m)$ sends a basis to a basis.
Sullivan models
===============
Definitions
-----------
Let $V$ be a graded vector space. Denote by $V^+=V^{\geq 1}$ the sub graded vector space of $V$ formed by the elements of $V$ of positive degrees: $V=V^0\oplus V^+$.
A *relative Sullivan model* (or *cofibration* in the category of cdgas) is a morphism of cdgas of the form $$(B,d_B)\hookrightarrow (B\otimes\Lambda V,d), b\mapsto b\otimes 1$$ where
$\bullet$ $H^0(B)\cong {{\mathbf{k}}}$,
$\bullet$ $V=V^{\geq 1}$,
$\bullet$ and $V$ is the direct sum of graded vector spaces $V(k)$: $$\forall n, V^n=\bigoplus_{k\in\mathbb{N}} V(k)^n$$ such that $d:V(0)\rightarrow B\otimes {{\mathbf{k}}}$ and $d:V(k)\rightarrow B\otimes \Lambda(V(<k))$. Here $V(<k)$ denotes the direct sum $V(0)\oplus\dots\oplus V(k-1)$.
Let $k\in{{\mathbb{N}}}$. Denote by $\Lambda^k V$ the sub graded vector space of $\Lambda V$ generated by elements of the form $v_1\wedge\dots\wedge v_k$, $v_i\in V$. Elements of $\Lambda^k V$ have by definition *wordlength* $k$. For example $\Lambda V={{\mathbf{k}}}\oplus V\oplus \Lambda^{\geq 2}V$ .
A relative Sullivan model $(B,d_B)\hookrightarrow (B\otimes\Lambda V,d)$ is *minimal* if $d:V\rightarrow B^+\otimes \Lambda V+ B\otimes \Lambda^{\geq 2}V$. A *(minimal) Sullivan model* is a (minimal) relative Sullivan model of the form $(B,d_B)=({{\mathbf{k}}},0)\hookrightarrow (\Lambda V,d)$.
[@Felix-Halperin-Thomas:ratht end of the proof of Lemma 23.1] Let $(\Lambda V,d)$ be cdga such that $V=V^{\geq 2}$. Then $(\Lambda V,d)$ is a Sullivan model.
[@Felix-Halperin-Thomas:ratht p. 144] Suppose that $d:V\rightarrow\Lambda^{\geq 2}V$. In this case, the $V(k)$ are easy to define: let $V(k):=V^k$ for $k\in N$. Let $v\in V^k$. By the minimality condition, $dv$ is equal to a sum $\sum_i x_iy_i$ where the non trivial elements $x_i$ and $y_i$ are both of positive length and therefore both of degre $\geq 2$. Since $\vert x_i\vert+\vert y_i\vert=\vert dv\vert=k+1$, both $x_i$ and $y_i$ are of degree less than k. Therefore $dv$ belongs to $\Lambda(V^{<k})=\Lambda(V(<k))$.
The composite of relative Sullivan models is again a Sullivan relative model.
Let $C$ be a cdga. A (minimal) *Sullivan model of* $C$ is a (minimal) Sullivan model $(\Lambda V,d)$ such that there exists a quasi-isomorphism of cdgas $(\Lambda V,d)\buildrel{\simeq}\over\rightarrow C$.
Let $\varphi:B\rightarrow C$ be a morphism of cdgas. A (minimal) *relative Sullivan model of* $\varphi$ is a (minimal) relative Sullivan model $(B,d_B)\hookrightarrow (B\otimes \Lambda V,d)$ such that $\varphi$ can be decomposed as the composite of the relative Sullivan model and of a quasi-isomorphism of cdgas: $$\xymatrix{
B\ar[r]^\varphi\ar[dr]
& C\\
& B\otimes\Lambda V\ar[u]_\simeq
}$$
Any morphism $\varphi:B\rightarrow C$ of cdgas admits a minimal relative Sullivan model if $H^0(B)\cong {{\mathbf{k}}}$, $H^0(\varphi)$ is an isomorphism and $H^1(\varphi)$ is injective.
This theorem is proved in general by Proposition 14.3 and Theorem 14.9 of [@Felix-Halperin-Thomas:ratht]. But in practice, if $H^1(\varphi)$ is an isomorphism, we construct a minimal relative Sullivan model, by induction on degrees as in Proposition 12.2. of [@Felix-Halperin-Thomas:ratht].
An example of relative Sullivan model {#exemple de
model relatif de Sullivan}
-------------------------------------
Consider the minimal Sullivan model of an odd sphere found in section \[modeles de Sullivan des spheres\]
$$(\Lambda v,0)\buildrel{\simeq}\over\rightarrow
A(S^{2n+1}).$$ Assume that $n\geq 1$. Consider the multiplication of $\Lambda v$: the morphism of cdgas $$\mu:(\Lambda v_1,0)\otimes (\Lambda v_2,0)\rightarrow (\Lambda v,0),
v_1\mapsto v, v_2\mapsto v.$$ Recall that $v$, $v_1$ and $v_2$ are of degree $2n+1$.
Denote by $sv$ an element of degree $\vert sv\vert =\vert s\vert+\vert
v\vert=-1+\vert v\vert$. The operator $s$ of degre $-1$ is called the *suspension*.
We construct now a minimal relative Sullivan model of $\mu$. Define $d(sv)=v_2-v_1$. Let $m:\Lambda(v_1,v_2,sv),d\rightarrow (\Lambda v,0)$ be the unique morphism of cdgas extending $\mu$ such that $m(sv)=0$. $$\xymatrix{
(\Lambda v_1,0)\otimes (\Lambda v_2,0)\ar[r]^-\mu\ar[dr]
& (\Lambda v,0)\\
& \Lambda(v_1,v_2,sv,d)\ar[u]_m
}$$
Let $A$ be a differential graded algebra such that $A^0={{\mathbf{k}}}$. The complex of indecomposables of $A$, denoted $Q(A)$, is the quotient $A^+/\mu(A^+\otimes A^+)$.
The complex of indecomposables of $(\Lambda v,0)$, $Q((\Lambda v,0))$, is $({{\mathbf{k}}}v,0)$ while $$Q(\Lambda(v_1,v_2,sv,d))=({{\mathbf{k}}}v_1\oplus{{\mathbf{k}}}v_2\oplus{{\mathbf{k}}}sv,d(sv)=v_2-v_1).$$ The morphism of complexes $Q(m): ({{\mathbf{k}}}v_1\oplus{{\mathbf{k}}}v_2\oplus{{\mathbf{k}}}sv,d(sv)=v_2-v_1)\rightarrow ({{\mathbf{k}}}v,0)$ map $v_1$ to $v$, $v_2$ to $v$ and $sv$ to $0$. It is easy to check that $Q(m)$ is a quasi-isomorphism of complexes.
By Proposition 14.13 of [@Felix-Halperin-Thomas:ratht], since $m$ is a morphism of cdgas between Sullivan model, $Q(m)$ is a quasi-isomorphim of if and only if $m$ is a quasi-isomorphism.
So we have proved that $m$ is a quasi-isomorphism and therefore $$(\Lambda v_1,0)\otimes (\Lambda v_2,0)\hookrightarrow\Lambda(v_1,v_2,sv,d)$$ is a minimal relative Sullivan model of $\mu$. Consider the following commutative diagram of cdgas where the square is a pushout $$\xymatrix{
&& \Lambda v,0\\
\Lambda (v_1,v_2),0\ar[urr]^\mu\ar[d]_\mu\ar[r]
& \Lambda(v_1,v_2,sv),d\ar[ur]_m^\simeq\ar[d]\\
\Lambda v,0\ar[r]
&\Lambda v,0\otimes_{\Lambda (v_1,v_2),0} \Lambda(v_1,v_2,sv),d
}$$ It is easy to check that the cdga $\Lambda v,0\otimes_{\Lambda (v_1,v_2),0} \Lambda(v_1,v_2,sv),d$ is isomorphic to $\Lambda(v,sv),0$. As we will explain later, we have computed in fact, the minimal Sullivan model $\Lambda(v,sv),0$ of the free loop space $(S^{2n+1})^{S^1}$. In particular, the cohomology algebra $H^*((S^{2n+1})^{S^1};{{\mathbf{k}}})$ is isomorphic to $\Lambda(v,sv)$. We can deduce easily that for $p\in{{\mathbb{N}}}$, $
\operatorname{dim} H^p((S^{2n+1})^{S^1})\leq 1
$. So we have shown that the sequence of Betti numbers of the free loop space on odd dimensional spheres is bounded.
The relative Sullivan model of the multiplication
-------------------------------------------------
[@Felix-Oprea-Tanre:algmodgeom Example 2.48]\[modele de Sullivan de la multiplication\] Let $(\Lambda V,d)$ be a relative minimal Sullivan model with $V=V^{\geq 2}$ (concentrated in degrees $\geq 2$). Then the multiplication $\mu: (\Lambda V,d)\otimes(\Lambda V,d)\twoheadrightarrow(\Lambda V,d)$ admits a minimal relative Sullivan model of the form $(\Lambda V\otimes \Lambda V\otimes\Lambda sV,D)$.
We proceed by induction on $n\in\mathbb{N^*}$ to construct quasi-isomorphisms of cdgas $\varphi_n:(\Lambda V^{\leq n}\otimes \Lambda V^{\leq n}\otimes\Lambda sV^{\leq n},D)
\buildrel{\simeq}\over\twoheadrightarrow(\Lambda V^{\leq n},d)$ extending the multiplication on $\Lambda V^{\leq n}$.
Suppose that $\varphi_n$ is constructed. We now define $\varphi_{n+1}$ extending $\varphi_n$ and $\mu$, the multiplication on $\Lambda V$. Let $v\in V^{n+1}$. Then $d(v)\in\Lambda^{\geq 2}(V^{\leq n})$ and $\varphi_n(dv\otimes 1\otimes 1-1\otimes dv\otimes 1)=0$. Since $\varphi_n$ is a surjective quasi-isomorphism, by the long exact sequence associated to a short exact sequence of complexes, $\text{Ker }\varphi_n$ is acyclic. Therefore since $dv\otimes 1\otimes 1-1\otimes dv\otimes 1$ is a cycle, there exists an element $\gamma$ of degree $n+1$ of $\Lambda V^{\leq n}\otimes \Lambda V^{\leq n}\otimes\Lambda sV^{\leq n}$ such that $D(\gamma)=dv\otimes 1\otimes 1-1\otimes dv\otimes 1$ and $\varphi_n(\gamma)=0$. For degree reasons, $\gamma$ is decomposable, i. e. has wordlength $\geq 2$. We define $D(1\otimes 1\otimes sv)=v\otimes 1\otimes 1-1\otimes v\otimes 1-\gamma$ and $\varphi_{n+1}(1\otimes 1\otimes sv)=0$. Since $D\circ D(1\otimes 1\otimes sv)=0$ and $d\circ\varphi_{n+1}(1\otimes 1\otimes sv)=\varphi_{n+1}\circ d(1\otimes 1\otimes sv)$, by Property \[proprietes universelles\], the derivation $D$ is a differential on $\Lambda V^{\leq n+1}\otimes \Lambda V^{\leq n+1}\otimes\Lambda sV^{\leq n+1}$ and the morphism of graded algebras $\varphi_{n+1}$ is a morphism of complexes.
The complex of indecomposables of $(\Lambda V^{\leq n+1}\otimes \Lambda V^{\leq n+1}\otimes\Lambda sV^{\leq n+1},D)$, $$Q((\Lambda V^{\leq n+1}\otimes \Lambda V^{\leq n+1}\otimes\Lambda sV^{\leq n+1},D)$$ is $(V^{\leq n+1}\oplus V^{\leq n+1}\oplus sV^{\leq n+1},d)$ with differential $d$ given by $d(v'\oplus v"\oplus sv)=v\oplus -v\oplus 0$ for $v'$, $v"$ and $v\in
V^{\leq n+1}$. Therefore it is easy to check that $Q(\varphi_{n+1})$ is a quasi-isomorphism. So by Proposition 14.13 of [@Felix-Halperin-Thomas:ratht], $\varphi_{n+1}$ is a quasi-isomorphism. Since $\gamma$ is of degree $n+1$ and $sV^{\leq n}$ is of degree $<n$, this relative Sullivan model is minimal. We now define $\varphi:(\Lambda V\otimes \Lambda V\otimes\Lambda sV,D)
\twoheadrightarrow(\Lambda V,d)$ as $$\displaystyle\lim_{\longrightarrow}\varphi_n=
\bigcup_{n\in\mathbb{N}}\varphi_n: \bigcup_{n\in\mathbb{N}} \left(\Lambda V^{\leq n}\otimes \Lambda V^{\leq n}\otimes\Lambda sV^{\leq n}\right)\rightarrow\bigcup_{n\in\mathbb{N}}\Lambda V^{\leq n}.$$ Since homology commutes with direct limits in the category of complexes [@Spanier:livre Chap 4, Sect 2, Theorem 7], $H(\varphi)=\displaystyle\lim_{\longrightarrow}H(\varphi_n)$ is an isomorphism.
Rational homotopy theory
========================
Let $X$ be a topological space. Denote by $S^*(X)$ the singular cochains of $X$ with coefficients in ${{\mathbf{k}}}$. The dga $S^*(X)$ is almost never commutative. Nevertheless, Sullivan, inspired by Quillen proved the following theorem.
[@Felix-Halperin-Thomas:ratht Corollary 10.10]\[quasi-isos entre A\_PL et les cochaines\] For any topological space $X$, there exists two natural quasi-isomorphisms of dgas $$S^*(X)\buildrel{\simeq}\over\rightarrow D(X)\buildrel{\simeq}\over\leftarrow A_{PL}(X)$$ where $A_{PL}(X)$ is commutative.
\[sur les reels formes de De Rham\] This cdga $A_{PL}(X)$ is called the algebra of *polynomial differential forms*. If ${{\mathbf{k}}}={{\mathbb{R}}}$ and $X$ is a smooth manifold $M$, you can think that $A_{PL}(M)$ is the De Rham algebra of differential forms on $M$, $A_{DR}(M)$ [@Felix-Halperin-Thomas:ratht Theorem 11.4].
[@Felix-Oprea-Tanre:algmodgeom Definition 2.34] Two topological spaces $X$ and $Y$ have the same *rational homotopy type* if there exists a finite sequence of continuous applications $$X\buildrel{f_0}\over\rightarrow Y_1\buildrel{f_1}\over\leftarrow Y_2
\dots
Y_{n-1}\buildrel{f_{n-1}}\over\leftarrow Y_n\buildrel{f_{n}}\over\rightarrow Y$$ such that the induced maps in rational cohomology $$\begin{gathered}
H^*(X;{{\mathbb{Q}}})\buildrel{H^*(f_0)}\over\leftarrow H^*(Y_1;{{\mathbb{Q}}})\buildrel{H^*(f_1)}\over\rightarrow H^*(Y_2;{{\mathbb{Q}}})
\dots
H^*(Y_{n-1}1;{{\mathbb{Q}}})\\\buildrel{H^*(f_{n-1})}\over\rightarrow H^*(Y_n;{{\mathbb{Q}}})\buildrel{H^*(f_{n})}\over\leftarrow H^*(Y;{{\mathbb{Q}}})\end{gathered}$$ are all isomorphisms.
\[modele minimal unique et groupes d’homotopie\] Let $X$ be a path connected topological space.
1\) (Unicity of minimal Sullivan models [@Felix-Halperin-Thomas:ratht Corollary p. 191]) Two minimal Sullivan models of $A_{PL}(X)$ are isomorphic.
2\) Suppose that $X$ is simply connected and $\forall n\in\mathbb{N}$, $H_n(X;{{\mathbf{k}}})$ is finite dimensional. Let $(\Lambda V,d)$ be a minimal Sullivan model of $X$. Then [@Felix-Halperin-Thomas:ratht Theorem 15.11] for all $n\in\mathbb{N}$, $V^n$ is isomorphic to $\text{Hom}_{{\mathbf{k}}}(\pi_n(X)\otimes_\mathbb{Z} {{\mathbf{k}}},{{\mathbf{k}}})\cong\text{Hom}_\mathbb{Z}(\pi_n(X),{{\mathbf{k}}})$. In particular [@Felix-Halperin-Thomas:ratht Remark 1 p.208], $\text{Dimension } V^n= \text{Dimension } \pi_n(X)\otimes_\mathbb{Z} {{\mathbf{k}}}< \infty$.
The isomorphim of graded vector spaces between $V$ and $\text{Hom}_{{\mathbf{k}}}(\pi_*(X)\otimes_\mathbb{Z} {{\mathbf{k}}},{{\mathbf{k}}})$ is natural in some sense [@Felix-Oprea-Tanre:algmodgeom p. 75-6] with respect to maps $f:X\rightarrow Y$. The isomorphism behaves well also with respect to the long exact sequence associated to a (Serre) fibration ([@Felix-Halperin-Thomas:ratht Proposition 15.13] or [@Felix-Oprea-Tanre:algmodgeom Proposition 2.65]).
[@Felix-Oprea-Tanre:algmodgeom Proposition 2.35][@Felix-Halperin-Thomas:ratht p. 139] Let $X$ and $Y$ be two simply connected topological spaces such that $H^n(X;{{\mathbb{Q}}})$ and $H^n(Y;{{\mathbb{Q}}})$ are finite dimensional for all $n\in {{\mathbb{N}}}$. Let $(\Lambda V,d)$ be a minimal Sullivan model of $X$ and let $(\Lambda W,d)$ be a minimal Sullivan model of $Y$. Then $X$ and $Y$ have the same rational homotopy type if and only if $(\Lambda V,d)$ is isomorphic to $(\Lambda W,d)$ as cdgas.
Sullivan model of a pullback
============================
Sullivan model of a product
---------------------------
Let $X$ and $Y$ be two topological spaces. Let $p_1:X\times Y\twoheadrightarrow Y$ and $p_2:X\times Y\twoheadrightarrow X$ be the projection maps. Let $m$ be the unique morphism of cdgas given by the universal property of the tensor product (Example \[example cdga\] 1)) $$\xymatrix{
& A_{PL}(Y)\ar[d]\ar@/^/[ddr]^{A_{PL}(p_2)}\\
A_{PL}(X)\ar[r]\ar@/_/[drr]_{A_{PL}(p_1)}
&A_{PL}(X)\otimes
A_{PL}(Y)\ar@{.>}[dr]|-{\exists!m}\\
&&A_{PL}(X\times Y).
}$$ Assume that $H^*(X;{{\mathbf{k}}})$ or $H^*(Y;{{\mathbf{k}}})$ is finite dimensional in all degrees. Then [@Felix-Halperin-Thomas:ratht Example 2, p. 142-3] $m$ is a quasi-isomorphism. Let $m_X:\Lambda V\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ be a Sullivan model of $X$. Let $m_Y:\Lambda W\buildrel{\simeq}\over\rightarrow A_{PL}(Y)$ be a Sullivan model of $Y$. Then by Künneth theorem, the composite $$\Lambda V\otimes \Lambda W\buildrel{m_X\otimes m_Y}\over\rightarrow
A_{PL}(X)\otimes A_{PL}(Y) \buildrel{m}\over\rightarrow
A_{PL}(X\times Y)$$ is a quasi-isomorphism of cdgas. Therefore we have proved that “the Sullivan model of a product is the tensor product of the Sullivan models”.
the model of the diagonal {#modele de la diagonale}
-------------------------
Let $X$ be a topological space such that $H^*(X)$ is finite dimensional in all degrees. Denote by $\Delta:X\rightarrow X\times X$, $x\mapsto (x,x)$ the diagonal map of $X$. Using the previous paragraph, since $A_{PL}(p_1\circ \Delta)=A_{PL}(p_2\circ \Delta)=A_{PL}({\operatorname{id}})={\operatorname{id}}$, we have the commutative diagram of cdgas. $$\xymatrix{
A_{PL}(X)\ar[r]\ar[dr]_{A_{PL}(p_1)}\ar@/_2pc/[ddr]_{{\operatorname{id}}}
&A_{PL}(X)\otimes A_{PL}(X)\ar[d]^{m}_\simeq
& A_{PL}(X)\ar[l]\ar[dl]^{A_{PL}(p_2)}\ar@/^2pc/[ddl]^{{\operatorname{id}}}\\
&A_{PL}(X\times X)\ar[d]^{A_{PL}(\Delta)}\\
&A_{PL}(X)
}$$ Therefore the composite $A_{PL}(X)\otimes A_{PL}(X)\buildrel{m}\over\rightarrow
A_{PL}(X\times X)\buildrel{A_{PL}(\Delta)}\over\rightarrow A_{PL}(X)$ coincides with the multiplication $\mu: A_{PL}(X)\otimes A_{PL}(X)\rightarrow A_{PL}(X)$. Therefore the following diagram of cdgas commutes $$\xymatrix{
A_{PL}(X)
&A_{PL}(X\times X)\ar[l]_{A_{PL}(\Delta)}\\
& A_{PL}(X)\otimes A_{PL}(X)\ar[ul]_{\mu}\ar[u]_{m}^\simeq\\
\Lambda V\ar[uu]^{m_X}_\simeq
&\Lambda V\otimes \Lambda V\ar[l]^{\mu}\ar[u]_{m_X\otimes m_X}^\simeq\\
}$$ Here $m_X:\Lambda V\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ denotes a Sullivan model of $X$. Therefore we have proved that “the morphism modelling the diagonal map is the multiplication of the Sullivan model”.
Sullivan model of a fibre product {#Sullivan model d'un produit fibre}
---------------------------------
Consider a pullback square in the category of topological spaces $$\xymatrix{
P\ar[r]^g\ar[d]_q
&E\ar[d]^p\\
X\ar[r]^f
&B
}$$
where
$\bullet$ $p:E\rightarrow B$ is a (Serre) fibration between two topological spaces,
$\bullet$ for every $i\in\mathbb{N}$, $H^i(X)$ and $H^i(B)$ are finite dimensional,
$\bullet$ the topological spaces $X$ and $E$ are path-connected and $B$ is simply-connected.
Since $p$ is a (Serre) fibration, the pullback map $q$ is also a (Serre) fibration. Let $A_{PL}(B)\otimes\Lambda V$ be a relative Sullivan model of $A(p)$. Consider the corresponding commutative diagram of cdgas $$\xymatrix{
&A_{PL}(B)\ar[r]^{A_{PL}(f)}\ar[d]\ar@/_2pc/[ddl]_{A_{PL}(p)}
&A_{PL}(X)\ar[d]\ar@/^2pc/[ddr]^{A_{PL}(q)}\\
&A_{PL}(B)\otimes\Lambda V\ar[r]\ar[dl]_m^\simeq
&A_{PL}(X)\otimes_{A_{PL}(B)}A_{PL}(B)\otimes\Lambda V\ar@{.>}[dr]|-{\exists!m'}\\
A_{PL}(E)\ar[rrr]^{A_{PL}(g)}
&&&A_{PL}(P)
}$$ where the rectangle is a pushout and $m'$ is given by the universal property. Explicitly, for $x\in A_{PL}(X)$ and $e\in A_{PL}(B)\otimes\Lambda V$, $m'(x\otimes e)$ is the product of $A_{PL}(q)(x)$ and $A_{PL}(g)\circ m(e)$.
Since $A_{PL}(B)\hookrightarrow A_{PL}(B)\otimes\Lambda V$ is a relative Sullivan model, the inclusion obtained via pullback $A_{PL}(X)\hookrightarrow A_{PL}(X)\otimes_{A_{PL}(B)}(A_{PL}(B)\otimes\Lambda V,d)\cong
(A_{PL}(X)\otimes\Lambda V,d)$ is also a relative Sullivan model (minimal if $A_{PL}(B)\hookrightarrow A_{PL}(B)\otimes\Lambda V$ is minimal).
By [@Felix-Halperin-Thomas:ratht Proposition 15.8] (or for weaker hypothesis [@Felix-Oprea-Tanre:algmodgeom Theorem 2.70]),
The morphism of cdgas $m'$ is a quasi-isomorphism.
We can summarize this theorem by saying that: “The push-out of a (minimal) relative Sullivan model of a fibration is a (minimal) relative Sullivan model of the pullback of the fibration.”
Since by [@Felix-Halperin-Thomas:ratht Lemma 14.1], $A_{PL}(B)\otimes\Lambda V$ is a “semi-free” resolution of $A_{PL}(E)$ as left $A_{PL}(B)$-modules, by definition of the differential torsion product, $$\text{Tor}^{A_{PL}(B)}(A_{PL}(X),A_{PL}(E)):=H(A_{PL}(X)\otimes_{A_{PL}(B)}(A_{PL}(B)\otimes\Lambda V).$$ By Theorem \[quasi-isos entre A\_PL et les cochaines\] and naturality, we have an isomorphim of graded vector spaces $$\text{Tor}^{A_{PL}(B)}(A_{PL}(X),A_{PL}(E))\cong \text{Tor}^{S^*(B)}(S^*(X),S^*(E)).$$ The Eilenberg-Moore formula gives an isomorphism of graded vector spaces $$\text{Tor}^{S^*(B)}(S^*(X),S^*(E))\cong H^*(P).$$ We claimed that the resulting isomorphism between the homology of $A_{PL}(X)\otimes_{A_{PL}(B)}(A_{PL}(B)\otimes\Lambda V)$ and $H^*(P)$ can be identified with $H(m)$. Therefore $m$ is a quasi-isomorphism.
Instead of working with $A_{PL}$, we prefer usually to work at the level of Sullivan models. Let $m_B:\Lambda B\buildrel{\simeq}\over\rightarrow A_{PL}(B)$ be a Sullivan model of $B$. Let $m_X:\Lambda X\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ be a Sullivan model of $X$. Let $\varphi$ be a morphism of cdgas such the following diagram commutes exactly
$$\xymatrix{
A_{PL}(B)\ar[r]^{A_{PL}(f)}
&A_{PL}(X)\\
\Lambda B\ar[r]^{\varphi}\ar[u]^{m_B}_\simeq
&\Lambda X\ar[u]^{m_X}_\simeq
}$$ Let $\Lambda B\hookrightarrow \Lambda B\otimes \Lambda V$ be a relative Sullivan model of $A_{PL}(p)\circ m_B$. Consider the corresponding commutative diagram of cdgas $$\label{diagram Sullivan model d'un produit fibre}
\xymatrix{
A_{PL}(B)\ar[dd]_{A_{PL}(p)}
&\Lambda B\ar[r]^{\varphi}\ar[d]\ar[l]_{m_B}^ \simeq
&\Lambda X\ar[d]\ar[r]^{m_X}_\simeq
&A_{PL}(X)\ar[dd]^{A_{PL}(q)}\\
&\Lambda B\otimes\Lambda V\ar[r]\ar[dl]_m^\simeq
&\Lambda X\otimes_{\Lambda B}(\Lambda B\otimes\Lambda V)\ar@{.>}[dr]|-{\exists!m'}\\
A_{PL}(E)\ar[rrr]^{A_{PL}(g)}
&&&A_{PL}(P)
}$$ where the rectangle is a pushout and $m'$ is given by the universal property. Then again, $\Lambda X\hookrightarrow \Lambda X\otimes_{\Lambda B}(\Lambda B\otimes\Lambda V)$ is a relative Sullivan model and the morphism of cdgas $m'$ is a quasi-isomorphism. The reader should skip the following remark on his first reading.
\[modele a homotopie pres\] 1) In the previous proof, if the composites $m_X\circ \varphi$ and $A_{PL}(f)\circ m_B$ are not strictly equal then the map $m'$ is not well defined. In general, the composites $m_X\circ \varphi$ and $A_{PL}(f)\circ m_B$ are only homotopic and the situation is more complicated: see part 2) of this remark.
2\) Let $m_B:\Lambda B\buildrel{\simeq}\over\rightarrow A_{PL}(B)$ be a Sullivan model of $B$. Let $m_X':\Lambda X'\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ be a Sullivan model of $X$. By the lifting Lemma of Sullivan models [@Felix-Halperin-Thomas:ratht Proposition 14.6], there exists a morphism of cdgas $\varphi':\Lambda B\rightarrow\Lambda
X'$ such that the following diagram commutes only up to homotopy (in the sense of [@Felix-Oprea-Tanre:algmodgeom Section 2.2]) $$\xymatrix{
A_{PL}(B)\ar[r]^{A_{PL}(f)}
&A_{PL}(X)\\
\Lambda B\ar[r]^{\varphi'}\ar[u]^{m_B}_\simeq
&\Lambda X'.\ar[u]^{m_X'}_\simeq
}$$ In general, this square is not strictly commutative. Let $\Lambda B\hookrightarrow \Lambda B\otimes \Lambda V$ be a relative Sullivan model of $A_{PL}(p)\circ m_B$. Then there exists a commutative diagram of cdgas $$\xymatrix{
A_{PL}(X)\ar[r]^{A_{PL}(q)}
& A_{PL}(P)\\
\Lambda X\ar[r]\ar[u]^ \simeq\ar[d]^ \simeq
& \Lambda X\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V)\ar[u]^ \simeq\ar[d]^ \simeq\\
\Lambda X'\ar[r]
& \Lambda X'\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V)}$$
Let $\Lambda B\buildrel{\varphi}\over\hookrightarrow \Lambda X\buildrel{\theta}\over\rightarrow \Lambda X'$ be a relative Sullivan model of $\varphi'$. Since the composites $m_{X}'\circ\theta\circ\varphi$ and $A_{PL}(f)\circ m_B$ are homotopic, by the homotopy extension property [@Felix-Oprea-Tanre:algmodgeom Proposition 2.22] of the relative Sullivan model $\varphi:\Lambda B\hookrightarrow \Lambda X$, there exists a morphism of cdgas $m_X:\Lambda X\rightarrow A_{PL}(X)$ homotopic to $m_{X}'\circ\theta$ such that $m_X\circ \varphi=A_{PL}(f)\circ m_B$. Therefore using diagram (\[diagram Sullivan model d’un produit fibre\]), we obtain the following commutative diagram of cdgas: $$\xymatrix{
A_{PL}(X)\ar[r]^{A_{PL}(q)}
& A_{PL}(P)
&A_{PL}(E)\ar[l]_{A_{PL}(g)}\\
\Lambda X\ar[r]\ar[u]^\simeq_{m_X}\ar[d]_\simeq^{\theta}
& \Lambda X\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V)\ar[u]^ \simeq_{m'}\ar[d]_\simeq^{\theta\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V)}
&\Lambda B\otimes \Lambda V\ar[u]^\simeq_{m}\ar[l]\\
\Lambda X'\ar[r]
& \Lambda X'\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V).
}$$ Here, since $\theta$ is a quasi-isomorphism, the pushout morphism $\theta\otimes_{\Lambda B} (\Lambda B\otimes \Lambda V)$ along the relative Sullivan model $\Lambda X\hookrightarrow \Lambda X\otimes_{\Lambda B}(\Lambda B\otimes\Lambda V)$ is also a quasi-isomorphism [@Felix-Halperin-Thomas:ratht Lemma 14.2].
Sullivan model of a fibration {#modele de Sullivan d'une fibration}
-----------------------------
Let $p:E\rightarrow B$ be a (Serre) fibration with fibre $F:=p^{-1}(b_0)$.
$$\xymatrix{
F\ar[r]^j\ar[d]
&E\ar[d]^p\\
b_0\ar[r]
&B
}$$ Taking $X$ to be the point $b_0$, we can apply the results of the previous section. Let $m_B:(\Lambda V,d)\buildrel{\simeq}\over\rightarrow A_{PL}(B)$ be a Sullivan model of $B$. Let $(\Lambda V,d)\hookrightarrow (\Lambda V\otimes\Lambda W,d)$ be a relative Sullivan model of $A_{PL}(p)\circ m_B$.
Since $A_{PL}(\{b_0\})$ is equal to $({{\mathbf{k}}},0)$, there is a unique morphism of cdgas $m'$ such that the following diagram commutes
$$\xymatrix{
A_{PL}(B)\ar[r]^{A_{PL}(p)}
&A_{PL}(E)\ar[r]^{A_{PL}(j)}
&A_{PL}(F)\\
(\Lambda V,d)\ar[r]\ar[u]_{m_B}^\simeq
&(\Lambda V\otimes\Lambda W,d)\ar[r]\ar[u]_\simeq
&(k,0)\otimes_{(\Lambda V,d)}(\Lambda V\otimes\Lambda W,d)\ar[u]_{m'}
}$$ Suppose that the base $B$ is a simply connected space and that the total space $E$ is path-connected. Then by the previous section, the morphism of cdga’s $$m':(k,0)\otimes_{(\Lambda V,d)}(\Lambda V\otimes\Lambda W,d)\cong
(\Lambda W,\bar{d})\buildrel{\simeq}\over\longrightarrow A_{PL}(F)$$ is a quasi-isomorphism:
“ The cofiber of a relative Sullivan model of a fibration is a Sullivan model of the fiber of the fibration.”
Note that the cofiber of a relative Sullivan model is minimal if and only if the relative Sullivan model is minimal.
Sullivan model of free loop spaces
----------------------------------
Let $X$ be a simply-connected space. Consider the commutative diagram of spaces $$\xymatrix{
X^{S^1}\ar[r]\ar[d]_{ev}
& X^I\ar[d]_{(ev_0,ev_1)}
& X\ar[dl]^{\Delta}\ar[l]^{\approx}_\sigma
\\
X\ar[r]_-{\Delta}
&X\times X
}$$ where the square is a pullback. Here $I$ denotes the closed interval $[0,1]$, $ev$, $ev_0$, $ev_1$ are the evaluation maps and the homotopy equivalence $\sigma:X\buildrel{\approx}\over\rightarrow X^I$ is the inclusion of constant paths. Let $m_X:\Lambda V\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ be a minimal Sullivan model of $X$. By Proposition \[modele de Sullivan de la multiplication\], the multiplication $\mu:\Lambda V\otimes\Lambda V\rightarrow \Lambda V$ admits a minimal relative Sullivan model of the form $$\Lambda V\otimes\Lambda V\hookrightarrow\Lambda V\otimes\Lambda V\otimes\Lambda sV.$$ Since $\mu$ is a model of the diagonal (Section \[modele de la diagonale\]) and since $\Delta=(ev_0,ev_1)\circ\sigma$, we have the commutative rectangle of cdgas $$\xymatrix{
A_{PL}(X\times X)\ar[rr]^{A_{PL}((ev_0,ev_1))}
&& A_{PL}(X^I)\ar[r]^{A_{PL}(\sigma)}
& A_{PL}(X)\\
\Lambda V\otimes\Lambda V\ar[u]^{m_{X\times X}}_\simeq\ar[rr]
&& \Lambda V\otimes\Lambda V\otimes\Lambda sV\ar[r]_-\simeq
&\Lambda V\ar[u]_{m_{X}}^\simeq
}$$ Since $\sigma$ is a homotopy equivalence, $S^*(\sigma)$ is a homotopy equivalence of complexes and in particular a quasi-isomorphim. So by Theorem \[quasi-isos entre A\_PL et les cochaines\] and naturality, $A_{PL}(\sigma)$ is also a quasi-isomorphism. Therefore, by the lifting property of relative Sullivan models [@Felix-Halperin-Thomas:ratht Proposition 14.6], there exists a morphism of cdgas $\varphi:\Lambda V\otimes\Lambda V\otimes\Lambda sV
\rightarrow A_{PL}(X^I)
$ such that, in the diagram of cdgas $$\xymatrix{
A_{PL}(X\times X)\ar[r]^{A_{PL}((ev_0,ev_1))}
& A_{PL}(X^I)\ar[r]^{A_{PL}(\sigma)}_\simeq
& A_{PL}(X)\\
\Lambda V\otimes\Lambda V\ar[u]^{m_{X\times X}}_\simeq\ar[r]
& \Lambda V\otimes\Lambda V\otimes\Lambda sV\ar[r]_-\simeq
\ar@{.>}[u]^{\varphi}_\simeq
&\Lambda V\ar[u]_{m_{X}}^\simeq
}$$ the left square commutes exactly and the right square commutes in homology. Therefore $\varphi$ is also a quasi-isomorphism. This means that $$\Lambda V\otimes\Lambda V\hookrightarrow\Lambda V\otimes\Lambda V\otimes\Lambda sV.$$ is a relative Sullivan model of the composite $$\Lambda V\otimes \Lambda V\buildrel{m_{X\times X}}\over\rightarrow
A_{PL}(X\times X)\buildrel{A_{PL}((ev_0,ev_1))}\over\longrightarrow
A_{PL}(X^I).$$ Here diagram (\[diagram Sullivan model d’un produit fibre\]) specializes to the following commutative diagram of cdgas $$\label{diagram Sullivan model lacets libres}
\xymatrix{
&\Lambda V\otimes\Lambda V\ar[r]^{\mu}\ar[d]
&\Lambda V\ar[d]\ar[r]^{m_X}_\simeq
&A_{PL}(X)\ar[dd]^{A_{PL}(ev)}\\
&\Lambda V\otimes\Lambda V\otimes\Lambda sV\ar[r]\ar[dl]_\varphi^\simeq
&\Lambda V\otimes_{\Lambda V\otimes\Lambda V}\Lambda V\otimes\Lambda V\otimes\Lambda sV\ar[dr]_{\simeq}\\
A(X^I)\ar[rrr]
&&&A(X^{S^1})
}$$ where the rectangle is a pushout. Therefore $$\Lambda V\hookrightarrow
\Lambda V\otimes_{\Lambda V\otimes\Lambda V}\left(\Lambda V\otimes\Lambda V\otimes\Lambda sV\right)\cong (\Lambda V\otimes\Lambda sV,\delta)$$ is a minimal relative Sullivan model of $A_{PL}(ev)\circ m_X$.
\[theoreme de Chen sur lacets libres\] Let $X$ be a simply-connected space. Then the free loop space cohomology of $H^*(X^{S^1};{{\mathbf{k}}})$ with coefficients in a field ${{\mathbf{k}}}$ of characteristic $0$ is isomorphic to the Hochschild homology of $A_{PL}(X)$, $HH_*(A_{PL}(X),A_{PL}(X))$.
Replacing $A_{PL}(X)$ by $A_{DR}(M)$ (Remark \[sur les reels formes de De Rham\]), this Corollary is a theorem of Chen [@Brylinski:loopchageo 3.2.3 Theorem] when $X$ is a smooth manifold $M$.
The quasi-isomorphism of cdgas $m_X:\Lambda V
\buildrel{\simeq}\over\rightarrow A_{PL}(X)$ induces an isomorphism between Hochschild homologies $$HH_*(m_X,m_X):HH_*(\Lambda V,\Lambda V)\buildrel{\cong}\over\rightarrow HH_*(A_{PL}(X), A_{PL}(X)).$$ By [@Felix-Halperin-Thomas:ratht Lemma 14.1], $\Lambda V\otimes\Lambda V\otimes\Lambda sV$ is a semi-free resolution of $\Lambda V$ as a $\Lambda V\otimes\Lambda V^{op}$-module. Therefore the Hochschild homology $HH_*(\Lambda V,\Lambda V)$ can be defined as the homology of the cdga $(\Lambda V\otimes\Lambda sV,\delta)$. We have just seen above that $H(\Lambda V\otimes\Lambda sV,\delta)$ is isomorphic to the free loop space cohomology $H^*(X^{S^1};{{\mathbf{k}}})$.
We have shown that a Sullivan model of $X^{S^1}$ is of the form $(\Lambda V\otimes \Lambda sV,\delta)$. The following theorem of Vigué-Poirrier and Sullivan gives a precise description of the differential $\delta$.
([@Vigue-Sullivan:homtcg Theorem p. 637] or [@Felix-Oprea-Tanre:algmodgeom Theorem 5.11]\[differentiel du modele de Sullivan des lacets libres\]) Let $X$ be a simply connected topological space. Let $(\Lambda V,d)$ be a minimal Sullivan model of $X$. For all $v\in V$, denote by $sv$ an element of degree $\vert v\vert -1$. Let $s:\Lambda V\otimes \Lambda sV\rightarrow \Lambda V\otimes \Lambda sV$ be the unique derivation of (upper) degree $-1$ such that on the generators $v$, $sv$, $v\in V$, $s(v)=sv$ and $s(sv)=0$. We have $s\circ s=0$. Then there exists a unique Sullivan model of $X^{S^1}$ of the form $(\Lambda V\otimes \Lambda sV,\delta)$ such that $\delta\circ s+s\circ \delta=0$ on $\Lambda V\otimes \Lambda sV$.
\[modele de la fibration des lacets libres\] Consider the free loop fibration $\Omega X\hookrightarrow X^{S^1}\buildrel{ev}\over\twoheadrightarrow X$. Since $(\Lambda V,d)\hookrightarrow (\Lambda V\otimes \Lambda sV,\delta)$ is a minimal relative Sullivan model of $A_{PL}(ev)\circ m_X$, by Section \[modele de Sullivan d’une fibration\], $${\Bbbk}\otimes_{(\Lambda V,d)}(\Lambda V\otimes \Lambda sV,\delta)\cong (\Lambda sV,\bar{\delta})$$ is a minimal Sullivan model of $\Omega X$. Let $v\in V$. By Theorem \[differentiel du modele de Sullivan des lacets libres\], $\delta(sv)=-s\delta v=-sdv$. Since $dv\in\Lambda^{\geq 2}V$, $\delta(sv)\in \Lambda^{\geq 1}V\otimes \Lambda^1 sV$. Therefore $\bar{\delta}=0$. Since $\Omega X$ is a $H$-space, this follows also from Theorem \[model H-space\] and from the unicity of minimal Sullivan models (part 1) of Theorem \[modele minimal unique et groupes d’homotopie\]).
Examples of Sullivan models
===========================
Sullivan model of spaces with polynomial cohomology
---------------------------------------------------
The following proposition is a straightforward generalisation [@Felix-Halperin-Thomas:ratht p. 144] of the Sullivan model of odd-dimensional spheres (see section \[modeles de Sullivan des spheres\]).
\[modele de Sullivan polynomial cohomology\] Let $X$ be a path connected topological space such that its cohomology $H^*(X;{{\mathbf{k}}})$ is a free graded commutative algebra $\Lambda V$ (for example, polynomial). Then a Sullivan model of $X$ is $(\Lambda V,0)$.
Odd-dimensional spheres $S^{2n+1}$, complex or quartenionic Stiefel manifolds [@Felix-Oprea-Tanre:algmodgeom Example 2.40] $V_k(\mathbb{C}^n)$ or $V_k(\mathbb{H}^n)$, classifying spaces $BG$ of simply connected Lie groups [@Felix-Oprea-Tanre:algmodgeom Example 2.42], connected Lie groups $G$ as we will see in the following section.
Sullivan model of an $H$-space
------------------------------
An *$H$-space* is a pointed topological space $(G,e)$ equipped with a pointed continuous map $\mu:(G,e)\times (G,e)\rightarrow (G,e)$ such that the two pointed maps $g\mapsto \mu(e,g)$ and $g\mapsto \mu(g,e)$ are pointed homotopic to the identity map of $(G,e)$.
[@Felix-Halperin-Thomas:ratht Example 3 p. 143]\[model H-space\] Let $G$ be a path connected $H$-space such that $\forall n\in\mathbb{N}$, $H_n(G;{{\mathbf{k}}})$ is finite dimensional. Then
1\) its cohomology $H^*(G;{{\mathbf{k}}})$ is a free graded commutative algebra $\Lambda V$,
2\) $G$ has a Sullivan model of the form $(\Lambda V,0)$, that is with zero differential.
1\) Let $A$ be $H^*(G;{{\mathbf{k}}})$ the cohomology of $G$. By hypothesis, $A$ is a connected commutative graded Hopf algebra (not necessarily associative). Now the theorem of Hopf-Borel in caracteristic $0$ [@DoldA:lecat VII.10.16] says that $A$ is a free graded commutative algebra.
2\) By Proposition \[modele de Sullivan polynomial cohomology\], 1) and 2) are equivalent.
Let $G$ be a path-connected Lie group (or more generally a $H$-space with finitely generated integral homology). Then $G$ has a Sullivan model of the form $(\Lambda V,0)$. By Theorem \[modele minimal unique et groupes d’homotopie\], $V^n$ and $\pi_n(G)\otimes_\mathbb{Z}{{\mathbf{k}}}$ have the same dimension for any $n\in\mathbb{N}$. Since $H_*(G;{{\mathbf{k}}})$ is of finite (total) dimension, $V$ and therefore $\pi_*(G)\otimes_\mathbb{Z}{{\mathbf{k}}}$ are concentrated in odd degrees. In fact, more generally [@Browder:torsionH-space Theorem 6.11], $\pi_2(G)=\{0\}$. Note, however that $\pi_4(S^3)=\mathbb{Z}/2\mathbb{Z}\neq \{0\}$.
Sullivan model of projective spaces
-----------------------------------
Consider the complex projective space $\mathbb{CP}^n$, $n\geq 1$. The construction of the Sullivan model of $\mathbb{CP}^n$ is similar to the construction of the Sullivan model of $S^2=\mathbb{CP}^ 1$ done in section \[modeles de Sullivan des spheres\]:
The cohomology algebra $H^*(A_{PL}(\mathbb{CP}^n))\cong H^*(\mathbb{CP}^n)$ is the truncated polynomial algebra $\frac{{{\mathbf{k}}}[x]}{x^{n+1}=0}$ where $x$ is an element of degree $2$. Let $v$ be a cycle of $A_{PL}(\mathbb{CP}^n)$ representing $x:=[v]$. The inclusion of complexes $({{\mathbf{k}}}v,0)\hookrightarrow A_{PL}(\mathbb{CP}^n)$ extends to a unique morphism of cdgas $m:(\Lambda v,0)\rightarrow
A_{PL}(\mathbb{CP}^n)$(Property \[proprietes universelles\]). Since $[v^{n+1}]=x^{n+1}=0$, there exists an element $\psi\in A_{PL}(\mathbb{CP}^n)$ of degree $2n+1$ such that $d\psi=v^{n+1}$. Let $w$ denote another element of degree $2n+1$. Let $d$ be the unique derivation of $\Lambda(v,w)$ such that $d(v)=0$ and $d(w)=v^{n+1}$. The unique morphism of graded algebras $m:(\Lambda(v,w),d)\rightarrow A_{PL}(\mathbb{CP}^n)$ such that $m(v)=v$ and $m(w)=\psi$, is a morphism of cdgas. In homology, $H(m)$ sends $1$, $[v]$, …, $[v^n]$ to $1$, $x$, …, $x^n$. Therefore $m$ is a quasi-isomorphism.
More generally, let $X$ be a simply connected space such that $H^*(X)$ is a truncated polynomial algebra $\frac{{{\mathbf{k}}}[x]}{x^{n+1}=0}$ where $n\geq 1$ and $x$ is an element of even degree $d\geq 2$. Then the Sullivan model of $X$ is $(\Lambda(v,w),d)$ where $v$ is an element of degree $d$, $w$ is an element of degree $d(n+1)-1$, $d(v)=0$ and $d(w)=v^{n+1}$.
Free loop space cohomology for even-dimensional spheres and projective spaces
-----------------------------------------------------------------------------
In this section, we compute the free loop space cohomology of any simply connected space $X$ whose cohomology is a truncated polynomial algebra $\frac{{{\mathbf{k}}}[x]}{x^{n+1}=0}$ where $n\geq 1$ and $x$ is an element of even degree $d\geq 2$.
Mainly, this is the even-dimensional sphere $S^d$ ($n=1$), the complex projective space $\mathbb{CP}^n$ ($d=2$), the quaternionic projective space $\mathbb{HP}^n$ ($d=4$) and the Cayley plane $\mathbb{OP}^2$ ($n=2$ and $d=8$).
In the previous section, we have seen that the minimal Sullivan model of $X$ is $(\Lambda(v,w),d(v)=0,d(w)=v^{n+1})$ where $v$ is an element of degree $d$ and $w$ is an element of degree $d(n+1)-1$. By the constructive proof of Proposition \[modele de Sullivan de la multiplication\], the multiplication $\mu$ of this minimal Sullivan model $(\Lambda(v,w),d)$ admits the relative Sullivan model $(\Lambda(v,w)\otimes \Lambda(v,w)\otimes \Lambda(sv,sw),D)$ where $$D(1\otimes 1\otimes sv)=v\otimes 1\otimes 1-1\otimes v\otimes 1\text{ and}$$ $$D(1\otimes 1\otimes sw)=w\otimes 1\otimes 1-1\otimes w\otimes 1-\sum_{i=0}^n v^i\otimes v^{n-i}\otimes sv.$$
Therefore, by taking the pushout along $\mu$ of this relative Sullivan model (diagram (\[diagram Sullivan model lacets libres\])), or simply by applying Theorem \[differentiel du modele de Sullivan des lacets libres\], a relative Sullivan model of $A_{PL}(ev)\circ m_X$ is given by the inclusion of cdgas $
(\Lambda(v,w),d)\hookrightarrow (\Lambda(v,w,sv,sw),\delta)
$ where $\delta(sv)=-sd(v)=0$ and $\delta(sw)=-s(v^{n+1})=-(n+1)v^nsv$. Consider the pushout square of cdgas $$\xymatrix{
(\Lambda(v,w),d)\ar[r]\ar[d]^\theta_\simeq
& (\Lambda(v,w,sv,sw),\delta)\ar[d]_\simeq^{\theta\otimes_{\Lambda (v,w)}\Lambda (sv,sw)} \\
(\frac{{{\mathbf{k}}}[v]}{v^{n+1}=0},0)\ar[r]
& \left(\frac{{{\mathbf{k}}}[v]}{v^{n+1}=0}\otimes \Lambda (sv,sw),\bar{\delta}\right).
}$$ Here, since $\theta$ is a quasi-isomorphism, the pushout morphism $\theta\otimes_{\Lambda (v,w)}\Lambda (sv,sw)$ along the relative Sullivan model $
\Lambda(v,w)\hookrightarrow \Lambda(v,w,sv,sw)
$ is also a quasi-isomorphism [@Felix-Halperin-Thomas:ratht Lemma 14.2]. Therefore, $H^*(X^{S^1};{{\mathbf{k}}})$ is the graded vector space $${\Bbbk}\oplus \bigoplus_{1\leq p\leq n,\; i\in\mathbb{N}} {\Bbbk}v^{p}(sw)^i
\oplus \bigoplus_{0\leq p\leq n-1,\; i\in\mathbb{N}} {\Bbbk}v^{p}sv(sw)^i.$$ (In [@MenichiL:cohrfl Section 8], the author extends these rational computations over any commutative ring.) Since for all $i\in\mathbb{N}$, the degree of $v(sw)^{i+1}$ is strictly greater than the degree of $v^n(sw)^i$, the generators $1$, $v^{p}(sw)^i$, $1\leq p\leq n$, $i\in\mathbb{N}$, have all distinct (even) degrees. Since for all $i\in\mathbb{N}$, the degree of $sv(sw)^{i+1}$ is strictly greater than the degree of $v^{n-1}sv(sw)^i$, the generators $v^{p}sv(sw)^i$, $0\leq p\leq n-1$, $i\in\mathbb{N}$, have also distinct (odd) degrees. Therefore, for all $p\in\mathbb{N}$, $\text{Dim
}H^p(X^{S^1};{{\mathbf{k}}})\leq 1$.
At the end of section \[exemple de model relatif de Sullivan\], we have shown the same inequalities when $X$ is an odd-dimensional sphere, or more generally for a simply-connected space $X$ whose cohomology $H^*(X;{{\mathbf{k}}})$ is an exterior algebra $\Lambda x$ on an odd degree generator $x$. Since every finite dimensional graded commutative algebra generated by a single element $x$ is either $\Lambda x$ or $\frac{{{\mathbf{k}}}[x]}{x^{n+1}=0}$, we have shown the following proposition:
\[monogene donne Betti bornes\] Let $X$ be a simply connected topological space such that its cohomology $H^*(X;{{\mathbf{k}}})$ is generated by a single element and is finite dimensional. Then the sequence of Betti numbers of the free loop space on $X$, $b_n:=\text{dim } H^n(X^{S^1};{{\mathbf{k}}})$ is bounded.
The goal of the following section will be to prove the converse of this proposition.
Vigué-Poirrier-Sullivan theorem on closed geodesics
===================================================
The goal of this section is to prove (See section \[proofofViguePoirrierSullivantheorem\]) the following theorem due to Vigué-Poirrier and Sullivan.
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
-----------------------------------------------------------------------
([@Vigue-Sullivan:homtcg Theorem p. 637] or [@Felix-Oprea-Tanre:algmodgeom Proposition 5.14]\[nombres de betti lacets libres pas bornes\]) Let $M$ be a simply connected topological space such that the rational cohomology of $M$, $H^*(M;\mathbb{Q})$ is of finite (total) dimension (in particular, vanishes in higher degrees).
If the cohomology algebra $H^*(M;\mathbb{Q})$ requires at least two generators then the sequence of Betti numbers of the free loop space on $M$, $b_n:=\text{dim } H^n(M^{S^1};\mathbb{Q})$ is unbounded.
\[Betti rationel sur le produit de spheres\](Betti numbers of $(S^3\times S^3)^{S^1}$ over $\mathbb{Q}$)
Let $V$ and $W$ be two graded vector spaces such $\forall n\in\mathbb{N}$, $V^n$ and $W^n$ are finite dimensional. We denote by $$P_{V}(z):=\sum_{n=0}^{+\infty}(\text{Dim }V^n) z^n$$ the sum of the *Poincaré serie* of $V$. If $V$ is the cohomology of a space $X$, we denote $P_{H^*(X)}(z)$ simply by $P_{X}(z)$. Note that $P_{V\otimes W}(z)$ is the product $P_V(z)P_W(z)$. We saw at the end of section \[exemple de model relatif de Sullivan\] that $H^*((S^3)^{S^1};\mathbb{Q})\cong \Lambda v\otimes \Lambda sv$ where $v$ is an element of degree $3$. Therefore $$P_{(S^3)^{S^1}}(z)=(1+z^3)\sum_{n=0}^{+\infty}z^{2n}=\frac{1+z^3}{1-z^2}.$$ Since the free loops on a product is the product of the free loops $$H^*((S^3\times S^3)^{S^1})\cong H^*((S^3)^{S^1})\otimes H^*((S^3)^{S^1}).$$ Therefore, since $\displaystyle{\frac{1}{1-z^2}=\sum_{n=0}^{+\infty}
(n+1) z^{2n}}$, $$P_{(S^3\times
S^3)^{S^1}}(z)=\left(\frac{1+z^3}{1-z^2}\right)^2=1+2z^2+\sum_{n=3}^{+\infty}
(n-1) z^n.$$ So the Betti numbers over $\mathbb{Q}$ of the free loop space on $S^3\times S^3$, $b_n:=\text{Dim }H^n((S^3\times
S^3)^{S^1};\mathbb{Q})$ are equal to $n-1$ if $n\geq 3$. In particular, they are unbounded.
\[conjecture geodesiques fermees\] The theorem of Vigué-Poirrier and Sullivan holds replacing $\mathbb{Q}$ by any field $\mathbb{F}$.
(Betti numbers of $(S^3\times S^3)^{S^1}$ over $\mathbb{F}$)
The calculation of Example \[Betti rationel sur le produit de spheres\] over $\mathbb{Q}$ can be extended over any field $\mathbb{F}$ as follows: Since $S^3$ is a topological group, the map $\Omega S^3\times
S^3\rightarrow (S^3)^{S^1}$, sending $(w,g)$ to the free loop $t\mapsto
w(t)g$, is a homeomorphism. Using Serre spectral sequence ([@Serre:suitespectrale Proposition 17] or[@Spanier:livre Chap 9. Sect 7. Lemma 3]) or Bott-Samelson theorem ([@SelickP:introhomot Corollary 7.3.3] or [@Husemoller:fibb Appendix 2 Theorem 1.4]), the cohomology of the pointed loops on $S^3$, $H^*(\Omega S^3)$ is again isomorphic (as graded vector spaces only!) to the polynomial algebra $\Lambda sv$ where $sv$ is of degree $2$. Therefore exactly as over $\mathbb{Q}$, $H^*((S^3)^{S^1};\mathbb{F})\cong \Lambda v\otimes \Lambda
sv$ where $v$ is an element of degree $3$. Now the same proof as in Example \[Betti rationel sur le produit de spheres\] shows that the Betti numbers over $\mathbb{F}$ of the free loop space on $S^3\times S^3$, $b_n:=\text{Dim }H^n((S^3\times
S^3)^{S^1};\mathbb{F})$ are again equal to $n-1$ if $n\geq 3$.
In fact, the theorem of Vigué-Poirrier and Sullivan is completely algebraic:
([@Vigue-Sullivan:homtcg] when $\mathbb{F}=\mathbb{Q}$, [@Halperin-Vigue:homfls Theorem III p. 315] over any field $\mathbb{F}$)\[nombres de Betti homologie de Hochschild\] Let $\mathbb{F}$ be a field. Let $A$ be a cdga such that $H^{<0}(A)=0$, $H^{0}(A)=\mathbb{F}$ and $H^*(A)$ is of finite (total) dimension. If the algebra $H^*(A)$ requires at least two generators then the sequence of dimensions of the Hochschild homology of $A$, $b_n:=\text{dim } HH_{-n}(A,A)$ is unbounded.
Generalising Chen’s theorem (Corollary \[theoreme de Chen sur lacets libres\]) over any field $\mathbb{F}$, Jones theorem [@JonesJ:Cycheh] gives the isomorphisms of vector spaces
$$H^n(X^{S^1};\mathbb{F})\cong HH_{-n}(S^*(X;\mathbb{F}), S^*(X;\mathbb{F})), \quad n\in\mathbb{Z}$$ between the free loop space cohomology of $X$ and the Hochschild homology of the algebra of singular cochains on $X$. But since the algebra of singular cochains $S^*(X;\mathbb{F})$ is not commutative, Conjecture \[conjecture geodesiques fermees\] does not follow from Theorem \[nombres de Betti homologie de Hochschild\].
A first result of Sullivan
--------------------------
In this section, we start by a first result of Sullivan whose simple proof illustrates the technics used in the proof of Vigué-Poirrier-Sullivan theorem.
[@Sullivan:conftokyo]\[cohomologie des lacets libres pas bornee\] Let $X$ be a simply-connected space such that $H^*(X;\mathbb{Q})$ is not concentrated in degree $0$ and $H^n(X;\mathbb{Q})$ is null for $n$ large enough. Then on the contrary, $H^n(X^{S^1};\mathbb{Q})\neq 0$ for an infinite set of integers $n$.
Let $(\Lambda V,d)$ be a minimal Sullivan model of $X$. Suppose that $V$ is concentrated in even degree. Then $d=0$. Therefore $H^*(\Lambda V,d)=\Lambda V$ is either concentrated in degree $0$ or is not null for an infinite sequence of degrees. By hypothesis, we have excluded theses two cases. Therefore $\text{dim }V^{odd}\geq 1$.
Let $x_1$, $x_2$, …, $x_m$, $y$, $x_{m+1}$, ..... be a basis of $V$ ordered by degree where $y$ denotes the first generator of odd degree ($m\geq 0$). For all $1\leq i\leq m$, $dx_i\in\Lambda x_{<i}$. But $dx_i$ is of odd degree and $\Lambda x_{<i}$ is concentrated in even degre. So $dx_i=0$. Since $dy\in \Lambda x_{\leq m}$, $dy$ is equal to a polynomial $P(x_1,\dots,x_m)$ which belongs to $\Lambda^{\geq 2}(x_1,\dots,x_m)$.
Consider $(\Lambda V\otimes\Lambda sV,\delta)$, the Sullivan model of $X^{S^1}$, given by Theorem \[differentiel du modele de Sullivan des lacets libres\]. We have $\forall 1\leq i\leq m$, $\delta(sx_i)=-sdx_i=0$ and $\delta (sy)=-sdy\in \Lambda^{\geq 1}(x_1,\dots,x_m)\otimes \Lambda^{1}(sx_1,\dots,sx_m)$. Therefore, since $sx_1$,…,$sx_m$ are all of odd degree, $\forall p\geq 0$, $$\delta (sx_1\dots sx_m(sy)^p)=\pm sx_1\dots sx_m p\delta(sy)(sy)^{p-1}=0.$$ For all $p\geq 0$, the cocycle $sx_1\dots sx_m(sy)^p$ gives a non trivial cohomology class in $H^*(X^{S^1};\mathbb{Q})$, since by Remark \[modele de la fibration des lacets libres\], the image of this cohomology class in $H^*(\Omega X;\mathbb{Q})\cong \Lambda V$ is different from $0$.
Dimension of $V^{odd}\geq 2$
----------------------------
In this section, we show the following proposition:
\[au moins deux generateurs de degree impair\] Let $X$ be a simply connected space such that $H^*(X;\mathbb{Q})$ is of finite (total) dimension and requires at least two generators. Let $(\Lambda V,d)$ be the minimal Sullivan model of $X$. Then $\text{dim }V^{odd}\geq 2$.
(Koszul complexes)\[complexes de Koszul a une variable\] Let $A$ be a graded algebra. Let $z$ be a central element of even degree of $A$ which is not a divisor of zero. Then we have a quasi-isomorphism of dgas $$(A\otimes\Lambda sz,d)\buildrel{\simeq}\over\twoheadrightarrow A/z.A\quad a\otimes 1\mapsto a, a\otimes sz\mapsto 0,$$ where $d(a\otimes 1)=0$ and $d(a\otimes sz)=(-1)^{\vert a\vert}az$ for all $a\in A$.
As we saw in the proof of Theorem \[cohomologie des lacets libres pas bornee\], there is at least one generator $y$ of odd degree, that is $\text{dim }V^{odd}\geq 1$. Suppose that there is only one. Let $x_1$, $x_2$, …, $x_m$, $y$, $x_{m+1}$,…be a basis of $V$ ordered by degree ($m\geq 0$).
First case: $dy=0$. If $m\geq 1$, $dx_1=0$. If $m=0$, $dx_1\in\Lambda^{\geq 2}(y)=\{0\}$ and therefore again $dx_1=0$. Suppose that for $n\geq 1$, $x_1^n$ is a coboundary. Then $x_1^n=d(yP(x_1,\dots))=yd(P(x_1,\dots))$ where $P(x_1,\dots)$ is a polynomial in the $x_i$’s. But this is impossible since $x_1^n$ does not belong to the ideal generated by $y$. Therefore for all $n\geq 1$, $x_1^n$ gives a non trivial cohomology class in $H^*(X)$. But $H^*(X)$ is finite dimensional.
Second case: $dy\neq 0$. In particular $m\geq 1$. Since $dy$ is a non zero polynomial, $dy$ is not a zero divisor, so by Property \[complexes de Koszul a une variable\], we have a quasi-isomorphism of cdgas $$\Lambda(x_1,\dots,x_m,y)\buildrel{\simeq}\over\twoheadrightarrow\Lambda(x_1,\dots,x_m)/(dy).$$ Consider the push out in the category of cdgas $$\xymatrix{
\Lambda(x_1,\dots,x_m,y)\ar[r]\ar[d]_\simeq
& \Lambda(x_1,\dots,x_m,y,x_{m+1},\dots),d\ar[d]\\
\Lambda(x_1,\dots,x_m)/(dy)\ar[r]
& \Lambda(x_1,\dots,x_m)/(dy)\otimes \Lambda(x_{m+1},\dots),\bar{d}
}$$ Since $\Lambda(x_1,\dots,x_m)/(dy)\otimes \Lambda(x_{m+1},\dots)$ is concentrated in even degrees, $\bar{d}=0$. Since the top arrow is a Sullivan relative model and the left arrow is a quasi-isomorphism, the right arrow is also a quasi-isomorphism ([@Felix-Halperin-Thomas:ratht Lemma 14.2], or more generally the category of cdgas over $\mathbb{Q}$ is a Quillen model category). Therefore the algebra $H^*(X)$ is isomorphic to $\Lambda(x_1,\dots,x_m)/(dy)\otimes \Lambda(x_{m+1},\dots)$. If $m\geq 2$, $\Lambda(x_1,\dots,x_m)/(dy)$ and so $H^*(X)$ is infinite dimensional. If $m=1$, since $\Lambda x_1/(dy)$ is generated by only one generator, we must have another generator $x_2$. But $\Lambda(x_1)/(dy)\otimes \Lambda(x_{2},\dots)$ is also infinite dimensional.
Proof of Vigué-Poirrier-Sullivan theorem {#proofofViguePoirrierSullivantheorem}
----------------------------------------
[@Vigue-Sullivan:homtcg Proposition 4]\[elimination generateur degree pair\] Let $A$ be a dga over any field such that the multiplication by a cocycle $x$ of any degre $A\rightarrow A$, $a\mapsto xa$ is injective (Our example will be $A=(\Lambda V,d)$ and $x$ a non-zero element of $V$ of even degree such that $dx=0$). If the Betti numbers $b_n=\text{dim } H^n(A)$ of $A$ are bounded then the Betti numbers $b_n=\text{dim } H^n(A/xA)$ of $A/xA$ are also bounded.
Since $H^n(xA)\cong H^{n-\vert x\vert}(A)$, the short exact sequence of complexes $$0\rightarrow xA\rightarrow A\rightarrow A/xA\rightarrow 0$$ gives the long exact sequence in homology $$\dots\rightarrow H^n(A)\rightarrow H^n(A/xA)\rightarrow H^{n+1-\vert x\vert}(A)\rightarrow\dots$$ Therefore $\text{dim }H^n(A/xA)\leq \text{dim } H^n(A)+\text{dim } H^{n+1-\vert x\vert}(A)$
Let $(\Lambda V,d)$ be the minimal Sullivan model of $X$. Let $(\Lambda V\otimes\Lambda sV,\delta)$ be the Sullivan model of $X^{S^1}$ given by Theorem \[differentiel du modele de Sullivan des lacets libres\]. From Proposition \[au moins deux generateurs de degree impair\], we know that $\text{dim }V^{odd}\geq 2$. Let $x_1$, $x_2$, …, $x_m$, $y$, $x_{m+1}$,…, $x_n$, $z=x_{n+1}$, … be a basis of $V$ ordered by degrees where $x_1$,…, $x_n$ are of even degrees and $y$, $z$ are of odd degrees. Consider the commutative diagram of cdgas where the three rectangles are push outs $$\xymatrix{
\Lambda(x_1,\dots, x_n)\ar[r]\ar[d]
&(\Lambda V,d)\ar[r]\ar[d]
&(\Lambda V\otimes\Lambda sV,\delta)\ar[d]\\
\mathbb{Q}\ar[r]
&\Lambda(y,z,\dots)\ar[r]\ar[d]
&(\Lambda(y,z,\dots)\otimes\Lambda sV,\bar{\delta})\ar[d]\\
&\mathbb{Q}\ar[r]
&(\Lambda sV,0)
}$$ Note that by Remark \[modele de la fibration des lacets libres\], the differential on $\Lambda sV$ is $0$.
For all $1\leq j\leq n+1$, $$\delta x_j=dx_j\in \Lambda^{\geq 2}(x_{<j},y)\subset
\Lambda^{\geq 1}(x_{<j})\otimes \Lambda y.$$ Therefore $$\delta (sx_j)=-s\delta x_j\in \Lambda x_{<j}\otimes\Lambda^1 sx_{<j}\otimes\Lambda y+\Lambda^{\geq 1}(x_{<j})\otimes \Lambda^1 sy.$$ Since $(sx_1)^2=\dots=(sx_{j-1})^2=0$, the product $$sx_1\dots sx_{j-1} \delta (sx_j)\in \Lambda^{\geq 1}(x_{<j})\otimes \Lambda^1 sy.$$ So $\forall 1\leq j\leq n+1$, $sx_1\dots sx_{j-1} \bar{\delta} (sx_j)=0$. In particular $sx_1\dots sx_{n} \bar{\delta} (sz)=0$. Similarly, since $dy\in\Lambda^{\geq 2} x_{\leq m}$, $sx_1\dots sx_m\delta(sy)=0$ and so $sx_1\dots sx_n\bar{\delta}(sy)=0$. By induction, $\forall 1\leq j\leq n$, $\bar{\delta}(sx_1\dots sx_j)=0$. In particular, $\bar{\delta}(sx_1\dots sx_n)=0$. So finally, for all $p\geq 0$ and all $q\geq 0$, $\bar{\delta}(sx_1\dots sx_n(sy)^p(sz)^q)=0$. The cocycles $sx_1\dots sx_n(sy)^p(sz)^q$, $p\geq 0$, $q\geq 0$, give linearly independent cohomology classes in $H^*(\Lambda(y,z,\dots)\otimes\Lambda sV,\bar{\delta})$ since their images in $(\Lambda sV,0)$ are linearly independent.
For all $k\geq 0$, there is at least $k+1$ elements of the form $sx_1\dots sx_n(sy)^p(sz)^q$ in degree $\vert sx_1\vert+ \dots+\vert sx_n\vert+k\cdot\text{lcm}(\vert sy\vert,\vert sz\vert)$ (just take $p=i\cdot\text{lcm}(\vert sy\vert,\vert sz\vert)/\vert sy\vert$ and $q=(k-i)\text{lcm}(\vert sy\vert,\vert sz\vert)/\vert sz\vert$ for $i$ between $0$ and $k$). Therefore the Betti numbers of $H^*(\Lambda(y,z,\dots)\otimes\Lambda sV,\bar{\delta})$ are unbounded.
Suppose that the Betti numbers of $(\Lambda V\otimes\Lambda sV,\delta)$ are bounded. Then by Lemma \[elimination generateur degree pair\] applied to $A=(\Lambda V\otimes\Lambda sV,\delta)$ and $x=x_1$, the Betti numbers of the quotient cdga $(\Lambda(x_2,\dots)\otimes\Lambda sV,\bar{\delta})$ are bounded. By continuing to apply Lemma \[elimination generateur degree pair\] to $x_2$, $x_3$, …, $x_n$, we obtain that the Betti numbers of the quotient cdga $(\Lambda(y,z,\dots)\otimes\Lambda sV,\bar{\delta}$ are bounded. But we saw just above that they are unbounded.
Further readings
================
In this last section, we suggest some further readings that we find appropriate for the student.
In [@Bott-Tu:difforms Chapter 19], one can find a very short and gentle introduction to rational homotopy that the reader should compare to our introduction.
In this introduction, we have tried to explain that rational homotopy is a functor which transforms homotopy pullbacks of spaces into homotopy pushouts of cdgas. Therefore after our introduction, we advise the reader to look at [@Hess:introrationalhtpy], a more advanced introduction to rational homotopy, which explains the model category of cdgas.
The canonical reference for rational homotopy [@Felix-Halperin-Thomas:ratht] is highly readable.
In the recent book [@Felix-Oprea-Tanre:algmodgeom], you will find many geometric applications of rational homotopy. The proof of Vigué-Poirrier-Sullivan theorem we give here, follows more or less the proof given in [@Felix-Oprea-Tanre:algmodgeom].
We also like [@TanreD:homrmc] recently reprinted because it is the only book where you can find the Quillen model of a space: a differential graded Lie algebra representing its rational homotopy type (instead of a commutative differential graded algebra as the Sullivan model).
|
---
abstract: 'I present a brief overview of the measurements of exclusive $B\to X_u\ell\nu$ transitions, with a focus on issues facing robust averaging of branching fractions and $|V_{ub}|$ from current and anticipated measurements.'
address: 'Cornell University, Ithaca, NY, USA'
author:
- L Gibbons
title: 'Towards Robust Averaging of Exclusive $B\to X_u\ell\nu$ Measurements'
---
Introduction
============
With the foreseen improvements in theoretical techniques for form factor calculations, measurement of exclusive $B\to X_u\ell\nu$ processes shows promise as the most robust route for determination of ${|V_{ub}|}$. The experiments BaBar, Belle and CLEO now have a variety of measurements, preliminary or published, of such processes. While the experiments employ a variety of strategies, the core techniques are similar and lead to potential correlations. With the current information available from the experiments, realistic evaluation of the correlations and, therefore, robust averaging of the results is very difficult. This review surveys areas in which correlated systematic uncertainties are likely to exist for the measurements. I pose a set of “homework” questions. These can hopefully serve as a starting point for discussion among the experiments aimed at standardized evaluation of uncertainties where correlations are likely to exist.
As discussed in any reference concerning exclusive ${b\to u\ell\nu}$ transitions, the major theoretical uncertainty in the extraction of ${|V_{ub}|}$ from these transitions arises from uncertainties in the hadronic form factors involved in the transitions (see, for example, [@cleo03]). For electron and muon transitions, one form factor dominates transitions to final states with a single pseudoscalar meson, and three form factors dominate those to a single vector meson. All of these form factors vary as a function of the momentum transfer $q^2 = M^2_{\ell\nu}$. I will review mechanisms leading from uncertainties in the form factors to uncertainty in the branching fractions and in ${|V_{ub}|}$. An excellent overview of the status of the theoretical work on the form factors can be found in the proceedings of the previous CKM workshop [@Battaglia:2003in].
Reference [@Battaglia:2003in] also makes a valiant start towards an average of the various experimental results. Better standardization among the experiments will be necessary to effect a more robust averaging procedure.
Overview of measurements
========================
General approach
----------------
Table \[tab:measurements\] summarizes the measurements of exclusive $|V_{ub}|$ channels that have been published or presented in preliminary form at conferences. All the measurements have made use of detector hermeticity to obtain an initial estimate the four momentum of the neutrino. At BaBar, Belle and CLEO, the initial four momentum of the $\Upsilon(4S)$ is known very well. Since the detectors cover most of the total $4\pi$ solid angle, the missing four momentum, defined via $$\begin{aligned}
{\vec{p}_{\mathrm{miss}}}& = & \vec{p}_{\Upsilon(4S)} - \sum_{{\stackrel{\mathrm{tracks,}}{{\tiny\rm showers}}}} \vec{p}_i\\ \nonumber
{E_{\mathrm{miss}}}& = & E_{\mathrm{\Upsilon(4S)}} - \sum_{{\stackrel{\mathrm{tracks,}}{{\tiny\rm showers}}}} E_i \nonumber,\end{aligned}$$ provides a good estimate of the total four momentum of all undetected particles in the event. For events with just one undetected neutrino from $B\to X_u\ell\nu$, the missing four momentum provides a good estimate of the neutrino’s four momentum.
One of the challenges facing “reconstruction” of the neutrino in this fashion lies in appropriate selection of the tracks and showers used in the sums. The BaBar ’01 analysis, for example, selects the subset of tracks $p_t > 100$ MeV$/c$, at least 12 hits in their drift chamber, and consistent with coming from the origin. The CLEO analyses have optimized track selection for hermeticity in two ways. For contributions to the missing momentum, CLEO attempts to identify one and only one reconstructed track per true particle from the origin (taking “poorly measured” in preference to “not measured”). Selection of tracks to be matched to hadronic showers in the calorimeter is performed separately. For example, particles with transverse momentum low enough to “curl” within the detector may result in several reconstructed tracks. CLEO groups these tracks and chooses the likeliest best representation of the original particle. For particles that decay in flight or suffer a hard scatter or a hadronic interaction, CLEO identifies the inner and outer tracks for the two different purposes. Similarly, the experiments have a range of selection criteria of showers in the electromagnetic calorimeter to separate true photons from showers originating with interactions of charged hadrons.
Measurement Modes studied
----------------------- ------------------------------------------------------------------------------------------------------------------------
Belle 2003 [@belle03] ${\omega\ell\nu}$
CLEO 2003 [@cleo03] ${\pi^{\pm}\ell\nu}$, ${\pi^{0}\ell\nu}$,${\rho^{\pm}\ell\nu}$, ${\rho^{0}\ell\nu}$,${\omega\ell\nu}$, ${\eta\ell\nu}$
BELLE 2002[@belle02] ${\pi^{\pm}\ell\nu}$, ${\rho^{0}\ell\nu}$
BaBar 2001 [@babar01] ${\rho^{\pm}\ell\nu}$, ${\rho^{0}\ell\nu}$, ${\omega\ell\nu}$
CLEO 2000 [@cleo00] ${\rho^{\pm}\ell\nu}$, ${\rho^{0}\ell\nu}$, ${\omega\ell\nu}$
CLEO 1996 [@cleo96] ${\pi^{\pm}\ell\nu}$, ${\pi^{0}\ell\nu}$,${\rho^{\pm}\ell\nu}$, ${\rho^{0}\ell\nu}$,${\omega\ell\nu}$
: Published and preliminary $B\to X_u\ell\nu$ measurements. The years correspond to year of publication, year of submission for publication, or year of presentation for identification purposes and are [*not*]{} indicative of the intellectual history. For example, the preliminary version of CLEO ’03 and of Belle ’02 appeared simultaneously.[]{data-label="tab:measurements"}
The missing momentum yields an accurate representation of a missing neutrino from a semileptonic decay only when that is the sole particle that has not been detected. In fact, the largest background contribution in these analyses, as in the inclusive ${b\to u\ell\nu}$ analyses, arises from events containing a ${b\to c\ell\nu}$ decay with at least one neutral particle missing in addition to the neutrino (see, for example, reference [@cleo96]). A number of the analyses introduce strict event selection criteria to bias against events containing multiple missing particles. For example, the CLEO ’96, ’03 and the Belle ’01, ’03 analyses require precisely one identified charged lepton (additional implying extra missing neutrinos), and a small reconstructed net charge to the event (nonzero implying at least one missing or double-counted particle). In addition, those analyses require the missing momentum to be consistent with a neutrino. The Belle analyses require a small ${M^2_{\mathrm{miss}}}$, while the CLEO analyses require a small ${M^2_{\mathrm{miss}}}/{E_{\mathrm{miss}}}$. Since the ${E_{\mathrm{miss}}}$ resolution is considerably worse than the ${\vec{p}_{\mathrm{miss}}}$ resolution (see, for example, reference [@cleo03]), the latter requirement is roughly constant in ${E_{\mathrm{miss}}}$ resolution. The missing momentum resolution is shown for CLEO and Belle in Figure \[fig:pmiss\].
to
The CLEO ’00 and BABAR ’01 $\rho\ell\nu$ analyses do not apply these strict event selection criteria. As a result, the analyses have a much higher signal efficiency. That efficiency, however, comes with a much fiercer ${b\to c\ell\nu}$ background. Hence both of those analyses are primarily sensitive to their signal in the region $p_\ell>2.3$ GeV$/c$, which lies beyond the kinematic endpoint for the ${b\to c\ell\nu}$ transitions. As we will discuss below, this approach shifts systematic uncertainties from detector-related uncertainties to signal form factor shape uncertainties.
As the $B$ factory datasets continue to grow, the exclusive ${b\to u\ell\nu}$ analyses will benefit from use of fully reconstructed (or “annealed”) samples used in the recent inclusive analyses (see talk by F. Muheim [@inclusive_review]). Such an approach should result in a significant reduction in background, allowing for selection criteria that can be made more uniform over phase space. As a result, systematic uncertainties both from detector and background modeling and from form factor uncertainties will be reduced.
to
After event selection, the analyses use a neutrino four momentum given by $p_\nu = (|{\vec{p}_{\mathrm{miss}}}|,{\vec{p}_{\mathrm{miss}}})$ because the missing momentum resolution is considerably better than the missing energy resolution. The experiments then consider a variety of kinematic variables related to a single ${b\to u\ell\nu}$ decay. The analyses either employ these for further background suppression or fit some combination of them for extraction of the signal yields. The variables $$\begin{aligned}
\Delta E & = & E_m + E_\ell + E_\nu - E_{\mathrm{beam}} \\
M_{\mathrm{beam}} & = & (E_\mathrm{beam}^2 - |\vec{p}_m+\vec{p}_\ell+\vec{p}_\nu)^{1/2} \end{aligned}$$ characterize energy conservation ($\Delta E=0$) and momentum conservation ($M_{\mathrm{beam}}=M_B$) in the decay, and therefore involve the “measured” $p_\nu$. Alternatively, one can calculate the angle between the meson+charged lepton system ($Y$) and the B meson, without recourse to $p_\nu$, via $$\cos \theta_{BY}=\frac{2 E_B E_{Y} - (M_B^2 + M_{Y}^2) c^4}
{2 |\vec{p}_B| |\vec{p}_{Y}| c^2}.$$ The variables $\cos\theta_{BY}$ and $M_{\mathrm{beam}} $ are strongly correlated: a candidate combination cannot have $M_{\mathrm{beam}}$ consistent with the $B$ meson mass without $\cos\theta_{BY}$ being within (or close to) its physical domain. All but the CLEO ’96 and CLEO ’03 analyses require an approximately physical $\cos\theta_{BY}$.
Extraction of ${b\to u\ell\nu}$ yields
--------------------------------------
to
To extract yields and thus obtain branching fractions, all experiments perform a multicomponent fit with Monte Carlo or data estimates of the ${b\to c\ell\nu}$ backgrounds, ${b\to u\ell\nu}$ signal and backgrounds (cross-feed among modes considered or feed down from modes not considered), and continuum backgrounds. The fits employ isospin (and quark model) constraints $$\begin{aligned}
\Gamma(B^0\to \pi^-\ell^+\nu) & = & 2\Gamma(B^+\to \pi^0\ell^+\nu) \\ \nonumber
\Gamma(B^0\to \rho^-\ell^+\nu) & = & 2\Gamma(B^+\to \rho^0\ell^+\nu) \\ \nonumber
& \approx & 2\Gamma(B^+\to \omega\ell^+\nu) \label{eq:isospin}\end{aligned}$$ when both the charged and neutral $B$ decay modes are measured in an analysis.
The CLEO ’00 and BaBar ’01 $\rho\ell\nu$ analyses fit the $m_{\pi\pi}$ versus $\Delta E$ distribution in coarse bins of $p_\ell$. Again, while these are relatively high efficiency analyses, they are primarily sensitive to signal in the region $p_\ell>2.3$ GeV/$c$ (in the $\Upsilon(4S)$ rest frame). The CLEO ’00 analysis made the first determination of $d\Gamma/dq^2$ for any exclusive ${b\to u\ell\nu}$ transition, albeit with large uncertainties. Figure \[fig:lange\_fits\] shows the fit results projected onto $m_{\pi\pi}$ and $\Delta E$ distributions for events with $p_\ell>2.3$ GeV/$c$.
The Belle analyses use a variety of approaches for extracting the yield. The Belle ’02 ${\pi^{+}\ell\nu}$ analysis fits $\Delta E$ versus $p_\ell$. After obtaining background and signal component normalizations from the fit, Belle has made a preliminary attempt to obtain $d\Gamma/dq^2$ by subtracting the background components from the reconstructed $d\Gamma/dq^2$ distribution and then correcting for resolution effects via an efficiency matrix. The Belle ’02 ${\rho^{0}\ell\nu}$ analysis requires $2.0\le p_\ell < 2.8$ GeV$/c$, then extracts yields using a fit to the $\Delta E$ versus $m_{\pi\pi}$ distribution.
Belle has observed the ${\omega\ell\nu}$ transition with their Belle ’03 analysis. After requiring $\Delta E$ and $M_{\mathrm{beam}}$ to be consistent with their signal, Belle performs a fit to $m_{\pi\pi\pi}$ versus $p_\ell$. As in the CLEO ’00 and BaBar ’01 analyses, the data are coarsely binned in $p_\ell$ (in the $\Upsilon(4S)$ rest frame). Similarly to those analyses, the analysis is primarily sensitive to signal with $p_\ell>2.4$ GeV$/c$. Figure \[fig:belle\_omega\] shows the $m_{\pi\pi\pi}$ distributions in two of the $p_\ell$ ranges.
to
to
The CLEO ’03 analysis design minimizes model dependence as much as possible. The $p_\ell$ requirements (in the $\Upsilon(4S)$ rest frame) are relatively loose: $p_\ell>1.0$ GeV$/c$ for pseudoscalar modes and $p_\ell>1.5$ GeV$/c$ for vector modes. The yields are extracted from a fit to $\Delta E$ versus $M_{\mathrm{beam}}$ distributions. The vector modes are also binned in the reconstructed $\pi\pi$ or $3\pi$ mass. Unlike in the above analyses (and in the CLEO ’96 analysis, which it supersedes), the signal rates are allowed to float independently in separate $q^2$ intervals. As discussed in the following section, this approach significantly reduces the uncertainty arising from the $q^2$–dependence of the form factors. The total branching fraction is obtained by summing the three individual rates obtained in the fit. By contrast, all other analyses obtain the branching fraction by integrating over the $q^2$ distribution predicted by theory+simulation and fitting the data for a single normalization. Figure \[fig:cleo\_pi\] shows the $M_{\mathrm{beam}}$ and the $\Delta E$ distributions for the ${\pi^{\ }\ell\nu}$ mode in the other’s signal region.
The branching fractions from all of the above measurements are summarized in Figure \[fig:branching\_fractions\]. The measurements agree quite well within their listed uncertainties, though the extent of correlation among the measurements has not been evaluated. Within the large experimental uncertainties, the independently measured $B^+$ rates agree well with isospin and quark model predictions (Eq. \[eq:isospin\]).
Averaging issues
================
Contribution $\delta{\cal B}_{\rho}/{\cal B}_{\rho}$ (%)
------------------------------------------------------------------------------------------- ---------------------------------------------
Tracking efficiency $\pm 5$
Tracking resolution $\pm 1$
$\pi^0$ efficiency $\pm 5$
$\pi^0$ energy scale $\pm 3$
$b\rightarrow c e \nu$ bkg. $+1.4,-1.7$
Resonant $b\rightarrow u e \nu$ bkg. $+6,-4$
Non-resonant $b\rightarrow u e \nu$ bkg. $\pm 9$
$B$ lifetime $\pm 1$
Number of $B\bar{B}$ pairs $\pm 1.6$
Misidentified electrons $<\pm 1$
Electron efficiency $\pm 2$
${\cal B}(\Upsilon(4S)\rightarrow B^+B^-)/{\cal B}(\Upsilon(4S)\rightarrow B^0\bar{B}^0)$ $<\pm 1$
Isospin and quark model symmetries $<\pm 1$
Fit method $+4,-6$
Total systematic uncertainty $\pm 14.4$
--------------------------------------------------------------------------------------------------------------
Systematic $\pi\ell\nu$ $\rho(\omega)\ell\nu$ $\eta\ell\nu$
------------------------------------------------------- -------------- ----------------------- ---------------
hermeticity 6.8 18.7 17.3
$B\to D/D^{*}/D^{**}/D^{\rm NR}X\ell\nu$ 1.7 2.0 5.5
$B\to X_u\ell\nu$‘w feed down &0.5 & 8.3& 1.6\
Continuum smoothing &1.0 & 3.0& 2.0\
Fakes &3.0 & 3.0& 3.0\
Lepton ID &2.0 & 2.0& 2.0\
$f_{+-}/f_{00}$ &2.4 & 0.0& 4.1\
$\tau_{B^+}/\tau_{B^0}$ &0.2 & 2.1& 1.4\
Isospin &0.0 & 2.4& 0.1\
Luminosity &2.0 & 2.0& 2.0\
Upper Total &[**8.6**]{} &[**21.4**]{}& [**19.3**]{}\
Non Resonant &– & -13&\
Lower Total &[**8.6**]{} &[**25.1**]{}& [**19.3**]{}\
& & &\
& & &\
--------------------------------------------------------------------------------------------------------------
To perform a robust average of branching fractions and of ${|V_{ub}|}$ extracted from the measured rates, careful consideration of correlated uncertainties will be necessary. As many of the results are quite new, the experiments have not yet done the work examining the correlations. As a result, I do not present an average in these proceedings. However, I summarize below the primary issues that the experiments must consider to effect a realistic average. Some of the effects may appear small on the scale of the statistical uncertainty of the current individual measurements. I anticipate, however, that a variety of results with improved precision will become available, and that averages with a statistical precision of 5% or better may finally become possible. We should anticipate this scenario, and begin examination of systematic contributions that are significant on this final scale, particularly if they may be largely correlated among the measurements.
Table \[tab:allsyst\] provides the systematics breakdown from the two most recent published analyses. Clearly hermeticity–related issues, modeling of the generic $B\to X_u\ell\nu$ background, and limiting the nonresonant $\pi\pi\ell\nu$ background contribute substantially in the measurements, particularly in the $\rho\ell\nu$ mode. What is not apparent in the table is that treatment of the uncertainties in the inclusive rate and modeling can exacerbate the hermeticity–related uncertainties. Changes of the detector response, for example, affect the best-fit normalizations for the $B\to X_u\ell\nu$ background, which in turn can cause a significantly larger variation in signal shape than obtained with a fixed $B\to X_u\ell\nu$ normalization. The following subsections discuss the systematic issues and potentially large correlations for the important categories.
Systematics related to technique
--------------------------------
variation $\pi^-\ell^+\nu$ $\rho^-\ell^+\nu$ $\eta\ell\nu$
------------------------------- ------------------ ------------------- --------------- -- -- -- -- -- --
$\gamma$ eff. 2.6 11.1 5.7
$\gamma$ resol. 4.1 2.9 9.6
$\mathbf{K_L}$ **showering** 1.3 6.0 2.7
Particle ID 1.9 8.2 0.2
**Splitoff rejection** 1.5 1.2 5.5
**track eff.** 3.7 8.6 9.5
track resol. 1.0 6.2 0.9
**splitoff sim.** 0.4 1.0 6.0
$\mathbf{K_L}$ **production** 0.2 0.1 0.1
$\mathbf{\nu}$ **production** 0.5 0.6 2.9
[**Total**]{} 6.8 18.7 17.3
: Systematics associated with the use of hermeticity in the CLEO ’03 analysis. The individual bold–faced items correspond to systematic effects that are likely to be at least partly correlated among the various experiments.[]{data-label="tab:hermsyst"}
All of the analyses performed to date rely on detector hermeticity to provide an estimate of the neutrino momentum and, as a result, more robust background suppression. The price to be paid is an enhancement of the systematic uncertainties from physics and detector simulation. A systematic uncertainty in the tracking efficiency, for example, acts coherently over all charged particles in an event. Since there are typically about ten charged particles per $B\bar{B}$ event at the $\Upsilon(4S)$), single track effects can get magnified dramatically. [*A priori*]{} it is difficult to predict how correlated the systematics will be and how they will vary from experiment to experiment. The systematic effects tend to affect both background levels and efficiency, so the effects could vary quite significantly among the measurements depending on the optimization of efficiency versus background level.
The hermeticity–related systematic categories considered by the CLEO ’03 analysis are tabulated in Table \[tab:hermsyst\]. Because detector simulation in the experiments tend to rely on a common GEANT base, or even more fundamentally, on a common pool of cross section measurements, many of the categories are likely to be at least partially correlated among the experiments. CLEO has found, for example, measurable differences between data and simulation for hadronic interactions of kaons in its Cesium–Iodide (CsI) calorimeter. These studies provide the input for the $K_L$ shower simulation listed in the table. While BaBar and Belle may be less susceptible to these effects because of their instrumented flux returns, this could remain a sizable, strongly correlated systematic. CLEO has also found that data and simulation disagree on the energy deposited in charged hadronic showers at distances too large to veto using proximity to a projected track. The resulting systematic contributions in the production of such “splitoffs” and in the variables used to reject them appear, fortunately, to be rather small. They are, however, also likely to be significantly correlated among experiments, so could result in a non-negligible contribution to an experimental average.
The CLEO tracking system and its associated uncertainties have been intensively studied at CLEO. As a result, the individual track–finding efficiency has been limited in studies at the $0.5\%$ level. The CLEO ’03 analysis has used $0.75\%$ per track in Table \[tab:hermsyst\] to be conservative. Even at this small rate, the track efficiency uncertainty becomes sizable when coherently amplified by all tracks in the event. At $0.5\%$ level, the uncertainty in the tracking efficiency studies that provided that limit are similar to the uncertainty in total number of interaction lengths in the tracking system because of uncertainties in the underlying cross sections.
At a smaller level in the CLEO analysis are the uncertainties in the $K_L$ production spectrum and rate and in the $\nu$ production spectrum and rate (particularly from the secondary production process $b\to c\to s\ell\nu$) from the $\Upsilon(4S)$. The production models tend to be strongly correlated among experiments through common event generators, so these categories still merit some attention.
Other systematics in the list may also be correlated, such as the photon resolution resulting from a common model of electromagnetic showers. Typically in measurements many of these effects are small. However, with the amplification from the use of hermeticity, the experiments should evaluate with some care the levels of uncertainty and the levels of correlation among experiments for a broad variety of detector effects.
> What is the complete set of systematics related to generic production model and detector simulation that are significant at the few percent level for the exclusive ${b\to u\ell\nu}$ analyses?
> What is the level of correlation among the experiments for these different systematic effects?
Anatomy of rate measurement dependence on form factors
------------------------------------------------------
The literature now teems with calculations of the form factors involved in the ${\pi^{\ }\ell\nu}$ and ${\rho^{\ }\ell\nu}$ transitions. For a relatively recent survey, see the references within reference [@cleo03]. This review will limit itself to those considered within the analyses to date. These include a variety of lattice QCD (LQCD) calculations from APE (APE ’00) [@Abada:2000ty], FNAL (FNAL ’01) [@El-Khadra:2001rv], JLQCD (JLQCD ’01) [@Aoki:2001rd], and UKQCD (UKQCD ’98 and UKQCD ’99) [@DelDebbio:1997kr; @Bowler:1999xn] and light-cone sum rules (LCSR) calculations from Ball and collaborators (Ball ’98 and Ball ’01) [@Ball:1998kk; @Ball:2001fp] and Khodjamirian and collaborators (KRWWY) [@Khodjamirian:2000ds]. Both approaches currently have uncertainties in the $15\%$ to $20\%$ range. Also incorporated into various analyses are the quark model calculations of Isgur [*et al*]{} (ISGW II) [@Scora:1995ty], Feldman and Kroll (SPD) [@Feldmann:1999sm], and the relativistic quark model calculation of Melikhov and Stech [@Melikhov:2000yu]. As they are purely models, it is difficult to assign a meaningful uncertainty to the quark model calculations. Finally, the analysis of Ligeti and Wise [@Ligeti:1995yz] based on study of heavy quark symmetry and measured $D\to K^*\ell\nu$ form factors is often considered.
to
A rate measurement suffers from significant theoretical uncertainties when the experimental efficiency varies significantly over the variables on which the form factor, or combination of form factors, depend. To clarify the problem, consider the ${\pi^{\ }\ell\nu}$ transition, for which a variety of form factor calculations are shown in Figure \[fig:pi\_models\]. The models were selected to indicate the span of $q^2$ dependence that arises in the literature: a significant variation in the case of ${\pi^{\ }\ell\nu}$. Since there is a single form factor that dominates the dynamics for this mode (in the limit of massless leptons), the $d\Gamma/dq^2$ distribution contains all of the dynamical information. If an analysis (a) integrates over a broad $q^2$ interval and (b) imposes selection criteria that cause a significant efficiency variation, then a significant variation in the form factors will cause a significant variation in the weighting of low–efficiency versus high–efficiency $q^2$ regions. As a result, the overall efficiency prediction will vary dramatically, leading to significant theoretical uncertainty.
A variety of effects can lead to a $q^2$–dependent efficiency. All analyses, for example, require a minimum lepton momentum, which results in the reconstruction efficiency falling as $q^2$ with $q^2$ (sloped light blue line in Figure \[fig:pi\_models\]. The higher the minimum momentum, the steeper the efficiency slope becomes.
to
A more subtle situation can arise from methods of continuum background suppression. The continuum background naturally reconstructs at low $q^2$. To achieve a large beam–constrained mass (near $M_B$), the kinematics favor the selection of a meson (in this case a pion) from one jet and a lepton pair from a semileptonic decay in the opposite jet (see Figure \[fig:topology\]). Interpreted as an $\Upsilon(4S)$ event, the $\ell\nu$ pair has a small invariant mass, and hence the ${\pi^{\ }\ell\nu}$ candidate decay appears to have low $q^2$. This topology, unfortunately, mimics that of a true $\Upsilon(4S)$ event containing a $B\to{\pi^{\ }\ell\nu}$ decay at low $q^2$ (Figure \[fig:topology\]). Continuum background suppression techniques based on variables that characterize the event [*shape*]{} generally introduce a significant bias into signal reconstruction efficiency versus $q^2$. Requiring the ratio of the second to zeroth Fox–Wolfram moments, $R2$, to be small (spherical topology) introduces such a bias. Variables characterizing the momentum flow relative to the lepton momentum direction, while providing powerful continuum suppression, severely bias the reconstruction efficiency. The efficiency curve that falls steeply at low $q^2$ in Figure \[fig:pi\_models\] characterizes the effect of lepton–based suppression. Fortunately, continuum suppression can be realized without introducing any bias into the reconstruction efficiency. The angle between the thrust or sphericity axes for the candidate event and the remainder of the event provides such suppression (see, for example, Ref. [@lkgAnnRev]).
The same considerations hold for other ${b\to u\ell\nu}$ decay modes. Decays to a vector meson cannot, of course, be characterized solely by the $d\Gamma/dq^2$ distribution because of the interference among the three form factors involved in those decays.
Analyses can, of course, be optimized to reduce effects from uncertainties in the form factors. Obviously, the looser the selection criteria on lepton momentum and event shape variables that bias the efficiency versus $q^2$, the smaller the bias. One can further reduce the form factor uncertainty via extraction of independent rates in separate $q^2$ intervals. The form factors are then only required to provide relative rates over each restricted $q^2$ interval, and not between the different $q^2$ intervals. As a bonus, the resulting $d\Gamma/dq^2$ distribution can be used as a test of the form factor calculations.
The CLEO ’03 analysis adopts both approaches, and the resulting $q^2$ distributions are shown in Figure \[fig:dgamma\], along with a variety of calculations. The ${\pi^{\ }\ell\nu}$ mode shows essentially no variation as the form factors are changed. The ${\rho^{\ }\ell\nu}$ mode still exhibits dependence on the form factor calculation. The branching fractions, obtained by summing the measured rates, show significantly smaller form factor dependence than other measurements (Figure \[fig:branching\_fractions\]) for both modes. The residual uncertainty in the ${\rho^{\ }\ell\nu}$ mode likely results from a cut on the angle between the lepton in the $W$ rest frame and the $W$ direction and the different effect this has on the three form factors for that decay.
to
Cross-feed among ${b\to u\ell\nu}$ modes also exhibits a strong $q^2$ dependence, with the high–$q^2$ regions seeing the most background contribution. This phenomenon is clearly visible in the cross-feed background contribution to ${\pi^{\ }\ell\nu}$ in Figure \[fig:cleo\_pi\]. Basically, at large $q^2$, which corresponds to soft mesons, there is a larger probability for random hadronic combinatorics to combine with the lepton pair from a non-signal ${b\to u\ell\nu}$ mode and satisfy the kinematic requirements for the signal mode. Most analyses have not studied variations of the form factors from specific background modes. The CLEO ’03 approach of extracting rates in independent $q^2$ intervals has reduced the sensitivity to the form factors relative to the CLEO ’96 analysis.
> What set of form factor calculations should all experiments use to evaluate the uncertainty of the signal rates on the signal form factors?
> Similarly, what sets should be used to evaluate cross-feed background from ${\pi^{\ }\ell\nu}$, ${\rho^{\ }\ell\nu}$, or ${\omega\ell\nu}$ into the signal ${b\to u\ell\nu}$ mode?
> Given the observed variations in rates, how should the uncertainties be assigned?
> Given the uncertainties, how should they be combined with other uncertainties in an averaging procedure?
Finally, we must consider evaluation of the systematics associated with simulation of generic ${b\to u\ell\nu}$ backgrounds. In principle, these uncertainties will be strongly correlated between analyses, but the broad variation in approaches to modeling the decays and evaluating the systematics make analysis of the correlations currently impossible. Unfortunately, these systematics make sizable contributions to the measurement of the vector modes in particular. The experiments should standardize on various aspects of simulation and systematic testing.
> What exclusive modes and inclusive parameters should be used in simulation, and how does one “marry” the exclusive and inclusive decays?
> What variation on exclusive modes and form factors and on inclusive parameters should be made?
> How should the “inclusive” decays be hadronized, and how should the hadronization be varied?
> How should potential contributions from nonresonant $\pi\pi\ell\nu$ (or other nonresonant final states) be evaluated, and potential correlations between experiments probed?
Extraction of $|V_{ub}|$
------------------------
to
Two methods for extraction of $|V_{ub}|$ have been used in the analyses. The first, used in most analyses, simply takes the measured partial rate for the signal mode obtained using a given form factor calculation and extracts ${|V_{ub}|}$ using that calculation’s predicted $\Gamma/{|V_{ub}|}^2$. The resulting values for ${|V_{ub}|}$ are summarized in the top portion of Figure \[fig:vub\_results\]. Note that the quoted theoretical uncertainties on ${|V_{ub}|}$ obtained by the experiments are [*NOT*]{} directly comparable. They are evaluated with rather different degrees of conservativeness and analyses with similar sensitivities will appear with differing uncertainties.
Independent extraction of rates in the three $q^2$ intervals in the CLEO ’03 analysis permits extraction of ${|V_{ub}|}$ with a method that minimizes unquantifiable modeling assumptions. The CLEO ’03 analysis restricts consideration to those calculations based firmly in QCD: LQCD and LCSR. However, extrapolation of those calculations outside of the $q^2$ range in which the techniques are valid to the full $q^2$ range involves modeling assumptions that cannot be quantified. CLEO therefore uses the measured rates only in the ranges of validity for each technique, thereby minimizing the modeling assumptions. Those results are also summarized in Figure \[fig:vub\_results\].
To provide a test of the various different form factor predictions over the full $q^2$ range, the CLEO ’03 analysis also fits the predicted $d\Gamma/dq^2$ from a particular calculation to the measured distribution obtained with that calculation. To obtain the optimal normalization, ${|V_{ub}|}$ is allowed to float. The quality of each fit then provides a test of each model. For ${\pi^{\ }\ell\nu}$, the fit quality for ISGW II is considerably poorer than for the other calculations tested (see Figure \[fig:pi\_models\]). Note that while similar results with similar numerical uncertainties for ${|V_{ub}|}$ are obtained with this technique. Because the results rely on more modeling assumptions and are therefore less robust than the other procedure, CLEO does not take these values as the primary results.
Summary
=======
We are entering an exciting phase for measurement of exclusive ${b\to u\ell\nu}$ branching fractions and extraction of ${|V_{ub}|}$. Many new measurements with new techniques have been presented recently, with more anticipated. The statistics are now sufficient to permit extraction of $d\Gamma/dq^2$ with reasonable precision in the decays, which in turn allows reduction of the uncertainty from theoretical form factor shapes on the rates. Measurement of $d\Gamma/dq^2$ also allows more robust extraction of ${|V_{ub}|}$.
The theoretical prediction on the overall form factor normalization continues to dominate extraction of ${|V_{ub}|}$ from the measured rates. I wait with great anticipation for the first unquenched LQCD calculations over a much broader range of $q^2$.
[99]{}
S. B. Athar [*et al.*]{} \[CLEO Collaboration\], arXiv:hep-ex/0304019. To appear in Phys. Rev. D.
M. Battaglia [*et al.*]{}, arXiv:hep-ph/0304132.
C. Schwanda [*et al.*]{}, for the Belle Collaboration. Presented at [*Workshop on the CKM Unitarity Triangle*]{}, IPPP Durham, April, 2003.
Y. Kwon [*et al.*]{}, for the Belle Collaboration. Presented at [*The 31st International Conference on High Energy Physics (ICHEP)*]{}, Amsterdam, July, 2002.
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F. Muheim, [*Experimental review of $|V_{ub}|$ inclusive*]{}, these proceedings.
A. Abada, D. Becirevic, P. Boucaud, J. P. Leroy, V. Lubicz, and F. Mescia, Nucl. Phys. B [**619**]{}, 565 (2001) \[arXiv:hep-lat/0011065\]. A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie, S. M. Ryan, and J. N. Simone, Phys. Rev. D [**64**]{}, 014502 (2001) \[arXiv:hep-ph/0101023\]. S. Aoki [*et al.*]{} \[JLQCD Collaboration\], Phys. Rev. D [**64**]{}, 114505 (2001) \[arXiv:hep-lat/0106024\]. L. Del Debbio, J. M. Flynn, L. Lellouch, and J. Nieves \[UKQCD Collaboration\], Phys. Lett. B [**416**]{}, 392 (1998) \[arXiv:hep-lat/9708008\]. K. C. Bowler [*et al.*]{} \[UKQCD Collaboration\], Phys. Lett. B [**486**]{}, 111 (2000) \[arXiv:hep-lat/9911011\]. P. Ball and V. M. Braun, Phys. Rev. D [**58**]{}, 094016 (1998) \[arXiv:hep-ph/9805422\]. P. Ball and R. Zwicky, JHEP [**0110**]{}, 019 (2001) \[arXiv:hep-ph/0110115\]. A. Khodjamirian, R. Rückl, S. Weinzierl, C. W. Winhart, and O. Yakovlev, Phys. Rev. D [**62**]{}, 114002 (2000) \[arXiv:hep-ph/0001297\]. D. Scora and N. Isgur, Phys. Rev. D [**52**]{}, 2783 (1995) \[arXiv:hep-ph/9503486\]. T. Feldmann and P. Kroll, Eur. Phys. J. C [**12**]{}, 99 (2000) \[arXiv:hep-ph/9905343\]. D. Melikhov and B. Stech, Phys. Rev. D [**62**]{}, 014006 (2000) \[arXiv:hep-ph/0001113\]. Z. Ligeti and M. B. Wise, Phys. Rev. D [**53**]{}, 4937 (1996) \[arXiv:hep-ph/9512225\]. L.K. Gibbons, Annu. Rev. Nucl. Part. Sci., [**48**]{}, 121 (1998).
|
---
abstract: 'This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2–dimensional cobordisms by taking into account their embedding into $\R^3$. Secondly, we extend the underlying cobordism category to a 2–category, where the usual relations hold up to 2–isomorphisms. The corresponding abelian 2–functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel–Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any EQFT based on a $\Z_2$–extension of the embedded cobordism category which coincides with Khovanov after reducing the coefficients modulo 2, gives rise to a link invariant homology theory isomorphic to those of Khovanov.'
address:
- 'Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland'
- 'Institut de Mathématiques de Bourgogne, Université de Bourgogne, UMR 5584 du CNRS, BP47870, 21078 Dijon Cedex, France'
author:
- Anna Beliakova
- Emmanuel Wagner
title: |
On link homology theories\
from extended cobordisms
---
Introduction {#introduction .unnumbered}
============
In his influential paper [@Kho], Khovanov constructed a link homology theory categorifying the Jones polynomial. During few years, this categorification was considered to be essentially unique, since the underlying $(1+1)$ TQFT was known to be determined by its Frobenius system and all rank two Frobenius systems were fully classified [@Kho1]. However, in [@Odd] Ozsvath, Rasmussen and Szabo came up with a new categorification of the Jones polynomial, which agrees with Khovanov’s one after reducing the coefficients modulo two. The underlying algebraic structure of the odd Khovanov homology can not be described in terms of the Frobenius algebra.
This fact attracts attention again to the question of description and classification of algebraic structures leading to link homology theories. In this paper, we provide an evidence to the fact that the appropriate algebraic structure is given by an extended quantum field theory (EQFT). A EQFT here is a 2–functor from a certain (semistrict) monoidal 2–category of cobordisms, called an [*extension*]{}, to an abelian category. Given a cobordism category by specifying its generators and relations, the 2–category is constructed by requiring the relations to be satisfied up to 2–isomorphisms. Furthermore, such a 2–category is called an extension of the original cobordism category, if the automorphism group of any 1–morphism is trivial. A simple example of an extension is a $\Z_2$–extension, where the 2–morphisms are just plus or minus the identity. Notice that extensions can be defined for both strict and semistrict monoidal 2–categories and the resulting EQFT will also be called strict and semistrict respectively.
The usage of the word “extension” in our setting is motivated by the fact that after replacing the original category by a group we will get a usual extension of that group. Those extensions are classified by the second cohomology classes of the group. Therefore, our approach can serve as a definition for the second cohomology of a category. A quite different notion of an extended topological field theory (ETFT) was introduced and studied in [@SP].
In this paper, we construct extensions of the category of 2–dimensional cobordisms $\Cob$ and of the category of embedded 2–cobordisms modulo the unknotting relation $\NC$. In the first case, we recover the Khovanov and the odd Khovanov homologies, as strict (trivial) and semistrict extensions respectively. In the second case, we construct so–called nested Khovanov homology, extracted by Stroppel and Webster [@CW] from the algebraic counterpart of the Seidel–Smith construction. In addition, we show that the last theory is equivalent to those of Khovanov. More precisely, for a given diagram $D$, let us denote by $\ll D\rr$ its Khovanov hypercube of resolutions. Applying the Khovanov TQFT, we get a complex $F_{\Kh}\ll D\rr$. On the other hand, using the nested Frobenius system, defined in Section \[NFS\], we get the complex $F\ll D\rr$.
\[main\] Given a diagram $D$ of a link $L$, the complexes $F_{\Kh} \ll D\rr$ and $F\ll D\rr$ are isomorphic.
Once the equivalence between the geometric construction of Seidel–Smith and the algebraic one of Cautis–Kamnitzer is established rigorously, this theorem can be used to finalize the proof of the Seidel–Smith conjecture.
A similar result was independently proved by a student of C. Stroppel.
The last result of the paper is the classification of all rank two strict $\Z_2$–extensions of $\NC$.
\[main1\] Any strict EQFT based on a $\Z_2$–extension of $\NC$, which agrees with Khovanov’s TQFT after reducing the coefficients modulo 2, gives rises to a link invariant homology theory isomorphic to those of Khovanov.
A challenging open problem is to classify all semistrict EQFTs based on $\NC$, which associate to a circle a rank two module. More generally, the problem is to compute the second cohomology of $\NC$ and construct cocycles restricting to the Schur cocycle of the symmetric group.
An interesting algebraic system underlying the categorification of the Kauffman skein module [@APS], [@TT] was proposed recently by Carter and Saito [@CS]. We wonder whether our approach could be extended to include their setting.
The paper is organized as follows. In the first sections we define the categories $\Cob$, $\NC$ and their extensions. Theorems \[main\] and \[main1\] are proved in Section 3. In the last section, odd Khovanov homology is realized as an extension of $\Cob$.
Acknowledgment {#acknowledgment .unnumbered}
--------------
The authors would like to thank Catharina Stroppel, Krzysztof Putyra, Alexander Shumakovitch, Christian Blanchet and Aaron Lauda for interesting discussions and to Dror Bar–Natan for the permission to use his picture of the Khovanov hypercube.
The category of 2–cobordisms and its extensions
===============================================
The category $\Cob$
-------------------
[The objects of $\Cob$ are finite ordered set of circles. The morphisms are isotopy classes of smooth 2–dimensional cobordisms. The composition is given by gluing of cobordisms. ]{}
The category $\Cob$ is a strict symmetric monoidal category with the monoidal product given by the ordered disjoint union and the identity given by the cylinder cobordism. In particular, we obtain a natural embedding of the symmetric group in $n$ letters into the automorphism group of $n$ circles.
By using Morse theory, one can decompose any 2–cobordism into pairs of pants, caps, cups and permutations, proving the following well–known presentation of $\Cob$ (see e.g. [@H])
\[rel-cob\] The morphisms of $\Cob$ are generated by
subject to the following relations:
\(1) Commutativity and co-commutativity relation
\(2) Associativity and coassociativity relations
\(3) Frobenius relations
\(4) Unit and Counit relations
\(5) Permutation relations
\(6) Unit-Permutations and Counit-Permutation relations
\(7) Merge-Permutation and Split-Permutation relations
For a commutative unital ring $R$, let $R$-$\mod$ be the category of finite projective modules over $R$. A (1+1)–dimensional topological quantum field theory (TQFT) is a symmetric (strict) monoidal functor from $\Cob$ to $R$-$\mod$. Such TQFTs are in $1:1$ correspondence with so–called Frobenius systems (compare [@Ko]).
One important application of Frobenius systems is Khovanov’s categorification of the Jones polynomial [@Kho].
In what follows we will assume that $\Cob$ is a pre–additive category. This means we supply the set of morphisms (between any two given objects) with the structure of an abelian group by allowing formal $\Z$–linear combinations of cobordisms and extend the composition maps bilinearly.
Extensions of categories
------------------------
In this section we use the language of 2–categories. A 2–category is a category where any set of morphisms has a structure of a category, i.e. we allow morphisms between morphisms called 2–morphisms. Given a 2–category $M$, the 2–morphisms of $M$ can be composed in two ways. For any three objects $a,b, c$ of $M$, the composition in the category $\Mor_{M}(a,b)$ is called vertical composition and the bifunctor $*:\Mor_M(a,b) \times \Mor_M(b,c) \to \Mor_M(a,c)$ is called horizontal composition. These compositions are required to be associative and satisfy an interchange law (see [@MacL] for more details).
Semistrict monoidal 2–categories can be considered as a weakening of monoidal 2–categories, where monoidal and interchange rules hold up to natural isomorphisms (compare [@Lauda Proposition 17]).
Assume $\cal C$ is a strict monoidal category, whose set of morphisms is given by generators and relations.
An extension of $\cal C$ is the semistrict monoidal 2–category $\BiC$, which has the same set of objects as $\cal C$. The 1–morphisms of $\BiC$ are compositions of generators of $\cal C$. The 2–morphisms are
- the identity automorphism of any 1–morphism of $\cal C$;
- a 2–isomorphism between any two 1–morphisms subject to a relation in $\C$.
This imposes a so–called “cocycle” condition on the set of 2–morphisms, since any composition of 2–morphisms going from a given 1–morphism to itself should be equal to the identity or any closed loop of 2–morphisms is trivial.
An example of an extension is given by a weak monoidal category $(M,\otimes, 1, \alpha, \lambda,\rho)$ where $\alpha$, $\lambda$ and $\rho$ are considered as 2–isomorphisms and the cocycle condition holds due to MacLane’s coherence theorem [@MacL Chapter VII]. In the case, when $\cal C$ is $\Cob$ restricted to connected cobordisms, (i.e. permutation is removed from the set of generators in Theorem \[rel-cob\] as well as relations (1),(5), (6) and (7)), then any pseudo Frobenius algebra, described in [@Lauda Proposition 25], defines an extension $\BiC$. The cocycle condition holds due to Lemmas 32, 33 in [@Lauda].
Providing $\cal C$ with a structure of a pre–additive category, we have a natural $\Z$–action on the set of 1–morphisms, restricting to $\Z_2=\{1,-1\}$, the group of two elements written multiplicatively, we can define a $\Z_2$–extension of $\cal C$, in which the 2–morphisms are just plus or minus the identity. Note that in this case $\BiC$ can be considered as a weak monoidal category, with the same set of generating 1–morphisms as $\cal C$, but with sign modified relations.
For any cobordism category $\cal C$, an [*extended quantum field theory*]{} (EQFT) based on $\cal C$ is a bifunctor from $\BiC$ to $R$-$\mod$, mapping 2–morphisms to natural transformations of $R$–modules. The EQFT is called strict if $\BiC$ is strict.
Embedded cobordisms
====================
Let $S^{\amalg a}$ be the disjoint union of $a$ copies of a circle smoothly embedded into a plane. Note that the embedding induces a partial order on the set of circles as follows. For two circles $c_1$ and $c_2$, we say $c_1< c_2$, if $c_1$ is inside $c_2$.
The objects of $\NC$ are finite collections of circles embedded into a plane. The morphisms are generated by
\
subject to the following sets of relations:
\(1) Frobenius type relations
\
\
\
\(2) Associativity type relations
\
\(3) Coassociativity type relations
\
\(4) Cancellation
\(5) Torus relation
In addition, the merge, the split, the birth, the death and the permutation are still subject to all the relations of Theorem \[rel-cob\]. \[nc\]
The relations of Definition \[nc\] can also be described as follows:
\
\
\
where the black circle corresponds to the starting configuration of circles, and the dashed arcs correspond to the operations which are performed. Notice that changing the order of operations produce the two different sides of the relations in Definition \[nc\]. In addition, associativity of the merge, the coassociativity of the split and the usual Frobenius relation are also depicted here.
The category $\NC$ is a symmetric strict monoidal category with a tensor product given by a partially ordered disjoint union, i.e. circles on the same level of nestedness are ordered. In particular, we obtain a natural embedding of the symmetric group into the automorphisms of any object, permuting circles not ordered by nestedness and at the same level of nestedness.\
Any morphism in $\NC$ is the composite of such a permutation and the tensor product of connected morphisms of $\NC$.
Any connected morphism in $\NC$ has the following normal form:
Assume that the boundary of our connected genus $g$ cobordism $C$ consists of $a$ incoming circles and $b$ outgoing ones. Let us suppose that $C$ is a composition of $B$ births, $D$ deaths, $M$ merges and $S$ splits. Then we have $$2-2g-a-b=B+D-M-S, \quad \quad a-M+B=b-S+D$$ or $$M=a+g-1+B,\quad \quad S=b+g-1+D$$ We arrive at the normal form if we will be able to push all merges (resp. splits) to the incoming (resp. outgoing) boundary of $C$. From the above formulas we see that $B$ merges and $D$ splits will cancel with the births and deaths, respectively, and $g$ splits and merges put together will create $g$ handles. The remaining $a-1$ merges commute with any split (nested or not nested one) due to the Frobenius type relations. Finally using the associativity type relations, we can commute nested and unnested merges (resp. splits) and arrive at the form in Figure \[inhandle\].
Furthermore applying the Torus relation, one can now reduce to the normal form.
Embedded cobordisms
-------------------
[A smoothly embedded 2–dimensional cobordism from $S^{\amalg a}$ to $S^{\amalg b}$ is a pair $(F,\phi)$, where $F$ a smooth 2–dimensional surface whose boundary consists of $a+b$ circles and $\phi: F \hookrightarrow \R^2 \times [0,1]$ is a smooth embedding, such that $\phi|_{\partial F} \cap \R^2\times \{0\}= S^{\amalg a}$ and $\phi|_{\partial F} \cap \R^2\times \{1\}= S^{\amalg b}$. ]{}
\[ec\]
The objects of $\EC$ are circles smoothly embedded into a plane. The morphisms are isotopy classes of smoothly embedded 2–dimensional cobordisms subject to the unknotting relation:
The composition is given by gluing along the boundary.
The category $\EC$ is again a symmetric strict monoidal category with a tensor product given by the partially ordered disjoint union and with the action of the permutation group depending on nestedness.
The category $\EC$ is isomorphic to the category $\NC$.
By [@H], any smooth 2–cobordism allows a pair of pants decomposition. Modulo the unknotting relation, there are two ways to embed a pair of pants into $\R^3$, providing the list of generators in Definition \[nc\]. The relations do not change the isotopy class of an embedded cobordism and allow to bring it into a normal form. It remains just to say, that the normal forms of two equivalent connected cobordisms coincide.
Strict $\Z_2$–extension
-----------------------
Assume $\NC$ is a pre–additive category.
Let $\NC_1$ be the strict monoidal 2–category obtained from $\NC$ by replacing the torus relation with
(T1)
$\NC_1$ is a $\Z_2$–extension of $\NC$. \[cocy\]
The only non–trivial 2–morphism corresponds to the torus relation. It remains to show that the automorphism group of any 1–morphism is trivial. By Bergman’s Diamond Lemma [@Berg], it suffices to check that any cube with T1 face has an even number of anticommutative faces. This is a simple case by case check. The 10 cubes to check are depicted in Figure \[cocycle\].
Notice that any element of $\NC_1$ does still have a normal form, which corresponds to the usual one plus the information of the parity of the number of inner 1–handles in Figure \[inhandle\].
Nested Frobenius system {#NFS}
-----------------------
In this section we construct a strict EQFT based on $\NC_1$, as proposed by Stroppel [@C-talk]. As in [@Kho], let us consider the 2–dimensional module $A:=\Z[t]\la{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }},X\ra$ over the polynomial ring $R:=\Z[t]$ in one variable. We denote by ${{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}$ the image of 1 under the embedding $\eta: R\to A$. For $\e\in \{0,1\}$, we define two kinds of a multiplication $m_\e:A\otimes A\to A$ as follows: m\_:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
&
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X& X\
X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& (-1)\^X\
XX& (-1)\^t
. Further, we define two comultiplications $\Delta_\e: A\to A\otimes A $ and a counit $\epsilon: A\to R$ as follows. \_:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
+ (-1)\^
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X\
X& XX + (-1)\^ t
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
. :{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& 0\
X& 1\
. The functor $F: \NC_1\to R$-$\mod$ maps any object $S^{\amalg a}$ to $A^{\otimes a}$ and is defined on the generating morphisms as follows: F(
[c]{}
)=m\_0, F(
[c]{}
)=\_0, F(
[c]{}
)=m\_1, F(
[c]{}
)=\_1, F(
[c]{}
)=, F(
[c]{}
)=,
The convention is that in $A\otimes A$ the first factor corresponds to the inner circle. It is easy to see that $F$ preserves all the relations listed in Definition \[nc\].
Let us introduce a grading on $A$ by putting $$\gr(t):=-4,\quad \gr(X):=-1,\quad \gr({{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}):=1$$ On the tensor product $A^{\otimes n}$ the grading is given by $\gr(a_1\otimes\dots \otimes a_n):=\gr(a_1)+\dots+\gr(a_n)$.
There exist a natural grading on $\NC$ given by the Euler characteristic of cobordisms. As in Khovanov’s case if $t=0$, $F$ is grading preserving.
Nested Khovanov homology
========================
Khovanov’s hypercube {#11}
--------------------
Suppose we have a generic diagram $D$ of an oriented link $L$ in $S^3$ with $c$ crossings. By resolving crossings of $D$ in two ways as prescribed by the Kauffman skein relations, one can associate to $D$ a $c$–dimensional cube of resolutions (compare [@Kho] or [@BN1]). The vertices of the cube correspond to the configurations of circles obtained after smoothing of all crossings in $D$. For any crossing, two different smoothings are allowed: the $0$– and the $1$–smoothings. Therefore, we have $2^c$ vertices. After numbering the crossings of $D$, we can label the vertices of the cube by $c$–letter strings of $0$’s and $1$’s, specifying the smoothing chosen at each crossing. The cube is skewered along its main diagonal, from $00...0$ to $11...1$. The number of 1 in the labeling of a vertex is equal to its ‘height’ $k$. The cube is displayed in such a way that the vertices of height $k$ project down to the point $r:=k-c_-$, where $c_\pm$ are the numbers of positive, resp. negative crossings in $D$ (see Figure \[Diag\]).
8
8
Two vertices of the hypercube are connected by an edge if their labellings differ by one letter. In Figure \[Diag\], this letter is labeled by $*$. The edges are directed (from the vertex where this letter is $0$ to the vertex where it is $1$). The edges correspond to a saddle cobordisms from the tail configuration of circles to the head configuration (compare Figure \[Diag\]). We denote this hypercube of resolutions by $\ll D\rr$, and would like to interpret it as a complex. The $r$th chain “space” $\ll D\rr^{r}$ is a formal direct sum of the $\frac{c!}{k!(c-k)!}$ “spaces” at height $k$ in the hypercube and the sum of “maps” with tails at height $k$ defines the $r$th differential. To achieve $\partial^2=0$, we assign a minus to any edge which has an odd number of 1’s before $*$ in its labeling.
Applying (1+1) TQFT $F_{Kh}$ to $\ll D\rr$, which sends any merge to $m_0$ and any split to $\Delta_0$, we get a complex of $R$-modules $(F_{\Kh}\ll D\rr, \partial_{\Kh})$. Its graded homology groups, known as Khovanov homology, are link invariants and the graded Euler characteristic is given by the Jones polynomial.
Nested homology
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Applying $F$ to the Khovanov hypercube, one can define a chain complex as follows. The $r$th chain group will be $F\ll D\rr^{r}$, the image of $\ll D\rr^{r}$ after applying the functor $F$, and the maps are defined by applying $F$ to the corresponding cobordisms. The main difference to the Khovanov case is that here not all faces are commutative. More precisely, the square corresponding to the Torus relation $(T1)$ is anti–commutative. However, by the definition of the $\Z_2$–extension, the 2–cochain $\psi\in H^2(B,\mathbb{Z}/2\mathbb{Z})$ ($B$ is the hypercube) which associates $1$ to any anticommutative face of the hypercube and $-1$ to any commutative one is a cocycle, i.e. it vanishes on the boundary of any cube. Since the cube is contractible, any cocycle is a coboundary. Consequently, there exists a function $\ep:{\cal E}\to\Z_2$ from the set of edges of the hypercube to $\Z_2$, called a sign assignment, such that $\ep(e_1)\ep(e_2)\ep(e_3)\ep(e_4)=\psi(D)$ for any four edges $e_1,\dots,e_4$ forming a square $D$. Hence, multiplying edges of the hypercube by the signs $\ep$, we get a chain complex $(F\ll D\rr, \partial_\ep)$. It is easy to see that this complex is independent on the choice of a sign assignment.
\[22\] Given two sign assignments $\ep$ and $\ep'$, the chain complexes $(F\ll D\rr, \partial_\ep)$ $(F\ll D\rr, \partial_{\ep'})$ are isomorphic.
The product $\ep\ep'$ is a 1–cocycle. Since the hypercube is contractible, this 1-cocycle is a coboundary of a 0–cochain $\eta: {\cal V}\to\Z_2$. The identity map times $\eta$ provides the required isomorphism.
In the case, when $t=0$, the homology groups of $(F\ll D\rr, \partial)$ are graded and the graded Euler characteristic coincides with the Jones polynomial. If $t\neq 0$, then $\gr$ defines a filtration on our chain complex, similar to the one considered by Lee [@Lee].
Our next aim is to show that the complex we just constructed is isomorphic to the Khovanov complex.
Proof of Theorem \[main\]
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We have to show that $(F\ll D\rr, \partial)$ and $(F_{\Kh}\ll D\rr, \partial_{\Kh})$ are isomorphic.
For any circle $c$ in $S^{\amalg k}$, we define $\deg(c)$ to be the number of circles in $S^{\amalg k}$ containing $c$ inside. Further, we define an endomorphism $\phi_k$ of $ F(S^{\amalg k})$ as follows: For a copy of $A$ associated with $c$, we put \_c:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
&
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
X& (-1)\^[(c)]{} X\
. Then $\phi_k$ is the composition of $\phi_c$ for all circles in $S^{\amalg k}$. By abuse of notation $\phi_k$ depends not only on $k$ but also on the configuration of circles in $S^{\amalg k}$.
Given a link diagram $D$ with $d$ crossings, consider two Khovanov’s hypercubes of resolutions associated with $D$. Apply $F$ to one of them and $F_{\Kh}$ to the other and do not use any sign assignment, i.e. all squares in $F_{\Kh} \ll D\rr$ are commutative. Further, observe that with each vertex of the hypercube, there is a copy of $A^{\otimes k}$, for a certain $k$, associated. Applying $\phi_{ k}$ to any such vertex, we get a map $\Phi$ with the source $(F\ll D\rr, \partial)$ and the target $(F_{\Kh}\ll D\rr, \partial_{\Kh})$, without any sign assignments. Our next goal is to see that there exists a sign assignment on the $(d+1)$–dimensional hypercube $\ll D\rr \times [0,1]$ making $\Phi$ to a chain map.
For this, it is enough to check that each $3$–dimensional cube in this $(d+1)$–dimensional hypercube contains an even number of anticommutative faces. Note that there are three different cases: (1) the cube is contained in the source hypercube, (2) the cube is contained in the target hypercube, (3) the cube contains exactly one face in the source hypercube and one face in the target hypercube. The first case follows from Lemma \[cocy\] and the second from the fact that $F_{\Kh}$ is a $(1+1)$ TQFT. The third case rely on a case by case check. Note that all faces in the source hypercube correspond to relations in $\NC_1$. Hence, we have to check the claim for any cube, whose upper face is a relation in $\NC_1$, the lower face is the corresponding Khovanov square and whose vertical edges are labeled by $\Phi$. In addition, since the map $\phi_c$ depends on $\deg(c)$ explicitly, we have to ensure that the claim holds after changing the nestedness of each circle by one. The tables below show that any cube of type (3) does have only commutative or anticommutative faces. It is left to the reader to check that all cubes of type (3) do have an even number of anticommutative faces. For this one has to consider all cubes where the upper face corresponds to one of the relations in Definition \[nc\]. Moreover, each cube should be checked twice for different nestedness modulo 2.\
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$\Delta_0$ $\phi\circ \Delta_0$ $\phi$ $\Delta_0\circ \phi$
---------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------- ---------------------------------------------- -------------------------------------------------------------
${{ \rm \setlength{\unitlength}{1em} $X \otimes {{ \rm \setlength{\unitlength}{1em} $-X\otimes {{ \rm \setlength{\unitlength}{1em} ${{ \rm \setlength{\unitlength}{1em} $X \otimes {{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}$ \end{picture} }}+{{ \rm \setlength{\unitlength}{1em} \end{picture} }}-{{ \rm \setlength{\unitlength}{1em} \end{picture} }}$ \end{picture} }}+{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}\otimes X$ \end{picture} }}\otimes X$ \end{picture} }}\otimes X$
$ X$ $X\otimes X +t{{ \rm \setlength{\unitlength}{1em} $X\otimes X +t {{ \rm \setlength{\unitlength}{1em} $-X$ $-X\otimes X -t{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}\otimes {{ \rm \setlength{\unitlength}{1em} \end{picture} }}\otimes {{ \rm \setlength{\unitlength}{1em} \end{picture} }}\otimes {{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1) \begin{picture}(0.75,1) \begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}} \put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}$ \end{picture} }}$ \end{picture} }}$
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
To finish, observe that the map $\phi$ composed with a sign assignment is clearly invertible, and hence, is the desired isomorphism.
Proof of Theorem \[main1\]
--------------------------
Let us search for further strict $\Z_2$–extensions of $\NC$ systematically. For $e_i \in \Z_2$ with $i=1,...,10$, we put m\_0:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_1
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X& e\_2 X\
X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_2 X\
XX& 0
. m\_1:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_5
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X& e\_6 X\
X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_7X\
XX& 0
.
\_0:{
[l@l]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_3 (X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
+
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X)\
X& e\_4 XX\
. \_1:{
[l@l]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& e\_8 X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
+ e\_9
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X\
X& e\_[10]{} XX\
. The relations in Definition \[nc\] should hold up to sign for any EQFT. They impose the following relations on $e_i$:
4\. row Frobenius type relations (1) $\Longrightarrow\;\;\;\;$ $e_6=e_5$, $e_9=e_{10}$;
3\. row Frobenius type relations (1) $\Longrightarrow\;\;\;\;$ $e_7 e_8=e_5 e_9$;
the ordinary Frobenius relation $\Longrightarrow\;\;\;\;$ $e_1=e_2$, $e_3=e_4$.
Modulo these identities, there are 5 free parameters, i.e. 32 cases to consider. It is a simple check that all of them produce the Khovanov or nested Khovanov Frobenius system, after changing the sign of one or two operations.
It remains to construct an isomorphism between, say, nested Khovanov complex and the one where $m_0$ is replaced by $-m_0$. Let us consider the map between two nested Khovanov hypercubes which is identity on all vertices, except of the tails of edges corresponding to $m_0$, at those edges the map is minus the identity. As in the proof of Theorem \[main\], the cone of this map is a hypercube of a dimension one bigger. Let convince our self that all 3–dimensional cubes of that hypercube have an even number of anticommutative faces. We have to check only cubes whose upper horizontal faces belong to the nested Khovanov complex, the bottom horizontal face to the nested Khovanov with $m_0$ replaced by $-m_0$, and whose vertical edges are given by our map. If the upper horizontal face has an even number of $m_0$ maps, then the cube has an even number of vertical anticommutative faces. If it has an odd number of $m_0$ maps, then there is an odd number of vertical anticommutative faces, but either the top or the bottom face is anticommutative. Hence, like in the proof of Theorem \[main\], there exists a sign arrangement on this hypercube providing the desired isomorphism.
Odd Khovanov homology
=====================
The extension $\Odd$
--------------------
The extension $\Odd$ of $\Cob$ is defined as follows:
The objects of $\Odd$ are finite ordered set of circles. The morphisms are generated by
subject to the following sets of relations:
\(1) Commutativity and co–commutativity relation
\(2) Associativity and coassociativity relations
\(3) Frobenius relations
\(4) Unit and Counit relations
\(5) Permutation relations
\(6) Unit–Permutations and Counit–Permutation relations
\(7) Merge–Permutation and Split–Permutation relations
\(8) Commutation relations
\
All the other commutation relations hold with plus sign.
\
\[odddef\]
All axioms of a semistrict monoidal 2–category are satisfied.
For another definition of the semistrict monoidal just described, we endow the morphisms with the following $\mathbb{Z}/2\mathbb{Z}$–grading: [sd]{}(
[c]{}
)=0, [sd]{}(
[c]{}
)=1, [sd]{}(
[c]{}
)=0, [sd]{}(
[c]{}
)=1. This grading is additive under composition and disjoint union. The monoidal structure $\boxtimes$ on $\Odd$ can be defined as follows:\
For any two generators $f$ and $g$ and the permutation $\rm{Perm}$, f:=f,fg:=(f)(g) where $\otimes$ denotes the disjoint union. The composition rule is modified as follows: (g)(f )= [(-1)]{}\^[[sd]{}(f)[sd]{}(g)]{} fg.
For an alternative description of $\Odd$, see Putyra’s Master Thesis [@Pu] using cobordisms with chronology.\
Let us check that $\Odd$ is indeed an extension.
The automorphism group of any 1–morphism is trivial.
The relations imply that all squares depicted on the RHS, resp. LHS, of Figure 2 in [@Odd] are commutative (resp. anticommutative). The result follows now from Lemma 2.1 in [@Odd], showing that any cube has even number of anticommutative faces and additional checks like the one in the relations satisfied by the 2–morphisms in Lemma 32 [@Lauda]. The result can also be checked completely by hand by proving that all the relations in Lemma 32 [@Lauda] are satisfied by the 2–morphisms in $\Odd$ which are only signs. Many of them are obvious, since many 2–morphisms in our case are just identities. The fact that this is enough still follows from Bergman’s Diamond lemma [@Berg].
Odd Frobenius system
--------------------
In [@Odd], an EQFT into the $\mathbb{Z}/2\mathbb{Z}$–graded abelian groups based on $\Odd$ is constructed.
Using Khovanov’s algebra $A_0=\Z[X]/X^2$, one can describe this EQFT $F_0: \Odd\to \Z$-$\mod$ as follows: $F_0$ maps a circle to $A$ where $A$ is $\mathbb{Z}/2\mathbb{Z}$–graded as follows: ${{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}$ is in degree $0$ and $X$ is in degree $1$. To $n$ circles, $F_0$ assigns $A^{\otimes n}$. To generating morphisms $F_0$ assigns the following maps: m:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
&
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X& X\
X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
& X\
XX& 0
. P:{
[l@r]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
&
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X& X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
\
X
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
&
(0.75,1) (0,0)[$1$]{}(0.34,0)[(0,1)[0.65]{}]{}
X\
XX& -XX
. $$\Delta:\left\{\begin{array}{l@{\quad\mapsto\quad}l}
{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}& X\otimes{{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}- {{ \rm \setlength{\unitlength}{1em}
\begin{picture}(0.75,1)
\put(0,0){$1$}\put(0.34,0){\line(0,1){0.65}}
\end{picture} }}\otimes X\\
X& X\otimes X \\
\end{array}
\right.$$ The maps $\e$ and $\eta$ are the same as in Khovanov case.
Due to the fact that $\Delta$ and $\e$ are of degree $1$, $F_0$ can not map disjoint union of cobordisms to the tensor product of maps assigned to them, since in this case relations (6) and (7) would not be satisfied. Instead, $F_0$ maps disjoint union to $\boxtimes$ defined as follows: f:=f,f:=[Perm]{}(f)[Perm]{},fg:=(f)(g) (g)(f )=[(-1)]{}\^[[deg]{}(f)[deg]{}(g)]{}fg The relations (6) and (7) hold now just by definition.
Applied to the Khovanov hypercube, this EQFT gives rise to a link homology theory, called odd Khovanov homology [@Odd].
[10]{} Asaeda, M.M Przytycki, J.S. Sikora. A.S [*Categorification of the Kauffman bracket skein module of $I$-bundle over surfaces,*]{} Algebr. Geom. Topology [**4**]{} (2004) 1177–1210. Bar–Natan, D.: [*On Khovanov’s categorification of the Jones polynomial*]{}, Algebr. Geom. Topology [**2**]{} (2002) 337–370 Bar–Natan, D.: [*Khovanov’s homology for tangles and cobordisms*]{}, Geometry and Topology [**9**]{} (2005) 1443–1499 Bergman, G.: [*The diamond lemma for ring theory*]{}, Advances in Mathematics [**29**]{} (1978) [**2**]{} 178–218 Carter, J.S., Saito, M.: [*Frobenius Module and Essential Surface Cobordisms*]{}, arXiv:0905:4475 Hirsch, M.W.: [*Differential topology*]{}, Grad. Texts in Math. [**33**]{} Springer–Verlag, New York, 1994 Khovanov, M.: [*A categorification of the Jones polynomial*]{}, Duke Math. J. [**101**]{} (2000) 359–426 Khovanov, M.: [*Link homology and Frobenius extensions*]{}, arXiv:math/0411447 Kock, J.: [*Frobenius algebras and 2D topological quantum field theories*]{}, London Mathematical Society Student Texts [**59**]{}, Cambridge University Press, 2004 Lauda, A.: [*Frobenius algebras and planar open string topological field theories*]{}, arXiv:math/0508349
Lee, E.: [*On Khovanov invariant for alternating links*]{}, arXiv:math.GT/0210213
Mac Lane, S.: [*Categories for working mathematician*]{} Grad. Texts in Math. 5, Springer 1998
Ozsváth, P., Rasmussen, J., Szabó, Z.: [*Odd Khovanov homology*]{}, arXiv:0710.4300
Putyra, K.: [*Cobordisms with chronologies and a generalization of the Khovanov complex,*]{} Master’s thesis.
Schommer–Pries, C.: [*The Classification of Two–Dimensional Extended Topological Field Theories*]{}, PhD thesis, Berkeley, 2009 (http://sites.google.com/site/chrisschommerpriesmath/Home)
Stroppel, C. Webster, B.: [*2–block Springer fibers: convolution algebras and coherent sheaves*]{}, arXiv:0802.1943
Stroppel, C.: [*Convolution algebra and Khovanov homology,*]{} Talk at the Swiss Knots Conference, March 2009
Turaev, V., Turner, P.: [*Unoriented topological quantum field theory and link homology,*]{} arXiv:math/0506229
|
---
abstract: 'Numerical solutions for asymptotically flat rotating black holes in the cubic Galileon theory are presented. These black holes are endowed with a nontrivial scalar field and exhibit a non-Schwarzschild behaviour: faster than $1/r$ convergence to Minkowski spacetime at spatial infinity and hence vanishing of the Komar mass. The metrics are compared with the Kerr metric for various couplings and angular velocities. Their physical properties are extracted and show significant deviations from the Kerr case.'
author:
- 'K. Van Aelst$^{1}$, E. Gourgoulhon$^{1}$, P. Grandclément$^{1}$, C. Charmousis$^{2}$'
bibliography:
- 'v1.bib'
nocite:
- '[@GW_first_detection; @GW170817; @GW170817_and_counterpart]'
- '[@GRAVITY_redshift_S2; @GRAVITY_motion_ISCO]'
- '[@EHT_Shape_SgrA; @EHT_Shadow_M87]'
title: Hairy rotating black holes in cubic Galileon theory
---
Laboratoire Univers et Théories, Observatoire de Paris, PSL Research University, CNRS, Université de Paris, Sorbonne Paris Cité, 92190 Meudon, France
Laboratoire de Physique Théorique, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
*Keywords:* modified gravity, cubic Galileon, hairy black hole, rotating black hole
Introduction
============
Increasingly strong regimes of gravity are tested by modern, highly accurate instruments. Over the last five years, the detectors of the LIGO/Virgo collaboration [@LIGO_Virgo_catalog_O1_O2], the instrument GRAVITY [@GRAVITY] and the Event Horizon Telescope [@EHT] have collected data from objects involved in high energy gravitational processes: coalescing compact objects [@GW_first_detection; @GW170817], stars and flares orbiting *Sgr A\** [@GRAVITY_redshift_S2; @GRAVITY_motion_ISCO], accretion disks and shadows of supermassive black holes [@EHT_Shape_SgrA; @EHT_Shadow_M87]. So far, all these observations are consistent with the black hole model of general relativity (GR), adding to the successes of the latter in weaker gravitational regimes [@pulsar_timing; @Will_review_tests_GR].
Yet, many alternative theories of gravitation are being investigated [@review_Clifton; @review_Berti; @review_Nojiri_cosmo; @review_Koyama_cosmo_tests] and this is important for at least two reasons. Firstly, identifying all the modifications that lead to theoretical pathologies or observational incompatibilies is a relevant approach to understand better why GR is successful. Secondly, GR actually suffers from several shortcomings or unresolved questions: on galactic and cosmological scales, it does not provide satisfactory explanations to the issues of dark matter and dark energy [@report_Saltas_cosmo; @review_Nojiri_cosmo], while it is expected to break down in the high energy or strong curvature regimes in view of its inadequacy to unify with the other fundamental interactions [@Kiefer_quantum_gravity; @gravity_unification_review; @review_Berti].
Modified gravity theories aim to provide answers or alternative solutions to such questions or shortcomings of GR in the ultra-violet and infra-red sectors of gravity. But then, the actual applicability of any modified theory of gravity is assessed from its compatibility with existing observational constraints and theoretical viability, e.g. well-posedness and stability. For example, regarding observational constraints in the dark energy sector, the gravitational wave detection GW170817 and its electromagnetic counterpart GRB170817A set uptight constraints on the speed of gravitational waves [@GW170817_and_counterpart] (see also [@deRham:2018red] for a critical approach on the interpretation of these constraints). Consequently, only restricted families of many modified theories of gravity turned out to be explicitly compatible with these constraints [@DE_after_GW_ahead]. Regarding the theoretical analysis, the so-called “no-hair” theorems [@scalar_hair_review; @hair_review] provide another kind of argument to appraise the relevance of a given modified theory. Such theorems state the equivalence between a modified theory and GR relative to black hole solutions. In other terms, these theorems single out conditions under which the black holes of a modified theory are as “hairless” as those of GR, i.e. belong to the Kerr-Newman family. A typical example is that of Brans-Dicke gravity that has identical black hole solutions to GR (see [@Hawking_no_hair] and more recently [@no_hair_Sotiriou_1] and references within).
Horndeski theories are the most general scalar-tensor theories leading to second-order field equations, which, as shown in [@Kobayashi:2011nu], coincide with the generalized covariant Galileon in four dimensions [@Galileon_original; @covariant_Galileon; @generalized_Galileons; @review_Galileon]). Within this context, the cubic Galileon theory is of particular interest among Horndeski theories [@Horndeski_original]. For a start, it is the simplest of Galileons with higher order derivatives. The cubic Galileon is also well-known for being related to the Dvali-Gabadadze-Porrati (DGP) braneworld model [@DGP_original], from which all (flat) Galileon theories originate [@Galileon_original; @review_Galileon; @review_Galileons_de_Rham]. More precisely, the DGP model is a 5-dimensional theory of gravity such that all non-gravitational fields are restricted to a 4-dimensional subspace (the usual spacetime), on which gravity is induced by a continuum of massive gravitons [@review_massive_de_Rham]. In this framework, an effective formulation of gravity on the 4-dimensional spacetime generates the scalar term corresponding to the cubic Galileon theory in the decoupling limit [@DGP_effective_action_2]. The DGP term, along with other covariant Galileons, also arises from Kaluza-Klein compactification of higher dimensional metric theories of gravity (see for example [@VanAcoleyen:2011mj; @Charmousis:2014mia]).
On the observational side, the cubic Galileon is compatible with the observed speed of gravitational waves [@GW_speed; @DE_after_GW_Vernizzi; @DE_after_GW_ahead; @DE_after_GW_revisited]. It should also be mentioned that constraints may be set on the free parameters of the theory based on cosmological data: such studies either assume convergence of cosmological Galileons to a common “tracker” solution [@Galileon_obs_status_Planck; @Galileon_ISW_CMB] or more agnostic scenarios [@Galileon_recent_cosmo_data; @Galileon_obs_status_cosmo_GW; @Galileon_lensing_voids].
On the theoretical side, various issues have been tackled within the framework of the cubic Galileon theory or larger theories including it: accretion onto a black hole [@Galileon_accretion; @stability_Galileon_accretion], types of coupling to matter [@Galileon_cosmo_paths], laboratory tests [@Galileon_labo], cosmological dynamics [@cubic_cosmo; @cubic_BH], structure formation [@cubic_Galileon_structure_formation], stability of cosmological perturbations [@Galileon_cosmo_viability; @Galileon_linear_cosmo_perturb], well-posedness [@well_posedness_Love_Horn; @stability_cosmo; @well_posed_cubic_Horn; @global_solutions_Horn]. Finally, it was found in [@no_hair_Galileon] (with important precisions given in [@hairy_BH_GB_Sotiriou_1; @hairy_BH_GB_Sotiriou_2; @Babichev:2016rlq]) that shift-symmetric Horndeski theory along with the cubic Galileon is subject to a no-hair theorem in the static and spherically symmetric case (see also [@slowly_rotating_no_hair] for an extension to slow rotation and [@Lehebel:2017fag] for stars). This could have removed the interest for black holes in this theory, but instead it was rapidly shown that slightly violating one of the hypotheses of the no-hair of [@no_hair_Galileon], namely the stationarity of the scalar field [@Babichev:2013cya], allowed to obtain static and spherically symmetric black holes different from GR solutions [@cubic_BH]. This indicated that rotating black holes in the cubic Galileon theory might significantly deviate from the Kerr solution, which motivated the work reported here.
In fact, several other cases, constructed by breaking one of the hypotheses of the no-hair theorem, were found for different sectors of Horndeski theory and beyond (see for example [@hairy_BH_GB_Sotiriou_1; @hairy_BH_GB_Sotiriou_2; @Babichev:2017guv]). Although these hairy solutions, with non trivial scalar field, are obtained for different higher order Horndeski terms, they can be separated in two generic classes: those in which spacetime is very close to that of GR, characterized by an additional parity symmetry of the action for the scalar ($\phi \leftrightarrow -\phi$) and often dubbed as stealth solutions; and those with no parity symmetry and significant departures from GR metrics. For the former case a rotating stealth black hole was recently analytically constructed [@Charmousis:2019vnf] making use of an analogy with geodesic congruences of Kerr spacetime [@Carter:1968rr]. In the latter class, on the other hand, belong the DGP and the Gauss-Bonnet black holes (see for example the recent works [@Antoniou:2017hxj], [@rotating_EdGB_1] and references within). In this paper we will concentrate on the case of rotating black holes for the DGP Galileon finding significant deviations from the GR Kerr spacetime.
The structure of the paper is as follows. Firstly, the field equations of the cubic Galileon theory are introduced in section \[section\_dynamics\] below. Based on these equations, the no-scalar-hair theorem which the cubic Galileon is subject to, and the minimal way to circumvent it, are reviewed in section \[section\_no\_hair\]. This method provided the ansatz used for the scalar field in the rotating case. It is described in section \[section\_ansatze\] along with the circular ansatz used for the metric. The rest of the numerical setup is presented in sections \[section\_equations\] to \[section\_accuracy\]. The numerical solutions are presented and analysed in sections \[section\_num\_sols\] and \[section\_phys\].
The cubic Galileon model {#section_Galileon}
========================
Dynamics {#section_dynamics}
--------
The vacuum action of the cubic Galileon involves the Einstein-Hilbert term (with a cosmological constant $\Lambda$) and the usual scalar kinetic term for the scalar field $\phi$ along with an additional nonstandard term: $$\begin{aligned}
\label{eq_action}
S\left[ g,\phi \right]
&= \int \left[
% \zeta (R^{(g)} - 2 \Lambda)
\zeta (R - 2 \Lambda)
- \eta (\partial \phi)^{2}
+ \gamma (\partial \phi)^{2} \Box\phi
\right]
\sqrt{\vert \det g \vert} d^{4}x,\end{aligned}$$ where $(\partial \phi)^{2} \equiv \nabla_{\mu} \phi \nabla^{\mu} \phi$ and $\zeta$, $\eta$ and $\gamma$ are coupling constants.
The scalar part of (\[eq\_action\]) is known to emerge from an effective formulation of the DGP model [@DGP_original]. In the DGP model, the usual spacetime is a timelike hypersurface of a 5-dimensional spacetime on which a metric alone is defined. But the effective dynamics of gravity on the 4-dimensional spacetime involves scalar terms including those appearing in (\[eq\_action\]) [@DGP_effective_action], which actually become the only relevant contributions in some physically consistent decoupling limit [@DGP_effective_action_2].
One may also note that action (\[eq\_action\]) enters the family of theories featuring “kinetic gravity braiding” [@KGB]. These are characterized by both scalar and metric equations of motions involving second-derivatives of both $g$ and $\phi$ in any conformal frame. Such “braided” coupling between $g$ and $\phi$ induces interesting cosmological phenomenology in regard of dark energy [@KGB].
Explicitly, the metric equations in the cubic Galileon theory take the form $$\begin{aligned}
\label{eq_metric}
G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T^{(\phi)}_{\mu\nu}\end{aligned}$$ where $$\begin{aligned}
\label{eq_T_phi_munu}
8\pi T^{(\phi)}_{\mu\nu} =
& \frac{\eta}{\zeta} \left( \partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}g_{\mu\nu}(\partial \phi)^{2} \right)
\nonumber \\
& \phantom{(}
+ \frac{\gamma}{\zeta}
\left(
\partial_{(\mu}\phi\partial_{\nu)}(\partial \phi)^{2} - \Box\phi \partial_{\mu}\phi\partial_{\nu}\phi
- \frac{1}{2}g_{\mu\nu}\partial^{\rho}\phi\partial_{\rho}[(\partial \phi)^{2}]
\right)\end{aligned}$$ does contain second derivatives of $\phi$.
The scalar field equation actually coincides with the current conservation associated with the shift-symmetry $\phi \rightarrow \phi + constant$ [^1] of action (\[eq\_action\]): $$\begin{aligned}
\label{eq_scalar}
\nabla_{\mu} J^{\mu} = 0,\end{aligned}$$ where $$\begin{aligned}
\label{eq_current}
J_{\mu} &= \partial_{\mu}\phi \left( \gamma \Box\phi - \eta \right) - \frac{\gamma}{2} \partial_{\mu}\left(\partial\phi\right)^2,\end{aligned}$$ which does generate second derivatives of the metric in (\[eq\_scalar\]).
No-scalar-hair theorem and hairy solutions {#section_no_hair}
------------------------------------------
One of the first no-scalar-hair theorems was proven by Hawking in 1972 for stationary black holes in Brans-Dicke theory [@Hawking_no_hair]. It was extended to a larger family of scalar-tensor theories by V. Faraoni and T. P. Sotiriou in 2012 using asymptotic flatness [@no_hair_Sotiriou_1; @no_hair_Sotiriou_2]. The result by L. Hui and A. Nicolis established in 2013 applies to another large family of scalar-tensor theories (shift-symmetric covariant Galileons) intersecting the former and including the cubic Galileon [@no_hair_Galileon]. But this theorem applies to a case more restricted than stationarity and asymptotic flatness, as is reviewed below.
Prior to this, one can see from the field equations (\[eq\_metric\]) and (\[eq\_scalar\]) that any solution of vacuum GR along with a constant scalar field[^2] is a solution to the cubic Galileon theory (see [@classification_Sotiriou] for general results on the theories featuring this property and their relations with other shift-symmetric theories).
The no-scalar-hair theorem stated in [@no_hair_Galileon] establishes the converse result in the case of an asymptotically flat, static, spherically symmetric black hole metric and a scalar field featuring the same symmetries and a standard kinetic term (i.e. $\eta \neq 0$ in (\[eq\_action\])): under such hypotheses, the solutions to the cubic Galileon theory can only be those of GR with a constant scalar field. The proof, and an extension to the case $\eta = 0$ (relevant for the work presented here, as detailed in section \[section\_BC\]), are given in \[appdx\_no\_hair\] in the restricted case of the cubic Galileon.
Yet the attractiveness of a given modified theory is to feature deviations away from GR at least in some circumstances, otherwise there would be no interest in studying its black holes. The solutions exhibited in [@cubic_BH] showed that such deviations exist in the cubic Galileon theory whenever the staticity of the scalar field is replaced by a linear time-dependence: $$\begin{aligned}
\label{eq_scalar_ansatz_sphe}
\phi = qt + \Psi(R),\end{aligned}$$ where $q$ is a constant, $t$ a time coordinate and $R$ a radial coordinate. The structure (\[eq\_scalar\_ansatz\_sphe\]) actually arises in a cosmological context from the assumption of a slow cosmological dynamics [@time_variation_qt], and it has been considered in several contexts [@Galileon_accretion; @cosmo_self_tun] due to the following interesting properties.
As the scalar field contributes to the field equations (\[eq\_metric\]) and (\[eq\_scalar\]) only through its derivatives, such a linear time-dependence does not bring any actual time-dependence into the equations, in which only the constant $q$ appears. Therefore, the scalar field does not share all the symmetries of spacetime while the latter may remain static and spherically symmetric. Furthermore, the ansatz (\[eq\_scalar\_ansatz\_sphe\]) does not spoil the self-consistency of the field equations in the static and spherically symmetric case; this means that one is left with as many unknown functions as independent ordinary differential equations [@cubic_BH], suited for numerical integration by a shooting method. Last but not least, it has been shown for cases where analytical expressions are known, [@Babichev:2013cya; @Babichev:2016rlq], that the linear time dependence (\[eq\_scalar\_ansatz\_sphe\]) renders the scalar field regular at the event horizon by precisely cancelling out the radial divergence in $\Psi(R)$. The existence of black hole solutions different from GR provided a path to hairy rotating solutions, whose numerical construction is now presented.
Numerical setup {#section_num_setup}
===============
Ansätze and assumptions {#section_ansatze}
-----------------------
The goal is to construct stationary, rotating (i.e. axisymmetric with a nonzero angular velocity), asymptotically flat black hole spacetimes. In addition, a simplifying assumption is made: the spacetime geometric structure is assumed to be circular, or “$t,\varphi$-orthogonal” (see [@Carter_killing_ortho_transitive; @Carter_Houches; @heusler_uniqueness_book; @intro_relat_stars; @Chandra_book; @Eric_Silvano_noncircular] for further details on the statements reported in this section). The accuracy of this hypothesis will be evaluated in section \[section\_accuracy\].
Denoting $\xi$ and $\chi$ the Killing vectors associated with stationarity and axisymmetry respectively, circularity amounts to requiring that there exists a coordinate system $(t, x^{1}, x^{2}, \varphi)$ such that $\xi = \partial_{t}$, $\chi = \partial_{\varphi}$ and the transformation $(t, \varphi) \mapsto (-t, -\varphi)$ leaves the metric unchanged. This is equivalent to complete integrability of the codistribution $(dt, d\varphi)$, i.e. the existence of a foliation of spacetime by 2-surfaces (called meridional surfaces) everywhere orthogonal to $\xi$ and $\chi$. Using Frobenius theorem, this property takes the form $$\begin{aligned}
\label{eq_Frob_circu}
d\xi \wedge \xi \wedge \chi = d\chi \wedge \xi \wedge \chi = 0,\end{aligned}$$ where the vectors are identified with their corresponding 1-form by metric duality.
Since the surfaces of transitivity (i.e. the orbits of the combined actions of $\xi$ and $\chi$) are orthogonal to the meridional surfaces, the metric components $(t x^{1})$, $(t x^{2})$, $(\varphi x^{1})$ and $(\varphi x^{2})$ vanish in coordinate systems having the aforementioned properties. A judicious choice of coordinates $(x^{1}, x^{2})$ within the meridional surfaces allows to cancel $g_{x^{1}x^{2}}$ as well so that the metric reads[^3] $$\begin{aligned}
\label{eq_circular}
ds^{2} = - N^2 dt^2 + A^2 \left( dr^2 + r^2 d\theta^2 \right) + B^2 r^2 \sin^2 \theta \left( d\varphi - \omega dt \right)^2,\end{aligned}$$ where $N$, $A$, $B$ and $\omega$ are only functions of the coordinates $r$ and $\theta$. Such a coordinate system is naturally called quasi-isotropic. In the case of spherical symmetry, the four functions only depend on $r$, while $\omega = 0$ and $A = B$ (so that the coordinates are merely called isotropic).
In a circular spacetime, Ricci-circularity holds, i.e. $$\begin{aligned}
\label{eq_Ric_circu}
Ric(\xi) \wedge \xi \wedge \chi = Ric(\chi) \wedge \xi \wedge \chi = 0,\end{aligned}$$ where $Ric$ is the Ricci tensor. In stationary, axisymmetric, asymptotically flat spacetimes, the converse result is true, i.e. (\[eq\_Ric\_circu\]) $\Rightarrow$ (\[eq\_Frob\_circu\]). Then, within GR, the Einstein equations allow to substitute the Ricci tensor with the energy-momentum tensor $T$[^4], so that an asymptotically flat black hole is circular if and only if the following holds (generalized Papapetrou theorem): $$\begin{aligned}
\label{eq_T_circu}
T(\xi) \wedge \xi \wedge \chi = T(\chi) \wedge \xi \wedge \chi = 0.\end{aligned}$$
This indicates that circularity may be interpreted in terms of the physical dynamics of matter rather than purely geometric statements. More precisely, the relations (\[eq\_T\_circu\]) indicate that the source of the gravitational field has purely rotational motion about the symmetry axis and no momentum currents in the meridional planes. Hence assuming circularity is very standard in numerical relativity to handle rapidly rotating stars since such objects have negligible convective meridional flows compared to rotation-induced circulation [@circular_rotating_relativistic_bodies]. For instance, circularity allowed to model rotating proto-neutron stars in general relativity [@circular_rotating_proto_neutron_stars]. In the case of a scalar field, circular rotating boson stars were also constructed numerically [@circular_rotating_boson_star]. Finally, circularity is very relevant to describe rotating black holes: the Kerr solution is circular[^5] and numerical metrics of rotating black holes were successfully computed in Einstein-Yang-Mills theory [@Einstein_SU2YM_1; @Einstein_SU2YM_2] and in the dilatonic Einstein-Gauss-Bonnet theory [@rotating_EdGB_1; @rotating_EdGB_2] based on circularity.
Regarding the scalar field, the successful ansatz (\[eq\_scalar\_ansatz\_sphe\]) is rehashed, with a mere additional angular dependence, in order to connect with the solutions of [@cubic_BH] in the non-rotating limit: $$\begin{aligned}
\label{eq_scalar_ansatz}
\phi = qt + \Psi(r,\theta).\end{aligned}$$
Equations in quasi-isotropic gauge {#section_equations}
----------------------------------
Injecting the ansätze (\[eq\_circular\]) and (\[eq\_scalar\_ansatz\]) into the metric equations (\[eq\_metric\]) yields eight nontrivial equations rather than ten since the components $(r,\varphi)$ and $(\theta,\varphi)$ of the three tensors appearing in (\[eq\_metric\]) all separately vanish. These eight metric equations are combined to form four coupled, independent equations adding to the scalar equation (\[eq\_scalar\]) to solve for the four metric functions $N$, $A$, $B$, $\omega$ and the scalar function $\Psi$.
Every quantity is then made dimensionless using the free parameters of the theory, which are the scalar velocity $q$, the cosmological constant $\Lambda$, the coupling constants $\zeta$, $\eta$ and $\gamma$, and the event horizon radial coordinate $r_{\mathcal{H}}$ (in quasi-isotropic coordinates, the event horizon is always located at a constant radial coordinate): $$\begin{aligned}
\bar{\Lambda} \equiv \Lambda r_{\mathcal{H}}^{2}, \hspace{0.1\textwidth}
&\bar{\eta} \equiv -q^{2} r_{\mathcal{H}}^{2} \frac{\eta}{\zeta}, \hspace{0.1\textwidth}
&\bar{\gamma} \equiv q^{3} r_{\mathcal{H}} \frac{\gamma}{\zeta}, \\
\bar{r} \equiv \frac{r}{r_{\mathcal{H}}},
&\bar{\omega} \equiv r_{\mathcal{H}} \omega,
&\bar{\Psi} \equiv \frac{\Psi}{q r_{\mathcal{H}}},\end{aligned}$$ and all the functions are manipulated as functions of $\bar{r}$.
Eventually, the four metric equations schematically take the form $$\begin{aligned}
N^{2} \Delta_{3} N = \mathcal{S}_{N}, \label{eq_metric_QI_N} \\
N^{3} \Delta_{2} [NA] = \mathcal{S}_{A}, \label{eq_metric_QI_NA} \\
N^{2} \Delta_{2} [NB\bar{r}\sin\theta] = \mathcal{S}_{B}, \label{eq_metric_QI_NB} \\
N \Delta_{3} [\bar{\omega} \bar{r} \sin \theta] = \mathcal{S}_{\bar{\omega}}, \label{eq_metric_QI_adom}\end{aligned}$$ where the right-hand side terms are explicitly given in \[appdx\_RHS\] and the following notations are used: $$\begin{aligned}
\Delta_{2} = \partial^{2}_{\bar{r}\bar{r}} + \frac{1}{\bar{r}} \partial_{\bar{r}} + \frac{1}{\bar{r}^{2}} \partial^{2}_{\theta\theta}, \\
\Delta_{3} = \partial^{2}_{\bar{r}\bar{r}} + \frac{2}{\bar{r}} \partial_{\bar{r}} + \frac{1}{\bar{r}^{2}} \partial^{2}_{\theta\theta} + \frac{1}{\bar{r}^{2} \tan\theta} \partial_{\theta}, \\
\tilde{\Delta}_{3} = \Delta_{3} - \frac{1}{\bar{r}^{2} \sin^{2}\theta}.\end{aligned}$$
Recall that the cubic Galileon theory features the shift-symmetry $\phi \rightarrow \phi + constant$, meaning that only the first derivatives of $\phi$ are physically meaningful. The numerical approach presented in section \[section\_numeric\] below concretely makes use of this fact: within the numerical code, the scalar field is only manipulated through its first derivatives $\bar{\Psi}' \equiv \partial_{\bar{r}} \bar{\Psi}$ and $\bar{\Psi}_{\theta} \equiv \partial_{\theta} \bar{\Psi}$. More precisely, $\bar{\Psi}'$ and $\bar{\Psi}_{\theta}$ are first introduced as independent functions, just like $N$, $A$, $B$ and $\bar{\omega}$. The fact that these functions actually arise from a common scalar field is then implemented through imposing $\partial_{\theta} \bar{\Psi}' = \partial_{\bar{r}} \bar{\Psi}_{\theta}$ (symmetry of second-derivatives) in addition to the equations $\{(\ref{eq_metric_QI_N})-(\ref{eq_metric_QI_adom}), (\ref{eq_scalar})\}$.
The complete set of equations to solve then is $$\begin{aligned}
N^{2} \Delta_{3} N = \mathcal{S}_{N}, \label{eq_metric_QI_N_bis} \\
N^{3} \Delta_{2} [NA] = \mathcal{S}_{A}, \label{eq_metric_QI_NA_bis} \\
N^{2} \Delta_{2} [NB\bar{r}\sin\theta] = \mathcal{S}_{B}, \label{eq_metric_QI_NB_bis} \\
N \Delta_{3} [\bar{\omega} \bar{r} \sin \theta] = \mathcal{S}_{\bar{\omega}}, \label{eq_metric_QI_adom_bis} \\
\partial_{\theta} \bar{\Psi}' = \partial_{\bar{r}} \bar{\Psi}_{\theta} \label{eq_Schwarz}, \\
\nabla_{\mu} \bar{J}^{\mu} = 0, \label{eq_scalar_bis}\end{aligned}$$ where the explicit expression of the scalar equation (\[eq\_scalar\_bis\]) is also given in \[appdx\_RHS\].
Of course, if a circular black hole exists in the cubic Galileon theory, then it satisfies the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]). But any solution to this system does not necessarily satisfy all the metric equations of motion (\[eq\_metric\]) since only four independent combinations of the latter are solved instead of eight. Hence each numerical solution to (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) was reinjected into the whole set of metric equations (\[eq\_metric\]) to assess the relevance of the circularity hypothesis a posteriori (see section \[section\_accuracy\]).
Boundary conditions {#section_BC}
-------------------
Equations (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) form a system of first (equation (\[eq\_Schwarz\])) and second order coupled partial differential equations (PDE) involving the six functions $N$, $A$, $B$, $\bar{\omega}$, $\bar{\Psi}'$ and $\bar{\Psi}_{\theta}$. It must then be provided with boundary conditions suitable for the search for black hole solutions with nontrivial scalar hair. More precisely, the system is defined on a meridional surface (all of them are equivalent due to circularity) between the intersections of the latter with the black hole event horizon and spacetime infinity. As mentioned in section \[section\_equations\], the event horizon is located at $\bar{r} = 1$, while spacetime infinity corresponds with the limit $\bar{r} \rightarrow \infty$. Boundary conditions must then be prescribed for both limits.
First, in quasi-isotropic coordinates, the function $N$ must vanish on the event horizon (see for instance [@quasi_isotropic_Kerr] for the case of Kerr). This induces an important alteration of the nature of the equations (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_metric\_QI\_adom\_bis\]) since all the second-order operators acting on the metric functions thus cancel at $\bar{r} = 1$. This kind of degeneracy actually reduces the required number of boundary conditions.
The other crucial condition at the horizon is the value of the function $\bar{\omega}$. The weak rigidity theorem states the existence of a constant $\Omega_{\mathcal{H}}$ such that $\xi + \Omega_{\mathcal{H}} \chi$ is (a Killing vector field) normal to the horizon [@Carter_Houches; @heusler_uniqueness_book]. On the horizon, the function $\bar{\omega}$ necessarily equals the constant $\bar{\Omega}_{\mathcal{H}} \equiv r_{\mathcal{H}} \Omega_{\mathcal{H}}$, called the dimensionless angular velocity of the horizon.
Regarding conditions at infinity, the only case considered in this paper is asymptotic flatness. This is a standard hypothesis made to study isolated black holes and establish no-hair theorems [@no_hair_Sotiriou_1]. Therefore, it should lead to hairy black holes that escape the no-hair theorem of [@no_hair_Galileon] in a minimal way. Yet, in the cubic Galileon theory, imposing asymptotic flatness in static and spherical symmetry is incompatible with the assumption $\eta \neq 0$.
To picture this, it is easier to consider the Schwarzschild-like coordinates $(t,R,\theta,\varphi)$ used in [@cubic_BH], with respect to which the static and spherically symmetric line element takes the form $$\begin{aligned}
\label{eq_metric_Schw_like}
ds^{2} = - h(R) dt^{2} + \frac{1}{f(R)} dR^{2} + R^{2} \left( d\theta^{2} + \sin^{2}\theta d\varphi^{2} \right).\end{aligned}$$
Using the scalar ansatz (\[eq\_scalar\_ansatz\_sphe\]), all the relevant equations are the following (the $(tR)$ equation[^6], a combination of the $(tR)$ and $(RR)$ equations, and a combination of the $(tR)$, $(RR)$ and $(tt)$ equations respectively): $$\begin{aligned}
\label{eq_4_1}
\gamma (R^{4}h)' fh \Psi'^{2} - \gamma q^{2} R^{4} h' - 2 \eta R^{4} h^{2} \Psi' = 0, \\
\label{eq_4_2}
\frac{\eta}{2\zeta} (fh\Psi'^{2} - q^{2}) + \frac{fh'}{R} + h \left( \frac{f-1}{R^{2}} + \Lambda \right)= 0, \\
\label{eq_4_3}
f\Psi'^{2} \left[ \eta R^{2} \sqrt{\frac{h}{f}} -\gamma \left( R^{2}\sqrt{fh} \Psi' \right)' \right] = 2\zeta Rh \left( \sqrt{\frac{f}{h}} \right)',\end{aligned}$$ where a prime denotes differentiation with respect to the unique variable $R$.
As mentioned in section \[section\_no\_hair\], one can note that the Schwarzschild-(Anti-)de Sitter metric along with $\Psi' = 0$ and $q = 0$ (i.e. $\phi = constant$) must be a solution to the system (\[eq\_4\_1\])-(\[eq\_4\_3\]) since it is a static and spherically symmetric vacuum solution of general relativity: $$\begin{aligned}
\label{eq_Schw_AdS}
h(R) = f(R) = 1 - \frac{\mu}{R} - \frac{\Lambda}{3} R^2,\end{aligned}$$ where $\mu$ appears as an integration constant
According to [@cubic_BH], injecting asymptotic expansions in powers of $1/R$ for $h$, $f$ and $\Psi$ into (\[eq\_4\_1\])-(\[eq\_4\_3\]) yields the following asymptotic behaviours if $\eta \neq 0$: $$\begin{aligned}
\label{eq_asympt_h}
h(R) = - \frac{\Lambda_{\rm eff}}{3} R^2 + 1 + O\left( \frac{1}{R} \right), \\
\label{eq_asympt_f}
f(R) = - \frac{\Lambda_{\rm eff}}{3} R^2 + c + O\left( \frac{1}{R} \right), \\
\label{eq_asympt_chi}
h(R) \Psi'(R) = \frac{\eta R}{3\gamma} + \frac{c'}{R} + O\left( \frac{1}{R^{2}} \right),\end{aligned}$$ where $c$ and $c'$ are some fixed constants and $\Lambda_{\rm eff}$ is an effective cosmological constant made from a combination of the bare cosmological constant $\Lambda$ and the kinetic coupling $\eta$.
Therefore, if $\Lambda_{\rm eff} \neq 0$, spacetime is asymptotically (anti-)de Sitter. Asymptotic flatness thus requires $\Lambda_{\rm eff} = 0$, which is impossible whenever $\eta \neq 0$ (see the relations (4.10) of [@cubic_BH]).
Then, setting $\eta$ to $0$ in (\[eq\_4\_2\]) yields $$\begin{aligned}
\label{eq_4_2_eta0}
f \left( \frac{h'}{Rh} + \frac{1}{R^{2}} \right) = \frac{1}{R^{2}} - \Lambda,\end{aligned}$$ while asymptotic flatness (i.e. vanishing Riemann tensor when $R \rightarrow \infty$) requires the following asymptotic behaviours: $$\begin{aligned}
\label{eq_asympt_flat_h}
\frac{h'}{h} = o\left( \frac{1}{R} \right), \\
\label{eq_asympt_flat_f}
f \longrightarrow 1,\end{aligned}$$ so that $\Lambda$ must be $0$ as well as $\eta$.
As mentioned in section \[section\_no\_hair\], it is shown in \[appdx\_no\_hair\] that, for the cubic Galileon, the no-hair theorem still holds if $\eta = 0$. Therefore, the asymptotically flat, static, spherically symmetric hairy solutions constructed in [@cubic_BH] with $\eta = \Lambda = 0$ evade the no-hair theorem in a minimal fashion since only the staticity of the scalar field is abandoned.
It is reasonable to think that asymptotic flatness requires vanishing $\eta$ and $\Lambda$ even in the rotating case, although there is no proof of such a claim. Regardless of the actual answer, $\eta$ and $\Lambda$ are set to zero in the numerical work exposed in this paper in order to connect with the solutions of [@cubic_BH] in the non-rotating limit.
Numerical treatment {#section_numeric}
-------------------
The numerical approach to solve the above problem comprises two steps implemented within the library *Kadath* [@Kadath]. First, the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) is discretized within the framework of spectral methods. This amounts to project each function $N$, $A$, $B$, $\bar{\omega}$, $\bar{\Psi}'$ and $\bar{\Psi}_{\theta}$ onto a set of basis functions defined as the products of (Legendre or Chebyshev) polynomials $T_{i}$ with trigonometric functions, e.g. for the function $A$: $$\begin{aligned}
A(r,\theta) = \sum_{i=0}^{m_{r}} \sum_{j=0}^{m_{\theta}} &\tilde{A}_{ij} T_{i}(r) \cos(2j\theta),\end{aligned}$$ where $m_{r}$ and $m_{\theta}$ are integers defining the resolution of the discretization[^7]. All the information about the unknown function $A$ is then encoded into the spectral coefficients $\tilde{A}_{ij}$. Moreover, the projection of any of its partial derivative is also given in terms of these coefficients. Applying this procedure to each unknown function $N$, $A$, $B$, $\bar{\omega}$, $\bar{\Psi}'$ and $\bar{\Psi}_{\theta}$ in the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) transforms the latter into a nonlinear algebraic system $S$, whose unknowns are suitable combinations of the spectral coefficients ensuring regularity conditions [@Kadath].
Secondly, the discretized system $S$ is solved with a Newton-Raphson algorithm. The vector $\tilde{X}$ gathering all the relevant combinations of the spectral coefficients should satisfy $S(\tilde{X}) = 0$. Starting with an initial guess $\tilde{X}^{(0)}$ and denoting $\tilde{X}^{(n)}$ the vector gathering the coefficients at step $n$, $\tilde{X}^{(n+1)}$ is built as the solution to $S(\tilde{X}^{(n)}) + dS_{\tilde{X}^{(n)}} ( \tilde{X}^{(n+1)} - \tilde{X}^{(n)} ) = 0$, which requires inverting the Jacobian matrix $dS_{\tilde{X}^{(n)}}$.
Under appropriate conditions, such an iterative process converges towards the exact solution of $S$. In particular, a good initial guess is an important condition of success. This merely means that the closest to the exact solution the process starts, the more chance it has to converge to the solution (while starting too far away from it induces risks to leave the neighbourhood of the solution after a few iterations and eventually diverge). This is the reason why the existence of the static and spherically symmetric black hole solutions of [@cubic_BH] is particularly useful since reconstructing these solutions numerically (for a fixed choice of the coupling constants) provides ideal initial guesses to reach slowly rotating solutions, which in turn serve as initial guesses to reach slighlty more rapidly rotating solutions and so on.
Finally, let us mention that, in the rotating cases, an additional condition was implemented in order to avoid a conical singularity [@intro_relat_stars]. It consists in imposing $A = B$ on the symmetry axis $\theta = 0, \pi/2$[^8], which guarantees that the metric could be regularly well-defined on an open chart containing the axis. For instance, this condition was also imposed in [@rotating_EdGB_1; @rotating_EdGB_2] to construct rotating black holes in the dilatonic Einstein-Gauss-Bonnet theory, but for rotating bosons stars [@circular_rotating_boson_star], the field equations alone imply $A = B$ on the symmetry axis.
Accuracy of the code {#section_accuracy}
--------------------
As explained in section \[section\_equations\], all the numerical solutions to the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) were reinjected into the whole set of metric equations (\[eq\_metric\]) in order to assess the validity of the code. Writing the metric equations (\[eq\_metric\]) as $E_{\mu\nu} = 0$, the error on each equation corresponds to its maximum spectral coefficent (in absolute value). Six out of the eight nontrivial[^9] metric equations feature a fast decrease of the error as the resolution increases, which confirms that these equations are properly solved numerically. Figure \[fig\_error\_evol\_21\_fixed\_adgamma\] illustrates this fact in the case of equation $E_{r\theta}$ for various angular velocities at fixed coupling $\bar{\gamma} = 1$.
On the other hand, the error on the two metric equations $E_{tr}$ and $E_{t\theta}$ is independent of the resolution, as illustrated on Fig \[fig\_error\_momentum\_1\_fixed\_adgamma\], revealing that there exists an actual violation of non numerical origin. The cause of this violation can be identified a bit more precisely. In quasi-isotropic coordinates, the components $(tr)$ and $(t\theta)$ of both the metric and Ricci tensors are zero. As a result, the metric equations $E_{tr} = E_{t\theta} = 0$ reduce to $T^{(\phi)}_{tr} = T^{(\phi)}_{t\theta} = 0$. Actually, these last two equations coincide with the two nontrivial circularity conditions provided by the generalized Papapetrou theorem (\[eq\_T\_circu\]). The latter can be applied to $T^{(\phi)}$ because the metric equations $E_{\mu\nu} = 0$ have an Einstein-like structure. This yields $$\begin{aligned}
\label{eq_T_circu_radial}
\left( T^{(\phi)}(\partial_{t}) \wedge \partial_{t} \wedge \partial_{\varphi} \right)_{tr\varphi}
= T^{(\phi)}_{t[t} \left( \partial_{t} \right)_{r} \left( \partial_{\varphi} \right)_{\varphi]}
\propto T^{(\phi)}_{tr}, \\
\left( T^{(\phi)}(\partial_{t}) \wedge \partial_{t} \wedge \partial_{\varphi} \right)_{t\theta\varphi}
= T^{(\phi)}_{t[t} \left( \partial_{t} \right)_{\theta} \left( \partial_{\varphi} \right)_{\varphi]}
\propto T^{(\phi)}_{t\theta}.\end{aligned}$$
Therefore the errors on $E_{tr}$ and $E_{t\theta}$ estimate the validity of the circularity hypothesis. More precisely, in the expression (\[eq\_T\_phi\_munu\]) of $T^{(\phi)}_{tr}$ (resp. $T^{(\phi)}_{t\theta}$), the only nontrivial terms are those proportional to $\partial_{t}\phi\partial_{r}\phi$ (resp. $\partial_{t}\phi\partial_{\theta}\phi$) and $\partial_{(t}\phi\partial_{r)}(\partial \phi)^{2}$ (resp. $\partial_{(t}\phi\partial_{\theta)}(\partial \phi)^{2}$) which are nonzero only if $\phi$ depends on both $t$ and $r$ (resp. $t$ and $\theta$). This means that non circularity is caused by combined time and radial, or time and angular, dependences of the scalar field. Yet the ansatz (\[eq\_scalar\_ansatz\_sphe\]) used in [@cubic_BH] to derive static and spherically symmetric solutions does feature both time and radial dependences. But these solutions were obtained taking advantage of the fact that $E_{tR}$ (in the Schwarzschild-like coordinates (\[eq\_metric\_Schw\_like\])) implies the scalar equation. Thus, solving $E_{tR} = 0$ instead of the scalar equation automatically fulfilled the circularity condition (\[eq\_T\_circu\_radial\]) since $E_{tR} \propto T^{(\phi)}_{tR} \propto T^{(\phi)}_{tr}$ (where the last relation holds because the transformation (\[eq\_change\_Schw\_QI\]) from Schwarzschild-like coordinates to quasi-isotropic coordinates relates only the coordinates $R$ and $r$ in spherical symmetry).
But as soon as one looks for rotating solutions and thus adds an angular dependence to all functions, including the scalar field according to the ansatz (\[eq\_scalar\_ansatz\]), the equations are too complex to benefit from a similar simplification. Therefore the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) based on the circular metric (\[eq\_circular\]) and the ansatz (\[eq\_scalar\_ansatz\]) is not exactly self-consistent. Yet, the violation of circularity in the dimensionless setup is less than $10^{-2}$, meaning that it is fairly small with respect to the scale given by the radial coordinate $r_{\mathcal{H}}$ of the event horizon in a dimensional physical configuration. In addition, Fig \[fig\_error\_momentum\_1\_fixed\_adgamma\] expectedly confirms that the violation continuously goes to zero with the angular velocity (since in this limit the solutions are exactly circular), so that it seems reasonable to believe that the solutions presented in the next sections still provide precise approximations to rotating black hole solutions of the cubic Galileon theory.
Black hole solutions {#section_num_sols}
====================
Static and spherically symmetric black holes {#section_stat_sphe}
--------------------------------------------
First, the existing static, spherically symmetric black hole solutions reported in [@cubic_BH] have been reconstructed in the quasi-isotropic gauge (instead of the Schwarzschild-like coordinates used in [@cubic_BH]) in order to later serve as initial guesses to compute rotating solutions. As mentioned in \[section\_no\_hair\], these solutions were obtained in [@cubic_BH] by numerical integration of the ordinary differential equations $(\ref{eq_4_1})-(\ref{eq_4_3})$. In addition, the value of $h'$ was prescribed at the horizon in order to obtain the desired asymptotic behaviour (shooting method). In this paper, these solutions are generated with the numerical treatment presented in section \[section\_numeric\], i.e. as solutions to the PDE system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]). In addition, boundary conditions are prescribed both at infinity and at the horizon; in particular, staticity is imposed by setting the dimensionless angular velocity $\bar{\Omega}_{\mathcal{H}}$ to zero. The resulting numerical solutions feature spherical symmetry ($A = B$, $\bar{\omega} = 0$ everywhere, and no angular dependence) although such symmetry is not imposed anywhere in the numerical process.
As explained in section \[section\_numeric\], the numerical process requires initial guesses. Conveniently, the test-field solution given in [@cubic_BH] (relations (4.12)-(4.13)) provides the very first of them. This configuration merely comes out from solving the scalar equation (\[eq\_scalar\]) on a Schwarzschild background metric with the scalar ansatz (\[eq\_scalar\_ansatz\_sphe\]), which physically amounts to neglecting the back-reaction of the scalar field onto the metric, i.e. taking the limit $\gamma \rightarrow 0$ ($\eta$ being already set to zero).
Actually, only the expression of $\Psi'$ is given for this test-field solution: $$\begin{aligned}
\label{eq_test_field}
\Psi'(R) = \frac{\pm q }{\left( 1 - \frac{R_{H}}{R} \right) \sqrt{\frac{4R}{R_{H}} - 3}},\end{aligned}$$ where $R_{H}$ is the Schwarzschild radius. As stated earlier, this is sufficient because, due to the shift-symmetry of the theory, only the first derivatives of $\phi$ are meaningful (and hence only $\Psi'$ in static and spherical symmetry).
One can see from the action (\[eq\_action\]) that flipping the sign of both $\gamma$ and $\phi$ of a given solution provides another solution to the theory. This fact holds true in the limit $\gamma \rightarrow 0$, which is why equation (\[eq\_test\_field\]) offers two test-field solutions with opposite signs. Moreover, it is thus sufficient to seek solutions for positive $\gamma$ only.
Once the Schwarzschild metric and the test scalar field (\[eq\_test\_field\]) are reexpressed in terms of the quasi-isotropic coordinates, the numerical process may converge to a solution of the system (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_scalar\_bis\]) in which the coupling $\bar{\gamma}$ is set to a slightly nonzero value. In turn, such solution serves as an initial guess to reach a solution with a slightly greater coupling $\bar{\gamma}$ and so on.
The resulting solutions are displayed in Fig. \[fig\_stat\_sphe\]. Due to spherical symmetry, one has $B = A$, $\bar{\omega} = 0$ and $\bar{\Psi}_{\theta} = 0$ everywhere, so that only the radial profiles of $N$, $A$ and $\bar{\Psi}'$ are non trivial. Actually, $Z \equiv N \bar{\Psi}'$ is plotted instead of $\bar{\Psi}'$ because the former is finite on the horizon contrary to the latter.
\
For the function $N$ (Fig. \[fig\_stat\_sphe\_N\]), the boundary values $N = 0$ at the horizon and $N = 1$ at infinity are enforced according to section \[section\_BC\]. On the contrary, the values of $A$ and $Z$ on the horizon are not imposed due to the degeneracy of the equations. Yet, it can be seen from the right-hand side (\[eq\_RHS\_NA\]) of equation (\[eq\_metric\_QI\_NA\_bis\]) that this degeneracy spontaneously imposes $A^{2} = Z^{2}$ on the horizon, which is manifest on figures \[fig\_stat\_sphe\_A\] and \[fig\_stat\_sphe\_Z\] (and confirmed numerically).
One can also note that the greater the coupling $\bar{\gamma}$ is, the faster the funtions $N$, $A$ and $Z$ converge towards their respective limits which correspond to a flat spacetime. Then, when travelling from the horizon towards infinity, spacetime looks flat more rapidly for stronger coupling values $\bar{\gamma}$. In other terms, the more the scalar field back-reacts on the metric, the more it hides the deformations induced by the black hole. This fact is further examined in section \[section\_mass\] below when discussing the extraction of a mass for these black hole solutions.
Rotating black holes {#section_rotating}
--------------------
The Kerr metric is usually parametrized by two parameters $M$ (the mass) and $a$ (the reduced angular momentum). The radial coordinate $r_{\mathcal{H}}$ of the event horizon may then be expressed in terms of these two parameters: $$\begin{aligned}
\label{eq_horizon_QI_radius}
r_{\mathcal{H}} = \frac{M}{2} \sqrt{1 - \left(\frac{a}{M}\right)^{2}}.\end{aligned}$$
Once $r_{\mathcal{H}}$ is used to make all the quantities dimensionless and all the metric components are expressed in terms of $\bar{r} \equiv r/r_{\mathcal{H}}$, the dimensionless Kerr solution is only parametrized by one quantity, which can be chosen to be $\bar{\Omega}_{\mathcal{H}}$. Of course, one such quantity might not be enough to parametrize the whole set of black hole solutions of the cubic Galileon theory with a scalar field structured as (\[eq\_scalar\_ansatz\]). Yet, the numerical approach employed here only reaches the solutions that continuously connect to Schwarzschild, by increasing $\bar{\gamma}$ first and then $\bar{\Omega}_{\mathcal{H}}$. This is why, once $\bar{\gamma}$ is fixed, $\bar{\Omega}_{\mathcal{H}}$ is also the only quantity that parametrizes the solutions presented here.
Figure \[fig\_rot\] displays the radial profiles of all six functions $N$, $A$, $B$, $\bar{\omega}$, $\bar{\Psi}'$ and $\bar{\Psi}_{\theta}$ at fixed $\bar{\gamma} = 1$ for various values of $\bar{\Omega}_{\mathcal{H}}$. For $\bar{\Omega}_{\mathcal{H}} = 0$ and $0.07$, the corresponding dimensionless Kerr solution is plotted for comparison: in the case of $N$, $A$ and $B$, the Kerr curve has the same color and linestyle as the Galileon curve corresponding to the same parameter $\bar{\Omega}_{\mathcal{H}}$, and in the case of $\bar{\omega}$, it is the thick dotted curve having the same value at the horizon with its Galileon analog. As for $Z$ and $\bar{\Psi}_{\theta}$, no Kerr analog is displayed since no test-field solutions are known in the rotating case (i.e. solutions to the scalar equation (\[eq\_scalar\]) on a Kerr background with the scalar ansatz (\[eq\_scalar\_ansatz\])) and such solutions could not be obtained numerically.
\
\
For the other values of $\bar{\Omega}_{\mathcal{H}}$ ($0.12$ and $0.18$), the Galileon solutions displayed in figure \[fig\_rot\] do not admit a Kerr analog. The reason is that the cubic Galileon admits solutions with dimensionless angular velocities greater than the maximum $\bar{\Omega}_{\mathcal{H}}$ that can be obtained from the Kerr metric. More precisely, at fixed mass $M$, the angular velocity $\Omega_{\mathcal{H}}$ of the Kerr black hole cancels at $a/M = 0$ and monotonically increases towards a finite value at $a/M = 1$, while the radial quasi-isotropic coordinate $r_{\mathcal{H}}$ of the event horizon is finite at $a/M = 0$ and monotonically decreases towards $0$ at $a/M = 1$ according to equation (\[eq\_horizon\_QI\_radius\]). Then, $\bar{\Omega}_{\mathcal{H}} = r_{\mathcal{H}} \Omega_{\mathcal{H}}$ is a positive function of the dimensionless ratio $a/M \in [0,1]$ cancelling both at $0$ and $1$: $$\begin{aligned}
\bar{\Omega}_{\mathcal{H}} = \frac{1}{4} \frac{ \frac{a}{M} }{1 + \left(1 - \left(\frac{a}{M}\right)^{2}\right)^{-\frac{1}{2}}},\end{aligned}$$ which is plotted on Fig. \[figure\_adOm\_vs\_asM\]. In particular, this function has a maximum value $\bar{\Omega}_{\mathcal{H},\rm max} \simeq 0.075$ at $a/M \simeq 0.8$, which actually turns out to be possible to exceed in the cubic Galileon theory. This will appear clearly in section \[section\_phys\] when extracting the angular momentum and the surface gravity of these black hole solutions.
![$\bar{\Omega}_{\mathcal{H}}$ with respect to $a/M$ for Kerr black holes.[]{data-label="figure_adOm_vs_asM"}](adOm_vs_asM.png){width="50.00000%"}
Going back to figure \[fig\_rot\], one first notes that, although the global behaviours are the same, there are non negligible gaps near the horizon between the Galileon solution and Kerr for any fixed dimensionless angular velocity. Naturally, for both the Galileon and Kerr, increasing $\bar{\Omega}_{\mathcal{H}}$ tends to slow the convergence towards the asymptotic values (at fixed radial coordinate, it is expected that spacetime looks less flat if the hole is rotating). One last remark to make is that, although these solutions feature quite rapid rotation ($\bar{\Omega}_{\mathcal{H}} = 0.07$ corresponds to $a/M \simeq 0.65$ for Kerr), the angular variations of the various functions are quite moderate for both the Galileon and Kerr; this is manifest on Fig. \[fig\_rot\_A\_angular\_horizon\] which displays the angular profile of the function $A$ on the horizon.
![Angular profile of $A$ at the horizon.[]{data-label="fig_rot_A_angular_horizon"}](A_fixed_adgamma_1_increase_adOm_angular_horizon_spectral.png){width="50.00000%"}
Physical properties {#section_phys}
===================
Mass {#section_mass}
----
The general definition of the Komar mass of an asymptotically flat stationary spacetime, equipped with a foliation $\left( \Sigma_{t} \right)_{t \in \mathbb{R}}$ by spacelike hypersurfaces, is [@3p1_formalism; @intro_relat_stars] $$\begin{aligned}
\label{eq_Komar_mass_general}
M_{\rm Komar} \equiv - \frac{1}{8\pi} \int_{\mathcal{S}} \ast d\xi,\end{aligned}$$ where $\xi$ is the stationary Killing vector here identified with its metric dual form, $\ast$ is the Hodge star and $\mathcal{S} \subset \Sigma_{t_{0}}$ (for some $t_{0} \in \mathbb{R}$) is a closed spacelike 2-surface containing the intersection of $\Sigma_{t_{0}}$ with the support of the energy-momentum tensor. In general relativity, the Einstein equations guarantee that the Komar mass does not depend on the choice of such 2-surface $\mathcal{S}$.
In practice, one then usually uses a 2-surface $\mathcal{S}$ lying at spatial infinity. In particular, in quasi-isotropic coordinates, the Komar mass may be computed from the following integral: $$\begin{aligned}
\label{eq_Komar_mass_QI}
M_{\rm Komar} = \frac{1}{2} \lim_{r \rightarrow \infty} \int_{0}^{\pi} \partial_{r}N\ r^{2} \sin\theta d\theta.\end{aligned}$$
Therefore, if $N$ has the following asymptotic behaviour: $$\begin{aligned}
\label{eq_N_asympt}
N = 1 + \frac{N_{1}}{r} + o\left( \frac{1}{r} \right),\end{aligned}$$ where $N_{1}$ is a constant, then $$\begin{aligned}
\label{eq_Komar_mass_N}
M_{\rm Komar} = - N_{1}.\end{aligned}$$
In the cubic Galileon theory, the contribution from the scalar field into equation (\[eq\_metric\]) does not allow to guarantee that the expression (\[eq\_Komar\_mass\_general\]) is independent of the 2-surface $\mathcal{S}$. Yet, as is usually done, one may try to extract a mass from the relation (\[eq\_Komar\_mass\_N\]). This can be done explicitly in the static and spherically symmetric case.
To do so, it is simpler to first switch back to the Schwarzschild-like coordinates (\[eq\_metric\_Schw\_like\]) used in section \[section\_BC\] to extract the asymptotic behaviours (\[eq\_asympt\_h\])-(\[eq\_asympt\_chi\]) when $\eta \neq 0$. Repeating the same procedure in the case of asymptotic flatness, i.e. injecting expansions in $1/R$ into (\[eq\_4\_1\])-(\[eq\_4\_3\]) with $\eta = \Lambda = 0$, one finds the following asymptotic behaviours: $$\begin{aligned}
\label{eq_asympt_chi_flat}
h(R) \Psi'(R) = \frac{d}{R^{2}} + O\left( \frac{1}{R^{5}} \right), \\
\label{eq_asympt_h_flat}
h(R) = 1 - \frac{d^{2}}{q^{2} R^{4}} + O\left( \frac{1}{R^{7}} \right), \\
\label{eq_asympt_f_flat}
f(R) = 1 - \frac{4 d^{2}}{q^{2} R^{4}} + O\left( \frac{1}{R^{7}} \right),\end{aligned}$$ where $d$ is some fixed constant. Note here that the test field approximation (\[eq\_test\_field\]) gives a wrong indication about the asymptotic behavior of $\Psi'(R)$ since it behaves as $1/\sqrt{r}$ although, according to equation (\[eq\_asympt\_chi\_flat\]), it behaves as $1/r^{2}$ as soon as the coupling $\gamma$ is nonzero, no matter how small. Yet this did not prevent the test-field solution from being useful as an initial guess in the numerical procedure.
Now, the change of coordinates from the Schwarzschild-like coordinates $(t,R,\theta,\varphi)$ to the quasi-isotropic coordinates $(t,r,\theta,\varphi)$ is merely given by the positive function $R(r)$ defined on $[r_{\mathcal{H}}, +\infty)$ such that $$\begin{aligned}
\label{eq_change_Schw_QI}
r R'(r) = R(r) \sqrt{f\left( R(r) \right)}.\end{aligned}$$
From this, one can infer the same types of asymptotic behaviours as (\[eq\_asympt\_chi\_flat\])-(\[eq\_asympt\_f\_flat\]) for the functions $Z$, $N$ and $A$: $$\begin{aligned}
\label{eq_asympt_Z_flat}
Z(r) = \frac{e}{r^{2}} + o\left( \frac{1}{r^{2}} \right), \\
\label{eq_asympt_N_flat}
N(r) = 1 + \frac{e'}{r^{4}} + o\left( \frac{1}{r^{4}} \right), \\
\label{eq_asympt_A_flat}
A(r) = 1 + \frac{e''}{r^{4}} + o\left( \frac{1}{r^{4}} \right),\end{aligned}$$ where $e$, $e'$ and $e''$ are some fixed constants.
One concludes that there is no term to the first inverse power of $r$ in the expansion (\[eq\_asympt\_N\_flat\]) of $N$, meaning that the Komar mass is zero according to the relation (\[eq\_Komar\_mass\_N\]). This fact may be checked numerically by extracting the asymptotic slope of $1-N$ in a log-log graph (Fig. \[fig\_stat\_1-N\_loglog\]), which corresponds to the asymptotically dominant power of $r$; the resulting numerical value is perfectly consistent with $-4$. The function $A-1$ does have a very similar log-log graph, and one may check on Fig. \[fig\_stat\_Z\_loglog\] that, for the function $Z$, the asymptotic slope is numerically consistent with $-2$.
Such asymptotic behaviours seem to be maintained in the rotating case although the dominant power for $N$ might not be exactly $-4$, but still smaller than $-3.5$, hence no mass term can be extracted either. One is thus led to conclude that the presence of a scalar field with structure (\[eq\_scalar\_ansatz\]) in the cubic Galileon theory generically hides the mass of an asymptotically flat black hole from infinity. Note that this could not be the case whenever asymptotic flatness is abandoned, i.e. nonzero $\Lambda$ and/or $\eta$, since the asymptotic expansions (4.17) of [@cubic_BH] require a standard mass term from the first inverse power of $r$.
Angular momentum {#section_angular_momentum}
----------------
Similarly to the definition (\[eq\_Komar\_mass\_general\]), the Komar angular momentum of an asymptotically flat axisymmetric spacetime is defined as $$\begin{aligned}
\label{eq_Komar_angul_general}
J_{\rm Komar} \equiv \frac{1}{16\pi} \int_{\mathcal{S}} \ast d\chi,\end{aligned}$$ where $\xi$ is the axisymmetric Killing vector.
Using the quasi-isotropic coordinates, the definition (\[eq\_Komar\_angul\_general\]) reexpresses as $$\begin{aligned}
\label{eq_Komar_angul_QI}
J_{\rm Komar} = - \frac{1}{8} \lim_{r \rightarrow \infty} \int_{0}^{\pi} \partial_{r}\omega\ r^{4} \sin^{3}\theta d\theta.\end{aligned}$$
Therefore, if $\omega$ has the following asymptotic behaviour: $$\begin{aligned}
\label{eq_omega_asympt}
\omega = \frac{\omega_{1}}{r^{3}} + o\left( \frac{1}{r^{3}} \right),\end{aligned}$$ where $\omega_{1}$ is a constant, then $$\begin{aligned}
\label{eq_Komar_omega}
J_{\rm Komar} = \frac{\omega_{1}}{2}.\end{aligned}$$
Again, one may try to extract a Komar angular momentum from the asymptotic expansion of $\omega$ although, in the cubic Galileon theory, such a value would have no reason to be common to all other 2-surfaces $\mathcal{S}$. Figure \[fig\_rot\_adomega\_loglog\] confirms that $\bar{\omega}$ has the asymptotic behaviour (\[eq\_omega\_asympt\]) (asymptotic slope equal to $-3$) so that the Komar angular momentum is nonzero.
Since only dimensionless quantities are processed numerically, one has $$\begin{aligned}
\label{eq_adomega_asympt}
\bar{\omega} \equiv r_{\mathcal{H}} \omega \sim \frac{2 \bar{J}_{\rm Komar}}{\bar{r}^{3}},\end{aligned}$$ where $\bar{J}$ is the dimensionless Komar angular momentum: $$\begin{aligned}
\label{eq_adJ}
\bar{J}_{\rm Komar} = \frac{J_{\rm Komar}}{r_{\mathcal{H}}^{2}}.\end{aligned}$$
The values of $\bar{J}_{\rm Komar}$ extracted for all the $\bar{\Omega}_{\mathcal{H}}$ that were reached for $\bar{\gamma} = 10^{-2}$ and $1$ are marked in figure \[fig\_adOm\_vs\_adJ\]. The relation between $\bar{\Omega}_{\mathcal{H}}$ and $\bar{J}_{\rm Komar}$ can be expressed explicitly in the case of the Kerr family, and it is represented by the solid red curve to highlight the deviations from GR.
As mentioned in section \[section\_rotating\], $r_{\mathcal{H}}$ tends to zero for the extremal Kerr solutions while $J_{\rm Komar}$ tends to the finite value $M^{2}$. Therefore $\bar{\Omega}_{\mathcal{H}}$ goes to zero while $\bar{J}_{\rm Komar}$ diverges according to the relation (\[eq\_adJ\]). This is why the curve corresponding to Kerr in Fig. \[fig\_adOm\_vs\_adJ\] is defined all over $\mathbb{R}_{+}$ and converges to zero at infinity. Since $\bar{\Omega}_{\mathcal{H}} = 0$ for $\bar{J}_{\rm Komar} = 0$ and $\bar{\Omega}_{\mathcal{H}}$ is positive, it must also have a maximum which is reached for $\bar{J}_{\rm Komar} \simeq 8$ according to Fig. \[fig\_adOm\_vs\_adJ\]. One can see that some cubic Galileon solutions exceed this maximum value, which clearly shows why it was not possible to provide a Kerr analog for the metric functions in Fig. \[fig\_rot\] for $\bar{\gamma} = 0.12$ and $0.18$.
Yet the existence of a maximum value for $\bar{\Omega}_{\mathcal{H}}$ in the Kerr case reveals that this quantity does not provide a bijective parametrization of the families of dimensionless black hole solutions. This represents a numerical difficulty: the solutions are gradually constructed by increasing the parameter $\bar{\Omega}_{\mathcal{H}}$ starting from the static and spherically symmetric solution $(\bar{\Omega}_{\mathcal{H}},\bar{J}_{\rm Komar}) = (0,0)$ (left part of the curve, i.e. located before the maximum). The algorithm no longer converges when the maximum value is reached. From then on, $\bar{\Omega}_{\mathcal{H}}$ should be lowered to explore more and more rapidly rotating solutions (right part of the curve). But numerically, using the “maximum” solution as initial guess to reach a solution with a smaller value of $\bar{\Omega}_{\mathcal{H}}$ will actually yield the less rapidly rotating solution (i.e. going backward on the left part of the curve) rather than the more rapidly rotating solution that has the same dimensionless angular velocity $\bar{\Omega}_{\mathcal{H}}$ but located to the right of the maximum.
Finding a way to “jump” over the maximum in order to explore the right part of the curve is a nontrivial issue: one must use another quantity, easily handled numerically, which does parametrize the black hole solutions in a bijective way at least in a neighborhood of the maximum, unlike $\bar{\Omega}_{\mathcal{H}}$. Attempts using the dimensionless surface gravity (discussed in the following section) and other parameters fulfilling this condition were unsuccessful so far. This is why the highest points marked on Fig. \[fig\_adOm\_vs\_adJ\] for $\bar{\gamma} = 0.12$ and $0.18$ represent the last solutions that could be reached, beyond which the numerical algorithm does not converge anymore, revealing the proximity of a maximum value.
Surface gravity {#section_surface_gravity}
---------------
In a circular spacetime, the zeroth law of black hole mechanics holds [@heusler_uniqueness_book; @Carter_Houches], i.e. the surface gravity is homogeneous on the horizons of stationary black holes. To check this for the solutions presented here, the dimensionless quantity $\bar{\kappa}$ corresponding to surface gravity $\kappa$ was extracted according to the following formula: $$\begin{aligned}
\label{eq_adkappa}
\bar{\kappa} \equiv r_{\mathcal{H}} \kappa = \frac{1}{A} \partial_{\bar{r}}N_{|_{1}}.\end{aligned}$$
In all solutions, the relative variations of $\bar{\kappa}$ on the horizon are smaller than $10^{-6}$, confirming that the surface gravity is numerically homogeneous on the horizon.
The relation between $\bar{\kappa}$ and $\bar{\Omega}_{\mathcal{H}}$ is represented on Fig. \[fig\_adOm\_vs\_adKappa\]. For each $\bar{\gamma}$, the static and spherically symmetric case corresponds to the only point such that $\bar{\Omega}_{\mathcal{H}} = 0$ but $\bar{\kappa} \neq 0$, while the origin of the graph, i.e. $(\bar{\Omega}_{\mathcal{H}},\bar{\kappa}) = (0,0)$ corresponds to extremal cases. The explicit case of Kerr is again represented by a solid red curve for comparison with GR.
![Angular velocity with respect to surface gravity.[]{data-label="fig_adOm_vs_adKappa"}](adOm_vs_adKappa.png){width="50.00000%"}
Ergoregion {#section_ergoregion}
----------
Locating the ergoregions of the rotating solutions provides another evidence of deviations from GR. Figure \[fig\_rot\_ergo\_increase\_adOm\] displays the ergoregions corresponding to various angular velocities $\bar{\Omega}_{\mathcal{H}}$ at fixed coupling $\bar{\gamma} = 1$ and Fig. \[fig\_rot\_ergo\_increase\_adOm\_vs\_Kerr\] compares two of them with Kerr (same color meaning same angular velocity). On both figures, the ergoregions are plotted in terms of Cartesian-like coordinates yet based on the quasi-isotropic coordinates: $(\bar{x},\bar{z}) = (\bar{r}\sin\theta, \bar{r}\cos\theta)$. This explains the irregularities observed at the poles even in the case of Kerr, although none is observed in the familiar Boyer-Lindquist coordinates: the change of coordinates from Boyer-Lindquist to quasi-isotropic coordinates is not regular at the poles.
The ergoregions of the cubic Galileon solutions generically have the same shape as Kerr: they coincide with the horizon at the poles and get thicker towards the equator. They grow as $\bar{\Omega}_{\mathcal{H}}$ increases, yet they are thinner than Kerr for a given angular velocity.
Conclusions
===========
Numerical configurations describing asymptotically flat hairy rotating black holes in the cubic Galileon theory have been presented. They are based on a scalar ansatz involving a linear time-dependence and a circular approximation of the metric. To realize asymptotic flatness, these Galileon solutions correspond to the special case of vanishing bare cosmological constant and kinetic coupling; they are thus dominated by the DGP term $(\partial \phi)^{2} \Box\phi$. The remaining coupling $\gamma$ induces significant deviations from the Kerr metric on different physical quantities such as surface gravity and angular momentum. In addition, these asymptotically flat solutions feature convergence towards Minkowski faster than Schwarzschild, which can be understood as a vanishing Komar mass at infinity.
Extreme angular velocities (and possibly extremal cases) were not reached yet but could be handled in future work, along with the search for asymptotically (anti-)de Sitter solutions (meaning nonzero $\Lambda$ and/or $\eta$) and the integration of the null and timelike geodesics around such black holes. The key to approach the first problem would be to find an initial guess for rapid rotation. One possible approach would be to take a Kerr background with a scalar stemming from the geodesic analogy similar to [@Charmousis:2019vnf]. Investigation on the latter point would allow to determine whether closed orbits are possible (and up to what distance to the black hole) in spite of the non-Schwarzschild asymptotics. Integration of the null geodesics would simulate the astrophysical imaging of an emitting accretion torus surrounding the Galileon black holes, to be compared with results obtained for other types of compact objects [@image_Kerr_scalar_hair; @image_boson; @image_regular_BH]. Such investigations now have a clear astrophysical relevance in regards of the observations from GRAVITY [@GRAVITY_redshift_S2; @GRAVITY_motion_ISCO] and the Event Horizon Telescope [@EHT_Shape_SgrA; @EHT_Shadow_M87] and we hope to be reporting on these issues in the near future.
No-scalar-hair theorem for the cubic Galileon {#appdx_no_hair}
=============================================
A static and spherically symmetric spacetime admits coordinates $(t,R,\theta,\varphi)$ with respect to which the metric can be written as (\[eq\_metric\_Schw\_like\]).
If the Galileon field features the same symmetries, it only depends on the radial coordinate $R$, and the $(tR)$ metric equation (i.e. equation (\[eq\_4\_1\]) in which $q$ is set to $0$) reads $$\begin{aligned}
\label{eq_tR_no_q}
\phi' \left[ f \phi' \left( \frac{h'}{h} + \frac{4}{R} \right) - \frac{2\eta}{\gamma} \right] = 0.\end{aligned}$$
The general no-hair theorem [@no_hair_Galileon; @slowly_rotating_no_hair] assumes the Galileon Lagrangian to contain a standard kinetic term, i.e. $\eta \neq 0$. Yet, for the cubic Galileon, the case $\eta = 0$ can be included in the theorem, or yields a nontrivial hairy solution if asymptotic flatness is abandoned (see below).
#### Case $\eta \neq 0$
The metric equation (\[eq\_4\_2\]) in which $q$ is set to $0$ gives $$\begin{aligned}
\phi'^{2} = -\frac{2\zeta}{\eta} \left[ \frac{h'}{Rh} + \frac{1}{f} \left( \frac{f-1}{R^{2}} + \Lambda \right) \right].\end{aligned}$$
Then, the asymptotic flatness requirements (\[eq\_asympt\_flat\_h\])-(\[eq\_asympt\_flat\_f\]) imply that $\phi'^{2} \longrightarrow - 2 \zeta \Lambda/\eta$. In particular, $\phi'$ is bounded at infinity, so that $$\begin{aligned}
f \phi' \left( \frac{h'}{h} + \frac{4}{R} \right) \longrightarrow 0.\end{aligned}$$
If the latter term was nonzero at some point, its absolute value would get smaller than e.g. $\eta/\gamma$ at some other point while remaining strictly positive, which would require $\phi' \neq 0$. This would contradict (\[eq\_tR\_no\_q\]), in which one could simplify the overall factor $\phi'$ while having no chance for $f \phi' (h'/h + 4/R) = 2\eta/\gamma$ to hold.
Therefore, $f \phi' (h'/h + 4/R)$ must vanish everywhere and equation (\[eq\_tR\_no\_q\]) finally implies that $\phi$ is trivial (up to a meaningless constant shift).
#### Case $\eta = 0$
A hairy solution would feature nonzero $\phi'$ on some interval $I$, which can be assumed to either extend to infinity, or to be such that $\phi'$ is zero beyond some upper bound. According to (\[eq\_tR\_no\_q\]) with $\eta = 0$, one would have $$\begin{aligned}
\label{eq_h_q0_eta0}
h = \frac{h_{1}}{R^{4}} \text{ over } I,\end{aligned}$$ where $h_{1}$ is an integration constant (whose sign must be the same as $f$ on $I$ for the metric to be Lorentzian). Yet the expression (\[eq\_h\_q0\_eta0\]) does not meet with the asymptotic behaviour (\[eq\_asympt\_flat\_h\]) so that $I$ cannot extend to infinity. This can also be seen from the metric equation (\[eq\_4\_2\]) in which $\eta$ is set to $0$: $$\begin{aligned}
\label{eq_f_q0_eta0_Lambda0}
f = \left( \frac{1}{R^{2}} - \Lambda \right) \left( \frac{h'}{Rh} + \frac{1}{R^{2}} \right)^{-1} = \frac{\Lambda R^{2} - 1}{3} \text{ over } I,\end{aligned}$$ which does not meet with the asymptotic behaviour (\[eq\_asympt\_flat\_f\]) either.
Therefore $\phi'$ should vanish at some point $R_{0}$ and remain zero up to infinity; whether this is possible to realize in a smooth way or not relies on the equation (\[eq\_scalar\_eta0\_nonflat\]) which provides the expression of $\phi'$ on $I$. But anyway, beyond $R_{0}$, $h$ and $f$ would become Schwarzschild, with no chance to match (\[eq\_h\_q0\_eta0\]) and (\[eq\_f\_q0\_eta0\_Lambda0\]) at $R_{0}$ in a smooth way: $$\begin{aligned}
h = f = 1 - \frac{4R_{0}}{3R},\ R \geq R_{0},\end{aligned}$$ so that only the Schwarzschild behaviour outside the event horizon located at $R = 4R_{0}/3$ and a trivial Galileon remain meaningful.
#### Solutions with $\eta = 0$
Abandoning asymptotic flatness allows us to use the expressions (\[eq\_h\_q0\_eta0\]) and (\[eq\_f\_q0\_eta0\_Lambda0\]) everywhere up to infinity, and thus inject them into equation (\[eq\_4\_3\]) in which $q$ and $\eta$ are set to $0$. The resulting equation takes the form $$\begin{aligned}
\left[ \left( R^{2} \sqrt{fh} \phi' \right)^{3} \right]' =
\frac{2\zeta}{\gamma} G_{\Lambda}',\end{aligned}$$ where $$\begin{aligned}
G_{\Lambda}' = \left( \frac{2}{R^{2}} - 3 \Lambda \right) \sqrt{ \frac{3 h_{1}^{3}}{\Lambda R^{2} - 1} },\end{aligned}$$ which integrates into
[1.8]{} $$\begin{aligned}
\hspace{-2.5cm}
G_{\Lambda} \hspace{-1mm} = \hspace{-1mm}
\left\{ \hspace{-2mm}
\begin{array}{l}
\sqrt{3 h_{1}^{3}} \left[ \frac{2\sqrt{\Lambda R^{2} - 1}}{R} - 3 \sqrt{\Lambda} \text{arcosh}\left( \sqrt{\Lambda} R \right) \right] \text{ if~$\Lambda > 0$, $R > \frac{1}{\sqrt{\Lambda}}$ and hence~$h_{1} > 0$}, \\
- \sqrt{3 \vert h_{1} \vert^{3}} \left[ \frac{2\sqrt{ 1-\Lambda R^{2} }}{R} + 3 \sqrt{\Lambda} \text{arcsin}\left( \sqrt{\Lambda} R \right) \right] \text{ if~$\Lambda > 0$, $R < \frac{1}{\sqrt{\Lambda}}$ and hence~$h_{1} < 0$}, \\
\sqrt{3 \vert h_{1} \vert^{3}} \left[ - \frac{2\sqrt{ 1-\Lambda R^{2} }}{R} + 3 \sqrt{\vert\Lambda\vert} \text{arsinh}\left( \sqrt{\vert\Lambda\vert} R \right) \right] \text{ if~$\Lambda \leq 0$ and hence~$h_{1} < 0$}.
\end{array}
\right.\end{aligned}$$
In any case, one finally has $$\begin{aligned}
\label{eq_scalar_eta0_nonflat}
\phi' = \sqrt{ \frac{3}{h_{1} \left( \Lambda R^{2} - 1 \right) } } \left( \frac{2\gamma}{\zeta} G_{\Lambda} + \alpha \right)^{1/3},\end{aligned}$$ where $\alpha$ is an integration constant.
If $\Lambda \leq 0$, then $t$ is a spacelike coordinate and $R$ is timelike, so that the expressions (\[eq\_h\_q0\_eta0\]), (\[eq\_f\_q0\_eta0\_Lambda0\]) and (\[eq\_scalar\_eta0\_nonflat\]) describe a time-dependent metric and a homogeneous, time-dependent scalar field. It is also the case if $\Lambda > 0$ and $R < 1/\sqrt{\Lambda}$, so that the time coordinate $R$ is bounded, like the interior Schwarzschild solution. Finally, if $\Lambda > 0$, the expressions (\[eq\_h\_q0\_eta0\]), (\[eq\_f\_q0\_eta0\_Lambda0\]) and (\[eq\_scalar\_eta0\_nonflat\]) describe the exterior domain of a hairy black hole spacetime with an event horizon located at $R = 1/\sqrt{\Lambda}$. Asymptotically, $\phi'$ converges to zero as $\ln(R)/R$.
Source terms and scalar equation {#appdx_RHS}
================================
The explicit expressions of the source terms and the scalar equation exposed below are justified in a Jupyter notebook based on the free software SageMath[^10]. The notebook is available at the following url: <https://share.cocalc.com/share/6cfa5f27-1564-4bd8-9b0c-fcb3c7d0f325/2019-09-29-155358/metric_and_scalar_equations_cubic_Galileon.ipynb?viewer=share>.
In the explicit expressions, the following notations are used for any functions $f$, $g$ and $h$ of $\bar{r}$ and $\theta$: $$\begin{aligned}
\partial f \partial g = \partial_{\bar{r}}f \, \partial_{\bar{r}}g + \frac{1}{\bar{r}^{2}} \partial_{\theta}f \, \partial_{\theta}g,
\\
\mathcal{H}^{(0)}_{f}[g,h] =
\left(
\begin{array}{c}
\partial_{\bar{r}}g \\
\frac{1}{\bar{r}} \partial_{\theta}g
\end{array}
\right)^{T}
\left(
\begin{array}{l r}
\partial^{2}_{\bar{r}\bar{r}}f & \frac{1}{\bar{r}} \partial^{2}_{\bar{r}\theta}f \\
\frac{1}{\bar{r}} \partial^{2}_{\bar{r}\theta}f & \frac{1}{\bar{r}^{2}} \partial^{2}_{\theta\theta}f
\end{array}
\right)
\left(
\begin{array}{c}
\partial_{\bar{r}}h \\
\frac{1}{\bar{r}} \partial_{\theta}h
\end{array}
\right),
\\
\mathcal{H}^{(1)}_{f}[g,h] =
\left(
\begin{array}{c}
\frac{1}{\bar{r}} \partial_{\theta}g \\
- \partial_{\bar{r}}g
\end{array}
\right)^{T}
\left(
\begin{array}{l r}
\partial^{2}_{\bar{r}\bar{r}}f & \frac{1}{\bar{r}} \partial^{2}_{\bar{r}\theta}f \\
\frac{1}{\bar{r}} \partial^{2}_{\bar{r}\theta}f & \frac{1}{\bar{r}^{2}} \partial^{2}_{\theta\theta}f
\end{array}
\right)
\left(
\begin{array}{c}
\frac{1}{\bar{r}} \partial_{\theta}h \\
- \partial_{\bar{r}}h
\end{array}
\right),
\\
\mathcal{H}^{(2)}_{f}[g,h] =
\left(
\begin{array}{c}
\partial_{\bar{r}}g \\
\frac{1}{\bar{r}} \partial_{\theta}g
\end{array}
\right)^{T}
\left(
\begin{array}{l r}
\partial^{2}_{\bar{r}\bar{r}}f & \frac{2}{\bar{r}} \partial^{2}_{\bar{r}\theta}f \\
\frac{2}{\bar{r}} \partial^{2}_{\bar{r}\theta}f & \frac{1}{\bar{r}^{2}} \partial^{2}_{\theta\theta}f
\end{array}
\right)
\left(
\begin{array}{c}
\partial_{\bar{r}}h \\
\frac{1}{\bar{r}} \partial_{\theta}h
\end{array}
\right).\end{aligned}$$
Then, the right-hand side terms of equations (\[eq\_metric\_QI\_N\_bis\])-(\[eq\_metric\_QI\_adom\_bis\]) read
$$\begin{aligned}
\label{eq_RHS_N}
\mathcal{S}_{N} =
&\frac{N \left( B\bar{r}\sin\theta \right)^{2}}{2} \partial\bar{\omega} \partial\bar{\omega}
- \frac{N^{2}}{B} \partial N \partial B
- N A^{2} \left( \bar{\eta} + \bar{\Lambda} N^{2} \right) \nonumber\\
&- \frac{\bar{\gamma}}{2} \left(1 + \frac{N^{2} \partial\bar{\Psi}\partial\bar{\Psi}}{A^{2}} \right)
\left( N \Delta_{3}\bar{\Psi} + \partial\bar{\Psi}\partial N + \frac{N \partial\bar{\Psi}\partial B}{B} \right),\end{aligned}$$
$$\begin{aligned}
\label{eq_RHS_NA}
\mathcal{S}_{A} =
&\frac{N^4}{A}\partial A \partial A
+ 2 N^3 \partial A \partial N
+ \frac{3 A (N B\bar{r}\sin\theta)^{2}}{4} \partial\bar{\omega} \partial\bar{\omega} \nonumber \\
& + \frac{\bar{\eta} N^{2} A}{2} \left( N^{2} \partial\bar{\Psi}\partial\bar{\Psi} - A^{2} \right)
- \bar{\Lambda} A^{3} N^{4} \nonumber \\
&- \bar{\gamma} \left(
N A \partial\bar{\Psi}\partial N
- \frac{N^{4} \partial\bar{\Psi}\partial\bar{\Psi} \ \partial\bar{\Psi}\partial A}{A^{2}} \right. \nonumber \\
& \hspace{1.2cm} \left. + \frac{1}{A} \left[
N^{4}\mathcal{H}^{(0)}_{\bar{\Psi}}[\bar{\Psi},\bar{\Psi}]
- \frac{N^{4} \partial_{r}\bar{\Psi}}{\bar{r}^{3}} \left(\partial_{\theta}\bar{\Psi}\right)^{2}
\right]
\right),\end{aligned}$$
$$\begin{aligned}
\label{eq_RHS_NB}
\mathcal{S}_{B} =
&- B \bar{r} \sin\theta \left[ N A^{2} \left( \bar{\eta} + 2 \bar{\Lambda} N^{2} \right) \right. \nonumber \\ %] to close \left opening square bracket not to spoil coloration
& \hspace{2.2cm} + \left. \frac{\bar{\gamma} N^2 \partial\bar{\Psi}\partial\bar{\Psi}}{A^{2}}
\left( N \Delta_{3}\bar{\Psi} + \partial\bar{\Psi}\partial N + \frac{N \partial\bar{\Psi}\partial B}{B} \right) \right],\end{aligned}$$
$$\begin{aligned}
\label{eq_RHS_adom}
\mathcal{S}_{\bar{\omega}} =
\frac{N \bar{\omega}}{\bar{r} \sin\theta}
+ \bar{r} \sin\theta \left( \partial\bar{\omega} \partial N
- \frac{3 N}{B} \partial \bar{\omega} \partial B \right)\end{aligned}$$
and the scalar equation takes the form $$\begin{aligned}
\hspace{-2.5cm} 0 =
& \hspace{-1.8cm} - \bar{\eta} N^{3} A^{2} \left( N\Delta_{3}\bar{\Psi} + \partial\bar{\Psi} \partial N + \frac{N \partial\bar{\Psi} \partial B}{B} \right) \nonumber\\
& \hspace{-1.8cm} + \bar{\gamma} \left\{ A^{2} \left( N\Delta_{3}N + \frac{N}{B} \partial N \partial B - 2 \partial N \partial N \right) \right. \nonumber\\
& \hspace{-0.7cm} + 2 N \left( N\Delta_{3}\bar{\Psi} + \partial\bar{\Psi} \partial N + \frac{N \partial\bar{\Psi} \partial B}{B} \right)
\left( \frac{N^{2} \partial\bar{\Psi}\partial A}{A} - N \partial\bar{\Psi}\partial N \right) \nonumber\\
& \hspace{-0.7cm} - 2 \left( N^{2} \Delta_{2}\bar{\Psi} + \frac{N^{2} \partial\bar{\Psi} \partial B}{B} \right)
\left( N^{2} \Delta_{3}\bar{\Psi} - \frac{N^{2} \partial_{\bar{r}}\bar{\Psi}}{\bar{r}} \right) \nonumber\\
& \hspace{-0.7cm} + \frac{2 N^{2}}{\bar{r}^{2}} \partial^{2}_{\theta\theta}\bar{\Psi} \left( N^{2} \Delta_{2}\bar{\Psi} - \frac{N^{2} \partial\bar{\Psi} \partial A}{A} \right) \nonumber\\
& \hspace{-0.7cm} - N^{3} \partial\bar{\Psi} \partial\bar{\Psi}
\left(\frac{N}{A} \left[ \Delta_{3}A - \frac{4}{\bar{r}}\partial_{\bar{r}}A \right]
+ \frac{N}{B} \Delta_{2}B
+ \frac{\partial N \partial A}{A}
- \frac{3N}{A^{2}} \partial A \partial A
+ \frac{N}{AB} \partial A \partial B
\right) \nonumber\\
& \hspace{-0.7cm} + 2 \left( N \partial\bar{\Psi} \partial N \right)^{2} \nonumber\\
& \hspace{-0.7cm} - N^{3} \, \mathcal{H}^{(0)}_{N}[\bar{\Psi},\bar{\Psi}] \nonumber\\
& \hspace{-0.7cm} + \frac{N^{4} \mathcal{H}^{(1)}_{B}[\bar{\Psi},\bar{\Psi}]}{B} \nonumber\\
& \hspace{-0.7cm} - \frac{2N^{4} \mathcal{H}^{(2)}_{\bar{\Psi}}[\bar{\Psi},A]}{A} \nonumber\\
& \hspace{-0.7cm} + 2 \left( \left[ N^{2} \partial^{2}_{\bar{r}\bar{r}}\bar{\Psi} \right]^{2} +
\left[ \frac{N^{2}}{\bar{r}^{2}}\partial_{\theta}\bar{\Psi} - \frac{N^{2} \partial_{\bar{r}\theta}\bar{\Psi}}{\bar{r}} \right]^{2} \right) \nonumber\\
& \hspace{-0.7cm} - \frac{2 N^{2} \partial_{\bar{r}}\bar{\Psi} \partial_{\bar{r}} A}{A}
\left( N^{2} \partial^{2}_{\bar{r}\bar{r}}\bar{\Psi} + \frac{2 N^{2} \partial_{\bar{r}}\bar{\Psi}}{\bar{r}} - \frac{N^{2}}{\bar{r}^{2}} \partial^{2}_{\theta\theta}\bar{\Psi} \right) \nonumber\\
& \hspace{-0.7cm} - \frac{N^{4} (\partial_{\bar{r}}\bar{\Psi})^{2}}{\bar{r}} \frac{ \partial_{\bar{r}} B}{B} \nonumber\\
& \hspace{-0.7cm} + \left. \frac{N^{2}}{\bar{r}^{3}} \partial_{\theta}\bar{\Psi}
\left( 2 N \partial_{\bar{r}}\bar{\Psi} \partial_{\theta}N - N \partial_{\theta}\bar{\Psi} \partial_{\bar{r}}N \right)
\right\}.\end{aligned}$$
The function $N$ could be factored out in many places but instead it is explicitly left everywhere it is needed to counterbalance divergences on the horizon. More precisely, it appears as a factor in front of all the quantities that involve the radial derivative of $\bar{\Psi}$, in order to form terms that remain finite on the horizon.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge valuable support from the CNRS project 80PRIME-TNENGRAV.
References {#references .unnumbered}
==========
[^1]: \[footnote\_shift\_sym\]Such symmetry is a remnant of the more general “Galilean” symmetry enjoyed by the action (\[eq\_action\]) on Minkowski space [@Galileon_original]: $\phi \rightarrow \phi + constant,\ \nabla\phi \rightarrow \nabla\phi + constant\ vector$.
[^2]: This is equivalent to cancel everywhere due to the shift-symmetry of the theory.
[^3]: When such a choice is made, $x^{1}$ and $x^{2}$ are rather denoted $r$ and $\theta$ respectively.
[^4]: Because one always has $g(\xi) \wedge \xi \wedge \chi = g(\chi) \wedge \xi \wedge \chi = 0$.
[^5]: The transformation from the usual Boyer-Lindquist coordinates to quasi-isotropic coordinates can be established explicitly [@quasi_isotropic_Kerr].
[^6]: In this context, the $(tR)$ equation implies the scalar equation [@cubic_BH].
[^7]: For class $C^{\infty}$ functions, the convergence of the spectral series towards the original function is exponential in the resolution.
[^8]: In the static and spherically symmetric cases, $A$ spontaneously equals to $B$ everywhere (as it should in spherical symmetry) through the numerical process without being imposed anywhere.
[^9]: See section \[section\_equations\].
[^10]: <http://www.sagemath.org/>
|
---
abstract: 'Let $X^{(\delta)}$ be a Wishart process of dimension $\delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes $\{\delta^{-1} X_t^{(\delta)}, t \leq 1 \}$ as $\delta$ tends to infinity. The process $X^{(\delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.'
author:
- 'C. Donati-Martin[^1]'
title: Large deviations for Wishart processes
---
[*Key Words:*]{} Wishart processes, large deviation principle
[*Mathematical Subject Classification (2000):*]{} 60F10, 60J60, 15A52
Introduction
============
Let $B$ be a $m \times m$ matrix valued Brownian motion. We consider a Wishart process $X_t$, solution of the following SDE, with values in ${{\mathcal S}}_m^{+}$, the set of $m\times m$ real symmetric non-negative matrices: $$\label{sdeW}
dX_t = \sqrt{X_t} \;dB_t + dB'_t \sqrt{X_t} + \delta I_m \;dt,
\quad X_0= x ,$$ where $x \in {{\mathcal S}}_m^{+}$ and $M'$ denotes the transpose of the matrix $M$.\
We recall the following existence theorem (see M.F. Bru [@Bru]):
- if $\delta \geq m+1$, and $x \in \widetilde{{{\mathcal S}}}_m^{+}$ (the set of positive definite symmetric matrices), then has a unique strong solution in $\widetilde{{{\mathcal S}}}_m^{+}$.
In fact, we can extend this result to a degenerate initial condition, and in the following, we shall allow $x=0$.
We shall look for a Large Deviation Principle for the $\widetilde{{{\mathcal S}}}_m^{+}$ valued diffusion with small diffusion coefficient: $$\label{eqW}
\left \{ \begin{array}{l}
dX^\epsilon_t = \epsilon ( \sqrt{X^\epsilon_t} \;dB_t + dB'_t \sqrt{X^\epsilon_t} )+ \delta I_m \;dt, \ t \leq T\\
X^\epsilon_0= x \end{array} \right.$$ with $\delta >0$. For $\epsilon$ small enough, according to the above existence result, has a unique solution $X^\epsilon_t \in \widetilde{{{\mathcal S}}}_m^{+}$ for $t >0$.\
Note that this problem is equivalent to look for a LDP for the family of processes $\displaystyle (\frac{1}{N} X_t^{(N \delta)}; t \leq 1)$ where $X_t^{(N \delta)}$ denotes a Wishart process of dimension $N \delta$, starting from $Nx$ as $N {\mathop{\longrightarrow}}\infty$.\
When $m=1$, is the equation for the squared Bessel process (BESQ) of dimension $\delta$.
In a previous paper [@DRYZ], we studied large deviations for BESQ and squared Ornstein-Uhlenbeck processes. Note that the diffusion coefficient in the BESQ equation is not Lipschitz and the Freidlin-Wentzell theory doesn’t apply directly (in the degenerate cases : $x = 0$ or $\delta =0$).We gave three approaches; the first one was based upon exponential martingales, the second one uses the infinite divisibility of the law of BESQ processes (and thus a Cramer theorem) and the third method is a consequence of the continuity of the Itô map for the Bessel equation (not square), a property proved by Mc Kean [@McK].\
We also refer to Feng [@Fe] for the study of a LDP for squares of Ornstein-Uhlenbeck processes.
In the matrix case, due to the restriction on the dimension $\delta$, the laws $Q_x^\delta$ of the Wishart processes are no more infinitely divisible. Moreover, we have no analogue of the Bessel equation for the square root of a Wishart process.\
Thus, we shall focus on the exponential martingale approach to extend the LDP in the matrix case. Since the delicate point is for a degenerate initial condition, we shall assume that $x = 0$.\
We denote by $C_0([0,T]; \widetilde{{{\mathcal S}}}_m^{+})$ the space of continuous paths $\varphi_t$ from $[0,T]$ to ${{\mathcal S}}_m^{+}$ such that $\varphi_0 = 0$ and $\varphi_t \in \widetilde{{{\mathcal S}}}_m^{+}$ for $t>0$.
The main result of the paper is:
\[theoprinc\] The family $P^\epsilon$ of distributions of $ (X^\epsilon_t ; t \in [0,T])$, solution of , satisfies a LDP in $C_0([0,T]; \widetilde{{{\mathcal S}}}_m^{+})$ with speed $\epsilon^2$ and good rate function $$\label{taux}
I(\varphi) = \frac{1}{8} \int_0^T {\operatorname{Tr}}(k_\varphi (s) \varphi(s) k_\varphi (s)) ds, \quad \varphi \in C_0([0,T];
\widetilde{{{\mathcal S}}}_m^{+})$$ where $k_\varphi(s)$ is the unique symmetric matrix, solution of $$\label{tauxdef}
k_\varphi (s) \varphi(s) + \varphi(s) k_\varphi(s)= 2 (\dot{\varphi}(s) - \delta I_m), \, s>0.$$
[**Remark:**]{} In the real case ($m=1$), we obtain (see [@DRYZ]), $$I(\varphi) = \frac{1}{8} \int_0^T \frac{(\dot{\varphi}(s) - \delta)^2}{\varphi(s)} ds.$$
The outline of the paper is the following. In Section 2, we prove an exponential tightness result for the distribution $P^\epsilon$ of $X^\epsilon$. In section 3, we prove Theorem 1.1 using the approach of exponential martingales. In Section 4, we discuss the Cramer’s approach, using the additivity of Wishart processes, when we put some restriction on the parameter $\delta$. In section 5, we give some applications of the contraction principle to obtain a LDP for some functionals of the Wishart process.
Exponential Tightness
=====================
We follow the same lines as in [@DRYZ Section 2], that is, we prove exponential tightness in the space $C_\alpha$ of $\alpha$-H" older continuous functions with $\alpha <1/2$. Let $\alpha <1/2$ and set $\Vert \varphi \Vert_\alpha = \sup_{0 \leq s \not= t \leq T} \frac{\Vert \varphi_t - \varphi_s \Vert}{\vert t- s \vert^\alpha}$ where $\Vert . \Vert$ is a norm on ${{\mathcal S}}_m^{+}$. Since all the norms are equivalent, we shall choose a suitable norm and we consider in this section $
\Vert M \Vert = \sum_{1\leq i,j \leq m} |M_{ij}|$.
The family of distributions $P_\epsilon$ of $X^\epsilon$ is exponentially tight in $C_\alpha$, in scale $\epsilon^2$, i.e. for $L >0$, there exists a compact set $K_L$ in $C_\alpha$ such that: $$\label{tensexpo}
\limsup_{\epsilon {\mathop{\longrightarrow}}0} \epsilon^2 \ln P( X^\epsilon \in K_L) \leq -L.$$
[**Proof:**]{} Let us fix $\alpha' \in (\alpha, 1/2)$ and $R>0$. The closed Hölder ball $B_{\alpha'}(0,R)$ is a compact set of $C^\alpha([0,1])$.\
Thus it’s enough to estimate $P( \Vert X^\epsilon \Vert_{\alpha'} \geq R)$. For simplicity, we assume $T=1$.
$$\Vert X^\epsilon \Vert_{\alpha'} \leq \Vert M^\epsilon \Vert_{\alpha' } + \delta m$$ where $M^\epsilon$ is the martingale defined by $$M^\epsilon_t = \epsilon ( \sqrt{X^\epsilon_t} \;dB_t + dB'_t \sqrt{X^\epsilon_t} ).$$ [**Bounds for $\Vert M^\epsilon \Vert_\alpha$**]{}. We shall use Garsia-Rodemich-Rumsey’s Lemma which asserts that if $$\int_0 ^1 \int_0 ^1 \Psi \Bigl(\frac{||M^{\epsilon}_t - M^{\epsilon}_s ||}{p(|t-s|)}\Bigr) ds dt \leq K$$ then $$||M^{\epsilon}_t - M^{\epsilon}_s|| \leq 8 \int_0 ^{\vert t-s \vert} \Psi^{-1} (4K/ u^2 ) dp(u) .$$ Take $\Psi(x) = e ^{c\epsilon^{-2}x} - 1$ for some $0<c<1/2$ and $p(x) = x^{1/2}$. So $\Psi^{-1}(y) = \frac{\epsilon^2}{c} \log (1 +y)$. This yields (see the same computations in [@DRYZ]): $$P\Big( \| M^{\epsilon} \|_{\alpha'} \geq R\Big)
\leq P\left(\int_0 ^1 \int_0 ^1
\exp \Bigl( c \epsilon^{-2}\frac{||M^{\epsilon}_t - M^{\epsilon}_s ||}{|t-s|^{1/2}}\Bigr) \ ds dt \geq K + 1 \right)$$ with $ K = \frac{1}{4} \Bigl( e^{\left(\frac{c\epsilon^{-2}R}{8} - K_2\right) - 4} - 1 \Bigr) $ and $K_2 = 2 \sup _{u \in [0,1]} u^{1/2 - \alpha'} \log \frac{1}{u}$.\
Now by Markov’s inequality, $$\label{MI}
P \Bigl( || M^{\epsilon}||_{\alpha} \geq R \Bigr) \leq
\frac{1}{K+ 1}\int_0 ^1 \int_0 ^1
E\left[\exp \Bigl(c\epsilon^{-2}\frac{||M^{\epsilon}_t - M^{\epsilon}_s ||}{|t-s|^{1/2}}\Bigr)
\right]ds dt .$$ Now, for a matrix $M$, $$\begin{aligned}
\exp(\lambda ||M||) = \prod_{i,j} \exp(\lambda |M_{ij}|) &\leq & \prod_{i,j} [\exp(\lambda M_{ij}) +
\exp(-\lambda M_{ij})] \\
&\leq & m^2 \max [\exp(\lambda M_{ij}) + \exp(-\lambda M_{ij})] \\
&\leq & m^2 \sum_{i,j} [\exp(\lambda M_{ij}) + \exp(-\lambda M_{ij})]\end{aligned}$$ Thus, $$\begin{aligned}
\lefteqn{
E[ \exp(\lambda ||M^{\epsilon}_t - M^{\epsilon}_s ||)] } \\
&&\leq m^2 \sum_{i,j}
\left(E[ \exp(\lambda (M^{\epsilon}_{i,j}(t) - M^{\epsilon}_{i,j}(s) )] +
E[ \exp(- \lambda (M^{\epsilon}_{i,j}(t) - M^{\epsilon}_{i,j}(s) )] \right)\\
&&\leq 2m^4 \max_{i,j} E[ \exp(2\lambda^2 \langle M^{\epsilon}_{i,j}\rangle_s^t)]\end{aligned}$$ where we use in the last inequality the exponential inequality for continuous martingales $$E[ \exp(\lambda Z_t)] \leq E[\exp(2 \lambda^2\langle Z \rangle_t)].$$ Now, $$\begin{aligned}
\langle M^\epsilon_{i,j} \rangle_s^t &= &\epsilon^2 \int_s^t (X_{ii}^\epsilon (u) + X_{jj}^\epsilon (u)) du \\
&\leq & \epsilon^2\int_s^t {\operatorname{Tr}}(X_u^\epsilon) du.\end{aligned}$$ Set $Y^\epsilon_u := {\operatorname{Tr}}(X^\epsilon_u)$, then, $ Y^\epsilon_u$ is a squared Bessel process, solution of the following SDE $$\label{BESQ}
\left\{ \begin{array}{l}
dY^\epsilon_u = 2 \epsilon \sqrt{Y_u^\epsilon} d\beta_u + \delta m \ dt\\
Y_0^\epsilon = 0
\end{array} \right.$$ with $\beta$ a real Brownian motion. Thus, we obtain: $$\begin{aligned}
E\left[\exp \left( c\epsilon^{-2} \frac{||M^{\epsilon}_t - M^{\epsilon}_s ||}{|t-s|^{1/2}}\right)\right]
&\leq& 2 m^4 \left\{E\left[ \exp
\left(\frac{2c^2 \epsilon^{-2}}{(t-s)}\int_s ^t Y^{\epsilon}_u du\right)\right]\right\}^{1/2} \nonumber\\
&\leq& 2m^4\Bigl\{\frac{1}{t-s} \int_s ^t E\left[\exp \left( 2c^2 \epsilon^{-2}Y^\epsilon_u
\right)\right] \ du\Bigr\}^{1/2}\end{aligned}$$ (by Jensen’s inequality). Thus, we obtain: $$\label{PI}
P(\Vert M^\epsilon \Vert_\alpha \geq R) \leq \frac{2m^4}{K+1} \left\{
\sup_{u \in [0,1]} E\left[\exp( 2 c^2\epsilon ^{-2} Y^\epsilon_u)\right]
\right\}^{1/2}\,,$$ where $K+1 = C \exp( cR\epsilon^{-2}/8)$ and $C$ a constant.\
Now, $$E[\exp( 2 c^2\epsilon ^{-2} Y^\epsilon_u)] =
Q_{0}^{m \delta \epsilon^{-2}}[\exp( 2 c^2 X_u)]$$ where $Q_x^\rho$ denotes the distribution of a squared Bessel process, starting from $x$, of dimension $\rho$. The Laplace transform of the BESQ is known ([@RY]) and we obtain: for $c <1/2$, $$Q_{0}^{m \delta \epsilon^{-2}}[\exp( 2 c^2 X_u)] = \left( 1-4c^2 u \right)^
{- \frac{m \delta\epsilon^{-2}}{2}} .$$ implying $$P(\Vert M^\epsilon \Vert_{\alpha'} \geq R) \leq C_m
A^{ m \delta \epsilon^{-2}} e^{- cR\epsilon^{-2}/8}$$ for a positive constant $A$. Thus, $$\lim_{R \rightarrow +\infty}\limsup_{\epsilon \rightarrow 0} \epsilon^2
\ln P(\Vert M^\epsilon \Vert_{\alpha'} \geq R) = - \infty . \quad \Box$$
Proof of Theorem \[theoprinc\]
==============================
From the previous section, we need to prove a weak LDP, that is to prove the upper bound for compact sets. We assume that $T = 1$. According to [@DZ], we shall prove:
- Weak upper bound: $$\label{weakupper}
\lim_{r \rightarrow 0} \limsup_{\epsilon \rightarrow 0} \epsilon^2 \ln P(X^\epsilon
\in B_r (\varphi))\leq - I(\varphi)$$ where $B_r (\varphi )$ denotes the open ball with center $\varphi \in {\cal
C}^{\alpha}_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$ and radius $r$.
- Lower bound : for all open set $O \subset {\cal
C}^{\alpha}_0([0,1],; \widetilde{{{\mathcal S}}}_m^{+})$, $$\label{lower}
\liminf_{\epsilon \rightarrow 0} \epsilon^2 \ln P(X^\epsilon
\in O) \geq - \inf_{\varphi \in O} I(\varphi)\,.$$
The upper bound
---------------
We denote by ${\cal M}_m$, resp. ${{\mathcal S}}_m$ the space of $m\times m$ matrices, resp. symmetric matrices, endowed with the scalar product: $$\langle A,B \rangle = {\operatorname{Tr}}(AB').$$ The corresponding norm is denoted by $ \Vert A \Vert_2$. Set $H = \{ h \in C([0,1]; {{\mathcal S}}_m): \dot h \in L^2([0,1]; {{\mathcal S}}_m) \} $. For $h \in H$ let $$M^{\epsilon,h}_t = \exp \left( \frac{1}{\epsilon^2} \{ \int_0^t {\operatorname{Tr}}(h(s)
(dX^\epsilon_s - \delta I _m \ ds) )\ - \frac{1}{2} \langle Z^\epsilon, Z^\epsilon \rangle_t\} \right), t \leq 1$$ where $$Z^\epsilon_t = \int_0^t {\operatorname{Tr}}(h(s) \sqrt{X^\epsilon_s}dB_s +h(s) dB'_s\sqrt{X^\epsilon_s}).$$ $$\langle Z^\epsilon, Z^\epsilon \rangle_t = 4 \int_0^t {\operatorname{Tr}}(h(s) X^\epsilon_s \ h(s)) ds.$$ $M^{\epsilon,h}$ is a positive, local martingale. In fact, using a Novikov’s type criterion (see [@RY Exercise VIII.1.40], [@DRYZ]), we can prove that $M^{\epsilon,h}$ is a martingale, then, $E(M^{\epsilon,h}_t) =1$.\
By an integration by parts, we can write: $$M^{\epsilon,h}_1 = \exp \left(\frac{1}{\epsilon^2} \Phi(X^\epsilon; h) \right)$$ with $$\Phi (\varphi; h) = G(\varphi; h) - 2 \int_0^1 {\operatorname{Tr}}(h(s) \varphi (s) h(s)) \ ds$$ and $$G(\varphi ; h) = {\operatorname{Tr}}(h_1 (\varphi_1 - \delta I_m)) - \int_0^1 {\operatorname{Tr}}((\varphi_s - \delta s I_m) \dot{h}_s) ds$$ for $\varphi \in C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$.\
[**Remark:**]{} If $\varphi$ is absolutely continuous, then, $$G(\varphi; h) = \int_0^1{\operatorname{Tr}}(h(s) (\dot{\varphi}_s - \delta I _m \ ds) ).$$ For $\varphi \in C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$, $h \in H$, $$\begin{aligned}
P \left(X^\epsilon \in B_r(\varphi)\right) &=& P \left(X^\epsilon \in
B_r(\varphi); \frac{M^{\epsilon, h}_1}{M^{\epsilon, h}_1}\right) \\
&\leq & \exp \left( - \frac{1}{\epsilon^2} \inf_{\psi \in B_r(\varphi)}
\Phi(\psi ; h) \right) E(M^{\epsilon,h}_1) \\
&\leq & \exp \left( - \frac{1}{\epsilon^2} \inf_{\psi \in B_r(\varphi)}
\Phi(\psi ; h) \right)\,,\end{aligned}$$ which yields : $$\limsup_{\epsilon \rightarrow 0} \epsilon^2 \ln P\left(X^\epsilon \in B_r(\varphi)\right)
\leq - \inf_{ \psi \in B_r(\varphi)} \Phi( \psi; h)\,.$$ For $h \in H$, the map $\varphi \longrightarrow \Phi(\varphi ;h)$ is continuous on ${\cal C}_0([0,1], {{\mathcal S}}_m^{+})$, so that $$\lim_{r \rightarrow 0} \limsup_{\epsilon \rightarrow 0} \epsilon^2 \ln P\left(X^\epsilon \in B_r(\varphi)\right)
\leq - \Phi(\varphi ; h).$$ Minimizing in $h \in H$, we obtain: $$\lim_{r \rightarrow 0} \limsup_{\epsilon \rightarrow 0} \epsilon^2 \ln P\left(X^\epsilon \in B_r(\varphi)\right)
\leq -\sup_{h \in H} \Phi(\varphi ; h).$$
For $\varphi \in C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$, $$\sup_{h \in H} \Phi(\varphi ; h) = I(\varphi)$$ where $I(\varphi)$ is defined by .
[**Proof:**]{} Since $\varphi \in C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$, $\int_0^1 {\operatorname{Tr}}(h_s \varphi_s h_s) ds >0$ for $h \not\equiv 0$. Replacing $h$ by $\lambda h$ for $\lambda \in {{\mathbb R}}$, we can see that $$J(\varphi) := \sup_{h \in H} \Phi(\varphi; h) = \frac{1}{8} \sup_{h \in H} \frac{G^2(\varphi;h)}{
\int_0^1 {\operatorname{Tr}}(h_s \varphi_s h_s) ds }.$$ We assume that $J(\varphi) < \infty$. We denote by $\Vert h \Vert_{L^2(\varphi)}$ the Hilbert norm on $C_0([0,1]; {{\mathcal S}}_m)$ given by $$\Vert h \Vert^2_{L^2(\varphi)} = \int_0^1 {\operatorname{Tr}}(h_s \varphi_s h_s) ds .$$ The linear form $G_\varphi \ : h {\mathop{\longrightarrow}}G(\varphi; h)$ can be extended to the space $L^2(\varphi)$ and by Riesz theorem, there exists a function $k_\varphi \in L^2(\varphi)$ such that $$G(\varphi; h) = \int_0^1 {\operatorname{Tr}}(h_s \varphi_s k_\varphi(s)) ds$$ Thus, $\varphi$ is absolutely continuous and we have $$\label{ac}
\int_0^1 {\operatorname{Tr}}(h_s (\dot{\varphi}_s - \delta I_m)) ds = \int_0^1 {\operatorname{Tr}}(h_s \varphi_s k_\varphi(s)) ds$$ for all symmetric matrix $h(s)$. Let $k_\varphi$ be given by . We refer to the Appendix for the existence of an unique solution of . Then, it is easy to see that is satisfied for all $h$ symmetric. Moreover, by Cauchy-Schwarz inequality, $$\begin{aligned}
\int_0^1 {\operatorname{Tr}}(h_s \varphi_s k_\varphi(s)) ds & \leq & \int_0^1 {\operatorname{Tr}}(h_s \varphi_s h_s)^{1/2}
{\operatorname{Tr}}(k_\varphi (s) \varphi_s k_\varphi(s))^{1/2} ds \\
& \leq & ( \int_0^1 {\operatorname{Tr}}(h_s \varphi_s h_s) ds )^{1/2}( \int_0^1 {\operatorname{Tr}}(k_\varphi (s) \varphi_s k_\varphi(s)) ds)^{1/2}
\end{aligned}$$ with equality for $h = k_\varphi$.\
Thus, $\displaystyle \frac{1}{8} \sup_{h \in L^2(\varphi)} \frac{G^2(\varphi;h)}{\Vert h \Vert^2_{L^2(\varphi)}} = I(\varphi)$.\
Now, the equality between $I(\varphi)$ and $J(\varphi)$ follows by density of $H$ in $L^2(\varphi)$. $\Box$
The lower bound
---------------
In order to prove the lower bound, we first prove $$\liminf_{\epsilon \rightarrow 0} \epsilon^2 \ln P(X^\epsilon
\in B_r (\varphi)) \geq - I(\varphi)$$ for all $r >0$ and for $\varphi$ in a subclass ${\cal H}$ of $C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$. Then, we shall show that this subclass is rich enough.
Set ${\cal H}$ the set of functions $\varphi $ such that $I(\varphi) < \infty$ and s.t. $k_\varphi $ defined by belongs to $H$.\
For $\varphi \in {\cal H}$, set $h_\varphi = \frac{1}{4} k_\varphi$. As in the previous subsection, we introduce the new probability measure $$\hat{P} := M^{\epsilon, h_\varphi}_1 P$$ where $P$ is the Wiener measure on $C([0,1]; {\cal M}_{m,m})$. Under $\hat{P}$, $$B_t = \hat{B}_t + \frac{2}{\epsilon} \int_0^t (\sqrt{X^\epsilon_s} \ h_\varphi (s) )\ ds$$ where $\hat{B}$ is a Brownian matrix on $\hat{P}$.\
Thus, under $\hat{P}$, $X^\epsilon$ solves the SDE $$dX^\epsilon_t = \epsilon ( \sqrt{X^\epsilon_t} d\hat{B}_t + d\hat{B}'_t \sqrt{X^\epsilon_t} ) +
(2( X^\epsilon_t h_\varphi (t) + h_\varphi (t) X^\epsilon_t) + \delta I_m) dt.$$ Under $\hat{P}$, $\displaystyle X^\epsilon_t {\mathop{\longrightarrow}}_{\epsilon \rightarrow 0}^{a.s.} \Psi_t$ solution of $$d\Psi_t = [2(\Psi_t h_\varphi (t) + h_\varphi (t) \Psi_t) + \delta I_m] dt$$ i.e. $$\dot{\Psi}_t - \delta I_m = 2(\Psi_t h_\varphi (t) + h_\varphi (t) \Psi_t) = \frac{1}{2} ( \Psi_t k_\varphi (t) + k_\varphi (t) \Psi_t).$$ Since $k_\varphi$ is continuous, this equation has $\varphi$ as a unique solution; thus $$X^\epsilon_t {\mathop{\longrightarrow}}_{\epsilon \rightarrow 0} \varphi_t \; \hat{P}\ \mbox{a.s.}$$ and $\displaystyle \lim_{\epsilon {\mathop{\longrightarrow}}0} \hat{P}(X^\epsilon \in B_r(\varphi)) = 1$ for every $r>0$. Now, $$\begin{aligned}
P \left(X^\epsilon \in B_r(\varphi)\right) &=& \hat{P} \left(X^\epsilon \in
B_r(\varphi) \frac{1}{M^{\epsilon, h_\varphi}_1}\right) \\
&\geq & \exp \left( - \frac{1}{\epsilon^2} \sup_{\psi \in B_r(\varphi)}
F(\psi ; h_\varphi) \right) \hat{P}(X^\epsilon \in B_r(\varphi)) \\\end{aligned}$$ which yields : $$\liminf_{\epsilon \rightarrow 0} \epsilon^2 \ln P\left(X^\epsilon \in B_r(\varphi)\right)
\geq - \sup_{ \psi \in B_r(\varphi)} F( \psi; h_\varphi)\,$$ and by continuity of $F(.,h)$, $$\lim_{r {\mathop{\longrightarrow}}0} \liminf_{\epsilon \rightarrow 0} \epsilon^2 \ln P\left(X^\epsilon \in B_r(\varphi)\right)
\geq F(\varphi; h_\varphi)\, = I(\varphi).$$ We now prove the:
For any $\varphi \in C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$ such that $I(\varphi ) < \infty$, there exists a sequence $\varphi_n$ of elements of ${\cal H}$ such that $\varphi_n {\mathop{\longrightarrow}}\varphi$ in $C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$ and $I(\varphi_n) {\mathop{\longrightarrow}}I(\varphi)$.
[**Proof:**]{} We follow the same lines as in the proof of the corresponding result for the scalar case in [@DRYZ].
a\) First, let us show that the condition $I(\varphi) < \infty$ implies that $$\lim_{t {\mathop{\longrightarrow}}0} \frac{\varphi_t}{t} = \delta I_m.$$ From the scalar case, we know that: $$\label{p1}
\lim_{t {\mathop{\longrightarrow}}0} \frac{{\operatorname{Tr}}(\varphi_t)}{t} = \delta m.$$ Indeed, ${\operatorname{Tr}}(X^\varepsilon_t)$ satisfies a LDP (see ) with rate function given by $J( g) = \frac{1}{8} \int_0^1 \frac{(\dot{g}(s) - \delta m)^2}{g(s)} ds$ and $J(g) < \infty$ implies that $ \lim_{t {\mathop{\longrightarrow}}0} \frac{g(t)}{t} = \delta m$. (see [@DRYZ], [@Fe]). From the upper bound, the condition $I(\varphi) < \infty$ implies that $J({\operatorname{Tr}}(\varphi)) < \infty$ and thus is satisfied.
Let us denote $|| A ||_1 = {\operatorname{Tr}}(|A|)$ and $||A||_2 = ({\operatorname{Tr}}(|A|^2))^{1/2}$ for a matrix $A$. $$\begin{aligned}
||\varphi_t - \delta t I_m||_1 &=& \Vert \int_0^t (\dot{\varphi}_s - \delta I_m) ds\, \Vert_1 \\
&=& \frac{1}{2} \Vert \int_0^t (\varphi_s k_\varphi (s) + k_\varphi (s)\varphi_s \, \Vert_1 \\
&\leq & \frac{1}{2}( \int_0^t \Vert \varphi_s k_\varphi (s)\Vert_1 ds + \int_0^t \Vert k_\varphi (s)\varphi_s \Vert_1 ds ) \\
&\leq & \int_0^t \Vert \sqrt{\varphi_s} \Vert_2 \ \Vert \sqrt{\varphi_s} k_\varphi (s)\Vert_2 ds \\
&= & \int_0^t ({\operatorname{Tr}}(\varphi_s))^{1/2} ({\operatorname{Tr}}(k_\varphi (s) \varphi_s k_\varphi (s)))^{1/2} ds \\
&\leq & \left( \int_0^t {\operatorname{Tr}}(\varphi_s) ds \right)^{1/2} \left( \int_0^t {\operatorname{Tr}}(k_\varphi (s) \varphi_s k_\varphi (s)) ds \right)^{1/2} \end{aligned}$$ Thus, $$||\frac{\varphi_t}{t} - \delta I_m ||_1 \leq \left( \frac{1}{t} \int_0^t \frac{ {\operatorname{Tr}}(\varphi_s)}{s} ds \right)^{1/2}
\left( \int_0^t {\operatorname{Tr}}(k_\varphi (s) \varphi_s k_\varphi (s)) ds \right)^{1/2} .$$ According to , the first term in the RHS is bounded and the second tends to 0 as $t$ tends to $0$ since $I(\varphi) < \infty$.
b\) As a second step, we approximate $\varphi$ by a function $\psi$ such that $k_\psi \in L^2([0, 1]; {\cal S}_m)$. Set $$\left\{ \begin{array}{ll}
\psi_r (t) = \delta t I_m, & t \leq r/2 \\
\psi_r (t) = \delta r/2\ I_m + (t - r/2) a_r, & r/2 \leq t \leq r \\
\psi_r (t) = \varphi(t), & t \geq r
\end{array} \right.$$ where the matrix $a_r$ is chosen such that $\psi$ is continuous in $r$. Let $k_\psi$ the solution of associated with $\psi$. Since $k_\psi(s) =0$ for $s \in [0, r/2]$, and $\psi(s)$ is invertible for $s>0$, $k_\psi \in L^2([0, 1]; {\cal S}_m)$. Obviously, $\displaystyle \psi_r {\mathop{\longrightarrow}}_{r\rightarrow 0} \varphi$ in $C_0([0,1]; \widetilde{{{\mathcal S}}}_m^{+})$.\
It remains to prove the convergence of $I(\psi_r)$ to $I(\varphi)$, or that $$\int_{r/2}^r {\operatorname{Tr}}(k_\psi (s) \psi(s) k_\psi (s)) ds {\mathop{\longrightarrow}}_{r\rightarrow 0} 0.$$ $$\begin{aligned}
\int_{r/2}^r {\operatorname{Tr}}(k_\psi (s) \psi(s) k_\psi (s)) ds &= &\int_{r/2}^r {\operatorname{Tr}}(k_\psi (s)(\dot{\psi} (s)- \delta I_m)) ds \\
&=& \int_{r/2}^r {\operatorname{Tr}}(k_\psi (s)(a_r- \delta I_m)) ds.
\end{aligned}$$ Note that $a_r$ and $k_{\psi_r} (s)$ for $s \in [r/2, r]$ are diagonalisable in the same basis with respective eigenvalues: $(a_i^{(r)})_i$ and $k_i (s) = \frac{a_i^{(r)} - \delta}{\delta r/2 + (s-r/2) a_i^{(r)}}$ and that, according to step a), $\displaystyle \lim_{r{\mathop{\longrightarrow}}0} a_i^{(r)} = \delta$. Thus, for $r$ small enough, $$\int_{r/2}^r {\operatorname{Tr}}(k_\psi (s) \psi(s) k_\psi (s)) ds = \int_{r/2}^r\sum_i \frac{(a_i^{(r)} - \delta)^2}{
\delta r/2 + (s-r/2)a_i^{(r)}} ds \leq 1/\delta \sum_{i=1}^m (a_i^{(r)} - \delta)^2$$ and the last quantity tends to 0 as $r$ tends to 0.
c\) Thanks to b), we must find an approximating sequence $\varphi^{(n)}$ of $\varphi$ in ${\cal H}$ for $\varphi$ satisfying $k_\varphi \in L^2 $.\
Let $k^{(n)}$ be a sequence of smooth functions with values in ${\cal S}_m$ such that $k^{(n)}$ converges to $k_\varphi$ in $L^2([0,1], {\cal S}_m)$. Let $\varphi^{(n)}$ be the unique solution of $$\left \{ \begin{array}{l}
\dot{\varphi}^{(n)}_t - \delta I_m = k^{(n)}_t \varphi^{(n)}_t + \varphi^{(n)}_t k^{(n)}_t \\
\varphi^{(n)}_0 = 0
\end{array} \right.$$ Since $$||\varphi^{(n)}_t || \leq \int_0^t \Vert \dot{\varphi}^{(n)}_s \Vert ds \leq 2 \int_0^t \Vert \varphi^{(n)}_s \Vert \ \Vert k^{(n)}_s \Vert ds
+ \delta,$$ the Gronwall inequality shows that $$\sup_n \sup_{t \in [0,1]} ||\varphi^{(n)}_t || < \infty$$ where we have chosen the operator norm on the set of matrices in the previous inequality. Another application of Gronwall’s inequality entails that: $$\sup_{t \in[0,1]} || \varphi_t - \varphi^{(n)}_t || {\mathop{\longrightarrow}}_{n \rightarrow \infty} 0.$$ Now, the convergence of $I(\varphi^{(n)})$ to $I(\varphi)$ follows from the convergence in $L^2$ of $k^{(n)}$ to $k_\varphi$ and the convergence in $L^\infty([0,1])$ of $\varphi^{(n)}$ to $\varphi$. $\Box$
The Cramer theorem
==================
Let $Q^\delta_x$ denote the distribution on $C({{\mathbb R}}, {\cal S}_m^+)$ of the Wishart process of dimension $\delta \geq m+1$, starting from $x \in {\cal S}_m^+$. We recall the following additivity property (see [@Bru]): $$Q^{\delta}_{x} \oplus Q^{ \delta'}_{y} = Q^{\delta+ \delta'}_{x+y}.$$ Let $\delta \geq m+1 $ and take $ \epsilon = \frac{1}{\sqrt{n}}$, then $X^\epsilon$, solution of , is distributed as $\frac{1}{n} \sum_{i=1}^n X_i$ where $X_i$ are independent copies of $Q^\delta_x$. From Cramer’s theorem ([@DZ] chap. 6), we obtain:
Let $\delta \geq m+1$. The family $P^\epsilon$ of distributions of $ (X^\epsilon_t ; t \in [0,T])$, solution of , satisfies a LDP in $C_0([0,T]; \widetilde{{{\mathcal S}}}_m^{+})$ with speed $\epsilon^2$ and good rate function: $$\label{Fenchel}
\Lambda^*(\varphi) = \sup_{ \mu \in {\cal M}([0,T], {\cal S}_m)} \left( \int_0^T {\operatorname{Tr}}(\varphi_t d\mu_t) \ -
\Lambda (\mu) \right),$$ where $$\label{loglaplace}
\Lambda(\mu) =\ln \left[Q^\delta_x \left( \exp( \int_0^T {\operatorname{Tr}}(X_s d\mu_s))\right)\right].$$
The Laplace transform of the $Q^\delta_x$ distribution can be computed explicitely in terms of Ricatti equation, extending to the matrix case, the well known result for the squared Bessel processes (see [@PY], [@RY Chap. XI]).
Let $\mu$ be a positive ${\cal S}_m^+$-valued measure on $[0,T]$. Then, $$\label{Laplace}
Q^\delta_x \left( \exp(- \frac{1}{2} \int_0^T {\operatorname{Tr}}(X_s d\mu_s))\right) = \exp(\frac{1}{2} {\operatorname{Tr}}(F_\mu (0) x))
\exp(\frac{\delta}{2}\int_0^T {\operatorname{Tr}}(F_\mu (s)) ds)$$ where $F_\mu(s)$ is the ${\cal S}_m$-valued, right continuous solution of the Riccati equation $$\label{Riccati}
\dot{F}+ F^2 = \mu, \quad F(T) = 0.$$
[**Proof:**]{} From Itô’s formula, $$\begin{aligned}
F_\mu(t) X_t &=& F_\mu(0) x + \int_0^t F_\mu(s) dX_s + \int_0^t dF_\mu(s) X_s\\
&=& F_\mu(0) x + \int_0^t F_\mu(s) dX_s + \int_0^t d\mu(s) X_s \ - \int_0^t F_\mu^2(s) X_s\ ds\end{aligned}$$ Consider the exponential local martingale $$Z_t = \exp \left( \frac{1}{2} \int_0^t {\operatorname{Tr}}(F_\mu(s) dM_s) - \frac{1}{2} \int_0^t {\operatorname{Tr}}(F_\mu(s) X_s F_\mu(s)) ds \right)$$ where $M_s = X_s - \delta I_m \ s $. Then, $$Z_t = \exp \left( \frac{1}{2} \left( {\operatorname{Tr}}(F_\mu(t) X_t) - {\operatorname{Tr}}( F_\mu(0) x) - \delta \int_0^t {\operatorname{Tr}}(F_\mu(s)) ds -
\int_0^t {\operatorname{Tr}}(X_s d\mu(s))\right)\right).$$ Now, $X_t$ is positive and $F_\mu(t)$ is negative[^2]. Thus, ${\operatorname{Tr}}(X_t F_\mu(t))
\leq 0$ and $Z_t$ is a bounded martingale. The Lemma follows from the equality $E(Z_0) = E(Z_T)$. $\Box$
[**Remarks:**]{}
1. The condition $F(T) = 0$ in is equivalent to $F(T-) = - \mu(\{T\})$.
2. Taking $d\mu_s = 2 \Theta \delta_1(ds) $ where $\Theta$ is a symmetric positive matrix, we find that $F_\mu (t) = -2\Theta(I_m + 2(1-t)\Theta)^{-1}, \; t < 1$, from which we recover (see [@Bru]): $$\label{LaplaceB}
Q^\delta_x \left( \exp(- {\operatorname{Tr}}(X_1 \Theta))\right) = \det(I_m + 2 \Theta)^{-\delta/2} \exp(-{\operatorname{Tr}}(x(I_m+2 \Theta)^{-1} \Theta)).$$ For $m=1$, this example is given in [@DRYZ], Subsection 8.3.
Let us try to make the correspondence between $\varphi$ and $\mu$ in . If $\mu$ is a negative measure, then, from , $$\int_0^T {\operatorname{Tr}}(\varphi_t d\mu_t) \ - \Lambda (\mu) = \int_0^T {\operatorname{Tr}}(\varphi_t d\mu_t) \ -
\frac{1}{2} {\operatorname{Tr}}(F_{-2\mu} (0) x) - \frac{\delta}{2}\int_0^T {\operatorname{Tr}}(F_{-2\mu} (s)) ds.$$ Since $d\mu(t) = - \frac{1}{2} (\dot{F}_t + F^2_t)$, an integration by parts gives: $$\label{eqF}
\int_0^T {\operatorname{Tr}}(\varphi_t d\mu_t) \ - \Lambda (\mu) = \frac{1}{2}\int_0^T {\operatorname{Tr}}(F_{-2\mu} (s)(\dot{\varphi}_s - \delta I_m)) ds - \frac{1}{2}\int_0^T {\operatorname{Tr}}(F^2_{-2\mu} (s) \varphi_s ) ds.$$ The optimal function $F(s)$ giving the supremum in solves the equation: $$\dot{\varphi}_s - \delta I_m = \varphi_s F(s) + F(s) \varphi_s$$ that is $F(s) = k_\varphi (s)/2$ where $k_\varphi$ is the solution of , and for this $F$, the RHS of is exactly $I(\varphi)$.
Some applications
=================
From the contraction principle, we can obtain a LDP for some continuous functionals of the Wishart process $X^\epsilon$.
The eigenvalues process
-----------------------
Let $(\lambda^\epsilon (t) = (\lambda^\epsilon_1 (t), \ldots, \lambda^\epsilon_m (t)); t \in [0,T])$ denote the process of eigenvalues of the process $X^\epsilon$.
\[propev\] The process $\lambda^\epsilon$ satisfies a LDP in $C_0([0,T], {{\mathbb R}}_+^m)$ with rate function: $$\label{tauxev}
J(x) = \frac{1}{8} \sum_{i=1}^m \int_0^T \frac{(\dot{x}_i (t) - \delta)^2}{x_i(t)} dt.$$
[**Remark:**]{} $(\lambda^\epsilon (t))_t$ is solution of the SDE (see [@Bru]): $$d\lambda^\epsilon_i (t) = 2 \epsilon \sqrt{\lambda^\epsilon_i (t)} d\beta_i (t) + \{ \delta + \epsilon^2
\sum_{k\not=i} \frac{ \lambda^\epsilon_i(t) + \lambda^\epsilon_k (t)}{\lambda^\epsilon_i (t) - \lambda^\epsilon_k(t)} \} dt$$ from which we can guess the form of the rate function $J$ in since the drift $b_\epsilon$ in the above equation satisfies $\displaystyle b_\epsilon (\lambda) {\mathop{\longrightarrow}}_{\epsilon \rightarrow 0} \delta$. Nevertheless, since the drift $b_\epsilon (\lambda)$ explodes on the hyperplanes $\{ \lambda_i = \lambda_j\}$ and the diffusion coefficient is degenerate, the classical results (see [@FW Theo. V.3.1]) do not apply.
[**Proof of the Proposition \[propev\]**]{} According to the contraction principle, $$J(x) = \inf\{ I(\varphi); e.v.(\varphi) = x \}.$$ Write $\varphi_t = P_t^{-1} \Lambda_t P_t$ where $\Lambda$ is the diagonal matrix of eigenvalues of $\varphi_t$ and $P_t$ is an orthogonal matrix. Then, $$\dot{\varphi}_t = P_t^{-1} \dot{\Lambda}_t P_t + \dot{P_t^{-1}} \Lambda_t P_t +
P_t^{-1} \Lambda_t \dot{P}_t.$$ We denote by $\tilde{k}_t $ the matrix $P_t k_\varphi(t) P_t^{-1}$ where $k_\varphi$ solves . Then, $${\operatorname{Tr}}(k_\varphi (t) \varphi(t) k_\varphi(t)) = {\operatorname{Tr}}(\tilde{k}_t \Lambda(t) \tilde{k}_t)$$ and $$\tilde{k}_{ij} (t) \lambda_i(t) + \tilde{k}_{ij} (t) \lambda_j(t) = 2(\dot{\lambda}_i(t) - \delta) \delta_{ij} +
R_{ij}(t)$$ where the matrix $R$ is defined by $$R(t) = P_t \dot{P_t^{-1}} \Lambda_t + \Lambda_t \dot{P}_t P^{-1}_t.$$ Now, it is easy to verify that $R_{ii} (t) =0$, thus: $${\operatorname{Tr}}(\tilde{k}_t \Lambda(t) \tilde{k}_t) = \sum_{i} \frac{(\dot{\lambda}_i (t) - \delta)^2}{\lambda_i(t)} +
\sum_{i \not= j} \frac{R_{ij}^2(t) \lambda_j(t)}{\lambda_i(t) + \lambda_j(t)}$$ and the infimum of the above quantity is obtained for $R\equiv 0$, corresponding to $P_t$ independent of $t$. For this choice, $I(\varphi) = J(\lambda)$ where $\lambda$ is the set of e.v. of $\varphi$. $\Box$
A LDP for the r.v. $X^\epsilon_1$
---------------------------------
The r.v. $X^\epsilon_1$ satisfies a LDP, in scale $\epsilon^2$, with rate function: $$\label{tauxrv}
K(M) = \frac{1}{2} {\operatorname{Tr}}(M) - \frac{\delta}{2}\ln(\det(M)) - \frac{m \delta}{2} + \frac{m\delta}{2} \ln(\delta), \; M \in {\cal S}_m^{+}.$$
[**Remark:**]{} For $m =1$, $$K(a) = \frac{1}{2} [ (a - \delta) - \ln(a/\delta)], \ a >0$$ which corresponds (for $\delta =1$) to the rate function obtained in the study of a LDP for a $\chi_2(n)$ distribution as $n {\mathop{\longrightarrow}}\infty$.
[**Sketch of proof:**]{}
i\) Since the application $\varphi {\mathop{\longrightarrow}}\varphi(1)$ is continuous, we must minimize $I(\varphi)$ under the constraint $\varphi(1) = M$. The optimal path $\varphi$ solves the Euler Lagrange equation (see [@L], Chap. 7), given in terms of $k_\varphi$ by: $$2 \dot{k}_\varphi (s) + k_\varphi^2(s) = 0, s \in (0,1).$$ This leads to $k_\varphi^{-1} (t) = \frac{t}{2} I_m + C$ and $\varphi(t) = \delta t I_m + t^2 A$ with a matrix $A$ determined by $\varphi(1) = M$. Note that this is the same path as in Section 4, Remark 2.\
Now, it is easy to verify that for $\varphi(t) = \delta t I_m +
t^2 (M - \delta I_m)$, $I(\varphi) = K(M)$ where $K$ is given by .
ii\) Of course, we can compute $K$ directly, using the Laplace transform (with $x=0$) and then, $$K(M) = \sup_{\Theta} \{{\operatorname{Tr}}(\Theta M) + \frac{\delta}{2} \ln(\det(I_m - 2 \Theta))\}.$$ The optimal $\Theta_0$ is given by $M = \delta(I_m - 2\Theta_0)^{-1}$. $\Box$
A LDP for the largest eigenvalue
---------------------------------
Let us denote by $\lambda^\epsilon_{max}$ the largest eigenvalue of the Wishart process $X^\epsilon$.
The process $\{ \lambda^\epsilon_{max}(t), t \in [0,T] \}$ satisfies a LDP in $C_0([0,T; {{\mathbb R}}_+)$ with rate function given by $$I_{max} (f) = \inf \{ J(x), x = (f, x_2, \ldots, x_m), x_i(t) \leq f(t) \mbox{ for } i= 2, \ldots m \}$$ where $J$ is given by .\
For $f$ belonging to a class of functions ${\cal F}$ to be defined in the proof, $$\label{tauxmax}
I_{max}(f) = \frac{1}{8} \left[ \int_0^T \frac{(\dot{f}_t - \delta)^2}{f_t} dt + (m-1) \int_0^T
\frac{(\dot{\underline{f}}_t - \delta)^2}{\underline{f}_t} \right]$$ where $\underline{f}(t) = \delta t + \inf_{s \leq t} (f(s) - \delta s)$.
[**Proof:**]{} We assume that the eigenvalues are given in decreasing order: $\lambda_1(t) \geq \lambda_2(t) \geq \ldots \geq \lambda_m (t)$.\
According to the contraction principle, $I_{max}$ is given by the minimium of : $$J(x) = \frac{1}{8} \sum_{i=1}^m \int_0^T \frac{(\dot{x}_i (t) - \delta)^2}{x_i(t)} dt$$ under the constraint $\{ x_i(t) \leq f(t), i = 2, \ldots m\}$ with $x_1 = f$ fixed.\
Set $$F(y) = \frac{1}{8} \int_0^T \frac{(\dot{y} (t) - \delta)^2}{y(t)} dt;$$ $F$ is a convex function on $C_0([0,T); {{\mathbb R}}_+)$ and introduce the convex function $G_f(y) = y -f \in C([0,T); {{\mathbb R}})$.\
The problem is to minimize $F(y)$ under the constraint $G(y) \leq 0$.
To $f$, we associate the measure $\mu_f$ associated to the Ricatti equation $$2 \mu_f = \dot{H} + H^2 \mbox{ on } (0,T), \ H(T) = - 2\mu_f(T)$$ with $\displaystyle H_t = \frac{(\dot{f} (t) - \delta)}{2f(t)} $.\
Then, we define the measure $d\tilde{\mu}_f (t) =
d\mu_f (t) 1_{(\underline{f} (t) = f(t))}$.\
Let ${\cal F} = \{ f; d\tilde{\mu}_f \mbox{ is a positive measure on $[0,T]$} \}$. For $f \in {\cal F}$, let us show that the Lagrangian $$L(y, \mu) = F(y) +\langle G_f(y), \mu \rangle$$ has a saddle point at $(\underline{f}, \tilde{\mu}_f)$, i.e., $$\label{saddle}
L(\underline{f}, \mu) \leq L(\underline{f}, \tilde{\mu}_f) \leq L(y, \tilde{\mu}_f).$$ for all $y \in C_0([0,T); {{\mathbb R}}_+)$ and all positive measure $\mu$.\
The first inequality follows from $$\langle G_f(\underline{f}), \mu \rangle \leq 0 = \langle G_f(\underline{f}), \tilde{\mu}_f \rangle$$ since $supp(\tilde{\mu}_f ) \subset \{t, f(t) = \underline{f}(t) \}$.\
For the second inequality, we must show that $\underline{f}$ minimize $F(y) + \langle G_f(y), \tilde{\mu}_f \rangle$. The optimal path of this problem of minimization solves the Euler- Lagrange equation (see [@DRYZ]): $$\label{EL}
\frac{d}{dt} \left( \frac{\partial g}{\partial b}(y, \dot{y}) \right) = \frac{\partial g}{\partial a}(y, \dot{y})
+ \tilde{\mu}_f \mbox{ on } (0,T), \ \left( \frac{\partial g}{\partial b}(y, \dot{y}) \right)_{t = T} = - \tilde{\mu}_f (T).$$ with $g(a,b) = \frac{(b- \delta)^2}{8a}$. The auxiliary function $H_t = \frac{(\dot{y} (t) - \delta)}{2y(t)} $ associated to the optimal path $y$ satisfies the Ricatti equation: $$2 \tilde{\mu}_f = \dot{H} + H^2; \; H(T) = - 2\tilde{\mu}_f(T).$$ By the choice of $\tilde{\mu}_f$, it is easy to see that $\underline{f}$ solves the Euler-Lagrange equation (or the associated Ricatti equation).
According to Luenberger (Theo2, Section 8.4), the existence of this saddle point implies that : $$\underline{f} \mbox{ minimize }F(y) \mbox{ under the constraint } G_f(y) \leq 0 . \; \Box$$
For a fixed time, we have the following result:
The r.v. $ \lambda^\epsilon_{max}(1)$ satisfies a LDP in $ {{\mathbb R}}_+$ with rate function given by $$\label{tauxmax1>}
K_{max}(a) =\frac{a}{2} - \frac{\delta}{2} \ln(a) - \frac{\delta}{2} + \frac{\delta}{2} \ln(\delta) \mbox{ if } a > \delta$$ $$\label{tauxmax1<}
K_{max} (a) =m\left( \frac{a}{2} - \frac{\delta}{2} \ln(a) - \frac{\delta}{2} + \frac{\delta}{2} \ln(\delta) \right) \mbox{ if } a \leq \delta$$
The proof is immediate from . We minimize $K(M)$ under the constraint $||M|| = a$, where $||.||$ denotes the operator norm.
Appendix
========
[**(A.1) On the equation AX+XA = B.**]{}\
Let $A$ and $B$ two symmetric matrices, with $A$ strictly positive. We are looking for a symmetric matrix $X$, solution of the equation ( see ): $$AX +XA = B \quad (*)$$ Since $A$ is symmetric, let $P$ and $D$ be orthogonal and positive diagonal matrices such that $A = P^{-1}DP$. Then, from $(*)$, the symmetric matrix $\tilde{X} = PXP^{-1}$ satisfies: $$D\tilde{X} + \tilde{X} D = PBP^{-1}:= \tilde{B}$$ that is: $$d_i \tilde{X}_{ij} + \tilde{X}_{ij} d_j = \tilde{B}_{ij}$$ and $\displaystyle \tilde{X}_{ij} = \frac{\tilde{B}_{ij}}{d_i + d_j}$. Thus, $X$ is uniquely determined.
[**(A.2) On the Riccati equation.**]{}\
We consider the Ricatti equation (see ): $$\label{Riccatibis}
\dot{F}+ F^2 = \mu, \quad F(T) = 0.$$ or $$F(t) = C + \mu(]0,t]) - \int_0^t F^2(s) ds$$ where $C$ is chosen that $F(T)= 0$. We diagonalize $F(t)$: $F_t = P^{-1}_t D_t P_t$ with $D_t$ the matrix of eigenvalues of $F_t$ and $P_t$ orthogonal. Then, the Ricatti equation can be written as: $$\dot{D} (t) + D^2(t) = P(t) \mu_t P^{-1}(t) + R_t$$ where $R$ is a matrix, whose diagonal entries are zeroes. Set $\nu = P \mu P^{-1}$, then $\nu$ is a positive ${\cal S}_m^+$-valued measure and the eigenvalues of $F$ satisfy the scalar Riccati equation: $$\dot{d_i}(t)+ d_i^2(t) = \nu_{ii}(t), \; d_i(T) = 0$$ with $\nu_{ii}$ a positive measure on \[0,T\]. We know (see [@RY], Chapter XI) that $d_i(t) \leq 0$ ($d_i$ is related to the decreasing solution of the Sturm Liouville equation $\phi''_i = \phi_i \nu_{ii}$). It follows that the matrix $F(t)$ is symmetric negative.
[99]{} Bru, M.-F.: Wishart processes. [*J*]{}. [*Theo*]{}. [*Probab*]{}., [**4**]{} (1991), 725–751. Dembo, A. and Zeitouni, O.: [*Large deviations techniques and applications*]{}. Second Edition, Springer, 1998. Donati-Martin, C.; Rouault, A.; Yor, M. and Zani, M.: Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes. [*Prob. Th. Rel. Fields*]{} [**129**]{} (2004) 261-289. Feng, S.: The behaviour near the boundary of some degenerate diffusions under random perturbations. In [*Stochastic models*]{} (Ottawa, ON, 1998), 115-123, Providence (2000) Amer. Math. Soc. Freidlin, M.I. and Wentzell A.D.: [*Random Perturbations of Dynamical Systems*]{}. Springer-Verlag, New York, 1984. Luenberger, D.G.: [*Optimization by vector space methods.*]{} John Wiley, 1969. Mc Kean, H. P. : The Bessel motion and a singular integral equation. [*Mem. Coll. Sci. Univ. Kyoto. Ser. A, Math.*]{} [**33**]{} (1960) 317-322. Pitman, J. and Yor, M.: A decomposition of Bessel bridges. [*Z. W*]{} [**59**]{} (1982) 425-457. Revuz, D. and Yor, M.: [*Continuous martingales and Brownian motion*]{}, [*3rd Ed*]{}., Springer, Berlin, 1999.
[^1]: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 6, Site Chevaleret, 13 rue Clisson, F-75013 Paris. email: donati@ccr.jussieu.fr
[^2]: See the Appendix (A.2)
|
---
abstract: 'This paper and its sequel prove that every Legendrian knot in a closed three-manifold with a contact form has a Reeb chord. The present paper deduces this result from another theorem, asserting that an exact symplectic cobordism between contact 3-manifolds induces a map on (filtered) embedded contact homology satisfying certain axioms. The latter theorem will be proved in the sequel using Seiberg-Witten theory.'
author:
- Michael Hutchings and Clifford Henry Taubes
title: Proof of the Arnold chord conjecture in three dimensions I
---
Introduction
============
The chord conjecture
--------------------
Let $Y$ be a closed oriented $3$-manifold (all $3$-manifolds in this paper will be assumed connected). Recall that a [*contact form*]{} on $Y$ is a $1$-form $\lambda$ on $Y$ with $\lambda\wedge d\lambda>0$ everywhere. The contact form $\lambda$ determines a [*contact structure*]{}, namely the oriented 2-plane field $\xi={{\operatorname}{Ker}}(\lambda)$. It also determines the [*Reeb vector field*]{} $R$ characterized by $d\lambda(R,\cdot)=0$ and $\lambda(R)=1$. A [*Legendrian knot*]{} in $(Y,\lambda)$ is a smooth knot ${{\mathcal K}}\subset Y$ such that $T{{\mathcal K}}\subset\xi|_{{\mathcal K}}$. A [*Reeb chord*]{} of ${{\mathcal K}}$ is a trajectory of the Reeb vector field starting and ending on ${{\mathcal K}}$, i.e. a path $\gamma:[0,T]\to Y$ for some $T>0$ such that $\gamma'(t)=R(\gamma(t))$ and $\gamma(0),\gamma(T)\in {{\mathcal K}}$. The main result of this paper is:
\[thm:cc\] Let $Y$ be a closed oriented $3$-manifold with a contact form $\lambda$. Then every Legendrian knot in $(Y,\lambda)$ has a Reeb chord.
This is a version of a conjecture of Arnold [@arnold]. For the 3-sphere with any contact form inducing the standard contact structure, and more generally for boundaries of subcritical Stein manifolds in any odd dimension, this was proved by Mohnke [@mohnke]. This was also proved by Abbas [@abbas] for Legendrian unknots in tight contact 3-manifolds satisfying certain assumptions.
The proof of Theorem \[thm:cc\] given here uses the relationship between embedded contact homology and Seiberg-Witten Floer cohomology. We now recall the relevant parts of this correspondence.
Embedded contact homology {#sec:ech}
-------------------------
We begin by briefly reviewing the definition of embedded contact homology. For more details see [@icm] and the references therein.
Let $Y$ be a closed oriented 3-manifold with a contact form $\lambda$. A [*Reeb orbit*]{} is a closed orbit of the Reeb vector field, i.e.a map $\gamma:{{\mathbb R}}/T{{\mathbb Z}}\to Y$ for some $T>0$ with $\gamma'(t)=R(\gamma(t))$, modulo reparametrization. The linearized Reeb flow along a Reeb orbit $\gamma$ defines an endomorphism $P_\gamma$ of the 2-dimensional symplectic vector space $(\xi_{\gamma(0)},d\lambda)$. A Reeb orbit $\gamma$ is [ *nondegenerate*]{} if $P_\gamma$ does not have $1$ as an eigenvalue. In this case either $P_\gamma$ has real eigenvalues, in which case we say that $\gamma$ is [*hyperbolic*]{}, or $P_\gamma$ has eigenvalues on the unit circle, in which case $\gamma$ is called [*elliptic*]{}. These notions do not depend on the parametrization of $\gamma$. We say that the contact form $\lambda$ is [ *nondegenerate*]{} if all its Reeb orbits are nondegenerate. A generic contact form has this property.
Assume now that the contact form $\lambda$ on $Y$ is nondegenerate. An [*orbit set*]{} is a finite set of pairs $\Theta=\{(\Theta_i,m_i)\}$ where the $\Theta_i$’s are distinct embedded Reeb orbits, and the $m_i$’s are positive integers which one can think of as “multiplicities”. The homology class of the orbit set $\Theta$ is defined by $$[\Theta] {\;{:=}\;}\sum_i m_i[\Theta_i] \in H_1(Y).$$ The orbit set $\Theta=\{(\Theta_i,m_i)\}$ is called [ *admissible*]{} if $m_i=1$ whenever $\Theta_i$ is hyperbolic. An admissible orbit set is also called an [ *ECH generator*]{}.
If $\Gamma\in H_1(Y)$, then the [*embedded contact homology*]{} $ECH_*(Y,\lambda,\Gamma)$ is the homology of a chain complex which is freely generated over ${{\mathbb Z}}/2$ by admissible orbit sets $\Theta$ with $[\Theta]=\Gamma$.
Although ECH is ordinarily defined over ${{\mathbb Z}}$, with the signs specified in [@obg2 §9], in this paper ECH is always defined with ${{\mathbb Z}}/2$ coefficients, because this is sufficient for the applications here and will allow us to avoid orientation headaches.
To define the chain complex differential one chooses a generic almost complex structure $J$ on ${{\mathbb R}}\times Y$ of the following type:
\[def:SA\] An almost complex structure $J$ on ${{\mathbb R}}\times Y$ is [ *symplectization-admissible*]{} if $J$ is ${{\mathbb R}}$-invariant, $J(\partial_s)=R$ where $s$ denotes the ${{\mathbb R}}$ coordinate, and $J$ sends $\xi$ to itself, rotating $\xi$ positively with respect to the orientation on $\xi$ given by $d\lambda$.
The reason for the terminology is that the noncompact symplectic manifold $({{\mathbb R}}\times Y,d(e^s\lambda))$ is called the [ *symplectization*]{} of $(Y,\lambda)$.
Given a symplectization-admissible $J$, we now consider (not necessarily embedded) $J$-holomorphic curves in ${{\mathbb R}}\times Y$ whose domains are (not necessarily connected) punctured compact Riemann surfaces. A [*positive end*]{} of such a holomorphic curve at a (not necessarily embedded) Reeb orbit $\gamma$ is an end which is asymptotic to the cylinder ${{\mathbb R}}\times\gamma$ as the ${{\mathbb R}}$ coordinate $s\to +\infty$. A [*negative end*]{} is defined analogously with $s\to-\infty$.
\[def:Jhol\] Given a symplectization-admissible $J$, and given (not necessarily admissible) orbit sets $\Theta=\{(\Theta_i,m_i)\}$ and $\Theta'=\{(\Theta'_j,m'_j)\}$, a “$J$-holomorphic curve from $\Theta$ to $\Theta'$” is a $J$-holomorphic curve in ${{\mathbb R}}\times Y$ as above with positive ends at covers of $\Theta_i$ with total multiplicity $m_i$, negative ends at covers of $\Theta'_j$ with total multiplicity $m'_j$, and no other ends.
Such a holomorphic curve may be multiply covered, but we are only interested in the corresponding current. In particular, let ${{\mathcal M}}^J(\Theta,\Theta')$ denote the moduli space of $J$-holomorphic curves from $\Theta$ to $\Theta'$, where two such curves are considered equivalent if they represent the same current in ${{\mathbb R}}\times
Y$, up to translation of the ${{\mathbb R}}$ coordinate.
Given ECH generators $\Theta$ and $\Theta'$ with $[\Theta]=[\Theta']=\Gamma$, the differential coefficient $\langle\partial\Theta,\Theta'\rangle\in {{\mathbb Z}}/2$ is the mod 2 count of $J$-holomorphic curves in ${{\mathcal M}}^J(\Theta,\Theta')$ with “ECH index” equal to $1$. The definition of the ECH index is not needed in this paper and may be found in [@ir]. If $J$ is generic, then $\partial$ is well-defined and $\partial^2=0$, as shown in [@obg1 §7]. In this case we denote the chain complex by $ECC_*(Y,\lambda,\Gamma;J)$. A symplectization-admissible almost complex structure that is generic in this sense will be called [ *ECH-generic*]{} here. It turns out that the curves counted by the ECH differential are embedded, except that they may include multiple covers of ${{\mathbb R}}$-invariant cylinders. The ECH index defines a relative ${{\mathbb Z}}/d(c_1(\xi)+2{\operatorname}{PD}(\Gamma))$ grading on the chain complex, where $d$ denotes divisibility in $H^2(Y;{{\mathbb Z}})/{\operatorname}{Torsion}$.
The isomorphism with Seiberg-Witten Floer cohomology {#sec:isoswf}
----------------------------------------------------
Although the ECH differential depends on the choice of $J$, the homology of the chain complex does not. This follows from a much stronger theorem of the second author [@e1; @e2; @e3; @e4] asserting that ECH is isomorphic to a version of Seiberg-Witten Floer cohomology as defined by Kronheimer-Mrowka. To be precise, there are three basic versions of Seiberg-Witten Floer cohomology, denoted by $\widehat{HM}^*$, $\check{HM}^*$, and $\overline{HM}^*$. The first of these is the one that is relevant to ECH; it assigns ${{\mathbb Z}}/2$-modules $\widehat{HM}^*(Y,{\mathfrak}{s})$ to each spin-c structure ${\mathfrak}{s}$ on $Y$, which have a relative ${{\mathbb Z}}/d(c_1({\mathfrak}{s}))$-grading.
In this paper, all Seiberg-Witten Floer cohomology is defined with ${{\mathbb Z}}/2$ coefficients (even though it can be defined over ${{\mathbb Z}}$, which is the default coefficient system in [@km]).
Recall that the set ${{\operatorname}{Spin}^c}(Y)$ of spin-c structures on $Y$ is an affine space over $H^2(Y;{{\mathbb Z}})$, and the contact structure $\xi$ determines a distinguished spin-c structure ${\mathfrak}{s}_\xi$. With this convention, the theorem is now that $$\label{eqn:echswf}
ECH_*(Y,\lambda,\Gamma) \simeq \widehat{HM}^{-*}(Y,{\mathfrak}{s}_\xi +
{\operatorname}{PD}(\Gamma)),$$ as relatively graded ${{\mathbb Z}}/2$-modules. (There is also an isomorphism with ${{\mathbb Z}}$ coefficients [@e3].)
It follows from scrutiny of the proof of , together with the invariance properties of $\widehat{HM}^*$, that the versions of $ECH_*(Y,\lambda,\Gamma)$ defined using different almost complex structures $J$ are canonically isomorphic to each other. This point is explained in detail in [@cc2]. Thus it makes sense to talk about $ECH_*(Y,\lambda,\Gamma)$ without referring to a choice of $J$. Moreover, under this identification, the isomorphism is canonical.
At times it is convenient to ignore the homology class $\Gamma$ in the definition of $ECH$, and simply define $$ECH_*(Y,\lambda) {\;{:=}\;}\bigoplus_{\Gamma\in
H_1(Y)}ECH_*(Y,\lambda,\Gamma).$$ This is the homology of a chain complex $ECC_*(Y,\lambda;J)$ generated by all admissible orbit sets, and by this homology is canonically isomorphic (as a relatively graded ${{\mathbb Z}}/2$-module) to $$\widehat{HM}^{-*}(Y){\;{:=}\;}\bigoplus_{{\mathfrak}{s}\in{{\operatorname}{Spin}^c}(Y)}\widehat{HM}^{-*}(Y,{\mathfrak}{s}).$$
The proof of Theorem \[thm:cc\] makes use of two additional structures on ECH: the action filtration and cobordism maps. We now explain these.
The action filtration {#sec:filtration}
---------------------
If $\Theta=\{(\Theta_i,m_i)\}$ is an orbit set, its [*symplectic action*]{} or [*length*]{} is defined by $$\label{eqn:length}
{{\mathcal A}}(\Theta) {\;{:=}\;}\sum_i m_i \int_{\Theta_i}\gamma.$$ The ECH differential for any (generic) symplectization-admissible $J$ decreases the action, i.e. if $\langle\partial\Theta,\Theta'\rangle\neq 0$ then ${{\mathcal A}}(\Theta)\ge{{\mathcal A}}(\Theta')$. This is because if $C\in{{\mathcal M}}^J(\Theta,\Theta')$, then $d\lambda|_C\ge 0$ everywhere[^1]. Thus for any real number $L$, it makes sense to define the [*filtered ECH*]{}, denoted by $
ECH_*^{L}(Y,\lambda) $, to be the homology of the subcomplex $ECC_*^L(Y,\lambda;J)$ of the ECH chain complex spanned by ECH generators with action less than $L$. It is shown in [@cc2] that $ECH_*^{L}(Y,\lambda)$, just like $ECH_*(Y,\lambda)$, does not depend on the choice of ECH-generic $J$. However $ECH_*^L(Y,\lambda)$, unlike the usual ECH, can change when one deforms the contact form $\lambda$.
For $L<L'$ there is a map $$\label{eqn:istar}
\imath^{L,L'}:ECH_*^{L}(Y,\lambda) \longrightarrow
ECH_*^{L'}(Y,\lambda)$$ induced by the inclusion of chain complexes (for some given $J$, although it is shown in [@cc2] that does not depend on $J$). The usual ECH is recovered as the direct limit $$\label{eqn:edr}
ECH_*(Y,\lambda) = \lim_{L\to\infty}ECH_*^{L}(Y,\lambda).$$ In particular, there is a natural map $$\label{eqn:iL}
\imath^L:ECH_*^L(Y,\lambda) \longrightarrow ECH_*(Y,\lambda),$$ again induced by an inclusion of chain complexes.
Cobordism maps in ECH {#sec:cobech}
---------------------
Let $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ be closed oriented 3-manifolds with nondegenerate contact forms. An [*exact symplectic cobordism*]{} from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ is a compact symplectic 4-manifold $(X,\omega)$ with boundary $\partial X = Y_+-Y_-$, for which there exists a $1$-form $\lambda$ on $X$ such that $d\lambda=\omega$ on $X$ and $\lambda|_{Y_\pm}=\lambda_\pm$. A $1$-form $\lambda$ as above is called a [*Liouville form*]{} for $(X,\omega)$. When we wish to specify a Liouville form (which we usually do), we denote the exact symplectic cobordism by $(X,\lambda)$, and we continue to write $\omega=d\lambda$.
Note that our designation of the cobordism as “from $Y_+$ to $Y_-$” is natural from the perspective of symplectic geometry, but opposite from the usual convention in Seiberg-Witten and Heegaard Floer homology. This is connected with the fact that embedded contact [ *homology*]{} is identified with Seiberg-Witten Floer [*cohomology*]{}.
Now let $(X,\lambda)$ be an exact symplectic cobordism as above. This cobordism, like any smooth cobordism, induces a map[^2] of ungraded ${{\mathbb Z}}/2$-modules from the Seiberg-Witten Floer cohomology of $Y_+$ to that of $Y_-$, which we denote by $$\label{eqn:swcob}
\widehat{HM}^*(X): \widehat{HM}^{*}(Y_+) \longrightarrow \widehat{HM}^{*}(Y_-).$$
\[def:PhiX\] Define $$\label{eqn:gencob}
\Phi(X):
ECH_*(Y_+,\lambda_+) \longrightarrow ECH_*(Y_-,\lambda_-)$$ to be the map on ECH obtained by composing the map on Seiberg-Witten Floer cohomology with the canonical isomorphism on both sides.
It is natural to expect that the map can be defined directly, without using Seiberg-Witten theory, by suitably counting holomorphic curves in the “completion” of $(X,\lambda)$. The latter is a noncompact symplectic manifold defined as follows. To start, one can find $\varepsilon>0$, a neighborhood $N_-$ of $Y_-$ in $X$, and an identification $N_-\simeq [0,\varepsilon)\times Y_-$, such that $\lambda=e^s\lambda_-$ on $N_-$, where $s$ denotes the $[0,\varepsilon)$ coordinate. The requisite map $[0,\varepsilon)\times Y_-\to X$ is obtained using the flow starting at $Y_-$ of the unique vector field $V$ on $X$ such that $\imath_V\omega=\lambda$. Likewise, a neighborhood $N_+$ of $Y_+$ in $X$ can be identified with $(-\varepsilon,0]\times Y_+$ so that $\lambda=e^s\lambda_+$ on $N_+$. Using these identifications, one can then glue symplectization ends to $X$ to obtain the [*completion*]{} $$\label{eqn:completion}
\overline{X} {\;{:=}\;}((-\infty,0]\times Y_-) \cup_{Y_-} X \cup_{Y_+}
([0,\infty)\times Y_+).$$ Note for reference later that the Liouville form $\lambda$ on $X$ canonically extends to a $1$-form on $\overline{X}$ which equals $e^s\lambda_\pm$ on the ends.
\[def:cobadm\] An almost complex structure $J$ on $\overline{X}$ is [ *cobordism-admissible*]{} if it is $\omega$-compatible[^3] on $X$, and if it agrees with symplectization-admissible almost complex structures $J_+$ for $\lambda_+$ on $[0,\infty)\times Y_+$ and $J_-$ for $\lambda_-$ on $(-\infty,0]\times Y_-$.
Given a cobordism-admissible $J$, and given (not necessarily admissible) orbit sets $\Theta^+=\{(\Theta_i^+,m_i^+)\}$ in $Y_+$ and $\Theta^-=\{(\Theta_j^-,m_i^-)\}$ in $Y_-$, we define a “$J$-holomorphic curve in $\overline{X}$ from $\Theta^+$ to $\Theta^-$” analogously to Definition \[def:Jhol\], and denote the moduli space of such curves by ${{\mathcal M}}^J(\Theta^+,\Theta^-)$, where two such curves are considered equivalent if they represent the same current in $\overline{X}$.
One would now like to define the map by choosing a generic cobordism-admissible $J$ and suitably counting $J$-holomorphic curves in $\overline{X}$ as above with ECH index 0, so as to define a chain map between the ECH chain complexes which induces the map on homology. (In general one also needs to include contributions from “broken” $J$-holomorphic curves, see §\[sec:exact\].) An important consequence of such a construction would be that the map respects the action filtrations, i.e. is induced by a chain map which does not increase the action filtration. The reason is that if $C$ is any holomorphic curve in ${{\mathcal M}}^J(\Theta^+,\Theta^-)$, then by Stokes’ theorem and the exactness of the cobordism we have $$\label{eqn:stokes}
{{\mathcal A}}(\Theta^+)-{{\mathcal A}}(\Theta^-) = \int_{C\cap [0,\infty)\times
Y_+}d\lambda_+ + \int_{C\cap X}\omega + \int_{C\cap
(-\infty,0]\times Y_-}d\lambda_-,$$ and all of the integrands on the right hand side are pointwise nonnegative by our assumptions on $J$.
Unfortunately it is not currently known how to define the map in terms of holomorphic curves as above. The difficulty is that, as explained in [@ir §5], the compactifications of the relevant moduli spaces of holomorphic curves can include broken curves with negative index multiply covered components, and it is not clear in general what these should contribute to the count (although examples show that such broken curves must sometimes make nonzero contributions). However we can still use Seiberg-Witten theory to show that the map respects the action filtrations (in a slightly weaker sense than above), and enjoys some other useful properties which would follow from a definition in terms of holomorphic curves. The precise statement uses filtered ECH and is given in Theorem \[thm:cob\] below.
The basic idea of the proof of Theorem \[thm:cob\] is to perturb the Seiberg-Witten equations on $\overline{X}$ using a large multiple of the symplectic form, much as in the proof of , and to show that with such a perturbation, Seiberg-Witten solutions that contribute to the cobordism map give rise to (possibly broken) holomorphic curves. The main analytical machinery is adapted from the proof of in [@e1; @e4]. Nonetheless the detailed proof is still long, so we have deferred it to the sequel [@cc2].
Legendrian surgery
------------------
Returning finally to the chord conjecture, let $(Y_0,\lambda_0)$ be a closed oriented 3-manifold with a contact form, and let ${{\mathcal K}}$ be a Legendrian knot in $(Y_0,\lambda_0)$. The contact structure determines a framing of ${{\mathcal K}}$, which we denote by $tb({{\mathcal K}})$. Let $Y_1$ denote the 3-manifold obtained by surgery on ${{\mathcal K}}$ with framing $tb({{\mathcal K}})-1$. The surgery procedure determines a smooth cobordism $X$ from $Y_1$ to $Y_0$. As was shown in [@weinstein] and as we review in §\[sec:ls\], the 3-manifold $Y_1$ has a natural contact structure, which can be expressed as the kernel of a contact form $\lambda_1$ such that $X$ has the structure of an exact symplectic cobordism from $(Y_1,\lambda_1)$ to $(Y_0,\lambda_0)$. Moreover, as is familiar from the work of Bourgeois-Ekholm-Eliashberg [@bee] on Legendrian surgery in contact homology, the contact form $\lambda_1$ can be chosen so that, modulo “long” Reeb orbits, one has:
[(\*)]{} The Reeb orbits of $\lambda_1$ correspond to the Reeb orbits of $\lambda_0$, together with cyclic words in the Reeb chords of ${{\mathcal K}}$.
In particular, if ${{\mathcal K}}$ has no Reeb chord, then $\lambda_1$ and $\lambda_0$ have the same “short” Reeb orbits.
The idea of the proof of the chord conjecture is to use the preceding observation, together with Theorem \[thm:cob\] regarding the properties of ECH cobordism maps, to show that if there is no Reeb chord then the ECH cobordism map $$\label{eqn:surcob}
\Phi(X): ECH_*(Y_1,\lambda_1) \longrightarrow ECH_*(Y_0,\lambda_0)$$ induced by the Legendrian surgery cobordism is an isomorphism. Note that this is what one would expect by analogy with a very special case of the aforementioned work of Bourgeois-Ekholm-Eliashberg.
But the map cannot be an isomorphism, because this would contradict results of Kronheimer-Mrowka, namely:
\[lem:ni\] If $Y_1$ is obtained from a closed oriented 3-manifold $Y_0$ by surgery along a knot ${{\mathcal K}}$, and if $X$ denotes the corresponding smooth cobordism from $Y_1$ to $Y_0$, then the induced map on Seiberg-Witten Floer cohomology with ${{\mathbb Z}}/2$ coefficients, $$\label{eqn:ni}
\widehat{HM}^*(X):\widehat{HM}^*(Y_1) \longrightarrow
\widehat{HM}^*(Y_0),$$ is not an isomorphism.
It follows from [@km Thm. 42.2.1], see also [@bloom; @kmos], that there is an exact triangle $$\cdots
\longrightarrow
\widehat{HM}^*(Y_2) \longrightarrow \widehat{HM}^*(Y_1)
\stackrel{\widehat{HM}^*(X)}{\longrightarrow} \widehat{HM}^*(Y_0)
\longrightarrow \widehat{HM}^*(Y_2) \longrightarrow \cdots$$ where $Y_2$ is obtained from $Y_0$ by a certain different surgery along ${{\mathcal K}}$. Note that the exact triangle was only proved over ${{\mathbb Z}}/2$, so it is fortunate that we are using ${{\mathbb Z}}/2$ coefficients everywhere. Now the exact triangle implies that if were an isomorphism, then $\widehat{HM}^*(Y_2)$ would vanish. However the latter is nontrivial, because it follows from [@km Cor.35.1.4] that for any 3-manifold $Y$, if ${\mathfrak}{s}$ is a torsion spin-c structure on $Y$ (these always exist), then $\widehat{HM}^*(Y,{\mathfrak}{s})$ is infinitely generated. This is proved in [@km] with ${{\mathbb Z}}$ coefficients, which immediately implies the statement with ${{\mathbb Z}}/2$ coefficients.
There are two wrinkles in the above argument. First, statement (\*) is true only for Reeb orbits whose action is not too large, where the definition of “large” depends on the details of the Legendrian surgery construction. However one can modify $\lambda_1$ so as to make the corresponding upper bound on the action arbitrary large, see Lemma \[lem:ro\] below. Moreover the different versions of $(Y_1,\lambda_1)$ fit into a sequence of exact cobordisms. As a result, by making appropriate use of the cobordism maps on filtered ECH, we can still show that if there is no Reeb chord then the ECH cobordism map induced by the Legendrian surgery is an isomorphism.
Second, the above argument only makes sense if the contact form $\lambda_0$ (and with it the contact form $\lambda_1$) is nondegenerate, so that its ECH chain complex is well-defined. A priori there could exist a degenerate contact form and a Legendrian knot with no Reeb chord, such that for any nondegenerate perturbation of the contact form the knot does have a Reeb chord. To deal with this issue, we will show that when $\lambda_0$ is nondegenerate, there exists a Reeb chord with an upper bound on its symplectic action, given by a quantitative measure of the failure of the cobordism map to be an isomorphism. The precise statement is given in Theorem \[thm:nondegenerate\] below. The aforementioned upper bound depends “continuously” on the contact form, as shown in Proposition \[prop:continuity\]. It then follows from a compactness argument that the chord conjecture holds in the degenerate case as well.
#### Contents of the rest of the paper.
In §\[sec:exact\] we give the precise statement of Theorem \[thm:cob\] on the existence and properties of maps on (filtered) ECH induced by exact symplectic cobordisms. In §\[sec:ls\]–§\[sec:ccdegenerate\] we use Theorem \[thm:cob\] as a “black box” to prove the chord conjecture. The formal proof of the chord conjecture is put together at the end of §\[sec:ccdegenerate\]. In the sequel [@cc2] we use Seiberg-Witten theory to prove Theorem \[thm:cob\].
#### Acknowledgments.
We thank Jonathan Bloom, Tobias Ekholm, Yasha Eliashberg, Ko Honda, Dusa McDuff, Tomasz Mrowka, and Ivan Smith for helpful discussions. The first author was partially supported by NSF grant DMS-0806037. The second author was partially supported by the Clay Mathematics Insitute, the Mathematical Sciences Research Institute, and the NSF. Both authors thank MSRI, where this work was carried out, for its hospitality.
ECH and exact symplectic cobordisms {#sec:exact}
===================================
We now state the theorem on the existence and properties of maps on (filtered) ECH induced by exact symplectic cobordisms.
We need some preliminary definitions. Below, let $(X,\lambda)$ be an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$, and assume that the contact forms $\lambda_\pm$ are nondegenerate. Fix a cobordism-admissible almost complex structure $J$ on $\overline{X}$ which restricts to symplectization-admissible almost complex structures $J_+$ on $[0,\infty)\times Y_+$ and $J_-$ on $(-\infty,0]\times Y_-$, as in Definition \[def:cobadm\].
#### Broken curves.
Let $\Theta^+$ and $\Theta^-$ be (not necessarily admissible) orbit sets in $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ respectively.
\[def:broken\] A [*broken $J$-holomorphic curve from $\Theta^+$ to $\Theta^-$*]{} is a collection of holomorphic curves $\{C_k\}_{1\le k\le N}$, and (not necessarily admissible) orbit sets $\Theta^{k+}$ and $\Theta^{k-}$ for each $k$, such that there exists $k_0\in\{1,\ldots,N\}$ such that:
- $\Theta^{k+}$ is an orbit set in $(Y_+,\lambda_+)$ for each $k\ge k_0$; $\Theta^{k-}$ is an orbit set in $(Y_-,\lambda_-)$ for each $k\le k_0$; $\Theta^{N+}=\Theta^+$; $\Theta^{1-}=\Theta^-$; and $\Theta^{k-}=\Theta^{k-1,+}$ for each $k>1$.
- If $k>k_0$ then $C_k\in{{\mathcal M}}^{J_+}(\Theta^{k+},\Theta^{k-})$; $C_{k_0}\in{{\mathcal M}}^J(\Theta^{k_0,+},\Theta^{k_0,-})$; and if $k<k_0$ then $C_k\in{{\mathcal M}}^{J_-}(\Theta^{k+},\Theta^{k-})$.
- If $k\neq k_0$ then $C_k$ is not ${{\mathbb R}}$-invariant (as a current).
Let $\overline{{{\mathcal M}}^J(\Theta^+,\Theta^-)}$ denote the moduli space of broken $J$-holomorphic curves from $\Theta^+$ to $\Theta^-$ as above.
Note that ${{\mathcal M}}^J(\Theta^+,\Theta^-)$ is a subset of $\overline{{{\mathcal M}}^J(\Theta^+,\Theta^-)}$ corresponding to broken curves as above in which $N=1$ (and it is perhaps a misnomer to call such curves “broken”).
#### Product cylinders.
If the cobordism $(X,\lambda)$ and the almost complex structure $J$ on $\overline{X}$ are very special, then $X$ may contain regions that look like pieces of a symplectization, in the following sense:
\[def:PR\] A [*product region*]{} in $X$ is the image of an embedding $[s_-,s_+]\times Z \to X$, where $s_-<s_+$ and $Z$ is an open 3-manifold, such that:
- $\{s_\pm\}\times Z$ maps to $Y_\pm$, and $(s_-,s_+)\times Z$ maps to the interior of $X$.
- The pullback of the Liouville form $\lambda$ to $[s_-,s_+]\times
Z$ has the form $e^s\lambda_0$, where $s$ denotes the $[s_-,s_+]$ coordinate, and $\lambda_0$ is a contact form on $Z$.
- The pullback of the almost complex structure $J$ to $[s_-,s_+]\times Z$ has the following properties:
- The restriction of $J$ to ${{\operatorname}{Ker}}(\lambda_0)$ is independent of $s$.
- $J(\partial/\partial s)=f(s)R_0$, where $f$ is a positive function of $s$ and $R_0$ denotes the Reeb vector field for $\lambda_0$.
Given a product region as above, the embedded Reeb orbits of $\lambda_\pm$ in $\{s_\pm\}\times Z$ are identified with the embedded Reeb orbits of $\lambda_0$ in $Z$. If $\gamma$ is such a Reeb orbit, then we can form a $J$-holomorphic cylinder in $\overline{X}$ by taking the union of $[s_-,s_+]\times \gamma$ in $[s_-,s_+]\times Z$ with $(-\infty,0]\times \gamma$ in $(-\infty,0]\times Y_-$ and $[0,\infty)\times\gamma$ in $[0,\infty)\times Y_+$.
\[def:PC\] We call a $J$-holomorphic cylinder as above a [*product cylinder.*]{}
#### Composition of cobordisms.
If $(X_1,\lambda_1)$ is an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_0,\lambda_0)$, and if $(X_2,\lambda_2)$ is an exact symplectic cobordism from $(Y_0,\lambda_0)$ to $(Y_-,\lambda_-)$, then we can compose them to obtain an exact symplectic cobordism $(X_2\circ X_1,\lambda)$ from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$. Here $X_2\circ X_1$ is obtained by gluing $X_1$ and $X_2$ along $Y_0$ analogously to , and $\lambda|_{X_i}=\lambda_i$ for $i=1,2$.
#### Homotopy of cobordisms.
Two exact symplectic cobordisms $(X,\omega_0)$ and $(X,\omega_1)$ from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ with the same underlying four-manifold $X$ are [*homotopic*]{} if there is a one-parameter family of symplectic forms $\{\omega_t\mid t\in[0,1]\}$ on $X$ such that $(X,\omega_t)$ is an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$ for each $t\in[0,1]$.
#### Scaling.
If $\lambda$ is a nondegenerate contact form on $Y$, and if $c$ is a positive constant, then there is a canonical “scaling” isomorphism $$\label{eqn:scaling}
s:ECH_*^{L}(Y,\lambda) \stackrel{\simeq}{\longrightarrow}
ECH_*^{cL}(Y,c\lambda).$$ To see this, observe that the chain complexes on both sides have the same generators. Moreover, an ECH-generic almost complex structure $J$ for $\lambda$ induces a symplectization-admissible almost complex structure $J^c$ for $c\lambda$, such that $J$ and $J^c$ agree when restricted to the contact planes $\xi$. The self-diffeomorphism of ${{\mathbb R}}\times Y$ sending $(s,y)\mapsto (cs,y)$ then induces a bijection between $J$-holomorphic curves and $J^c$-holomorphic curves. So with these choices, the canonical identification of generators is an isomorphism of chain complexes. Moreover, it is shown in [@cc2] that the resulting isomorphism does not depend on $J$ (under the canonical isomorphisms between the versions of ECH defined using different almost complex structures).
\[thm:cob\] Let $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ be closed oriented 3-manifolds with nondegenerate contact forms. Let $(X,\lambda)$ be an exact symplectic cobordism from $(Y_+,\lambda_+)$ to $(Y_-,\lambda_-)$. Then there exist maps (of ungraded ${{\mathbb Z}}/2$-modules) $$\label{eqn:PhiL}
\Phi^L(X,\lambda): ECH_*^{L}(Y_+,\lambda_+) \longrightarrow
ECH_*^{L}(Y_-,\lambda_-)$$ for each real number $L$, such that:
[(Homotopy Invariance)]{} The map $\Phi^L(X,\lambda)$ depends only on $L$ and the homotopy class of $(X,\omega)$.
[(Inclusion)]{} If $L<L'$ then the following diagram commutes: $$\begin{CD}
ECH_*^{L}(Y_+,\lambda_+) @>{\Phi^L(X,\lambda)}>> ECH_*^{L}(Y_-,\lambda_-) \\
@VV{\imath^{L,L'}}V @VV{\imath^{L,L'}}V \\
ECH_*^{L'}(Y_+,\lambda_+) @>{\Phi^{L'}(X,\lambda)}>> ECH_*^{L'}(Y_-,\lambda_-).
\end{CD}$$
[(Direct Limit)]{} $$\lim_{L\to\infty}\Phi^L(X,\lambda) = \Phi(X): ECH_*(Y_+,\lambda_+) \longrightarrow
ECH_*(Y_-,\lambda_-),$$ where $\Phi(X)$ is as in Definition \[def:PhiX\].
[(Composition)]{} If $(X,\lambda)$ is the composition of $(X_2,\lambda_2)$ and $(X_1,\lambda_1)$ as above with $\lambda_0$ nondegenerate, then $$\Phi^L(X_2\circ X_1,\lambda) = \Phi^L(X_2,\lambda_2) \circ
\Phi^L(X_1,\lambda_1).$$
[(Scaling)]{} If $c$ is a positive constant then the following diagram commutes: $$\begin{CD}
ECH_*^{L}(Y_+,\lambda_+) @>{\Phi^L(X,\lambda)}>> ECH_*^{L}(Y_-,\lambda_-) \\
@VV{s}V @VV{s}V \\
ECH_*^{cL}(Y_+,c\lambda_+) @>{\Phi^{cL}(X,c\lambda)}>> ECH_*^{cL}(Y_-,c\lambda_-).
\end{CD}$$
[(Holomorphic Curves)]{} Let $J$ be a cobordism-admissible almost complex structure on $\overline{X}$ such that $J_+$ and $J_-$ are ECH-generic. Then there exists a (noncanonical) chain map $$\hat{\Phi}^L : ECC_*^L(Y_+,\lambda_+,J_+) \longrightarrow
ECC_*^L(Y_-,\lambda_-,J_-)$$ inducing $\Phi^L(X,\lambda)$, such that if $\Theta^+$ and $\Theta^-$ are ECH generators for $(Y_+,\lambda_+)$ and $(Y_-,\lambda_-)$ respectively with action less than $L$, then:
[(i)]{} If there are no broken $J$-holomorphic curves in $\overline{X}$ from $\Theta^+$ to $\Theta^-$, then $\langle \hat{\Phi}^L\Theta^+,\Theta^-\rangle=0$.
[(ii)]{} If the only broken $J$-holomorphic curve in $\overline{X}$ from $\Theta^+$ to $\Theta^-$ is a union of covers of product cylinders, then $\langle \hat{\Phi}^L\Theta^+,\Theta^-\rangle=1$.
For any three-manifold with a nondegenerate contact form, the empty set of Reeb orbits is a cycle in the ECH chain complex (whose homology class in ECH corresponds to the “contact invariant” in Seiberg-Witten Floer cohomology, see [@e5]). If $\emptyset_\pm$ denotes the empty set of Reeb orbits, regarded as a generator of the ECH chain complex for $Y_\pm$, then it follows from the Holomorphic Curves axiom that $$\Phi^L(X,\lambda): [\emptyset_+] \longmapsto [\emptyset_-].$$ The reason is that for any cobordism-admissible almost complex structure $J$, by there is a unique $J$-holomorphic curve with no positive end, namely the empty holomorphic curve.
Theorem \[thm:cob\] has applications beyond the chord conjecture, for example to symplectic embedding obstructions [@qech]. The Scaling axiom is not needed for the proof of the chord conjecture, but is useful in these other applications.
Legendrian surgery {#sec:ls}
==================
We now explain the details of the Legendrian surgery construction we will use. In particular we define a sequence of Legendrian surgeries, related to each other by exact symplectic cobordisms, in which the Reeb vector field is increasingly well-behaved.
To begin, recall that a [*Liouville vector field*]{} on a symplectic manifold $(X,\omega)$ is a vector field $V$ such that ${{\mathcal L}}_V\omega = \omega$. A Liouville vector field $V$ is equivalent to a $1$-form $\lambda$ such that $d\lambda=\omega$, via the equation $\lambda=\imath_V\omega$.
If $Y$ is a hypersurface in $(X,\omega)$ transverse to a Liouville vector field $V$, then $\lambda_Y{\;{:=}\;}\lambda|_Y$ is a contact form on $Y$. Now let $Y'$ be another hypersurface transverse to $V$, and suppose that the time $t$ flow of $V$ defines a diffeomorphism $\phi:Y\to Y'$, where $t$ is some function on $Y$. Then the contact forms on $Y$ and $Y'$ are related by $$\label{eqn:et}
\phi^*\lambda_{Y'} = e^t\lambda_Y.$$
With the above preliminaries out of the way, consider now a closed oriented 3-manifold $Y_0$ with a nondegenerate contact form $\lambda_0$. Let ${{\mathcal K}}$ be a Legendrian knot in $(Y_0,\lambda_0)$. Let $Y_1$ be the 3-manifold obtained by surgery along ${{\mathcal K}}$ with framing $tb({{\mathcal K}})-1$.
\[prop:ls\] There exist:
- a nondegenerate contact form $\lambda_1$ on $Y_1$,
- an exact symplectic cobordism $(X,\lambda)$ from $(Y_1,\lambda_1)$ to $(Y_0,\lambda_0)$; let $V$ denote its associated associated Liouville vector field;
- a compact hypersurface $Y_{1/n}$ in $X$ transverse to $V$ for each positive integer $n$,
- and neighborhoods $U_{1/n}$ of ${{\mathcal K}}$ in $Y_0$ with $U_{1/n}\supset U_{1/(n+1)}$ and $\bigcap_{n=1}^\infty
U_{1/n}={{\mathcal K}}$,
with the following properties:
[(a)]{} The induced contact form $\lambda_{1/n}$ on $Y_{1/n}$ is nondegenerate.
[(b)]{} The negative time flow of $V$ induces a diffeomorphism $Y_{1/n}\stackrel{\simeq}{\to} Y_{1/(n+1)}$. (The flow time varies over $Y_{1/n}$.)
[(c)]{} The time $-1/n$ flow of $V$, call it $\phi_{-1/n}$, is defined on all of $Y_{1/n}$. There is a subset $\widetilde{U}_{1/n}$ of $Y_{1/n}$ such that $$\phi_{-1/n}(Y_{1/n}\setminus \widetilde{U}_{1/n}) = Y_0\setminus
U_{1/n}.$$
[(d)]{} The Reeb vector field on $(Y_{1/n},\lambda_{1/n})$ has no closed orbits contained entirely within $\widetilde{U}_{1/n}$.
The idea for building the cobordism $(X,\lambda)$ is to start with the exact symplectic cobordism $([0,1]\times Y_0,e^s\lambda_0)$, where $s$ denotes the $[0,1]$ coordinate. One then attaches a 2-handle with an appropriate Liouville form to $\{1\}\times Y_0$ in a neighborhood of $\{1\}\times{{\mathcal K}}$. We proceed in four steps.
[*Step 1.*]{} We first describe a model for the handle attachment, following [@weinstein]. Consider ${{\mathbb R}}^4$ with coordinates $q_1,q_2,p_1,p_2$ and the symplectic form $\omega=\sum_{i=1}^2
dp_i\, dq_i$. Define a Liouville vector field on ${{\mathbb R}}^4$ by $$V = \sum_{i=1}^2 \left(-p_i\frac{\partial}{\partial p_i} +
2q_i\frac{\partial}{\partial q_i}\right).$$ Consider the hypersurface $$Y = (p_1^2+p_2^2=1)\subset {{\mathbb R}}^4,$$ regarded as the boundary of $(p_1^2+p_2^2\ge 1)$. The Liouville vector field $V$ is transverse to $Y$ and so induces a contact form on $Y$. With respect to this contact form, the circle $$C=(q_1=q_2=0,p_1^2+p_2^2=1)$$ is a Legendrian knot. To the region $(p_1^2+p_2^2\ge 1)$ we now attach the $2$-handle consisting of the subset of ${{\mathbb R}}^4$ where $$p_1^2+p_2^2\le 1, \quad \quad q_1^2+q_2^2\le \varepsilon$$ for some $\varepsilon>0$. The boundary of the region with the 2-handle attached has a corner where $p_1^2+p_2^2=1$ and $q_1^2+q_2^2
= \varepsilon$. To round the corner, we replace the boundary hypersurface $q_1^2+q_2^2=\varepsilon$ of the handle with a nearby hypersurface, staying within the region $\varepsilon/2<q_1^2+q_2^2\le\varepsilon$, and defined by an equation of the form $$\label{eqn:fqp}
f(q_1^2+q_2^2,p_1^2+p_2^2)=0,$$ where at each point on the zero set of $f$ we have $\partial
f/\partial x > 0$ and $\partial f/\partial y \le 0$. The boundary of the region with the 2-handle attached is then a smooth hypersurface $Y'$ which is transverse to the Liouville vector field $V$.
[*Step 2.*]{} We now pass from the model case to the case of interest. By [@weinstein Prop. 4.2], there is a diffeomorphism of a neighborhood $N$ of $C$ in ${{\mathbb R}}^4$ with a neighborhood of $\{1\}\times {{\mathcal K}}$ in the symplectization ${{\mathbb R}}\times Y_0$, which respects the symplectic forms and Liouville vector fields and locally identifies the hypersurface $Y$ in ${{\mathbb R}}^4$ with the hypersurface $\{1\}\times Y_0$ in ${{\mathbb R}}\times Y_0$. If $\varepsilon>0$ is sufficiently small, then $N\cap Y$ will contain the region in $Y$ to which the $2$-handle in ${{\mathbb R}}^4$ is attached. We then use the above diffeomorphism to attach the $2$-handle described above in ${{\mathbb R}}^4$, with its symplectic form and Liouville vector field, to $[0,1]\times
Y_0$. We now provisionally define $X$ to be the resulting exact symplectic cobordism, and $(Y_1,\lambda_1)$ to be its positive boundary with the induced contact form. It is not hard to check that as a smooth 3-manifold, $Y_1$ is obtained from $Y_0$ by surgery on ${{\mathcal K}}$ with framing $tb({{\mathcal K}})-1$. We also define $\widetilde{U}_1$ to be the part of $Y_1$ in the handle, and $U_1=Y_0\setminus \phi_{-1}(Y_1\setminus\widetilde{U}_1)$. That is, $U_1\subset Y_0$ corresponds to the subset of $\{1\}\times Y_0$ to which the handle is attached.
[*Step 3.*]{} We now check that the Reeb vector field $R_1$ on $Y_1$ has the required properties. We first show that $R_1$ has no closed orbit contained in $\widetilde{U}_1$. On $\widetilde{U}_1$, in terms of the coordinates on ${{\mathbb R}}^4$, the Reeb vector field $R_1$ is parallel to the Hamiltonian vector field associated to the function . Thus $$gR_1 = \frac{\partial f}{\partial
x}\left(q_1\frac{\partial}{\partial p_1} +
q_2\frac{\partial}{\partial p_2}\right) -
\frac{\partial f}{\partial y}\left(p_1\frac{\partial}{\partial q_1}
+ p_2\frac{\partial}{\partial q_2}\right),$$ where $g$ is some positive function on $\widetilde{U}_1$. Now define another function $h$ on $\widetilde{U}_1$ by $$h {\;{:=}\;}p_1q_1 + p_2q_2.$$ We then compute that $$\label{eqn:Rh}
R_1(h)>0$$ on all of $\widetilde{U}_1$. It follows immediately that $R_1$ has no closed orbit contained in $\widetilde{U}_1$.
Next we consider nondegeneracy of $\lambda_1$. By construction, $\lambda_1$ is a constant multiple of $\lambda_0$ outside of $\widetilde{U}_1$. Since $\lambda_0$ was assumed nondegenerate, it follows that any Reeb orbit for $\lambda_1$ that avoids the region $\widetilde{U}_1$ is nondegenerate as well. Consequently we can make $\lambda_1$ nondegenerate by perturbing it (specifically, multiplying it by a positive function close to $1$) in $\widetilde{U}_1$. By equation , such a perturbation of $\lambda_1$ can be effected by perturbing the hypersurface $\widetilde{U}_1$ in the definition of $X$. If this perturbation is sufficiently $C^1$-small, then will still hold, so the Reeb vector field of $\lambda_1$ will now have all of the required properties.
[*Step 4.*]{} The hypersurface $Y_{1/n}\subset X$ is now defined to be the positive boundary of the region obtained by starting with $[0,1/n]\times Y_0$ and attaching a taller and thinner $2$-handle. This handle is obtained by starting with the subset of ${{\mathbb R}}^4$ where $$p_1^2+p_2^2 \le e^{2(1-1/n)}, \quad\quad q_1^2+q_1^2 \le
2^{1-n}\varepsilon,$$ then rounding corners as before and perturbing if necessary to make $\lambda_{1/n}$ nondegenerate. Finally, one defines $\widetilde{U}_{1/n}$ to be the part of $Y_{1/n}$ in the handle, and $U_{1/n}=Y_0\setminus\phi_{-1/n}(Y_1\setminus\widetilde{U}_1)$.
A basic consequence of the above construction is the following:
\[lem:ro\] Suppose ${{\mathcal K}}$ has no Reeb chord with action $\le L$. Then for all $n$ sufficiently large:
[(a)]{} The Reeb orbits of $\lambda_{1/n}$ with action $<e^{1/n}L$ avoid the region $\widetilde{U}_{1/n}$.
[(b)]{} $\phi_{-1/n}$ defines a bijection from the Reeb orbits of $\lambda_{1/n}$ with action $<e^{1/n}L$ to the Reeb orbits of $\lambda_0$ with action $<L$.
Suppose $\{n_k\}_{k=1,2,\ldots}$ is an increasing sequence of positive integers and that for each $k$ there exists a Reeb orbit $\gamma_k$ for $\lambda_{1/n_k}$ of action $<e^{1/n_k}L$ intersecting $\widetilde{U}_{1/n_k}$. Then for each $k$, the set $\phi_{-1/n_k}(\gamma_k\cap(Y_{1/n_k}\setminus\widetilde{U}_{1/n_k}))$ is a union of Reeb trajectories of $\lambda_0$ starting and ending on the boundary of $U_{1/n_k}$ with total action less than $L$. Picking one of these trajectories for each $k$, we can pass to a subsequence so that these trajectories converge to a Reeb chord with action $\le L$. This proves (a). Similarly, if $n$ is sufficiently large then the Reeb orbits of $\lambda_0$ with action $<L$ avoid the region $U_{1/n}$. This together with (a) implies (b).
The chord conjecture: nondegenerate case {#sec:ccnondegenerate}
========================================
We now prove the chord conjecture, Theorem \[thm:cc\], in the case when the contact form $\lambda_0$ is nondegenerate. Below, we use the notation from the Legendrian surgery construction in Proposition \[prop:ls\]. Also, to shorten the notation we write $H_*(Y,\lambda)$ to denote $ECH_*(Y,\lambda)$, and $H_*^L(Y,\lambda)$ to denote $ECH_*^{L}(Y,\lambda)$.
Observe that by the construction in §\[sec:ls\], the exact symplectic cobordism $(X,\lambda)$ contains an exact symplectic cobordism from $(Y_{1/n},\lambda_{1/n})$ to $(Y_0,\lambda_0)$, call this $X_{1/n}$. The main lemma is now:
\[lem:main\] Let $L>0$. Suppose that ${{\mathcal K}}$ has no Reeb chord of action $\le L$. Then for all $n$ sufficiently large, the cobordism map $$\label{eqn:cob1}
\Phi^{e^{1/n}L}(X_{1/n},\lambda): H_*^{e^{1/n}L}(Y_{1/n},\lambda_{1/n}) \longrightarrow
H_*^{e^{1/n}L}(Y_0,\lambda_0)$$ is the composition of an isomorphism $$\label{eqn:iso1}
H_*^{e^{1/n}L}(Y_{1/n},\lambda_{1/n}) \stackrel{\simeq}{\longrightarrow}
H_*^L(Y_0,\lambda_0)$$ with the inclusion-induced map $\imath^{L,e^{1/n}L}: H_*^L(Y_0,\lambda_0)\to
H_*^{e^{1/n}L}(Y_0,\lambda_0)$.
By Lemma \[lem:ro\], if $n$ is sufficiently large, then the Reeb orbits for $\lambda_{1/n}$ of action less than $e^{1/n}L$ correspond via $\phi_{-1/n}$ to the Reeb orbits for $\lambda_0$ of action less than $L$, and the latter stay outside of the neighborhood $U_{1/n}$ of ${{\mathcal K}}$. Let $n$ be so large.
By Proposition \[prop:ls\](c), the flow of the Liouville vector field $V$ starting on $Y_{1/n}$ for times in the interval $[-1/n,0]$ defines an embedding of $[-1/n,0]\times Y_{1/n}$ into $X_{1/n}$. Let $X_{1/n}^0$ denote the image of this embedding. We identify $X_{1/n}^0$ with $[0,1/n]\times Y_{1/n}$ such that $Y_{1/n}$ is identified with $\{1/n\}\times Y_{1/n}$, and the Liouville vector field $V=\partial/\partial s$, where $s$ denotes the $[0,1/n]$ coordinate. Then $\{0\}\times Y_{1/n}$ defines a hypersurface in $X_{1/n}$ which includes $Y_0\setminus U_{1/n}\subset \partial
X_{1/n}$, and which also passes into the interior of $X_{1/n}$. Let $X_{1/n}^1$ denote $X_{1/n}\setminus X_{1/n}^0$. We can now decompose the completed cobordism $\overline{X_{1/n}}$ as $$\label{eqn:X1n}
\overline{X_{1/n}} = ((-\infty,0]\times Y_0) \cup X_{1/n}^1
\cup ([0,\infty)\times Y_{1/n}),$$ where $X_{1/n}^0$ corresponds to $[0,1/n]\times Y_{1/n}$ in .
We now choose a cobordism-admissible almost complex structure $J$ on $\overline{X_{1/n}}$ in four steps as follows. First, let $J_+$ be an almost complex structure on ${{\mathbb R}}\times Y_{1/n}$ which is symplectization-admissible with respect to $\lambda_{1/n}$ and ECH-generic. Require $J$ to agree with $J_+$ on $[1/n,\infty)\times
Y_{1/n}$. Second, extend $J$ over $[0,1/n]\times Y_{1/n}$ by setting $J=J_+$ on ${{\operatorname}{Ker}}(\lambda_{1/n})$, and $J(\partial_s)=f(s)R_{1/n}$, where $R_{1/n}$ denotes the Reeb vector field associated to $\lambda_{1/n}$, and $f:[0,1/n]\to{{\mathbb R}}$ is a positive function which equals $1$ near $s=1/n$ and which equals $e^{1/n}$ near $s=0$. Third, extend $J$ over $(-\infty,0]\times Y_0$ so that it agrees with an almost complex structure $J_-$ on ${{\mathbb R}}\times Y_0$ which is symplectization-admissible for $\lambda_0$ and ECH-generic. Note that one can arrange for $J_-$ to be ECH-generic without disturbing the previous choices because $J_+$ is ECH-generic. To complete the construction of $J$, choose an arbitrary $\omega$-compatible extension of $J$ over $X_{1/n}^1$.
With the above choices, $[0,1/n]\times
(Y_{1/n}\setminus\widetilde{U}_{1/n})$ is a product region in the sense of Definition \[def:PR\]. In particular, let $\Theta$ be an ECH generator for $\lambda_{1/n}$ of action less than $e^{1/n}L$. Since the Reeb orbits in $\Theta$ stay out of the region $\widetilde{U}_{1/n}$, there is a union of covers of product cylinders (see Definition \[def:PC\]) in $\overline{X_{1/n}}$ from $\Theta$ to $\phi_{-1/n}(\Theta)$.
We claim that if $C$ is any other $J$-holomorphic curve in $\overline{X_{1/n}}$ from the above $\Theta$ to an ECH generator $\Theta'$ for $\lambda_0$, then $$\label{eqn:actionClaim}
e^{-1/n}{{\mathcal A}}(\Theta)>{{\mathcal A}}(\Theta'),$$ where ${{\mathcal A}}$ denotes the symplectic action. To prove , observe that the Liouville form $\lambda$ on $\overline{X_{1/n}}$ agrees with $e^s\lambda_0$ on $(-\infty,0]\times
Y_0$ and agrees with $e^{s-1/n}\lambda_{1/n}$ on $[0,\infty)\times
Y_{1/n}$ in . Using Stokes’ theorem, we obtain $$\begin{split}
e^{-1/n}{{\mathcal A}}(\Theta) - {{\mathcal A}}(\Theta') = & \int_{C\cap ([0,\infty)\times
Y_{1/n})}d\left(e^{-1/n}\lambda_{1/n}\right)\\
&+ \int_{C\cap X_{1/n}^1}d\lambda\\
& + \int_{C\cap ((-\infty,0]\times Y_0)}d\lambda_0.
\end{split}$$ The first and third integrals on the right are pointwise nonnegative, and zero only where $C$ is tangent to $\partial_s$. The second integral on the right is pointwise positive. We conclude that $e^{-1/n}{{\mathcal A}}(\Theta)-{{\mathcal A}}(\Theta')\ge 0$, with equality if and only if $C$ is a union of covers of product cylinders.
Consider now the chain map $\hat{\Phi}^{e^{1/n}L}(X_{1/n},\lambda)$ inducing provided by the Holomorphic Curves axiom in Theorem \[thm:cob\]. It follows from that this chain map is a composition of chain maps $$ECC_*^{e^{1/n}L}(Y_{1/n},\lambda_{1/n};J_+)\longrightarrow
ECC_*^L(Y_0,\lambda_0;J_-) \longrightarrow ECC_*^{e^{1/n}L}(Y_0,\lambda_0;J_-),$$ where the map on the right is the inclusion, and the map on the left is triangular with respect to the identification of generators induced by $\phi_{-1/n}$. In particular the left map is an isomorphism of chain complexes, and hence induces an isomorphism on homology.
To proceed, we now define quantitative measures of the failure of the cobordism map to be an isomorphism.
\[def:AB\]
[(a)]{} Define $A$ to be the infimum of the set of real numbers $L$ such that the image of the inclusion-induced map $$\imath^L: H_*^L(Y_0,\lambda_0) \longrightarrow H_*(Y_0,\lambda_0)$$ is not contained in the image of the cobordism map .
[(b)]{} Define $B$ to be the infimum of the set of real numbers $L$ such that the kernel of the cobordism map $$\Phi^L(X,\lambda):
H_*^L(Y_1,\lambda_1) \longrightarrow H_*^L(Y_0,\lambda_0)$$ is not contained in the kernel of the inclusion-induced map $$\imath^L:
H_*^L(Y_1,\lambda_1) \longrightarrow H_*(Y_1,\lambda_1).$$
\[lem:AB\]
[(a)]{} If is not surjective, then $A<\infty$.
[(b)]{} If is not injective, then $B<\infty$.
\(a) If is not surjective then there exists an element of $H_*(Y_0,\lambda_0)$ which is not in the image; and by , any given element of $H_*(Y_0,\lambda_0)$ comes from $H_*^L(Y_0,\lambda_0)$ for some $L$.
\(b) If is not injective, then there exists a nonzero element $H_*(Y_1,\lambda_1)$ which maps to zero in $H_*(Y_0,\lambda_0)$. We can represent the former by a chain $\zeta$ of action less than some $L$, and its image under the cobordism chain map is the boundary of a chain with action less than some $L'$. We may assume that $L'\ge L$. It then follows from the Inclusion axiom in Theorem \[thm:cob\] that $\imath^{L,L'}[\zeta]$ is in the kernel of the cobordism map $\Phi^{L'}(X,\lambda)$. But $\imath^{L,L'}[\zeta]$ is not in the kernel of the inclusion-induced map $\imath^{L'}$, because $\imath^{L'}\imath^{L,L'}[\zeta]=\imath^L[\zeta]\neq 0$ in $H_*(Y_1,\lambda_1)$. Thus $B\le L'$.
In view of Lemma \[lem:ni\] and Definition \[def:PhiX\], the chord conjecture in the nondegenerate case now follows from:
\[thm:nondegenerate\] Let $Y_0$ be a closed oriented 3-manifold with a nondegenerate contact form $\lambda_0$, and let ${{\mathcal K}}$ be a Legendrian knot in $(Y_0,\lambda_0)$.
[(a)]{} If is not surjective, then ${{\mathcal K}}$ has a Reeb chord of action $\le A$.
[(b)]{} If is not injective, then ${{\mathcal K}}$ has a Reeb chord of action $\le B$.
To prove part (a), it is enough to show that given $L>0$, if there is no Reeb chord of action $\le L$, then $A\ge L$. (Because then if $A$ is finite, then there exists a Reeb chord of action $\le A+1/n$ for every positive integer $n$, so a compactness argument shows that there exists a Reeb chord of action $\le A$.) To show that $A\ge L$, it is enough to show that the image of the inclusion-induced map $$\imath^L: H_*^{L}(Y_0,\lambda_0) \longrightarrow H_*(Y_0,\lambda_0)$$ is contained in the image of the cobordism map . By the Inclusion and Direct Limit axioms in Theorem \[thm:cob\] we have a commutative diagram $$\begin{CD}
H_*^{e^{1/n}L}(Y_{1/n},\lambda_{1/n})
@>{\Phi^{e^{1/n}L}(X_{1/n},\lambda)}>> H_*^{e^{1/n}L}(Y_0,\lambda_0) \\
@VV{\imath^{e^{1/n}L}}V @VV{\imath^{e^{1/n}L}}V \\
H_*(Y_{1/n},\lambda_{1/n})
@>{\Phi(X_{1/n})}>> H_*(Y_0,\lambda_0)
\end{CD}$$ If $n$ is sufficiently large as in Lemma \[lem:main\], then it follows that we have a commutative diagram $$\begin{CD}
& & H_*^{e^{1/n}L}(Y_{1/n},\lambda_{1/n}) @>{\simeq}>> H_*^L(Y_0,\lambda_0) \\
& & @VV{\imath^{e^{1/n}L}}V @VV{\imath^L}V \\
H_*(Y_1,\lambda_1) @>{\simeq}>> H_*(Y_{1/n},\lambda_{1/n}) @>{\Phi(X_{1/n})}>>
H_*(Y_0,\lambda_0).
\end{CD}$$ Here the lower left arrow is induced by the cobordism $\overline{X
\setminus X_{1/n}}$ from $Y_1$ to $Y_{1/n}$; this map is an isomorphism because the cobordism $\overline{X\setminus X_{1/n}}$ is diffeomorphic to the product $[0,1]\times Y_1$, and product cobordisms induce isomorphisms on Seiberg-Witten Floer cohomology. In addition, the composition of the lower two arrows is the cobordism map , by the Composition axiom in Theorem \[thm:cob\] (or by the composition property for $\widehat{HM}^*$). The statement we need to prove now follows by chasing the diagram.
The proof of part (b) is similar to the proof of part (a). It is enough to show that if there is no Reeb chord of action $\le L$, then the kernel of the cobordism map $H_*^L(Y_1,\lambda_1)\to
H_*^L(Y_0,\lambda_0)$ is contained in the kernel of the inclusion-induced map $H_*^L(Y_1,\lambda_1)\to H_*(Y_1,\lambda_1)$. To do so, let $n$ be sufficiently large as in Lemma \[lem:main\] (with $L$ replaced by $e^{-1/n}L$). Then by the Inclusion and Direct Limit axioms in Theorem \[thm:cob\], we have a commutative diagram $$\begin{CD}
H_*^L(Y_1,\lambda_1) @>{\Phi^L(\overline{X\setminus X_{1/n}} ,
\,\lambda)}>> H_*^L(Y_{1/n},\lambda_{1/n}) @>{\simeq}>>
H_*^{e^{-1/n}L}(Y_0,\lambda_0) \\
@VV{\imath^L}V @VV{\imath^L}V @VV{\imath^{e^{-1/n}L,L}}V\\
H_*(Y_1,\lambda_1) @>{\simeq}>> H_*(Y_{1/n},\lambda_{1/n}) & &
H_*^L(Y_0,\lambda_0),
\end{CD}$$ where the composition of the two rightmost arrows is $\Phi^L(X_{1/n},\lambda)$. By the Composition axiom in Theorem \[thm:cob\], the composition of the three arrows from $H_*^L(Y_1,\lambda_1)$ to $H_*^L(Y_0,\lambda_0)$ is $\Phi^L(X,\lambda)$. Now suppose $\zeta\in H_*^L(Y_1,\lambda_1)$ maps to zero in $H_*^L(Y_0,\lambda_0)$. Since the latter is the direct limit of $H_*^{e^{-1/n}L}(Y_0,\lambda_0)$ as $n\to \infty$, it follows that if $n$ is chosen sufficiently large then $\zeta$ maps to zero in $H_*^{e^{-1/n}L}(Y_0,\lambda_0)$. Chasing the diagram then shows that $\zeta$ maps to zero in $H_*(Y_1,\lambda_1)$, as required.
The chord conjecture: degenerate case {#sec:ccdegenerate}
=====================================
We now use Theorem \[thm:nondegenerate\] for the nondegenerate case to deduce the chord conjecture when $\lambda_0$ is degenerate.
When $\lambda_0$ is degenerate, one can repeat the surgery construction from §\[sec:ls\], to obtain an exact symplectic cobordism $(X,\lambda)$ as before, now with degenerate contact forms $\lambda_{1/n}$ on the hypersurfaces $Y_{1/n}$. However by one can perturb these hypersurfaces slightly, as well as the boundary hypersurfaces $Y_0$ and $Y_1$, in the completion $\overline{X}$, so as to make the induced contact forms on them nondegenerate. In particular, for each positive integer $k$ we can find functions $f_{1/n}^{(k)}$ on $Y_{1/n}$ and $f_0^{(k)}$ on $Y_0$ with $$\|f_{1/n}^{(k)}\|_{C^1},\; \|f_0^{(k)}\|_{C^1} <1/k$$ such that we have an exact cobordism $X^{(k)}$ as in §\[sec:ls\], contained in $\overline{X}$ with the same Liouville form $\lambda$ (which we henceforth omit from the notation), with nondegenerate contact forms $\lambda_{1/n}^{(k)}=e^{f_{1/n}^{(k)}}\lambda_{1/n}$ on $Y_{1/n}$ and $\lambda_0^{(k)}=e^{f_0^{(k)}}\lambda_0$ on $Y_0$. We can also assume that $$\label{eqn:convenient}
f_0^{(1)}>f_0^{(k)}, \quad\quad f_1^{(1)}>f_1^{(k)},$$ which will be convenient below.
Now let $A(k)$ and $B(k)$ denote the upper bounds on the action of a Reeb chord coming from the cobordism $X^{(k)}$ from $(Y_1,\lambda_{1}^{(k)})$ to $(Y_0,\lambda_{0}^{(k)})$. By Theorem \[thm:nondegenerate\] and a compactness argument for Reeb chords explained at the end of this section, to prove the chord conjecture for $\lambda_0$ it is enough to show that $A(k)$ and $B(k)$ stay bounded as $k\to\infty$. In fact we have:
\[prop:continuity\] $A(k)\le A(1)$ and $B(k)\le B(1)$.
By the first part of , there is a subset $X_+^{(k)}$ of $\overline{X}$, diffeomorphic to $[0,1]\times Y_1$, which defines an exact symplectic cobordism from $(Y_1,\lambda_1^{(1)})$ to $(Y_1,\lambda_1^{(k)})$. Likewise, by the second part of , there is a subset $X_-^{(k)}$ of $\overline{X}$, diffeomorphic to $[0,1]\times Y_0$, which is an exact symplectic cobordism from $(Y_0,\lambda_0^{(1)})$ to $(Y_0,\lambda_0^{(k)})$. Let $X_0^{(k)}$ denote compact subset of $\overline{X}$ bounded by the negative boundary of $X_+^{(k)}$ and the positive boundary of $X_-^{(k)}$. This is an exact symplectic cobordism from $(Y_1,\lambda_1^{(k)})$ to $(Y_0,\lambda_0^{(1)})$, and we have the compositions $X_0^{(k)}\circ X_+^{(k)} = X^{(1)}$ and $X_-^{(k)}\circ
X_0^{(k)} = X^{(k)}$.
Now fix $L$ and consider the diagram $$\begin{CD}
H_*^L(Y_1,\lambda_1^{(1)}) @>{\Phi^L(X_+^{(k)})}>>
H_*^L(Y_1,\lambda_1^{(k)}) @>{\Phi^L(X_0^{(k)})}>>
H_*^L(Y_0,\lambda_0^{(1)}) @>{\Phi^L(X_-^{(k)})}>>
H_*^L(Y_0,\lambda_0^{(k)}) \\
@VV{\imath^L}V
@VV{\imath^L}V
@VV{\imath^L}V
@VV{\imath^L}V
\\
H_*(Y_1,\lambda_1^{(1)}) @>{\Phi(X_+^{(k)})}>{\simeq}>
H_*(Y_1,\lambda_1^{(k)}) @>{\Phi(X_0^{(k)})}>>
H_*(Y_0,\lambda_0^{(1)}) @>{\Phi(X_-^{(k)})}>{\simeq}>
H_*(Y_0,\lambda_0^{(k)}).
\end{CD}$$ This diagram commutes by the Inclusion axiom, and the maps $\Phi(X_\pm^{(k)})$ are isomorphisms because the cobordisms $X_\pm^{(k)}$ are diffeomorphic to products, which induce isomorphisms on Seiberg-Witten Floer cohomology. By the Composition axiom, the composition of the two upper left horizontal arrows is $\Phi^L(X^{(1)})$, the composition of the two upper right horizontal arrows is $\Phi^L(X^{(k)})$, the composition of the two lower left horizontal arrows is $\Phi(X^{(1)})$, and the composition of the two lower right horizontal arrows is $\Phi(X^{(k)})$.
To prove that $A(k)\le A(1)$, it is enough to show that if the image of $\imath^L:H_*^L(Y_0,\lambda_0^{(k)})\to H_*(Y_0,\lambda_0^{(k)})$ is contained in the image of $\Phi(X^{(k)})$, then the image of $\imath^L: H_*^L(Y_0,\lambda_0^{(1)})\to H_*(Y_0,\lambda_0^{(1)})$ is contained in the image of $\Phi(X^{(1)})$. This follows immediately by chasing the above diagram.
To prove that $B(k)\le B(1)$, it is enough to show that if the kernel of $\Phi^L(X^{(k)})$ is contained in the kernel of $\imath^L:H_*^L(Y_1,\lambda_1^{(k)})\to H_*(Y_1,\lambda_1^{(k)})$, then the kernel of $\Phi^L(X^{(1)})$ is contained in the kernel of $\imath^L:H_*^L(Y_1,\lambda_1^{(1)})\to H_*(Y_1,\lambda_1^{(1)})$. This also follows immediately from the above diagram.
To conclude, we have:
Let $Y_0$ be a closed oriented 3-manifold with a contact form $\lambda_0$, and let ${{\mathcal K}}$ be a Legendrian knot in $(Y_0,\lambda_0)$.
If $\lambda_0$ is nondegenerate, then it follows from Lemma \[lem:ni\] and Definition \[def:PhiX\] that the map is not an isomorphism. Let $A,B\in[0,\infty]$ be the numbers in Definition \[def:AB\]. By Lemma \[lem:AB\] we have $\min(A,B)<\infty$, and by Theorem \[thm:nondegenerate\] the knot ${{\mathcal K}}$ has a Reeb chord of length at most $\min(A,B)$.
If $\lambda_0$ is degenerate, let $\{\lambda_0^{(k)}\}_{k=1,2,\ldots}$ be a sequence of nondegenerate perturbations of $\lambda_0$ as described at the beginning of this section, and let $A(k),B(k)$ denote the corresponding quantities from Definition \[def:AB\]. By the nondegenerate case, ${{\mathcal K}}$ has a Reeb chord $\gamma_k$ for $\lambda_0^{(k)}$ of length at most $\min(A(k),B(k))<\infty$. By Proposition \[prop:continuity\], the length of $\gamma_k$ has a $k$-independent upper bound. Thus we can pass to a subsequence such that the lengths of the Reeb chords $\gamma_k$ converge to a real number $L$. We can also pass to a subsequence so that the starting and ending points of the Reeb chords $\gamma_k$ converge to points $y_0,y_L\in{{\mathcal K}}$. Now as $k\to\infty$, the $1$-form $\lambda_0^{(k)}$ converges to $\lambda_0$ in $C^1$, and so the Reeb vector field for $\lambda_0^{(k)}$ converges to the Reeb vector field for $\lambda_0$ in $C^0$. Consequently there is a Reeb chord for $\lambda_0$ of length $L$ from $y_0$ to $y_L$.
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[^1]: In fact if $\langle\partial\Theta,\Theta'\rangle\neq 0$ then the strict inequality ${{\mathcal A}}(\Theta)>{{\mathcal A}}(\Theta')$ holds, because $d\lambda$ vanishes identically on $C$ if and only if the image of $C$ is ${{\mathbb R}}$-invariant, in which case $C$ has ECH index zero and cannot contribute to the differential.
[^2]: Kronheimer-Mrowka define this map on the “completed” Seiberg-Witten Floer cohomology $\widehat{HM}^{\bullet}$. However for $\widehat{HM}$, the completed and uncompleted cohomologies are the same. Completion only makes a difference for the alternate versions $\check{HM}^*$ and $\overline{HM}^*$ of Seiberg-Witten Floer cohomology.
Note also that if one uses coefficients in ${{\mathbb Z}}$ instead of ${{\mathbb Z}}/2$, then the signs in the cobordism map on $\widehat{HM}$ depend on a choice of “homology orientation” of $X$. However one expects to be able to define cobordism maps on $ECH$ over ${{\mathbb Z}}$ without making such a choice, cf. [@lw Lem. A.14]. Presumably an exact symplectic cobordism has a canonical homology orientation which makes the signs agree.
[^3]: Everything we describe below should still be possible if one weakens the $\omega$-compatible condition here to $\omega$-tame. However, because the papers relating Seiberg-Witten Floer cohomology to ECH use compatible almost complex structures, we will stick with the latter to avoid confusion.
|
---
abstract: 'One of the proposed channels of binary black hole mergers involves dynamical interactions of three black holes. In such scenarios, it is possible that all three black holes merge in a so-called hierarchical merger chain, where two of the black holes merge first and then their remnant subsequently merges with the remaining single black hole. Depending on the dynamical environment, it is possible that both mergers will appear within the observable time window. Here we perform a search for such merger pairs in the public available LIGO and Virgo data from the O1/O2 runs. Using a frequentist p-value assignment statistics we do not find any significant merger pair candidates. Assuming no observed candidates in O3/O4, we derive upper limits on merger pairs to be $\sim11-110\ {\rm year^{-1}Gpc^{-3}}$, corresponding to a rate that relative to the total merger rate is $\sim 0.1-1.0$. From this we argue that both a detection and a non-detection within the next few years can be used to put useful constraints on some dynamical progenitor models.'
author:
- Doğa Veske
- Zsuzsa Márka
- Andrew Sullivan
- Imre Bartos
- 'K. Rainer Corley'
- Johan Samsing
- Szabolcs Márka
bibliography:
- 'Refs.bib'
title: 'Have hierarchical three-body mergers been detected by LIGO/Virgo?'
---
=1
Introduction {#sec:Introduction}
============
The LIGO Scientific Collaboration and the Virgo Collaboration have publicly announced properties of 10 binary black hole (BBH) mergers from the first and second observing runs (O1 and O2) in the gravitational wave (GW) catalog GWTC-1 [@Abbott_2019]. Individual groups have also performed searches on the open data from O1 and O2 and found additional merger candidates [@venumadhav2019new; @zackay2019detecting; @Nitz_2019]. The set of confirmed events have been used to constrain e.g. general relativity and its possible modifications [e.g. @2019PhRvD.100j4036A]; however, how and where the BBHs form in our Universe are still major unsolved questions. There are several plausible formation scenarios, including field binaries [@2012ApJ...759...52D; @2013ApJ...779...72D; @2015ApJ...806..263D; @2016ApJ...819..108B; @2016Natur.534..512B; @2017ApJ...836...39S; @2017ApJ...845..173M; @2018ApJ...863....7R; @2018ApJ...862L...3S; @10.1093/mnras/stz359; @10.1093/mnras/sty1999; @2017MNRAS.472.2422M], chemically homogeneous binary evolution [@10.1093/mnras/stw1219; @10.1093/mnras/stw379; @refId0], dense stellar clusters [@2000ApJ...528L..17P; @2010MNRAS.402..371B; @2013MNRAS.435.1358T; @2014MNRAS.440.2714B; @2015PhRvL.115e1101R; @2016PhRvD..93h4029R; @2016ApJ...824L...8R; @2016ApJ...824L...8R; @2017MNRAS.464L..36A; @2017MNRAS.469.4665P], active galactic nuclei (AGN) discs [@2017ApJ...835..165B; @2017MNRAS.464..946S; @2017arXiv170207818M; @2019ApJ...876..122Y], galactic nuclei (GN) [@2009MNRAS.395.2127O; @2015MNRAS.448..754H; @2016ApJ...828...77V; @2016ApJ...831..187A; @2016MNRAS.460.3494S; @2017arXiv170609896H; @2018ApJ...865....2H], very massive stellar mergers [@Loeb:2016; @Woosley:2016; @Janiuk+2017; @DOrazioLoeb:2017], and single-single GW captures of primordial black holes [@2016PhRvL.116t1301B; @2016PhRvD..94h4013C; @2016PhRvL.117f1101S; @2016PhRvD..94h3504C]. The question is; how do we observationally distinguish these merger channels from each other? Recent work have shown that the measured BH spin [@2016ApJ...832L...2R], mass spectrum [@2017ApJ...846...82Z; @2019PhRvL.123r1101Y], and orbital eccentricity [@2014ApJ...784...71S; @2017ApJ...840L..14S; @2018ApJ...853..140S; @2018PhRvD..97j3014S; @2018ApJ...855..124S; @2018MNRAS.tmp.2223S; @2019ApJ...871...91Z; @2018PhRvD..98l3005R; @2019arXiv190711231S; @2019PhRvD.100d3010S] can be used. In addition, indirect probes of BH populations have also been suggested; for example, stellar tidal disruption events can shed light on the BBH orbital distribution and corresponding merger rate in dense clusters [e.g. @2019PhRvD.100d3009S], or spatial correlations with host galaxies [@2017NatCo...8..831B].
In this paper we perform the first search for a feature we denote ‘hierarchical merger chains’ that are unique to highly dynamical environments [e.g. @2018MNRAS.476.1548S; @2019MNRAS.482...30S]. The most likely scenario of a hierarchical merger chain is the interaction of three BHs, $\{ BH_{1}, BH_{2}, BH_{3}\}$, that undergo two subsequent mergers; the first between $\{ BH_{1}, BH_{2}\}$ and the second between $\{BH_{12},BH_{3}\}$, where $BH_{12}$ is the BH formed in the first merger. Such hierarchical merger chains have been shown to form in e.g. globular clusters (GCs) as a result of binary-single interactions. In this case, the first merger happens during the three-body interaction when the BHs are still bound to each other, which makes it possible for the merger remnant to subsequently merge with the remaining single BH [@2018MNRAS.476.1548S; @2019MNRAS.482...30S]. Fig. \[fig:DMfig\] illustrates schematically this scenario. Such few-body interactions are not restricted to GCs, but can also happen in e.g. AGN discs [e.g. @2019arXiv191208218T]. Interestingly, under certain orbital configurations, both the first and the second merger can show up as detectable GW signals within the observational time window [e.g. @2019MNRAS.482...30S]. The hierarchical merger chain scenario can therefore be observationally constrained, and can as a result be used to directly probe the dynamics leading to the assembly of GW sources.
With this motivation, we here look for hierarchical merger pair events in the public O1 and O2 data from LIGO and Virgo. For this, we present a new algorithm to identify merger pairs, the simplest example of a hierarchical merger chain, and use it to search for such events in the public GWTC-1 catalogue.
The paper is organized as follows. In Section \[sec:search\] we describe our search method, and corresponding results are given in Section \[sec:results\]. Finally, we conclude our work in Section \[sec:conclusion\].
Search {#sec:search}
======
In this section we describe our methods for searching for GW merger pairs originating from three-body interactions, as the one shown in Fig. \[fig:DMfig\].
Parameters
----------
Our search is based on a frequentist p-value assignment by using a test statistic (TS). As Neyman-Pearson’s lemma suggests [@doi:10.1098/rsta.1933.0009], we choose our TS to be the ratio of the likelihood of the signal hypothesis to the likelihood of the null hypothesis; where we define our null hypothesis $H_0$ as having two unrelated mergers, and our signal hypothesis $H_s$ as having two related mergers originating from a three-body interaction. We use 3 parameters of the BBH mergers for calculating the likelihood ratio:
- [*Mass estimates:*]{} One of the initial BH masses in the second merger should agree with the final mass of the BH formed in the first merger.
- [*Correct time order:*]{} The first merger, as defined by the mass difference, should happen before the second merger.
- [*Localization:*]{} Both the first and the second merger must originate from the same spatial location.
Using these three parameters our TS is $$TS=\begin{cases}\frac{\mathcal{L}(M_f,m_{1,s},m_{2,s},V_f,V_s|H_s)}{\mathcal{L}(M_f,m_{1,s},m_{2,s},V_f,V_s|H_0)} &, t_f<t_s\\
0 &, t_f\geq t_s
\end{cases}$$ where $\mathcal{L}$ represents the likelihoods of the parameters for each hypothesis, $M$ represents the final mass estimate, $m_1$ and $m_2$ represent the mass estimates of the merging BHs, $V$ represents the spatial localization, and $t$ represents the merger times. Subscripts $f$ and $s$ represent the first and second merger, respectively. We do not use the spins of the BHs due to large uncertainties in the spin measurements [e.g. @Abbott_2019]; however, we do hope this becomes possible later, as spin adds an additional strong constraint (the BH formed in the first merger must appear in the second merger with a spin of $\sim 0.7$ [e.g. @2007PhRvD..76f4034B; @2017ApJ...840L..24F]).
For writing down the likelihoods we assume that the individual BH masses in the first merger follow a power law distribution with index -2.35 between 5-50$M_{\odot}$ (denoted as $\mathscr{M}_i$) [@PhysRevX.6.041015]. We further assume 5% of the total initial BH mass is radiated during merger, as suggested by previous detections and theory [e.g. @Abbott_2019]. Hence, for BHs which are a result of a previous merger the corresponding mass spectrum is the self-convolution of the $\mathscr{M}_i$ mass spectrum (denoted as $\mathscr{M}_c$) with its values reduced by 5%. We are well aware of that different dynamical channels predict different BH mass distributions; however, we do find that our results do not strongly depend on the chosen model. The full expression for the likelihood ratio is given in the Appendix.
![Illustration of a hierarchical merger chain, where two subsequent BBH mergers form from a single three-body interaction. The interaction progresses from left to right, where the BH tracks are highlighted with black thin lines. As seen, the initial configuration is a binary interacting with an incoming single (grey dots). During the interaction, two of three BHs merge, after which the product merges with the remaining single [@2019MNRAS.482...30S]. In this paper we search for such BBH merger pairs.[]{data-label="fig:DMfig"}](DMDoga.pdf){width="0.9\columnwidth"}
Generating the background distribution {#bg}
--------------------------------------
Our significance test is based on a frequentist p-value assignment via comparison with a background distribution. In order to have the background distribution, we perform BBH merger simulations and localize them with BAYESTAR [@PhysRevD.93.024013; @Singer_2016]. The simulations assume that the mass of BHs that are not a result of a previous merger is drawn independently from our assumed initial BH mass distribution $\mathscr{M}_i$. The mergers are distributed uniformly in comoving volume, and the orientation of their orbital axes are uniformly randomized. We assume the BH spins to be aligned with the orbital axis and we don’t include precession [@2019MNRAS.488.4459C]. We use the reduced-order-model (ROM) SEOBNRv4 waveforms [@PhysRevD.95.044028], and the cosmological parameters from the nine-year WMAP observations [@2013ApJS..208...19H]. The simulated detection pairs are made at O2 sensitivity for different detector combinations corresponding to first and second merger detected by either the LIGO Hanford-LIGO Livingston (HL) combination or the LIGO Hanford-LIGO Livingston-Virgo (HLV) combination. We denote the pairs that are both detected by HL as HL-HL, both by HLV as HLV-HLV, first by HL and second by HLV as HL-HLV, and first by HLV and second by HL as HLV-HL.
In order to construct the background distributions for the likelihood ratios, we need the same inputs as real detections. For this, we first assume that there is 5% mass loss in the merger to have a central value for the final mass. Second, in order to include realistic detection uncertainties, we broaden the exact masses to triangular distributions whose variances depend on the signal-to-noise ratio (SNR) of the detections and the distributions’ modes are the exact masses. We use the triangular distributions for imitating the asymmetry of the estimates in the real detections around the median[@Abbott_2019]. For determining the upper and lower bounds of the triangular likelihood distributions of masses we use a linear fit whose parameters are obtained by fitting a line to the relative 90% confidence intervals of the mass estimate likelihoods of real detections (which is obtained by dividing the posterior distribution to prior distribution from the parameter estimation samples) as a function of detection SNR. This fit is done separately for both component masses and the final masses. The minimum relative uncertainty is bounded at 5% which is the lowest uncertainty from real detections [@Abbott_2019].
Results and Discussion {#sec:results}
======================
In this section we show and discuss our results for the 10 published BBH mergers from O1 and O2 – We do not use the new mergers found by individual groups [@venumadhav2019new; @zackay2019detecting], as their localization and parameter estimation samples have not been publicly shared [@2019arXiv191111142G]. Of these, 6 were localized by the HL detector combination, and 4 were localized by the HLV detector combination [@Abbott_2019]. These 10 mergers give us a total of 45 possible hierarchical merger pair combinations. Considering the time order of the detected mergers; 15 of them are HL-HL pairs, 20 of them are HL-HLV pairs, 4 of them are HLV-HL pairs, and 6 of them are HLV-HLV pairs.
![The consecutive merger scenarios for the three most significant event pairs with their individual p-values.[]{data-label="fig:triples"}](triple2.png){width="\columnwidth"}
Event Pair Significance {#sec:Event Pair Significance}
-----------------------
In Fig. \[fig:triples\] we show the 3 most significant event pairs from our search. The most significant merger pair GW151012 (first merger) and GW170729 (second merger) has an individual p-value of 1.8%, meaning that only 1.8% of the background event pairs are more significant than this. Having the event GW170729 in the list seems exciting at first, as its primary mass exceeds the (hypothesized) pair-instability mass limit suggesting it could be the result of a previous merger [@Abbott_2019; @2019PhRvL.123r1101Y]. However, the significance of this event pair comes due to the mass matching of the lighter mass in GW170729 with the final mass of GW151012, which are both $\sim 35 M_{\odot}$ [@Abbott_2019]. The projected spatial localizations of the two events are shown in Fig. \[fig:skymap\], to show their spatial overlap. For an independent study of GW170729 see [@2020RNAAS...4....2K]. Finally, as the number of events increases, we will inevitably have low p-value event pairs. To account for this, one has to include a ‘multiple hypothesis correction’, which in our case brings a factor of 45 (the number of merger pairs) to the individual p-values. After this correction, none of the event pairs can be considered significant.
Limits on hierarchical triple merger rates {#sec:Limits on hierarchical triple merger rates}
------------------------------------------
We start by estimating the upper limits on the rate density of hierarchical merger pairs given the absence of an observed pair during O1 and O2. For this we assume that the first mergers in the hierarchical chain scenario are Poisson point processes with a uniform rate density per comoving volume, $R$, and that the temporal difference between the two mergers, $t_{12}$, follows a power law distribution $P(t_{12}<T) \propto (T/t_{max})^\alpha$, where $t_{\max}$ ($T \leq t_{\max}$) and $\alpha$ ($\alpha > 0$) are parameters that are linked to the underlying dynamical process [e.g. @10.1093/mnras/sty2249]. We further assume the duty cycle of each given time period is the same during the observing runs, i.e., we do not consider the non-uniformity of running times during the runs. The at least 2 detector duty cycle during O1 is 42.8% and during O2 is 46.4% [@Vallisneri_2015; @Abbott_2019]. Studies have shown that about half of all BBH mergers forming during three-body interactions will appear with an eccentricity $e>0.1$ at 10 Hz [@samsing2019gravitationalwave; @Rodriguez_2018]. However, current matched filter search template banks only include circular orbits [@LIGOeccentric] (except a recent study on binary neutron star mergers [@alex2019search]). Non-template based searches are able to recover eccentric binaries [@2019PhRvD.100f4064A], but with somewhat lower sensitivity compared to that of template based searches for circular binaries for the masses considered here. Hence, for simplicity, we consider a 50% loss of efficiency as well. Together with this loss, we denote the overall duty cycles as $\kappa_1$ and $\kappa_2$, respectively for O1 and O2, and the O1 duration by $\Delta t_1$, the O2 duration by $\Delta t_2$, and the time in between O1 and O2 by $\Delta t_0$ (O1 lasted about 4 months, O2 lasted about 9 months and they had about 10 months in between). The search comoving volumes are denoted for O1 and O2 by $C_1$ and $C_2$, respectively. These two volumes, $C_1$ and $C_2$, we estimate by (i) using the ratios of the ranges of the LIGO instruments in the O1, O2 and O3 runs; (ii) the search comoving volume for the O3 run in [@collaboration2013prospects]; (iii) neglecting the contribution to the search comoving volume in O3 by Virgo (due to having less than the half range of LIGO detectors), and (iv) assuming independent 70% duty cycles for the LIGO detectors in O3 [@collaboration2013prospects]. We estimate $C_1$ to be 0.07 ${\rm Gpc^3year/year}$ and $C_2$ to be 0.14 ${\rm Gpc^3year/year}$. Following this model we then calculate the probability $\mathcal{P}$ of not seeing a hierarchical merger pair during O1 and O2 (The full expression for $\mathcal{P}$ is found in the Appendix). Results are presented in Fig. \[fig:rate\], which shows the frequentist 90% upper limit for the rate density $R$ that satisfies $\mathcal{P}=0.1$, for different values of $t_{max}$ and $\alpha$. As seen, the upper rate density varies between $\sim 150-210\ {\rm year^{-1}Gpc^{-3}}$ for our chosen range of values.
We now investigate the expected future limits for triple hierarchical mergers assuming a null result when the third observing run of LIGO and Virgo (O3), and planned fourth observing run (O4) with KAGRA [@PhysRevD.88.043007], also are included in our search. O3 started on April 1st, 2019, and is planned to have 12 months of observing duration, with a one month break in October 2019. Although O4 dates remain fluid, it is estimated to be in between 2021/2022-2022/2023 [@collaboration2013prospects]. For our study we assume O3 and O4 to last for a year, with O4 starting in January 2022. The comoving search volumes in O3 and O4 are estimated to be 0.34 ${\rm Gpc^3year/year}$ and 1.5 ${\rm Gpc^3year/year}$, respectively. Although it will be more accurate to include the contribution from Virgo to these volumes, we here neglect its contribution to the duty cycles in a conservative manner and assume 70% independent duty cycles for the LIGO detectors [@collaboration2013prospects]. We adopt the median expected BBH merger detection counts from [@collaboration2013prospects], which are 17 and 79 for O3 and O4 respectively. Our derived lowest limits with the inclusion of O3 and O4 is shown in Fig. \[fig:rate\]. As seen, the rate densities are now $\sim 11-110\ {\rm year^{-1}Gpc^{-3}}$.
We end our analysis by investigating the upper limits for the fractional contribution from the first mergers of the hierarchical triple mergers to the total BBH merger rate. For the detection number and duration of the O1 and O2 runs, then at 90% confidence, the upper limits of the fractional contribution for the model parameters we consider in Fig. \[fig:rate\] are all $\approx 1$. We get more informative upper limits when we consider absence of merger pairs in the O3 and O4 runs as illustrated in the lower panel of Fig. \[fig:rate\]. As seen, the upper limits now vary between $\sim 0.1-1$.
Finally, we stress that our rate estimates from this section are associated with large uncertainties, mainly due to unknowns in the underlying dynamical model. For example, the functional shape of our adopted $P(t_{12} < T)$-model from Section \[sec:Limits on hierarchical triple merger rates\], depends in general on both the BH mass hierarchy, the exact underlying dynamics, the initial mass function, as well as on the individual spins of the BHs [e.g. @2019MNRAS.482...30S]; all of which are unknown components. Another aspect is how the rate limit depends on other measurable parameters, such as orbital eccentricity and BH spin. For example, in [@2019MNRAS.482...30S] it was argued that most hierarchical three-body merger chains are associated with high eccentricity; a search for eccentric BBH mergers, as the one performed in [@2019MNRAS.490.5210R], can therefore be used to put tight constraints on this scenario. Another example, is the effective spin parameter, $\chi_{\rm eff}$, which was used to argue that the primary BH of GW170729 is likely not a result of a previous BBH merger despite its relative high mass and spin [@2020RNAAS...4....2K]. However, we are actively working on improving our search algorithm both through the inclusion of eccentricity and spin. Having a fast and accurate pipeline searching for correlated events might also be useful for putting constraints on gravitationally lensed events.
Conclusion {#sec:conclusion}
==========
We presented a search method (Section \[sec:search\]) for detecting hierarchical GW merger pair events resulting from binary-single interactions (see Fig. \[fig:DMfig\]), and applied it to the public available O1/O2 data from the LIGO and Virgo collaborations. Using a frequentist p-value assignment statistics we do not find any significant GW merger candidates in the data that originate from a hierarchical binary-single merger chain (Section \[sec:Event Pair Significance\]). Using a simple model for describing the time between the first and second merger (Section \[sec:Limits on hierarchical triple merger rates\]), we estimated the upper limit on the rate of hierarchical mergers from binary-single interactions from the O1/O2 runs to be $\sim150-210\ {\rm year^{-1}Gpc^{-3}}$ for varying parameter values of our time-difference model. Assuming no significant merger pairs in the O3/O4 runs we find the upper limit reduces to $\sim11-110\ {\rm year^{-1}Gpc^{-3}}$, corresponding to a rate that relative to the total merger rate is $\sim 0.1-1.0$. The theoretical predicted rate of hierarchical GW merger pair events is highly uncertain; however, we have argued and shown that both a detection and a non-detection of merger pairs can provide useful constraints on the origin of BBH mergers. In future work we plan on including both eccentricity and BH spin parameters in our search for hierarchical GW merger pair events. Moreover considering the expectancy of such events happening in dense environments, known GC and AGNs can also be used to correlate with the spatial reconstruction of the events in the search.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful for the useful feedback of Christopher Berry and Jolien Creighton. The authors thank the University of Florida and Columbia University in the City of New York for their generous support. The Columbia Experimental Gravity group is grateful for the generous support of the National Science Foundation under grant PHY-1708028. DV is grateful to the Ph.D. grant of the Fulbright foreign student program. AS is grateful for the Columbia College Science Research Fellowship. JS acknowledges support from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 844629. The authors are grateful to Leo Singer of GSFC for the BAYESTAR package and his valuable help with our use case scenario. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
Appendix
========
Likelihood ratio
----------------
All BBH mergers are assumed to be uniformly distributed in comoving volume. In this case the likelihood ratio becomes
$$\begin{gathered}
\frac{\mathcal{L}(M_f,m_{1,s},m_{2,s},V_f,V_s|H_s)}{\mathcal{L}(M_f,m_{1,s},m_{2,s},V_f,V_s|H_0)}=\frac{\int P(M_f,m_{1,s},m_{2,s}|m',H_s)P(m'|H_s)dm' \int P(V_f,V_s|r,H_s)P(r|H_s)drd\mathbf{\Omega}}{\int P(M_f,m_{1,s},m_{2,s}|H_0)dm' \int P(V_f,V_s|H_0)drd\mathbf{\Omega}} \\
=\frac{ \int \frac{P(r|V_f)P(r|V_s)}{r^2}drd\mathbf{\Omega} \sum_{\substack{x,y=1,2 \\ x\neq y}}\int \frac{P(m'|M_f)}{P_f(m')}\frac{P(m'|m_{x,s})}{P_{x,s}(m')}\mathscr{M}_c(m')dm'\int \frac{P(m'|m_{y,s})}{P_{y,s}(m')}\mathscr{M}_i(m')dm'}{\int \frac{P(m'|M_f)}{P_f(m')}\mathscr{M}_c(m')dm'\int \frac{P(m'|m_{1,s})}{P_{1,s}(m')}\mathscr{M}_i(m')dm'\int \frac{P(m'|m_{2,s})}{P_{2,s}(m')}\mathscr{M}_i(m')dm'}\end{gathered}$$
where $m'$, $r$ and $\mathbf{\Omega}$ are the integration variables for mass, distance and sky location. $P_f(m')$, $P_{1,s}(m')$ and $P_{2,s}(m')$ are the mass priors used in the parameter estimation. We take these from the parameter estimation sample released in GWTC-1 [@parameter_estimation]. The integrals over the spatial localization in the denominator equals unity and are therefore not written. The summed terms in the numerator represent either of the BHs in the second merger resulting from the first merger.
Probability $\mathcal{P}$
-------------------------
To proceed we first write the probability $\mathcal{P}$ of not seeing a hierarchical merger pair for the parameters $R$, $t_{max}$, $\alpha$, $\kappa_1$, $\kappa_2$, $\Delta t_1$, $\Delta t_2$, $ \Delta t_0$, and with the number of seen events, $n_i$, during O1 ($n_1 = 3$) and O2 ($n_2 = 7$). The condition of not seeing a pair of hierarchical mergers is to see at most one of the mergers in the pair.
$$\begin{gathered}
\mathcal{P}=\bigg[\sum_{i=0}^{n_1} Poisson(i,\kappa_1 R\Delta t_1 C_1) \frac{i!}{{\Delta t_1}^i} \\ \int_0^{\Delta t_1}\int_{0}^{\tau_i}...\int_{0}^{\tau_2}[1-\kappa_2(\frac{\Delta t_1+\Delta t_2+\Delta t_0-\tau_1}{t_{max}})^\alpha+\kappa_2(\frac{\Delta t_1+\Delta t_0-\tau_1}{t_{max}})^\alpha-\kappa_1(\frac{\Delta t_1-\tau_1}{t_{max}})^\alpha]\times \\... \times [1-\kappa_2(\frac{\Delta t_1+\Delta t_2+\Delta t_0-\tau_{i-1}}{t_{max}})^\alpha+\kappa_2(\frac{\Delta t_1+\Delta t_0-\tau_{i-1}}{t_{max}})^\alpha-\kappa_1(\frac{\Delta t_1-\tau_{i-1}}{t_{max}})^\alpha] \\ \times [1-\kappa_2(\frac{\Delta t_1+\Delta t_2+\Delta t_0-\tau_i}{t_{max}})^\alpha+\kappa_2(\frac{\Delta t_1+\Delta t_0-\tau_i}{t_{max}})^\alpha-\kappa_1(\frac{\Delta t_1-\tau_i}{t_{max}})^\alpha]d\tau_1...d\tau_{i-1}d\tau_i\bigg] \\
\times \bigg[\sum_{i=0}^{n_2} Poisson(i,\kappa_2 R\Delta t_2 C_2) \frac{i!}{{\Delta t_2}^i} \\ \int_0^{\Delta t_2}\int_{0}^{\tau_i}...\int_{0}^{\tau_2}[1-\kappa_2(\frac{\Delta t_2-\tau_1}{t_{max}})^\alpha] \times ... \times [1-\kappa_2(\frac{\Delta t_2-\tau_{i-1}}{t_{max}})^\alpha]\times [1-\kappa_2(\frac{\Delta t_2-\tau_i}{t_{max}})^\alpha]d\tau_1...d\tau_{i-1}d\tau_i\bigg]
\label{eq:long}\end{gathered}$$
with $Poisson(n,k)$ being the probability of seeing $n$ events from the Poisson point process with mean $k$. $\frac{i!}{\Delta t}$ is the value of joint probability distribution of Poisson arrival times given that there are $i$ events in time interval $\Delta t$. The integrals give the probability of not seeing any of the second mergers of $i$ observed first mergers during the observation times. The first term in Eq. gives the probability of not seeing an hierarchical merger pair whose first event can happen during O1 and second event can happen during O1 or O2. The second term gives the probability of not seeing an hierarchical merger pair whose both mergers can happen during O2. Multiplication of them gives us the probability of not seeing an hierarchical pair during O1 and O2. We use the integral identity
$$\int_0^a\int_0^{\tau_i}...\int_0^{\tau_2}f(\tau_1)\times ... \times f(\tau_{i-1})\times f(\tau_i)d\tau_1...d\tau_{i-1}d\tau_i = (\int_0^a f(\tau_1)d\tau_1)^i\frac{1}{i!}$$
to simplify the expression for $\mathcal{P}$. $$\begin{gathered}
\mathcal{P}=\bigg[\sum_{i=0}^{n_1} Poisson(i,\kappa_1 R\Delta t_1 C_1) \frac{1}{{\Delta t_1}^i}\\ \big[\int_0^{\Delta t_1} [1-\kappa_2(\frac{\Delta t_1+\Delta t_2+\Delta t_0-\tau_1}{t_{max}})^\alpha+\kappa_2(\frac{\Delta t_1+\Delta t_0-\tau_1}{t_{max}})^\alpha-\kappa_1(\frac{\Delta t_1-\tau_1}{t_{max}})^\alpha] d\tau_1\big]^i\bigg] \\ \times \bigg[\sum_{i=0}^{n_2} Poisson(i,\kappa_2 R\Delta t_2 C_2) \frac{1}{{\Delta t_2}^i}\big[\int_0^{\Delta t_2}[1-\kappa_2(\frac{\Delta t_2-\tau_1}{t_{max}})^\alpha]d\tau_1\big]^i\bigg]
\label{eq:short}\end{gathered}$$
|
---
abstract: 'The single top quark production has an electroweak nature and provides an additional to the top pair production source of the top quarks. The processes involving single top have unique properties, they are very interesting from both theoretical and experimental view points. Short review of the single top quark production processes is given in the paper.'
address: 'Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics (MSU SINP), Leninskie gory, Moscow 119991, Russia'
author:
- 'E. Boos[^1], L. Dudko[^2]'
title: The Single Top Quark Physics
---
Introduction
============
The top quark has been discovered at the proton-antiproton collider Tevatron in 1995 by two collaborations CDF[@top_cdf_1995_ttbar] and D0[@top_d0_1995_ttbar]. This discovery was a triumph of the Standard Model (SM) since the top quark was found in the mass interval predicted before from a detail comparison of LEP measurements and SM computations obtained at quantum loop level of accuracy[@Blondel:1996wm]. More than 10 years later, the top quark was found in so-called single top production process which according to SM has a cross section only 2.5 times smaller than pair production. The direct single top observation by the Tevatron experiments[@Abazov:2009ii; @Aaltonen:2009jj] was one more important confirmation of our understanding of SM as a quantum gauge field theory describing the Nature at extremely small distances of the order of $10^{-17}$ cm. Why single top is specially interesting, why it took so long to discover it at the Tevatron, how it was discovered at Tevatron and rediscovered at the LHC? What may tell us the study of the single top about possible physics beyond the Standard Model? This review is aimed to answer these questions in some details.
In SM the top quark is the spin-${1\over 2}$ fermion with the electric charge $Q_{em}^t={2\over 3 }\mid e \mid$, the weak isospin partner of the $b$ quark, and a color triplet. Top quark is needed in SM to ensure a cancellation of chiral anomaly[^3] and therefore to ensure a consistency of SM as a quantum theory. All the couplings and charges of the top quark are predicted in SM to be exactly the same as for other two up-type quarks, u-quark and c-quark. A natural question one may ask is why then the top quark is special and interesting.
The difference with the other quarks comes from two experimental facts, namely, a very large top quark mass comparing to masses of all other quarks and very small mixing to quarks of the first and second generations. The measured value of the top quark mass is known now with a precision better than 1% $M_{top} = 173.3 \pm 0.6({\rm stat}) \pm 0.9({\rm syst})$ GeV[@1007.3178:1900yx] being the most precisely known quark mass.
The top quark is the heaviest elementary particle found so far with a mass slightly less than the mass of the gold nucleus. In various respects the top quark is a very unique object. Top Yukawa coupling $\lambda_{t} = 2^{3/4}G_F^{1/2}m_{t}$ is very close to unit. The quark mixing in SM is encoded in matrix elements of the Cabibbo-Kobayashi-Maskawa matrix. The matrix element $V_{tb}$ is close to one while the elements $V_{ts}$ and $V_{td}$ are significantly smaller than one. These two experimental facts, large mass ans small mixing, lead to the conclusion that in SM the top quark decays to W-boson and b-quark with a probability close to 100 %. The width of the top quark being calculated in SM at the NLO level[@Jezabek:1993wk] is about 1.4 GeV. From one side, the top width is much smaller than its mass, and therefore the top quark is a narrow resonance (top decay width is proportional to the third power of its mass). From the other side the top width is significantly larger than a typical QCD scale $\Lambda$ 200 MeV. As a result the top quark life time ($\tau_t \approx 5\times 10^{-25}$s) as predicted by SM is much smaller than a typical time for formation of QCD bound states ($\tau_{\rm QCD}\approx
1/\Lambda_{\rm QCD} \approx 3\times 10^{-24}$s). Therefore, the $t$-quark decays long before it can hadronize and hence top quark containing hadrons do not exist [@Bigi:1986jk]. The top quark provides a very clean source for fundamental information.
Since the top quark decays before hadronization its spin properties are not spoiled. Therefore the spin correlation in top production and decays is an interesting issue of the top quark physics.
In the single top quark $t$–channel and $s$–channel production processes the top quark is produced in SM through the left-handed interaction. The production is very similar to the top decay process turned backward in time. For the polarized top decay, it is well-known that the charged lepton tends to point along the direction of top spin[@Jezabek:1994zv]. In the production process this is the direction of the initial $\bar{d}$-quark for the $s$-channel, and the dominant direction of the final $d$-quark for the $t$-channel. Therefore, in the top quark rest frame there is strong correlation in the angle of produced lepton with respect to one of the above directions[@Mahlon:1996pn; @Boos:2002xw]: $$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^*_{\ell}} = \frac{1}{2}(1+\cos\theta^*_{\ell}).
\label{cos_theta}$$ Spin properties in the $tW$ production process are more involved. Here, one can find a kinematic region in which top quarks are produced with the polarization vector preferentially close to the direction of the charged lepton or the $d,s$-quark momentum from the associated $W$ decay. In this kinematic region, the direction of the produced charged lepton or the $d,s$-quark should be as close as possible to the direction of the initial gluon beam in the top quark rest frame[@Boos:2002xw].
One can also measure several others quantum numbers of the top quark. One can extract the electric charge by measuring the process $t\bar{t}\gamma$ where photon radiates off the top. The weak isospin of top would be confirmed by looking on the Wtb vertex structure via top decay in pair production and via single top production. The confirmation that the top quark is a color triplet follows from precision measurements of the top pair production cross section.
Since the top is so heavy and point-like at the same time one might expect a possible deviations from the SM predictions more likely in the top sector. Top quark physics will be a very important part of research programs for all future hadron and lepton colliders including studies of top quark properties, various new physics via the top quark, and kinematical characteristics of top quark events as significant backgrounds to a number of other processes. In particular, the single top production plays a special role here due to its unique properties. Many details of theoretical studies and experimental analysis of single top production and decay properties could be found in a number of review papers Ref. –.
Computation and modeling of processes with single top quark
============================================================
At hadron and lepton colliders top quarks are produced either in pairs or singly. The representative diagrams for the single top production at the Tevatron and LHC are shown in Fig. \[diag\_lhc\].
Three mechanisms of the single top production are distinguished by the virtuality $Q^2_W$ of the W-boson involved: $t$–channel ($Q^2_W < 0$), $s$–channel ($Q^2_W > 0$), associated tW ($Q^2_W = M^2_W$).
The single top quark production at hadron colliders was considered for the first time in Ref. and later in Ref. –. The authors of Ref. studied the most complete tree-level set of processes in the SM that contribute to the single top quark production. QCD NLO corrections to various single top production processes have been calculated in several papers –. In particular, NLO corrections to kinematic distributions were presented[@Harris:2002md]. The influence of NLO corrections not only to the production but also to the subsequent top quark decay has been studied in Ref. . Potentially important corrections at the threshold region have been resummed up to NNLL accuracy[@Kidonakis:2010tc; @Kidonakis:2010ux; @Kidonakis:2011wy]. Monte-Carlo (MC) analyses of the production processes of the single top quark allowing to extract it from main backgrounds were performed in Ref. .
The NLO cross sections including NNLL soft gluon threshold correction resummation for the main single top production processes at hadron colliders are collected in the Table \[tab:nlocs\]:
The processes of the single top-quark production were simulated using MC event generators such as ONETOP[@Carlson:1993dt] and TopReX[@Slabospitsky:2002ag], and MC generators based on more generic packages such as MadGraph[@Herquet:2008zz], CompHEP[@Boos:2004kh], PYTHIA[@Sjostrand:2006za], AcerMC[@Kersevan:2004yg], MC@NLO[@Frixione:2005vw], and POWHEG[@Alioli:2009je]. There are several problems associated with the correct and precise simulation of the single top quark production processes. Some of these problems are listed below.
- The combination of events corresponding to the diagram in Fig. \[fig:nlo\_diag\](a) allowing for the parton showers in the initial state (ISR) and to the diagrams in Figs. \[fig:nlo\_diag\] (b), (c), and (d) results in double counting. Such a double counting takes place in a soft $P_T$ region of produced b-quark originated from the ISR to the diagram (a) where gluon splits to $b\bar b$ pair and the diagram (d). One may subtract the first term in the gluon splitting part to remove the double counting and this procedure gives the correct production cross section. However, the direct application of the subtraction procedure for MC event generation for the process results in a negative weight for part of the events.
- Matching procedure for matrix elements and parton showers should be included into event generation. The modern standards require LHEF format for generated events to be useful in experimental analyses.
- As emphasized in Ref.[@Mahlon:1996pn; @Boos:2002xw], the top quark is produced in electroweak processes with significant polarization owing to the $(V-A)$ structure of the Wtb vertex in the SM. As a result, spin correlations between the production and the decay of the top quark appear. Therefore, the correct MC generator should include these correlations.
- The single top-quark production processes are sensitive to various new physics contributions[@Tait:2000sh] such as anomalous contributions to the Wtb vertex[@Kane:1991bg; @Boos:1999dd; @AguilarSaavedra:2008zc; @Zhang:2010dr], FCNC couplings[@Beneke:2000hk] and additional scalar and vector bosons. In order to study such extensions of the SM, MC generators should include the corresponding anomalous contributions in the production as well in the subsequent top quark decay.
- At the LHC collider, $t$ and $\bar t$ quarks are produced with different cross sections. The corresponding asymmetry in the kinematic distributions is useful for reducing the systematic errors in the measurement of the top quark parameters[@Boos:1999dd]. Therefore, it is necessary to have the possibility to separate the production models for $t$ and $\bar t$ quarks at the level of the MC generator.
- In case of $tW$-associated production channel one should carefully split the electroweak contribution from QCD top quark pair production[@Tait:1999cf; @Belyaev:2000me; @Campbell:2005bb].
The mentioned MC event generators try to solve the above problems. However, non of the generators resolve all the problems completely.
In the analysis of the Tevatron data leading to the first observation of the single top production by D0 and CDF collaborations, an effective NLO approach[@Boos:2006af] for event generation was used. The method was first implemented in the analysis of physics prospects of the CMS experiment and described in Ref. . The method is realized in the SingleTop MC generator[@Boos:2006af] for the analysis in the D0 experiment and in the generator based on MadGraph/MadEvent package[@Herquet:2008zz] in the CDF experiment. This method of simulation helps in modeling of t- and tW-channels. Special analysis has shown that for the s-channel one can simply generate LO events with NLO k-factor. In this case, all of the kinematic distributions obtained from simulated events are in a complete agreement with the same distributions obtained by NLO computations[@Sullivan:2004ie]. The simulation of the $t$-channel process was performed in the five-flavor scheme in which the $2\to 2$ diagram Fig. \[fig:nlo\_diag\](a) with the b-quark in the initial state is the leading order contribution. Diagram \[fig:nlo\_diag\](b) represents one of the loop NLO contribution while diagrams \[fig:nlo\_diag\](c) and \[fig:nlo\_diag\](d) represent tree NLO contributions. One should stress that the diagram \[fig:nlo\_diag\](d) gives the NLO $P_T(b)$ spectrum of produced b-quark at high $P_T(b)$ region. One can reproduce low $P_T(b)$ region by switching on ISR corrections to the diagram \[fig:nlo\_diag\](a). All loop and radiation corrections (diagrams \[fig:nlo\_diag\](b) and \[fig:nlo\_diag\](c)) do not change high $P_T(b)$ spectrum since they are not involved produced b-quark. They affect a renormalization of very soft $P_T(b)$ region and therefore can be included numerically by a proper normalization. Such a normalization is performed by summing up hard $P_T(b)$ region as it follows from exact tree NLO computation and soft $P_T(b)$ region multiplying it by some coefficient. This coefficient and phase space slicing parameter in $P_T(b)$ which separates hard and soft regions are determined from two requirements, the first is that the sum of two contributions should be equal to the total NLO cross section and the second is that the $P_T(b)$ distribution should be smooth. In this way, one can combine generated events in the soft and hard regions to one event sample with correct NLO rate and all smooth distributions without negative weights (see examples of such distributions in Fig. \[distributions\_lhc\]).
In SingleTop generator, the CompHEP[@Boos:2004kh] package is used for complete tree-level computations and NLO rate is calculated by means of the MCFM tool[@Campbell:2004ch]. In this way, all spin correlations between production and subsequent top and W boson decays, particle masses and nontrivial decay widths are taken into account. The use of CompHEP allows one to generate events for different extensions of the SM such as mentioned above anomalous Wtb couplings, FCNC couplings and new scalar or vector resonances.
Results of a comparison of various kinematic distributions obtained from the events generated with the SingleTop and computed by means of NLO codes ZTOP[@Sullivan:2004ie] and MCFM[@Campbell:2004ch] are shown in Figs. \[distributions-ztop\], \[distributions-mcfm\] and demonstrate very good agreement.
The best spin correlation variable $\cos\theta^*_{\ell}$ (eq. \[cos\_theta\]) is of a special interest as was mentioned in the introduction, because of unique polarization properties of the single top processes. Therefore, it is important to show the influence of the NLO corrections to the distribution obtained from the generated events. It is easy to show that for the $s$–channel process the spin projection axis corresponding to the maximum polarization is the momentum direction of the $\bar{d}$–quark from the initial state in the rest frame of the top quark[@Mahlon:1996pn]. Owing to the correspondence between the decay and production diagrams of the top quark (the diagrams are topologically equivalent), the best probe for the top quark spin is its decay-product lepton[@Boos:2002xw]. Thus, the best variable to observe spin correlations in the $s$–channel process is the cosine of the angle between the momenta of the initial $\bar{d}$–quark and the lepton in the rest frame of the top quark. Spin correlations can be numerically characterized by the coefficient $R_{\rm spin}$ of $\cos{\theta^*_{e^+,\bar{d}}}$ in the normalized distribution $$\frac{1}{\sigma}\frac{d\sigma}{d\cos{\theta^*_{e^+,\bar{d}}}} =
\frac{1+R_{spin}(\bar{s})\cos{\theta^*_{e^+,\bar{d}}}}2.$$ Then, $R_{\rm spin}(\bar{p}_d)=1$ (or 100%) for the $s$–channel process. Since the NLO approximation is manifested only in the $K$–factor in this process, we do not expect any significant reduction of $R_{\rm spin}$ owing to the inclusion of NLO corrections. The diagram of the $t$–channel process in the LO approximation is also topologically equivalent to the decay diagrams of the $s$–channel process. Thus, the top quark is polarized, and the axis of the maximum polarization is the momentum of the final light quark in the rest frame of the top quark. The dotted histogram in Fig. \[spin\_lhc\] corresponds to LO events. The first-order polynomial fit to the distribution gives $R_{\rm spin}(\bar{p}_d)_{\rm LO}=0.98\pm0.02$, which indicates the maximum polarization of the top quark in the LO approximation. In the NLO approximation, a significant contribution comes from the real correction with the additional b-quark. In this process, the top quark can be produced in the QCD vector interaction vertex with the gluon, which reduces the polarization of the top quark. However, this reduction is not strong because the main contribution to the $pp\to tqb$ process comes from the diagram with the Wtb production vertex of the top quark. The solid histogram in Fig. \[spin\_lhc\] shows the distribution of the NLO events in $\cos{\theta^*_{l^+,d}}$ The straight-line fit of the distribution gives $R_{\rm spin}(\bar{p}_d)_{\rm NLO}=0.89\pm0.02$, which indicates the reduction of the polarization value[@Boos:2006af].
The $tW$–process requires special consideration. The LO diagrams are shown on the Fig. \[fig:tw\_diag\](a,b). The diagrams (c,d) are the representative diagrams for NLO loop and tree corrections involving gluons and light quarks. Diagram (e) is one of the tree NLO contribution with additional b-quark produced, similar to the diagram in Fig. \[fig:nlo\_diag\](d) for the t-channel process. Diagrams \[fig:tw\_diag\](f) and \[fig:tw\_diag\](g) contain the top pair contribution with a subsequent decay of the second top to $W$ and b. There are many other subleading diagrams mentioned in Ref. making the result exactly electroweak gauge invariant. The problem of double counting of contributions of diagrams (a,b) and (e) is very similar to that discussed here for t-channel case and is resolved in the same manner by slicing phase space on hard and soft $P_T(b)$ regions. However, in $tW$ case there is another problem of how to split single top and top pair contributions leading to the same final states. In order to do that, several methods have been proposed. In the paper[@Tait:1999cf], the top pair part was removed by subtraction top pairs on shell. This procedure called the diagram subtraction scheme is obviously gauge invariant and later was realised on the generator level in MC@NLO code[@Frixione:2005vw]. The procedure includes all the interferences of single and top pair contributions into the single top part leading, however, to negative weights for some fraction of generated events. In another scheme called the diagram removal scheme also implemented in MC@NLO, diagrams Fig. \[fig:tw\_diag\](f,g) are removed from the complete set of the diagrams. In this approach, all the interferences between single and top pair are removed leading, however, to a small violation of the electroweak gauge invariance. In the paper[@Belyaev:2000me], the phase space removal scheme was used in which the part of events with the $Wb$ invariant mass around the top quark pole was removed. In this way all the interferences are removed, however, there is an ambiguity in the size of the removal $Wb$ invariant mass region. In the paper[@Campbell:2005bb], an approach with veto on the $P_T(b)$ larger than some $P_T^{\rm veto}(b)$ separation parameter was used. In this approach, some small part of top pair contribution might still belong to single top part and also there is some ambiguity in choosing of the separation parameter. However, in practice the veto is realisable only in some part of the phase space region where the b-quark could be observed as a b-jet.
Spin correlations in $tW$-channel requires special consideration. The leading order Feynman diagrams for this process include two diagrams: in Fig. \[fig:tw\_diag\](a), the top quark is produced in the QCD interaction vertex, and in another one Fig. \[fig:tw\_diag\](b) it is produced in the electroweak interaction vertex. Both contributions are comparable in rate. The first diagram leads to unpolarized top quark production while from the second diagram the top is produced highly polarized. If the top quark would be produced only from electroweak diagram Fig. \[fig:tw\_diag\](b), its spin direction in the rest frame would fully correlated with a direction of charged lepton momentum coming from the $W$ boson decay. However, this property is spoiled by the contribution of the first diagram Fig. \[fig:tw\_diag\](a). As was shown in[@Boos:2002xw], one can apply some kinematic cuts in order to suppress the contribution of the diagram Fig. \[fig:tw\_diag\](a) making the spin correlation property more pronounced. In this region, the outgoing charged lepton should be as close as possible to the direction of one of the initial beam in the top quark rest frame. It was shown that one can increase the polarization of the top quark from the initial 24 (without any cuts) to 80 – 90% applying some reasonable cuts.
Single top observation at the Tevatron
======================================
Two years before the top quark was discovered, D0 collaboration had organized single top research group to perform a search for the single electroweak top quark production. The cross section of the single top processes is only about two times smaller than the top pair production, but the number of final jets is smaller and therefore the backgrounds are significantly higher. During the Run I data taking, D0 detector did not have vertex detector and could not identify b-quarks with high efficiency (CDF detector had the part). Since the top quark decays almost 100% with production of b-quark, this feature of the detector is very important in top physics. The small ratio of signal to backgrounds and lack of the vertex detector were the main reasons to find sophisticated methods to increase signal to background ratio and achieve sensible result. It was a powerful stimulation for the analyzers to implement and develop multivariate analysis techniques. The main strategy in D0 analysis was to apply loose initial selection to keep the signal statistics and apply multivariate analysis technique to distinguish the signal events. CDF strategy included rather tight initial selection to cut the contribution of pair production by the cuts on number of jets (only two jet events) and number of $b$-quarks (only one $b$-quark in event). Due to the complexity of the analysis and lack of experimental statistics the first results[@Abazov:2001ns] were published in about 7 years after top quark was discovered. The available statistics in Run I analysis did not allow to observe the single top processes and both collaborations set upper limits on the production cross section. Because of multivariate technique in the analysis, D0 collaboration succeeded in achieving the same sensitivity as CDF collaboration with the use of vertex detector. Both the collaborations set 95% C.L. upper limits on the production cross section for s-channel process 17 pb is the D0 results and 18 pb is the CDF result; for the t-channel process 13 pb is the CDF result and 22 pb is the D0 result.
During the second run of the Tevatron (Run II), the luminosity has increased significantly and collaborations upgraded their detectors. The first evidence of the single top quark production was reported by D0 collaboration in December of 2006[@Abazov:2006gd] and later by CDF collaboration[@Aaltonen:2008sy], with the first measurement of the production cross section and the first direct measurement of the $V_{tb}$ CKM matrix element. After the luminosity has reached 2-3 $fb^{-1}$ both of the collaborations reported $5\sigma$ observation of the single top production[@Abazov:2009ii]. Based on the full Run II statistics, both collaborations are able to distinguish $s-$ and $t$-channel processes and measure their cross sections separately. This measurement significantly increases the sensitivity for the possible BSM contribution. The current results of the SM measurements in D0[@Abazov:2011pt] with integrated luminosity of 5.4 $fb^{-1}$ are the $\sigma({{\mbox{$p\bar{p}$}}}{{\mbox{ $\rightarrow$ }}}tb+X, tqb+X) = 3.43\pm^{0.73}_{0.74}\rm pb$ the corresponding measurement of the CKM matrix element is $0.79 < |V_{tb}| \leq 1$ at the 95% C.L. The separate measurements of $s-$ and $t$-channel are the following $\sigma({{\mbox{$p\bar{p}$}}}{{\mbox{ $\rightarrow$ }}}tb+X) = 0.68\pm^{0.38}_{0.35}\rm pb$ and $\sigma({{\mbox{$p\bar{p}$}}}{{\mbox{ $\rightarrow$ }}}tqb+X) = 2.86\pm^{0.69}_{0.63} \rm pb$. The current results of the SM measurements in CDF[@Aaltonen:2010jr] with integrated luminosity of 3.2 $fb^{-1}$ and $5\sigma$ statistical significance are $\sigma({{\mbox{$p\bar{p}$}}}{{\mbox{ $\rightarrow$ }}}tb+X, tqb+X) = 2.3\pm^{0.6}_{0.5}$(stat+sys) pb, the measured CKM matrix element value $|Vtb|=0.91\pm^{0.11}_{0.11} {\rm (stat+sys)} \pm 0.07$(theory) with a lower 95% C.L. limit $0.71<|Vtb|$.
Single top evidence and observation at the LHC
==============================================
The relative contribution of different single top production channels significantly differs for LHC than at the Tevatron. All the processes with initial gluon are significantly larger than the contribution of the processes with initial quarks. Therefore, the main mechanisms of single top production at the LHC are $t$- and $tW$-channels, but the $s$-channel cross section is significantly lower (Table \[tab:nlocs\]). High luminosity and relatively high cross section result in possible very high statistics of single top events at the LHC and the main limitation for the measurements is the systematic uncertainty. During the first round of analysis at $\sqrt{s}=7$ TeV, CMS and ATLAS collaborations measured cross section of the $t-$ and $tW-$channel processes and set the first upper limits for $s$-channel production cross section. CMS collaboration reported[@Chatrchyan:2011vp] the first evidence of $t$-channel single top production at the LHC and measure the cross section $83.6 \pm 29.8 {\rm (stat.+syst.)} \pm 3.3 $ (lumi.) pb with 3.5$\sigma$ significance at 36 $\rm pb^{-1}$ of integrated luminosity and 95% C.L. limit for CKM matrix element $0.68<|Vtb|$. This result was improved with the higher statistics[@CMS-t-channel]: $70.2 \pm 5.2 {\rm (stat.)} \pm 10.4 {\rm (syst.)}\pm 3.4 $ (lumi.) pb. CMS collaboration made the first measurement of the $tW$-channel cross section[@CMS-tw]$\sigma(tW)=16\pm^{5}_{4}{\rm (stat.+syst.)}$ with 4$\sigma$ of the observed significance with 4.9 $fb^{-1}$ of integrated luminosity. The ATLAS collaboration has reported[@ATLAS-t] observation of $t$-channel single top production with cross section $90\pm 9 {\rm (stat.)} \pm^{31}_{20} {\rm (syst.)}$ pb at 7.6$\sigma$ significance with 0.7 $\rm fb^{-1}$ of integrated luminosity. The first evidence of $tW$-channel production (at 2.05 $\rm fb^{-1}$) has reported by ATLAS[@:2012dj] with $3.3\sigma$ significance and measured cross section $\sigma(tW) = 16.8 \pm 2.9 {\rm (stat)} \pm 4.9 {\rm (syst)}$ pb, translated to measurement of $|V_{tb}| = 1.03^{+0.16}_{-0.19}$. ATLAS has set the first LHC $s-$channel limits[@ATLAS-s] $\sigma(s-{\rm channel})<26 pb$. Both of the collaborations have started the searches for the “New Physics” effects in single top production.
Single top at future linear colliders
=====================================
For completeness we discuss in this section single top quark production at future lepton colliders. In $e^+e^-$ collisions, the top quarks can be produced in pairs or singly similarly to hadronic collisions. However, an important difference is that at lepton collider both pair and single production processes have the same electroweak origin while at hadron colliders the top quark pair is a strong production process. Due to the fact of the electroweak nature, both processes may simultaneously give contributions to the same final states. Therefore, a special care should be paid to split these two contributions correctly.
To illustrate this, let us consider for simplicity the case when one of the top remains stable. In this case, the contributing diagrams for both pair and single top quark production in $e^+e^-$ collisions are shown in Figs. \[eetops\] and \[eetopt\] for $e \nu_{e} b t$ final state[@Boos:2001sj]. The diagrams form so-called CC20[@Bardin] set of diagrams which splits into two gauge invariant subsets of 10 diagrams, s-channel and t-channel (see[@boos-ohl]. The s-channel subset contains only two diagrams (diagrams 1,2 in Fig. \[eetops\]) with top pair production and subsequent decay. All other diagrams in s-channel and t-channel subsets contribute to the single top production. The other possible final states correspond to the top pair production and follow from possible decay modes of $W$-boson $e^+e^- \to t\bar{t} \to W W b \bar{b}, ~~~~~W \to f\bar{f^{\prime}}$,\
where e.g. for $W^+$ $f = u,c,\nu_{e},\nu_{\mu},\nu_{\tau}\nu_{\mu}$; $f^{\prime} = d,s,e,\mu,\tau$
figs/fd\_entb\_s.tex
figs/fd\_entb\_t.tex
One can compute the single top cross section as the difference of the complete tree-level (CTL) contribution and the Breit-Wigner (BW) resonance contribution obtained from a proper fit of the invariant mass distribution computed from the gauge invariant complete set of tree-level diagrams: $\int dM_{e \nu b} \, (d\sigma^{CTL}/dM_{e \nu b}-d\sigma^{BW}/dM_{e \nu
b})$. One can also use a cut on the $e \nu b$ invariant mass around the top quark pole[@sensitivity; @Belyaev:2000me] as an equivalent of the BW subtraction procedure $$\sigma=\int\limits_{M_{min}}^{m_{top}-\Delta} dM_{e \nu b}
\frac{d\sigma^{CTL}}{dM_{e \nu b}}
+\int\limits^{M_{max}}_{m_{top}+\Delta} dM_{e \nu b}
\frac{d\sigma^{CTL}}{dM_{e \nu b}},$$ where the value of $\Delta$ is taken to be 20 GeV. With such a value cross sections in both ways of computation agree very well. This value of $\Delta$ is much larger than an intuitively expected one of the order of the top quark width, which would lead to large contributions of surviving $t \bar t$ events. Obviously, the procedure applied is gauge invariant.
In case of $\gamma\gamma$ collisions there are no nontrivial gauge invariant subsets Fig. \[aatopt\] Ref.[@boos-ohl]. A situation is similar to single top at the LHC in $Wt$ mode. The top pair and single top contributions could be separated in a gauge invariant way as above.
figs/aa\_diagr.tex
In $\gamma e$ collisions the top quarks could be produced only singly Fig. \[aetopt\], the corresponding diagrams are shown in the following figure.
figs/fd\_ge\_ntb
This is one of so called “gold plated” processes in $\gamma e$ collision mode of ILC.
Table \[tab:lc\_cs\] shows single top production cross sections in various collision modes for unpolarized and polarized beams expected at a linear collider[@Boos:2001sj].
One can see from the Table 1 that both t-channel and s-channel parts contribute to the single top production rate. However the t-channel contribution grows with energy and dominates for very high energies. The NLO QCD corrections to the single top quark production in an effective $W\gamma$ approximation (EWA) have been computed in Ref in the case of $\gamma e$ and in Ref. for the case of $e^+ e^-$ collisions showing the corrections are of the order of $10\%$.
Cross sections of various processes in different collision modes at a linear collider are collected in Fig. \[fig:LC\_xsections\] [@Weiglein:2004hn]. One can stress the single top quark production in $\gamma e$ collisions is smaller than the top-pair rate in $e^+e^-$ only by a factor of 1/8 at 500–800 GeV energies, and it becomes the dominant LC process for the top production at a multi-TeV LC like CLIC. The $|V_{tb}|$ matrix element can be measured at a LC, especially in $\gamma e$ collisions, significantly more accurate compared to the LHC.
Similar to single top production in $tW$ more at hadron colliders there is interesting spin correlation between top production and decay in $\gamma e$ collisions. Here the directions of the initial photon and the electrons play the role of the gluon and lepton direction in the $tW$ process at a hadron collider. The directions of $\gamma$ and electron beams are close to the top-quark rest frame since the top is moving slowly here. So one would expect that the top quark is strongly polarized in the direction of the initial electron beam. Indeed, the angular distribution for the angle between the lepton from the top decay and the initial electron beam shows about 90% correlation [@Boos:2002xw] (see Fig. \[fig:spincorr\_LC.eps\]).
“New Physics” via single top.
==============================
Single top processes having a significant production rate at both the Tevatron and the LHC colliders are important backgrounds for various searches beyond the Standard Model (BSM). However, the processes are extremely interesting for different types of possible manifestations of “New Physics”. In general, two main situations are possible depending on relations between characteristic collision energy and thresholds of possible new states. If the collision energy is smaller than the production threshold $E_{\rm collisions} < E_{threshold}$ new states can not be produced directly and “New Physics” may manifest as deviations in production cross sections and kinematic distributions due to possible anomalous couplings or the interference of new resonances with the SM contribution below thresholds. In case of single top, the anomalous couplings could be anomalous $Wtb$ and/or FCNC couplings. If the collision energy is greater than the production threshold $E_{\rm collisions} > E_{threshold}$ new states can be produced directly in the production or in the decay of the top quark. Such states could be an additional vector ($W^\prime$) or scalar ($H^+$) bosons predicted in many extensions of SM.
Generic parametrisation of the anomalous couplings follows from the effective Lagrangian approach[@Buchmuller:1985jz]. There are number of dimension 6 effective operators which preserve the SM gauge invariance[@AguilarSaavedra:2008zc; @Zhang:2012cd] and lead to modifications of $Wtb$ vertex and appearance of additional FCNC couplings of the top quark with $u-$ and $c$-quarks.
The effective $Wtb$ vertex has the following form[@Kane:1991bg]: $$\begin{aligned}
\mathcal{L}&=& \frac{g}{\sqrt{2}}\bar{b} \gamma^\mu V_{tb}
\nonumber (f^L_V P_L + f^R_V P_R) t W_{\mu}^{-}\\
& &-\frac{g}{\sqrt{2}} \bar{b} \frac{i\sigma^{\mu\nu} q_{\nu}V_{tb}}{M_W}
(f^L_T P_L + f^R_T P_R) t W_{\mu}^{-} + h.c. \, ,
\label{anom_wtb_eq_lagrangian}\end{aligned}$$ where $M_W$ is the mass of the $W$ boson, $q$ is its four-momentum, $V_{tb}$ is the Cabibbo-Kobayashi-Maskawa matrix element and $P_{L}=(1 - \gamma_5)/2$ ($P_{R}=(1 + \gamma_5)/2$) is the left-handed (right-handed) projection operator. The anomalous couplings $f^L_V,f^R_V,f^L_T,f^R_T$ are related to the constants in front of the effective operators[@AguilarSaavedra:2008zc] in the following way: $$\begin{aligned}
\label{eq:operator}
|f^L_V| &=& 1 + |C_{\phi q}^{(3,3+3)}| \frac{v^2}{V_{tb} \Lambda^2} \; ,\nonumber \\
|f^R_V| &=& \frac{1}{2} |C_{\phi \phi}^{33}| \frac{v^2}{V_{tb} \Lambda^2} \; ,\nonumber \\
|f^L_T| &=& \sqrt{2} |C_{dW}^{33}| \frac{v^2}{V_{tb} \Lambda^2} \; ,\nonumber \\
|f^R_T| &=& \sqrt{2} |C_{uW}^{33}| \frac{v^2}{V_{tb} \Lambda^2} \; , \end{aligned}$$ One should mentioned that the couplings $C$ are naturally of the order of unity. Therefore, one may expect the natural size of $f$ couplings is of the order of $\frac{v^2}{\Lambda^2}$ which is about 0.05 or less. The theoretical estimations performed in Refs.[@Boos:1999dd; @Tsuno:2005qb; @AguilarSaavedra:2008gt; @Bernreuther:2008us] have shown the Tevatron and LHC collider potentials showing the natural size of the parameters could be achieved. However, the expected limits could not be much better because uncertainties are dominated by systematics. Expected bounds could be improved by a factor of $2 \div 3$ at a Linear Collider specially if the $e\gamma$-collision mode will be realised[@Boos:2001sj; @Boos:1997rd]. One should mentioned that in the single top processes the $Wtb$ anomalous couplings contribute to the production, to the subsequent decay of produced top quark and change the total width of the top quark correspondingly. The resulting dependence of the signal process on anomalous parameters is more complicated than a simple polinom structure. Such a dependence affects spin correlations between production and decay and can be exploited in experimental analysis. In the pair top quark production processes, anomalous operators are detectable in the W boson helicity fractions in the decay of top quarks[@Chen:2005vr]. The recent and most tight direct limits[^4] to the anomalous couplings in $Wtb$ vertex comes from combination of the measurements in single and pair top quark production processes[@:2012iwa]. The current limits are[@:2012iwa]: $|f_V^R|^2 < 0.30 $, $|f_T^L|^2 < 0.05 $, $|f_T^R|^2 < 0.12 $. Assuming the scale $\Lambda=1$ TeV these limits translate to the corresponding limits on anomalous operators to be $|C_{\phi q}^{(3,3+3)}| < 14.7$, $|C_{\phi \phi}^{33}|< 18.0$, $|C_{dW}^{33}|< 2.5$, and $|C_{uW}^{33}|< 4.1$.
FCNC top quark anomalous couplings can be probed in their production or in their rear decays[@Beneke:2000hk]. In particular, $tug$ and $tcg$ FCNC couplings may affect the single top production rate and it was exploited at the Tevatron and the LHC to set the corresponding limits. The gauge invariant effective Lagrangian describing these anomalous couplings has the following form $$\label{eq:fcnc_lagrangian}
{\cal L}_{\rm FCNC} = \frac{\kappa_{tgq}}{\Lambda} g_s \bar{q} \sigma^{\mu\nu} \frac{\lambda^a}{2} t G^a_{\mu\nu} ,$$ where $q$ = $u$ or $c$, with $u$, $c$ and $t$ representing the quark fields; $\kappa_{tgq}$ defines the strength of the $tgu$ or $tgc$ couplings; $g_s$ and $\lambda^a$ are the strong coupling constant and color matrices; $\sigma^{\mu\nu}=i/2(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})$ and $G^a_{\mu\nu}$ are the Dirac tensor and the gauge field tensor of the gluon. From the above effective Lagrangian one easily obtains the partial rare decay widths $t\to qg$: $$\begin{aligned}
\Gamma({ t \to q g}) \,\, &=& \left( \frac{\kappa^g_{tq}}{\Lambda} \right)^2
\frac{8}{3} \alpha_s m_t^3 \quad \quad , \quad \quad
\label{anomeq:fcnc_br} \end{aligned}$$ The partial width is directly proportional to the ratio $\left( \frac{\kappa^g_{tq}}{\Lambda} \right)^2 $, therefore various authors and collaborations present limits either in terms of the constants $\kappa$ or in terms of corresponding branching ratio. The representative diagrams contributing to various processes[@Han:1998tp; @Liu:2005dp; @Gao:2009rf] of single top production due to presence of FCNC interactions are shown in Fig. \[fig:fcnc\_feynman\].
![The representative leading order Feynman diagrams for FCNC gluon coupling between a charm quark and a top quark (for a u-quark the diagrams are the same). \[fig:fcnc\_feynman\] ](figs/fcnc_feynman.eps){width="49.90000%"}
The model independent analysis based on the signal diagrams in Fig. \[fig:fcnc\_feynman\] corresponds to the case when extra hard jet is detected in association with the top quark. The most recent results on FCNC anomalous coupling limits in terms of both couplings and branchings are presented in Table \[tab:obslim\]. The accumulated luminosities for each of the analysis are listed explicitly in the table. From the table one can see that the current LHC limits at 7 TeV obtained by the ATLAS collaboration are already significantly tighter than the Tevatron limits by D0 and CDF experiments with almost the same luminosity. One should stress the best current limits are far above very small SM value for the decay branching ratio $Br^{SM}(t\to cg) \approx 5 \times 10^{-11}$, however, being in the interesting range for some extensions of the SM[@Beneke:2000hk].
As was mentioned, if the achievable characteristic collision energy $E_{\rm collisions}$ is grater than a production threshold of possible new particles $E_{\rm threshold}$ one may detect these particles directly. In our case of single top production, such particles could be either newly charged vector or scalar bosons produced as $s-$channel resonances decaying to the top quark. We consider only single top production processes and, therefore, we refer only to $W^\prime$ and charged scalar decaying to top and bottom quarks and do not discuss the processes with leptonic decays of $W^\prime$ where the observed limits are much stronger in some cases. Such models like Non-Commuting Extended Technicolor [@Chivukula:1995gu], Composite[@Kaplan:1983fs; @Georgi:1984af] and Little higgs models[@Arkani:2001nc; @Kaplan:2003uc; @Schmaltz:2005ky], models of composite gauge bosons[@composites], Supersymmetric top-flavor models[@Batra:2003nj], Grand Unification[@guts], Superstring theories[@Cvetic:1996mf; @Pati:2006nw; @superstrings], and Left-Right symmetric models[@Pati:1974yy]-[@Langacker:1989xa], represent examples where extension of gauge group lead to appearing of $W^{\prime}$. $W^{\prime}$ appears also in the Universal extra-dimension[@Datta:2000gm; @extradimensions] type of models. Charged scalar (pseudo-scalar) bosons naturally present in many SM extensions as well, in particular, in two-Higgs-doublet models (2HDMs) of various types two charged Higgs bosons appear.
In searches for these new bosons in a model independent way, the effective Lagrangian approach is used. Corresponding effective Lagrangians have the following forms \[eq:wprime\_lagrangian\] and \[eq:higgs\_lagrangian\] neglecting higher-dimensional structures.
$$\begin{aligned}
\label{eq:wprime_lagrangian}
{\cal L} = \frac{V_{q_iq_j}}{2\sqrt{2}} g_w \overline{q}_i\gamma_\mu
\bigl( a^R_{q_iq_j} (1+{\gamma}^5) + a^L_{q_iq_j}
(1-{\gamma}^5)\bigr) W^{\prime\mu} q_j
+ \mathrm{H.c.} \,, \end{aligned}$$
where $a^R_{q_iq_j}, a^L_{q_iq_j}$ - left and right couplings of $W^\prime$ to quarks, $g_w = e/(s_w)$ is the SM weak coupling constant and $V_{q_iq_j}$ is the SM CKM matrix element. The notations are taken such that for so-called SM-like $W^{\prime}$ $a^L_{q_iq_j}=1$ and $a^R_{q_iq_j}=0$.
$$\begin{aligned}
\label{eq:higgs_lagrangian}
\mathcal{L} & = & \frac{g_w V_{q_i q_j}}{2\sqrt{2}} H^+ \bar{q}_i \left[
g^{ij}_L \left( \frac{1 - \gamma^5}{2} \right) + g^{ij}_R \left( \frac{1 + \gamma^5}{2}
\right) \right] q_j \end{aligned}$$
Since there are no charge scalars in SM the couplings $g_L^{ij}$, $g_R^{ij}$ are obviously equal to zero in SM.
For the $W^\prime$ searches the limits on $W^\prime$ mass depend on both left and right couplings $a^L, a^R$ and for the case of right-handed interaction of $W^\prime$ on a relation between the $W^\prime$ mass and the mass of possible right-handed neutrino $\nu_R$. If the decay of $W^\prime$ to right-handed neutrino is kinematically allowed the branching ratio of $W^\prime$ decay to top and bottom quarks is smaller and corresponding mass limits are expected to be worse. The NLO corrections to the $s-$channel $W^\prime$ production with subsequent decay were computed in Ref.[@Sullivan:2002jt] and an influence of $W^\prime$–$W^{SM}$ interference was demonstrated in Ref.[@Boos:2006xe]. In case the $W^\prime$ is a first KK state in models with extra deminsions, an additional interference with rest of KK tower should be taken into account Ref.[@Boos:2011ib]. The 95% C.L. limits obtained by the Tevatron at $\sqrt{s}=1.96$ TeV and the LHC collaborations at $\sqrt{s}=7$ TeV are collected in the Table \[tab:wprime\] for the cases where the couplings $a^L, a^R$ are equal to $1, 0$ (Left), $0, 1$ (Right) and $1, 1$ (Left, Right). One can see the ATLAS limit with about two times smaller statistics is already better than the Tevatron limits.
The available experimental limits for the charged scalar Higgs particle decaying to top and bottom quarks are published by D0 collaboration[@Abazov:2008rn]. The mass limits are given in the mass range grater than 180 GeV ($M_{H^+}>M_{top} + M_{b}$) since lower mass range is covered with better precision from the top pair production with the top decay to the charged Higgs and b-quark. The limits strongly depend on the ratio of two Higgs vacuum expectation values $\tan\beta$ and the type of 2HDMs (Type I, Type II and Type III).
Conclusions and outlook
=======================
The top quark, being the heaviest elementary particle discovered so far, has been observed in pair first and later in single production mode. Single top quark registration at the Tevatron and then at the LHC is an important confirmation of the structure of the electroweak part of the Standard Model. Measured production cross sections in combination of $t-$ and $s-$channel modes at the Tevatron and in the $t-$channel only at the Tevatron and the LHC are in a good agreement with the SM computations at NLO level including part of NNLO corrections. $tW-$channel is very small to be measured at the Tevatron while it was measured for the first time at the LHC. Within rather large experimental uncertainties the results for the $tW-$channel are in an agreement with SM as well. The $s-$channel cross section at the LHC is much smaller than the $t-$channel making an observation of this channel at the LHC very challenging especially because the $t-$channel gives a huge irreducible background to the $s-$channel in a kinematic range where an additional light jet is undetectable. However, one expects the detection of the $s-$channel with increased analyzed statistic. Having small rates, these processes are interesting to search for deviations from the SM predictions.
The performed studies of the single top production allowed to measure directly for the first time the CKM mixing matrix parameter $|V_{tb}|$. Published results are in agreement with expected unity, however, still at about 20% level of accuracy.
The top is very heavy and as the result is very special fermion in SM. The single top production is a power tool to search for delicate deviations from the SM. No such deviations have been observed yet and limits on the anomalous $Wtb$ and FCNC top quark couplings as well as on parameters of new vector and scalar bosons decaying to top quark were extracted. The mass limits for new resonances are expected to be much higher (in few TeV range) with increasing LHC energy. While the limits on anomalous couplings will be dominated by systematic uncertainties and therefore might be improved by a factor of two or so. Much tighter limits at a percentage level of accuracy could be achieved at a future Linear Collider.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are grateful to many theory colleagues and colleagues from single top groups of D0 and CMS collaborations for clarifying discussions and joined studies. The work was supported in part by the RFBR grant 12-02-93108-CNRSL−a and the grant of Russian Ministry of Education and Science NS-3920.2012.2.
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[^1]: boos@theory.sinp.msu.ru
[^2]: dudko@sinp.msu.ru
[^3]: For the cancellation of the anomaly in each generation the sum of electric charges of leptons should be equal with the opposite sign to the sum quark charges: $(Q_{top}+Q_{b})\times N_c + Q_{/tau} = 0$
[^4]: Indirect limits on the anomalous $Wtb$ couplings follow from measurements of the $b \rightarrow s\gamma$ decays[@Drobnak:2011aa], however, the limits are obtained with the assumption that there are no other BSM contribution in the loop.
|
---
abstract: 'We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.'
address: |
Department of Mathematics\
North Dakota State University\
Fargo, North Dakota\
USA
author:
- 'Benton L. Duncan'
title: Operator algebras associated to modules over an integral domain
---
Background on semicrossed products
==================================
Although the study of semicrossed products (as objects of study) could be traced to [@Peters] and its precedents [@Arveson; @ArvesonJosephson] the theory, as studied now was revived by [@DavidsonKatsoulis] where semicrossed products by $ \mathbb{Z}_+$ were shown to be complete invariants for topological dynamics. This recent interest in semicrossed products as a class of operator algebras has seen significant recent growth, see for example [@Simple; @Envelope; @dilation; @KK]. The primary focus of many of these new results is in the expansion of what semigroups are acting on the underlying $C^*$-algebra.
In particular, in [@multivariable] an attempt was made to understand actions of the free semigroup on $n$-generators acting on a compact Hausdorff space. Other research focused on abelian semigroups, see [@DuncanPeters] and [@Fuller]. The paper [@DFK] formalized many of the different ideas used in the theory by combining constructions and different viewpoints into one overarching theme: studying the dilation theory of different classes of semicrossed products that are as much a function of the semigroup as the action on the underlying $C^*$-algebra.
During this same time period the author in [@Duncan] introduced a construction, similar in spirit to [@CuntzLi] although different in perspective. Focusing on the non-selfadjoint theory this class of algebras was recognized as a semicrossed product [@Duncan Section 3]. Here we take the algebraic notion to the next stage. Here we consider a semicrossed product associated to a module over an integral domain $D$. Of course [@Duncan] is a special case of this as any ring is a module over itself.
Here we let a domain “act” on the group $C^*$-algebra associated to the module, via multiplication. As we view the group as a discrete group the actions are continuous and give rise to $*$-endomorphisms of the group $C^*$-algebras. For the most part we are undertaking this study to provide examples and motivation for questions in the broader area of semicrossed products.
For these purposes we first consider a general approach to operator algebras associated to a module over an integral domain, defining those properties that such an operator algebra should have. We then justify the connection to semicrossed products by showing that an associated Fock semicrossed product will satisfy the list of requirements for an operator algebra associated to a module over an integral domain. This allows us to identify the $C^*$-envelope of the algebra.
We then use this connection to motivate and begin the study of questions for general semicrossed product and (often) the underlying dynamics. In the first such example we consider short exact sequences of modules and look at structures that arise in the operator algebra context. This leads us to propose looking at semicrossed products from a categorical point of view. Another example that we investigate is the notion of finite generation of a module over an integral domain. This gives rise to a similar notion of semicrossed products. We close with some results relating finite generation in semicrossed products to results on dynamics, in limited contexts. We also point out the difficulties inherent in considering this notion for arbitrary semicrossed products. We do, however, expect that these approaches will provide examples of results to look for in investigating more general semicrossed products.
In what follows we assume familiarity with the results and notation of [@DFK].
Operator algebras of modules over integral domains
==================================================
Let $R$ be an integral domain and $M$ be an $R$-module. The multiplicative semigroup of $R$ (call it $R^{\times}$) acts on $M$ via the mapping $r(m) = r\cdot m$. If we view $M$ as a topological group with the discrete metric then this action will induce an action of $R^{\times}$ on $C^*(M) = C(\widehat{M})$ (this latter is the continuous functions on the compact Pontryagin dual of $M$). However this action is not necessarily as homeomorphisms and hence semicrossed products are a natural approach to understanding this action using operator algebras. In particular the action $ \sigma_r: C^*(M) \rightarrow C^*(M)$ is invertible if and only if $r$ is invertible in $R$.
For this reason we define the operator algebra of a module over an integral domain to be the semicrossed product $ C^*(M) \times_{F} R^{\times}$. In general one can consider a more general construction (which parallels the semicrossed product constructions of [@DFK]). We first consider a specific representation. Consider the Hilbert space $ \mathcal{H} := \ell^2(M) \otimes \ell^2(R^{\times})$. For each $r \in R^{\times}$ we define $ S_r: \mathcal{H} \rightarrow \mathcal{H}$ on elementary tensors by $ S_r( v_m \otimes u_s) = v_{m} \otimes u_{rs}$ and then by extension we have $S_r \in B(\mathcal{H})$. At the same time for each $m \in M$ we define $U^m(v_n \otimes u_r) = v_{mr+n} \otimes u_r$, and again extending to all of $ \mathcal{H}$.
We notice a few facts about the operators $S_r$ and $U^m$.
Let $M$ be an $R$ module and consider $m, n \in M$ and $r,s \in R^{\times}$. Then the following are true:
1. \[unitaries\] $U^m$ is a unitary with $U^{-m} = (U^m)^*$.
2. \[isometries\] $S_r $ is an isometry and if $r$ is invertible then $S_r$ is a unitary with $S_r^* = S_{r^{-1}}$.
3. \[grouprepn\] The map $ \mu: M \rightarrow B(\mathcal{H})$ induced by $ \mu(m) = U^m$ is a group representation with $ C^*(\mu(M)) \cong C_r^*(M)$ (the latter being the group $C^*$-algebra associated to $M$).
4. \[semigrouprepn\] The map $ \rho: R^{\times} \rightarrow B(\mathcal{H})$ given by $ \rho(r) = S_r$ is a semigroup representation.
5. \[respectsmodule\] The operators respect the module structure as follows:
- $U^mS_r = S_rU^{rm}$,
- $U^{m}S_{r+s} = S_{r+s}U^{rm}U^{sm}$,
- $U^{m+n}S_r = S_rU^{rm}U^{rn}$, and
- $S_1 = U^0= 1_{\mathcal{H}}$,
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1. We calculate that $$\langle U^n(v_m \otimes u_r), v_l \otimes u_s \rangle = \begin{cases} 1 & r=m+nr \mbox{ and } r=s \\ 0 & \mbox{ otherwise} \end{cases}$$ and that $$\langle v_m \otimes u_r, U^{-n} (v_l \otimes u_s) = \begin{cases} 1 & m = l-ns \mbox{ and } s = r \\ 0 & \mbox{ otherwise} \end{cases}.$$ It follows using linearity that $(U^n)^* = U^{-n}$ and then calculating we see that $U^nU^{-n} = 1_{B(\mathcal{H}} = U^{-n}U^n$ which tells is that each $U^n$ is a unitary.
2. We begin by calculating the adjoint of $S_r$. To do this we first write $H_r$ to be the subspace of $\mathcal{H}$ spanned by elements of the form $ v_m \otimes u_{rt}$ where $ m \in M $ and $ t \in R^{\times}$. We define $T_r( v_m \otimes u_{rt}) = v_m \otimes u_t$ on $H_r$ and $T_r $ is zero on the orthgonal complement of $H_r$.
Now $$\begin{aligned}
\langle S_r( v_m \otimes u_s), v_n \otimes u_t \rangle & = \langle v_m \otimes u_{sr}, v_n \otimes u_t \rangle \\ & = \begin{cases} 1 & \mbox{ if } t = sr \\ 0 & \mbox{ otherwise} \end{cases} \end{aligned}$$ and similarly we have $$\langle v_m \otimes u_s, T_r(v_n \otimes u_t) \rangle = \begin{cases} 1 & \mbox{ if } t = rs \\ 0 & \mbox{ otherwise} \end{cases}$$ and hence $T_r$ is the adjoint of $S_r$.
We then can calculate that $ T_rS_r ( u_n \otimes v_s) = T_r(u_n \otimes v_{rs}) = u_n \otimes v_s$ and hence $S_r$ is an isometry with range equal to $H_r$. Notice that if $r$ is invertible then for any $ m \in M$ and $ t \in R^{\times}$ then $ v_m \otimes u_t = v_m \otimes u_{rr^{-1}t}$ and hence $H_r = \mathcal{H}$ which tells us that $S_r$ is surjective and hence is a unitary with $T_r$. In this case a simple calculation will show us that $S_{r^{-1}} = T_r$.
3. Notice that this representation induces a unitary representation of $M$ acting on $B( \mathcal{H})$. Also if we consider the subspace spanned by $ v_m \otimes u_1$ for all $ m \in M$ then the restriction of our representation to this subspace will give the left regular representation of $M$ acting on a Hilbert space isomorphic to $\ell^2(M)$. Since $M$ is abelian and hence amenable it follows that the representation must be a faithful representation of $C^*(M)$.
4. This follows by noting that $ S_{rs} (v_m \otimes u_t) = (v_{r} \otimes u_{rst} ) = S_r( v_{m} \otimes u_{st}) = S_r(S_s)(v_m \otimes u_t)$ and hence $S_{rs} = S_rS_s$ yielding a semigroup representation.
5. We verify the first property. First we see that $$\begin{aligned}
U^mS_r(v_n \otimes u_s) & = U^m( v_{n} \otimes u_{sr}) \\ & = v_{n+mrs} \otimes u_{sr} \\ & = S_r(v_{n+mrs} \otimes u_s) \\ & = S_rU^{rm} (v_n \otimes u_s )\end{aligned}$$ and then extending to all of $B( \mathcal{H})$ we have the indicated identity. The second and third properties here follow from the first. The final is a simple calculation.
Given a Hilbert space $H$ and two collections of operators $ \mathcal{U} = \{ u^m: m \in M \}$ and $ \mathcal{S} = \{ s^r: r \in R^{\times} \}$ we define a pair of maps $\mu: M \rightarrow B(\mathcal{H})$ by $ \mu(m) = u^m$ and $ \rho: R^{\times} \rightarrow B(\mathcal{H})$ by $ \rho(r) = s_r$. We say that the pair $ (\mu, \rho)$ is an isometric representation of $M$ with respect to $R$ if (1-5) of the above proposition are satisfied for the collection $ \mathcal{U}$ and $ \mathcal{S}$, it is said to be unitary if the family $ \mathcal{S}$ consists of unitaries. The previous proposition gives us a canonical isometric representation of $M$ with respect to $R$, which we call the Fock representation.
As in [@DFK] we can consider other semicrossed products associated to the action of $ R^{\times}$ acting on $C^*(M)$. In particular there are three standard operator algebras associated to such an action: the Fock algebra which we denote $C^*(M) \times_F R^{\times}$, the isometric semicrossed product which we denote $C^*(M) \times_i R^{\times}$ and the unitary semicrossed product which we denote by $C^*(M) \times_u R^{\times}$.
The algebra $C^*(M) \times_F R^{\times}$ is the norm closed algebra inside $B(\ell^2(M) \otimes \ell^2(R^{\times}))$ generated by the families $ \{ S_r \}$ and $\{ U^m \}$. This algebra will, for the most part, be the focus of this paper due to its concrete realization as acting on a Hilbert space. The algebras $C^*(M) \times_i R^{\times}$ and $C^*(M) \times_u R^{\times}$ can be constructed by considering the algebra $A_0 $ consisting of finite sums of the form $ \sum s_ra_r$ such that $a_r \in C^*(M)$ with a convolution multiplication that respects $a s_r= s_r\alpha_r(a)$, where $\alpha_r$ is the action on $C^*(M)$ induced by $r$ acting on $M$. One notices that any isometric (unitary) representation of $M$ with respect to $R$ gives rise to a representation of the algebra $A_0$ acting on the same Hilbert space. It follows that one can then norm $A_0$ by taking the supremum over all isometric (unitary) representations of $M$ with respect to $R$. Completing the algebras with respect to the induced norms gives rise to the semicrossed products $C^*(M) \times_i R^{\times}$ and $C^*(M) \times_u R^{\times}$, respectively.
The Fock semicrossed product $C^*(M) \times_F R^{\times}$ is isomorphic to the unitary semicrossed product $C^*(M) \times_u R^{\times}$ if and only if $M$ is torsion free.
The forward direction of this result follows from [@DFK Theorem 3.5.4] by noting that if $M$ is torsion free then the action of $r$ on $M$ is injective for every $ r \in R^{\times}$. Hence $R^{\times}$ acts injectively on $C^*(M)$ and the result applies. For the alternative, again from [@DFK] it is noted that if the action of $R^{\times}$ is not injective (i.e. has torsion) then $C^*(M) \times_u R^{\times}$ does not contain $C^*(M)$ but rather only a quotient of $C^*(M)$. However $C^*(M) \times_F R^{\times}$ always contains a copy of $C^*(M)$ and hence the two algebras are not isomorphic.
It is not the case that the isometric semicrossed product is isomorphic to the other two semicrossed products, even in the case that the module is torsion free, since the action of $R^{\times}$ on $M$ acts as automorphisms if and only if $R$ is a field. In this latter case [@DFK Theorem 3.5.6] we get that these algebras are isomorphic. In this case the algebra $C^*(M) \times_F(R^{\times})$ is a crossed product by an abelian (and hence amenable) group which means that the universal crossed product and the reduced crossed products are isomorphic. When $R$ is not a field it is straightforward to see that $C^*(M) \times_F R^{\times}$ is not a $C^*$-algebra and hence one important consideration is the $C^*$-algebra it generates. There is a canonical $C^*$-algebra in which an operator algebra embeds, this algebra is the $C^*$-envelope and is thought of as the “smallest” $C^*$-algebra which the nonselfadjoint algebra generates. Following [@DFK] we consider the $C^*$-envelopes of the algebra $C^*(M) \times_F R^{\times}$ in the case that $M$ is torsion free. The only added complexity is verifying that the construction still produces an $R$-module.
Let $R$ be an integral domain and $M$ be an $R$-module. There exists a $Q(R)$-module $N$ and an $R$-module injection $i: M \rightarrow N$ such that $C^*(N) \times_F Q(R)^{\times}$ is the $C^*$-envelope of $C^*(M) \times_F R^{\times}$
We mimic the construction from [@DFK Section 3.2] working at the level of $R$-modules rather than $C^*$-algebras.
We note first that $Q(R)^{\times}$ is the enveloping group for $R^{\times}$. We put a partial ordering on $R^{\times}$ given by $s \leq t$ if in $Q(R)^{\times}$ we have that $ts^{-1} \in R^{\times}$. Then for $r \in R^{\times}$ we let $M_r = M$ and define connecting maps $ \alpha_{t,s}: M_s \rightarrow M_t$ when $ s \leq t$ by $ \alpha_{t,s} (m) = ts^{-1}(M)$. Then $N:= \lim_{\rightarrow} (M_r, \alpha_{t,s})$ is an $R$-module such that the $r$ action is now surjective and injective on $N$, hence we can define the $r^{-1}$ action on $N$ by inverting the $r$-action giving an action of $Q(R)^{\times}$ on $N$.
Noting that the functor from abelian groups to $C^*$-algebras preserves direct limits it follows that $C^*(N)$ is the algebra constructed in [@DFK] and the result now follows from [@DFK Theorem 3.2.3].
Since the $C^*$-algebras in our semicrossed products arise from group $C^*$-algebras we can use information about group $C^*$-algebras to improve our analysis. In the following section we consider some results that will be helpful.
Some useful results on groups, semigroups, and semicrossed products
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Let $M$ and $N$ be discrete abelian groups with $N$ a normal subgroup of $M$. This relationship induces a short exact sequence $$1 \rightarrow N \rightarrow M \rightarrow M/N \rightarrow 1.$$
While it is not the case that a short exact sequence of groups gives rise to a short exact sequence of the associated $C^*$-algebras, the relationship between groups does give us a relationship between the associated $C^*$-algebras. We build up to the result with some preliminary lemmas, the first is a combination of Propositions 2.5.8 and 2.5.9 of [@BrownOzawa].
Given an inclusion of groups $N \subseteq M$ there is an inclusion of $C^*$-algebras $ C^*(N) \subseteq C^*(M)$ and $C^*_r(N) \subseteq C^*_r(M)$.
In the reduced case the proof involves noticing that $ \ell^2(N)$ is a subspace of $\ell^2(M)$ and the left regular action of $M$ acting on $\ell^2(M)$ gives rise to the left regular action of $N$ acting on the subspace $\ell^2(N)$. Hence the reduced $C^*$-algebra $C_r^*(N)$ can be viewed as sitting inside $C_r^*(M)$. The proof for the universal algebra is more subtle. We refer the reader to [@BrownOzawa] for the details.
Given a surjection of groups $\pi: M \rightarrow M/\ker\pi$ there is a unital surjective $*$-homomorphism $\pi: C^*(M) \rightarrow C^*(M/ \ker \pi)$.
This is a straightforward application of the universal property of $C^*(M)$ (i.e. any representation of $C^*(M/ \ker \pi)$ induces a representation of $C^*(M)$ via composition with the map $ \pi$).
Notice that if we know that the groups are amenable then the reduced $C^*$-algebra will satisfy the universal property as in the preceding lemma and hence both results will apply.
Given a $C^*$-algebra $A$ with $C^*$-subalgebra $B$ we say that a unital representation $\pi: A \rightarrow C$ trivializes $B$ if $B = \pi^{-1}(1_C)$. We consider this in the context of an exact sequence of groups.
\[exact\] Let $1 \rightarrow N \rightarrow M \rightarrow M/N \rightarrow 1$ be a short exact sequence of discrete abelian groups. Then the natural surjection $\pi: C^*(M) \rightarrow C^*(M/N)$ trivializes the subalgebra $C^*(N)$ inside $C^*(M)$.
By definition the generators for $C^*(N)$ inside $C^*(M)$ will be mapped to $1_{C^*(M/N)}$ by the natural surjection and hence the range of $C^*(N)$ under the natural surjection is $\{ \mathbb{C} 1_{C^*(M/N)} \}$.
Now consider $U_g \in C^*(M)$ and consider $ \pi(U_g)( \zeta_{xN})$. If $ \pi(U_g)$ is a multiple of the identity then $ \zeta_{xN} = \lambda \zeta_{gxN}$ for some $ \lambda$. It follows that $ \lambda$ must equal $1$ and $gxN = xN$ for all $x$. Then there is some $n \in N$ such that $gx = xn$. Canceling the $x$ tells us that $ g \in N$ and hence the only $U_g$ which $\pi$ maps to the identity are those inside $C^*(N)$.
It follows that any polynomial in the generators of $C^*(M)$ the same result will apply. Specifically $ \sum_{i=1}^n \alpha_{g_i}U^{g_1} \mapsto \mathbb{C}$ implies that $ g_i \in N$ for all $i$ and hence by continuity $\pi^{-1}(\mathbb{C} 1_{C^*(M/N)}) = C^*(N)$.
In the preceding proof it is not necessary that the groups be abelian (amenable would suffice). However, for our purposes abelian will be more than enough.
Consider the case of $C^*(\mathbb{Z}) \cong C(\mathbb{T})$. Notice that any subgroup of $ \mathbb{Z}$ is of the form $ m\mathbb{Z}$. Then $ \mathbb{Z} / m \mathbb{Z} \cong \mathbb{Z}_m$ and there is a natural representation $\pi: C(\mathbb{T}) \rightarrow \oplus_{m} \mathbb{C}$ given by $\pi(f(z)) = ( f(e^0), f(e^{\frac{2\pi i}{m}}), f(e^{\frac{ 4 \pi i }{m}}), \cdots , f(e ^{ \frac{(2m-2) \pi i}{m}}))$. Notice that $ \pi(z^k) = (1,0,0, \cdots, 0)$ if and only if $ m|k$. Then if we set $A = \{ f: \pi(f) = ( \lambda, 0, 0, \cdot, 0 ) \}$ then $A$ is a $C^*$-subalgebra of $C(\mathbb{T})$ generated by $ \{ z^km: k \in \mathbb{Z} \}$. Notice that $\pi$ trivializes $A$ and in fact $A \cong C^*(m \mathbb{Z})$.
On the other hand notice that the representation $ \pi: C(\mathbb{T}) \rightarrow \mathbb{C}$ given by $ \pi(f(z)) = f(i)$ trivializes the $C^*$-subalgebra generated by $ \{ z^4 \}$. However, $ \pi^{-1}(\mathbb{C} 1) )= C^*(\mathbb{Z})$.
In fact we can go further. If $K $ is a subgroup of $G$ then there is a universal trivializing algebra through which all trivializing representations for $C^*(K) \subseteq C^*(G)$ factor. This representation is, in fact, the standard representation $ \pi: C^*(G) \rightarrow C^*(G/K)$.
\[trivializing\] Let $A$ be a $C^*$-subalgebra of $C^*(G)$. There is a subgroup $K \subset G$ such that $A \cong C^*(K)$ if and only if $$A = \bigcap \{\pi^{-1}( \mathbb{C}1): \pi \mbox{ trivializes }C^*(K) \} .$$
The forward direction is described before the statement of the proposition. We consider here the converse. So let $\pi: C^*(G) \rightarrow B(\mathcal{H})$ be a representation which trivializes $A$. Now let $N_{\pi} = \{ g \in G: \pi(U^g) = 1 \}$. Checking we see that $N_{\pi}$ is a subgroup of $G$, since if $g, h \in N$ then $\pi(U^{-g}) = \pi((U^{g})^*) = \pi(U^g)^* = 1^* = 1$ and $ \pi(U^{g+h}) = \pi(U^gU^h) = \pi(U^g)\pi(U^h) = 1$ so that $N_{\pi}$ is closed with respect to inverses and the group operation. Then we define $K = \cap \{ N_{\pi}: \pi \mbox{ trivializes } A \}$.
It remains to prove that $A = C^*(K)$. Certainly we know that $C^*(K)$ can be viewed as sitting inside $A$. Next we consider the natural map $ \pi: C^*(G) \rightarrow C^*(G/K)$ which trivializes $C^*(K)$ and hence $A \subseteq C^*(K)$.
Next we consider similar results for semicrossed products.
\[invariance\] Let $A$ be a closed subalgebra of $B$ and let $S$ be a discrete abelian semigroup acting on $B$ via completely contractive endomorphisms. If $\sigma_s(A) \subseteq A$ for all $ s \in S$ then $A\times_{F} S \subseteq B \times_{F} S$.
Let $ \pi: B \rightarrow B(\mathcal{H})$ be a faithful representation, then notice that $ \pi|_A: A \rightarrow B(\mathcal{H})$ is a faithful representation. Further the canonical Fock representation of $B \times_F S \rightarrow B(\mathcal{H} \otimes \ell^2(S))$ restricts to, since $\sigma_s(A) \subseteq A$ for all $s \in A$, a Fock representation of $A \times_F S$ acting on $ B(\mathcal{H} \otimes \ell^2(S))$ and the result now follows, since this representation will be unitarily equivalent to the standard canonical Fock representation of $A \times_F S$.
Let $\pi: B \rightarrow C$ be a completely contractive quotient map and assume that an abelian discrete semigroup $S$ acts on $B$ (via $\sigma$) and on $C$ (via $\tau$) as completely contractive endomorphisms. If $ \tau_s \circ \pi = \pi \circ \sigma_s$ for all $s$ then there is a completely contractive quotient map $\tilde{\pi}: B \times_F S \rightarrow C \times_F S$ such that $ \tilde{\pi}(b) = \pi(b)$ for all $b \in B$ and $\tilde{\pi}(S_s) = T_s$ for all $s \in S$.
This proof is a simple outcome of the Fock construction since the map $\pi: B \rightarrow C$ with the associated representation of $S_s \mapsto T_s$ gives rise to a Fock representation of the pair $(B, \sigma)$. By definition [@DFK Definition 3.5.1] the Fock algebra $B \times_F S$ is universal for Fock representations we get the indicated homomorphism.
Let $X$ be a compact Hausdorff space and consider an abelian semigroup $S$ acting on $X$ by continuous maps $ \{\sigma_s: s \in S \}$. If $S = M \times N$ then there is a natural action of $N$ on $C(X) \times_F M$ given on the generators of $C(X) \times_FM$ by $\sigma_n(f(x)) = f( \sigma_n(x))$ and $\sigma_n(S_m) = S_m$ for all $m \in M$; it is straightforward to that verify the covariance relationship is preserved.
Let $S = M\times N$ be a direct product of abelian semigroups acting via $*$-endomorphisms on a $C^*$-algebra $A$. Then $A \times_F S \cong ( A \times_F M) \times_F N \cong ( A \times_F N) \times_F M$.
For $s \in S$ we denote by $ \sigma_s: A \rightarrow A$ the associated $*$-endomorphism. For $ n \in N$ we define an action on the generators of $A \times_F M$ by $ \alpha_n ( a) = \sigma_n(a)$ for all $ a \in A$ and $ \alpha_n(S_m) = S_m$ for all $ m \in M$. This yields a Fock representation of $A\times_F M$ since the actions of $M$ and $N$ on $A$ commute and hence it induces a completely contractive representation of $A \times_F M$.
One then considers the canonical Fock representation of $(A \times_F M) \times N$ and notes that this yields a Fock representation of $A \times_F (M\times N)$. Similarly the canonical Fock representation of $A \times_F (M \times N)$ yields a Fock representation of $(A \times_F M) \times_F N$. A similar result holds for the alternative with $M$ and $N$ changing place and hence the result holds.
Clearly this extends via induction to any finite product of abelian semigroups.
Notice that this is true (as a special case) for a crossed product by the product of two abelian groups, since abelian groups are amenable and hence the canonical Fock representation is faithful in this context.
As an application of this result, if we denote by $R_u$ the group of units in $R$ then one can see that $C^*(M) \times_F R^{\times} \cong (C^*(M) \times_F R^u) \times (R^{\times}/R^u)$. This allows us to identify (as in [@Duncan]) the diagonal of the algebra $C^*(M) \times_FR^{\times}$ as the crossed product $C^*(M) \times R^u$.
In our extended context however submodules are more than just subgroups of $M$. We thus extend the previous results in the following section.
Submodules and Quotients
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In [@BrownOzawa Corollary 2.5.12] it is shown that if $ N$ is a subgroup of $M$ then there is a conditional expectation $E: C^*(M) \rightarrow C^*(N)$ such that for $ m\in M$ we have $E(U^m) = \begin{cases} U^m: m \in N \\ 0 \end{cases}$ (this is true for both the universal and reduced $C^*$-algebras). A partial converse of this can be found as a corollary of results in [@Choda; @LOP]; the former containing a version of the result for the reduced $C^*$-algebra for a discrete group and the latter containing a version of the result for abelian group $C^*$-algebras (in which the universal and reduced $C^*$-algebras are isomorphic). Specifically, we have the following proposition.
Let $A \subseteq C^*(M)$ be a subalgebra. Then there is a subgroup $N \subseteq M$ such that $A \cong C^*(N)$ if and only if there is a conditional expectation $E: C^*(M) \rightarrow A$ with for any $ m \in M$ $E(U^m) = \begin{cases} U^m &: U^m \in A \\ 0 & \mbox{ otherwise}. \end{cases}$.
Let $R$ be a ring and $M$ be an $R$-module. A submodule $N \subseteq M$ is a subgroup of $M$ which is closed under the action of $R^{\times}$. To study submodules and quotient modules in the context of operator algebras we use the results of the preceding section and the preceding result in the context of the Fock algebra associated to a module over an integral domain. The following is just a combination of the preceding result and Proposition \[invariance\].
Let $A$ be a subalgebra of $C^*(M) \times_F R^{\times}$. There is a submodule $N \subseteq M$ with $A \cong C^*(N) \times_F R^{\times}$ if and only if there is a conditional expectation $E: C^*(M) \rightarrow A \cap C^*(M)$ such that $ E(U^m) = \begin{cases} U^m & U^m \in A \\ 0 & \mbox{ otherwise} \end{cases}$ and $\sigma_s(A) \subseteq A$ for all $ s \in S$.
We look now to find an alternative that does not necessarily require the latter condition. Specifically assume that $A$ is a subalgebra of $C^*(M) \times_F R^{\times}$ and that $N$ is a subgroup of $M$. Then we consider the Hilbert space $ \mathcal{H}_N:=\ell^2(M/N) \otimes \ell^2(R^{\times}$, which we call the quotient Hilbert space for the pair $(M,N)$. For every $m \in M$ we define $U^m: \mathcal{H}_N \rightarrow \mathcal{H}_N$ on elementary tensors by $U^m (v_{g+N} \otimes u_r) = v_{(mr+g) +N} \otimes u_r$ and extending by linearity to all of $B( \mathcal{H}_N)$. Similarly for $t \in R^{\times}$ we define $S_t(v_{g+N} \otimes u_r) =v_{g+N} \otimes u_{rt}$ and extending to all of $B(\mathcal{H}_N)$. We call the collection of operators $\{ U^m, S_r \}$ the quotient operators for the pair $(M,N)$.
Let $N$ be a subgroup of the $R$-module $M$. Then $N$ is a submodule if and only if the quotient operators for the pair $(M,N)$ give rise to a covariant representation of $C^*(M) \times_F R^{\times}$.
If $N$ is a submodule for $N$ then the quotient operators are acting on the canonical Hilbert space for $C^*(M/N) \times_F R^{\times}$ and hence they will give rise to a covariant representation of $C^*(M) \times_F R^{\times}$, via the quotient map $q: C^*(M) \rightarrow C^*(M/N)$.
For the reverse direction notice that $N$ is an $R$-submodule if and only if $rn \in N$ for every $r \in R^{\times}$ and $ n \in N$. So, if $M$ is not an $R$-submodule then there must be some $r \in R^{\times}$ and $ n \in N$ such that $rn \not\in N$. Notice that $U^n = 1$ by definition but $U^{rn} \neq 1$ but the covariance condition would imply that $S_rU^{rn} = U^{n} S_r$ for any $r$. This would force the conclusion that $S_r U^{rn} = S_r$ but since $S_r$ is an isometry it follows that $U^{rn} = 1$ which is a contradiction.
As was noted in the proof notice that if $N$ is an $R$-submodule of $M$ then $C^*(N)$ is trivialized by the representation induced by the quotient operators.
We then have the simple corollary concerning quotient modules.
Let $M$ be an $R$-module and $\pi: C^*(M) \times_F R^{\times} \rightarrow A \times_F R^{\times}$ be a completely contractive homomorphism. Then $A \cong C^*(M/N)$ for some $R$-submodule $N$ if and only if $\pi^{-1}(\mathbb{C}) = C^*(N)$.
The forward direction is just an implication of Theorem \[exact\]. For the backward direction notice that once we know that $\pi^{-1}(\mathbb{C}) = C^*(N)$ then we only need to verify that the quotient operators induce a completely contractive representation. But that follows from the fact that $ \pi$ is a completely contractive representation of $C^*(M) \times_F R^{\times}$.
This example now motivates similar concepts for general semicrossed products. Given two semicrossed products $C(X) \times_{\mathcal{F}} S$ and $C(Y) \times_{\mathcal{F}} S$ which are generated by $ \{ C(X) \} \cup \{ S_t \}$ and $\{ C(Y) \} \cup \{ T_t \}$. We say that the latter is a quotient semicrossed product of the former if there is a covariant representation $ \pi: C(X) \times_{\mathcal{F}} S \rightarrow C(Y) \times_{\mathcal{F}} S$ which is surjective and satisfies $\pi(S_t) = T_t$ for all $t \in S$. In effect we are requiring a surjective $*$-homomorphism $ \pi: C(X) \rightarrow C(Y)$ such that $ T_t\pi( f(x)) = \pi(\alpha_t(f(x))) T_t$ for all $t \in S$.
Next we say that $C(Y) \times_{\mathcal{F}}S$ is a sub-semicrossed product of $C(Z) \times_{\mathcal{F}} S$ if there is a quotient semi-crossed product $C(X) \times_{\mathcal{F}} S$ such that the composition sends $C(Z)$ to the identity in $C(X)$ and fixes the semigroup generators.
Using Gelfand theory we can readily see that the following are required for the existence of a sub-semicrossed product.
- A continuous injection $ \pi: X \rightarrow Z$.
- A continuous surjection $\tau: Y \rightarrow X$.
- A continuous surjection $\sigma: X \rightarrow N$ where $N$ is the zero set of the ideal $J$ corresponding to the quotient $C(Y) /J \cong C(X)$.
- The continuous map $\pi \circ \tau$ must be a constant map.
- The action of $S$ commutes with $\pi$ and $ \tau$ (i.e. $\pi \circ \alpha_s = \beta_s \circ \tau$ for all $s$.
Notice that the construction for submodules and quotient module operator algebras are special cases of this construction for general semi-crossed products.
Although the above list gives necessary conditions there is no claim that these conditions are sufficient. A useful categorical approach to this question would necessitate a complete set of necessary and sufficient conditions. Such an approach would also likely require considering the different universal semicrossed products to give rise to a reasonable theory.
Finite generation
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One view of the collection of algebras considered in this paper is as a class of readily understood semicrossed products that can motivate questions in the general case of semicrossed products. We consider an example of this type of process in this section. We will focus on finitely generated modules over an integral domain $R$. We remind the reader that a module $M$ is finitely generated as an $R$-module if there exists some $m_1, m_2, \cdots, m_n \in M$ such that for every $m \in M$ there is $r_1, r_2, \cdots, r_n \in R$ such that $ m = r_1m_1 + r_2m_2 + \cdots + r_nm_n$. The set $ \{ m_1, m_2, \cdots, m_n \}$ is called a generating set for $M$. In the case that $M$ has a generating set of size $1$ we say that $M$ is cyclic.
We now consider how this translates to a property of operator algebras, specifically we will focus first on the $C^*$-algebra generated by $C^*(M) \times_F R^{\times}$ inside $B(\ell^2(M) \otimes \ell^2(R^{\times}))$ to motivate the general definition.
If $M$ is cyclic as an $R$-module then there is some $m \in M$ such that for any $n \in M$ there is $r \in R$ such that $rm = n$. Inside the $C^*$-envelope of the semicrossed product we then have that $S_r^* U_mS_r = U^n$. It follows that $ C^*(M)$ is the closure of the unitization of the subspace generated by $ \{ S_r^*U^mS_r: r \in R^{\times} \}$. We can take this as a general definition for a semicrossed product. Let $C(X) \times_{{\mathcal{F}}} S$ be a Fock algebra associated to an action of $S$ (an abelian discrete semigroup) on $C(X)$ acting as completely contractive endomorphisms. We will call the the closure of the unitization of the subspace spanned by $\{ S_r^*f(x)S_r: r \in R^{\times} \}$ the cyclic subspace generated by $ f$ and will denote it by $ \langle \langle f \rangle \rangle$. We can say that the action is cyclic if there is some $f \in C(X)$ such that $C(X) = \langle \langle f \rangle \rangle$.
Let $\sigma: X \rightarrow X$ be the identity map. Then $ S = \mathbb{Z}_+$ and $S$ acts on $C(X)$ by iterating the map $\sigma$. In this case $C(X) \times S$ is cyclic if and only if $X$ consists of one or two points. To see this notice that that for any $f(x)$ we have $ S_n^*f(x) S_n = f(x)$ for all $n$ and hence $$\langle \langle f \rangle \rangle = \begin{cases} {\rm span} \{ 1, f(x) \} \cong \mathbb{C}^2 &: \mbox{ if } f(x) \neq \lambda\cdot 1 \mbox{ for any } \lambda \in \mathbb{C} \\ \mathbb{C} & \mbox{ otherwise} \end{cases}.$$ Which happens if and only if $X$ consists of at most two points.
Let $X = \{ 0, 1, \cdots, n-1 \}$ and $ \sigma: X \rightarrow X$ be given by $\sigma(i) = \begin{cases} i+1 &: i \leq n-2 \\ 0 &: i = n-1 \end{cases}$. Let $\mathbb{Z}_+$ act on $X$ by iterating $\sigma$. Then $C(X) \times S$ is cyclic since $ f(x) = \chi_{1} \in C(X)$ and $S_r^*f(x)S_r = \chi_{r \mod n}$ so that $C(X) = {\rm span} \{ S_r^*f(x)S_r: 1 \leq r \leq n \} \subseteq \langle \langle f(x) \rangle \rangle$.
This points to a general result. We denote by $\mathcal{O}(x) = \overline{ \{ \sigma_n(x) : n \in \mathbb{Z}_+ \}}$.
Let $X$ be a totally disconnected compact set and $\sigma: X \rightarrow X$ be a continuous map. Then if there is some $z$ such that $ X \setminus \mathcal{O}(z) \subseteq \{ y \}$ for some point $y$ then $C(X) \times \mathbb{Z}_+$ is cyclic.
As $X$ is totally disconnected it is generated by projections of the form $ \chi_{\{ x \}}$ such that $ x \in X$. Let $f(x) = \chi_{\{ z \}}$ then $ S_n^*f(x)S_n = \chi_{\{ \sigma^n(z) \}}$ and notice that ${\rm span} \{ S_n^*f(x)S_n \}$ is a subalgebra of $C(X)$. The unitization of this algebra separates the points of $X$ and is unital and hence by the Stone-Weierstrass Theorem we have that $\langle \langle f(x) \rangle \rangle = C(X)$.
Considering arbitrary compact sets $X$ points out two problems that can arise for a fixed $f$:
- $\langle \langle f \rangle \rangle$ need not be an algebra.
- $\langle \langle f \rangle \rangle$ need not be self-adjoint.
If we know a-priori these two facts then the semicrossed product is cyclic if and only if the set $ \langle \langle f \rangle \rangle$ separates points (via the Stone-Weierstrass Theorem). In general this is not going to be true.
Let $X = [-1,1]$ and $\sigma(t) = t^2$, and $f(t) = t$. Then $\langle \langle f \rangle \rangle$ is not an algebra since $t^3 = f^3$ but $f(\sigma^n)$ is an even function for all $ n \geq 1$. Now calculus tells us that $t^3 \neq \lambda_1 + \lambda_2 t + g(t)$ for any even function $g(t)$ since the second derivative of $t^3 $ is an odd function but the second derivative of the latter term is an even function. It follows that $ \langle \langle f(t) \rangle \rangle$ is not an algebra.
Of course, for any cyclic module over a commutative ring $R$, the associated semicrossed product algebra is cyclic (by definition). It follows that one can easily construct examples of cyclic semicrossed products. The reason this appears to work, however, is that the semigroup is “large” which means that $C^*(M)$ will “move” a lot when we consider all actions of $R^{\times}$ on it.
The general case of $n$-generation is similar. We translate the commutative algebra definition into the context of $C^*(M) \times_{\mathcal{F}} R^{\times}$. Then we tweak the definition into the more general semicrossed product context. For brevity we will proceed right to the general definition. We denote by $ \langle \langle f_1, f_2, \cdots, f_n \rangle \rangle$ to be the closure of the span of all elements of the form $ g_1\cdot g_2\cdot \cdots \cdot g_n$ where $ g_i \in \langle \langle f_i \rangle \rangle$. Notice that since each of the sets $ \langle \langle f_i \rangle \rangle$ is unital then so is $ \langle \langle f_1, f_2, \cdots, f_n \rangle \rangle$ and we also have that $ \langle \langle f_i \rangle \rangle \subseteq \langle \langle f_1, f_2, \cdots, f_n \rangle \rangle$ for all $i$.
If $X$ is finite then $C(X) \times S$ is finitely generated by no more than $|X|$ elements of $C^*(X)$.
This requires no action by $S$ since is $X = \{ x_1, x_2, \cdots, x_n \}$ then $C(X)$ is spanned by $ f_i := \chi_{\{ x_i \}}$ and hence $C(X) = \langle \langle f_1, f_2, \cdots, f_n \rangle \rangle$.
We can often do with fewer generating elements.
Let $X$ be finite then $C(X) \times \mathbb{Z}_+$ is finitely generated by $k$ elements where $k$ is the number of components in the orbit of the action. To see this, for each component of the orbit of $\sigma$ choose a single element $x_i$. Then if we set $ g_i = \chi_{ \{ x_i \}}$ then $\langle \langle g_i, \rangle \rangle$ is the unitization of $\{ f(x): \mbox{ the support of } f(x) \subseteq \mathcal{O}(x_i ) \}$. The result now follows.
Notice that these constructions work for any representation which is faithful on $C^*(M)$ and sends the $S_g$ to an isometry satisfying the covariance conditions.
We have the following proposition.
Assume that $C(X) \times_{\mathcal{F}} \times S$ is finitely generated and there is a surjective homomorphism $\pi: C(X) \times_{\mathcal{F}} S \rightarrow C(Y) \times_{\mathcal{F}} S$. Then $C(Y) \times_{\mathcal{F}} S$ is finitely generated.
This relies on the fact that $ \langle \langle \pi(f_1), \pi(f_2), \cdots, \pi(f_n) \rangle \rangle = \pi ( \langle \langle f_1, f_2, c\dots, f_n \rangle \rangle) = C(Y)$ if $\pi$ is surjective.
It is well known [@Roman Chapter 4] that finite generation is not inherited by submodules. We do not, however, know that a module is finitely generated if the associated semicrossed product is finitely generated. If we has such a result then we would have an example where a sub-semicrossed product does not inherit the finitely generated property.
[00]{}
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---
abstract: 'Let ${\ensuremath{\mathcal{P}}}$ be a simple polygon with $n$ vertices. The *dual graph* ${\ensuremath{T}}^*$ of a triangulation ${\ensuremath{T}}$ of ${\ensuremath{\mathcal{P}}}$ is the graph whose vertices correspond to the bounded faces of ${\ensuremath{T}}$ and whose edges connect those faces of ${\ensuremath{T}}$ that share an edge. We consider triangulations of ${\ensuremath{\mathcal{P}}}$ that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in $O(n^3\log n)$ time using dynamic programming. If ${\ensuremath{\mathcal{P}}}$ is convex, we show that any minimizing triangulation has dual diameter exactly $2\cdot\lceil\log_2(n/3)\rceil$ or $2\cdot\lceil\log_2(n/3)\rceil -1$, depending on $n$. Trivially, in this case any maximizing triangulation has dual diameter $n-2$. Furthermore, we investigate the relationship between the dual diameter and the number of *ears* (triangles with exactly two edges incident to the boundary of ${\ensuremath{\mathcal{P}}}$) in a triangulation. For convex ${\ensuremath{\mathcal{P}}}$, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of $n$ points there are triangulations with dual diameter in $O(\log n)$ and in $\Omega(\sqrt n)$.'
author:
- 'Matias Korman[^1] $^,$[^2]'
- 'Stefan Langerman[^3]'
- 'Wolfgang Mulzer[^4]'
- 'Alexander Pilz[^5]'
- 'Maria Saumell[^6]'
- Birgit Vogtenhuber
bibliography:
- 'bibliography.bib'
title: 'The Dual Diameter of Triangulations$^*$'
---
Foreword {#foreword .unnumbered}
========
Research on this topic was initiated at the *Brussels Spring Workshop on Discrete and Computational Geometry*, which took place May 20–24, 2013. The authors would like to thank all the participants in general and Ferran Hurtado in particular. Ferran participated in the early stages of the discussion, but modestly decided not to be an author of this paper. To us he has been a teacher, supervisor, advisor, mentor, colleague, coauthor, and above all: a friend. We are very grateful that he was a part of our lives.
Introduction
============
Let ${\ensuremath{\mathcal{P}}}$ be a simple polygon with $n>3$ vertices. We regard ${\ensuremath{\mathcal{P}}}$ as a closed two-dimensional subset of the plane, containing its boundary. A *triangulation* ${\ensuremath{T}}$ of ${\ensuremath{\mathcal{P}}}$ is a maximal crossing-free geometric (i.e., straight-line) graph whose vertices are the vertices of ${\ensuremath{\mathcal{P}}}$ and whose edges lie inside ${\ensuremath{\mathcal{P}}}$. Hence, ${\ensuremath{T}}$ is an outerplanar graph. Similarly, for a set $S$ of $n$ points in the plane, a *triangulation* ${\ensuremath{T}}$ of $S$ is a maximal crossing-free geometric graph whose vertices are exactly the points of $S$. It is well known that in both cases all bounded faces of ${\ensuremath{T}}$ are triangles. The *dual graph* ${\ensuremath{T}}^*$ of ${\ensuremath{T}}$ is the graph with a vertex for each bounded face of ${\ensuremath{T}}$ and an edge between two vertices if and only if the corresponding triangles share an edge in ${\ensuremath{T}}$. If all vertices of ${\ensuremath{T}}$ are incident to the unbounded face, then ${\ensuremath{T}}^*$ is a tree. An *ear* in a triangulation of a simple polygon is a triangle whose vertex in the dual graph is a leaf (equivalently, two out of its three edges are edges of ${\ensuremath{\mathcal{P}}}$). We call the diameter of the dual graph ${\ensuremath{T}}^*$ the *dual diameter (of the triangulation ${\ensuremath{T}}$)*. In the following, we will study combinatorial and algorithmic properties of *minimum* and *maximum dual diameter triangulations* for simple polygons and for planar point sets ([$\mathrm{minDT}$]{}s and [$\mathrm{maxDT}$]{}s for short). Note that both triangulations need not to be unique.
#### Previous Work
Shermer [@s-cbtt-91] considers *thin* and *bushy* triangulations of simple polygons, i.e., triangulations that minimize or maximize the number of ears. He presents algorithms for computing a thin triangulation in time $O(n^3)$ and a bushy triangulation in time $O(n)$. Shermer also claims that bushy triangulations are useful for finding paths in the dual graph, as is needed, e.g., in geodesic algorithms. In that setting, however, the running time is not actually determined by the number of ears, but by the dual diameter of the triangulation. Thus, bushy triangulations are only useful for geodesic problems if there is a connection between maximizing the number of ears and minimizing the dual diameter. While this holds for convex polygons, we show that, in general, there exist polygons for which no ${\ensuremath{\mathrm{minDT}}}$ maximizes the number of ears. Moreover, we give examples where forcing a single ear into a triangulation may almost double the dual diameter, and the dual diameter of any bushy triangulation may be quadratic in the dual diameter of a ${\ensuremath{\mathrm{minDT}}}$.
The dual diameter also plays a role in the study of edge flips: given a triangulation $T$, an *edge flip* is the operation of replacing a single edge of $T$ with another one so that the resulting graph is again a valid triangulation. In the case of convex polygons, edge flips correspond to rotations in the dual binary tree [@sleator_tarjan_thurston]. For this case, Hurtado, Noy, and Urrutia [@hurtado_noy_urrutia; @urrutia_flip_talk] show that a triangulation with dual diameter $k$ can be transformed into a fan triangulation by a sequence of most $k$ parallel flips (i.e., two edges not incident to a common triangle may be flipped simultaneously). They also obtain a triangulation with logarithmic dual diameter by recursively cutting off a linear number of ears.
While we focus on the dual graph of a triangulation, distance problems in the primal graph have also been considered. For example, Kozma [@kozma] addresses the problem of finding a triangulation that minimizes the total link distance over all vertex pairs. For simple polygons, he gives a sophisticated $O(n^{11})$ time dynamic programming algorithm. Moreover, he shows that the problem is strongly NP-complete for general point sets when arbitrary edge weights are allowed and the length of a path is measured as the sum of the weights of its edges.
#### Our Results
In Section \[sec\_ears\], we present several properties of the dual diameter for triangulations of simple polygons. Among other results, we calculate the exact dual diameter of ${\ensuremath{\mathrm{minDT}}}$s and ${\ensuremath{\mathrm{maxDT}}}$s of convex polygons, which can be obtained by maximizing and minimizing the number of ears of the triangulation, respectively. On the other hand, we show that there exist simple polygons where the dual diameter of any ${\ensuremath{\mathrm{minDT}}}$ is $O(\sqrt{n})$, while that of any triangulation that maximizes the number of ears is in $\Omega(n)$. Likewise, there exist simple polygons where the dual diameter of any triangulation that minimizes the number of ears is in $O(\sqrt{n})$, while the maximum dual diameter is linear. In Section \[sec\_poly\], we present efficient algorithms to construct a ${\ensuremath{\mathrm{minDT}}}$ and a ${\ensuremath{\mathrm{maxDT}}}$ for any given simple polygon.
Finally, in Section \[sec\_points\] we consider the case of planar point sets, showing that for any point set in the plane in general position there are triangulations with dual diameter in $O(\log n)$ and in $\Omega(\sqrt{n})$, respectively.
The Number of Ears and the Diameter {#sec_ears}
===================================
The dual graph of any triangulation ${\ensuremath{T}}$ has maximum degree $3$. In this case, the so-called *Moore bound* implies that the dual diameter of ${\ensuremath{T}}$ is at least $\log_2(\frac{t+2}{3})$, where $t$ is the number of triangles in ${\ensuremath{T}}$ (see, e.g., [@moore_survey]). For convex polygons, we can compute the minimum dual diameter exactly.
\[prop\_convex\] Let ${\ensuremath{\mathcal{P}}}$ be a convex polygon with $n \geq 3$ vertices, and let $m \geq 1$ such that $n \in \{3\cdot 2^{m-1} + 1, \dots, 3 \cdot 2^m\}$. Then any [$\mathrm{minDT}$]{} of ${\ensuremath{\mathcal{P}}}$ has dual diameter $2\cdot\lceil\log_2(n/3)\rceil -1$ if $n \in \{3\cdot 2^{m-1} + 1, \dots, 4 \cdot 2^{m-1}\}$, and $2\cdot\lceil\log_2(n/3)\rceil$ if $n \in \{4 \cdot 2^{m-1} + 1, \dots, 3\cdot 2^m\}$, for some $m \geq 1$.
The dual graph of any triangulation of ${\ensuremath{\mathcal{P}}}$ is a tree with $n-2$ vertices and maximum degree $3$; see \[fig\_convex\_tree\] for an example. Furthermore, every tree with $n-2$ vertices and maximum degree $3$ is dual to some triangulation of ${\ensuremath{\mathcal{P}}}$.
For the upper bound, suppose first that $n = 3\cdot 2^m$, for some $m \geq 1$. We define a triangulation ${\ensuremath{T}}_1$ as follows. It has a central triangle that splits ${\ensuremath{\mathcal{P}}}$ into three sub-polygons, each with $2^m$ edges on the boundary. For each sub-polygon, the dual tree for ${\ensuremath{T}}_1$ is a full binary tree of height $m-1$ with $2^{m-1}$ leaves; see \[fig\_convex\_full\]. The leaves of ${\ensuremath{T}}_1^*$ correspond to the ears of ${\ensuremath{T}}_1$. The shortest path between any two ears in two different sub-polygons has length exactly $2(m-1) + 2 = 2 \log_2(n/3)$. The shortest path between any two ears in the same sub-polygon has length at most $2(m-1)$. Thus, the dual diameter of ${\ensuremath{T}}_1$ is $2 \log_2(n/3)$.
Now let $n \in \{3 \cdot \frac{4}{3} \cdot 2^{m-1} + 1, \dots,
3\cdot 2^m-1\},$ and consider the triangulation ${\ensuremath{T}}_2$ of ${\ensuremath{\mathcal{P}}}$ obtained by cutting off $3\cdot 2^m -n \leq 2 \cdot 2^{m-1} - 1$ ears that are consecutive in the convex hull from ${\ensuremath{T}}_1$. Then ${\ensuremath{T}}_2^*$ is a subtree of ${\ensuremath{T}}_1^*$. Since ${\ensuremath{T}}_1$ has $3\cdot 2^{m-1}$ ears, ${\ensuremath{T}}_2$ has at least $2^{m-1}+1$ ears in common with ${\ensuremath{T}}_1$. Two of them lie in different sub-polygons, so the dual diameter remains $2m = 2\cdot\lceil\log_2(n/3)\rceil$.
Finally, for $n \in \{3\cdot 2^{m-1} + 1, \dots, 3 \cdot \frac{4}{3} \cdot 2^{m-1}\}$, if we remove $3\cdot 2^m - n \leq 3\cdot2^{m-1}-1$ ears from ${\ensuremath{T}}_1$ such that all ears in two of the sub-polygons are removed, we get a triangulation with dual diameter $2m-1=2\cdot\lceil\log_2(n/3)\rceil-1$; see \[fig\_convex\_sparse\].
For the lower bound, assume there is a tree ${\ensuremath{T}}^*$ with $n-2$ vertices, maximum degree $3$, and diameter $k$ strictly smaller than in the proposition. Consider a longest path $\pi$ in ${\ensuremath{T}}^*$ and a vertex $v$ on $\pi$ for which the distances to the endpoints of $\pi$ differ by at most one. By adding vertices, we can turn ${\ensuremath{T}}^*$ into a tree with $n' -2 > n-2$ vertices, diameter $k$, and the same structure as ${\ensuremath{T}}_1^*$ or ${\ensuremath{T}}_2^*$ for a convex polygon with $n'$ vertices (with $v$ as central vertex). Since the upper bound on the dual diameter grows monotonically, this means that the triangulation $T_1$ or $T_2$ for a convex polygon with $n$ vertices has diameter $k$, a contradiction.
As the dual graph of a triangulation of *any* simple polygon has maximum degree $3$, the proof of Proposition \[prop\_convex\] yields the following corollary.
\[cor\_lb\] Let ${\ensuremath{\mathcal{P}}}$ be a simple polygon with $n \geq 3$ vertices, and let $m \geq 1$ such that $n \in \{3\cdot 2^{m-1} + 1, \dots, 3 \cdot 2^m\}$. The dual diameter of any triangulation of ${\ensuremath{\mathcal{P}}}$ is at least $2\cdot\lceil\log_2(n/3)\rceil -1$ if $n \in \{3\cdot 2^{m-1} + 1, \dots, 4 \cdot 2^{m-1}\}$, and $2\cdot\lceil\log_2(n/3)\rceil$ if $n \in \{4 \cdot 2^{m-1} + 1, \dots, 3\cdot 2^m\}$.
Proposition \[prop\_convex\] also shows that if ${\ensuremath{\mathcal{P}}}$ is convex, there exists a ${\ensuremath{\mathrm{minDT}}}$ with a maximum number of ears. Next, we show that this does not hold for general simple polygons. Hence, any approach that tries to construct ${\ensuremath{\mathrm{minDT}}}$s by maximizing the number of ears is doomed to fail.
\[prop\_leaves\] For arbitrarily large $n$, there exist simple polygons with $n$ vertices in which any ${\ensuremath{\mathrm{minDT}}}$ minimizes the number of ears.
Let $k\geq 1$ and consider the polygon ${\ensuremath{\mathcal{P}}}$ with $n = 4k + 8$ vertices sketched in \[fig\_ears1\]. Any triangulation of ${\ensuremath{\mathcal{P}}}$ has either $4$ or $5$ ears. The triangulation in \[fig\_ears1\_5\] is the only triangulation with 5 ears, and it has dual diameter $4k+2$. However, as depicted in \[fig\_ears1\_4good\], omitting the large ear at the bottom allows a triangulation with $4$ ears and dual diameter $2k+3$. Thus, forcing even one additional ear may nearly double the dual diameter.
Figure \[fig\_ears1\_4bad\] shows a triangulation of ${\ensuremath{\mathcal{P}}}$ with $4$ ears and almost twice the diameter as in \[fig\_ears1\_4good\]. Thus, neither for minimizing the diameter nor for maximizing the number of ears this triangulation is desirable. However, it has the nice property that the two top ears are connected by a dual path with four interior vertices. Below, this property will be useful when making a larger construction.
\[thm\_leaves\] For arbitrarily large $n$, there is a simple polygon with $n$ vertices that has minimum dual diameter $O(\sqrt{n})$ while any triangulation that maximizes the number of ears has dual diameter $\Omega(n)$.
Let $c$ be a parameter to be determined later, and let ${\ensuremath{\mathcal{P}}}'$ be the polygon constructed in Proposition \[prop\_leaves\]. We construct a polygon ${\ensuremath{\mathcal{P}}}$ by concatenating $c$ copies of ${\ensuremath{\mathcal{P}}}'$ as in \[fig\_ears2\]. ${\ensuremath{\mathcal{P}}}$ has $n=c(4k+4)+4$ vertices. Using the triangulation from \[fig\_ears1\_5\] for each copy, we obtain a triangulation with the maximum number $3c+2$ of ears and dual diameter $c(4k+1)+1$ (the curved line in \[fig\_ears2\] indicates a longest path). On the other hand, using the triangulation from \[fig\_ears1\_4good\] for the leftmost and rightmost part of the polygon and the one from \[fig\_ears1\_4bad\] for all intermediate parts yields a triangulation with dual diameter $4c+4k-3$ that has only $2c+2$ ears. For $c=k$, we obtain $c,k=\Theta(\sqrt{n})$. Thus, the dual diameter for the triangulation with maximum number of ears is $\Theta(n)$, while the optimal dual diameter is $O(\sqrt{n})$.
Similarly, for maximizing the dual diameter, we can give examples where the dual diameter is suboptimal when the number of ears is minimized.
For arbitrarily large $n$, there is a simple polygon with $n$ vertices that has maximum dual diameter $\Omega(n)$ while any triangulation that minimizes the number of ears has dual diameter $O(\sqrt{n})$.
\[fig:fig\_ears\_max\] shows a triangulation of a part of a simple polygon. We suppose that the indicated dual path $\pi$ is the only one of maximum length. In addition to the ears at the endpoints of $\pi$, there are two ears at the vertices $v_p$ and $v_q$. If we want to have at most one ear in this part of the polygon, at least one of $v_p$ and $v_q$ must be connected to a non-neighboring vertex by a triangulation edge. For this, the only possibilities are $v_p v_a$ and $v_q v_a$. But then there cannot be any edge between the bottommost vertex $v_b$ and the $k$ vertices between $v_q$ and $v_a$. In particular, that part must be triangulated as shown to the right of \[fig:fig\_ears\_max\]. Here, there is only one ear, but the dual diameter is reduced by $k$ (assuming the remainder of the polygon is large enough). As in the proof of Theorem \[thm\_leaves\], we concatenate $\Theta(\sqrt{n})$ copies of this construction and choose $k = \Theta(\sqrt{n})$. The parts are independent in the sense that they are separated by *unavoidable* edges (i.e., edges that are present in any triangulation of the resulting polygon).[^7] Hence, while the dual diameter of a [$\mathrm{maxDT}$]{} is linear in $n$, it is in $O(\sqrt{n})$ for any triangulation that minimizes the number of ears.
![Two triangulations of a part of a polygon where the dual diameter is locally decreased by $k$ when minimizing the number of ears.[]{data-label="fig:fig_ears_max"}](fig_ears_max)
It is easy to construct polygons for which the dual graph of any triangulation is a path, forcing minimum dual diameter $\Omega(n)$. The other direction is slightly less obvious.
For any $n$, there exists a simple polygon ${\ensuremath{\mathcal{P}}}$ with $n$ vertices such that the dual diameter of any [$\mathrm{maxDT}$]{} of ${\ensuremath{\mathcal{P}}}$ is in $\Theta(\log n)$.
We incrementally construct ${\ensuremath{\mathcal{P}}}$ by starting with an arbitrary triangle $t$. See \[fig:fig\_max\_log\] for an accompanying illustration. We replace every corner of $t$ by four new vertices so that two of them can see only these four new vertices. This means that the edge between the other two newly added vertices is unavoidable. We repeat this construction recursively in a balanced way. If necessary, we add dummy vertices to obtain exactly $n$ vertices. The unavoidable edges partition ${\ensuremath{\mathcal{P}}}$ into convex regions, either hexagons or quadrilaterals. The dual tree of this partition is balanced with diameter $\Theta(\log n)$. Since every triangulation of ${\ensuremath{\mathcal{P}}}$ contains all unavoidable edges, the maximum possible dual diameter is $O(\log n)$.
![The convex vertices of a polygon are incrementally replaced by four new vertices, resulting in unavoidable edges (dotted).[]{data-label="fig:fig_max_log"}](fig_max_log)
Optimally Triangulating a Simple Polygon {#sec_poly}
========================================
We now consider the algorithmic question of constructing a ${\ensuremath{\mathrm{minDT}}}$ and a ${\ensuremath{\mathrm{maxDT}}}$ of a simple polygon ${\ensuremath{\mathcal{P}}}$ with $n$ vertices. Let $v_1,\ldots, v_n$ be the vertices of ${\ensuremath{\mathcal{P}}}$ in counterclockwise order. The segment $v_iv_j$ is a *diagonal* of ${\ensuremath{\mathcal{P}}}$ if it lies completely in ${\ensuremath{\mathcal{P}}}$ but is not part of the boundary of ${\ensuremath{\mathcal{P}}}$. For a diagonal $v_iv_j$, $i < j$, we define ${\ensuremath{\mathcal{P}}}_{i,j}$ as the polygon with vertices $v_i, v_{i+1},\dots, v_{j-1}, v_j$; see \[fig\_pij\]. Observe that ${\ensuremath{\mathcal{P}}}_{i,j}$ is a simple polygon contained in ${\ensuremath{\mathcal{P}}}$. If $v_iv_j$ is not a diagonal, we set ${\ensuremath{\mathcal{P}}}_{i,j}=\emptyset$.
![Any triangulation of ${\ensuremath{\mathcal{P}}}_{i,j}$ (gray) has exactly one triangle adjacent to $v_iv_j$ (dark gray).[]{data-label="fig_pij"}](fig_pij){width=".5\columnwidth"}
\[thm\_min\_alg\] For any simple polygon ${\ensuremath{\mathcal{P}}}$ with $n$ vertices, we can compute a ${\ensuremath{\mathrm{minDT}}}$ in $O(n^3\log n)$ time.
We use the classic dynamic programming approach [@klincsek], with an additional twist to account for the non-local nature of the objective function. Let $v_iv_j$ be a diagonal. Any triangulation ${\ensuremath{T}}$ of ${\ensuremath{\mathcal{P}}}_{i,j}$ has exactly one triangle $t$ incident to $v_iv_j$; see \[fig\_pij\]. Let $f({\ensuremath{T}})$ be the maximum length of a path in ${\ensuremath{T}}^*$ that has $t$ as an endpoint.
For $d>0$ and $i,j = 1, \dots, n$, with $i < j$, let $\mathcal{T}_d(i,j)$ be the set of all triangulations of ${\ensuremath{\mathcal{P}}}_{i,j}$ with dual diameter at most $d$ (we set $\mathcal{T}_d(i,j)=\emptyset$ if $v_iv_j$ is not a diagonal of ${\ensuremath{\mathcal{P}}}$). We define $M_d[i,j]=\min_{{\ensuremath{T}}\in\mathcal{T}_d(i,j)}f({\ensuremath{T}})+1$, if $\mathcal{T}_d(i,j)\neq \emptyset$, or $M_d[i,j] = \infty$, otherwise. Intuitively, we aim for a triangulation that minimizes the distance from $v_iv_j$ to all other triangles of ${\ensuremath{\mathcal{P}}}_{i,j}$ while keeping the dual diameter below $d$ (the value of $M_d[i,j]$ is the smallest possible distance that can be obtained). Let $\mathcal{V}(i,j)$ be all vertices $v_l$ of ${\ensuremath{\mathcal{P}}}_{i,j}$ such that the triangle $v_iv_jv_l$ lies inside ${\ensuremath{\mathcal{P}}}_{i,j}$. We claim that $M_d[i,j]$ obeys the following recursion:
$$M_d[i, j] = \left\{
\begin{array}{l}
0, \hfill \mbox{if $i+1 = j$,}\\
\infty, \hfill \mbox{if $v_iv_j$ is not a diagonal,}\\
\min_{v_l\in\mathcal{V}(i,j)} I_d[i,j,l], \hfill \quad \mbox{otherwise,}
\end{array}
\right.$$ where $$I_d[i,j,l] = \left\{
\begin{array}{l}
\infty, \hfill \mbox{if $M_d[i,l] + M_d[l,j] > d$,} \\
\max\{M_d[i,l], M_d[l,j]\}+1, \quad \mbox{otherwise.}
\end{array} \right.$$
We minimize over all possible triangles $t$ in ${\ensuremath{\mathcal{P}}}_{i,j}$ incident to $v_iv_j$. For each $t$, the longest path to $v_iv_j$ is the longer of the paths to the other edges of $t$ plus $t$ itself. If $t$ joins two longest paths of total length more than $d$, there is no valid solution with $t$. Thus, we can decide in $O(n^3)$ time whether there is a triangulation with dual diameter at most $d$, i.e., if $M_d[1,n] \neq \infty$. Since the dual diameter is at most $n-3$, a binary search gives an $O(n^3 \log n)$ time algorithm.
We can use a very similar approach to obtain some [$\mathrm{maxDT}$]{}.
For any simple polygon ${\ensuremath{\mathcal{P}}}$ with $n$ vertices, we can compute a ${\ensuremath{\mathrm{maxDT}}}$ in $O(n^3\log n)$ time.
The proof is similar to the one of Theorem \[thm\_min\_alg\]. This time, we are looking for triangulations that have dual diameter at least $d$. Let $f(T)$ be defined as before, and let $\mathcal{T}(i,j)$ be the set of all triangulations of ${\ensuremath{\mathcal{P}}}_{i,j}$ (this time, we do not need the third parameter). We define $M_d[i,j]$ in the following way. If $\mathcal{T}(i,j) = \emptyset$, then $M_d[i,j] = -\infty$. If $\mathcal{T}(i,j)$ contains a triangulation with diameter at least $d$, $M_d[i,j] = \infty$. Otherwise, let $M_d[i,j] = \max_{T \in \mathcal{T}(i,j)} f(T) + 1$. Clearly, there is a triangulation with diameter at least $d$ if and only if $M_d[1,n] = \infty$. With $\mathcal{V}(i,j)$ defined as before, the recursion for $M_d[i,j]$ is
$$M_d[i, j] = \left\{
\begin{array}{l}
0, \hfill \mbox{if $i+1 = j$,}\\
-\infty, \hfill \mbox{if $v_iv_j$ is not a diagonal,}\\
\max_{v_l\in\mathcal{V}(i,j)} I_d[i,j,l], \hfill \quad \mbox{otherwise,}
\end{array}
\right.$$ where $$I_d[i,j,l] = \left\{
\begin{array}{l}
-\infty, \hfill \mbox{if $M_d[i,l]$ or $M_d[l,j]$ is $-\infty$,} \\
\infty, \quad \hfill \mbox{if $M_d[i,l] + M_d[l,j] \geq d$,} \\
\max\{M_d[i,l], M_d[l,j]\}+1, \quad \hfill \mbox{otherwise.}
\end{array} \right.$$ For the given diagonal $v_i v_j$, we maximize over all possible triangles. If at some point the triangle $t$ at $v_i v_j$ closes a path of length at least $d$, we are basically done, as any triangulation of the remainder of the polygon results in a triangulation with dual diameter at least $d$. If the triangulation of ${\ensuremath{\mathcal{P}}}_{i,j}$ does not contain such a long path, we store the longer one to $v_i v_j$, as before. Again, we can find the optimal dual diameter via a binary search, giving an $O(n^3 \log n)$ time algorithm.
Bounds for Point Sets {#sec_points}
=====================
We are now given a set $S$ of $n$ points in the plane in general position, and we need to find a triangulation of $S$ whose dual graph optimizes the diameter. Since the dual graph has maximum degree $3$, it is easy to see that the $\Omega(\log n)$ lower bound for simple polygons extends for this case. It turns out that this bound can always be achieved, as we show in Section \[sec\_min\_point\_set\]. In Section \[sec\_max\_point\_set\], we find a triangulation of $S$ that has dual diameter in $\Omega(\sqrt{n})$.
Minimizing the Dual Diameter {#sec_min_point_set}
----------------------------
\[theo\_pointset\] Given a set $S$ of $n$ points in the plane in general position, we can compute in $O(n \log n)$ time a triangulation of $S$ with dual diameter $\Theta(\log n)$.
Let ${\ensuremath{\mathcal{P}}}$ be a convex polygon with $n$ vertices and ${\ensuremath{T}}'$ a triangulation of ${\ensuremath{\mathcal{P}}}$ with dual diameter $\Theta(\log n)$ (e.g., the triangulation from Proposition \[prop\_convex\]). The triangulation ${\ensuremath{T}}'$ is an outerplanar graph. Any outerplanar graph of $n$ vertices has a plane straight-line embedding on any given $n$-point set [@pgmp-eptvsp-91]. Furthermore, such an embedding can be found in $O(nd)$ time and $O(n)$ space, where $d$ is the dual diameter of the graph [@b-oeopgps-02].
Let ${\ensuremath{T}}_S$ be the embedding of ${\ensuremath{T}}'$ on $S$. In general, ${\ensuremath{T}}_S$ does not triangulate $S$; see \[fig\_pointset\].
![When computing a ${\ensuremath{\mathrm{minDT}}}$ of a point set $S$, we first view it as if it were in convex position and construct a ${\ensuremath{\mathrm{minDT}}}$ (left image). Then, we embed $T_S$ into the actual point set (solid edges in the right image). (The correspondence is marked by the central triangle and the thick boundary edge.) All remaining untriangulated pockets (highlighted region in the figure) are triangulated arbitrarily (dashed edges). []{data-label="fig_pointset"}](fig_pointset2){width="0.7\columnwidth"}
Consider the convex hull of ${\ensuremath{T}}_S$ (which equals the convex hull of $S$). The untriangulated *pockets* are simple polygons. We triangulate each pocket arbitrarily to obtain a triangulation ${\ensuremath{T}}$ of $S$. We claim that the dual diameter of ${\ensuremath{T}}$ is $O(\log n)$.
\[lem\_logpoly\] The dual distance from any triangle in a pocket to any triangle in ${\ensuremath{T}}_S$ is $O(\log n)$.
Let $Q$ be a pocket, and ${\ensuremath{T}}_Q$ a triangulation of $Q$. Since $Q$ is a simple polygon, the dual ${\ensuremath{T}}_Q^*$ is a tree with maximum degree $3$. A triangle $t$ of ${\ensuremath{T}}_Q$ not incident to the boundary of ${\ensuremath{T}}_S$ either has degree $3$ in ${\ensuremath{T}}_Q^*$, or it is the unique triangle in ${\ensuremath{T}}_Q$ that shares an edge with the convex hull of $S$. We perform a breadth-first-search in ${\ensuremath{T}}_Q^*$ starting from $t$, and let $k$ be the maximum number of consecutive layers from the root of the BFS-tree that do not contain a triangle incident to the boundary. By the above observation, all vertices in the first $k-1$ levels have degree three in ${\ensuremath{T}}_Q^*$. Thus, each vertex of level $k-1$ or lower has two children. In particular, at each level the number of vertices must double (except at the topmost level where the number of vertices is tripled), hence $k = O(\log n)$.
Given Lemma \[lem\_logpoly\] and the fact that ${\ensuremath{T}}_S$ has dual diameter $O(\log n)$, Theorem \[theo\_pointset\] is now immediate.
Obtaining a Large Dual Diameter {#sec_max_point_set}
-------------------------------
We now focus our attention on the problem of triangulating $S$ so that the dual diameter is maximized.
\[theo\_pointslargeDiam\] Given a set $S$ of $n \geq 3$ points in the plane in general position, we can compute in $O(n \log n)$ time a triangulation of $S$ with dual diameter at least $\sqrt {n-3}$.
Naturally, the triangulation ${\ensuremath{T}}$ must contain the edges of the convex hull of $S$. Let $v_1,v_2,\ldots,v_h$ be the vertices of the convex hull of $S$ in clockwise order. We connect $v_1$ to the vertices $v_3,v_4,\ldots,v_{h-1}$; see \[fig:fig\_points\_max\] (left). In order to complete this set of edges to a triangulation, it suffices to consider the triangular regions $v_1v_iv_{i+1}$ (for $2\leq i \leq h-1$) with at least one point of $S$ in their interior.
![Left: $v_1$ is connected to all remaining vertices in the convex hull. Right: additional edges added inside $\Delta_i$. We connect the points of an increasing subsequence of $v_{j_1},v_{j_2},\ldots,v_{j_{n_i}}$ to both $v_1$, $v_i$ as well as the predecessor and successor in the subsequence.[]{data-label="fig:fig_points_max"}](fig_points_max)
Let $\Delta_i= v_1v_iv_{i+1}$ be such a triangular region, $S_i \subset S$ the points in the interior of $\Delta_i$, and $n_i=|S_i|$. Let $w_1, w_2, \dots, w_{n_i}$ denote the points in $S_i$ sorted in clockwise order with respect to $v_1$, and $w_{j_1}, w_{j_2}, \dots,
w_{j_{n_i}}$ denote the same points sorted in counterclockwise order with respect to $v_i$. By the Erdős-Szekeres theorem [@es_subsequence], the index sequence $j_k$ contains an increasing or decreasing subsequence $\sigma_i$ of length at least $\sqrt{n_i}$.
If $\sigma_i$ is increasing, we connect all points of $\sigma_i$ to both $v_1$ and $v_i$. In addition, we connect each point of $\sigma_i$ to its predecessor and successor in $\sigma_i$; see \[fig:fig\_points\_max\] (right). Since the clockwise order with respect to $v_1$ coincides with the counterclockwise order with respect to $v_i$, the new edges do not create any crossing.
If $\sigma_i$ is decreasing, we claim that the corresponding point sequence is in counterclockwise order around $v_{i+1}$. Indeed, let $w$ and $w'$ be two vertices of $S_i$ whose indices appear consecutively in $\sigma_i$ (with $w$ before $w'$). By definition, the segment $v_1w'$ crosses the segment $v_iw$. Moreover, points $v_1$ and $v_i$ are on the same side of the line through $w$ and $w'$; see \[fig:fig\_points\_subsequence\] (left). Since $w$ and $w'$ are contained in $\Delta_i$, we conclude that $v_{i+1}$ must lie on the opposite side of the line. Thus, $v_i,w,w',v_1$ form a counterclockwise sequence around $v_{i+1}$, and we can connect each point of $\sigma_i$ to $v_1$, $v_{i+1}$, and its predecessor and successor in $\sigma_i$ without crossings. Finally, we add arbitrary edges to complete the resulting graph inside $\Delta_i$ into a triangulation ${\ensuremath{T}}_i$.
![Left: if $\sigma_i$ generates a decreasing sequence, the same sequence must be increasing when we view the angles with $v_{i+1}$ instead. Right: any path between $p$ and $q$ in the dual graph must visit all triangles $\Delta_i$ and at least $\sqrt{n_i}$ additional triangles between the crossing of segments $v_1v_i$ and $v_1v_{i+1}$.[]{data-label="fig:fig_points_subsequence"}](fig_points_subsequence)
We claim that, regardless of how we complete the triangulation, there are two triangles whose distance in the dual graph is at least $\sqrt{n-3}$. Let $p$ and $q$ be the triangles of ${\ensuremath{T}}$ incident to edges $v_1v_2$ and $v_1v_{h}$, respectively (since both segments are on the convex hull, $p$ and $q$ are uniquely defined). Let $\pi$ be the shortest path from $p$ to $q$. Clearly, $\pi$ must cross each segment $v_1v_i$, for $i\in \{3, \ldots h-1\}$, exactly once and in increasing order. This gives $h-3$ steps (one step for each triangle incident in clockwise order around $v_1$ on an edge $v_1v_i$, $i \in \{3, \dots, h-1\}$). In addition, at least $\sqrt{n_i}$ additional triangles must be traversed between the segments $v_1v_i$ and $v_1v_{i+1}$ (for all $i \in \{2, \ldots, h-1\}$): indeed, for each vertex $w \in \sigma_i$, the edges $v_1w$ and either $wv_i$ or $wv_{i+1}$ (depending on whether $\sigma_i$ was increasing or decreasing) disconnect $p$ and $q$., Hence at least one of the two must be crossed by $\pi$, and the triangles following these edges are pairwise distinct and distinct from the triangles following the segments $v_1v_i$. Summing over $i$, we get $$|\pi| \geq
h - 3 + \sum_{i=2}^{h-1}\sqrt{n_i} \geq
h+\sqrt{n-h} - 3 \geq
3+\sqrt{n-3}-3 =
\sqrt{n - 3}.
$$ In the second inequality we used the fact that $\sum_{i=2}^{h-1} \sqrt{n_i} \geq
\sqrt{\sum_{i=2}^{h-1} n_i} = \sqrt{n - h}$. (since a point is either on the convex hull or in its interior), the third inequality follows from $h\geq 3$ (and the fact that the expression is minimized when $h$ is as small as possible).
Finding a longest increasing (or decreasing) subsequence of $n_i$ numbers takes $O(n_i \log n_i)$ time, which is optimal in the comparison model [@fredman]. Hence, all subsequences, as well as the whole triangulation can be computed in $O(n \log n)$ time, where the last part also uses the fact that point location in a triangulation on $\leq n$ vertices can be done in $O(\log n)$ time after $O(n \log n)$ preprocessing.
\[prp:convex\_subset\] Any set of $n$ points in the plane in general position with $k$ points in convex position has a triangulation with dual diameter in $\Omega(k)$.
See \[fig:fig\_convex\_k\_set\] for an accompanying illustration. Let $S$ be such a point set with $C \subseteq S$ being the convex subset of size $k$. First, triangulate only $C$ by a zig-zag chain of edges: for the convex hull of $C$ being defined by the sequence $(c_1, \dots, c_k)$, add the edges $c_i c_{k-i}$ and $c_i c_{k-i-1}$ for $1 \leq i < \lfloor k/2 \rfloor$, as well as the boundary of the convex hull of $C$. Then, add the extreme points of $S$ and triangulate the convex hull of $S$ without $C$ such that each added edge is incident to a point of $C$ (this is not necessary to obtain the result, but it makes our arguments simpler). The resulting triangulation has dual diameter $\Omega(k)$, as is witnessed by the triangles at $c_k$ and $c_{\left \lfloor k/2 \right \rfloor}$ inside $C$: if we label each triangle with the index of the incident point in $C$ that is closest to $c_{\left \lfloor{k/2} \right \rfloor}$, then this index can change by at most 1 along a step in any dual path between the triangles inside $C$ incident to $c_k$ and $c_{\left \lfloor k/2 \right \rfloor}$. The dual diameter does not decrease when adding the remaining points of $S$ and completing the triangulation arbitrarily.
![Left: zig-zag triangulation of a convex subset of size $k$. Right: adding the edges to the extreme points implies a labeling of the triangles that relate to their distance to $c_{\left \lfloor k/2 \right \rfloor}$.[]{data-label="fig:fig_convex_k_set"}](fig_convex_k_set)
Conclusions {#sec_conclusion}
===========
The proof of Corollary \[cor\_lb\] (lower bound for simple polygons) is essentially based on fundamental properties of graphs (i.e., bounded degree) rather than geometric properties. Since the bound is tight even for the convex case, it cannot be tightened in general. However, we wonder if, by using geometric tools, one can construct a bound that depends on the number of reflex vertices of the polygon (or interior points for the case of sets of points). Another natural open problem is to extend our dynamic programming approach for simple polygons to general polygonal domains (or even sets of points).
It is open whether Theorem \[theo\_pointslargeDiam\] is tight. That is: does there exist a point set $S$ such that the diameter of the dual graph of any triangulation of $S$ is in $O(\sqrt{n})$? From Proposition \[prp:convex\_subset\], we see that any such point set can contain at most $O(\sqrt{n})$ points in convex position. Thus, the point set must have $\Theta(\sqrt n)$ convex hull layers, each with $\Theta(\sqrt{n})$ points. We suspect that some smart perturbation of the grid may be an example, but we have been unable to prove so.
Acknowledgements {#acknowledgements .unnumbered}
================
Research for this work was supported by the ESF EUROCORES programme EuroGIGA–ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306. M. K. was supported in part by the ELC project (MEXT KAKENHI No. 24106008). S.L. is Directeur de Recherches du FRS-FNRS. W.M. is supported in part by DFG grants MU 3501/1 and MU 3501/2. Part of this work has been done while A.P. was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. M.S. was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports, and by the project NEXLIZ CZ.1.07/2.3.00/30.0038, which was co-financed by the European Social Fund and the state budget of the Czech Republic.
[^1]: National Institute of Informatics (NII), Tokyo, Japan. [korman@nii.ac.jp]{}
[^2]: JST, ERATO, Kawarabayashi Large Graph Project.
[^3]: Université Libre de Bruxelles (ULB), Brussels, Belgium. [stefan.langerman@ulb.ac.be]{}. Directeur de Recherches du FRS-FNRS.
[^4]: Institut für Informatik, Freie Universität Berlin, Germany. [mulzer@inf.fu-berlin.de]{}
[^5]: Institute for Software Technology, Graz University of Technology, Austria, [@ist.tugraz.at]{}
[^6]: Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Czech Republic, [saumell@kma.zcu.cz]{}
[^7]: Unavoidable edges are defined by segments between two vertices s.t. no other edge crosses them. Hence, they have to be present in every triangulation. Unavoidable edges of point sets have been investigated by Karoly and Welzl [@karolyi_welzl] (as “crossing-free segments”), and Xu [@xu] (as “stable segments”).
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---
abstract: 'In this paper the formation mechanisms of the femtosecond laser-induced periodic surface structures (LIPSS) are discussed. One of the most frequently-used theories explains the structures by interference between the incident laser beam and surface plasmon-polariton waves. The latter is most commonly attributed to the coupling of the incident laser light to the surface roughness. We demonstrate that this excitation mechanism of surface plasmons contradicts to the results of laser-ablation experiments. As an alternative approach to the excitation of LIPSS we analyse development of hydrodynamic instabilities in the melt layer.'
address:
- 'Chair of Applied Laser Technology, Ruhr-Universität Bochum, Universitätsstraße 150, 44801 Bochum, Germany'
- 'Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany'
author:
- 'E. L. Gurevich'
- 'S. V. Gurevich'
title: 'Laser Induced Periodic Surface Structures Induced by Surface Plasmons Coupled via Roughness.'
---
\#1\#2
LIPSS ,plasmon ,ripples ,self-organization
Introduction
============
Laser-induced periodic surface structures (LIPSS) appear on dielectric, semiconductor, polymer and metal surfaces exposed to single or multiple short and ultrashort laser pulses, see, e.g., [@Birnbaum; @AcuosticWave; @MyPRE; @Straub; @Rebollar; @Bonse; @SPRLIPSS]. This pattern can be considered as one of the examples of self-organization phenomena on nano- and micrometer scale. The nanoscale pattern formation is observed in different physical, chemical and biological systems [@FacskoScience1999; @Grzybowski2005; @LiSmall2012; @Purrucker2005]. One of the most common structures is the periodic self-organised stripe pattern, which can be found, e.g., on the fracture surface of brittle glasses [@WangPRL2007] or silicon wafers [@MenPRL2002]. The periodic stripes are also observed by welding of metallic alloys [@Takalo1979]. During the welding the surface layers of the processed metals are melted. After the solidification, periodic stripes were observed on the surface. Periodic ripples are also found by ablation of solids induced by water jet cutting. In this case the periodic surface structures can be explained in frames of the Kuramoto-Sivashinsky model [@FriedrichPRL2000]. In this paper we analyse ablation of solid surfaces by ultrashort laser pulses.
After a metal surface is exposed to an ultrashort laser pulse, the following chain of processes takes place [@Anisimov]. The laser light is absorbed by electrons, which temperature increases during the laser pulse irradiation, while the lattice remains at the initial temperature. The system is driven out of thermal equilibrium and consists now of two subsystems at different temperatures: electron at the temperature of the order of one electron volt and the lattice at the room temperature. The thermal equilibrium between the lattice and electrons is established on the picosecond time scale. If the energy of the laser pulse is sufficient, the surface melts and remains in the melted state for up to a nanosecond. The depth of the melt is of the order of one or several hundreds of nanometers. After the resolidification, the self-organized patterns are frozen into the surface and can be observed, e.g., by means of scanning electron microscopy, see image in Fig. \[Pattern\].
![LIPSS on the gold surface exposed to a single Ti:sapphire laser pulse, the wave length $\lambda\approx 800\,nm$, pulse duration $\tau_p\approx10^{-13}\,s$, laser fluence $F\approx3.3\,J/cm^2$. The average LIPSS period is $\Lambda\approx 0.76\,\mu m$.[]{data-label="Pattern"}](AuLIPSS.eps){width="8cm"}
Although the basic physical processes at the laser ablation are well understood and moreover, the effect of the LIPSS formation is interesting as well for fundamental physics as for practical applications, the mechanisms of the pattern formation are still not completely clear. There are two theoretical approaches, which try to explain the laser-induced periodic structures: (1) theories based on interference, i.e., a purely optical approach; (2) theories involving hydrodynamic instabilities, which result in self-organisation effects. Patterns explained in frames of the optical theories are referred to as *coherent structures*, whereas patterns explained by hydrodynamic-like theories are referred to as *non-coherent structures* [@Bauerle]. In this paper we analyze applicability of both these approaches for LIPSS induced by femtosecond laser pulses.
Interference with Surface Plasmons
==================================
As mentioned above, the interference-based theory (*coherent structures*) describes the LIPSS formation as an interference between the incident laser beam and a surface electromagnetic wave excited on the surface during the laser ablation [@Bonse; @Sipe; @Husinsky]. The nature of this wave is not clear, but obviously its excitation time scale must be well below the pulse duration, hence, only surface plasmons can be taken into account. The interference between the laser light and the surface plasmons gives reasons for the periodicity of the induced structures. The pattern period can be estimated and the orientation of the stripes can be explained in some experiments. However, there are contradictions between the experimental observations and the predictions made in frames of this plamonic theory reported, e.g., in [@AcuosticWave; @SPRLIPSS]; the validity of the plasmon excitation conditions by laser ablation is also debatable [@SPRLIPSS].
Indeed, the plasmon excitation via light is described by the dispersion curves of the both waves (plasmons and photons) and the excitation conditions are defined by their intersection, see Fig. \[DispCurves\]. The dispersion curves of photons and plasmons are described by the equations $$k=\frac{\omega}{c}\qquad \mathrm{and}\qquad k=\frac{\omega}{c}\sqrt{\varepsilon_1\varepsilon_2/(\varepsilon_1+\varepsilon_2)},
\label{DispCurvesEq}$$ respectively. Here the $\omega$ is the frequency, $k$ - the wave vector, $\varepsilon_1=1$ - dielectric constant of the surrounding medium (air), $\varepsilon_2=1-\omega_p^2/\omega^2$ - dielectric constant of the metal with the plasma frequency $\omega_p$. Typical values for the plasma frequencies for gold and copper are comparable $\omega_p\approx1.4\cdot10^{16}\,s^{-1}$ [@Nanoplasmonics]. It is useful to note that that of surface plasmons (denoted as [*SP*]{}) starts saturating at approximately $k_c=\frac{\omega_p}{c}\approx5\cdot10^7\,m^{-1}$.
From Fig. \[DispCurves\] one can see that there is only one intersection point between the dispersion curve of surface plasmons and that of free photons (denoted as [*Photons*]{}) and this intersection corresponds to the zero frequency. Due to this reason the plasmonic theory has difficulties to explain the excitation mechanism of the surface plasmons upon laser ablation. Thus, direct optical excitation of surface plasmons is impossible [@Nanoplasmonics; @Zayats]. There are two methods how to avoid this limitation [@sambles]: (1) decrease in the slope of the photon dispersion line, i.e., to decrease the phase velocity of light (see line $sl$ in Fig. \[DispCurves\]); (2) shift of the dispersion curve. The second important issue is the energy of the excited plasmonic wave. The amplitude of the plasmonic electric field must be comparable to that of the incident light, since for any sort of interference one needs two nearly coherent waves with comparable amplitudes. Hence at least half of the energy of the incident laser light should excite the oscillations of the electrons on the surface. The practical realization and difficulties of these plasmon excitation scenarios are discussed in the two following subsections.
Excitation via Slow Light
-------------------------
The phase velocity of photons can be slowed down to fit the velocity of the propagating plasmon-polariton wave by choosing appropriate incident angle and dielectric constants of materials. This is realized in the Otto and in the Kretschmann configurations [@Kretchmann; @otto] by illuminating the surface through a dielectric prism and choosing the incidence angle so high, that the total internal reflection of the incident light takes place on the dielectric interface. This method is used in biology and medicine and allows coupling of up to nearly whole laser energy to plasmons: Depending on the laser wavelength and incidence angle, the reflection can be varied from approximately 100% to approximately 1% [@SashaSPR; @SPR_SAB]. The method requires illumination of the metal surface at a fixed incidence angle $\alpha\neq 0^\circ$ through a dielectric material with high refractive index [@Nanoplasmonics; @Zayats; @sambles; @Kretchmann; @otto] to achieve the necessary condition for the total internal reflection.
In common laser-ablation experiments, in which the formation of LIPSS is observed, the surface is exposed at the incidence angle of $\alpha=0^\circ$ through the air. In some experiments silicon surfaces are exposed through a layer of liquid and a thin layer of silicon oxide [@straub2012], but the conditions for the total internal reflection remain unsatisfied there. In the theoretical studies one can also consider the electron plasma excited under the oxide layer in silicon and analyze a layered system, for which some of the excitations condition are easier to satisfy [@straub2012]. The most fundamental problem of the plasmon excitation in laser ablation is the incident angle of the laser light, which is typically $\alpha=0^\circ$. The momentum of the incident wave is perpendicular to the sample surface, hence the excitation of a wave in the surface plane (i.e., with the momentum perpendicular to the momentum of the exciting wave) is difficult without involving other physical mechanisms.
That is, the excitation of the surface plasmons by reducing the phase velocity of light is not realized by laser ablation, since it requires an exact fitting the incident angle of the light to the optical constants of the material. The LIPSS appear at similar laser ablation parameters in metals, semiconductors, dielectrics and polymers [@Birnbaum; @AcuosticWave; @MyPRE; @Straub; @Rebollar; @Bonse; @Henyk], i.e., in conductive and in dielectric materials, which have a broad range of dielectric constants. This fact suggests that other ways of the plasmon excitation, independent on the material properties, must be looked for.
Excitation via Surface Morphology
---------------------------------
In 1902 R. W. Wood observed drops in the optical spectrum produced by the diffraction grating, which he could not explain [@Wood]. The positions of these drops depend on the incidence angle. This observation proves that plasmons can be excited on a periodically patterned surface [@Nanoplasmonics], e.g., on a diffraction grating. A coupling between the surface plasmon wave and the incident light may also happen due to surface roughness (corrugation), which spectrum contains frequencies, at which the coupling is effective [@Bauerle; @Sipe; @Akhmanov]. On the language of the dispersion curves presented in Fig. \[DispCurves\], a surface pattern with the wave number $k_n,\quad n\in\mathbb{N}$ shifts the dispersion curves by $\pm k_n$, see the line marked as $n$ in the Fig. \[DispCurves\].
(0,0)node\[below\][0]{}–(0,6)node\[left\][$\omega$]{}; (0,0)–(7,0)node\[right\][$k$]{}; (0,0)..controls(1.2,3.2)..(7,4)node\[above\][ $SP$]{}; (0,0)–(2,5.5) node\[above\][$Photons$]{}; (0,0)–(5.4,4.5) node\[above\][$sl$]{}; (3,0)–(5,5.5) node\[above\][$n$]{}; at (1.5,-0.2)[$k_c$]{}; at (3,-0.2)[$k_n$]{}; (4.4,3.0)–(4.4,0.7); (4.4,0)–(6.4,5.5) node\[above\][$n+1$]{}; at (4.4,-0.2)[$k_{n+1}$]{}; (5.8,3.0)–(5.8,0.7); (5.8,0)–(6.4,1.5) node\[above\][$n+2$]{}; at (5.8,-0.2)[$k_{n+2}$]{}; at (-0.3,2.8)[$\omega_l$]{}; (-0.1,2.8)–(0,2.8);
The possibility of the plasmonic excitation via the surface roughness on a polished surface is difficult to evaluate experimentally since the roughness amplitude is small and the spectrum is broad. But we can solve the problem from the other side and estimate, which wave number in the spectrum of the surface roughness we need in order to be able to couple the incident light of the frequency $\omega_l\approx 10^{15}\,s^{-1}$ to the metal with the plasmonic frequency $\omega_p\approx1.4\!\cdot\!10^{16}\,s^{-1}$. Substituting $\omega=\omega_l$ into the dispersion equation of plasmons (\[DispCurvesEq\]) we calculate that the wave vector of the surface roughness must be $k_0\approx10^4\,m^{-1}$. This corresponds to the period of approximately 0.6mm, which is one order of magnitude larger than the diameter of the laser crater on the sample surface. Although this estimation demonstrates that the excitation of surface plasmons via surface roughness cannot be effective, let us suppose for the following analysis that this low efficiency allows the excitation anyway and consider multiple pulse exposure.
Here, we do not discuss a possible origin of the LIPSS formation via the *first* laser pulse. We assume that the plasmons can be excited by *coupling on a periodic structure generated by the previous pulses* and compare the theoretical predictions of this model to the experimental observations. If a contradiction will be found, we can conclude that the excitation of surface plasmons upon laser ablation either doesn’t happen or plays a secondary role for the LIPSS formation even in the ideal situation of a periodically patterned surface. Consequently the coupling via the surface roughness of a polished surface, which has no distinctive spatial period, is even less probable and should not be taken into account by the explanation of the LIPSS nature.
The difficulty of the surface roughness spectrum definition vanishes after the surface is exposed to the *first* laser shot, since the LIPSS appears (see Fig. \[Pattern\]) and the period of the structure $\Lambda$ can be easily measured. This is the new period of the surface corrugation; it defines the wave vector corresponding to the maximum of the power spectrum of the roughness after the first laser pulse $k_1=2\pi/\Lambda_1$. Now if the sample is exposed to multiple pulses, the *second* (and every next) laser shot interacts with the periodically patterned surface and the coupling conditions are fulfilled due to the periodic surface corrugation with the period $\Lambda_n$ and the corresponding wave number $k_n=2\pi/\Lambda_n$, see the line labeled as $n$ in Fig. \[DispCurves\]. Here, the index $n$ denotes the number of laser shots.
On the other hand, from the figure \[DispCurves\] one can see that the excitation of the surface plasmons in the second laser shot is impossible, while the incident light has a fixed frequency $\omega_l\approx 10^{15}\,s^{-1}$, but the intersection of the dispersion curves shifts towards higher $\omega_l$ from pulse to pulse. Hence, if the plasmon excitation at the given laser light frequency was possible in the first pulse, in the next one the frequency of the incident light must be increased to fit the resonance conditions. However, even if we neglect this limitation and suppose some sort of light spectrum modification [@WLprl2013] or Raman scattering [@sakabe] by the ablation or a very broad spectrum of the incident laser pulses, we anyway come to the contradiction, which is described below.
In frames of our assumption, that the LIPSS are formed by the surface plasmons excited due to the surface corrugation, the wave vector $k_n$ of the surface pattern after the $n^{th}$ laser pulse should depend on the number of pulses $n$. Indeed, after the first laser pulse the surface is periodically patterned with the period $\Lambda_1\approx0.76\,\mu m$, see Fig. \[Pattern\] for gold or $\Lambda_1\approx0.69\,\mu m$, see Fig. \[Cu1000\] (A) for copper. The corresponding wave vector $k_1$ can be calculated as $k_1=2\pi/\Lambda_1\approx10^7\,m^{-1}$.
The slope of the plasmonic dispersion curve is always smaller than the slope of the dispersion line of photons. Hence after every laser shot the wave vector of the surface pattern $k_n$ must grow, as it is schematically shown in figure \[DispCurves\]. The iterative map described schematically in the figure \[DispCurves\] can be represented as a function, which establishes relation between the pattern wave vectors after previous $n$ laser pulses $k_n$ and in the next following laser pulse $k_{n+1}$: $$\frac{k_{n+1}^2}{(k_{n+1}-k_{n})^2}=\frac{(k_{n+1}-k_{n})^2-\omega_p^2/c^2}{2(k_{n+1}-k_{n})^2-\omega_p^2/c^2}
\label{iterative}$$
![Experimentally measured (blue points) and theoretically predicted (solid red line) dependencies of the LIPSS period on copper surface $\Lambda$ on the number of laser shots $n$.[]{data-label="WLexp"}](WLexpTh1.eps){width="8cm"}
Analytical analysis of the equation (\[iterative\]) is difficult and we first solve it numerically and plot the solution in the terms of the pattern wavelength as the solid line in the Fig. \[WLexp\]. As one can see from the plot, the numerically calculated $\Lambda_n$ (the solid line) must decay rapidly to zero. In addition, we also analyse the asymptotic behaviour for large number of pulses analytically. After approximately $n\sim10^2$ laser pulses, the wave vector $k_n$ should considerably exceed the $k_c$ and the dispersion curve of surface plasmons saturates. In this case it can be approximated by the constant $\omega=\omega_p/\sqrt{2}$. Hence, after every laser shot the wave number must grow with a constant increment of $\Delta k=\omega_p/c\sqrt{2}$. This enables us to derive a simple analytical relation for the period of the surface structure $\Lambda_{n+m}$ as a function of the number of shots $m+n$ if the period after $n$ shots $\Lambda_{n}$ is known: $$\Lambda_{n+m}=\frac{\Lambda_{n}}{1+m\myfrac{\Lambda_{n}\omega_p}{2\pi c\sqrt{2}}}.
\label{LambdaM}$$
According to the equation (\[LambdaM\]), the LIPSS wavelength for large number of pulses $\Lambda_{n+m}$ decreases and tends to zero. However this result contradicts to the results reported in the literature and to our experimental observations. In Fig. \[WLexp\] one can see dependence of the LIPSS period measured experimentally on a copper surface exposed to multiple femtosecond laser pulses as a function of the number of pulses $n$ (see blue spots). The difference between the predictions based on the plasmonic theory and the experimental results are so large that we had to plot the results in the semi-logarithmic plot and break the $\Lambda$-axis.
The period of the pattern observed in our experiments after the first pulse ($\Lambda_1\approx 0.69\pm0.02\,\mu m$) is slightly larger than after $n\gtrsim 10^2$ pulses ($\Lambda_{250}\approx 0.59\,\mu m$) and does not change at least in the range from 250-to-1 till 8000-to-1 pulses. An example of the copper surface exposed to 1000 pulses can be found in Fig. \[Cu1000\] (B). Notice that results reported by other groups also disagree with the results of the plasmonic theory: in [@BonseKrueger] authors also observed a decrease in the pattern period in silicon but after approximately 100 laser pulses the period saturates, which is in agreement with our experiments. The period of the ripples on the ZnO surface was almost independent on the pulse number [@Dufft]. The discrepancy between the plasmonic theory and experimental results are even larger if insulating materials are ablated. In [@Rebollar2012; @Hoehm] authors observed an increase in the period of the surface pattern with the number of laser pulses in polymers and in quartz.
We conclude from the comparison between the experimentally measured and theoretically predicted dependencies of the pattern period on the number of laser pulses that the plasmonic theory cannot explain the pattern wavelength on metallic and semiconductor surfaces. Thus the surface plasmons probably do not play the key role in the LIPSS formation.
![Copper surface (A) exposed to $n=1$ and (B) exposed to $n=1000$ laser shots.[]{data-label="Cu1000"}](Cu_AB.eps){width="14cm"}
Here, we notice that Huang and coauthors [@Huang] also tried to explain the decrease in the LIPSS period with the number of laser pulses by the assumption that the laser light is coupled to the LIPSS on the surface. However, in the Huang theory, the change in the plasmon wavelength was induced by the shift in the Brillouin zones due to the deepening of the grooves.
Hydrodynamic Instabilities
==========================
The LIPSS may develop in the liquid melt phase, which appears on the sample surface after the laser ablation due to some hydrodynamic instabilities. The instabilities develop if (1) the conditions for the excitation of the instability are fulfilled; (2) the resolidification of the surfaces takes place on the time scale $\tau_{melt}$ larger than the characteristic time scale, at which the instability develops $\tau_i=\gamma^{-1}$ with $\gamma$ - the growth rate of the instability. Numerical simulation show that $h\sim 10^{-7}\,m$ and $\tau_{melt}\lesssim 10^{-9}\,s$ [@Zhigilei]. If the melt depth $h$ is smaller than the period of the LIPSS, the growth rates can be calculated in the frames of the lubrication approximation [@Oron].
The thickness of the melt film $h$ is close to the range, at which the Van der Waals forces come into play. The latter can destabilize the liquid layer and induce the instability, which growth rate $\gamma (k)$ is described by the dispersion relation (\[VdW\]) [@Oron] $$\gamma(k)= \left(\myfrac{A}{6\pi h}-\myfrac{\sigma h^3 k^2}{3}\right)\myfrac{k^2}{\eta}.
\label{VdW}$$ Here, $A\sim10^{-21}$ is the Hamaker constant of gold calculated for the solid-melt-vacuum interface system [@Hamaker], the viscosity $\eta=5.5\!\cdot\!10^{-3}\,Pa\, s$, surface tension $\sigma=1.2\,Nm^{-1}$ [@Iida]. If we substitute the experimentally observed wavelength of the pattern into Eq. \[VdW\], we will see that the growth rate is negative, hence the pattern with such a wavelength cannot develop due to this mechanism. However, the wavelength of the high-amplitude pattern observed in the experiments may differ from that of the low-amplitude instability, for which the equation (\[VdW\]) is valid. But as one can calculate from this equation, the minimal period of the pattern, at which the instability can be observed is $\Lambda_c\approx 6\!\cdot\!10^{-3}\,m$, which is four orders of magnitude smaller than the experimentally observed pattern period. Instabilities with the wave periods smaller than that critical value are suppressed by the surface tension.
Similarity between the patterns generated upon the laser ablation (cells [@MyPRE; @Guo], LIPSS) and the convection patterns [@Cross] suggests that convection may be the physical reason of the LIPSS generation. Moreover, the patterns in these both systems are induced by heating of a thin liquid layer. On the other hand this mechanism can be excluded due to the following two reasons: (1) the temperature gradient for both Marangoni and Rayleigh convection mechanisms should be directed from the surface into the bulk, i.e., the surface temperature must be lower than the temperature of the bottom of the melt layer. Numerical simulations of the laser ablation always indicate the opposite direction of the temperature gradient [@Anisimov; @Zhigilei]. (2) The Marangoni number $\mathcal{M}n$ and the Rayleigh number $\mathcal{R}a$ are several orders of magnitude lower than the critical values needed for the onset of the convection.
The physical parameters of liquid gold [@Iida] used for the calculations are: density $\rho=17\,g \!\cdot\! cm^{-3}$, variation of the surface tension with the temperature $\sigma'_T=-0.3 \cdot 10^{-3}\,N\,m^{-1} K^{-1}$, thermal expansivity $\alpha=10^{-4}\,K^{-1}$, $g=9.8\,s^{-2}m$. The thermal diffusivity was estimated as $\chi\approx 2\cdot 10^{-4}\,m^2s^{-1}$. With these values we can calculate the Rayleigh number $\mathcal{R}a=\myfrac{\alpha\Delta T h^3\rho g}{\eta \chi}\approx 10^{-11}$ and Marangoni number $\mathcal{M}n=\myfrac{|\sigma'_T|\Delta T h}{\eta \chi}\approx0.03$. For the calculations we used the temperature difference across the layer $\Delta T\approx 10^3\,K$. The calculated Rayleigh and Marangoni numbers are both much lower than the corresponding critical values $\mathcal{R}a_c\sim 10^3$ and $\mathcal{M}n_c\sim 10^2$. This leads us to the conclusion that the convection cannot develop in experiments on single shot femtosecond laser ablation.
Here it makes sense to estimate whether the LIPSS formation due to the melt redistribution is principally possible or not. From Fig. \[Pattern\] one can see that the modulation of the surface profile is large and a considerable part of the melt is moved on the distance $\ell\approx\Lambda/4$. Since this mass redistribution is only possible in the liquid state, i.e., during the time $\tau_{melt}$, we can estimate the flow velocity as $v\approx\Lambda/(4\tau_{melt})\sim10^2\,m\!\cdot\!s^{-1}$. This value is one order of magnitude lower than the speed of sound in liquid metals [@Iida], which demonstrates that the LIPSS can principally be induced by flow in the liquid metal.
Conclusion
==========
In this paper we analysed different possible mechanisms of the LIPSS formation upon femtosecond laser ablation with single and multiple pulses. At first we tested the possibility of the plasmon excitation due to the surface roughness. We compared the predictions based on this theory with our experimental observations and observations made by other groups. The surface plasmons cannot explain the observed surface patterns since the excitation conditions are not fulfilled and the predictions based on the plasmonic theory contradict to experimental observations. The theory that the surface plasmons can be coupled by the surface roughness would imply the rapid decrease of the LIPSS period with the number of laser pulses in multi-shot laser ablation experiments. The decrease in $\Lambda$ is really observed in our experiments after the first pulses, but it saturates rapidly.
Our analysis demonstrates that the physical background of the LIPSS formation cannot be explained in the frames of pure plasmonic theory. The analysis of most promising hydrodynamic instabilities demonstrated that they, in principle, can be used to explain the formation of the periodic structures. However, the basic hydrodynamic instabilities like the instabilities induced by convection or Van der Waals forces cannot explain the pattern.
Temperature-driven hydrodynamic instabilities can develop if the temperature distribution on the surface is inhomogeneous [@Oron]. Moreover, a periodically modulated temperature on the surface can modulate the evaporation velocity and hence cause a periodic pattern in the surface profile. The conditions for the spontaneous destabilisation of the temperature profile will be analysed elsewhere.
We gratefully acknowledge many fruitful discussions with the late Professor Rudolf Friedrich.
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|
---
author:
- |
Jiaming Song\
Stanford University\
`tsong@cs.stanford.edu`\
Shengjia Zhao\
Stanford University\
`zhaosj12@cs.stanford.edu`\
Stefano Ermon\
Stanford University\
`ermon@cs.stanford.edu`\
bibliography:
- 'references.bib'
title: 'A-NICE-MC: Adversarial Training for MCMC'
---
Introduction
============
Notations and Problem Setup
===========================
Adversarial Training for Markov Chains {#sec:mgan}
======================================
Adversarial Training for Markov Chain Monte Carlo {#sec:mcmc}
=================================================
Experiments {#sec:experiments}
===========
Discussion
==========
|
---
abstract: 'Let $\phi$ be a positive unital normal map of a von Neumann algebra $M$ into itself. It is shown that there exists a largest Jordan subalgebra $C_\phi$ of $M$ such that the restriction of $\phi$ to $C_\phi$ is a Jordan automorphism and each weak limit point of $(\phi^n (a))$ for $a\in M$ belongs to $C_\phi$.'
author:
- 'Erling St[ø]{}rmer'
date:
title: Multiplicative properties of positive maps
---
Dedicated to the memory of Gert K. Pedersen
1. Introduction {#introduction .unnumbered}
===============
In the study of positive linear maps of $C^*$-algebras the multiplicative properties of such maps have been studied by several authors, see e.g.[@K2],[@B],[@C1],[@C2],[@ER]. If $\phi\colon A \to B$ is a positive unital map between $C^*$-algebras $A$ and $B$ an application of Kadison’s Schwarz inequality,[@K1] to the operators $a+a^*$ and $i(a-a^*)$ yields the inequality [@S] $$\label{e1}
\phi(a\circ a^*)\geq \phi(a)\circ\phi(a)^*, a\in A,$$ where $a\circ b={{\frac{1}{2}}}(ab+ba)$ is the Jordan product. Thus one obtains an operator valued sesquilinear form $$\label{e2}
<a,b>=\phi(a\circ b^*)-\phi(a)\circ\phi(b)^*, a,b\in A.$$ If we apply the Cauchy-Schwarz inequality to $\omega(<a,b>)$ for all states $\omega$ of $B$ it was noticed in [@ER] that if $\phi(a\circ a^*)= \phi(a)\circ\phi(a)^*$ then $<a,b>=0$ for all $b\in A$. We call the set $$A_\phi =\{a\in A:\phi(a\circ a^*)= \phi(a)\circ\phi(a)^*\}$$ the *definite set* of $\phi$. It is a Jordan subalgebra of $A$, and if $a\in A_\phi$ then $\phi(a\circ b)=\phi(a)\circ
\phi(b)$ for all $b\in A$.
In the present paper we shall develop the theory further. We first study positive unital normal, i.e. ultra weakly continuous, maps $\phi\colon M \to M$, where $M$ is a von Neumann algebra. We mainly study properties of the definite set $M_\phi$ and some of its Jordan subalgebras of $M$ plus convergence properties of the orbits $(\phi^n (a))$ for $a\in M$. We shall show that when there exists a faithful family ${{\cal F}}$ of $\phi$-invariant normal states there is a largest Jordan subalgebra $C_\phi$ of $M $ called the multiplicative core of $M$, on which $\phi$ acts as a Jordan automorphism. Furthermore if $a\in M$ then every weak limit point of the orbit $(\phi^n (a))$ lies in $C_\phi$, and if $\rho(a\circ
b)=0$ for all $b\in C_\phi$, then $\phi^n (a)\rightarrow 0$ weakly.
Much of the above work was inspired by a theorem of Arveson, [@A]. In the last section we study the $C^*$-algebra case and the relation of our discussion with Arveson’s work. Then $\phi
\colon A\to A$ is a positive unital map, and we assume the orbits $(\phi^n (a))$ with $a\in A$ are norm relatively compact and that there exists a faithful family ${{\cal F}}$ of $\phi$-invariant states. It is then shown that the multiplicative core $C_\phi$ of $\phi$ equals the set of main interest in [@A], namely the norm closure of the linear span of all eigenoperators $a\in A$ with $\phi(a)=\lambda a, |\lambda|=1$, and that $
\lim_{n\rightarrow\infty} \|\phi^n (a)\|=0$ if and only if $\rho(a\circ b)=0$ for all $b\in C_\phi$ and $\rho\in {{{\cal F}}}$.
2. Maps on von Neumann algebras {#maps-on-von-neumann-algebras .unnumbered}
===============================
Throughout this section $M$ denotes a von Neumann algebra, $\phi\colon M\to M$ is a positive normal unital map. $ M_\phi$ denotes the definite set of $\phi$ and $< , >$ the operator valued sesquilinear form $<a,b>=\phi(a\circ b^*)-\phi(a)\circ\phi(b)^*,
a,b\in A$.
\[lem 2.1\] Let assumptions be as above, and suppose $(a_\alpha)$ is a bounded net in $M$ which converges weakly to $a\in M$. If $<a_\alpha,a_\alpha>\rightarrow 0$, then $a\in M_\phi$, and $\phi(a\circ b)= \phi(a)\circ \phi(b)$ for all $b\in M.$
[[*Proof.*]{} ]{}Let $\omega$ be a normal state on $M$. By the Cauchy-Schwarz inequality, if $b,c\in M$ we have $$|\omega(<b,c>)|^2 \leq\omega(<b,b>)\omega(<c,c>).$$ By assumption, if $a_\alpha$ and $a$ are as in the statement of the lemma, and $b\in M$ then $$\begin{aligned}
|\omega(<a,b>)|^2 &=& \lim_\alpha |\ \omega(<a_\alpha,b>)|^2\\
&\leq &\lim_\alpha \omega(<a_\alpha,a_\alpha>)\omega(<b,b>) =0.\end{aligned}$$ Since this holds for all normal states $\omega$, $<a,b>=0$, completing the proof.
In analogy with the definition of G-finite for automorphism groups we introduce
\[def 2.2\] With $\phi$ as above we say $M$ is $\phi$-*finite* if there exists a faithful family ${{\cal F}}$ of $\phi$-invariant normal states on the von Neumann algebra generated by the image $\phi(M)$.
\[lem 2.3\] Assume $M$ is $\phi$-finite. Then for $a\in M$ we have Every weak limit point of the orbit $(\phi^n (a))$ of $a$ belongs to $M_\phi$. If $\rho(\phi^n (a)\circ b)=0$ for all $b\in M_\phi, \rho \in {{\cal F}}$, then $\phi^n (a)\rightarrow 0$ weakly.
[[*Proof.*]{} ]{}If $\rho\in {{\cal F}}$ denote by $\| . \|_\rho$ the seminorm $\|x\|_\rho =\rho(x\circ x^*)^{{\frac{1}{2}}}$. Then by the inequality (1) $$\begin{aligned}
\|\phi^{n+1}(a)\|_\rho ^2 &=&
\rho(\phi^{n+1}(a)\circ\phi^{n+1}(a)^*)\\
&\leq& \rho(\phi(\phi^n (a)\circ\phi^n (a)^*))\\
&=&\|\phi^n (a)\|_\rho ^2.\end{aligned}$$ Thus the sequence $\|\phi^n (a)\|_\rho ^2$ is decreasing, hence $\|\phi^n (a)\|_\rho ^2 - \|\phi^{n+1}(a)\|_\rho ^2 \rightarrow
0$. We have $$\begin{aligned}
\rho(<\phi^n (a),\phi^n (a)>)&=&\rho(\phi(\phi^n (a))\circ\phi^n
(a)^*)-\phi(\phi^n (a))\circ\phi(\phi^n (a)^*))\\
&=&\rho(\phi^n (a)\circ\phi^n(a)^* -\phi^{n+1}
(a)\circ\phi^{n+1}(a)^*)\\
&=&\|\phi^n (a)\|_\rho ^2 - \|\phi^{n+1}(a)\|_\rho ^2 \rightarrow
0.\end{aligned}$$ Since this hold for all $\rho\in {{\cal F}}$ and ${{\cal F}}$ is faithful, $<\phi^n (a),\phi^n (a)>\rightarrow 0$ weakly. By Lemma \[lem 2.1\], if $a_0$ is a weak limit point of $(\phi^n (a))$ then $a_0
\in M_\phi$, proving (i).
To show (ii) suppose $\rho(\phi^n (a)\circ b)=0$ for all $b\in
M_\phi, \rho\in {{\cal F}}$. Let $a_0$ be a weak limit point of $(\phi^n
(a))$. Then $\rho(a_0 \circ b)=0$ for all $b\in M_\phi$, in particular by part (i) $\rho(a_0\circ a_0)=0$. Since ${{\cal F}}$ is faithful on the von Neumann algebra generated by $\phi(M)$, $a_0
=0$. Thus 0 is the only weak limit point of $(\phi^n (a))$, so $\phi^n (a)\rightarrow 0$ weakly. The proof is complete.
It is not true in general that $\phi(M_\phi)\subseteq M_\phi$. We therefore introduce the following auxiliary concept. If $\phi\colon A\to A$ is positive unital with A a $C^*$-algebra, then $A_{\Phi}=\{a\in A_{\phi} : {\phi}^k (a)\in A_{\phi}, k\in
{{\mathbb N}}\}$.
\[lem 2.4\] Let $M$ be $\phi$-finite and $M_\Phi$ defined as above. Then $M_\Phi$ is a weakly closed Jordan subalgebra of $M_\phi$ such that $\phi(M_{\Phi})\subseteq M_{\Phi}$, and if $a\in M$ then every weak limit point of $(\phi^n (a))$ belongs to $M_\Phi$. Furthermore, if $\rho(\phi^n (a)\circ b)=0$ for all $b\in M_\Phi, \rho\in {{\cal F}}$, then $\phi^n (a)\rightarrow 0$ weakly.
[[*Proof.*]{} ]{}Since $M$ is weakly closed and $\phi$ is weakly continuous on bounded sets $M_\Phi$ is weakly closed. Since $\phi$ and its powers $\phi^k$ are Jordan homomorphisms on $M_phi$ it is straightforward to show $M_\Phi$ is a Jordan subalgebra of $M$. Furthermore it is clear from its definition that $\phi(M_\Phi)\subseteq M_\Phi$.
If $a\in M$ and $a_0$ is a weak limit point of $(\phi^n (a))$, then $a_0\in M_\phi$ by Lemma 3. Then $\phi(a_0)$ is a weak limit point of $(\phi^{n+1}(a))$, hence belongs to $M_\phi$, again by Lemma 3. Iterating we have $\phi^k (a_0)\in M_\phi$ for all $k\in{{\mathbb N}}$. Thus $a_0\in M_\Phi$. The last statement follows exactly as in Lemma 3. The proof is complete.
It is not true that $\phi(M_\Phi)=M_\Phi$. To remedy this problem we introduce yet another Jordan subalgebra.
\[def 2.5\] Let $\phi\colon A\to A$ be positive unital with $A$ a $C^*$-algebra. The *multiplicative core* of $\phi$ is the set $$C_\phi=\bigcap_{n=0}^{\infty}\phi^n (A_\Phi).$$
\[\] $C_{\phi}$ satisfies the following: $C_{\phi}$ is a Jordan subalgebra of $A$. $\phi(C_{\phi})=C_{\phi}$. Suppose there exists a family ${{\cal F}}$ of $\phi$-invariant states which is faithful on the $C^*$-algebra generated by $\phi(A)$. Then we have The restriction of $\phi$ to $C_{\phi}$ is a Jordan automorphism. $C_{\phi}$ is the largest Jordan subalgebra of $A$ on which the restriction of $\phi$ is a Jordan automorphism.
[[*Proof.*]{} ]{}
As in Lemma 4 $C_\phi$ is clearly a Jordan subalgebra of $A$ such that $\phi(C_\phi)\subseteq C_\phi$ and is weakly closed in the von Neumann algebra case. Furthermore, since $\phi(A_\Phi)\subseteq A_\Phi$, we have $\phi^n(A_\Phi)\subseteq
\phi^{n-1}(A_\Phi)$, so that the sequence $(\phi^n(A_\Phi))$ is decreasing. Thus $$C_\phi =\bigcap_{n=0}^{\infty}\phi^{n+1}(A_\Phi)=\phi(C_\phi),$$ so (i) and (ii) are proved.
We next show (iii), and let ${{\cal F}}$ be as in the statement of the lemma. By (ii) the restriction of $\phi$ to $C_\phi$ is a Jordan homomorphism of $C_{\phi}$ onto itself. In particular ${{\cal F}}$ is faithful on $C_{\phi}$, so that $\phi$ is faithful on $C_{\phi}$, hence is a Jordan automorphism of $C_\phi$, proving (iii).
To show (iv) let $B$ be a Jordan subalgebra of $A$ such that $\phi|_B$ is a Jordan automorphism of $B$. Then clearly $B\subseteq A_\Phi$, and $\phi^n (B)=B$, so that $$B=\bigcap_{n=0}^{\infty}\phi^n (B)
\subseteq\bigcap_{n=0}^{\infty}\phi^n (A_\Phi)=C_\phi.$$ The proof is complete.
We can now prove our main result.
\[Thm 2.5\] Let $M$ be $\phi$-finite, and ${{\cal F}}$ a set of normal $\phi$-invariant states which is faithful on the von Neumann algebra generated by $\phi(M)$. Let $a\in M$. Then we have Every weak limit point of $(\phi^n (a))$ lies in $C_\phi$. If $\rho(a\circ b)=0$ for all $b\in C_\phi, \rho\in {{\cal F}}$, then $\phi^n (a) \rightarrow 0$ weakly.
[[*Proof.*]{} ]{}Ad(i). Let $a_0$ be a weak limit point of $(\phi^n (a))$. By Lemma 4 $a_0\in M_\Phi$. Choose a subnet $(\phi^{n_\alpha}(a))$ which converges weakly to $a_0$. Let $k\in {{\mathbb N}}$, and let $(\phi^{m_\beta} (a))$ be a subnet of $(\phi^{n_{\alpha} -k}(a))$ which converges weakly to $a_1\in M_\Phi$ (again using Lemma 4, since $(\phi^{m_\beta} (a))$ will be a subnet of $(\phi^n (a))$). Each $m_\beta$ is of the form $n_{\alpha_j}-k$. The net $(\phi^{n_{\alpha_j}}(a))$ converges to $a_0$, since it is a subnet of the converging net $(\phi^{n_\alpha}(a))$. Thus we have $$\begin{aligned}
\phi^k (a_1)&=& \lim \phi^k (\phi^{m_\beta}(a))\\
&=& \lim \phi^{k+(n_{\alpha_{j}}-k)} (a)\\
&=& \lim \phi^{n_{\alpha_j}}(a)\\
&=& a_0.\end{aligned}$$ Thus $a_0\in \phi^k (M_\Phi)$ for all $k\in {{\mathbb N}}$, hence $a_0\in
C_\phi$.
To show (ii) suppose $\rho(a\circ b)=0$ for all $\rho\in {{\cal F}},
k\in{{\mathbb N}}$. Since $\phi^k (C_\phi)=C_\phi$ there exists $c\in
C_\phi$ such that $b=\phi^k (c)$. Thus $$\begin{aligned}
\rho(\phi^k (a)\circ b) &=& \rho(\phi^k (a)\circ \phi^k (c))\\
&=& \rho(\phi^k (a\circ b))\\
&=& \rho(a\circ b) = 0.\end{aligned}$$ By part (i) every weak limit point $a_0$ of $(\phi^n (a))$ lies in $C_\phi$, so it follows by the above that $\rho(a_0 \circ b)=0$ for all $b\in C_\phi$. In particular $\rho(a_0 \circ a_0)=0$, so by faithfulness of ${{\cal F}}$, $a_0=0$, hence $\phi^n (a)\rightarrow 0$ weakly. The proof is complete.
One might believe that the converse of part (ii) in the above theorem is true. This is false. Indeed, let $M_0$ be a von Neumann algebra with a faithful normal tracial state $\tau_0$. Let $M_i=M_0, \tau_i =\tau_0, i\in {{\mathbb Z}}$, and let $M=
\bigotimes_{-\infty}^{\infty} (M_i,\tau_i)$. Let $\phi$ be the shift to the right. Then $C_\phi =M$. However, if $a= ...1\otimes
a_0 \otimes 1...\in M$ with $a_0\in M_0$, then $\lim_{n\rightarrow\infty} \phi^n (a)= \tau_0 (a_0)1$, so if $\tau_0(a_0)=0$, then the weak limit is 0. But $\tau(a\circ b)\neq
0$ for some $b\in M =C_\phi$.
If we assume convergence in the strong-\* topology then the converse holds, as we have
\[prop2.6\] Let $M$ be $\phi$-finite. Let $a\in M$ and suppose the sequence $(\phi^n (a))$ converges in the strong-\* topology. Then $\rho(a\circ b)=0$ for all $b\in C_\phi, \rho\in {{\cal F}}$ if and only if $\phi^n (a)\rightarrow 0$ \*-strongly.
[[*Proof.*]{} ]{}If $\rho(a\circ b)=0$ for all $b\in C_\phi, \rho\in {{\cal F}}$ then $\phi^n (a)\rightarrow 0$ weakly by the theorem. Since the sequence converges \*-strongly the limit must be 0.
Conversely, if $\phi^n (a)\rightarrow 0$ \*-strongly, then for all $b\in C_\phi, \rho\in {{\cal F}}$ $$\rho(a\circ b)=\rho(\phi^n (a\circ b))=\rho(\phi^n (a)\circ\phi^n
(b))\rightarrow 0,$$ since multiplication is \*-strongly continuous on bounded sets. The proof is complete.
We have not in general found a nice description of the complement of $C_\phi$ in $M$, i.e. a subspace $D$ such that $M$ is a direct sum of $C_\phi$ and $D$. In the finite case with a faithful normal $\phi$-invariant trace this can be done.
\[prop 2.7\] Suppose $M$ has a faithful normal $\phi$-invariant tracial state. Then there exists a faithful normal positive projection $P\colon
M\to C_\phi$ which commutes with $\phi$. Let $D=\{a-P(a): a\in
M\}$. Then $M=C_\phi + D$ is a direct sum, and if $a\in D$ then $\phi^n (a)\rightarrow 0$ weakly.
[[*Proof.*]{} ]{}Since $M$ is finite the same construction as that of trace invariant conditional expectations onto von Neumann subalgebras yields the existence of a faithful trace invariant positive normal projection $P \colon M\to C_\phi$, see [@HE]. Let $\tau$ be the trace alluded to in the proposition. Since $\tau$ is faithful and $\phi$-invariant, $\phi$ has an adjoint map $\phi^* \colon
M\to M$ defined by $\tau(a \phi^*(b))=\tau(\phi(a)b)$ for $a,b\in
M$. Clearly $\phi^*$ is $\tau$-invariant, positive, unital, and normal, and its extension $\bar\phi^*$ to an operator on $L^2(M,\tau)$ is the usual adjoint of the extension $\bar\phi$ of $\phi$. Since the restriction of $\bar\phi$ to the closure $C_\phi^-$ of $C_\phi$ in $L^2(M,\tau)$ is an isometry of $C_\phi^-$ onto itself, so is $\bar\phi^*$. It follows that $\phi
P=P\phi P=(P \phi^* P)^*=(\phi^* P)^* =P\phi$.
It is clear that $M=C_\phi + D$ is a direct sum. Suppose $a\in D$, i.e. $P(a)=0$. Then $\tau(a\circ b)=0$ for all $b\in C_\phi$. If we let ${{\cal F}} =\{\tau|_{C_\phi}\circ P\}$ then, since $P$ commutes with $\phi$, ${{\cal F}}$ is a faithful family of normal $\phi$-invariant states. By Theorem 6 $\phi^n (a)\rightarrow 0$ weakly, proving the proposition.
3. Maps of C\*-algebras {#maps-of-c-algebras .unnumbered}
=======================
Arveson [@A] proved the following result.
\[Thm.3.1\](Arveson) Let $A$ be a $C^*$-algebra, $\phi\colon A\to A$ a completely positive contraction such that the orbit $(\phi^n (a))$ is norm relative compact for all $a\in A$. Then there exists a completely positive projection $P\colon A\to A$ onto the norm closed linear span $E_\phi$ of the eigenoperators $a\in A$ with $\phi(a)=\lambda a$, with $|\lambda|=1$, and $\alpha
=\phi|_{E_\phi}$ is a complete isometry of $E_\phi$ onto itself. We have $$\lim_{n\rightarrow\infty}\|\phi^n (a) - (\alpha\circ P)^n (a)\|=0,$$ and $A$ is the direct sum of $E_\phi$ and the set $\{a\in A:\lim_n
\|\phi^n (a)\|=0\}$.
We shall now show how our previous results yield a result which is in a sense complementary to Arveson’s theorem.
\[thm 3.2\] Let $A$ be a unital $C^*$-algebra and $\phi\colon A\to A$ a positive unital map such that the orbit $(\phi^n (a))$ is norm relative compact for all $a\in A$. Let $C_\phi$ be the multiplicative core for $\phi$ in $A$, and let $E_\phi$ denote the set of eigenoperators $a\in A$ such that $\phi(a)=\lambda a$, with $|\lambda|=1$. Assume there exists a set ${{\cal F}}$ of $\phi$-invariant states which is faithful on the $C^*$-algebra generated by $\phi(A)$. Then we have $E_\phi=C_\phi$ is a Jordan subalgebra of $A$. The restriction $\phi|_{E_\phi}$ is a Jordan automorphism of $E_\phi$. Let $a\in A$. Then $\rho(a\circ b)=0$ for all $\rho\in{{\cal F}}, b\in
C_\phi$ if and only if $\lim_{n\rightarrow \infty}\|\phi^n
(a)\|=0$.
[[*Proof.*]{} ]{}We first show (ii). If $\phi(a)=\lambda a$ then $\phi(a^*)=\bar\lambda a^*$, so $E_\phi$ is self-adjoint. Furthermore by inequality (1) $$\phi(a\circ a^*)\geq \phi(a)\circ\phi(a^*)=\lambda a\circ \bar
\lambda a^*=a\circ a^*.$$ Composing by $\rho\in{{\cal F}}$ and using that ${{\cal F}}$ is faithful on $C^*(\phi(A))$ it follows that $\phi(a\circ
a^*)=\phi(a)\circ\phi(a^*)$, so $a\in A_\phi$, the definite set of $\phi$. Since $a\in E_\phi$ is an eigenoperator, so is $a^2$, hence $E_\phi$ is a Jordan subalgebra of $A_\phi$. Note that if $\phi(a)=\lambda a$ then $\phi(\phi(a))=\phi(\lambda a)=\lambda \phi(a))
$, so $\phi(a)\in E_\phi$. Thus $\phi\colon E_\phi \to E_\phi$. If $a=\sum \mu_i a_i \in E_\phi$ where $\phi(a_i)=\lambda_i a_i$, then $a=\sum \mu_i \bar\lambda{_i} \phi(a_i) \in \phi(E_\phi)$, so by density of such $a'$s, $\phi(E_\phi)=E_\phi$. Thus by faithfulness of ${{\cal F}}$ the restriction $\phi|_{E_\phi}$ is a Jordan automorphism, proving (ii).
It follows from Lemma 6 that $E_\phi\subseteq C_\phi$. To show the converse inclusion we use that the orbit $(\phi^n (a))_{n\in{{\mathbb N}}}$ is norm relative compact for all $a\in A$. By Lemma 6 the restriction of $\phi$ to $C_\phi$ is a Jordan automorphism, hence in particular an isometry. We assert that if $a\in C_\phi$ then the orbit $(\phi^n (a))_{n\in{{\mathbb Z}}}$ is relative norm compact. For this it is enough to show that the set $(\phi^{-n}
(a))_{n\in{{\mathbb N}}}$ is relative norm compact, or equivalently that each sequence $(\phi^{-n_k} (a))$ has a convergent subsequence. By assumption $(\phi^{n_k} (a))$ has a convergent subsequence $(\phi^{m_l} (a))$. Since this sequence is Cauchy, and $$\|\phi^{-n} (a) -\phi^{-m}
(a)\|=\|\phi^{n+m}(\phi^{-n}(a)-\phi^{-m} (a))\|=\|\phi^n (a)
-\phi^m (a)\|,$$ it follows that $(\phi^{-m{_l}} (a))$ is Cauchy, and therefore converges. Thus the set $(\phi^{-n} (a))_{n\in{{\mathbb N}}}$ is relative norm compact, as is $(\phi^n (a))_{n\in{{\mathbb Z}}}$. By a well known result on almost periodic groups, see e.g. Lemma 2.8 in [@A], $\phi|_{C_\phi}$ has pure point spectrum. Thus $C_\phi \subseteq
E_\phi$, proving (i).
It remains to show (iii). As in the proof of Lemma \[lem 2.3\] we find that every norm limit point $a_0$ of $(\phi^n (a))$ belongs to $A_\phi$, and by the proof of Lemma \[lem 2.4\] $a_0\in A_\Phi =\{x\in A_\phi : \phi^k (x)\in A_\phi, k\in {{\mathbb N}}$}. A straightforward modification of the proof of Theorem \[Thm 2.5\](i), replacing weak by norm, shows that $a_0\in C_\phi$. Let $a\in A$ satisfy $\rho(a\circ b)=0$ for all $ b\in C_\phi, \rho\in
{{\cal F}}$. Then by the proof of Theorem \[Thm 2.5\](ii), every norm limit point of $(\phi^n (a))$ is 0. Thus there is a subsequence $(\phi^{n_k}(a))$ of $(\phi^n (a))$ such that for all $\varepsilon
>0$ there is $k_0$ such that $\|\phi^{n_k}\|\leq \varepsilon$ when $k\geq k_0$. But then $n> n_k$ for $k\geq k_0$ implies $$\|\phi^n (a)\|=\|\phi^{n-n_k}(\phi^{n_k}(a)\|\leq
\|\phi^{n_k}\|<\varepsilon.$$ Thus $\|\phi^n (a)\|\rightarrow 0$.
Conversely, if $\|\phi^n (a)\|\rightarrow 0$ then for $a\in A,
b\in C_\phi, \rho\in {{\cal F}}$ $$\rho(a\circ b)=\rho(\phi^n (a\circ b))=\rho(\phi^n (a)\circ \phi^n
(b))\rightarrow 0,$$ for $n\rightarrow\infty$. Thus $\rho(a\circ
b)=0$, completing the proof of the theorem.
It was shown in [@ES] that if $A$ is a $C^*$-algebra, and $P\colon A\to A$ is a faithful positive unital projection then the image $P(A)$ is a Jordan subalgebra of $A$. The following corollary proves more.
\[cor 3.3\] Let $A$ be a $C^*$-algebra and $P\colon A\to A$ a faithful positive unital projection. Then $E_P =C_P =P(A)$, Hence $P(A)$ is in particular a Jordan subalgebra of $A$.
[[*Proof.*]{} ]{}Since $P^2 =P$ the orbit of each $a\in A$ is finite, so compact. Since $P$ is faithful the set of states ${{\cal F}}=\{\omega|_{P(A)}\circ P\}$ with $ \omega$ a state on $A$, is a faithful family of $P$-invariant states. Thus by Theorem \[thm 3.2\] we have $E_P = C_P$. Since $P$ is a projection the only nonzero eigenvalue of $P$ is 1, and the corresponding eigen operators are the elements in $P(A)$. Thus $E_P =P(A)$, proving the corollary.
[999]{}
W. Arveson, *Asymptotic stability I: completely positive maps*, Int.J.Math. 15(3) (2004), 289-312.
B.M.Broise, letter to the author (1967).
M.-D. Choi, *Positive linear maps on $C^*$-algebras*, Thesis, University of Toronto (1972).
M.-D. Choi, *Positive linear maps of $C^*$-algebras*, Canad. J. Math. 24 (1972), 520-529.
E.Effros and E.St[ø]{}rmer, *Positive projections and Jordan structure in operator algebras*, Math.Scand.45 (1979),127-138.
D.Evans and R.H[ø]{}egh-Krohn, *Spectral properties of positive maps on $C^*$-algebras*, J.London Math.soc. 17(1978), 345-355.
U.Haagerup and E.St[ø]{}rmer, *Positive projections of von Neumann algebras onto JW-algebras*, Reports on Math.Phys.36 (1995),317-330.
R.V.Kadison, *A general Schwarz inequality and algebraic invariants for operator algebras*, Ann.Math.56(1952),494-503.
R.V.Kadison, *The trace in finite von Neumann algebras*, Proc.Amer.Math.Soc.12(1961),973-977.
E.St[ø]{}rmer, *Positive linear maps of operator algebras*, Acta Math.110(1963),233-278.
Department of Mathematics, University of Oslo, 0316 Oslo, Norway.
e-mail erlings@math.uio.no
|
---
abstract: 'Lithium abundances are presented and discussed for 70 members of the 50 Myr old open cluster $\alpha$ Per. More than half of the abundances are from new high-resolution spectra. The Li abundance in the F-type stars is equal to its presumed initial abundance confirming previous suggestions that pre-main sequence depletion is ineffective for these stars. Intrinsic star-to-star scatter in Li abundance among these stars is comparable to the measurement uncertainties. There is marginal evidence that the stars of high projected rotational velocity ([*v*]{} sin [*i*]{}) follow a different abundance vs temperature trend to the slow rotators. For stars cooler than about 5500 K, the Li abundance declines steeply with decreasing temperature and there develops a star-to-star scatter in the Li abundance. This scatter is shown to resemble the well documented scatter seen in the 70 Myr old Pleiades cluster. The scatter appears to be far less pronounced in the 30 Myr clusters which have been studied for Li abundance.'
author:
- |
Suchitra C. Balachandran$^{1}$, Sushma V. Mallik$^{2}$[^1], David L. Lambert$^{3}$\
$^{1}$Astronomy Department, University of Maryland, College Park, MD 20742-2421, USA; suchitra.balachandran@gmail.com\
$^{2}$Indian Institute of Astrophysics, Bangalore 560034, India; sgvmlk@iiap.res.in\
$^{3}$W. J. McDonald Observatory. The University of Texas at Austin. 1 University Station, C1400.\
Austin, TX 78712$-$0259, USA; dll@astro.as.utexas.edu\
title: 'Lithium Abundances in the $\alpha$ Per Cluster'
---
\[firstpage\]
open clusters: individual ($\alpha$ Per) — stars: abundances
Introduction
============
Abundance measurements of lithium in stellar atmospheres have long been an active pursuit for observers and theoreticians alike. Much of the activity is directed at understanding the depletion of the atmospheric lithium from its abundance, often an inferred quantity, acquired at birth. Open clusters serve as astrophysical laboratories in which to investigate the internal depletion of lithium because a given cluster provides a close approximation to a sample of coeval stars of a common age and initial composition including that of lithium but spanning a range of masses and other properties such as rotation. And, crucially, the suite of clusters spans a large range of ages for a small range in composition. Three principal episodes of lithium depletion are recognized: (i) depletion through destruction of lithium at the base of the convective envelope of the pre-main sequence star, (ii) continued depletion by destruction in the main sequence phase, and (iii) depletion by a combination of diffusion and destruction in F-type main sequence stars in the narrow effective temperature range of about 6400–6900 K (the so-called Li-dip). Sestito & Randich (2005) assemble Li abundance data for 20 open clusters with ages from 5 Myr to 8 Gyr to confront theories for Li depletion with observations.
The cluster $\alpha$ Per was among the sample of 20 clusters with Li observations drawn from Balachandran, Lambert, & Stauffer (1988, 1996 – hereafter BLS) and Randich et al. (1998). In this paper, we obtain and analyze high-resolution spectra from which Li abundances are obtained for about 50 stars. When the BLS and this new sample are combined in a uniform manner and a reconsideration made of the cluster membership of the stars, Li abundances are provided for 70 cluster members.
Observations of Li in $\alpha$ Per were made initially by Boesgaard et al. (1988) who analyzed high-resolution spectra of six F-type stars to show that the Li-dip (Boesgaard & Tripicco 1986) has not yet developed in this young (age of about 50 Myr) cluster. Our principal goal was not to define the run of Li abundance along the main sequence because that is already well known for F, G, and K-type stars (Boesgaard et al. 1988; BLS; Randich et al. 1998) and M-type stars (García López et al. 1994; Zapatero Osorio et al. 1996). Rather we sought to determine if the Li abundance at a given effective temperature has an intrinsic scatter.
Such a star-to-star variation in apparent Li abundances has been reported for the Pleiades, a cluster only slightly older than $\alpha$ Per (Butler et al. 1987; Soderblom et al. 1993; King et al. 2000). This variation appears for stars with effective temperatures less than about 5300 K and extends to the useful limit of the sample at about 4000 K. The peak-to-peak variation is about 1.5 dex in apparent Li abundance. Stars with the stronger Li[i]{} 6707 Åfeature at a given temperature have higher projected rotational velocities ([*v*]{} sin [*i*]{}). The debate is ongoing as to whether the variation in Li[i]{} line strength in the Pleiades and other clusters reflects a real abundance difference or differences in atmospheric structure not modelled by classical atmospheres.
Randich et al. (1998) studied Li abundances in 18 very active, X-ray selected members of $\alpha$ Per enlarging the original sample of BLS in the 5500 $-$ 3900 K range of effective temperature. Randich et al. (1988) suggested that, at T$_{eff}$ $\le$ 5300K, there was indeed a significant dispersion in Li abundances in stars at the same temperature. They further suggested that rapid rotators had more Li and exhibited a smaller dispersion than slow rotators at the same T$_{eff}$. They inferred from these observations a likely relationship between Li, chromospheric activity and the rotational history of stars.
Examination of BLS’s lithium observations as reanalyzed by Randich et al. (1998) led Xiong & Deng (2005) to suggest that star-to-star variations were also present among $\alpha$ Per members at effective temperatures of about 4700 K and that the variations primarily arose from atmospheric effects and not a real abundance variation. With our larger sample of cluster members, we reexamine the question of star-to-star variation in lithium abundance.
In Section 2, we discuss selection of the newly observed stars. In Section 3, we describe the new high-resolution spectra of $\alpha$ Per stars. Section 4 presents the stellar parameters with emphasis on the effective temperature. The abundance analysis is introduced in Section 5. The run of Li abundance with effective temperature and the star-to-star variations are discussed in Section 6. The paper concludes with general remarks in Section 7.
Construction of the final sample
================================
In referring to members of the cluster, we follow the convention of ‘WEBDA’, a website devoted to stellar clusters[^2]. Heckmann et al. (1956) and Heckmann & Lübeck (1958) introduced a numbering scheme preceded by the letters ‘He’. Later, Stauffer et al. (1985, 1989a) and Prosser (1992) identified fainter stars with the letters ‘Ap’. In WEBDA, the label He is replaced by \#, thus He 12 becomes \#12. In the case of the Ap stars, the numbering is increased by 1500 and Ap replaced by \#, thus Ap 79 becomes \#1579.
When the observations (see below) made for this paper are combined with those reported by Balachandran et al. (1988, 1996), we have spectra for 86 stars. Since our primary goal is to determine whether the run of Li abundance down the main sequence of the cluster exhibits scatter at a given effective temperature, it is vital to sort cleanly the cluster members from the non-members and also to separate out suitable from unsuitable (i.e., doubled-lined spectroscopic binaries) members.
As long recognized, clean separation of members from non-members is not an easy task for $\alpha$ Per because it is at low Galactic latitude and has a small relative proper motion. In making the separation, we have called upon a variety of publications that have previously attempted the task. The primary source and the one used in selecting stars for observation was the seminal study of the cluster by Prosser (1992). He considered a variety of membership indicators among which the primary ones were proper motions and radial velocities of the stars.
Makarov (2006) reanalysed the cluster’s proper motions using astrometry and photometry from the Tycho-2 Catalogue and the Second USNO CCD Astrometric Catalog (UCAC2). Makarov’s table of ‘High-Fidelity’ members lists 139 stars with a V magnitude brighter than about 11.5; no table of non-members is provided. Of the stars in our sample with V$<$ 11.5 and designated as members according to Prosser, all but ten are among Makarov’s high-fidelity members. Four of the ten stars not listed as members by Makarov are categorized as non-members by Mermilliod et al. (2008) (see below). It is unclear whether Makarov studied all stars from Prosser’s list with V $<$ 11.5, and therefore the absence of a star in Makarov’s table is not necessarily an indication that it is not a member. We have included the remaining six stars in our sample in Table 1.
Mermilliod et al. (2008) undertook a radial velocity program to check for spectroscopic binaries in the cluster. Their criteria for membership were threefold: proper motions (from UCAC2), radial velocity and location in the color-magnitude diagram. These criteria were applied independently of Prosser’s and Makarov’s efforts at membership determination. Fifty four of our 86 stars were in Mermilliod et al.’s program. Of these only four were identified as non-members in that program: \# 143, 573, 1100, and 1181, with the first two shown to be spectroscopic binaries. We adopt Mermilliod et al.’s view that this quartet are non-members and list these stars in Table 2. Some members were shown to be spectroscopic binaries. Binaries not yet shown to be double-lined are included in the list of 70 members and identified in the final column in Table 1; all seven fall near the main sequence locus in a color-magnitude diagram suggesting the secondary star contributes very little to the composite spectrum.
Patience et al. (2002) report on an imaging search for close binaries among known cluster members; these authors made no independent determinations of membership. A large fraction of our stars was examined by Patience et al. with the majority reported not to have a companion that would have contributed to our spectrum which we have assumed is that of a single star. Four stars were excluded as unsuitable for analysis on the basis of the reported imaging; these have companions separated by less than 0.5 arc seconds and rather similar masses. The stars are \# 696 (also known as \#1538), 935, 1541 and 1598.
In summary, 70 of the 86 stars are considered to be cluster members (Table 1). Information provided in Table 1 is as follows: the WEBDA \# is in column 1, the adopted stellar parameters are in columns 2, 3, and 4. The projected rotational velocity ([*v*]{} sin [*i*]{}) in column 7 is primarily taken from Prosser (1992). The equivalent width of the Li[i]{} 6707 Å feature is given for stars with low [*v*]{} sin [*i*]{} in column 8 and the derived Li abundance is in column 9. Columns 10, 11, and 12 summarize the membership status of the star as given by Prosser (1992), Makarov (2006) and Mermilliod et al. (2008). The final column identifies the seven stars that are spectroscopic binaries. These seven are members and, it is assumed, that the secondary star is too faint to contribute to the spectrum. They are therefore included with the single stars in Table 1 and in our analyses.
Sixteen stars, originally classified as members by Prosser (1992), have subsequently been identified as non-members, single-lined or double-lined spectroscopic binaires, or close double stars. The nature of these stars and the source of the revised information is listed in Table 2 which has the same format as Table 1. The stars are not rejected outright from our sample. Rather, temperatures, rotational velocities and Li and Fe abundances were determined as for the members and the results are discussed with caveats and questions in Sections 6.2 and 6.3.
Our sample of 70 certain members and 16 stars possibly of questionable status represents the largest selection to date for which lithium abundances are available in $\alpha$ Per.
------ -------------------------- ------------- ------------- --------------------------- ---------------------------- --------------------- --------------------- -------------- ------ ----------- --------- -------
Star [*$T_{\rm eff}(V-K)$*]{} $\log g$ $\xi_t$ [*$T_{\rm eff}(spec)$*]{} [*$T_{\rm eff}(\beta)$*]{} [*v*]{} sin [*i*]{} $W_\lambda$(Li)$^c$ $\log$ N(Li) Notes$^b$
\# (K) cm s$^{-2}$ km s$^{-1}$ (K) (K) km s$^{-1}$ (mÅ) Pr Ma Me
12 6663 4.5 1.5 6953 49 101 3.27 Y Y Y SB$?$
56 5703 4.5 0.8 5600 7 69 2.35 Y Y $\dots$
92 6683 4.5 1.5 23 109 3.31 Y $\dots$ $\dots$
93 5764 4.5 1.5 5880 25 119 2.66 Y $\dots$ $\dots$
94 5703 4.5 1.5 65 225 2.94 Y Y $\dots$
135 6903 4.5 1.5 6714 16 70 3.21 Y Y Y
174 4928 4.5 1.5 5000 5319 12 196 2.25 Y Y Y
270 6742 4.5 1.5 6491 33 96 3.24 Y Y Y SB
299 6036 4.5 1.3 6200 15 106 2.99 Y Y Y
309 6903 4.5 1.5 6448 65 62 3.15 Y Y $\dots$
334 6511 4.5 1.7 6400 7040 19 105 3.23 Y Y Y
338 6606 4.5 1.5 56 111 3.25 Y Y Y
350 5673 4.5 1.5 5893 42 215 2.91 Y Y Y
361 7113 4.5 1.5 6740 30 59 3.18 Y Y Y
421 6761 4.5 1.5 6935 90 58 3.05 Y Y $\dots$
490 7005 4.5 1.5 6821 17 76 3.31 Y Y Y
520 5405 4.5 1.5 5468 91 317 2.87 Y $\dots$ $\dots$
588 6205 4.5 1.5 6532 120 76 2.75 Y Y $\dots$
621 6862 4.5 1.5 6613 28 76 3.25 Y Y Y
632 7007 4.5 1.5 6632 160 76 3.31 Y Y $\dots$
660 6310 4.5 1.5 38 90 2.92 Y$?$ Y Y
709 5873 4.5 1.5 59 187 3.01 Y Y $\dots$
750 6437 4.5 1.5 26 170 3.34 Y Y Y
767 6222 4.5 1.3 6100 10 139 3.27 Y Y Y
799 7244 4.5 1.5 6622 49 70 3.30 Y Y $\dots$
828 5503 4.5 1.5 12 157 2.68 Y Y Y
833 6702 4.5 1.5 6491 27 144 3.46 Y $\dots$ Y
841 6530 4.5 1.5 65 86 3.04 Y Y $\dots$
917 6003 4.5 1.5 5841 40 191 3.11 Y $\dots$ Y
968 6474 4.5 1.5 30 200 3.51 Y Y Y
972 6455 4.5 1.5 6491 87 93 3.08 Y Y $\dots$
1086 5749 4.3 1.4 5900 6122 12 151 2.96 Y Y Y
1101 5540 4.5 1.5 5387 35 377 3.11 Y Y $\dots$
1180 6761 4.5 1.5 6522 45 99 3.32 Y Y Y
1185 5718 4.5 1.2 6000 5669 7 132 2.91 Y Y Y SB
1514 5503 4.3 1.0 5400 8 187 2.94 Y $\dots$ Y
1519 5417 4.5 1.5 50 293 2.81 Y Y $\dots$
1525 5187 4.3 1.6 5300 12 193 2.64 Y Y Y
1528 4757 4.3 1.5 4900 12 70 1.32 Y $\dots$ Y
1532 6419 4.5 1.5 65 148 3.26 Y Y Y SB
1533 4889 4.5 1.5 $<$ 10 143 1.79 Y $\dots$ Y
1537 5008 4.5 1.7 5200 20 83 1.77 Y $\dots$ Y
1543 4695 4.5 1.5 72 539 2.50 Y $\dots$ $\dots$
1551 6400 4.5 1.5 65 92 3.00 Y Y $\dots$
1556 4757 4.5 1.5 110 299 2.11 Y $\dots$ $\dots$
1565 4832 4.5 0.9 4800 10 61 1.27 Y $\dots$ Y
1570 4908 4.3 1.9 5300 7 169 2.26 Y $\dots$ Y
1572 5018 4.5 1.4 5100 10 110 2.03 Y $\dots$ Y
1575 4072 3.8 2.2 4900 11 43 0.17 Y $\dots$ Y SB
1578 4804 4.3 2.0 5200 13 162 1.94 Y $\dots$ $\dots$
1589 5381 4.3 1.0 5900 8 77 2.24 Y $\dots$ Y$?$
1590 5970 4.5 1.5 12 148 3.08 Y Y Y
1591 4822 4.5 1.5 25 232 2.22 Y $\dots$ $\dots$
1593 4785 4.5 1.5 75 365 2.26 Y $\dots$ $\dots$
1597 5232 4.5 1.5 10 196 2.70 Y Y Y SB
1600 4315 4.5 1.5 205 69 0.55 Y $\dots$ $\dots$
1601 4286 4.5 1.5 $<$ 10 60 0.48 Y $\dots$ Y
1604 5598 3.8 1.1 5900 8 45 2.06 Y $\dots$ Y
1606 4889 4.0 2.2 8 158 1.90 Y $\dots$ Y
1607 4948 4.5 1.5 9 129 2.00 Y $\dots$ Y
1610 5187 4.3 1.5 5200 8 205 2.62 Y $\dots$ Y
------ -------------------------- ------------- ------------- --------------------------- ---------------------------- --------------------- --------------------- -------------- ------ ----------- --------- -------
------ -------------------------- ------------- ------------- --------------------------- ---------------------------- --------------------- --------------------- -------------- ---- ----------- --------- --
Star [*$T_{\rm eff}(V-K)$*]{} $\log g$ $\xi_t$ [*$T_{\rm eff}(spec)$*]{} [*$T_{\rm eff}(\beta)$*]{} [*v*]{} sin [*i*]{} $W_\lambda$(Li)$^c$ $\log$ N(Li) Notes$^b$
\# (K) cm s$^{-2}$ km s$^{-1}$ (K) (K) km s$^{-1}$ (mÅ) Pr Ma Me
1612 4405 4.5 1.5 13 10 $-$0.65 Y $\dots$ $\dots$
1614 4524 4.5 1.5 12 120 1.30 Y $\dots$ Y
1617 4712 4.5 1.5 83 469 2.35 Y $\dots$ $\dots$
1618 5133 4.5 1.5 160 307 2.51 Y $\dots$ $\dots$
1621 5405 4.3 1.3 5500 10 159 2.67 Y $\dots$ Y$?$
1669 4557 4.0 1.6 4800 8 54 0.86 Y $\dots$ Y
1697 4767 4.3 1.7 5000 10 78 1.49 Y $\dots$ Y
1731 4228 4.5 0.8 4500 25 56 0.38 Y $\dots$ $\dots$
1735 4651 4.3 2.2 4900 11 24 0.34 Y $\dots$ Y
------ -------------------------- ------------- ------------- --------------------------- ---------------------------- --------------------- --------------------- -------------- ---- ----------- --------- --
$^a$ Pr = Prosser (1992), Ma = Makarov (2006), Me = Mermilliod et al. (2008) $^b$ All SB and SB$?$ designations from Mermilliod et al. (2008) except for \#727 from Prosser (1992) $^c$ For stars with [*v*]{} sin [*i*]{} $>$ 25 km s$^{-1}$, EQWs were not measured but derived from the Li abundance determined from spectrum synthesis.
------ -------------------------- ------------- ------------- ------------------------- ---------------------------- --------------------- --------------------- -------------- ---- ----------- --------- -------------
Star [*$T_{\rm eff}(V-K)$*]{} $\log g$ $\xi_t$ [*$T_{\rm eff}(sp)$*]{} [*$T_{\rm eff}(\beta)$*]{} [*v*]{} sin [*i*]{} $W_\lambda$(Li)$^c$ $\log$ N(Li) Notes$^b$
\# (K) cm s$^{-2}$ km s$^{-1}$ (K) (K) km s$^{-1}$ (mÅ) Pr Ma Me
143 5873 4.0 1.0 5700 6243 10 83 2.62 Y $\dots$ N SB1O
(Pr,Me)
407 5937 28 29 2.01 Y $\dots$ $\dots$
573 6549 4.0 0.6 6600 6782 12 $<$5 1.69 Y $\dots$ N SB
(Me)
715 6903 6522 110 104 3.41: Y Y $\dots$ SB2$?$
(Pr,Ma)
848 6346 4.5 1.3 6500 16 95 3.15: Y Y Y SB2O
(Pr,Ma,Me)
935 6119 56 176 3.13 Y Y $\dots$ Double
(Ma)
1100 5528 4.5 0.8 5800 8 59 2.06 Y $\dots$ N
1181 6205 4.0 1.1 5700 6034 7 58 2.72 Y $\dots$ N
1234 5658 4.5 1.6 6000 10 90 2.56: Y Y Y SB2
(Me)
1538 5613 4.5 1.5 5700 10 193 2.95 Y Y Y Double
1541 5288 4.3 1.5 5400 8 200 2.76 Y $\dots$ Y Double
1598 4938 10 199 2.30 Y $\dots$ Y Double
1602 5381 4.3 1.0 5600 11 141 2.56: Y Y Y SB2
(this work)
1625 5358 4.3 1.8 5700 48 108 2.33: Y $\dots$ $\dots$ SB2
(this work)
1656 5311 4.3 0.8 5600 8 103 2.28: Y $\dots$ Y SB2
(Me)
1713 5243 4.3 0.7 5500 5 18 1.04: Y $\dots$ Y SB2
(Pr,Me)
------ -------------------------- ------------- ------------- ------------------------- ---------------------------- --------------------- --------------------- -------------- ---- ----------- --------- -------------
$^a$ Pr = Prosser (1992), Ma = Makarov (2006), Me = Mermilliod et al. (2008)
$^b$ Classifications as SB from various sources : Mermilliod et al., Makarov, Prosser and our observations. Double denotes a close binary reported by Patience et al. (2002). \#407 is a non-member according to Fresneau (1980) and unusually reddened (Trullols et al. 1989).
$^c$ For stars with [*v*]{} sin [*i*]{} $>$ 25 km s$^{-1}$, EQWs were not measured but derived from the Li abundance determined from spectrum synthesis.
Observations
============
High-resolution spectra were obtained between 1992 and 1994. Observations were made during December 1992 and November 1993 for 30 stars at the 2.7m telescope at the W.J. McDonald Observatory with the Robert G. Tull cross-dispersed echelle spectrograph (Tull et al. 1995) at a resolving power of about 60,000 with exposure times chosen to provide a S/N ratio of 100 or higher. In January 1994, observations were carried out for 21 stars at the 4m telescope at KPNO with the Casségrain echelle spectrograph, the red long-focus camera and the Tex 2048 x 2048 CCD chip to give a 2-pixel resolution of 0.16 Å (R $\sim$ 40,000). Integration times were chosen to provide a S/N ratio close to 150 for most stars and even higher in a few cases.
Data reduction was carried out following standard IRAF procedures.[^3] The frames were trimmed and overscan corrected. Bias frames were combined and subtracted from the raw spectrum. The spectrum was divided by the normalized flat field image to account for the pixel to pixel sensitivity difference of the detector and then corrected for scattered light. No sky subtraction was done as the sky signal was negligible in all cases.
Nineteen and twenty-four echelle orders were extracted respectively from the McDonald and the KPNO data. The wavelength scale for all the orders was derived using the Thorium-Argon spectrum. The wavelength calibrated spectrum was then normalized to a continuum of one.
The measured equivalent widths (EQW) of the Li[i]{} line at 6707.8 Å are given in Tables 1 and 2 for stars with low projected rotational velocities ([*v*]{} sin [*i*]{} $<$ 25 km s$^{-1}$). The EQWs include the contribution of the Fe I blend at 6707.435 Å. The contribution of the Fe I blend was removed by the program MOOG (Sneden 1973) during the derivation of the Li abundance. In the slow rotators ([*v*]{} sin [*i*]{} $<$ 25 km s$^{-1}$), the uncertainty in the EQW, largely caused by the placement of the continuum, was 2-3 mÅ at $\sim$ 15 mÅ, 6 mÅ at $\sim$ 130 mÅ, and 10 mÅ at $\sim$ 200 mÅ. In the spectra of the more rapidly rotating stars in which lines were still measurable, the EQW uncertainty was estimated to be as large as $\sim$ 15 mÅ. EQWs were not measured in these stars, rather Li abundances were determined by spectral synthesis, and the EQWs listed in Tables 1 and 2 for stars with [*v*]{} sin [*i*]{} $>$ 25 km s$^{-1}$ were calculated from the derived abundance using MOOG (Sneden 1973).
The analysis of the 6707 Å line was done for all the stars with spectrum synthesis fits to the observed spectrum. For the 36 slowly rotating stars for which spectroscopic analysis was possible, the Li abundance was determined in addition from the Li EQW. The match between the two measurements of Li abundance was in excellent agreement.
Stellar Parameters
==================
The Li abundance determined from the 6707 Å feature, is primarily sensitive to the adopted effective temperature $T_{\rm eff}$. An error of $\pm$ 200 K in $T_{\rm eff}$, results in an uncertainty in $\log$ N(Li) between $\pm$ 0.28 to $\pm$ 0.14 over the 4500 K to 6500 K temperature range. Thus, we devoted considerable effort to a determination of $T_{\rm eff}$. The Li abundance is quite insensitive to the adopted surface gravity; a variation in $\log g$ of $\pm$0.5 dex results in a change in Li abundance by less than $\pm0.02$ dex. The adopted microturbulence has a small influence on Li when the 6707 Å feature is strong.
Effective Temperature
---------------------
The effective temperature is derived from photometry, primarily the $(V-K)$ index, and checked by use of the Strömgren $\beta$ index and spectroscopy.
### Photometry
Our principal photometric indicator of effective temperature is the (V-K) colour index which is available for all the stars. All of the observed stars have a K$_{\rm s}$ magnitude in the 2MASS catalogue.[^4] The K$_{\rm s}$ were transformed to Johnson K magnitudes by the Koornneef transformations (Carpenter 2001). The V magnitudes were taken from Prosser (1992).
The literature contains various estimates of the reddening affecting the cluster. Several authors refer to a variable reddening across the cluster. Cluster members are slightly reddened but there is little solid evidence that the reddening is significantly different from star-to star. BLS adopted E(B$-$V) = 0.08 (Mitchell 1960) for all their stars and remarked that Crawford & Barnes (1974) suggested a range from 0.04 to 0.21. BLS note that the extremities of the range correspond to effective temperatures lower by 100 K and hotter by 450 K. Thus, the larger reddenings are a concern in the search for the origin of a scatter in Li abundances.
Inspection of Crawford & Barnes (1974) shows, however, little evidence for a variation in reddening. In their Table III, they list measurements of $E(b-y)$ for 21 F-type stars. Fifteen stars are members and six are non-members according to Makarov (2006). (Twelve of the 15 members are in Table 1.) The mean $E(b-y)$ for the 15 is 0.054$\pm0.017$ with extremes of 0.032 and 0.091. In contrast, larger reddening is seen among non-members with the six non-members exhibiting a range in $E(b-y)$ from 0.023 to 0.148. Crawford & Barnes did note that 18 A-type and 31 B-type cluster members gave higher and similar reddening: fourteen A-type stars, members according to Makarov (2006), give a mean $E(b-y)$ = 0.089$\pm0.036$ with extreme values of 0.038 and 0.139. The factor of 1.7 between the mean values for the F- and A-type stars points to an issue with the calibration. Possibly, the larger standard error in the A stars may indicate non-uniform colors resulting perhaps from diffusion or metallicity effects. Trullols et al. (1989) provide $E(b-y)$ for 12 F star members: the mean $E(b-y)$ is 0.065$\pm$0.012 where the standard error again indicates little if no variation from star-to-star. Peña & Sareyan (2006) provide Strömgren photometry for cluster stars from a combination of their own measurements and published values and obtained a mean reddening from 169 stars of $E(b-y)$ = 0.073$\pm0.038$. However, for the 15 F stars in common, their reddening ($E(b-y)$ = 0.086$\pm0.029$) differs from that of Crawford & Barnes, suggesting calibration differences.
Crawford (1975) used the available uvbyH$\beta$ photometry for bright stars and cluster members to calibrate H$\beta$ in terms of intrinsic colour $(b-y)$ and other indices, applicable for $\beta$ between 2.59 and 2.72. Using this calibration, we made fresh estimates of $E(b-y)$ for 40 stars with the available Stromgren photometry (from WEBDA which essentially includes the observations of Crawford & Barnes (1974) and Trullols et al. (1989)) in the above range of $\beta$ taking care that the sample contained no binaries or possible non-members. We found that $E(b-y)$ ranged from 0.02 to 0.12, very similar to previous studies, with an average $E(b-y)$ of 0.075 and a standard deviation of the measurements (standard error) of $\pm$0.04. For the normal interstellar reddening law, $A_V$ = 0.32, this translates to $E(V-K)$ = 0.284 $\pm$ 0.15. The standard error may be used as one estimate of the reddening uncertainty for each star.
Prosser (1992) derived $E(V-I)$ for about 75 M cluster dwarfs from a somewhat unusual process. Low-dispersion spectra provided spectral types which with a color-spectra type relation gave the intrinsic $(V-I)$ color of a star. Comparison of intrinsic and observed $(V-I)$ gave a star’s reddening. The mean $E(V-I)$ $\simeq 0.18$ corresponds to $E(b-y)$ $\simeq 0.08$ and $E(V-K)$ $\simeq 0.3$, values consistent with other reddening measures from traditional techniques applied to earlier spectral types. Given that the reddenings are described by Prosser as ‘preliminary values’, one may attach little weight to his suggestion that the reddening is not uniform for cluster members.
The intrinsic $(V-K)$ colour was calculated assuming thus an average interstellar reddening of $E(V-K)$ = 0.284. The photometric calibrations $(V-K)$ - $T_{\rm eff}$ of Alonso et al. (1996) given below were used to derive the $(V-K)$ effective temperature $T_{\rm eff}(V-K)$ for all the stars. This procedure was applied to all stars including those previously analysed by BLS.
$$\theta_{\rm eff} = 0.555 + 0.195(V-K) + 0.013(V-K)^2$$
for 0.4 $\le$ $(V-K)$ $\le$ 1.6 and
$$\theta_{\rm eff} = 0.566 + 0.217(V-K) - 0.003(V-K)^2$$
for 1.6 $\le$ $(V-K)$ $\le$ 4.1
where $\theta_{\rm eff}$ = 5040/$T_{\rm eff}$. We denote this photometric temperature by $T_{\rm eff}(V-K)$ (see Table 1 for cluster members). The photometric error in K is 0.02 mag.[^5] and in V it is 0.058 mag. (Prosser 1992), yielding a standard deviation in $(V-K)$ of 0.06 mag. An error in $(V-K)$ of 0.06 results in an error of 125K at 6600 K, 50K at 4500K and 75K at 5500K. We caution that the presence of a cool companion may increase the $(V-K)$ color of the primary component of a binary and therefore the temperature derived for these stars (Table 2) may be systematically low. We note here that the standard deviation of 0.15 in $E(V-K)$ derived from the reddening calculation is substantially higher than the uncertainty in the photometric measurement determined above and translates to temperature uncertainties of 325 K at 6600 K, 180 K at 5500 K, and 125 K at 5500 K.
Other colour indices might be considered as thermometers. Several previous studies of $\alpha$ Per (and other clusters) have employed $(B-V)$ for which measurements are available for all but five of our stars. However, the V vs. $(B-V)$ colour-magnitude diagrams for young clusters are fundamentally different from the V vs. $(V-I)$ plots when overlaid by theoretical isochrones. The best-fitting isochrone in the V vs. $(B-V)$ diagram for young clusters such as $\alpha$ Per follows the observed main sequence down to the late-K stars which lie conspicuously to the left of the theoretical main sequence; they are consistently fainter and bluer. This is seen in the Pleiades and $\alpha$ Per (S.V. Mallik, private communication). Stauffer et al. (2003) first pointed out this blue anomaly in the Pleiades and suggested it may be due to a flux contribution (larger in B than in V) not represented by the model atmospheres. This so-called $(B-V)$ anomaly is predominant in the younger clusters but is absent in clusters as old as Praesepe: $\alpha$ Per is about 0.05 Gyr to Praesepe’s 0.6 Gyr. If this interpretation rather than a more deep-seated deficiency in the models providing the isochrones is correct, the anomaly is likely related to stellar surface activity that decays as stars age. This has two obvious consequences for deriving and interpreting Li abundances. First, $(B-V)$ may be a poorer temperature indicator for the cooler stars than $(V-K)$; the colour-temperature calibration is likely dependent on a star’s age but may also vary from star-to-star with changes in stellar activity. Then, these dependencies may provide a star-to-star scatter in Li abundance.
### The Strömgren $\beta$ index
As a reddening-free index, the Strömgren $\beta$ index is a useful measure of effective temperature for stars hotter than about 5000 K. Measures of $\beta$ are taken from Crawford & Barnes (1974) and Trullols et al. (1989). The $\beta$ vs $T_{\rm eff}$ calibration is taken from Alonso et al. (1986 - see also Castelli & Kurucz 2006). $T_{\rm eff}(\beta)$ is listed in column 6 of Table 1 for 24 cluster members and the comparison $T_{\rm eff}(\beta)$ vs $T_{\rm eff}(V-K)$ is shown in Figure 1. $T_{\rm eff}(\beta)$ is also listed for four stars of questionable status in Table 2 and included in Figure 1.
While there is good agreement between $(V-K)$ and H$\beta$ temperatures at the cool end of the useful range of the $\beta$ index, there is a surprisingly larger scatter, at the warm end, with temperature differences as large as 500K for the same star from the two calibrations. The difference is also surprising because the two calibrations come from a common paper. We have no leads on whether this difference is due to calibration issues or photometry errors but we surmise it is unlikely to be caused by variable reddening as it is confined to a small temperature range. The non-members and SB2s lie within the scatter defined by the members.
{height="9cm" width="8cm"}
### Spectroscopy
Our spectroscopic temperature is based on the usual condition that the Fe[i]{} lines in the observed spectra return the same Fe abundance independent of a line’s lower excitation potential. The McDonald spectra provide about 30 Fe[i]{} lines spanning about 4 eV in the lower excitation potential. The KPNO spectra with their greater wavelength coverage yield about 130 Fe[i]{} lines. These numbers pertain to slowly rotating stars ([*v*]{} sin [*i*]{} $\le$ 20 km s$^{-1}$); more rapidly rotating stars have broader lines that lead to blending and a difficulty in measuring EQWs accurately, especially of weak lines. The $T_{\rm eff}$ determination has to be made simultaneously with that for the microturbulence $\xi_t$. For this exercise in determining $T_{\rm eff}$ and $\xi_t$, we used astrophysical $gf$-values for Fe[i]{} and Fe[ii]{} lines. The $gf$-values were determined using MOOG (Sneden 1973) with measurements of Fe[i]{} and Fe[ii]{} equivalent widths from the high resolution digital solar atlas (Delbouille et al. 1990), the Kurucz solar model atmosphere with no convective overshoot (Castelli, Gratton & Kurucz 1997) and requiring the lines to yield Fe/H=7.50 at $\xi_t$ = 0.8 km s$^{-1}$.
We derived spectroscopic temperatures for 24 cluster members including eight stars observed by BLS for which we were able to retrieve their spectra (Table 1), and for 12 binaries, doubles or non-members (Table 2). The microturbulence and spectroscopic temperatures are listed in columns 4 and 5 respectively. A 100 K change in effective temperature resulted in a significant non-zero slope of Fe I vs. lower excitation potential to allow us to constrain temperatures to $\pm 100 K$. Errors in equivalent width measurement, gf-values and microturbulence would result in random and systematic temperature errors and we feel that $\pm$ 200 K conservatively constrains the error in our derived $T_{\rm eff}(spec)$. The $\xi_t$ is determined to about $\pm$ 0.1 km s$^{-1}$.
A comparison of $T_{\rm eff}(V-K)$ and the spectroscopic temperature $T_{\rm eff}(spec)$ is presented in Figure 2. On average, $T_{\rm eff}(V-K)$ is cooler than $T_{\rm eff}(spec)$ by about 250 K. The temperature difference appears to vanish for the hotter stars, say $T > 6000$ K. This level of agreement is consistent with the estimated uncertainty from the analysis of the Fe[i]{} lines and lends support to the assertion that reddening is not very variable across the cluster. The visual doubles, the SB2s and three of the four non-member stars lie within the scatter defined by the cluster members. Only \#1181 has a much cooler $T_{\rm eff}(spec)$ compared to $T_{\rm eff}(V-K)$.
![Comparison of the effective temperatures derived from the $(V-K)$ index and the Fe[i]{} lines ($T_{\rm eff}(spec)$) The line corresponds to perfect correspondence between the two measurements. The symbols are described in the key.](apervkspmixtemp.ps){height="9cm" width="8cm"}
Surface gravity
---------------
With the inclusion of Fe[ii]{} lines in the spectroscopic analysis, it is possible to determine the surface gravity $\log g$. Spectra of 18 stars provide an adequate number of eight to ten Fe[ii]{} lines. A $\log g$ determination requires the same Fe abundance from Fe[i]{} and Fe[ii]{} lines and this is possible to an accuracy of $\pm$ 0.25 dex. For the stars without a spectroscopically determined $\log g$, we adopt the $\log g$ = 4.5 for the $T_{\rm eff}$ determination from the Fe[i]{} lines. This value was adopted for all stars without a spectroscopic determination of surface gravity.
Microturbulence
---------------
The microturbulence is taken either from the analysis of the Fe[i]{} lines or a value of 1.5 km s$^{-1}$ was assumed.
Abundance analysis
==================
Lithium Abundances
------------------
The abundance analysis from which we extract the Li abundance takes the standard form.
Model atmospheres were generated in 100 K intervals in temperature and 0.1 dex intervals in gravity using the program ATLAS9, written and supplied by R. L. Kurucz. Standard solar opacity distibution functions were used with overshoot turned off (see Castelli, Gratton, & Kurucz 1997). The appropriate model was chosen for each star according to the stellar parameters listed in Tables 1 and 2.
The line analysis program MOOG (Sneden 1973) was used to convert EQWs of the 6707 Å Li[i]{} resonance doublet to an abundance. Throughout the assumption of local thermodynamic equilibrium (LTE) is adopted. The $gf$-values and wavelengths of the fine- and hyperfine-structure components of the Li[i]{} feature were taken from Andersen, Gustafsson & Lambert (1984). The Li abundance was chosen by the best fit of a synthetic spectrum to a region around the 6707 Å feature with the line list adopted by BLS. It is most unlikely there is any $^6$Li in stars where $^7$Li is even slightly depleted. Therefore, $^6$Li was included in the line list only for stars with $log$ N(Li) $>$ 3.0.
Lithium abundances were computed for the model parameters listed in Tables 1 and 2. An error of $\pm$100K in $T_{\rm eff}$, $\pm$0.25 dex in $\log g$, $\pm$0.1 kms$^{-1}$ in $\xi_t$ and 5 mÅ in Li I EQW results in Li abundance errors of $\pm$0.1, $\pm$0.01, $\pm$0.00 and $\pm$0.09 respectively. As these estimates show, the two principal sources of uncertainty affecting the derived lithium abundances arise from the effective temperature and the measured equivalent width. The effect of a 200 K spread in effective temperature is shown in Figure 3 by the shaded area at the bottom of the figure. At the lowest temperatures where the stars of the same effective temperature can show lithium lines of quite different strengths, the uncertainty in measurement of the equivalent width may have a larger effect on the derived abundance than the temperature uncertainty, especially for those few stars where the lithium line is weak. The reddening uncertainty estimated from H$\beta$ increases the temperature error over 200K only at the hottest temperatures; the uncertainty in $E(V-K)$ of 0.15 translates to a temperature uncertainty of 325 K at 6600 K. This uncertainty would merely increase the Li abundance uncertainty at 6600 K to roughly the same magnitude as in the cooler stars (Figure 3), and would not affect the discussion on Li abundance dispersion that follows in Section 6. In the temperature and gravity range of our sample, non-LTE corrections to the Li abundance are estimated to be small (Carlsson et al. 1994); non-LTE corrections would lower the LTE abundances by 0.019 at the cool end of our sample and by 0.009 dex at the hot end. Again, not incorporating these relatively small corrections would not affect our discussion on the dispersion in Li abundance that follows.
Li abundances for the single-lined binaries in Table 1 and the non-members and double stars in Table 2 were determined as for the single stars. The Li abundances of non-members should have the same accuracy as the remainder of our sample and the Li abundance errors on the doubles is unknown. However if cool companion lowers the temperature estimated for single-lined and double-lined binaries, those Li abundances will be proportionately lowered. In addition, double-lined binaries may have weaker Fe I and Li I lines due to continuum dilution. We have therefore marked the Li abundances of the SB2s as uncertain in Table 2. If anything, the Li abundances of these stars are likely to be larger than our estimates. A stronger Li I line may result if a neighboring feature line from the companion falls on the Li I line but we have measured the wavelength separation of the two components and are certain that the Li I feature is not contaminated in any of our SB2s.
Iron abundance
--------------
Iron abundances were determined for 25 cluster members (Table 3). The typical measurement uncertainty in the EQW of a 40-60 mÅ Fe I line is $\pm$5 mÅ. An error of $\pm$100K in $T_{\rm eff}$, $\pm$0.25 dex in $\log g$, $\pm$0.1 kms$^{-1}$ in $\xi_t$ and $\pm$5 mÅ in Fe I EQW results in Fe abundance errors of $\pm$0.07, $\pm$0.02, $\pm$0.02 and $\pm$ 0.05 respectively. When these uncertainties are combined, the resulting error in the Fe abundance is $\pm$ 0.09.
The mean Fe abundance is 7.40$\pm0.08$ dex where this standard error is comparable to the estimate of the precision of a single determination. There may be a slight decrease in the derived Fe abundance with decreasing temperature; stars with $T_{\rm eff} > 5500$ K give a mean that is 0.09 dex higher than stars with $T_{\rm eff} < 5500$ K. A similar suggestion of a temperature dependence was made by BLS. Since our Fe abundance is based on astrophysical $gf$-values for Fe[i]{} and Fe[ii]{} lines and is derived using the solar abundance of $\log$N(Fe) = 7.50, the mean Fe abundance may be quoted as \[Fe/H\] = $-0.10$, with \[Fe/H\] = $-0.04$ for $T_{\rm eff} > 5500$ K and \[Fe/H\] = $-0.13$ for stars with $T_{\rm eff} < 5500$ K. BLS obtained a mean Fe abundance about 0.13 dex higher with astrophysical $gf$-values calculated from the empirical Holweger-Müller model (1974) and a microturbulence of 1.2 km s$^{-1}$. Our result is in good agreement with Boesgaard & Friel’s (1990) spectroscopic determination by an essentially equivalent technique including the use of the Kurucz grid, though with fewer (15) Fe[i]{} lines. They obtained \[Fe/H\]$ = -0.054\pm0.046$ from six stars with $T_{\rm eff}$ from 6415 K to 7285 K; our result from four stars hotter than 6000 K is \[Fe/H\] $ = -0.04\pm0.08$.
Also listed in Table 3 are the Fe abundance of the non-members, visual doubles and binaries from Table 2 for which spectroscopic analysis was possible. The mean Fe abundance of the four non-members is \[Fe/H\]=$-$0.13$\pm$0.15, of the two double stars is \[Fe/H\] = $-$0.07$\pm$0.04, and of the six double-lined spectroscopic binaries is \[Fe/H\]=$-$0.14$\pm$0.21. The mean Fe abundances are not very different from that of the cluster members; the standard errors are slightly larger than for the cluster mean.
-------------------------- --------------- ------------- ------------- ---------------- -----------
Star Model [*\[Fe/H\]*]{} Notes$^a$
\# $T_{\rm eff}$ $\log g$ $\xi_t$
(K) cm s$^{-2}$ km s$^{-1}$
Cluster Members
56 5600 4.5 0.8 + 0.06
174 5000 4.3 2.0 $-$ 0.25
299 6200 4.5 1.3 0.00
334 6400 4.5 1.7 $-$ 0.18
767 6100 4.5 1.3 + 0.03
1086 5900 4.3 1.4 $-$ 0.12
1185 6000 4.5 1.2 $-$ 0.03 SB
1514 5400 4.3 1.0 $-$ 0.18
1525 5300 4.3 1.6 $-$ 0.15
1528 4900 4.3 1.5 $-$ 0.03
1537 5200 4.5 1.7 $-$ 0.08
1538 5700 4.5 1.5 $-$ 0.03
1565 4800 4.5 0.9 $-$ 0.02
1570 5300 4.3 1.9 $-$ 0.07
1572 5100 4.5 1.4 $-$ 0.05
1575 4900 3.8 2.2 $-$ 0.26 SB
1578 5200 4.3 2.0 $-$ 0.06
1604 5900 3.8 1.1 $-$ 0.09
1606 4800 4.0 2.2 $-$ 0.14
1610 5200 4.3 1.5 $-$ 0.15
1621 5500 4.3 1.3 $-$ 0.04
1669 4800 4.0 1.6 $-$ 0.19
1697 5000 4.3 1.7 $-$ 0.14
1731 4500 4.5 0.8 $-$ 0.13
1735 4900 4.3 2.2 $-$ 0.13
Non-members and Binaries
143 5700 4.0 1.0 $-$ 0.19 NM, SB1O
573 6600 4.0 0.6 $-$ 0.23 NM, SB
848 6500 4.5 1.3 $-$ 0.10 SB2O
1100 5800 4.5 0.8 + 0.09 NM
1181 5700 4.0 1.1 $-$ 0.19 NM
1234 6000 4.5 1.6 + 0.18 SB2
1538 5700 4.5 1.5 $-$ 0.04 Double
1541 5400 4.3 1.5 $-$ 0.10 Double
1602 5600 4.3 1.0 $-$ 0.26 SB2
1625 5700 4.3 1.8 $-$ 0.33 SB2
1656 5600 4.3 0.8 $-$ 0.33 SB2
1713 5500 4.3 0.7 + 0.02 SB2
-------------------------- --------------- ------------- ------------- ---------------- -----------
$^a$ Classifications as in Tables 1 and 2. Here NM denotes a non-member.
The Li abundance vs. Temperature relation
=========================================
The general nature of the relation between lithium abundance and effective temperature was discussed previously by BLS and Randich et al. (1998).
In Figure 3, we show the Li vs $T_{\rm eff}(V-K)$ relation for the 70 stars in Table 1 where the symbol’s size reflects [*v*]{} sin [*i*]{} as depicted in the legend on the figure. The shaded region at the bottom of the figure displays the effect of a correction to $T_{\rm eff}$ of 200 K across the range from 6400–4500 K.
![The effective temperature vs lithium abundance relation for $\alpha$ Per. The projected rotational velocity ([*v*]{} sin [*i*]{}) of the stars is represented as in the legend. The shaded strip at the bottom of the figure shows the Li abundance spread resulting from an effective temperature uncertainty of 200 K. Several stars are labelled by the membership number for easy reference.](figb.eps){height="9cm" width="12cm"}
The Hot Stars
-------------
--------------------- ----------------- -------------
$T_{\rm eff}$ range Mean $N_{stars}$
(K) $\log$ N(Li)
Hot Stars
$>$ 7000 3.28 $\pm$ 0.06 4
6750-7000 3.20 $\pm$ 0.10 5
6500-6750 3.25 $\pm$ 0.12 7
6250-6500 3.19 $\pm$ 0.22 6
Middle Third
6000-6250 3.09 $\pm$ 0.13 4
5750-6000 2.97 $\pm$ 0.23 3
5500-5750 2.72 $\pm$ 0.34 9
--------------------- ----------------- -------------
The Li vs $T_{\rm eff}(V-K)$ relation asymptotically approaches a constant abundance at the high temperature end. In analysing this approach, we calculate mean abundances in four temperature bins for stars hotter than 6250 K (Table 4). Each bin has roughly the same number of stars and the mean Li abundance in the four bins is essentially the same within the errors. In the three hottest bins, the dispersion within each bin can be easily accounted for by a combination of temperature errors ($\pm 200$K corresponds to $\pm 0.14$ dex), S/N of the spectra, and small reddening variations. Whether the slightly larger dispersion is the coolest of the four bins is significant, cannot be determined from our relatively small sample size. Within our errors, there appears to be no dispersion in the Li distribution or significant change in mean Li value of these hottest stars.
Within these stars, there is an indication from Figure 3 that the most massive rapidly rotating stars (at $T_{\rm eff}(V-K)$ $> 6400$ K) have a slightly lower Li abundance than the slow rotators. Taken as a whole, the results in Figure 3 may suggest that the rapidly rotating stars provide a relation with a shallower slope for $T_{\rm eff} > 5500$ K than the slowly rotating stars.
The Li abundance in the hottest stars is equal to the meteoritic value ($\log$N(Li) = 3.25$\pm$0.06 according to Grevesse et al. 2007). It is this value that has often been taken as a fair representation of the initial value for young open clusters like $\alpha$ Per. Some authors also quote a very similar Li abundance derived from T Tauri stars (see, for example, Magazzu et al. 1992 and Martín et al. 1994). These two data points suggest that the local value of the Galactic Li abundance has changed little in the last 4.5 Gyrs.
The Middle Third
----------------
Three temperature bins define the middle third of the sample between 5500 K and 6250 K (Table 4). The mean Li trend declines by 0.5 dex in this range. The dispersion in lithium appears to be larger than can be accounted for by the uncertainties in the stellar parameters and the S/N of our spectra. In order to understand this dispersion, we examined four cluster members, \#1589, \#1604, \#56 and \#93, with $T_{\rm eff}$ between 5400 K and 5750 K. These appear to be outliers to what would otherwise be a fairly narrow mean Li trend similar to that seen in the hotter stars; in the absence of these stars, the decline in the mean Li trend between 6250 K and 5500 K would be 0.3 dex and the dispersion in the coolest bin be only $\pm0.13$.
It is worth noting that these four stars have much lower Li EQWs compared to other stars in the same temperature range. We therefore begin by examining three possible explanations for these outliers: (i) their assigned effective temperature is in error, (ii) the stars are non-members that have experienced normal Li depletion for their age, (iii) the dispersion in Li is not real but a reflection of differences in chromospheric activity levels which affect the formation of the Li I line and thereby the the equivalent widths of line, and (iv) an unusual amount of Li depletion has occurred in these cluster members. We comment on each of these in turn.
Errors in the estimated temperatures appear to be the least likely cause of the outlier stars. An increase in effective temperature would increase the estimated Li abundance of the outlier, but as the mean Li trend increases with increasing temperature, the required temperature increase is larger than that indicated simply by the temperature difference between the outlier abundance and the mean trend at that temperature. Consider the case of \#1589 which is about 0.4 dex below the mean relation. A 500 K increase in effective temperature eliminates this deficit but at the new temperature of 5900 K the star remains about 0.3 dex below the mean relation. A similar problem arises if the effective temperature is lowered. The temperature change required to meet the mean Li trend is even larger in \#56 and it cannot be reconciled with the errors we have derived for our estimated temperatures. For example, the estimated temperatures of \#56, $T_{\rm eff}$(spec) = 5600 K and $T_{\rm eff}(V-K)$ = 5703 K, are in good agreement and we see no reason to consider them to be in error by 700 K or larger. The other outliers would require similar and unacceptably large increases or decreases in temperature to put them on the mean Li trend. An added constraint against such a large increase in temperature is the measured Fe abundance. The 700 K increase in effective temperature required for \#56 would increase \[Fe/H\] from the measured value of +0.06 to an extraordinarily high value of +0.42. Similarly, \#1604 has a measured metallicity of \[Fe/H\]=$-$0.09, consistent with the cluster mean, and the even larger increase in effective temperature increase would result in an unbelievable Fe abundance. An explanation for the quartet in terms of an error in their effective temperatures is therefore not credible.
Possibly, these outliers are not in fact cluster members. In the case of \#56, Prosser (1992) assigned it a questionable status as a member on the basis of its radial velocity but full status on the basis of proper motion. If \#56 is an interloper star with normal Li depletion for its age, it should be roughly the age of the Hyades cluster (600 Myr) (Thorburn et al. 1993, Boesgaard & Tripicco 1986, Boesgaard & Budge 1988). Similarly, at roughly the same temperature, \#1604 with a slightly lower Li abundance, would be a somewhat older star. However, neither \#56 nor \#1604 could be as old as NGC 752 (2.4 Gyr) or M67 (4.5 Gyr) because by that age solar-temperature stars have Li abundances of logN(Li)= 1.5 or lower (see Balachandran 1995 and references therein). The likelihood of Hyades-age interlopers in the field of view of the $\alpha$ Per cluster and at the distance of the $\alpha$ Per cluster is small. Although Prosser (1992) assigned cluster membership to \#93 without a radial velocity measurement, the moderate rotational velocity of the star ([*v*]{} sin [*i*]{} = 25 km s$^{-1}$) increases the likelihood that it is a young star and therefore a cluster member; G stars are observed to have spun down by the age of the Pleiades (Stauffer et al. 1984). As noted in Table 1, Mermilliod et al. (2008) questioned the cluster membership of \#1589. The relatively high Li abundance (log N(Li)=2.24) relative to field stars at the same temperature suggests that the star is young; the abundance of Li is of order log N(Li)=1.0 at 5300 K even in a cluster as young as the Hyades. Therefore, even if the quartet are rejected as members on this flimsy evidence, it is obviously no simple matter to account for their Li abundance as field stars.
The effect of chromospheric activity, in particular surface inhomogenieties in the form of spots and plages, on the formation of the Li I line, and the subsequent effect on the equivalent width of the line has been the focus of several studies (Randich 2001; Hünsch et al. 2004, King & Schuler 2004, Xiong & Deng 2006, King et al. 2010). Typically the K I resonance line, that is formed in the same part of the atmosphere as Li I, is measured for comparison. Although a spread in K I equivalent widths has been observed in stars of the same temperature in the Pleiades (Jeffries 1999) and IC 2602 (Randich 2001), the authors state that while there is a need to understand K I differences, there is no conclusive evidence that the spread in Li abundances in these young clusters can be attributed to differences in chromospheric activity alone. In a recent study of the high resolution spectra of 17 cool Pleiades dwarfs, King et al. (2010) found that the Li I line strengths had a larger scatter than the K I $\lambda$ 7699 Å line strengths. They concluded that there must be a true abundance component to the Pleiades Li dispersion and suggested that it may be due to differences in pre-MS Li burning caused by the effects of surface activity on stellar structure. Here we add a few nuggets to that discussion. Our KPNO spectra contain the 7699 Å K I resonance line at the edge of one of the echelle orders. We were able to measure this K I feature in 14 stars. The data show the expected increase in EQW with decreasing temperature but the sparsity of the data preclude a detailed analysis. In addition to the two outliers, \#1604 and \#1589, we were able to measure the K I EQWs of two normal stars at the same temperature \#1086 and \#1185. The data are shown in Table 5. There are two findings of relevance. First, while the Li I EQW of \#1604 is a factor of three smaller than that of \#1086 and \#1185, the K I EQWs of all three stars are within about 15 percent of each other. Second, comparing the K I lines in the two stars with low Li, \#1589 and \#1604, we find the ratio of their KI EQWs is 1.6, perhaps reflecting the lower $T_{\rm eff}(V-K)$ of \#1589. The Li I EQW ratio of the two stars is 1.7 and mirrors the K I EQW ratio. Thus, we are able to discern no reason to attribute the low Li abundances to \#1589 and \#1604 to the effects of chromospheric activity.
Since a convincing case cannot be made for either explanation (i), (ii), or (iii), the so-called outliers must be accepted as cluster members with an above-average depletion of lithium. With their inclusion as members, we conclude that a dispersion in lithium is clearly present between 5500 K and 5750 K and, as will be discussed in the next sub-section, this dispersion persists at cooler temperatures.
------------ ------------------------------ ------------------------------- ----------------------- ------------------------ --------------
Star \# [*$T_{\rm eff}(V-K)$*]{} (K) [*$T_{\rm eff}(spec)$*]{} (K) $W_\lambda$(K I) (mÅ) $W_\lambda$(Li I) (mÅ) $\log$ N(Li)
Warm Stars
1086 5750 5900 192 151 2.96
1185 5718 6000 194 132 2.91
1589 5381 5900 350 77 2.24
1604 5600 5900 221 45 2.06
Cool Stars
174 4928 5000 308 196 2.25
1697 4767 5000 391 78 1.49
------------ ------------------------------ ------------------------------- ----------------------- ------------------------ --------------
The Cool Stars
--------------
Stars cooler than about 5500 K appear to fall in a widening band of declining lithium with decreasing temperature (Figure 3). There seems to be a lower envelope, defined by low [*v*]{} sin [*i*]{} stars, running from a Li abundance of about $\log$ N(Li) = 2.5 at 5500 K to $\log$ N(Li) = $-$0.4 at 4500 K, and an upper envelope running through high [*v*]{} sin [*i*]{} stars K with a Li abundance of around $\log$ N(Li) = 3.0 at 5500 K and then falling to a Li abundance of $\log$ N(Li) = 0.0 at 4200 K. The width of the band at temperatures less than 4700 K is about 1.5 dex, a width resembling that for the Pleiades, a cluster about 20 to 30 Myr older than $\alpha$ Per (Soderblom et al. 1993; Sestito & Randich 2005)). Xiong & Deng (2005) in a discussion on the Li abundances provided by the BLS sample drew attention to a scatter appearing around the colour index (V$-$I$_c$) = 1.03 or about 4700 K. This was about the cool end of BLS’s sample. Our expanded sample shows that the scatter begins at a somewhat warmer temperature around 5600 K and extends to cooler temperatures.
The Li distribution band is sketched in Figure 4. Accepting the outliers discussed in the previous section as members, the Li distribution band in the cooler stars can be extended to warmer temperatures with the upper and lower bands asymptoting to the Li plateau value at 7000 K. The dispersion in Li may begin at 6500 K, though additional stars are required to define this spread, and gradually widen in the cooler stars.
{height="9cm" width="12cm"}
Randich et al. (1998) provided Li abundances for 18 X-ray selected members of the cluster. An additional five stars were analysed but declared to be non-members. The spectra were at a resolution of 1Å but lines blended with the Li[i]{} doublet were taken into account in the analysis. The adopted $T_{\rm eff}$ scale is in good agreement with ours. A comparison with their and our temperatures for the BLS sample indicates a mean difference (Us$-$Them) of only 6$\pm67$K when two wildly discrepant stars are excluded. This suggests that we may add these X-ray selected stars to our sample. Furthermore, we note that one star - \#1601 (AP101) - is a common star: we find $\log$ N(Li) = 0.48 and Randich et al. give 0.68, an unimportant difference given the spread at the 4300 K temperature of the star. In Figure 4, abundances from Randich et al. are included along with ours and lines drawn to represent the possible upper and lower envelopes to the Li abundance variation with effective temperature. These additional stars at $T_{\rm eff} < 5000$ K tend to populate the upper half of the band between our suggested upper and lower envelopes. Unfortunately, the new points provide few high [*v*]{} sin [*i*]{} objects.
Scatter at temperatures below about 5500 K cannot be attributed to the standard sources of uncertainty (incorrect effective temperature, uncertainties in measuring the 6707 Å feature, contamination of the sample by nonmembers, etc.). Several ideas have been suggested linking the Li abundance scatter at least in part to the failure of classical model atmospheres (as used here) to represent the real atmospheres of these young late-type dwarfs. In Section 6.2 we discussed our K I data for four stars around 5900 K. Our data include K I equivalent widths for two additional stars at 5000 K, \#174 and \#1697, both bonafide cluster members (Table 5). These data also do not provide any support for a link between high Li abundance and chromospheric activity. Rather the larger K I EQW corresponds to the star with the smaller Li abundance.
Doubles, Binaries and Non-members
---------------------------------
In Figure 4, the stars designated as double stars and double-lined spectroscopic binaries in Table 2 are shown by symbols of different colors as indicated in the legend accompanying the figure.
The four double stars with separations of 0.5 arc seconds or less lie well within the Li distribution band of the normal stars. The spectra of these stars appear to not have been significantly contaminated by the presence of the nearby star and future analyses may simply include them as member stars.
With the exception of \#1713, the double-lined spectroscopic binaries (green triangles) also lie well within the Li band of the cluster members. Their measured Li abundances may be regarded as lower limits to the value that would be obtained if the continuum contamination of the secondary was properly accouted for. Any further interpretation of their abundances would require a rigorous analysis that takes into account the continuum and line spectrum of both stars.
Surprisingly three of the four stars identified as non-members by Mermilliod et al. (2008): \#143, \#1181 and \#1100, have Li abundances that are entirely compatible with the mean cluster trend. The star \#1181 is coincident with \#588, a rapidly rotating cluster member, \#143 lies on the lower Li envelope of the cluster, and \#1100 is in the proximity of \#1604, the outlier star that we found no reason to exclude from the sample. The three stars were deemed to be non-members by Mermilliod et al. (2008) on the basis of their radial velocity and proper motion alone; all three lie on the cluster’s color-magnitude diagram and are therefore at the distance of the cluster. This combination of facts makes the three stars rather enigmatic. The relatively large lithium abundances of these stars compared to field stars would make them not much older than a few hundred Myr; the likelihood of relatively young interloper stars in the field of view of the cluster and at the distance of the cluster must be rather small.
The two remaining stars identified as non-members, \#573 and \#407, may be field stars. The former appears to lie in the region of the Li-dip and the latter has a low enough Li abundance to be consistent with field star values (Chen et al. 2001). We note, however, that \#407 has a rather large rotational velocity ([*v*]{} sin [*i*]{} = 28 km s$^{-1}$), which is unusual in a field G star. In summary, all of the five stars categorized as non-members may warrant closer scrutiny.
Concluding remarks
==================
In the Introduction, we referred to the powerful role that is played by open clusters in placing observational constraints on lithium depletion in pre-main sequence (PMS) and main sequence (MS) stars. Perhaps, the principal outstanding questions about lithium depletion concern the onset of the depletion and the star-to-star spread in (apparent) lithium abundances at low masses. (There remains too the incompletely understood Li-dip in warm older stars.) In order to determine when lithium depletion at low masses develops, and how it evolves with time, depletion of Li must be traced from the earliest PMS phases to the age of the $\alpha$ Per cluster and beyond.
PMS lithium depletion is now mappable by looking at the very youngest of clusters and associations; the stars are faint but accessible with large telescopes. In addition to the uncertainty of defining membership in clusters, associations and moving groups, there are several problems associated with interpreting their Li abundance trends. First, since young PMS stars of different masses tend to lie in the same temperature range between 3000 K to 4000 K as they evolve down the HR diagram, and older PMS stars rapidly increase their temperature as they evolve towards the main sequence, the temperature of the PMS star is not a sufficient indication of its mass and determination of stellar mass requires the use of theoretical evolutionary tracks which continue to differ from author to author. This difficulty may be offset partly by the result that theoretical prediction of a cluster’s age from the location of the lithium depletion boundary (LDB) is not very dependent on which set of PMS evolutionary tracks is chosen (Jeffries & Oliveira 2005). Second, because the young PMS stars are cool, analysis of their spectra is complicated by molecular features and the derived Li abundance has a larger uncertainty than in warmer main sequence stars.
For these reasons and because the available data are limited with respect both to the number of young clusters, associations, and moving groups and to the number of stars per cluster, we defer a detailed search for the onset of lithium depletion at low masses. It is worth noting that an interesting set of Li data in a range of young clusters and associations has been accumulated ([*e.g.*]{} Sestito & Randich 2005, Mentuch et al. 2008). In their Table 1, Sestito & Randich (2005) list the ’classical’ ages of clusters, i.e., those determined from isochrone fitting. For four young clusters in their sample, IC 2391, NGC 2547, $\alpha$ Per and the Pleiades, new independent estimates of the ages have been obtained based on the position of LDB (Stauffer et al. 1998, 1999; Barrado y Navascues et al. 2004, Jeffries & Oliviera 2005) which are, in general, higher than the classical ages. Although the LDB technique is less model dependent than MS fitting, Sestito & Randich chose to adopt the classical ages for uniformity through the entire sample. In our discussion of clusters chosen from Sestito & Randich (2005), including $\alpha$ Per and the Pleiades, we have adopted these same classical ages.
On the other hand, the ages of the young associations studied by Mentuch et al. (2008), that we compare the results of $\alpha$ Per and the Pleiades to, are derived by comparing the dependence of Li abundance on temperature with isochrones from pre-MS evolutionary tracks. Mentuch et al. state that these ages are consistent with the earlier estimates based on isochrone fitting or other methods. We will not analyze the strengths and weaknesses of cluster age determinations in our discussions. Rather, we note that the crucial point is that the ordering of the clusters according to age is robust. When comparing the results of Li scatter in these young associations with those of clusters from Sestito & Randich (2005) with classical ages, the chronological order of ages is not disturbed even if we adopt the higher LDB ages for the 4 young clusters, namely, NGC 2547, IC 2391, $\alpha$ Per and the Pleiades. It is therefore worth examining the star samples of these associations with those of $\alpha$ Per and other clusters.
Sestito & Randich list NGC 2264 at 5 Myr as their youngest cluster with the survey of Li abundances from Soderblom et al. (1999). The Li abundance for the warmer stars in NGC 2264 is about 3.2, a value consistent with our result for the hotter stars in $\alpha$ Per and also with the canonical value for an initial Li abundance for young stars. Stars between 0.5 and 1.0 M$_\odot$ in NGC 2264 show only a mild (0.4 dex) Li dispersion. Given measurement and analyses uncertainties, one may conclude that PMS depletion has possibly not begun in these young stars. Mild PMS depletion may be seen in the 12 Myr old $\eta$Cha cluster and TW Hydrae association (Mentuch et al. 2008). By 20 Myr, the $\beta$ Pic moving group and by 27 Myr the Tucanae-Horologium association show nearly a 3.0 dex range in Li abundance (see Figure 8 in Mentuch et al. 2008). However, in the absence of reliable mass determinations, the presence or absence of an abundance disperson at a particular mass cannot be deciphered.
A clearer view of Li dispersion at a particular mass may be obtained once the cluster is on the main sequence. Sestito & Randich (2005) list IC 2602, IC 2391, IC 4665, and NGC 2547 as main sequence clusters younger than $\alpha$ Per. Impression of a smaller star-to-star scatter in Li abundances in clusters younger than $\alpha$ Per is conveyed by results for IC 2602 with an age of 30 Myr (Randich et al. 1997, 2001). Randich et al. (2001) define a regression curve to represent Li abundances from 3900 K to 6900 K. This curve is above the upper envelope in Figure 4 for $T_{\rm eff} < 4400$ K and coincident with it for higher temperatures. To effect a fair comparison, a correction would need to be made for the mass-dependent temperature change between an age of 30 Myr and 50 Myr. Little additional Li depletion is predicted in this interval. The point of interest here is that the scatter about the regression curve is at most $\pm0.5$dex, often much less, and less than exhibited in Figure 3. Indeed, most points in the equivalent plot to Figure 3 (Randich et al.’s (2001) Figure 4) touch the regression curve with their error bars. A similar conclusion applies to IC 2391, also 30 Myrs old, from inspection of the same Figure 4 which assembles Li abundances from that paper and Stauffer et al. (1989b). For IC 4665 at 35 Myr, the available Li abundances (Martín & Montes 1997; Jeffries et al. 2009) are too few at low temperatures to define the Li abundance trend with temperature and certainly not to detect a star-to-star variation. For NGC 2547 also at 35 Myr, there is evidence of a variation approaching that seen in Figure 3 (Jeffries et al. 2003) with a lower envelope to the Li abundances resembling that of the upper envelope in Figure 4. One may speculate from these comparisons that the dispersion in Li at a given mass develops and strengthens between 30 and 50 Myr, that is between the ages of IC 2391 and IC 2602 and the age of $\alpha$ Per.
This is further corroborated by observations of the AB Doradus moving group by Mentuch et al. (2008) which has an age of 45 Myr, very similar to that of $\alpha$ Per. With the additional caveats that the sample is small and there are membership issues in defining a moving group, we compare our $\alpha$ Per sample with that of AB Dor. The comparison is frustrated because Mentuch et al. (2008) systematically find an abundance $\log$N(Li) $\simeq 3.8$ in their samples for stars that are unaffected by PMS depletion. This ‘initial’ value is about 0.6 dex greater than our value for the hotter stars. We have adopted the view that the high initial abundance is a consequence of a systematic overestimate of the Li abundance but we have no basis for knowing if this overestimate carries over to lower temperatures. Between 5300 K and 4900 K, four AB Dor stars have Li abundances between 3.4 and 1.3; the range is comparable to that seen in $\alpha$ Per and larger than that seen in the Sestio & Randich (2005) survey. Below 4900 K, five AB Dor stars are coincident with the upper envelope of $\alpha$ Per stars. There may be some indication in this limited sample that AB Dor exhibits a larger Li spread than the slightly younger clusters discussed in Sestito & Randich (2005) but very similar to what is seen in $\alpha$ Per.
For clusters older than $\alpha$ Per, we restrict comparison to the well-sampled Pleiades (age of 70 Myr) where we have taken Pleiades data from Soderblom et al. (1993, see also King et al. 2000). Perhaps, a fairer comparison would be to take data for both clusters from Sestito & Randich (2005) who undertook a uniform analysis of these and other open clusters. The mean relations and their scatter are very similar but for two minor differences when compared in the abundance-effective temperature plane; the evolution in $T_{\rm eff}$ over the 20 Myr age difference is very small and ignored. First, the $\alpha$ Per cool outliers – \# 1612 and \#1735 – have no counterparts in the Pleiades. Second, and more prominently, the Pleiades has four stars with undepleted lithium ($\log$N(Li) $\simeq 3.2$) at $T_{\rm eff} \simeq 5000$ K with no counterparts in $\alpha$ Per where $\log$N(Li) $\simeq$ 2.5 at this temperature.
In observed clusters older than Pleiades, main sequence depletion begins to reduce the Li abundances in the coolest stars noticeably. This is certainly apparent for M34 with an age of 250 Myr where stars have been observed down to about 4200 K (Jones et al. 1997). Here, the star-to-star scatter remains similar to that of the Pleiades and $\alpha$ Per but the mean abundances are smaller. By the age of the Hyades, only upper limits to the Li I equivalent width are measurable in stars $\leq$ 5000 K (Soderblom et al. 1995).
In summary, the Li abundances for $\alpha$ Per fit the pattern provided by observations of clusters both younger and older than it. The star-to-star spread appears to develop after about 20 Myr. The spread survives up to 250 Myr and its demise is hidden from observers as main sequence lithium depletion removes any inequalities in lithium abundance from observers’ view.
Inspection of Figures 3 and 4 shows a relative dearth of measurements at temperatures lower than about 4700 K. Additional members of the $\alpha$ Per cluster are to be found in Prosser (1992). Although expansion of the sample at lower temperatures would be informative, perhaps the most useful benefit from an enlarged sample, would be an application of the best techniques of quantitative stellar spectroscopy to pairs of stars with maximum and minimum Li abundance but similar observed properties such as colour and rotation period. If such a study discovers differences only for lithium, then atmospheric effects may truly be eliminated as the cause of the Li dispersion.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has made use of the WEBDA database maintained at the Institute for Astronomy of the University of Vienna. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. We thank the referee for several constructive comments on the manuscript. SCB is pleased to acknowledge support from NSF grant AST-0407057. On the eve of her departure from 25 years of research in Astronomy, SCB would like to thank colleagues and collaborators who have made it a rich and satisfying experience. DLL wishes to thank the Robert A. Welch Foundation of Houston, Texas for support through grant F-634. The authors would like to thank the referee for helpful remarks that improved the manuscript.
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[^1]: E-mail: sgvmlk@iiap.res.in
[^2]: http://www.univie.ac.at/webda/
[^3]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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[^5]: http://www.ipac.caltech.edu/2mass/releases/allsky/
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---
abstract: 'Prediction performance of a risk scoring system needs to be carefully assessed before its adoption in clinical practice. Clinical preventive care often uses risk scores to *screen* asymptomatic population. The primary clinical interest is to predict the risk of having an event by a pre-specified *future* time $t_0$. Prospective accuracy measures such as positive predictive values have been recommended for evaluating the predictive performance. However, for commonly used continuous or ordinal risk score systems, these measures require a subjective cut-off threshold value that dichotomizes the risk scores. The need for a cut-off value created barriers for practitioners and researchers. In this paper, we propose a threshold-free summary index of positive predictive values that accommodates time-dependent event status. We develop a nonparametric estimator and provide an inference procedure for comparing this summary measure between competing risk scores for censored time to event data. We conduct a simulation study to examine the finite-sample performance of the proposed estimation and inference procedures. Lastly, we illustrate the use of this measure on a real data example, comparing two risk score systems for predicting heart failure in childhood cancer survivors.'
author:
- |
YAN YUAN$^{\ast \dag}$\
[School of Public Health, University of Alberta, Edmonton, AB T6G1C9, Canada]{}\
- |
QIAN M. ZHOU$^\dag$\
Department of Statistics and Actuarial Science\
Simon Fraser University, Burnaby, B.C. V5A1S6, Canada\
Department of Mathematics and Statistics\
Mississippi State University, Starkville, Mississippi 39762, USA\
- |
BINGYING LI\
Department of Statistics and Actuarial Science\
Simon Fraser University, Burnaby, B.C. V5A1S6, Canada\
- |
HENGRUI CAI\
School of Public Health, University of Alberta, Edmonton, AB T6G1C9, Canada\
- |
ERIC J. CHOW\
Fred Hutchinson Cancer Research Center\
Seattle Children’s Hospital, University of Washington, Seattle, Washington, USA\
- |
GREGORY T. ARMSTRONG\
Department of Epidemiology and Cancer Control, Division of Neuro-Oncology\
St. Jude Children’s Research Hospital, Memphis, TN 38105, USA\
bibliography:
- 'AP.bib'
title: 'A Threshold-free Prospective Prediction Accuracy Measure for Censored Time to Event Data'
---
[Y. Yuan, Q. Zhou and others]{} [Time-dependent AP]{}
Censored event time; Positive predictive value; Precision-recall curve; Risk prediction; Screening; Time-dependent prediction accuracy
Introduction {#sec:intro}
============
Clinical medicine is facing a paradigm shift from current diagnosis and treatment practices to prevention through earlier intervention based on risk prediction [@wright2014conceptual]. Diagnosis and treatment approaches help individual patients seek relief from their symptoms. However, evidence is mounting that health interventions may be more effective in improving long-term health outcomes when they target asymptomatic individuals who are predicted to be at high risk for the condition of interest [@espeland2007reduction; @james2010early]. The condition of interest typically has the following characteristics: 1) its seriousness may result in a high risk of mortality or significantly affect the quality of life; 2) early detection/intervention can make a difference in disease prognosis; and importantly but subtly 3) its event rate is low. A prevention approach to medicine relies on the development of risk scores to stratify individuals into different risk groups. Early intervention strategies are typically recommended to subjects who are in the high-risk group.
In the prevention paradigm, the use of risk scores as population *screening* tools is increasingly advocated in clinical practices, e.g. [@goff20142013]. For example, one systematic review identified forty-six algorithms that predict the risk of type 2 diabetes [@buijsse2011risk]. Another study established several risk score systems to predict congestive heart failure for childhood cancer survivors who are at an elevated risk due to treatment toxicity [@chow2015individual]. One of the defining characteristics of screening is a low event rate in the targeted asymptomatic population. Taking the aforementioned two diseases as an example, the crude prevalence of undiagnosed type 2 diabetes, a common disease, was low at 3.5% in 1987 and 5.7% in 1992 [@lindstrom2003diabetes], while the cumulative event rate of congestive heart failure by 35 years post childhood cancer diagnosis was 4.7% [@chow2015individual]. The event rate is much lower for other notable conditions such as cancer, multiple sclerosis, AIDS, dementia, COPD and perinatal conditions. A low event rate and a focus on prevention necessitate the development of *screening* tools such as risk scores.
Before a risk scoring system is adopted for clinical screening, evaluation of its predictive accuracy is critical. The most popular accuracy metric used in the clinical literature is the area under the receiver operating characteristic (ROC) curve (AUC). The AUC is a summary index of two accuracy metrics - sensitivity and specificity - which are both retrospective metrics. Thus, the AUC does not reflect the prospective predictive accuracy of risk score systems. Indeed, one influential article criticized these retrospective metrics as being of little use for clinicians because clinical interest almost always focuses on prediction [@grimes2002uses]. In contrast, a prospective accuracy measure, such as positive predictive value (PPV), can prospectively answer the question: “Can risk scores be trusted?” Unfortunately, a risk score with high sensitivity and specificity, and thus a high AUC, can have poor PPV when applied to low-prevalence populations. This limitation is often overlooked by clinicians and biomedical researchers. Despite its popularity, studies confirm that the AUC is insensitive in evaluating risk prediction models. For example, including a marker with a risk ratio of 3.0 showed little improvement on the AUC, while it could shift the predicted 10-year disease risk for an individual patient from 8% to 24% [@cook2007use]. This range would result in different recommendations on follow-up/intervention strategies.
Compared to the AUC, the PPV provides a more appropriate assessment of the prospective prediction performance of the risk score [@moskowitz2004quantifying], making the PPV a superior metric for risk score systems used as screening tools. The PPV is calculated with data from a prospective cohort, where the risk scores are computed using baseline information and the outcome is followed prospectively. Originally, the PPV was defined for a dichotomous test. Moskowitz and Pepe (2004) extended the definition of PPV for a continuous risk score [@moskowitz2004quantifying]. Assuming that the higher the risk score, the greater the individual risk, the PPV is defined as the probability of having the disease when the risk score value is larger than a given cut-off value $z$,
$$\label{equ:PPV-ContMarker-BinOutcome}
\ppv(z) = Pr\{D=1\mid Z \geq z\}\quad \hbox{and}\quad \npv(z) = Pr\{D=0\mid Z < z\},$$
where $D=1$ indicates the presence of the disease, and $D=0$ indicates the absence of the disease. Zheng et al. (2008) further generalized the definition to accommodate the censored event time outcome [@zheng2008time]. Since the PPV is threshold dependent, as seen in (\[equ:PPV-ContMarker-BinOutcome\]), it is often evaluated at either several fixed specificities or several fixed quantiles of the . Such evaluations allow the comparison across different risk score systems [@moskowitz2004quantifying; @wald2014area]. The selection of specificities or quantiles can be subjective, and it is possible that different systems could outperform others, depending on the cut-off points selected [@zheng2010semiparametric].
For the above reasons, a threshold-free summary metric for the PPV is needed to facilitate its clinical usage. Two curves of PPV have been investigated in the literature. Raghavan et al. (1989) and Zheng et al. (2010) considered a curve of PPV versus quantiles of the risk score [@zheng2010semiparametric; @raghavan1989critical]. However, they did not provide a summary index of the proposed PPV curve. A second curve is called the precision-recall (PR) curve, which was proposed in the information retrieval community [@raghavan1989critical; @manning1999foundations], where precision is equivalent to the PPV and recall is equivalent to the sensitivity. The relationship of PR and ROC curves and the area under them has been discussed in Davis and Goadrich [@davis2006]. They showed that the PR curve of a risk score system dominates that of another system if its ROC curve is also dominant. However, such a relationship does not exist for the area under these two curves [@davis2006; @su2015relationship]. Two recent papers illustrated the advantage of using the area under the PR curve over the AUC for predicting low prevalence diseases [@yuan2015threshold; @ozenne2015precision]. We refer to the summary metric for the area under the PR curve as the average positive predictive value (AP) [@yuan2015threshold]. These previous research on the area under the PR curve have only considered binary outcomes. However, for many clinical applications, the outcome is time to event.
We make three contributions in the assessment of risk scoring systems for clinical screening. First, we define a time-dependent AP, $AP_{t_0}$ for censored event time outcomes. We propose a robust nonparametric estimator of $\ap_{t_0}$ without modeling assumptions on the relationship between the risk score and event time. Secondly, we provide a statistical inference procedure to compare the $\ap_{t_0}$ two risks scores regarding . Thirdly, we provide an R package to implement our method. The paper is organized as follows. In Section \[sec:def\], we introduce the definition and interpretation of $AP_{t_0}$. In Section \[sec:estimator\], we present the inference procedures for estimating $AP_{t_0}$ of a single risk score as well as the comparison between two competing risk scores. In Section \[sec:simulation\], we conduct a simulation study to investigate the performance of the proposed estimation and inference procedures in finite samples. In Section \[sec:data-analysis\], we illustrate the proposed metric $AP_{t_0}$ by analyzing two risk score systems with data from the Childhood Cancer Survival Study [@robison2009childhood]. We conclude with a discussion and suggestions for future work in Section \[sec:discussion\].
Time-dependent Average Positive Predictive Values {#sec:def}
=================================================
Consider a continuous risk score $Z$. Let $T$ be the time to the event of interest. Time-dependent PPV and TPF [@zheng2008time; @heagerty2000time] are defined as $$\label{equ:PPV-ContMarker-EventTime}
\ppv_{t_0}(z) = Pr\{T<t_0\mid Z \geq z\}\quad \hbox{and}\quad \tpf_{t_0}(z) = Pr\{ Z \geq z \mid T<t_0\}.$$ In the above setting, the event status is time-dependent, i.e., $D_{t_0}=I(T<t_0)$, where $I(\cdot)$ is an identity function. Consequently, the PPV and TPF are also functions of $t_0$.
Following [@yuan2015threshold], we define $\ap_{t_0}$, as the area under the time-dependent PR curve $\{(\tpf_{t_0}(z) ,\ppv_{t_0}(z)), z\in \Rcal\}$, $$\label{equ:TimeDependent-AP}
\ap_{t_0} = \int_{\mathcal{R}} \ppv_{t_0}(z) d \tpf_{t_0}(z).$$ Note that the TPF describes the distribution function of $Z$ in “cases" who experience the event by time $t_0$, i.e. $T<t_0$. It can be shown that $\ap_{t_0} = E_{Z_1}\left\{\ppv_{t_0}(Z_1)\right\}$, where $Z_1$ denotes the risk score in cases. In the real data example of Section \[sec:data-analysis\], we will show that AP is estimated to be 0.114 at $t_0=35$ years for a risk score system. That is, by 35 years post diagnosis, we expect that on average 11.4% of the subjects with a high risk score (compared to the risk score of a randomly selected case) will experience the event of interest.
In addition, $\ppv_{t_0}(z)$ can be written as $\ppv_{t_0}(z) = P( Z \geq z \mid T<t_0)P(T < t_0) / P(Z \geq z) = \pi_{t_0}\left\{1-F_1(z)\right\}/\left\{1-F(z)\right\}$, where $F_1(z)=Pr(Z< z \mid T<t_0)=P(Z_1<z)$ is the distribution function of the risk score $Z_1$ for cases, $F(z)=P(Z < z)$ is the distribution function of the risk score $Z$ for the target population, and $\pi_{t_0}=Pr(T < t_0)$ is the event rate by time $t_0$ in the target population. Thus, the AP can be written as $$\label{equ:TimeDepedent-AP-alt}
\ap_{t_0} = \pi_{t_0} \int_{\mathcal R} \frac{1-F_1(z)}{1-F(z)} d F_1(z).$$ A perfect risk score system would always assign higher values to cases, individuals with $T<t_0$, compared to those controls, individuals with $T\geq t_0$, i.e. $P(Z \geq Z_1\mid T \geq t_0) = 0$. This leads to $\ap_{t_0}=1$ from equation (\[equ:TimeDepedent-AP-alt\]). A non-informative risk score system would randomly assign risk scores to both cases and controls. i.e., $P(Z \geq z\mid T\geq t_0) = P(Z\geq z \mid T<t_0)$ for each $z$, which leads to $\ap_{t_0}=\pi_{t_0}$. Thus, the theoretical range of $AP_{t_0}$ is $[\pi_{t_0},1]$.
Estimating and Comparing $\ap_{t_0}$ {#sec:estimator}
====================================
Nonparametric Estimator of $\ap_{t_0}$ for a single risk score {#sec:single}
--------------------------------------------------------------
Often, the event times of some subjects are censored due to the end of the study or loss to follow up. Due to censoring, one can only observe $X=\min\{T,C\}$ where $C$ is the censoring time, and $\delta = I(T<C)$. Let $\{(X_i,\delta_i,Z_i),i=1,\cdots,n\}$ be $n$ independent realizations of $(X,\delta,Z)$.
In the presence of censoring, event status at $t_0$, $I(T_i<t_0)$, may not be observed for some subjects. We suggest using the inverse probability weighting (IPW) [@uno2007evaluating; @lawless2010estimation] to account for censoring. The time-dependent PPV and TPF are estimated by $$\ppvhat_{t_0}(z) = \frac{\sum_{i=1}^n \what_{t_0,i} I(Z_i \geq z)I(X_i < t_0)}{\sum_{i=1}^n I(Z_i\geq z)}\quad \hbox{and}\quad \tpfhat_{t_0}(z) = \frac{\sum_{i=1}^n \what_{t_0,i} I(Z_i\geq z)I(X_i < t_0)}{\sum_{i=1}^n \what_{t_0,i}I(X_i < t_0)},$$ where $\what_{t_0,i}$ is the inverse of the estimated probability that the time-dependent event status $I(T_i < t_0)$ is observed, specifically $$\what_{t_0,i} = \frac{I(X_i < t_0)\delta_i}{\Gcalhat(X_i)} + \frac{I(X_i \geq t_0)}{\Gcalhat(t_0)},$$ where $\Gcalhat(c)$ is a consistent estimator of the survival function of the censoring time, $\Gcal(c) = Pr(C \geq c)$. Note that the proposed estimator does not imposes any assumptions on the relationship between the risk score $Z$ and the event time $T$. Under the assumption of independent censoring, i.e., the censoring time $C$ is independent of both the event time $T$ and the risk score $Z$, $\Gcal(c)$ can be obtained by the nonparametric Nelson-Aalen or Kaplan-Meier estimator. If the censoring time $C$ depends on the risk score $Z$, additional model assumptions might be required. For example, a proportional hazards (PH) model could be fit to estimate $\Gcal_z(t) = Pr(C \geq c \mid Z=z)$.
Based on the estimated $\ppv_{t_0}(z)$ and $\tpf_{t_0}(z)$, $\ap_{t_0}$ can be estimated by $$\label{ap-est}
\widehat{\ap}_{t_0} = \frac{\sum_{j=1}^nI(X_j \leq t_0)\hat{w}_{t_0,j} \ppvhat_{t_0}(Z_j)}{\sum_{j=1}^nI(X_j \leq t_0)\hat{w}_{t_0,j}}.$$ [@uno2007evaluating] shows that $\ppvhat_{t_0}(z)$ and $\tpfhat_{t_0}(z)$ are both consistent estimators, and thus, $\widehat{\ap}_{t_0}$ is also a consistent estimator of $\ap_{t_0}$ for any given value of $t_0$.
In practice, we often deal with discrete risk scores, where tied risk scores are common. To accommodate risk scores with ties, following [@pepe2003statistical], we modify the above estimator (\[ap-est\]) by replacing $\ppvhat_{t_0}(Z_j)$ with $$\ppvtilde_{t_0}(Z_j) = \frac{\sum_{i=1}^n \what_{t_0,i} \left\{I(Z_i > Z_j) + \frac{1}{2}I(Z_i=Z_j)\right\}I(X_i < t_0)}{\sum_{i=1}^n \left\{I(Z_i > Z_j) + \frac{1}{2}I(Z_i=Z_j)\right\}}.$$ To construct confidence intervals, we suggest the nonparametric bootstrap [@efron1979bootstrap] method. Specifically, let $\widehat{\mbox{AP}}^{\mathbbm B}_{t_0} =\left\{\widehat{\mbox{AP}}^b_{t_0}, b=1,2,\cdots,B\right\}$ denote the estimated $\ap_{t_0}$ obtained from $B$ bootstrape resamples. A 95% confidence interval (CI) for the $\ap_{t_0}$ is given as $(\widehat{\mbox{AP}}_{t_0}^{\mathbbm B,0.025}, \widehat{\mbox{AP}}_{t_0}^{\mathbbm B,0.975})$, where $\widehat{\mbox{AP}}_{t_0}^{\mathbbm B,0.025}$ and $\widehat{\mbox{AP}}_{t_0}^{\mathbbm B,0.975}$ are the 2.5% and 97.5% empirical percentiles of the $\widehat{\mbox{AP}}^{\mathbbm B}_{t_0}$, respectively.
Comparing two risk scores
-------------------------
We consider comparing two risk scores $Z_1$ and $Z_2$ on the $\ap_{t_0}$ scale. In many studies, both risk scores $Z_1$ and $Z_2$ are calculated for each individual. With the paired data, we can quantify the relative predictive performance of $Z_1$ vs. $Z_2$, using the ratio of their respective time-dependent AP, $$\text{rAP}_{t_0}=\ap_{Z_1,t_0}/\ap_{Z_2,t_0},$$ where $\ap_{Z_1,t_0}$ and $\ap_{Z_2,t_0}$ denote the time-dependent AP for $Z_1$ and $Z_2$ at $t_0$ respectively. For a single risk score $Z$, the ratio $AP_{Z,t_0}/\pi_{t_0}$ can be regarded as the relative predictive performance of $Z$ compared to a non-informative risk score.
The AP ratio $\text{rAP}_{t_0}$ can be estimated by $\widehat{\text{rAP}}_{t_0}=\widehat{\ap}_{Z_1,t_0}/\widehat{\ap}_{Z_2,t_0}$, where $\widehat{\ap}_{Z_1,t_0}$ and $\widehat{\ap}_{Z_2,t_0}$ are the the nonparametric estimator $\widehat{\ap}_{t_0}$ in (\[ap-est\]) of $Z_1$ and $Z_2$ respectively. The standard percentile bootstrap method descried in Section \[sec:single\] can be used to construct a CI for $\text{rAP}_{t_0}$ or test $H_0: \text{rAP}_{t_0}=1$ for any given time point $t_0$. Specifically, the CI could be obtained based on the empirical percentiles of the $B$ bootstrap counterparts of of $\widehat{\text{rAP}}_{t_0}$, denoted by $\widehat{\text{rAP}}^b_{t_0}=\widehat{\ap}_{Z_1,t_0}^b/\widehat{\ap}_{Z_2,t_0}^b$, where $\widehat{\ap}_{Z_1,t_0}^b$ and $\widehat{\ap}_{Z_2,t_0}^b$ are the estimated $\ap_{t_0}$ for $Z_1$ and $Z_2$ based on the same bootstrap resample, $b=1,\cdots,B$.
Simulation study {#sec:simulation}
================
We conducted a simulation study to examine the performance of the time-dependent AP estimator in finite samples. In this simulation study, we considered two risk scores $U_{1}$ and $U_{2}$. They were generated from a standard normal distribution $N(0, 1)$. The event time associated with both risk scores for the ith subject was generated from the following model $$\log(T_i) = 7.2 - 1.1 U_{i1} - 2.5 U_{i2} - 1.5 log(U_{i1}^2) + \epsilon_T,$$ where $\epsilon_T \sim N(0, 1.5)$. This setting provides an example where the ROC curves of the two risk scores cross at time $t_0=8$, shown in Figure \[fig:1\], with $\auc_{U_1, t_0}$ and $\auc_{U_2, t_0}$ are similar in values. On the other hand, the PR curve of $U_1$ dominates that of $U_2$ over the most range of the TPR with $\ap_{t_0}$ of $U_1$ greater than that of $U_2$.
The censoring time $C_i$ was generated following $C_i = \min (A_i, B_i+1)$ where, $A_i \sim Uniform(0, 50)$, and $B_i \sim Gamma(25,0.75)$. This configuration results in about 50% of censoring overall. Let $X_i = \min(T_i, C_i)$, $\delta_i = I(T_i \leq C_i)$. In this setting, the censoring time is independent of both the event time and risk scores.
![The ROC curves in the left panel and the precision-recall curves in the right panel for the two risk scores $U_1$ and $U_2$ at $t_0=8$ when the event rate is 5%. The numbers shown in graph correspond to the AUC and the AP values.[]{data-label="fig:1"}](figure1.eps){width="5.0in"}
We considered three prediction time points $t_0$ where the corresponding event rates are, $r=P(T_{i} < t_{0})$, 0.01, 0.05 and 0.1, respectively. To allow a reasonable number of events by $t_0$, we generated the data $\{(X_i, \delta_i, U_{1i}, U_{2i}), i = 1,...,n\}$ with sample size $n$ being 2000 and 5000 (Tables \[table:1\] and \[table:2\]). In each table, we report the following summary statistics based on 1000 repetitions: bias, empirical standard error (ESE) of the estimator, average standard errors from bootstrap ($ASE^b$), and the empirical coverage probability ($ECOVP^b$) of 95% confidence intervals obtained from 1000 bootstrap resamples as described in Section \[sec:estimator\].
These results show that the estimators of both time-dependent AP and ratio of AP, $AP_1/AP_2$, between the two risk scores had small biases in all $t_0$ values and different sample sizes. The bias decreases with increasing event rate and increasing sample size. Also, the standard errors $ASE^b$ obtained from bootstrap were close to the empirical standard errors. Thus, the confidence intervals attained the nominal coverage probabilities for both smaller sample size 2000 and larger sample size 5000.
We remark that this simulation provides an illustrative example of the relationship between ROC curve and PR curve as well as the relationship between the AUC and the AP investigated in [@davis2006]. When the ROC curves of two competing risk scores cross, the PR curves cross too. In situations like this, the AUC and the AP may rank the risk scores differently. In our simulation setting, $U_2$ outperforms $U_1$ according to the AUC, which indicates that $U_2$ is better at discriminating between cases and controls. On the other hand, $U_1$ outperforms $U_2$ according to the AP, which suggests that $U_1$ is a better screening tool for stratifying subjects into different risk groups.
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[cm[1in]{}<ccccccc]{}
$t_0$ & Event rate & & TRUE & BIAS & ESE & $ASE^b$ & $ECOVP^b(\%)$\
0.5 & 0.0101 & $\ap_1$ & 0.182 & 0.0365 & 0.0810 & 0.0795 & 92.3\
& & $\ap_2$ & 0.124 & 0.0339 & 0.0689 & 0.0678 & 93.0\
& & $\text{rAP}$ & 1.47 & 0.4890 & 1.5300 & 1.7600 & 95.1\
8 & 0.0495 & $\ap_1$ & 0.364 & 0.0096 & 0.0527 & 0.0516 & 92.5\
& & $\ap_2$ & 0.266 & 0.0129 & 0.0452 & 0.0450 & 93.4\
& & $\text{rAP}$ & 1.37 & 0.0140 & 0.3290 & 0.3320 & 95.7\
36 & 0.0991 & $\ap_1$ & 0.462 & 0.0098 & 0.0534 & 0.0558 & 95.9\
& & $\ap_2$ & 0.375 & 0.0118 & 0.0493 & 0.0501 & 94.5\
& & $\text{rAP}$ & 1.23 & 0.0135 & 0.2310 & 0.2420 & 94.9\
=4.25pt
[cm[1in]{}<ccccccc]{}
$t_0$ & Event rate & & TRUE & BIAS & ESE & $ASE^b$ & $ECOVP^b(\%)$\
0.5 & 0.0101 & $\ap_1$ & 0.182 & 0.0185 & 0.0500 & 0.0504 & 93.1\
& & $\ap_2$ & 0.124 & 0.0155 & 0.0416 & 0.0417 & 94.8\
& & $\text{rAP}$ & 1.47 & 0.1550 & 0.7060 & 0.7600 & 93.8\
8 & 0.0495 & $\ap_1$ & 0.364 & 0.0042 & 0.0337 & 0.0333 & 92.9\
& & $\ap_2$ & 0.266 & 0.0049 & 0.0291 & 0.0288 & 93.7\
& & $\text{rAP}$ & 1.37 & 0.0060 & 0.2160 & 0.2100 & 95.4\
36 & 0.0991 & $\ap_1$ & 0.462 & 0.0034 & 0.0354 & 0.0346 & 95.5\
& & $\ap_2$ & 0.375 & 0.0037 & 0.0310 & 0.0313 & 94.1\
& & $\text{rAP}$ & 1.23 & 0.0051 & 0.1490 & 0.1510 & 95.0\
Data Analysis {#sec:data-analysis}
=============
In this section, we illustrate the use of AP$_{t_0}$ metric with a data set from the Childhood Cancer Survivor Study [@robison2009childhood]. This cohort follows children who were initially treated for cancer at 26 US and Canada institutions between 1970 and 1986 and who survived at least 5 years after their cancer diagnosis. Among the survivors, cardiovascular disease has been recognized as a leading contributor to morbidity and mortality [@oeffinger2006chronic]. To inform future screening and intervention strategy for congestive heart failure (CHF) in this population, [@chow2015individual] developed several risk score systems using the CCSS data and validated them on external cohorts. For the purpose of illustration, we chose two of these risk scores and evaluated their predictive performance using the proposed AP$_{t_0}$.
We included 11,457 subjects in our analysis from the CCSS study who met the original study inclusion criteria and had both risk scores. In this data, a total of 248 subjects experienced the CHF. Between the two risk scoring systems we focused on in this data analysis, the simpler model used information on age at cancer diagnosis, sex, whether the patient was exposed to chest radiotherapy, and whether the patient was exposed to a particular chemotherapy agent. We refer to this model as the simple model. The more elaborate model, known as the heart dose model, included detailed clinical information on the average radiation dose to the heart and the cumulative dose of the specific chemotherapy agent, along with age at diagnosis and sex. This is an example where a simple risk score system utilizes minimum treatment information and can be used for any patient by virtually all clinicians, while the more complex risk score system demands specific dose information which may not be readily available to clinicians providing long-term follow-up care. We used the original risk scores of the simple model and the heart dose model from the reference study [@chow2015individual]. Briefly, these scores were constructed via linear combinations of the corresponding covariates, where the regression coefficients were obtained from Poisson regression models.
Table \[tab:CCSS\] reports the estimated AP$_{t_0}$ with 95% CIs for both the simple model (denoted by $\ap_{s,t_0}$) and heart dose model (denoted by $\ap_{h,t_0}$) at $t_0$ = 20 and 35 years post-diagnosis where the corresponding estimated event rates were 1.3% and 4.7% respectively. These two models were compared using the ratio of AP, i.e. $\text{rAP}_{t_0} = \ap_{h,t_0}/\ap_{s,t_0}$. In addition, we also provided the estimated time-dependent AUC ($\auc_{t_0}$) at these two time points as well as the difference of AUCs between these two models $\Delta\auc_{t_0}=\auc_{h, t_0} - \auc_{s, t_0}$. To illustrate the time-varying performance for each model as well as the comparison between these two models over time, $\ap_{t_0}$, $\auc_{t_0}$, $\text{rAP}_{t_0}$ and $\Delta\auc_{t_0}$ versus $t_0$ were plotted in Figure \[fig:CCSS\]. Note that we assumed independent censoring in estimating the AP and the AUC.
The results in Table \[tab:CCSS\] show that the heart dose model outperforms the simple model at both time points. For example, the estimated $\ap_{20}$ of the heart dose model is 0.075, which indicates that by 20 years post-diagnosis, using the risk score from the heart dose model, we expect that on average 7.5% subjects with a high risk score (compared to the risk score of a randomly selected case) will experience heart failure. This AP is almost six times of the event rate 1.3%, which corresponds to the AP of a non-informative risk score system. In contrast, the estimated $\ap_{20}$ for the simple model is 0.039, roughly half of that of the heart dose model ($r\ap_{20}$=1.96, 95%CI:1.41-2.88). At 35 years post diagnosis, the heart dose model is significantly better than the simple model with $r\ap_{35}$ = 1.45 (95%CI: 1.26 - 1.70). Indeed, The plots (c) and (d) in Figure \[fig:CCSS\] show that in terms of the $\ap_{t_0}$, the heart dose model outperforms the simple model at identifying the high risk subjects from the targeted population at all time points considered. On the other hand, the AUCs are similar between these two models, as seen in plot (d). Especially, $\Delta\auc$ is not significantly different from 0 at the beginning and towards the end of the time period that were considered, For example $\Delta\auc_{35} = 0.01$ (95% CI: -0.02 - 0.03, p=0.47), shown in Table \[tab:CCSS\]. It suggests that according to the AUC, there is not much performance difference between the two models. If, due to incorporating more information, the heart dose model is indeed superior to the simple model in terms of identifying the high risk individuals, the results in Table \[tab:CCSS\] and Figure \[fig:CCSS\] implies that the AP is a better metric for discriminates the risk prediction performance than the AUC does.
=4.25pt
$t_0$ Event rate Risk score system AP AUC
---------- ------------ ------------------- ---------------------- --------------------
20 years 0.013 Simple 0.039 (0.029, 0.052) 0.79 (0.75, 0.82)
Heart dose 0.077 (0.050, 0.127) 0.82 (0.78, 0.86)
Comparison 1.96 (1.41, 2.88) 0.03 (0.01, 0.05)
35 years 0.047 Simple 0.079 (0.066, 0.095) 0.81 (0.78, 0.84)
Heart dose 0.114 (0.092, 0.144) 0.82 (0.78, 0.85)
Comparison 1.45 (1.26, 1.70) 0.01 (-0.02, 0.03)
: Estimated $\ap_{t_0}$ and AUC$_{t_0}$ with 95% CIs for two risk scoring systems at $t_0$ = 20 and 35 years, respectively. The comparison of APs is measured by $\text{rAP}$ and the comparison of AUCs is measured by $\Delta\auc$. []{data-label="tab:CCSS"}
![CCSS Data analysis: panel (a) shows the estimates of the time-dependent AP for the simple model $\ap_{s,t_0}$ and heart dose model $\ap_{h,t_0}$; panel (b) shows the estimates of the time-dependent AUC for the simple model $\auc_{s,t_0}$ and heart dose model $\auc_{h,t_0}$; panel (c) shows the estimates of $\text{rAP}_{t_0}=\ap_{h,t_0}/\ap_{s,t_0}$, the ratio of the time-dependent AP of the heart dose model over that of the simple model; panel (d) shows the estimates of $\Delta \auc_{t_0}=\auc_{h,t_0}-\auc_{s,t_0}$, the difference of the time-dependent AUC between the heart dose model and the simple model. The dotted lines in panels (c) and (d) represent the pointwise 95%CI for $\text{rAP}_{t_0}$ and $\Delta \auc_{t_0}$, respectively.[]{data-label="fig:CCSS"}](figure2.eps){width="5.0in"}
Discussion {#sec:discussion}
==========
One of the main goals of clinical risk prediction is to screen the asymptomatic population and to stratify them for tailored intervention. *Prospective* accuracy measures such as PPV$_{t_0}$ is preferred for this purpose. However, the calculation of PPV$_{t_0}$ demands a threshold for continuous risk scores, which can create practical difficulties for evaluating risk score systems, especially when more than two systems are compared. In this paper, we defined and interpreted $\ap_{t_0}$, which is the area under the time-dependent precision-recall curve, for event time data. We proposed a nonparametric estimator of $\ap_{t_0}$ and a ratio estimator of $\ap_{t_0}$ for comparing two competing risk score systems. We suggested the use of the bootstrap method for inference, which is broadly applicable in practical settings. We also developed an R package `APtools` for download available in CRAN which implements our method for binary and survival outcomes.
The AUC has been the most widely used performance metric in the clinical research community. A number of authors have pointed out that the AUC is informative on the classification performance and discrimination power [@gail2005criteria; @zheng2008time; @yuan2015threshold], but not an appropriate metric for assessing the prospective accuracy performance [@moskowitz2004quantifying; @zheng2008time]. Consistent with the criticism on the insensitivity of the AUC in evaluating risk prediction models (See [@cook2007use]), our data analysis illustrated that using the AUC as the metric, the performance of the simple model and the heart dose model appears close. However,based on the AP, the heart dose model outperforms the simple model and could be preferred in clinical screening. Thus the conclusion reached based on the AUC in this example may mislead researchers and clinicians.
It should be noted that when comparing different risk score systems, the ranking of their AUCs and APs are not necessarily concordant; our simulation study in Section 4 gives such an example. As the purpose of this article is to introduce a time-dependent $\ap_{t_0}$, we refer readers to [@davis2006] for insight on the relationship between the ROC curve and PR curve and to [@su2015relationship] where the relationship between the AUC and the AP is illuminated. Thus, risk scores which perform well in separating cases from controls may perform poorly in identifying a higher risk subpopulation, which is the goal of screening. Compared to the AUC, the AP as the summary metric of PPV is better suited in evaluating the usefulness of the risk scores and comparing the *prospective* prediction performance among competing risk scores, when the objective is *screening* through risk stratification.
[@zheng2008time; @zheng2010semiparametric] proposed to use the curves of PPV$_{t_0}$ versus risk score quantiles as an assessment tool for quantifying prospective prediction accuracy. One curve corresponds to one particular value of $t_0$, which limits its ability to assess the accuracy across time points. In contrast, plotting AP$_{t_0}$ against time could facilitate visualizing the performance of different risk score systems over time in one single plot.
Unlike the AUC, the AP is event rate dependent and should be estimated in a prospective cohort or population-based study. AP cannot be estimated from a case-control study; the estimate will be of very little use because the prevalence rate is artificially fixed by the study design. While the range of the AUC is always between 0.5 and 1, the range of AP is between the event rate and 1. While AP’s wide range could be advantageous in differentiating risk score systems, caution is needed when re-evaluating risk score systems in other study populations for the same outcome. This is because the underlying event rate may differ among populations. Thus, it is possible that AP will select different risk score systems as superior for the same outcome in different study populations.
For future work, we will consider estimating the time-dependent AP with multiple markers, as well as the incremental value of AP by adding new markers on top of an existing risk profile. In addition, similar to the partial AUC, partial AP could be defined as the area over a certain range of interest, such as those at the low values of TPF where PPV is typically high. Competing risk in the context of AP is another topic that needs to be addressed for event time data.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Yan Chen for data support. Dr. Zhou’s research is supported by the Natural Sciences and Engineering Research Council of Canada. Dr. Yuan’s research is supported by the Canadian Institutes of Health Research. The CCSS was supported by the National Cancer Institute (CA55727, G.T. Armstrong, Principal Investigator).
|
---
abstract: |
We show that the stochastic Schrödinger equation for the filtered state of a system, with linear free dynamics, undergoing continual non-demolition measurement or either position or momentum, or both together, can be solved explicitly within a class of Gaussian states which we call extended coherent states. The asymptotic limit yields a class of relaxed states which we describe explicitly. Bellman’s principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic performance criteria is treated as the important special case and solved exactly for the class of relaxed states.
PACS numbers: 07.55.Ge, 42.50.Lc, 03.65.Ta, 05.45.Mt
author:
- |
John Gough[^1]\
Department of Computing & Mathematics\
Nottingham-Trent University, Burton Street,\
Nottingham NG1 4BU, United Kingdom.
title: Optimal Quantum Feedback Control for Canonical Observables
---
Introduction
============
Quantum noise was originally developed to model irreversible quantum dynamical systems, where it played an external and secondary role, however, the realization that it could be measured and the results used to influence the system evolution has had a profound effect on its physical status [@Belavkin89],[@CollettGardiner],[@Belavkin99]. The great leap forward since then has been made by experimentalists who have made the practical implementation of quantum state estimation and adaptive feedback control a reality. With this, has come new problems that have received intense interest in the physics community \[4-10\].
In this paper, we wish to treat the problem of how to describe the quantum evolution of a system with linear free dynamics when we perform non-demolition measurements of, typically both, canonical position and momentum. The problem where position measurements only are made has been of historical importance. In this situation, the model is the one considered by Ghirardi, Rimini and Weber [@GRW], who also obtained the asymptotic form for the state. The asymptotic solution, with explicit reference to the stochastic Schrödinger equation within the Itô formulation, was first given by Diósi [@Diosi(a)], see also Belavkin and Staszewski [@BS]. Essentially, the solution to the stochastic Schrödinger equation could be understood as an randomly parameterized Gaussian state. The parameters being mean position, mean momentum and a complex inverse variance. We shall show that the same class of states, which we term extended coherent states, suffice for the stochastic Schrödinger equation describing simultaneous monitoring of position and momentum.
The problem of optimal quantum feedback control can then be tackled at this point. Bellman equations have been derived previously for the optimal cost of controlling a qubit system [@BoutenEdwardsBelavkin]. In fact, the general problem can be understood as a classical control problem on the space of quantum states [@GSB] if one exploits the separation of quantum estimation component from the control component: here we may construct a, typically infinite dimensional, Hamilton Jacobi Bellman theory and are then faced with the problem of finding a sufficient parameterization of states for particular situation. In the case of non-demolition position and momentum measurements, we have that the extended coherent states offer a sufficient parameterization. The quadratic performance problem is the important special case and has been treated by Doherty and Jacobs [@DJ]for feedback from measuring one quadrature of a Bosonic mode. We show that this problem is solvable when both canonical observables are measured.
Stochastic Schrödinger Equation
-------------------------------
Consider a quantum system evolving with free Hamiltonian $H$ while undergoing continual diffusive interaction with several independent apparatuses, each coupling to the system in a Markovian manner with coupling operator $L_{j}$ for the $j$-th apparatus. (The $\left\{ L_{j}\right\} $ do not generally need to be either commuting or self-adjoint.) The state, $\psi
_{t}$, of the system continually updated using the output of the apparatuses, will then satisfy a stochastic Schrödinger equation of the type [@Diosi],[@Belavkin99],[@GutaBoutenMaassen], $$\begin{aligned}
\left| d\psi _{t}\right\rangle &=&\frac{1}{i\hbar }H\left| \psi
_{t}\right\rangle \,dt-\frac{1}{2}\sum_{j}\left( L_{j}^{\dagger
}L_{j}-2\lambda _{j}\left( t\right) L_{j}+\lambda _{j}^{2}\left( t\right)
\right) \left| \psi _{t}\right\rangle \,dt \notag \\
&&+\sum_{j}\left( L_{j}-\lambda _{j}\left( t\right) \right) \left| \psi
_{t}\right\rangle \,dW_{t}^{\left( j\right) }. \label{SSE}\end{aligned}$$ where $\lambda _{j}\left( t\right) =\func{Re}\left\langle \psi
_{t}|L_{j}\,\psi _{t}\right\rangle $ and $\left\{ W^{\left( j\right)
}\right\} $ is a multi-dimensional Wiener process with $dW_{t}^{\left(
j\right) }dW_{t}^{\left( k\right) }=\delta _{jk}dt$. This equation was first postulated in the context of filtering by Belavkin where the apparatuses are separate Bose fields and the $W_{t}^{\left( j\right) }$ are innovations processes obtained by de-trending the output processes.
The stochastic Schrödinger equation for measurement of canonically conjugate observables, $\hat{q}$ and $\hat{p}$, has been derived from first principles by Scott and Milburn [@SM]. They considered a discrete time model with simultaneous measurement of position and momentum by separate apparatuses, and considered the continuous time limit of progressively more imprecise and frequent measurements. Taking $L_{1}=\sqrt{\dfrac{\kappa }{2}}\hat{q}$ and $L_{2}=\sqrt{\dfrac{\tilde{\kappa}}{2}}\hat{p}$ and denoting the innovations by $W_{t}^{\left( 1\right) }=W_{t}$ and $W_{t}^{\left(
2\right) }=\tilde{W}_{t}$, their particular stochastic Schrödinger equation reads as $$\begin{aligned}
\left| d\psi _{t}\right\rangle &=&\left( \frac{1}{i\hbar }H-\frac{\kappa }{4}\left( \hat{q}-\left\langle \hat{q}\right\rangle _{t}\right) ^{2}-\frac{\tilde{\kappa}}{4}\left( \hat{p}-\left\langle \hat{p}\right\rangle
_{t}\right) ^{2}\right) \left| \psi _{t}\right\rangle \,dt \notag \\
&&+\sqrt{\frac{\kappa }{2}}\left( \hat{q}-\left\langle \hat{q}\right\rangle
_{t}\right) \left| \psi _{t}\right\rangle \,dW_{t}+\sqrt{\frac{\tilde{\kappa}}{2}}\left( \hat{p}-\left\langle \hat{p}\right\rangle _{t}\right) \left|
\psi _{t}\right\rangle \,d\tilde{W}_{t}. \label{SSEqp}\end{aligned}$$
The equation involves the expectations $\left\langle \hat{q}\right\rangle
_{t}=\left\langle \psi _{t}|\hat{q}\,\psi _{t}\right\rangle $ and $\left\langle \hat{p}\right\rangle _{t}=\left\langle \psi _{t}|\hat{p}\,\psi
_{t}\right\rangle $ and is therefore non-linear in the state $\psi _{t}$. Here the constants $\kappa $ m$^{-2}$s$^{-1}$ and $\tilde{\kappa}$ N$^{-2}$s$^{-3}$ are positive and describe the measurement strength for the two apparatuses. In general, $\kappa $ and $\tilde{\kappa}$ has units of inverse variance of position, respectively momentum, per unit time. In [@GS], the limiting procedure was revisited and, as an alternative to increasingly imprecise measurements, one could use increasingly weak interaction between the apparatuses and the system. The scaling between the imprecision of measurement, or weakness of interaction with the apparatus, and the rate at which the discrete measurements is made must be such as to allow a general central limit effect to take place. In principle, it is possible, to set up the apparatuses to obtain desired values of $\kappa $ and $\tilde{\kappa}$.
The purpose of [@SM] was to consider nonlinear dynamics, however, we shall only deal with quadratic Hamiltonians of the type $H=H\left(
f,v\right) $$$H=\frac{1}{2m}\hat{p}^{2}+\frac{1}{2}\hbar \mu \hat{q}^{2}-f\hat{q}+v\hat{p}.
\label{H}$$ Here $f$ and $v$ are external fields which will later be replaced with control functions. We shall show that it is possible to find a general solution for the stochastic state $\psi _{t}$, with initial condition being that we start in a coherent state, realized as a random wave function taking values in a special class of wave functions, termed extended coherent states.
Extended Coherent States
========================
Let $L^{2}\left( \mathbb{R}\right) $ be the Hilbert space of square integrable functions of position coordinate $x$ with standard Schrödinger representation of the canonical observables $\hat{q}$ and $\hat{p}$. By an *extended coherent state*, we mean a wave function $\psi \left( \bar{q},\bar{p},\eta \right) $, parameterized by real numbers $\bar{q},\bar{p}$ and a complex number $\eta =\eta ^{\prime }+i\eta ^{\prime
\prime } $ where $\eta ^{\prime }>0$, taking the form $$\left\langle x|\psi \left( \bar{q},\bar{p},\eta \right) \right\rangle
=\left( \frac{\eta ^{\prime }}{2\pi }\right) ^{1/4}\exp \left\{ -\frac{\eta
}{4}\left( x-\bar{q}\right) ^{2}+i\frac{\bar{p}}{\hbar }x\right\} .$$ When $\eta $ is real $\left( \eta ^{\prime \prime }=0\right) $, the vectors are just the well-known coherent states [@Louisell] The distribution of the canonical variables in extended coherent state $\psi \left( \bar{q},\bar{p},\eta \right) $ is Gaussian with characteristic function $$\left\langle \exp \left\{ ir\hat{q}+is\hat{p}\right\} \right\rangle _{\bar{q},\bar{p},\eta }=\exp \left\{ ir\bar{q}+is\bar{p}-\frac{1}{2}\left(
C_{qq}r^{2}+2C_{qp}rs+C_{pp}s^{2}\right) \right\} , \label{Weyl expectation}$$ where $$C_{qq}=\frac{1}{\eta ^{\prime }},\;C_{qp}=-\frac{\hbar \eta ^{\prime \prime }}{2\eta ^{\prime }},\;C_{pp}=\frac{\hbar ^{2}}{4}\left( \eta ^{\prime }+\frac{\eta ^{\prime \prime 2}}{\eta ^{\prime }}\right) . \label{covariances}$$ The mean values of the position and the momentum in an extended coherent state are evidently $\left\langle \hat{q}\right\rangle =\bar{q}$ and $\left\langle \hat{p}\right\rangle =p$ respectively. We have that $C_{qq}$ is the variance of $\hat{q}$, $C_{pp}$ is the variance of $\hat{p}$, while $C_{qp}=\frac{1}{2}\left\langle \hat{q}\hat{p}+\hat{p}\hat{q}\right\rangle
-\left\langle \hat{p}\right\rangle \left\langle \hat{q}\right\rangle $ is the covariance of $\ \hat{q}$ and $\hat{p}$.
Derivation of the Characteristic Function
-----------------------------------------
To establish $\left( \ref{Weyl expectation}\right) $, let us first recall that coherent states may be constructed from creation/annihilation operators $a^{\pm }=\dfrac{1}{2}\sqrt{\eta ^{\prime }}\hat{q}\pm \dfrac{1}{i\hbar
\sqrt{\eta ^{\prime }}}\hat{p}$ by identifying $\psi \left( \bar{q},\bar{p},\eta ^{\prime }\right) $ as the eigenstate of $a^{-}$ with eigenvalue $\alpha =\dfrac{1}{2}\sqrt{\eta ^{\prime }}\bar{q}-\dfrac{1}{i\hbar \sqrt{\eta ^{\prime }}}\bar{p}$. In particular, if $\Omega $ denotes the zero-eigenstate of $a^{-}$ then $$\psi \left( \bar{q},\bar{p},\eta ^{\prime }\right) =D_{\alpha }\,\Omega$$ where $D_{\alpha }=\exp \left\{ \alpha a^{+}-\alpha ^{\ast }a^{-}\right\} $ is a Weyl displacement unitary. Next observe that we may obtain extended coherent states from coherent states by the simple application of a unitary transformation: $$\psi \left( \bar{q},\bar{p},\eta ^{\prime }+i\eta ^{\prime \prime }\right)
\equiv V\;\psi \left( \bar{q},\bar{p},\eta ^{\prime }\right)$$ with $V=\exp \left\{ -\frac{i}{4}\eta ^{\prime \prime }\left( \hat{q}-\bar{q}\right) ^{2}\right\} $. (This transformation is, in fact, linear canonical.) We may introduce new canonical variables $\hat{q}^{\prime }$ and $\hat{p}^{\prime }$ by $\hat{q}^{\prime }=V^{\dagger }\hat{q}V\equiv \hat{q}$ and $\hat{p}^{\prime }=V^{\dagger }\hat{p}V=\hat{p}-\frac{1}{2}\hbar \eta
^{\prime \prime }\left( \hat{q}-\bar{q}\right) $. We note that $\exp \left\{
ir\hat{q}+is\hat{p}\right\} =D_{z}$ where $z=-\dfrac{1}{2}\hbar \sqrt{\eta
^{\prime }}s+i\dfrac{1}{\sqrt{\eta ^{\prime }}}r$ and $$V^{\dagger }D_{z}V=\exp \left\{ ir\hat{q}^{\prime }+is\hat{p}^{\prime
}\right\} =D_{w}\,e^{\frac{1}{2}i\hbar \eta ^{\prime \prime }\bar{q}s}$$ where $w=-\dfrac{1}{2}\hbar \sqrt{\eta ^{\prime }}s+i\dfrac{1}{\sqrt{\eta
^{\prime }}}\left( r-\frac{1}{2}\hbar \eta ^{\prime \prime }s\right) $. Using well-known properties for Weyl displacement operators [@Louisell] and their $\Omega $-state averages, we find $$\begin{aligned}
\left\langle \exp \left\{ ir\hat{q}+is\hat{p}\right\} \right\rangle _{\bar{q},\bar{p},\eta } &=&\left\langle \Omega |D_{\alpha }^{\dagger }V^{\dagger
}D_{z}VD_{\alpha }\,\Omega \right\rangle \\
&=&\left\langle \Omega |D_{\alpha }^{\dagger }D_{w}D_{\alpha }\,\Omega
\right\rangle e^{\frac{1}{2}i\hbar \eta ^{\prime \prime }\bar{q}s} \\
&=&e^{w\alpha ^{\ast }-w^{\ast }\alpha -\frac{1}{2}|w|^{2}}e^{\frac{1}{2}i\hbar \eta ^{\prime \prime }\bar{q}s}\end{aligned}$$ and substituting in for $\alpha $ and $w$ gives the required result.
Weyl Independence
-----------------
We say that the canonical variables are Weyl independent for a given state $\left\langle \,\cdot \,\right\rangle $, not necessarily pure, if we have the following factorization $$\left\langle \exp \left\{ ir\hat{q}+is\hat{p}\right\} \right\rangle
=\left\langle \exp \left\{ ir\hat{q}\right\} \right\rangle \,\left\langle
\exp \left\{ is\hat{p}\right\} \right\rangle$$ for all real $r$ and $s$. If the state possesses moments to all orders, then Weyl independence means that symmetrically (Weyl) ordered moments factor according to $\left\langle :f\left( \hat{q}\right) g\left( \hat{p}\right)
:\right\rangle =\left\langle f\left( \hat{q}\right) \right\rangle
\left\langle g\left( \hat{p}\right) \right\rangle $, for all polynomials $f,g
$. By inspection, we see that coherent states leave the canonical variables Gaussian and Weyl-independent. However, the $\eta ^{\prime \prime }\neq 0$ extended states do not have this Weyl-independence property.
Stochastic Wave Function
========================
We now return to the equation $\left( \ref{SSEqp}\right) $ for the conditioned state $\psi _{t}$. Let $\left\langle X\right\rangle
_{t}=\left\langle \psi _{t}\right| X\left| \psi _{t}\right\rangle $, for a general operator $X$, then we have the following stochastic Ehrenfest equation
$$\begin{gathered}
d\left\langle X\right\rangle =\left\{ \frac{1}{i\hbar }\left\langle \left[
X,H\right] \right\rangle -\frac{\kappa }{4}\left\langle \left[ \left[ X,\hat{q}\right] ,\hat{q}\right] \right\rangle -\frac{\tilde{\kappa}}{4}\left\langle \left[ \left[ X,\hat{p}\right] ,\hat{p}\right] \right\rangle
\right\} \,dt \notag \\
+\sqrt{\frac{\kappa }{2}}\left( \left\langle X\hat{q}+\hat{q}X\right\rangle
-\left\langle \hat{q}\right\rangle \left\langle X\right\rangle \right)
\,dW_{t}+\sqrt{\frac{\tilde{\kappa}}{2}}\left( \left\langle X\hat{p}+\hat{p}X\right\rangle -\left\langle \hat{p}\right\rangle \left\langle
X\right\rangle \right) \,d\tilde{W}_{t}. \label{filter X}\end{gathered}$$
For $X=\hat{q},\hat{p}$, we find $$\begin{aligned}
d\left\langle \hat{q}\right\rangle &=&\left( \frac{1}{m}\left\langle \hat{p}\right\rangle +v\right) \,dt+\sqrt{2\kappa }C\left( \hat{q},\hat{q}\right)
\,dW_{t}+\sqrt{2\tilde{\kappa}}C\left( \hat{q},\hat{p}\right) \,d\tilde{W}_{t}, \notag \\
d\left\langle \hat{p}\right\rangle &=&\left( -\hbar \mu \left\langle \hat{q}\right\rangle +f\right) \,dt+\sqrt{2\kappa }C\left( \hat{q},\hat{p}\right)
\,dW_{t}+\sqrt{2\tilde{\kappa}}C\left( \hat{p},\hat{p}\right) \,d\tilde{W}_{t}. \label{filter q,p}\end{aligned}$$ where $C\left( \hat{q},\hat{q}\right) =\left\langle \hat{q}^{2}\right\rangle
-\left\langle \hat{q}\right\rangle ^{2},C\left( \hat{p},\hat{p}\right)
=\left\langle \hat{p}^{2}\right\rangle -\left\langle \hat{p}\right\rangle
^{2},$ and $C\left( \hat{q},\hat{p}\right) =\frac{1}{2}\left\langle \hat{p}\hat{q}+\hat{q}\hat{p}\right\rangle -\left\langle \hat{p}\right\rangle
\left\langle \hat{q}\right\rangle $. In the following, we wish to investigate the dynamical evolution of the random state $\psi $ starting from an initial coherent state. It turns out however that we do not remain within the class of coherent states: if we did, then $\hat{q}$ and $\hat{p}$ would remain Weyl-independent and, in particular, $C\left( \hat{q},\hat{p}\right) $ would vanish, along with the noise term in the $\left\langle \hat{p}\right\rangle $-equation of $\left( \ref{filter q,p}\right) $ above and this would lead to an inconsistent system of equations. Fortunately, it turns out that it is possible to think of $\psi $ as evolving as a random state taking values amongst the extended coherent states. Explicitly, we make the ansatz that the state $\psi _{t}$ takes the form $$\psi _{t}=\psi \left( \bar{q}_{t},\bar{p}_{t},\eta _{t}\right)
\label{ansatz}$$ where $\bar{q}_{t}$ and $\bar{p}_{t}$ are real-valued diffusion processes satisfying and $\eta _{t}$ is a complex-valued deterministic function. Our assumption that we start from a coherent state is equivalent to asking that $\eta \left( 0\right) =\sigma ^{-2}>0$, with $\sigma $ having the interpretation as the initial dispersion in position.
We shall now show that $\bar{q}$, $\bar{p}$ satisfy the diffusion equations $\left( \ref{filter q,p}\right) $, while $\eta $ satisfies the Riccati equation $$\frac{d}{dt}\eta =2\kappa +i2\mu -\frac{1}{2}\left( \tilde{\kappa}\hbar
^{2}+i\frac{\hbar }{m}\right) \eta ^{2}. \label{Riccati}$$
Consistency with the Statistical Evolution
------------------------------------------
Let $r,s$ be fixed real parameters and set $D=\exp \left\{ ir\hat{q}+is\hat{p}\right\} $. We shall investigate the evolution through the characteristic function $$G_{t}=\left\langle \psi _{t}\right| D\left| \psi _{t}\right\rangle
=\left\langle D\right\rangle _{t}.$$ Observing that $\left[ D,\hat{q}\right] =\hbar sD,\;\left[ D,\hat{p}\right]
=-\hbar rD$ we find $$\begin{gathered}
dG=\left\{ \frac{ir}{2m}\left\langle \hat{p}D+D\hat{p}\right\rangle -\frac{is\hbar \mu }{2}\left\langle \hat{q}D+D\hat{q}\right\rangle +\left( ifs+ivr-\frac{\hbar ^{2}\left( \kappa s^{2}+\tilde{\kappa}r^{2}\right) }{4}\right)
G\right\} \,dt \\
+\sqrt{\frac{\kappa }{2}}\left( \left\langle D\hat{q}+\hat{q}D\right\rangle
-\left\langle \hat{q}\right\rangle G\right) \,dW+\sqrt{\frac{\tilde{\kappa}}{2}}\left( \left\langle D\hat{p}+\hat{p}D\right\rangle -\left\langle \hat{p}\right\rangle G\right) \,d\tilde{W}.\end{gathered}$$ The identity $e^{ir\hat{q}+is\hat{p}}=e^{\frac{1}{2}irs\hbar }e^{ir\hat{q}}e^{is\hat{p}}=e^{-\frac{1}{2}irs\hbar }e^{is\hat{p}}e^{ir\hat{q}}$ (Baker Campbell Hausdorff formula) then allows us to compute that $$\left\langle \hat{q}D\right\rangle =e^{\frac{1}{2}irs\hbar }\frac{1}{i}\frac{\partial }{\partial r}\left( e^{-\frac{1}{2}irs\hbar }G\right) =\left( \bar{q}+i\left( C_{qq}^{2}r+C_{qp}^{2}s\right) +\frac{1}{2}s\hbar \right) G,$$ and likewise $$\begin{aligned}
\left\langle D\hat{q}\right\rangle &=&\left( \bar{q}+i\left(
C_{qq}r+C_{qp}s\right) -\frac{1}{2}s\hbar \right) G, \\
\left\langle \hat{p}D\right\rangle &=&\left( \bar{p}+i\left(
C_{qp}r+C_{pp}s\right) +\frac{1}{2}r\hbar \right) G, \\
\left\langle D\hat{p}\right\rangle &=&\left( \bar{p}+i\left(
C_{qp}r+C_{pp}s\right) -\frac{1}{2}r\hbar \right) G.\end{aligned}$$ Hence $$\begin{aligned}
dG &=&\frac{ir}{m}\left\{ \bar{p}+i\left( C_{qp}r+C_{pp}s\right) \right\}
G\,dt-is\hbar \mu \left\{ \bar{q}+i\left( C_{qq}r+C_{qp}s\right) \right\}
G\,dt \notag \\
&&+\left( ifs+ivr-\frac{\kappa \hbar ^{2}s^{2}}{4}-\frac{\tilde{\kappa}\hbar
^{2}r^{2}}{4}\right) G\,dt \notag \\
&&+i\sqrt{2\kappa }\left( C_{qq}r+C_{qp}s\right) G\,dW+i\sqrt{2\tilde{\kappa}}\left( C_{qp}r+C_{pp}s\right) G\,d\tilde{W}. \label{dG(1)}\end{aligned}$$ Under our ansatz $\left( \ref{ansatz}\right) $, we should also have, by the Itô rule, $$\begin{aligned}
dG &=&\frac{\partial G}{\partial \bar{q}}d\bar{q}+\frac{\partial G}{\partial
\bar{p}}d\bar{p}+\frac{1}{2}\frac{\partial ^{2}G}{\partial \bar{q}^{2}}\left( d\bar{q}\right) ^{2}+\frac{\partial ^{2}G}{\partial \bar{q}\partial
\bar{p}}\left( d\bar{q}d\bar{p}\right) +\frac{1}{2}\frac{\partial ^{2}G}{\partial \bar{p}^{2}}\left( d\bar{p}\right) ^{2} \notag \\
&&+\frac{\partial G}{\partial \eta ^{\prime }}d\eta ^{\prime }+\frac{\partial G}{\partial \eta ^{\prime \prime }}d\eta ^{\prime \prime } \notag
\\
&=&irGd\bar{q}+isGd\bar{p}-\frac{1}{2}r^{2}G\left( d\bar{q}\right)
^{2}-rsG\left( d\bar{q}d\bar{p}\right) -\frac{1}{2}s^{2}G\left( d\bar{p}\right) ^{2} \notag \\
&&+\left( \frac{1}{2\eta ^{\prime 2}}r^{2}-\frac{\hbar \eta ^{\prime \prime }}{2\eta ^{\prime }}rs-\frac{\hbar ^{2}}{8}\left( 1-\frac{\eta ^{\prime
\prime 2}}{\eta ^{\prime 2}}\right) s^{2}\right) d\eta ^{\prime } \notag \\
&&+\left( \frac{\hbar }{2\eta ^{\prime }}rs-\frac{1}{4}\frac{\hbar ^{2}\eta
^{\prime \prime }}{\eta ^{\prime }}s^{2}\right) d\eta ^{\prime \prime }.
\label{dG(2)}\end{aligned}$$ Equating the coefficients of $\left( \ref{dG(1)}\right) $ and $\left( \ref
{dG(2)}\right) $ gives the system of equations $$\begin{aligned}
r &:&d\bar{q}=\left( \dfrac{1}{m}\bar{p}+v\right) \,dt+\sqrt{2\kappa }C_{qq}\,dW+\sqrt{2\tilde{\kappa}}C_{qp}\,d\tilde{W}, \\
s &:&d\bar{p}=\left( -\hbar \mu \bar{q}+f\right) \,dt+\sqrt{2\kappa }C_{qp}\,dW+\sqrt{2\tilde{\kappa}}C_{pp}\,d\tilde{W}, \\
r^{2} &:&\left( d\bar{q}\right) ^{2}-\dfrac{1}{\eta ^{\prime 2}}d\eta
^{\prime }=\dfrac{1}{m}C_{qp}dt+\dfrac{\tilde{\kappa}\hbar ^{2}}{2}\,dt, \\
s^{2} &:&\left( d\bar{p}\right) ^{2}+\dfrac{\hbar ^{2}}{4}\left( 1-\dfrac{\eta ^{\prime \prime 2}}{\eta ^{\prime 2}}\right) \,d\eta ^{\prime }+\frac{1}{2}\dfrac{\hbar ^{2}\eta ^{\prime \prime }}{\eta ^{\prime }}\,d\eta
^{\prime \prime }=-2\hbar \mu C_{qp}\,dt+\dfrac{\kappa \hbar ^{2}}{2}\,dt, \\
rs &:&\left( d\bar{q}d\bar{p}\right) +\dfrac{\hbar \eta ^{\prime \prime }}{2\eta ^{\prime 2}}\,d\eta ^{\prime }-\dfrac{\hbar }{2\eta ^{\prime }}\,d\eta
^{\prime \prime }=\frac{1}{m}C_{pp}\,dt-\hbar \mu \,C_{qq}dt.\end{aligned}$$ The first two of these agree exactly with $\left( \ref{filter q,p}\right) $, while the next three are entirely consistent with the pair of real equations $$\left\{
\begin{array}{c}
\dfrac{d}{dt}\eta ^{\prime }=2\kappa +\dfrac{\hbar }{m}\eta ^{\prime }\eta
^{\prime \prime }-\frac{1}{2}\tilde{\kappa}\hbar ^{2}\left( \eta ^{\prime
2}-\eta ^{\prime \prime 2}\right) , \\
\\
\dfrac{d}{dt}\eta ^{\prime \prime }=2\mu -\dfrac{\hbar }{2m}\left( \eta
^{\prime 2}-\eta ^{\prime \prime 2}\right) -\tilde{\kappa}\hbar ^{2}\eta
^{\prime }\eta ^{\prime \prime }.
\end{array}
\right.$$ Together, they are equivalent to the single complex Riccati equation $\left(
\ref{Riccati}\right) $.
Asymptotic States
-----------------
The Riccati equation $\left( \ref{Riccati}\right) $ is to be solved in the half plane $\eta ^{\prime }>0$ of physical solutions and has the unique, globally attractive, fixed point $$\eta _{\infty }=\frac{2}{\hbar }\sqrt[+]{\frac{\kappa +i\mu }{\tilde{\kappa}+\dfrac{i}{m\hbar }}}.$$ (Here $\sqrt[+]{\cdot }$ denotes the complex root having positive real part.)
In the case of a harmonic oscillator of frequency $\omega $, we have $\mu =\dfrac{m\omega ^{2}}{\hbar }>0$ and we may achieve a coherent state ($\eta
_{\infty }$ real) as the limit state if we tune the measurement strengths such that $\kappa \equiv m^{2}\omega ^{2}\,\tilde{\kappa}$. In this case, $\eta _{\infty }\equiv \dfrac{2m\omega }{\hbar }$, corresponding to a coherent state with position uncertainty $\sigma _{\infty }=\sqrt{\dfrac{\hbar }{2m\omega }}$. Otherwise the limit state will be an extended coherent state.
We should remark that $\sqrt{\dfrac{\kappa }{\tilde{\kappa}}}$ corresponds to the squeezing parameter $s$ introduced in [@SM] to describe the bias in favor of the $\hat{q}$ or $\hat{p}$ coupling.
Optimal Quantum Feedback Control
================================
We fix a terminal time $T>0$ and let $\left\{ f_{t}:0<t<T\right\} $ and $\left\{ v_{t}:0<t<T\right\} $ be prescribed functions which we refer to as control policies. Let $\psi _{t}=\psi \left( \bar{q}_{t},\bar{p}_{t},\eta
_{t}\right) $ be the solution to the stochastic Schrödinger equation with time-dependent free Hamiltonian $H=H\left( f_{t},v_{t}\right) $ and initial state being an extended state $\psi \left( \bar{q}_{0},\bar{p}_{0},\eta _{0}\right) $ at time $t_{0}$ somewhere in the time interval $\left[ 0,T\right] $.
We wish to grade the control policies $\left\{ f_{t}\right\} $ and $\left\{
v_{t}\right\} $ over the time interval $\left[ t_{0},T\right] $ and do so by assigning a cost $J=J\left[ \left\{ f_{t}\right\} ,\left\{ v_{t}\right\}
;t_{0},T;\bar{q}_{0},\bar{p}_{0},\eta _{0}\right] $ taking the general form $$J\left[ \left\{ f_{t}\right\} ,\left\{ v_{t}\right\} ;t_{0},T;\bar{q}_{0},\bar{p}_{0},\eta _{0}\right] =\int_{t_{0}}^{T}\ell \left( s;f_{s},v_{s};\bar{q}_{s},\bar{p}_{s},\eta _{s}\right) ds+g\left( \bar{q}_{T},\bar{p}_{T},\eta
_{T}\right) .$$ Here $\ell $ is a function of time, the current control policy values, and current state parameters. The function $g$, known as a target or bequest function in control theory, is a function of the state parameters at termination. We assume that both are continuous in their arguments.
The cost $J$ will vary from one experimental trial to another, and must be thought of as a random variable depending on the measurement output. The aim of this section is to evaluate the minimum average cost over all possible control policies, which we denote as $$S\left( t_{0},T;\bar{q}_{0},\bar{p}_{0},\eta _{0}\right) =\min_{\left\{
f_{t}\right\} ,\left\{ v_{t}\right\} }\mathbb{E}\left\{ J\left[ \left\{
f_{t}\right\} ,\left\{ v_{t}\right\} ;t_{0},T;\bar{q}_{0},\bar{p}_{0},\eta
_{0}\right] \right\} .$$
Bellman Optimality Principle
----------------------------
For simplicity, let us write $z\equiv \left( \bar{q},\bar{p},\eta \right) $ and $u=\left( f,v\right) $ and $S\equiv S\left( t_{0};z_{t_{0}}\right) $, etc. Taking $t_{0}<t_{0}+\Delta t<T$, we have that $$S\left( t_{0};z_{t_{0}}\right) =\min_{\left\{ f_{t}\right\} ,\left\{
v_{t}\right\} }\mathbb{E}\left\{ \int_{t_{0}}^{t_{0}+\Delta t}\ell \left(
s;u_{s};z_{s}\right) ds+J\left[ \left\{ u_{t}\right\} ;t_{0}+\Delta
t,T;z_{0}+\Delta z\right] \right\}$$ where $\Delta z=z_{t}-z_{t_{0}}$ is, of course the random change in the state parameters from time $t_{0}$ to $t_{0}+\Delta t$. We have that $$\int_{t_{0}}^{t_{0}+\Delta t}\ell \left( s;u_{s};z_{s}\right) ds=\ell \left(
t_{0},u_{t_{0}},z_{t_{0}}\right) \,\Delta t+o\left( \Delta t\right)$$ up to terms that are small of order in $\Delta t$. Likewise, assuming that $S $ will be sufficiently differentiable, $$\begin{aligned}
&&S\left( t_{0}+\Delta t;z_{0}+\Delta z\right) \\
&=&S\left( t_{0};z_{0}\right) +\left. \frac{\partial S}{\partial t}\right|
_{0}\Delta t+\left. \frac{\partial S}{\partial z}\right| _{0}\Delta z+\frac{1}{2}\Delta z^{\prime }\left. \frac{\partial ^{2}S}{\partial z^{2}}\right|
_{0}\Delta z+o\left( \Delta t\right) \\
&=&S\left( t_{0};z_{0}\right) +\left. \frac{\partial S}{\partial t}\right|
_{0}\Delta t+\left. \frac{\partial S}{\partial \bar{q}}\right| _{0}\left(
\frac{1}{m}\bar{p}+v_{t}\right) \Delta t+\left. \frac{\partial S}{\partial
\bar{p}}\right| _{0}\left( -\hbar \mu \bar{q}+f_{t}\right) \Delta t \\
&&+\left. \frac{\partial S}{\partial \eta ^{\prime }}\right| _{0}\frac{d\eta
^{\prime }}{dt}\Delta t+\left. \frac{\partial S}{\partial \eta ^{\prime
\prime }}\right| _{0}\frac{d\eta ^{\prime \prime }}{dt}\Delta t \\
&&+\frac{1}{2}\left. \frac{\partial ^{2}S}{\partial \bar{q}^{2}}\right| _{0}\left[ 2\kappa C_{qq}^{2}+2\tilde{\kappa}C_{qp}^{2}\right] \Delta t+\frac{1}{2}\left. \frac{\partial ^{2}S}{\partial \bar{p}^{2}}\right| _{0}\left[
2\kappa C_{qp}+2\tilde{\kappa}C_{pp}\right] \Delta t \\
&&+\left. \frac{\partial ^{2}S}{\partial \bar{q}\partial \bar{p}}\right|
_{0}2\sqrt{\kappa \tilde{\kappa}}\left[ C_{qq}^{2}+C_{pp}^{2}\right]
C_{qp}\Delta t+o\left( \Delta t\right) .\end{aligned}$$ (On the right hand side, we are evaluating at $t_{0},\bar{q}_{0},\bar{p}_{0},\eta _{0}$.)
The Bellman principle of optimality [@Bellman], see also [@Davis] for instance, states that if $\left\{ u_{t}^{\ast }\right\} $ is an optimal control policy exercised over the time interval $\left[ t_{0},T\right] $ for a given start state at time $t_{0}$, then if we operated this policy up to time $t_{0}+\Delta t$ then the remaining component of the policy will be optimal for the control problem over $\left[ t_{0}+\Delta t,T\right] $ with start state being the current (random) state at time $t_{0}+\Delta T$. If we assume the existence of such an optimal policy, then, within the above approximations, as $\Delta t\rightarrow 0^{+}$, we are lead to the partial differential equation (Hamilton Jacobi Bellman equation, or just Bellman equation) for $S=S\left( t;\bar{q},\bar{p},\eta \right) $
$$\begin{gathered}
0=\frac{\partial S}{\partial t}+\mathcal{H}\left( t;\bar{q},\bar{p},\eta ;\frac{\partial S}{\partial \bar{q}},\frac{\partial S}{\partial \bar{p}}\right) +\frac{\partial S}{\partial \eta ^{\prime }}\frac{d\eta ^{\prime }}{dt}+\frac{\partial S}{\partial \eta ^{\prime \prime }}\frac{d\eta ^{\prime
\prime }}{dt} \notag \\
+\frac{\partial ^{2}S}{\partial \bar{q}^{2}}\left[ \kappa C_{qq}^{2}+\tilde{\kappa}C_{qp}^{2}\right] +2\frac{\partial ^{2}S}{\partial \bar{q}\partial
\bar{p}}\sqrt{\kappa \tilde{\kappa}}\left[ C_{qq}+C_{pp}\right] C_{qp}+\frac{\partial ^{2}S}{\partial \bar{p}^{2}}\left[ \kappa C_{qp}^{2}+\tilde{\kappa}C_{pp}^{2}\right] \label{Bellman equation}\end{gathered}$$
where we introduce
$$\mathcal{H}\left( t;\bar{q},\bar{p},\eta ;y_{q},y_{p}\right)
:=\min_{f,v}\left\{ y_{q}\left( \frac{1}{m}\bar{p}+v\right) +y_{p}\left(
-\hbar \mu \bar{q}+f\right) +\ell \left( t;f,v;\bar{q},\bar{p},\eta \right)
\right\} .$$
It should perhaps be stressed that the derivation of this equation is entirely classical. The key feature of the Bellman equation is that the minimum is now taken pointwise: that is we look for the optimal scalar values $f,v$ at a single instant of time. The equation is to be solved subject to the terminal condition $\lim_{t\rightarrow T^{-}}S\left( t,\bar{q},\bar{p},\eta \right) =g\left( \bar{q},\bar{p},\eta \right) $.
In principle, once a minimizing solution $f^{\ast }=f\left( t;\bar{q},\bar{p},\eta \right) ,v^{\ast }=v^{\ast }\left( t;\bar{q},\bar{p},\eta \right) $ is known, it may be used as a Markov control for closed loop feedback: that is, the control policies are taken as these functions of the current state parameters.
The Bellman equations arising in quantum feedback control have so far proved to be highly nonlinear and prohibitively hard to solve as a rule. Our equation $\left( \ref{Bellman equation}\right) $ is no exception, however, the nonlinearities are in due to the $\eta $ variable. We remark that if we assume that we start off in a state relaxed at the equilibrium value $\eta
=\eta _{\infty }$, then the coefficients of the $\eta ^{\prime },\eta
^{\prime \prime }$ derivatives vanish exactly, and we may take the covariances $C_{qq}$, $C_{qp}$ and $C_{pp}$ at their relaxed value determined from $\left( \ref{covariances}\right) $ evaluated at the asymptotic value $\eta _{\infty }$. As the relaxation time is typically small, we may justify this for large times $T$ in comparison. This ignores any $\eta $-transient contribution to the cost, but at least opens up the possibility of solving the Bellman equation and finding optimal Markov control policies. We give the fundamental class of interest, quadratic performance criteria, next.
Linear Quantum Stochastic Regulator
-----------------------------------
We consider the following quadratic control problem not involving any costs on the $\eta $ parameter. In particular, we make the assumption that the starting state is an asymptotic state $\left( \eta =\eta _{\infty }\right) $ and so we ignore $\eta $ as a variable. We set $x=\left( \bar{q},\bar{p}\right) $ and $u=\left( f,v\right) $ and take the specific choices $$\begin{aligned}
\ell \left( t,u,x\right) &=&\frac{1}{2}x^{\prime }A_{t}x+\frac{1}{2}u^{\prime }E_{t}u, \\
g\left( x\right) &=&\frac{1}{2}x^{\prime }Rx,\end{aligned}$$ where $A_{t},E_{t}$ and $R$ are $2\times 2$ symmetric matrices with $E_{t}$ being invertible. The free Heisenberg equations are linear and can be written as $\dot{x}_{t}=F_{t}x_{t}+M_{t}u$. The control problem is now essentially the same as the classical stochastic regulator [@Davis]. In this case we introduce a dual variable $y$ to $x$ and obtain the $\mathcal{H}
$-function $$\begin{aligned}
\mathcal{H}\left( t,x,y\right) &=&\min_{u}\left\{ \ell \left( t,u,x\right)
+y^{\prime }\left( F_{t}x+M_{t}u\right) \right\} \\
&=&\frac{1}{2}x^{\prime }A_{t}x+y^{\prime }F_{t}x+\min_{u}\left\{ \frac{1}{2}u^{\prime }E_{t}u+y^{\prime }M_{t}u\right\} \end{aligned}$$ with the minimum attained at $$u^{\ast }=-E_{t}^{-1}M_{t}^{\prime }y$$ and we find $$\mathcal{H}\left( t,x,y\right) =\frac{1}{2}x^{\prime }A_{t}x+y^{\prime
}F_{t}x-\frac{1}{2}y^{\prime }M_{t}E_{t}^{-1}M_{t}^{\prime }y$$ Seeking an $\eta $-independent solution, the Bellman equation $\left( \ref
{Bellman equation}\right) $ reduces to $$0=\frac{\partial S}{\partial t}+\mathcal{H}\left( t,x,\nabla S\right) +\frac{1}{2}K_{ij}\frac{\partial ^{2}S}{\partial x_{i}\partial x_{j}}.$$ Here $K$ is the matrix of the second order coefficients in $\left( \ref
{Bellman equation}\right) $ and these will be determined by the covariances $\left( \ref{covariances}\right) $ determined at the asymptotic value $\eta
_{\infty }$. As is well known [@Davis], the solution takes the form $S\left( t,x\right) =\dfrac{1}{2}x^{\prime }\Sigma _{t}x+a_{t}$ where $\Sigma
_{t}$ satisfies the matrix Riccati equation $$\frac{d\Sigma _{t}}{dt}=-\Sigma _{t}F_{t}-F_{t}^{\prime }\Sigma _{t}+\Sigma
_{t}M_{t}E_{t}^{-1}M_{t}^{\prime }\Sigma _{t}-A_{t},\qquad \Sigma _{T}=R,$$ while $a_{t}$ satisfies $$\frac{da_{t}}{dt}=-tr\left\{ K\Sigma _{t}\right\} ,\qquad a_{T}=0.$$
The optimal control policy is therefore given by $$u^{\ast }\left( t,x\right) =-E_{t}^{-1}M_{t}^{\prime }\nabla
S=E_{t}^{-1}M_{t}^{\prime }\Sigma _{t}x.$$
Commentary
----------
The sufficiency property of the extended coherent states means that the results above are of importance to the corresponding filtering problem. Indeed this allows us to implement a quantum analogue of the Kalman filter for state estimation amongst the class of extended coherent states. The Kalman filter is of considerable conceptual and practical importance in classical control theory and plays a crucial role in optimal feedback control. In fact, the matrix Riccati equation occurring in linear stochastic regulator also appears in a dual formulation as a Kalman filtering problem [@Davis]. Unfortunately, the solution to the fully parameterized Bellman equation, that is, when we do not start from the equilibrium value $\eta
=\eta _{\infty }$, seems to be disappointingly difficult even in the linear regulator example as the matrix $K$ will be quartic in $\eta $. (Such difficulties seem to be sadly the norm in applications to optimal quantum control as a whole, so far.) The control problem is however tractable for the class of relaxed coherent states and corresponds to the linear regulator model for quadratic performance and this at least gives us some insight into possible applications.
**Acknowledgment:** The author is indebted to Luc Bouten for a critical reading of the article.
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[^1]: john.gough@ntu.ac.uk
|
---
abstract: 'Numerical simulations for the merger of binary neutron stars are performed in full general relativity incorporating both nucleonic and hyperonic finite-temperature equations of state (EOS) and neutrino cooling. It is found that for the nucleonic and hyperonic EOS, a hyper massive neutron star (HMNS) with a long lifetime $(t_{\rm life}\gtrsim 10~{\rm ms})$ is the outcome for the total mass $\approx 2.7~M_\odot$. For the total mass $\approx 3~M_\odot$, a long-lived (short-lived with $t_{\rm life}\approx 3~{\rm ms}$) HMNS is the outcome for the nucleonic (hyperonic) EOS. It is shown that the typical total neutrino luminosity of the HMNS is $\sim 3$ – $6 \times 10^{53}~{\rm erg /s}$ and the effective amplitude of gravitational waves from the HMNS is $1$ – $4\times 10^{-22}$ at $f\approx 2$ – $3.2~{\rm kHz}$ for a source of distance of 100 Mpc. During the HMNS phase, characteristic frequencies of gravitational waves shift to a higher frequency for the hyperonic EOS in contrast to the nucleonic EOS in which they remain constant approximately. Our finding suggests that the effects of hyperons are well imprinted in gravitational wave and its detection will give us a potential opportunity to explore the composition of the neutron star matter. We present the neutrino luminosity curve when a black hole is formed as well.'
address: 'Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan'
author:
- 'Kenta Kiuchi, Yuichiro Sekiguchi, Koutarou Kyutoku, Masaru Shibata'
title: 'Gravitational waves, neutrino emissions, and effects of hyperons in binary neutron star mergers'
---
Introduction
============
Coalescence of binary neutron stars (BNS) is drawing attention of researchers in various fields because it is one of most promising source for next generation kilo-meter-size gravitational-wave (GW) detectors [@LIGO] and a possible candidate for the progenitor of short-hard gamma-ray bursts [@SGRB] as well as a high-end laboratory of the nuclear theory [@Lattimer].
BNSs evolve due to gravitational radiation reaction and eventually merge. Before the merger sets in, each neutron star (NS) is cold, (i.e., thermal energy of constituent nucleons is much smaller than the Fermi energy), because thermal energy inside the NSs is significantly reduced by neutrino and photon coolings due to the long-term evolution (typically $\gtrsim~10^8$ yrs) until the merger. By contrast, after the merger, shocks are generated by hydrodynamic interactions, which heat a merged object, hyper massive neutron star (HMNS), up to $\sim 30-50$ MeV, and hence, copious neutrinos are emitted. The rest mass density $\rho$ of the merged object will go well beyond the nuclear matter density $\rho_{\rm nuc}\approx 2.8\times 10^{14}{\rm g/cm^3}$ after the merger. In such a high density region, properties of the NS matter are still poorly understood. In recent years, the influence of non-nucleonic degrees of freedom, such as hyperons, meson condensations, and quarks, on properties of the NS matter has been discussed extensively. Among exotic particles, $\Lambda$ hyperon are believed to appear first in (cold) NS around the rest mass density of $\rho\sim2-3~\rho_{\rm nuc}$ [@Lattimer; @Schaffner:1995th]. The presence of such the exotic particle results in softening of equation of state (EOS). Hence, if it exists or appears in the course of BNS mergers, it will affect the dynamics of BNS mergers and be imprinted in emitting GWs as a result.
Motivated by these facts, numerical simulations have been extensively performed for the merger of BNS in the framework of full general relativity in the past decade since the first success in 2000 [@Shibata:1999wm]. In particular, it is mandatory to do numerical relativity simulations of BNS mergers implementing a finite temperature EOS as well as a neutrino cooling. In this proceeding, we present the results based on Refs. [@Sekiguchi:2011zd; @Sekiguchi:2011mc], in which the NSs are modeled with a nuclenoic or hyperonic EOS and the cooling is properly taken into account. Specifically, we adopt Shen-EOS for the nucleonic EOS [@Shen:1998by] and a finite-temperature EOS including contributions of $\Lambda$ hyperons as the hyperonic EOS [@Shen:2011qu]. In the following, we report a comparison between the models with these EOSs, particularly focused on the dynamics of BNS merger, emitted GWs, and neutrinos.
Framework
=========
Formulation and EOS
-------------------
Numerical simulations in full general relativity are performed using the following formulation and numerical schemes: Einstein’s equations are solved in the original version of Baumgarte-Shapiro-Shibata-Nakamura formulation implementing the puncture method [@BSSN]; a fourth-order finite differencing in space and a fourth-order Runge-Kutta time integration are used; a conservative shock capturing scheme with third-order accuracy in space and fourth-order accuracy in time is employed for solving hydrodynamic equations [@Sekiguchi:2011zd; @Sekiguchi:2011mc]. We solve evolution equations for neutrino $(Y_\nu)$, electron $(Y_e)$, and total lepton $(Y_l)$ fractions per baryon, taking into account weak interaction processes and neutrino cooling employing a general relativistic leakage scheme for electron type $(\nu_e)$, electron anti-type $(\bar{\nu}_e)$, and other types $(\mu/\tau)$ of neutrinos $(\nu_x)$ [@Sekiguchi:2010ep; @Sekiguchi:2010ja]. We implement Shen-EOS [@Shen:1998by], tabulated in terms of the rest-mass density $(\rho)$, temperature $(T)$, and $Y_e$ or $Y_l$, as the nucleonic EOS and a finite-temperature EOS including contributions of $\Lambda$ hyperons (Hyp-EOS) as the hyperonic EOS [@Shen:2011qu]. These EOSs produce a maximum mass of zero-temperature spherical NS of $\approx 2.2 M_\odot$ for Shen-EOS and $\approx 1.8 M_\odot$ for Hyp-EOS. Although the maximum mass of Hyp-EOS is slightly smaller than the observational constraint of PSR J1614-2230 $(M_{J1614-2230}=1.97 \pm 0.04 M_\odot)$ [@Demorest], it deserves employing to explore the impact of hyperons on the BNS merger because it can be a viable of EOS of the NS matter not in extremely high densities and hence for studying the evolution of the HMNS that do not have the extremely high densities for most of their lifetime. It should be also noted that it is possible to produce a more massive neutron star with hyperons by fine-tuning the parameters (see e.g, in Ref. [@exotic2]).
Numerical issues and set up
---------------------------
Numerical simulations are performed preparing a non-uniform grid as in Ref. [@Sekiguchi:2011zd; @Sekiguchi:2011mc; @Kiuchi:2009jt; @Kiuchi:2010ze]. The inner domain is composed of a finer uniform grid and the outer domain of a coarser nonuniform grid. The grid resolution in the inner zone is typically chosen to cover the major diameter of each NS in the inspiral orbit by 80 grid points. Outer boundaries are located in a local wave zone at $\approx 560-600 {\rm km}$, which is longer than gravitational wavelengths in the inspiral phase. As a monitor of the accuracy, we check the conservation of baryon rest-mass, total gravitational mass, and total angular momentum and find that they are preserved within the accuracy of 0.5$\%$, 1$\%$, and 3$\%$, respectively.
References [@Sekiguchi:2011zd; @Sekiguchi:2011mc] focus on the merger of equal mass BNS with total mass ranging from $2.7~M_\odot$ to $3.2~M_\odot$. In this proceeding, we report the models with total mass $\approx 2.7~M_\odot$ and $\approx 3~M_\odot$. There are two possible fates of BNS: If its total mass $M$ is greater than a critical mass $M_{\rm c}$, a black hole (BH) will be formed soon after the onset of the merger, while a differentially rotating HMNS will be formed for $M<M_{\rm c}$. $M_{\rm c}$ for Shen-EOS (Hyp-EOS) is $2.8$ – $2.9~M_\odot$ ($2.3$ – $2.4~M_\odot$). Therefore, for the binaries of $M \approx 2.7~M_\odot$, we expect that Shen-EOS model will form a HMNS and Hyp-EOS model will collapse to a BH. The models with $M \approx 3~M_\odot$ are anticipated to collapse to the BH, irrespective of the EOSs. We name Shen-EOS (Hyp-EOS) model with $M \approx 2.7~M_\odot$ S135 (H135) and those with $M \approx 3~M_\odot$ S15 (H15). It should be noted that these figures of the mass are motivated by the narrow mass distribution of observed binary neutron stars.
Numerical results
=================
Figure \[fig1\] plots the evolution of maximum rest-mass density, $\rho_{\rm max}$, maximum temperature, $T_{\rm max}$, and maximum mass fraction of hyperons, $X_{\Lambda, {\rm max}}$ as functions of $t-t_{\rm merge}$ where $t_{\rm merge}$ is an approximate onset time of the merger. Before the merger ($t<t_{\rm
merge}$), $\rho_{\rm max}$ and $T_{\rm max}$ for H135 (H15) agree well with those for the corresponding S135 (S15), because $X_{\Lambda, {\rm max}}$ in this phase is small as $O(10^{-2})$ and effects of hyperons on dynamics are not significant. After the merger sets in ($t>t_{\rm merge}$), on the other hand, $X_{\Lambda,{\rm max}}$ increases to be $\gtrsim 0.1$ in accordance with the increase in $\rho_{\rm max}$, and hyperons play a substantial role in the post-merger dynamics. The panel (b) in Fig. \[fig1\] tells us that $T_{\rm max}$ in the HMNSs reach to $50$ – $60$ MeV just after the first contact, irrespective of the models. For H15, it rapidly increases to a high value of $130$ – $140$ MeV just before the collapse to a BH. For H135 and Shen-EOS models, it decreases due to the neutrino cooling in the subsequent evolution and then relaxes to $\sim 25$ – $40$ MeV. We observe the rapid enhancement of $T_{\rm max}$ just before a BH formation in H135.
Although the total mass, $M$, is larger than the maximum mass of the zero-temperature spherical NS for all the models, a HMNS is formed after the merger, supported by the centrifugal force and thermal contribution to the pressure [@Sekiguchi:2011zd; @Hotokezaka:2011dh]. In particular, the enhancement of the critical mass $M_{\rm c}$ due to the thermal effect is prominent in S15, which does not collapse to a BH during the simulation contrary to our expectation. The HMNSs subsequently contract by emission of GWs, which carry energy and angular momentum from the HMNS; $\rho_{\rm max}$ increases in the gravitational radiation timescale. For Hyp-EOS models, they collapse to the BH at $t=t_{\rm BH}$ where $t_{\rm BH}-t_{\rm merge} \approx 11.0$ ms for H135 and $\approx 3.1$ ms for H15. After the BH formation of Hyp-EOS models, an accretion torus is formed around the BH and gradually settles down to a quasi-steady state. The torus mass is $\approx$ 0.082, and 0.035 $M_\odot$ for H135 and H15, respectively [@Sekiguchi:2011mc]. The torus mass of H135 is greater than that of H15 due to the longer transient HMNS phase, in which the angular momentum is transported outwardly. Note that the lifetime of HMNS for H135 is longer than that for H15. At $t-t_{\rm merge}\sim 20~{\rm ms}$ for Shen-EOS models, the degree of its nonaxial symmetry becomes low enough that the emissivity of GWs is significantly reduced. Because no dissipation process except for the neutrino cooling is present, the HMNS has not yet collapsed to a BH. We expect it will be alive at least for a cooling time, $t_{\rm cool} \equiv E_{\rm th}/L_{\nu} \sim 2$–3 s, where $E_{\rm th}$ is total thermal energy and $L_{\nu}$ is total neutrino luminosity [@Sekiguchi:2011zd].
Figure \[fig2\] plots neutrino luminosities as functions of time for three flavors ($\nu_e,\bar{\nu}_e$, and sum of $\nu_x$). It is found that electron anti neutrinos are dominantly emitted for any model. The reason for this is as follows: The HMNS has a high temperature, and hence, electron-positron pairs are efficiently produced from thermal photons, in particular in its envelope. The positron capture $n+e^+\to p + \bar{\nu}_e$ proceeds more preferentially than the electron capture $p+e^-\to n +\nu_e$ because the proton fractions is much smaller than the neutron fraction. Such hierarchy in the neutrino luminosities was reported also in Ref. [@Ruffert]. These features in neutrino luminosities are quantitatively the same for Hyp-EOS and Shen-EOS models, and hence, it would be difficult to extract information of the NS matter only from the neutrino signal. Soon after the BH formation for Hyp-EOS models, $\mu/\tau$ neutrino luminosity steeply decreases because high temperature regions are swallowed into the BH, while luminosities of electron neutrinos and anti neutrinos decrease only gradually because these neutrinos are emitted via charged-current processes from the massive accretion torus.
The anti neutrino luminosity for the long-lived HMNS for Shen-EOS models is $L_{\rm \bar{\nu}}\sim 2$ – $3\times 10^{53}{\rm erg/s}$ with small time variability. It is greater than that from the protoneutron stars found after supernovae [@Sumiyoshi:2005ri]. Averaged neutrino energy is $\epsilon_{\bar{\nu}}\sim 20-30~{\rm MeV}$. The sensitivity of water-Cherenkov neutrino detectors such as Super-Kaminokande and future Hyper-Kamiokande (HK) have a good sensitivity for such high-energy neutrinos in particular for electron anti neutrinos. The detection number for electron neutrinos is approximately estimated by $\sigma \Delta T L_{\bar{\nu}}/(4\pi D^2 \epsilon_{\bar{\nu}})$ where $\sigma$ is the total cross section of the detector against target neutrinos, $\Delta T$ is the lifetime of the HMNS, and D is the distance to the HMNS. For one-Mton detector such as HK, the expected detection number is $\gtrsim 10$ for $D \lesssim 5$ Mpc with $\Delta T\sim 2$ – 3 s, neutrinos from the HMNS may be detected and its formation may be confirmed. Note that GWs from the HMNS will be simultaneously detected for such a close event, reinforcing the confirmation of the HMNS formation. For Hyp-EOS models, the detection of anti neutrino is less likely because of the short lifetime of the HMNSs. However, as discussed below, the gravitational waves could constrain the hyperonic or nucleonic EOS. Figure \[fig3\] plots the plus and cross mode ($h_{+,\times}$) of GWs with $l=|m|=2$ as a function of $t_{\rm ret}-t_{\rm merge}$ where $t_{\rm ret}$ is the retarded time $t_{\rm ret}=t-D-2M\log(D/M)$, extracted from the metric in the local wave zone, i.e., at $\approx$ 550 km. The waveforms are composed of the so-called chirp waveform, which is emitted when the BNS is in an inspiral motion (for $t_{\rm ret}\lesssim t_{\rm merge}$), and the merger waveform (for $t_{\rm ret}\gtrsim t_{\rm merge}$). The GW amplitude is $|h_{+,\times}| \lesssim 2\times 10^{-22}$ for a source at a distance $D=100$ Mpc with the direction perpendicular to the orbital plane. GWs from the inspiral phase (for $t_{\rm ret}
\lesssim t_{\rm merge}$) agree well with each other for the models with Hyp-EOS and Shen-EOS with the same mass. On the other hand, quasi-periodic GWs from the HMNS (for $t_{\rm ret} \gtrsim t_{\rm
merge}$) show differences. Quasi-periodic GWs is suddenly shut down at the BH formation for Hyp-EOS models, $t_{\rm ret}-t_{\rm merge} \approx 11$ ms for H135 and $t_{\rm ret}-t_{\rm merge}\approx 3$ ms for H15. This is because the HMNS collapses to a BH due to the softening of the EOS before relaxing to a stationary spheroid. The quasi normal mode is excited after the BH formation, which is imprinted in the gravitational wave spectra at $\sim 7$ kHz (see Fig. \[fig4\](b)). For Shen-EOS models, the long-lived HMNSs emit the gravitational waves whose amplitude gradually decreases indicating that they approach to the axi-symmetric quasi-stationary state.
The characteristic GW frequency, $f_{\rm GW}$, increases with time for Hyp-EOS models. These facts are well imprinted in the effective amplitude (see Fig. \[fig4\](a)) defined by $h_{\rm eff}
\equiv 0.4 f |h(f)|$ where $h(f)$ is the Fourier transform of $h_+ -i
h_{\times}$ and the factor 0.4 comes from taking average in terms of direction to the source and rotational axis of the HMNS. The peak amplitudes of $h_{\rm eff}(f)$ in Hyp-EOS models are smaller than those in Shen-EOS models due to a shorter lifetime of the HMNS. The prominent peak in the GW spectrum for Hyp-EOS models is broadened because of the shift of the characteristic frequency. This frequency shift is well imprinted in the evolution of the characteristic frequency $f_{\rm GW}$ in Fig. \[fig4\] (b), where $f_{\rm GW} \equiv d\phi_{\rm GW}/dt/2\pi$ with the Weyl scalar $\Psi_4=A {\rm e}^{-i \phi_{\rm GW}}$. For Shen-EOS models, the central value of the frequency remains almost constant after the merger $(t_{\rm ret}\gtrsim t_{\rm merge})$ and it corresponds to the peak frequency of gravitational wave spectrum in Fig. \[fig4\] (a). By contrast, Hyp-EOS models exhibit a gradual increase in $f_{\rm GW}$. Specifically, for H135, $f_{\rm GW}$ grows from $\approx 2 $ kHz at $t_{\rm ret}-t_{\rm merge}=2$ ms to $\approx 2.5$ kHz at $t_{\rm ret}-t_{\rm merge}=10$ ms. For H15, it increases from $\approx 2 $ kHz at $t_{\rm ret}-t_{\rm merge}=2$ to $\approx 3.4$ kHz at $t_{\rm ret}-t_{\rm merge}=2.5$ ms though the lifetime of HMNS is short. This feature in the frequency shift is explained as follows. For Shen-EOS models, the gravitational waves carry away the angular momentum and then the HMNS slightly contracts. For Hyp-EOS models, the hyperon fraction $X_{\Lambda}$ increases with the contraction of the HMNS, resulting in the relative reduction of the pressure. This leads further contraction of the HMNS. Recent studies suggest that $f_{\rm GW}$ is associated with the f-mode which is approximately proportional to $\sqrt{M_{\rm NS}/R_{\rm NS}^3}$ with $M_{\rm NS}$ and $R_{\rm NS}$ being the mass and radius of HMNS [@Stergioulas:2011gd; @Bauswein:2011tp]. Applying this result to our case indicates that Hyp-EOS models exhibit the frequency shift due to the run-away contraction, i.e., decreasing the radius of the HMNSs, and the Shen-EOS models emit the gravitational waves with the coherent frequency due to the slight contraction. In other words, the frequency $f_{\rm GW}$ for Shen-EOS models evolves on a timescale of gravitational radiation reaction. The effective amplitude $1$ – $4\times 10^{-22}$ for $D=100$ Mpc suggests that for a specially-designed version of advanced GW detectors such as broadband LIGO, which has a good sensitivity for a high-frequency band, GWs from the HMNS oscillations may be detected with ${\rm S}/{\rm N}=5$ if $D\lesssim 20$ Mpc or the source is located in an optimistic direction [@Sekiguchi:2011zd].
Summary
=======
We have reported the results of numerical-relativity simulation performed incorporating both a finite-temperature nucleonic EOS and hyperonic EOS as well as neutrino cooling effect. We showed that for both the nucleonic and hyperonic EOS, HMNS is the canonical outcome and BH is not promptly formed after the onset of the merger for the canonical value of the NS mass $1.35~M_\odot$ and for more massive case of $1.5M_\odot$. The primary reason is that the thermal pressure plays an important role for sustaining the HMNS. For the nucleonic EOS, the lifetime of the formed HMNS is much longer than its dynamical timescale, $\gg 10{\rm ms}$, and will be determined by the timescale of neutrino cooling. For the hyperonic EOS, the HMNS subsequently collapse to a BH due to a softening of EOS as a consequence of increase of $\Lambda$ hyperon. Neutrino luminosity of the HMNS was shown to be high as $\sim 3$ – $6\times 10^{53} {\rm erg/s}$. The effective amplitude of GWs is $1$ – $4\times 10^{-22}$ at $f_{\rm peak}\approx 2$ – $3.2~{\rm kHz}$ for a source distance of 100 Mpc.
We further found that the characteristic frequency of gravitational waves, $f_{\rm GW}$, from the HMNS increases with time for the hyperonic EOS in contrast to the nucleonic EOS in which $f_{\rm GW}$ approximately remains constant. This time-dependent characteristic frequency leads to the broaden gravitational wave spectrum for the hyperonic EOS. For the nucleonic EOS, the spectrum has a sharp peak around the less time-dependent frequency. This result suggests that the emergence of hyperons may be captured from the evolution of characteristic frequency and the peak width of the gravitational waves spectra. If the BNS merger happens at a relatively short distance or is located in an optimistic direction, such GWs may be detected and HMNS formation will be confirmed. In particular, to increase the chance of the detection of such GWs, we need for the detectors with high sensitivities at high frequencies of several kilohertz such as the Einstein Telescope [@ET].
![Maximum rest-mass density, maximum matter temperature, and maximum mass fraction of hyperons as functions of time for H135 (solid-red), S135 (dashed-green), H15 (short-dashed blue), and S15 (dotted-magenta). The vertical thin solid (short-dashed) line shows the time at which a BH is formed for H135 (H15). \[fig1\]](fig1.eps)
![Neutrino luminosities for H135, S135, H15 and S15. The solid (short-dashed) curves are for Hyp-EOS (Shen-EOS) models. The dashed vertical lines show the time at which a BH is formed for Hyp-EOS models and $a{\nu}_e$ represents electron anti neutrino. \[fig2\]](fig2.eps)
[cc]{}
![GWs observed along the axis perpendicular to the orbital plane for the hypothetical distance to the source $D=100$ Mpc for (a) H135, (b) S135, (c) H15, and (d) S15. \[fig3\]](fig3a.eps)
![GWs observed along the axis perpendicular to the orbital plane for the hypothetical distance to the source $D=100$ Mpc for (a) H135, (b) S135, (c) H15, and (d) S15. \[fig3\]](fig3b.eps)
[cc]{}
![ (a) The effective amplitude of GWs defined by $0.4f|h(f)|$ as a function of frequency for $D=100$ Mpc. The noise amplitudes of a broadband configuration of Advanced Laser Interferometer Gravitational wave Observatories (bLIGO), Large-scale Cryogenic Gravitational wave Telescope (LCGT) and Einstein Telescope (ET) are shown together. (b) The frequency of gravitational wave in the HMNS phase for H135 (solid-red), S135 (dashed-green), H15 (short-dashed blue), and S15 (dotted-magenta). The thin vertical lines are BH formation time for Hyp-EOS models. \[fig4\]](fig4a.eps)
![ (a) The effective amplitude of GWs defined by $0.4f|h(f)|$ as a function of frequency for $D=100$ Mpc. The noise amplitudes of a broadband configuration of Advanced Laser Interferometer Gravitational wave Observatories (bLIGO), Large-scale Cryogenic Gravitational wave Telescope (LCGT) and Einstein Telescope (ET) are shown together. (b) The frequency of gravitational wave in the HMNS phase for H135 (solid-red), S135 (dashed-green), H15 (short-dashed blue), and S15 (dotted-magenta). The thin vertical lines are BH formation time for Hyp-EOS models. \[fig4\]](fig4b.eps)
Numerical simulations are performed on SR16000 at YITP of Kyoto University and SX9 and XT4 at CfCA of NAOJ. This work was supported by Grant-in-Aid for Scientific Research (21018008, 21105511, 21340051, 22740178, 23740160), Grant-in-Aid on Innovative Area (20105004), and HPCI Stragetic Program in Japan MEXT.
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---
abstract: '[We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov–Oberlin–Thiele, and (ii) a variation-norm Rademacher–Menshov theorem of Lewko–Lewko.]{}'
address:
- 'Department of Mathematics, Yale University, New Haven, CT 06511, USA'
- 'Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA'
- 'Department of Mathematics, University of Rochester, Rochester, NY 14627-0251, USA'
author:
- Yen Do Richard Oberlin Eyvindur Ari Palsson
bibliography:
- 'variationaltree.bib'
title: Variational bounds for a dyadic model of the bilinear Hilbert transform
---
Introduction
============
In this paper we consider a variation-norm analog of the following maximal operator $$H^*[f_1,f_2](x) = \sup_k |\sum_{|I_P| \geq 2^k} |I_P|^{-1/2}\<f_1,\phi_{P_1}\>\<f_2,\phi_{P_2}\>\phi_{P_3}(x)|$$ where we sum over $P$ in a collection ${\overline}{\P}$ of dyadic rectangles in $\mathbb R^+ \times \mathbb R^+$ of area four (also known as quartiles) and $\phi_{P_1},\phi_{P_2},\phi_{P_3}$ denote dyadic wave packets adapted to appropriate subsets of $P$, see Section \[termsection\] for details. Note that we have suppressed the notational dependency on ${\overline}{\P}$ for simplicity (and all implicit constants in this paper shall be independent of the underlying collection of quartiles). The non-maximal variant of $H^*$ is known as the quartile operator and was introduced in [@thiele95tfa] as a discrete model of the bilinear Hilbert transform. The operator $H^*$ serves as a dyadic model for both the maximal bilinear Hilbert transform and the bilinear maximal function [@lacey00bmf] (cf. [@thiele01mqo; @demeter07pce]). See also the discussion after .
Our aim here is to bound the operator formed by replacing the $\ell^{\infty}$ norm in the definition of $H^*$ by a stronger variation-semi-norm. Given an exponent $r \geq 1$ write $$\|g\|_{V^r_k} = \sup_{N, k_0 < \cdots < k_N} (\sum_{j = 1}^N|g(k_j) - g(k_{j-1})|^r)^{1/r}$$ where the supremum is over all strictly increasing finite-length sequences of integers. Setting $$H^r[f_1,f_2](x) = \|\sum_{|I_P| \geq 2^k} |I_P|^{-1/2}\<f_1,\phi_{P_1}\>\<f_2,\phi_{P_2}\>\phi_{P_3}(x)\|_{V^r_k},$$ we will prove
\[maintheorem\] Suppose $r > 2$, and $p_1, p_2, q$ satisfy $$\frac{1}{q} = \frac{1}{p_1} + \frac{1}{p_2}, \ \ \ \ \frac{2}{3} < q < \infty, \ \ \ \ 1 < p_1, p_2 \le \infty,$$ then for some constant $C=C(p_1,p_2,r)<\infty$ we have $$\label{maintheoremeq}
\|H^r[f_1,f_2]\|_{L^q} \leq C \|f_1\|_{p_1} \|f_2\|_{p_2}.$$
We became interested in bounds for $H^r$ while studying the following bilinear operator $$\label{e.variationBHT}
B^r[f_1,f_2](x) = \|\frac{1}{2t}\int_{-t}^tf_1(x + y)f_2(x-y)\ dy \|_{V^r_t}.$$ The simpler maximal variant of $B^r$ is the bilinear maximal function studied in [@lacey00bmf]. An oscillation-norm variant[^1] of $B^r$ was also considered in [@demeter07pce]. Bounds for the simpler linear version of $B^r$, i.e. the variation-norm analog of the centered Hardy-Littlewood maximal function, proved in [@bourgain89pes], can be used to strengthen the Birkhoff ergodic theorem on the pointwise convergence of linear ergodic averages. Similarly, bounds on the more delicate $B^r$ and its oscillation-norm variants are useful for studies of pointwise convergence of bilinear ergodic averages, see e.g. [@demeter07pce]. While the oscillation-norm estimates in [@demeter07pce] are just enough for this purpose, bounds on $B^r$ give more quantitative information about the related rate of convergence. In this paper, only the dyadic variant $H^r$ of $B^r$ will be considered, which is technically simpler than the continuous setting and therefore allows for a relatively clear and accessible illustration of the main ideas, which we expect to be useful in forthcoming study of $B^r$.
Structure of the paper
----------------------
We essentially follow the framework of [@lacey00bmf], [@thiele01mqo], although the argument is slightly reorganized and simplified to reflect the modern language of time-frequency analysis. The main new ingredients in the proof are variational extensions of several maximal theorems, including a variation-norm extension of the Rademacher–Menshov theorem obtained in [@lewko12esv] and an extension of a lemma of Bourgain [@bourgain89pes] to the variation-norm setting obtained in [@nazarov10czd] (cf. [@oberlin11awm]). These auxiliary results and other background materials are summarized in Sections \[termsection\] and \[auxsection\]. Several technical lemmas are proven in Sections \[sbsection\] and \[sisection\], and we show how they imply Theorem \[maintheorem\] in Section \[poftsection\].
Notational conventions
----------------------
We use $|\cdot|$ to denote Lebesgue measure, cardinality, or an understood norm depending on context. The indicator function of a set $E$ is written $1_E.$ Dyadic intervals are half-open on the right, i.e. of the form $[n2^k, (n+1)2^k)$ for integers $n,k.$
Acknowledgement {#acknowledgement .unnumbered}
---------------
This work was initiated while the authors were visiting the University of California, Los Angeles in Winter 2012, and the visit was supported in part by the AMS Math Research Communities program. The authors would like to thank the MRC and Christoph Thiele for their generous support, hospitality, and useful conversations.
Terminology {#termsection}
===========
In this paper, a *tile* is a dyadic rectangle in ${\mathbb{R}}^+ \times {\mathbb{R}}^+$ of area one. A *quartile* is defined analogously, except with area four instead of one. Each quartile $P=I_P \times \omega_P$ can be written as the disjoint union of four tiles $P_1, P_2, P_3, P_4$ where $P_i = I_{P_i} \times \omega_{P_i},$ $I_{P_i} = I_P$, and $\omega_{P_i}$ is the i’th dyadic grandchild of $\omega_P$, in increasing order from left to right.
The Walsh wave-packet $\phi_p$ associated to a tile $p$ can be defined as follows. First, if $p = I \times [0,2^{n})$ then $\phi_{p}(x) = 2^{-n/2}1_{I}(x).$ To extend the definition to all tiles, we use the following recursive formulas where the subscripts $l$ and $r$ denote the left and right halves of a dyadic interval: $$\label{recursive}
\phi_{I \times \omega_r} = \frac{1}{\sqrt{2}}(\phi_{I_l \times \omega} - \phi_{I_r \times \omega}), \ \ \phi_{I \times \omega_l} = \frac{1}{\sqrt{2}}(\phi_{I_l \times \omega} + \phi_{I_r \times \omega}).$$ It is not hard to see that $\phi_p$ and $\phi_{p'}$ are orthogonal if $p \cap p' = \emptyset$.
Given a collection of quartiles $T$, a “top frequency” $\xi_T \in {\mathbb{R}}^+$, and a dyadic “top interval” $I_T \subset {\mathbb{R}}^+$, we say that $(T,\xi_T, I_T)$ form a *tree* if for every $P \in T$, $I_P \subset I_T$ and $\xi_T \in \omega_P$. Letting $\omega_T$ be the dyadic interval of length $|I_T|^{-1}$ containing $\xi_T$, we write $p_T$ for the tile $I_T \times \omega_T.$
For $i=1, \ldots, 4$ a tree is said to be *$i$-overlapping* if for every $P \in T$, $\xi_T \in \omega_{P_i}.$ We will say that a tree is *$i$-lacunary* if it is $j$-overlapping for some $j \neq i.$ One can check that if $T$ is $i$-lacunary then the tiles $\{P_i\}_{P \in T}$ are pairwise disjoint.
To define a notion of size that is compatible with $H^r$, we will need to linearize and dualize the variation-norm. For each $x$ consider an increasing integer-valued sequence $\{k_j(x)\}_{j = -\infty}^{\infty}$, and a sequence $\{a_j(x)\}_{j = -\infty}^{\infty}$ such that $\sum_{j = -\infty}^{\infty}|a_j(x)|^{r'} \leq 1$. Then an appropriate choice of such sequences guarantees that, for every $x$, $$\label{modelop}
\sum_{P \in {\overline}{\P}} |I_P|^{-1/2}\<f_1,\phi_{P_1}\>\<f_2,\phi_{P_2}\>\phi_{P_3}(x)a_P(x) \geq \frac{1}{2} H^r[f_1,f_2](x)$$ where $a_P(x) := a_m(x)$ if $m=m(P,x)$ is the (clearly unique) integer satisfying $2^{k_{m-1}(x)} \leq |I_P| < 2^{k_{m}(x)}$, and $0$ if such $m$ does not exist.
Thus, to prove , it suffices to give a corresponding bound for the left side above which is independent of the choice of sequences. Fixing these sequences once and for all, we write $${\tilde{\phi}}_{P_3}(x) = a_P(x) \phi_{P_3}(x),$$ and when $i\neq3$ write ${\tilde{\phi}}_{P_i}(x) = \phi_{P_i}(x).$[^2] For collections of quartiles $\P$ and functions $f$ on ${\mathbb{R}}^+$ we define $$\label{sizedef}
{\mathrm{size}}_i(\P,f) = \sup_{T \subset \P} (\frac{1}{|I_T|}\sum_{P \in T}|\<f,{\tilde{\phi}}_{P_i}\>|^2)^{1/2}$$ where the supremum is over all $i$-lacunary trees contained in $\P.$
Let $A_k$ denote the dyadic averaging operator $$A_k[f](x) = \frac{1}{|I|} \int_{I} f(y)\ dy$$ where $I$ is the unique dyadic interval of length $2^k$ containing $x$. Sums of wave packets in lacunary trees can be truncated using $A_k$ as follows:
\[averageclaim\] Suppose that $T$ is a $i'$-overlapping tree. Then for $i \neq i'$ there is $\nu\in \{0,1\}$ such that for any coefficients $\{c_P\}_{P \in T}$ $$\label{trunctree}
\sum_{P \in T: \, \, |I_P| > 2^{k + \nu} } c_P \phi_{P_i} = {\mathop{\mathrm{sgn}}}(\phi_{p_T}) A_{k}[{\mathop{\mathrm{sgn}}}(\phi_{p_T} ) \sum_{P \in T} c_P \phi_{P_i}].$$ Moreover, $\nu = 0$ when $\{i,i'\}=\{1,2\}$ or $\{3,4\}$, and $\nu = 1$ otherwise.
If $\nu = 0$ one can check using that $${\mathop{\mathrm{sgn}}}(\phi_{p_T})\phi_{P_i} = \pm|I_P|^{-1/2}(1_{(I_{P})_l} - 1_{(I_{P})_r})$$ where choice of sign $\pm$ is uniform over $x$. If $\nu = 1$ then similarly $${\mathop{\mathrm{sgn}}}(\phi_{p_T}) \phi_{P_i} = \pm|I_P|^{-1/2}\left((1_{((I_{P})_l)_l} - 1_{((I_{P})_l)_r}) \pm(1_{((I_{P})_r)_l} - 1_{((I_{P})_r)_r})\right).$$ The claim then follows by inspection of averages.
A consequence of is that for each $x$ $$\label{vartrunc}
\|\sum_{P \in T: \, \, |I_P| \geq 2^{k} } c_P \phi_{P_i}(x)\|_{V^r_k} = \|A_k[{\mathop{\mathrm{sgn}}}(\phi_{p_T} ) \sum_{P \in T} c_P \phi_{P_i}](x)\|_{V^r_k}.$$
Auxiliary estimates {#auxsection}
===================
We will start by recalling three variation-norm bounds which will be of later use. The first is a special case of a theorem of Lépingle [@lepingle76lvd].
\[leplemma\] Suppose $r > 2$ and $1 < t < \infty$. Then $$\|A_k[f](x)\|_{L^t_x(V^r_k)} \leq C_{r,t} \|f\|_{L^t}.$$
Now, let $\Xi$ be any finite subset of ${\mathbb{R}}^+$, and for each integer $k$ let $\Omega_k$ be the set of dyadic intervals of length $2^{-k}$ which intersect $\Xi.$ Let $$\label{mfprojection}
\Delta_k[f] = \sum_{\omega \in \Omega_k} \sum_{I : \,\, |I| = 2^k} \<f,\phi_{I \times \omega}\>\phi_{I \times \omega}.$$ Note that while the definition of $\Delta_k[f]$ involves an infinite sum, for each $x$ only finitely many terms are nonzero. Equivalently, $$\Delta_k[f] = \sum_{\omega \in \Omega_k} (1_{\omega} \hat{f})\check{\ }.$$ The following Lemma follows from [@oberlin11awm Lemma 9.2], see also [@nazarov10czd], which is a variation-norm extension of a lemma of Bourgain [@bourgain89pes].
\[vblemma\] Suppose $r > 2$ and $\epsilon > 0.$ Then $$\|\Delta_k[f](x)\|_{L^2_x(V^r_k)} \leq C_{r,\epsilon} |\Xi|^{\epsilon} \|f\|_{L^2}.$$
Below, we have a variation-norm Rademacher-Menshov theorem which was proven in [@lewko12esv], see also the proof of Theorem 4.3 in [@nazarov10czd].
\[vnrmlemma\] Let $X$ be a measure space and $f_1, \ldots, f_N$ be orthogonal functions on $X$. Then $$\|\sum_{j = 1}^n f_j(x)\|_{L^2_x(V^2_n)} \leq C (1 + \log(N)) (\sum_{j = 1}^{N}\|f_j\|^2_{L^2})^{1/2}.$$
Finally we will need the following John-Nirenberg type lemma. See, for example, the proof of Lemma 4.2 in [@muscalu04leb].
\[jnlemma\] Let $\{c(P)\}_{P \in \P}$ be a collection of coefficients. Denote $$A_{1,\infty} = \sup_{T \subset \P} \frac{1}{|I_T|}\|(\sum_{P \in T}|c(P)|^2\frac{1_{I_P}}{|I_P|})^{1/2}\|_{L^{1,\infty}},$$ $$A_2 = \sup_{T \subset \P} \frac{1}{|I_T|^{1/2}}\|(\sum_{P \in T}|c(P)|^2\frac{1_{I_P}}{|I_P|})^{1/2}\|_{L^{2}}$$ where both supremums are over (say) all $i$-lacunary trees. Then $$\label{jnconc}
A_2 \leq C A_{1,\infty}.$$
A variation-norm size bound {#sbsection}
===========================
Let $M^t$ denote the dyadic $L^t$-Hardy-Littlewood maximal operator $$\label{e.dyadicHL}
M^t[f](x) = \sup_k (A_k[|f|^t](x))^{1/t}.$$
\[propositionvnsb\] Let $\lambda > 0, r > 2,$ and $1 < t < \infty$. Suppose that $\P$ is a collection of quartiles such that for each $P \in \P$ $$\label{notbadtileeq}
I_P \not\subset \{M^t[f] > \lambda\}.$$ Then for each $i$ $${\mathrm{size}}_i(\P,f) \leq C_{r,t} \lambda.$$
Using Lemma \[jnlemma\], it follows from that $${\mathrm{size}}_i(\P,f) \leq C \sup_{T \subset \P} |I_T|^{-1/t} \|(\sum_{P \in T}|\<f,{\tilde{\phi}}_{P_i}\>|^2\frac{1_{I_P}}{|I_P|})^{1/2}\|_{L^t}$$ which, by the usual Rademacher function argument, is $$\leq C_t \sup_{T \subset \P} \sup_{\{b_P\}_{P \in \P}}|I_T|^{-1/t} \|\sum_{P \in T} b_P \<f,{\tilde{\phi}}_{P_i}\> \phi_{P_i} \|_{L^t}$$ where the right supremum is over all sequences $\{b_P\}_{P \in \P}$ of $\pm1$’s.
Let $t' = t/(t - 1)$. For any nonempty $i$-lacunary tree $T\subset \P$ and any binary sequence $\{b_P\}$, by duality we have $$\begin{aligned}
&|I_T|^{-1/t} \|\sum_{P \in T} b_P \<f,{\tilde{\phi}}_{P_i}\> \phi_{P_i} \|_{L^t} \\
&\leq |I_T|^{-1/t} \|1_{I_T}f\|_{L^t} \sup_{g : \|g\|_{L^{t'} = 1}} \|\sum_{P \in T} b_P \<g,\phi_{P_i}\> {\tilde{\phi}}_{P_i} \|_{L^{t'}} \\
&\leq \lambda \sup_{\|g\|_{L^{t'} = 1}} \|\sum_{P \in T} b_P \<g,\phi_{P_i}\> {\tilde{\phi}}_{P_i} \|_{L^{t'}} \qquad \qquad \text{(using \eqref{notbadtileeq})} \\
&\leq \lambda \sup_{\|g\|_{L^{t'} = 1}} \|\sum_{P \in T: \,\, |I_P| \geq 2^k} b_P \<g,\phi_{P_i}\> \phi_{P_i}(x) \|_{L^{t'}_x(V^r_k)} \qquad \text{(by def. of ${\tilde{\phi}}_{P_i}$)}\\
&= \lambda \sup_{\|g\|_{L^{t'} = 1}} \|A_k[{\mathop{\mathrm{sgn}}}(\phi_{p_T})\sum_{P \in T} b_P \<g,\phi_{P_i}\> \phi_{P_i}](x) \|_{L^{t'}_x(V^r_k)} \qquad \text{(by \eqref{vartrunc})} \\
&\leq C_{r,t'}\lambda \sup_{\|g\|_{L^{t'} = 1}} \|\sum_{P \in T} b_P \<g,\phi_{P_i}\> \phi_{P_i}\|_{L^{t'}} \qquad \text{(by Lemma \ref{leplemma})} \\
&\leq C_{r,t'} \lambda \qquad \text{(by standard dyadic Calder\'{o}n-Zygmund theory).} \end{aligned}$$ Note that much of the argument above is superfluous unless $i=3$.
A variation-norm size lemma {#sisection}
===========================
The main result in this section is Proposition \[vnsprop\], and its proof requires Propositions \[mainpropositionstrong\] and \[mainpropositionweak\]. We assume throughout this section that $\epsilon>0$ and $r>2$, and all implicit constants are allowed to depend on $\epsilon$ and $r$.
\[vnsprop\] Let $\P$ be a finite collection of quartiles. Suppose $|f| \leq 1_E.$ Then for each $\alpha$ satisfying $${\mathrm{size}}_i(\P, f)^2 \leq \alpha$$ we can find a collection of trees $\T$, each contained in $\P$, satisfying $${\mathrm{size}}_i(\P \setminus \bigcup_{T \in \T}T, f)^2 \leq \frac{1}{4} \alpha,$$ $$\label{vnslconcNeq}
\sum_{T \in \T} |I_T| \leq C \alpha^{-(1+ \epsilon)} |E|.$$
Let $j_1, j_2, j_3$ be an enumeration of $\{1,2,3,4\} \setminus \{i\}$ and $\P_0^1 = \P$. If there is a $j_1$-overlapping tree $S \subset \P^1_0$ satisfying $$\label{biglac}
\frac{1}{|I_{S}|}\sum_{P \in S}|\<f,{\tilde{\phi}}_{P_i}\>|^2 \geq \frac{1}{4} \alpha$$ then let $S_1^1$ be such a tree, chosen in the following manner:\
(i) If $j_1 < i$ then we pick such $S_1^1$ with $\inf \omega_{S_1^1}$ maximal.\
(ii) If $j_1 > i$ then we pick such $S_1^1$ with $\inf \omega_{S_1^1}$ minimal.\
We then let $T_1^1$ be the maximal (with respect to inclusion) tree contained in $\P^1_0$ with top data $(I_{S_1^1},\xi_{S_1^1})$.
Now, consider $\P_1^1 = \P_0^1 - T_1^1$ and iterate the above selection process until no trees satisfying can be found (the process must stop in finite time due to the assumption that $\P$ is finite), we obtain trees $T_1^1, \ldots, T_{n_1}^1$ and $S_1^1, \ldots, S_{n_1}^1$ where $S_1^1, \ldots, S_{n_1}^1$ are $j_1$-overlapping trees and satisfy $\eqref{biglac}$, and with $S_{i}^1 \subset T_{i}^1$ for each $i$ .
We then consider the remaining tile collection $\P_0^2 = \P_0^1 - T_1^1 - \dots - T_{n_1}^1$ and repeat the same process as above, but choosing $j_2$-overlapping trees instead of $j_1$-overlapping trees. This gives $S_1^2, \ldots, S_{n_2}^2$ which are inside more general trees $T_1^2, \ldots, T_{n_2}^2$. Finally we select $j_3$-overlapping trees from the remaining tile collection and obtain $j_3$-overlapping trees $S_1^3, \ldots, S_{n_3}^3$ which are inside trees $T_1^3, \ldots, T_{n_3}^3$.
By construction we have $${\mathrm{size}}_i(\Q, f)^2 \leq \frac{1}{4} \alpha
\qquad \text{where} \qquad \Q = \P \setminus \bigcup_{k = 1,2,3} \bigcup_{l = 1}^{n_k}T_{l}^k.$$ Thus, it remains to show that for each $k$ $$\sum_{l=1}^{n_k} |I_{S_l^k}| \leq C \alpha^{-(1 + \epsilon)} |E|.$$ To verify this estimate, first note that the sets $\bigcup_{P \in S_l^k} P_i$ indexed by $l$ are pairwise disjoint. Indeed, suppose $P_i \cap P'_i \neq \emptyset$, $P \in S_l^k,$ $P' \in S_{l'}^k$, and $l < l'$. By geometry, the maximality/minimality of $\inf \omega_{S_l^k}$ guarantees that $P' \in T_l^k$, contradicting the fact that $P' \in S_{l'}^k.$
Now, if $i\ne 3$ then by orthogonality of the ${\tilde{\phi}}_{P_i}$ and we have $$\sum_{l=1}^{n_k} |I_{S_l^k}| \leq C \alpha^{-1} \|f\|_{2}^{2}$$ which implies (the assumption that $f \leq 1_E$ guarantees that any ${\mathrm{size}}_i(\P,f) \leq 1$ and so we may assume that $\alpha \leq 4$).
If $i=3$, orthogonality between ${\tilde{\phi}}_{P_3}$ is not available, and we apply Proposition \[mainpropositionstrong\] below.
\[mainpropositionstrong\] Suppose that $\T$ is a finite collection of 3-lacunary trees such that the elements of $\{P_3:\, P\in \bigcup_{T \in \T} T\}$ are pairwise disjoint and furthermore for each $T \in \T$ $$\label{largesizeineq}
\frac{1}{|I_T|} \sum_{P \in T } |\<f,{\tilde{\phi}}_{P_3}\>|^2 \geq \alpha.$$ Then for $N := \sum_{T \in \T}1_{I_T}$ we have $$\label{mainpropstrongconc}
\|N\|_{L^1} \leq C \alpha^{-(1+\epsilon)} \|f\|^{2 + 2\epsilon}_{L^{2 + 2\epsilon}}.$$
We’ll show that if $c>0$ is sufficiently small then for $\lambda \geq 1$ $$\label{goodlambda}
|\{N > \lambda\}| \leq |E_\lambda| + \frac{1}{100} |\{N > \lambda/4\}|\ , \qquad \text{where}$$ $$E_\lambda:=\{M^2 f > c \alpha^{1/2} \lambda^{1/(2 + 2\epsilon)}\},$$ and $M^2f$ is the $L^2$ dyadic Hardy-Littlewood maximal function (see ). Once this is done, we can integrate both sides of , $$\begin{aligned}
\|N\|_{L^1} &\leq \int |E_\lambda| \ d\lambda + \int \frac{1}{100} |\{N > \lambda/4\}| d\lambda \\
&= C\, \alpha^{-(1 + \epsilon)}\|M^2 f \|_{2 + 2\epsilon}^{2 + 2\epsilon} + \frac{1}{25} \|N\|_{L^1} \\
&\leq C\, \alpha^{-(1 + \epsilon)}\|f\|_{2 + 2\epsilon}^{2 + 2\epsilon} + \frac{1}{25} \|N\|_{L^1},\end{aligned}$$ and obtain the desired claim .
Let ${\mathcal{I}}$ be the collection of maximal dyadic intervals contained in $\{N > \lambda/4\}$. This collection clearly covers $\{N > \lambda\}$. Thus, will follow if for any $I \in {\mathcal{I}}$ that intersects the set $E_\lambda$ it holds that $$|\{N > \lambda \} \cap I| \leq \frac{1}{100}|I|.$$ To see this, take $I$ be such an interval. Then $$\|1_I f\|_{L^2} \le |I|^{1/2} \inf_{x\in I} M^2[f](x)$$ $$\label{e.localL2}
\leq |I|^{1/2} c \alpha^{1/2} \lambda^{1/(2 + 2\epsilon)}.$$ It follows from the maximality of $I$ that $$\{N > \lambda\} \cap I \subset \{N_I > \lambda/4\} \qquad \text{where} \qquad N_I := \sum_{T \in \T: \, \, I_T \subset I} 1_{I_T} .$$ Finally, applying Proposition \[mainpropositionweak\] with $1_If$ in place of $f$ and $\{T \in \T : I_T \subset I\}$ in place of $\T$, we obtain for some $C'$ depends on $r,\epsilon'$: $$\begin{aligned}
|\{N_I \geq \lambda/4\}| & \leq C' \, \alpha^{-1} \lambda^{-(1-\epsilon')} \|1_If\|_{L^2}^2 \\
&\leq C' \, \alpha^{-1} \lambda^{-(1-\epsilon')} \,\, c^2 \alpha \lambda^{\frac{1}{1 + \epsilon}} \,\, |I| \qquad \text{(by \eqref{e.localL2})} \\
&\leq |I|/100\end{aligned}$$ where the last inequality follows by choosing $\epsilon' = \frac{\epsilon}{1 + \epsilon}$, and a sufficiently small choice of $c$ depending on $C'$.
\[mainpropositionweak\] Suppose that $\T$ and $N$ are as in the hypotheses of Proposition \[mainpropositionstrong\]. Then $$\label{mainpropconc}
|\{x : N(x) > \lambda\}| \leq C \alpha^{-1} \frac{\|f\|^2_{L^2}}{\lambda^{1 - \epsilon}}.$$
We may assume $\lambda \geq 1$ as $N$ is integer valued. We first estimate $$\label{suminl}
|\{x : N(x) > \lambda\}| \leq \sum_{l \geq 0} |\{2^l \lambda < N \leq 2^{l+1} \lambda\}|.$$ It is clear that trees $T$ with $I_T \subset \{N > 2^{l+1} \lambda\}$ make no contribution to the $l$’th set in the display above. Thus, letting $$\T_l := \{T \in \T : I_T \not\subset \{N > 2^{l+1} \lambda\}\}$$ and $N_l = \sum_{T \in \T_l}1_{I_T}$, we have $$\{2^l \lambda < N \leq 2^{l+1} \lambda\} \subset \{N_l > 2^l \lambda\}.$$ Note that in the evaluation of $N_l$ at each point, only a nested sequence of top intervals are involved, and the smallest of them intersects $\{N\le 2^{l+1}\lambda\}$. It follows that $$\label{linftyNl}
\|N_{l}\|_{L^{\infty}} \leq 2^{l+1} \lambda.$$
By Chebyshev and , with $\P_l = \bigcup_{T \in \T_l} T$ we have $$|\{N_l > 2^l \lambda\}| \leq (\alpha 2^l \lambda )^{-1} \sum_{P \in \P_l}|\<f,{\tilde{\phi}}_{P_3}\>|^2.$$ Consequently, together with , it will suffice for to show $$\|\sum_{P \in \P_l}\<f,{\tilde{\phi}}_{P_3}\>\phi_{P_3}\|_{L^2} \leq C (2^l\lambda)^{\epsilon} \|f\|_{L^2}.$$ Invoking duality and unravel the definition of ${\tilde{\phi}}_{P_3}$, this follows from $$\label{unlinearized}
\|\sum_{P \in \P_l: \,\, |I_P| \geq 2^k} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_k)} \leq C (2^l\lambda)^{\epsilon} \|f\|_{L^2}.$$ In the rest of the proof, we show .
We will repeatedly use a “long-jump/short-jump” decomposition to estimate the variation-norm. Namely, if $\{k_j\}$ is any strictly increasing sequence of integers, then for $r \geq 2$ we have $$\label{longshortbound}
\|g(k)\|_{V^r_k} \leq C(\|g(k_j)\|_{V^r_j} + \|g(k)\|_{\ell^2_j(V^r_{k_j \leq k < k_{j+1}})})$$ where the notation $\|\cdot\|_{V^r_{a \leq k < b}}$ indicates that, in the variation-norm, we only consider sequences lying between $a$ and $b$ (in the case that $b=a+1$, we set the value to $0$).
The first step in proving is the following decomposition of $\P_l$.\
(i) Let ${\mathcal{J}}_1$ be the collection of top intervals of elements of $\T_l$.\
(ii) For $m \geq 1$ let ${\mathcal{I}}_m$ be the set of maximal intervals in ${\mathcal{J}}_m$.\
(iii) Let ${\mathcal{J}}_{m+1} = {\mathcal{J}}_m \setminus {\mathcal{I}}_m.$\
(iv) Let $\P_{l,m} \subset \P_l$ contains those quartiles $P$ such that there exists an element of ${\mathcal{I}}_m$ that contains $I_P$, but no such exists in ${\mathcal{I}}_{m+1}$.
It then follows from that ${\mathcal{J}}_m = {\mathcal{I}}_m = \emptyset$ for $m > 2^{l+1} \lambda$. Furthermore, every $x$ that contributes to the left hand side of is contained in a chain $I_1 \supset \cdots \supset I_{M(x)}$ of intervals, where $I_i \in {\mathcal{I}}_i$. Forming a sequence $\{k_j(x)\}$ based on the lengths of these intervals, and applying pointwise, it follows that the left side of is $$\label{unlinearizedsplit}
\leq C \|\sum_{m = 1}^n \sum_{P \in \P_{l,m}} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_{n \leq 2^{l+1} \lambda})}$$ $$\label{unlinearizedsplit2}
+ C \|\sum_{P \in \P_{l,m}: \,\, |I_P| \geq 2^k} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(\ell^2_m(V^r_k))}.$$ By Lemma \[vnrmlemma\], we can bound by $$\leq C (1 + \log(2^{l}\lambda)) (\sum_{m=1}^{[2^{l+1}\lambda]} \| \sum_{P \in \P_{l,m}} \<f,\phi_{P_3}\>\phi_{P_3}\|_{L^2}^2)^{1/2}.$$ It then follows from disjointness of $P_3$’s that the above display is controlled by the right hand side of .
It remains to bound . For each $I\in {\mathcal{I}}_m$ let $\P(I)$ be the set of elements of $\P_{l,m}$ whose time interval is inside $I$. Note that these collections form a partition of $\P_{l,m}$. Using Fubini and spatial orthogonality, we can rewrite as $$\label{unlinearizedsplit3}
C (\sum_{m=1}^{2^{l+1} \lambda} \sum_{I \in {\mathcal{I}}_m}\|\sum_{P \in \P(I): \,\, |I_P| \geq 2^k} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_k)}^2)^{1/2}.$$ It suffices to show that, for each $m$ and each $I\in {\mathcal{I}}_m$, $$\|\sum_{P \in \P(I): \,\, |I_P| \geq 2^k} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_k)}$$ $$\label{localunlinearized}
\le C(2^l\lambda)^\epsilon \|\sum_{P \in \P(I)} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x}$$ (the desired bound for then follows from disjointness of $P_3$’s).
Let $\T(I)$ be the set of trees in $\T_l$ that intersect $\P(I)$, then $$\label{localtreecount}
|\T(I)| \leq 2^{l+1}\lambda.$$ Indeed, the top interval of any $T \in \T_I$ must contain $I$, since otherwise it would be contained in some element of ${\mathcal{I}}_{m+1}$ and so $T\cap\P_{l,m}=\emptyset$, contradiction. Consequently, $|\T(I)|$ is equal to the value of $\sum_{T \in \T(I)}1_T$ on $I$. The above estimate then follows from .
Now, by splitting $\T(I)$ and absorbing a factor, we may assume that there is an $i' \ne 3$ such that every element of $\T(I)$ is $i'$-overlapping. Let $\nu$ be as in Claim \[averageclaim\] with $i=3$.
Form $\Omega_j$ and $\Delta_k$ as in using the collection $\Xi$ of top frequencies of elements of $\T(I)$. By we have $|\Xi| \leq 2^{l+1}\lambda$.
Let $k_1, \ldots, k_N$ be the increasing enumeration of those $k$’s such that $$|\Omega_{k+4}| > |\Omega_{k-4}|.$$ Since $|\Omega_k|\le |\Xi|$, it follows that $N \leq 8|\Xi|\le 2^{l+4}\lambda$.
*Below and for the rest of the current proof, all quartiles are in $\P(I)$, in particular in the summations.* Applying we have $$\|\sum_{|I_P| \geq 2^k} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_k)} \leq C \|\sum_{|I_P| \geq 2^{k_j}} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_j)}$$ $$\label{longshortagain}
\qquad + \, C\, \|\sum_{2^k \leq |I_P| < 2^{k_{j+1}}} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(\ell^2_j(V^r_{k_{j} \leq k < k_{j+1}}))}.$$ Since $N \le 2^{l+2}\lambda$, it is clear from Lemma \[vnrmlemma\] that the first term on the right of is controlled by the right hand side of .
We ’ll show that $k_j \leq k < k_{j+1}$ then $$\label{insertdelta}
\sum_{ 2^k \leq |I_P| < 2^{k_{j+1}}} \<f,\phi_{P_3}\>\phi_{P_3} = \Delta_{k-(\nu + 1)}[\sum_{2^{k_{j}} \leq |I_P| < 2^{k_{j+1}}} \<f,\phi_{P_3}\>\phi_{P_3}].$$ Since $|\Xi|\le 2^{l+1}\lambda$, it follows from and Lemma \[vblemma\] that, for each $j$, $$\begin{aligned}
\|\sum_{ 2^k \leq |I_P| < 2^{k_{j+1}}} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2_x(V^r_{k_{j} \leq k < k_{j+1}})}
\\
\leq C (2^l \lambda)^{\epsilon} \|\sum_{2^{k_j} \leq |I_P| < 2^{k_{j+1}}} \<f,\phi_{P_3}\>\phi_{P_3}(x)\|_{L^2}. \end{aligned}$$ Taking $\ell^2$ sum over $j$, the second term on the right of is clearly controlled by the right hand side of .
It remains to show . It is clear from the definition of $\nu$ that if $k \le \log_2 |I_P|$ then $\Delta_{k-\nu-1}[\phi_{P_3}] = \phi_{P_3}$. Therefore it remains to show $$\Delta_{k-\nu-1}[\phi_{P_3}] =0$$ if $2^{k_j}\le |I_P|<2^k$. Assume that this is not the case. Then one interval in $\Omega_{k-\nu-1}$ must intersect $\omega_{P_3}$. Since $k> n:=\log_2|I_P|$, one interval in $\Omega_{n-\nu}$ must also intersect $\omega_{P_3}$. By definition of $\nu$, this interval and the dyadic interval of length $2^{\nu-n}$ containing $\xi_T$ are siblings, i.e. they share the same dyadic parent, here $T$ is the tree where $P$ lives. Thus, $$|\Omega_{n-\nu}| > |\Omega_{n-\nu-1}|$$ therefore $|\Omega_{n+5}|>|\Omega_{n-3}|$. But $k_j< n+1 \le k < k_{j+1}$ so this violates the choice of $\{k_j\}$.
Proof of Theorem \[maintheorem\] {#poftsection}
================================
Recall from that our aim is to prove that the operator $$\label{linearizedundualized}
\sum_{P \in {\overline}{\P}} |I_P|^{-1/2}\<f_1,{\tilde{\phi}}_{P_1}\>\<f_2,{\tilde{\phi}}_{P_2}\>{\tilde{\phi}}_{P_3}(x)$$ is bounded from $L^{p_1} \times L^{p_2}$ into $L^q$ whenever $\frac{2}{3} < q < \infty$ and $1 < p_1,p_2 \le \infty.$ Without loss of generality, we can assume that ${\overline}\P$ is finite, provided that the estimates are uniform in ${\overline}{\P}.$
Despite the possibility that $q < 1$, the “restricted type” interpolation method of [@muscalu02mlo] allows one to deduce bounds on from certain estimates for $$\Lambda(f_1,f_2,f_3) = \sum_{P \in {\overline}{\P}} |I_P|^{-1/2} \prod_{i = 1}^3\<f_i,{\tilde{\phi}}_{P_i}\>.$$ Specifically, our desired bounds follows from Proposition \[restrictedtypeproposition\] and its symmetric variants (whose proofs are analogous). Below, we say $H \subset G$ is a major subset if $|H|\ge |G|/2$.
\[restrictedtypeproposition\] Let $r > 2$ and $E_1,E_2,E_3$ be subsets of ${\mathbb{R}}^+$ of positive measures. Assume $|f_i| \le 1_{E_i}$ for every $i$. Then there exists a major subset $\tilde{E_1}$ of $E_1$ such that in any neighborhood of $(-\frac 1 2, \frac 1 2, 1)$ we can find an $\alpha=(\alpha_1,\alpha_2,\alpha_3)$ with $\alpha_1+\alpha_2 +\alpha_3=1$ satisfying $$|\Lambda(f_11_{\tilde{E}_1},f_2,f_3)| \leq C_{r,\alpha} \prod_{i=1}^3 |E_i|^{\alpha_i}.$$
By (dyadic) dilation symmetry we can assume $|E_1| \in [1/2, 1)$. Fix $q>1$ close to $1$ to be chosen later. Then we choose $\tilde{E}_1 = E_1 \setminus F$ where $$F = \bigcup_{i=1}^3 \{M^q[1_{E_i}] \geq C |E_i|^{1/q}\}$$ with $C$ is chosen sufficiently large to guarantee that $|F| \leq \frac{1}{4}$.
Now, without loss of generality assume that $f_1$ is supported in $\tilde{E}_1$. Then the quartiles that contribute to $\Lambda$ belong to $\P = \{P \in {\overline}{\P} : I_P \not\subset F\}$. Note that by Proposition \[propositionvnsb\] we have $$\label{universalsizebound}
S_i:= {\mathrm{size}}_{i}(\P, f_i) \le C |E_i|^{1/q}.$$ Here $C$ and other implicit constants below can depend on $r$, $q$, and $\beta_i$ (defined below).
Applying Proposition \[vnsprop\] repeatedly, we obtain a decomposition of $\P$ into collections of trees $(\T_n)_{n\in \mathbb Z}$ with $$\label{treeboundinproof}
\sum_{T \in \T_n} |I_T| \leq C 2^n,$$ and furthermore for any $T\in \T_n$ we have $$\label{sizeboundinproof}
{\mathrm{size}}_{i}(T,f_i) \leq C 2^{-n/(2q)}|E_i|^{1/(2q)}.$$
Now, for any tree $T$ we have $$\label{treeestimate}
\sum_{P \in T} |I_P|^{-1/2} \prod_{i = 1}^3|\<f_i,{\tilde{\phi}}_{P_i}\>| \leq 4 |I_T|\prod_{i = 1}^3 {\mathrm{size}}_i(T,f_i).$$ To show , by further decomposing $T$ we can assume that it is $i$-overlapping for some $i \in \{1, 2,3, 4\}$. If $i \ne 4$ we will estimate for every $P \in T$ $$|I_P|^{-1/2}|\<f_i,{\tilde{\phi}}_{P_i}\>| \leq {\mathrm{size}}_i(T,f_i)$$ and apply Cauchy-Schwarz to estimate the remaining bilinear sum by $$|I_T| \prod_{j \in \{1,2,3\} \setminus \{i\}}{\mathrm{size}}_j(T, f_j).$$ The case $i =4$ is even simpler, one can apply the above $\ell^\infty \times \ell^2 \times \ell^2$ estimate in any order.
Applying , , , we obtain $$\begin{aligned}
|\Lambda(f_1, f_2, f_3)| &\leq C\sum_{n} 2^n \prod_{i=1}^3 \min(S_i, 2^{-n/(2q)}|E_i|^{1/(2q)}).\end{aligned}$$ For any $\beta_1,\beta_2,\beta_3\in [0,1]$ we can further estimate by $$\le C\, S_1 S_2 S_3 \sum_{n} 2^n \min \Big(1, 2^{-n\frac{\beta_1+\beta_2 + \beta_3}{2q}} \prod_{i=1}^3 |E_i|^{\frac{\beta_i}{2q}} S_i^{-\beta_i} \Big).$$ The above estimate is a two sided geometric series if we choose $\beta_i$’s such that $\beta_1+\beta_2+\beta_3 > 2q$ (which is possible if $q$ is close to $1$). We obtain $$|\Lambda(f_1, f_2, f_3)| \le C\prod_{i=1}^3 S_i^{1-\gamma_i} |E_i|^{\gamma_i/(2q)} \ , \qquad \gamma_i := 2q\beta_i/(\beta_1+\beta_2+\beta_3),$$
$$\le C \Big(\prod_{i=1}^3 |E_i|^{1 -\frac{\gamma_i}{2}}\Big)^{1/q} \qquad \text{(using \eqref{universalsizebound} )}.$$ Since $|E_1| \sim 1$, we can ignore its contribution in the above estimate. Now, by sending $(q,\beta_1,\beta_2,\beta_3)$ to $(1,1,1,0)$ inside the region $\{\beta_1+\beta_2+\beta_3>2q\} \cap \{0\le \beta_1,\beta_2,\beta_3\le 1<q\}$, we obtain the desired claim.
[^1]: More precisely, [@demeter07pce] proves estimates on a finite sum of oscillations, however the implicit constant depends on the number of oscillations being measured.
[^2]: At first glance this notation might seem slightly abusive, but since each tile is contained in a unique quartile, $\tilde{\phi}_p$ is defined implicitly for any tile $p$.
|
---
abstract: 'We present a technique to produce arrangements of lines with nice properties. As an application, we construct $(22_4)$ and $(26_4)$ configurations of lines. Thus concerning the existence of geometric $(n_4)$ configurations, only the case $n=23$ remains open.'
address: 'Michael Cuntz, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany'
author:
- 'M. Cuntz'
title: '$(22_4)$ and $(26_4)$ configurations of lines'
---
Enumerating arrangements
========================
There are several ways to enumerate arrangements of lines in the real plane. For instance, one can enumerate all wiring diagrams and thus oriented matroids. However, without a very strong local condition on the cell structure, such an enumeration is feasible only for a small number of lines. In any case, most types of interesting arrangements of more than say $20$ lines can probably not be enumerated completely (nowadays by a computer).
A much more promising method is (as already noted by many authors) to exploit symmetry. In fact, most relevant examples in the literature have a non-trivial symmetry group. Symmetry reduces the degrees of freedom considerably and allows us to compute examples with many more lines. The following (very simple) algorithm is a useful tool to produce “interesting” examples of arrangements with non-trivial symmetry group:
[[Enumerate arrangements]{}[($q$,$P$)]{}]{}\[Enumerate arrangements\]
[*Look for matroids with $P$ which are realizable over $\CC$*]{}.
[**Input:**]{} a prime power $q$, a property $P$
[**Output:**]{} matroids of arrangements of lines in $\CC\PP^2$ with $P$
[**.**]{}
Depending on $P$, choose a small set of lines $\Ac_0\subseteq \PFq$ and an $n\in\NN$.
For every group $H\le \operatorname{PGL}_3(\FF_q)$ with $|H|=n$, compute the orbit $\Ac:=H\Ac_0$.
If $\Ac$ has property $P$, then compute its matroid $M$. Print $M$ if it is realizable over $\CC$.
1. If $q$ is not too big, then it is indeed possible to compute all the subgroups $H$ with $|H|=n$. However, if $q$ is too small, then only very few matroids $M$ will be realizable in characteristic zero.
2. If we are looking for arrangements with $m = nk$ lines, then it is good to choose $\Ac_0$ with approximately $k$ lines.
3. This algorithm mostly produces matroids that are not orientable. Thus it is a priori not the best method if one is searching for arrangements in the real projective plane. On the other hand, most “interesting” arrangements will define a matroid that is realizable over many finite fields, such that these matroids will certainly appear in the enumeration.
4. Realizing rank three matroids with a small number of lines, depending on the matroid maybe up to 70 lines, is not easy but works in most cases (see for example [@p-C10b]).
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(400,400)(100,200) (359.561670289725574667590751435,380.146109903424808444136416189) (461.275246333884696436032456233,380.146109903424808444136416189) (229.289321881345247559915563789,380.146109903424808444136416189) (345.282284107614971997974010011,329.289321881345247559915563789) (410.418458311805135551811603833,329.289321881345247559915563789) (280.146109903424808444136416189,329.289321881345247559915563789) (319.853890096575191555863583811,470.710678118654752440084436211) (189.581541688194864448188396167,470.710678118654752440084436211) (254.717715892385028002025989989,470.710678118654752440084436211) (370.710678118654752440084436211,419.853890096575191555863583811) (240.438329710274425332409248566,419.853890096575191555863583811) (138.724753666115303563967543768,419.853890096575191555863583811) (319.853890096575191555863583811,238.724753666115303563967543768) (280.146109903424808444136416189,459.561670289725574667590751435) (229.289321881345247559915563789,510.418458311805135551811603833) (370.710678118654752440084436211,354.717715892385028002025989989) (280.146109903424808444136416189,561.275246333884696436032456233) (319.853890096575191555863583811,340.438329710274425332409248566) (229.289321881345247559915563789,445.282284107614971997974010011) (370.710678118654752440084436211,289.581541688194864448188396167) (499.012117842188477954921988695,380.146109903424808444136416189)(100.987882157811522045078011305,380.146109903424808444136416189) (487.082869338697069279187436616,329.289321881345247559915563789)(112.917130661302930720812563384,329.289321881345247559915563789) (487.082869338697069279187436616,470.710678118654752440084436211)(112.917130661302930720812563384,470.710678118654752440084436211) (499.012117842188477954921988695,419.853890096575191555863583811)(100.987882157811522045078011305,419.853890096575191555863583811) (411.714493677880360916003934651,565.891144737429637376841996797)(309.038263942426231078593425378,200.204329914517322033677434938) (459.874685152285089959924033189,279.833095040865293151803134434)(179.833095040865293151803134434,559.874685152285089959924033188) (420.757884611552781857849372664,240.571228116933348049012040785)(264.555410330033193795717058860,596.834146080215671435389586479) (499.795670085482677966322565062,390.961736057573768921406574622)(134.108855262570362623158003204,288.285506322119639083996065349) (335.444589669966806204282941140,203.165853919784328564610413521)(179.242115388447218142150627336,559.428771883066651950987959215) (280.146109903424808444136416189,599.012117842188477954921988695)(280.146109903424808444136416189,200.987882157811522045078011305) (468.424587950951963797615435626,292.142880747017693320801326074)(131.575412049048036202384564374,507.857119252982306679198673925) (420.166904959134706848196865566,240.125314847714910040075966811)(140.125314847714910040075966811,520.166904959134706848196865566) (318.054302078973365566221202889,599.183438509433253558117242933)(100.816561490566746441882757067,381.945697921026634433778797111) (499.183438509433253558117242933,418.054302078973365566221202889)(281.945697921026634433778797111,200.816561490566746441882757067) (496.834146080215671435389586479,364.555410330033193795717058860)(140.571228116933348049012040785,520.757884611552781857849372664) (319.853890096575191555863583811,599.012117842188477954921988695)(319.853890096575191555863583811,200.987882157811522045078011305) (459.428771883066651950987959215,279.242115388447218142150627336)(103.165853919784328564610413521,435.444589669966806204282941140) (407.857119252982306679198673925,231.575412049048036202384564374)(192.142880747017693320801326074,568.424587950951963797615435627) (370.710678118654752440084436211,587.082869338697069279187436616)(370.710678118654752440084436211,212.917130661302930720812563384) (465.891144737429637376841996797,511.714493677880360916003934651)(100.204329914517322033677434938,409.038263942426231078593425378) (229.289321881345247559915563789,587.082869338697069279187436616)(229.289321881345247559915563789,212.917130661302930720812563384) (290.961736057573768921406574622,599.795670085482677966322565062)(188.285506322119639083996065349,234.108855262570362623158003204)
(400,400)(100,200) (367.082039324993690892275210062,377.639320225002103035908263313) (180.541887076558077005776535340,439.819370974480640998074488220) (419.458112923441922994223464660,360.180629025519359001925511780) (232.917960675006309107724789938,422.360679774997896964091736687) (220.361258051038718003851023560,377.639320225002103035908263313) (277.639320225002103035908263313,377.639320225002103035908263313) (481.638163672920460956389689567,377.639320225002103035908263313) (322.360679774997896964091736687,479.638741948961281996148976440) (118.361836327079539043610310433,422.360679774997896964091736687) (277.639320225002103035908263313,467.082039324993690892275210062) (322.360679774997896964091736687,422.360679774997896964091736687) (198.000578276040821039759286873,501.999421723959178960240713127) (401.999421723959178960240713127,298.000578276040821039759286873) (277.639320225002103035908263313,581.638163672920460956389689567) (322.360679774997896964091736687,332.917960675006309107724789938) (322.360679774997896964091736687,218.361836327079539043610310433) (260.180629025519359001925511780,519.458112923441922994223464660) (277.639320225002103035908263313,320.361258051038718003851023560) (339.819370974480640998074488220,280.541887076558077005776535340) (379.638741948961281996148976440,422.360679774997896964091736687) (489.736659610102759919933612666,336.754446796632413360022129111)(110.263340389897240080066387334,463.245553203367586639977870889) (498.746069143517905200435966577,377.639320225002103035908263313)(101.253930856482094799564033423,377.639320225002103035908263313) (461.054672240373943871890584238,518.580742743270597912524823203)(100.759952976983799440071255927,417.418486222008307367796127633) (483.061814743914600319592527135,319.449568699697882651735716776)(150.254423113298622607231348655,532.575496238441461918269182668) (425.540275315678140229647971569,244.309154817427579652556050738)(270.544837448586885134247112298,597.819092605010779433045861937) (455.690845182572420347443949262,274.459724684321859770352028431)(102.180907394989220566954138063,429.455162551413114865752887702) (332.424950472861671010762344439,202.645946109960873452770295163)(134.310098098285437281942152403,512.012751094686559050400929605) (449.745576886701377392768651345,267.424503761558538081730817333)(116.938185256085399680407472865,480.550431300302117348264283224) (497.354053890039126547229704837,367.575049527138328989237655561)(187.987248905313440949599070395,565.689901901714562718057847597) (412.012751094686559050400929605,234.310098098285437281942152403)(102.645946109960873452770295163,432.424950472861671010762344439) (499.240047023016200559928744073,382.581513777991692632203872367)(138.945327759626056128109415762,281.419257256729402087475176797) (322.360679774997896964091736687,598.746069143517905200435966577)(322.360679774997896964091736687,201.253930856482094799564033423) (277.639320225002103035908263313,598.746069143517905200435966577)(277.639320225002103035908263313,201.253930856482094799564033423) (498.746069143517905200435966577,422.360679774997896964091736687)(101.253930856482094799564033423,422.360679774997896964091736687) (432.575496238441461918269182667,250.254423113298622607231348655)(219.449568699697882651735716776,583.061814743914600319592527134) (418.580742743270597912524823203,561.054672240373943871890584238)(317.418486222008307367796127633,200.759952976983799440071255927) (329.455162551413114865752887702,202.180907394989220566954138063)(174.459724684321859770352028431,555.690845182572420347443949262) (497.819092605010779433045861937,370.544837448586885134247112298)(144.309154817427579652556050738,525.540275315678140229647971569) (363.245553203367586639977870889,210.263340389897240080066387334)(236.754446796632413360022129111,589.736659610102759919933612666) (465.689901901714562718057847597,287.987248905313440949599070395)(267.575049527138328989237655561,597.354053890039126547229704837) (282.581513777991692632203872367,599.240047023016200559928744073)(181.419257256729402087475176797,238.945327759626056128109415762) (380.550431300302117348264283224,216.938185256085399680407472865)(167.424503761558538081730817333,549.745576886701377392768651345)
(400,400)(100,200) (131.060716568171734311280321889,456.313094477276088562906559370) (306.932458726091498077071311518,456.313094477276088562906559370) (268.377223398316206680011064556,456.313094477276088562906559370) (287.654841062203852378541188037,368.377223398316206680011064556) (218.996587647131616194175816704,368.377223398316206680011064556) (356.313094477276088562906559370,368.377223398316206680011064556) (243.686905522723911437093440630,431.622776601683793319988935444) (381.003412352868383805824183296,431.622776601683793319988935444) (312.345158937796147621458811963,431.622776601683793319988935444) (331.622776601683793319988935444,343.686905522723911437093440630) (468.939283431828265688719678111,343.686905522723911437093440630) (293.067541273908501922928688481,343.686905522723911437093440630) (243.686905522723911437093440630,393.067541273908501922928688481) (356.313094477276088562906559370,231.060716568171734311280321889) (268.377223398316206680011064556,318.996587647131616194175816704) (331.622776601683793319988935444,412.345158937796147621458811963) (356.313094477276088562906559370,406.932458726091498077071311518) (243.686905522723911437093440630,568.939283431828265688719678111) (268.377223398316206680011064556,387.654841062203852378541188037) (331.622776601683793319988935444,481.003412352868383805824183296) (491.908403647139374806395530474,456.313094477276088562906559370)(108.091596352860625193604469526,456.313094477276088562906559370) (497.484176581314990174384610437,368.377223398316206680011064556)(102.515823418685009825615389563,368.377223398316206680011064556) (497.484176581314990174384610437,431.622776601683793319988935444)(102.515823418685009825615389563,431.622776601683793319988935444) (491.908403647139374806395530474,343.686905522723911437093440630)(108.091596352860625193604469526,343.686905522723911437093440630) (455.444300125756513269905714769,274.154580701506128944891118697)(111.635276040120426406204119997,467.221505245853949799845218093) (373.412906734483218844461093281,213.960904310964604029725787978)(113.960904310964604029725787978,473.412906734483218844461093281) (498.525159788499978712131880586,375.756218715914980280545111807)(109.539479100724373218822407106,461.031057493513842858803943657) (425.845419298493871055108881303,244.555699874243486730094285232)(232.778494754146050200154781907,588.364723959879573593795880003) (490.460520899275626781177592894,338.968942506486157141196056343)(101.474840211500021287868119414,424.243781284085019719454888193) (356.313094477276088562906559370,591.908403647139374806395530475)(356.313094477276088562906559370,208.091596352860625193604469526) (486.306417630942395563794016946,472.731827623983686748180653471)(113.693582369057604436205983054,327.268172376016313251819346529) (486.039095689035395970274212022,326.587093265516781155538906719)(226.587093265516781155538906719,586.039095689035395970274212022) (463.939692032901194685372519229,514.559056281716604199537271376)(185.440943718283395800462728624,236.060307967098805314627480771) (414.559056281716604199537271376,563.939692032901194685372519229)(136.060307967098805314627480771,285.440943718283395800462728624) (361.031057493513842858803943657,209.539479100724373218822407106)(275.756218715914980280545111807,598.525159788499978712131880586) (243.686905522723911437093440630,591.908403647139374806395530475)(243.686905522723911437093440630,208.091596352860625193604469526) (324.243781284085019719454888193,201.474840211500021287868119414)(238.968942506486157141196056343,590.460520899275626781177592894) (372.731827623983686748180653471,586.306417630942395563794016945)(227.268172376016313251819346529,213.693582369057604436205983054) (331.622776601683793319988935444,597.484176581314990174384610437)(331.622776601683793319988935444,202.515823418685009825615389563) (367.221505245853949799845218093,211.635276040120426406204119997)(174.154580701506128944891118697,555.444300125756513269905714769) (268.377223398316206680011064556,597.484176581314990174384610437)(268.377223398316206680011064556,202.515823418685009825615389563) (488.364723959879573593795880003,332.778494754146050200154781906)(144.555699874243486730094285232,525.845419298493871055108881304)
(400,400)(100,200) (406.191270323980923959719363144,368.610100216678785379156576753) (193.808729676019076040280636856,431.389899783321214620843423248) (242.264973081037423549085121950,417.066343621697132887961061847) (357.735026918962576450914878050,382.933656378302867112038938153) (317.066343621697132887961061847,342.264973081037423549085121950) (282.933656378302867112038938153,457.735026918962576450914878050) (331.389899783321214620843423248,293.808729676019076040280636856) (268.610100216678785379156576753,506.191270323980923959719363144) (242.264973081037423549085121950,342.264973081037423549085121950) (357.735026918962576450914878050,342.264973081037423549085121950) (357.735026918962576450914878050,457.735026918962576450914878050) (389.124926702283791071758301298,400.000000000000000000000000000) (210.875073297716208928241698702,400.000000000000000000000000000) (193.808729676019076040280636856,506.191270323980923959719363144) (193.808729676019076040280636856,293.808729676019076040280636856) (406.191270323980923959719363144,293.808729676019076040280636856) (406.191270323980923959719363144,506.191270323980923959719363144) (242.264973081037423549085121950,457.735026918962576450914878050) (300.000000000000000000000000000,310.875073297716208928241698702) (300.000000000000000000000000000,489.124926702283791071758301298) (491.796076874067532766006639931,343.305512651433408089006258135)(108.203923125932467233993360069,456.694487348566591910993741865) (356.694487348566591910993741865,208.203923125932467233993360069)(243.305512651433408089006258135,591.796076874067532766006639931) (491.485421551267621995020382274,342.264973081037423549085121950)(108.514578448732378004979617726,342.264973081037423549085121950) (456.696264548597265936791884181,275.716933267174245861507966481)(280.884905558704872643770902211,599.084437273485463477367973416) (319.115094441295127356229097789,200.915562726514536522632026584)(143.303735451402734063208115819,524.283066732825754138492033519) (193.808729676019076040280636856,569.479833924214146761498507626)(193.808729676019076040280636856,230.520166075785853238501492374) (406.191270323980923959719363144,569.479833924214146761498507626)(406.191270323980923959719363144,230.520166075785853238501492374) (242.264973081037423549085121950,591.485421551267621995020382274)(242.264973081037423549085121950,208.514578448732378004979617726) (355.941343290630579660618079662,207.982901514370806724901045967)(169.257286168709711000505980441,551.348415188363749712145137829) (430.742713831290288999494019559,248.651584811636250287854862171)(244.058656709369420339381920338,592.017098485629193275098954033) (357.735026918962576450914878050,591.485421551267621995020382274)(357.735026918962576450914878050,208.514578448732378004979617726) (492.017098485629193275098954033,344.058656709369420339381920338)(148.651584811636250287854862171,530.742713831290288999494019559) (492.017098485629193275098954033,455.941343290630579660618079662)(148.651584811636250287854862171,269.257286168709711000505980441) (451.348415188363749712145137829,269.257286168709711000505980441)(107.982901514370806724901045967,455.941343290630579660618079662) (499.084437273485463477367973415,419.115094441295127356229097789)(175.716933267174245861507966481,243.303735451402734063208115819) (491.485421551267621995020382274,457.735026918962576450914878050)(108.514578448732378004979617726,457.735026918962576450914878050) (456.696264548597265936791884181,524.283066732825754138492033519)(280.884905558704872643770902211,200.915562726514536522632026584) (430.742713831290288999494019559,551.348415188363749712145137829)(244.058656709369420339381920338,207.982901514370806724901045967) (469.479833924214146761498507626,293.808729676019076040280636856)(130.520166075785853238501492374,293.808729676019076040280636856) (355.941343290630579660618079662,592.017098485629193275098954033)(169.257286168709711000505980441,248.651584811636250287854862171) (451.348415188363749712145137829,530.742713831290288999494019559)(107.982901514370806724901045967,344.058656709369420339381920338) (424.283066732825754138492033519,243.303735451402734063208115819)(100.915562726514536522632026584,419.115094441295127356229097789) (499.084437273485463477367973415,380.884905558704872643770902211)(175.716933267174245861507966481,556.696264548597265936791884181) (424.283066732825754138492033519,556.696264548597265936791884181)(100.915562726514536522632026584,380.884905558704872643770902211) (469.479833924214146761498507626,506.191270323980923959719363144)(130.520166075785853238501492374,506.191270323980923959719363144) (319.115094441295127356229097789,599.084437273485463477367973416)(143.303735451402734063208115819,275.716933267174245861507966481)
(400,400)(100,200) (381.649658092772603273242802490,355.607978004894860689352196086) (255.607978004894860689352196086,318.350341907227396726757197510) (300.000000000000000000000000000,331.472523394971209596277955014) (344.392021995105139310647803914,481.649658092772603273242802490) (300.000000000000000000000000000,468.527476605028790403722044986) (218.350341907227396726757197510,444.392021995105139310647803914) (300.000000000000000000000000000,526.041680087877742583890606404) (381.649658092772603273242802490,550.177134697801393676964847476) (149.822865302198606323035152524,481.649658092772603273242802490) (300.000000000000000000000000000,273.958319912122257416109393596) (218.350341907227396726757197510,249.822865302198606323035152524) (450.177134697801393676964847476,318.350341907227396726757197510) (426.041680087877742583890606404,400.000000000000000000000000000) (450.177134697801393676964847476,481.649658092772603273242802490) (381.649658092772603273242802490,249.822865302198606323035152524) (173.958319912122257416109393596,400.000000000000000000000000000) (149.822865302198606323035152524,318.350341907227396726757197510) (218.350341907227396726757197510,550.177134697801393676964847476) (381.649658092772603273242802490,444.392021995105139310647803914) (344.392021995105139310647803914,318.350341907227396726757197510) (368.527476605028790403722044986,400.000000000000000000000000000) (231.472523394971209596277955014,400.000000000000000000000000000) (218.350341907227396726757197510,355.607978004894860689352196086) (255.607978004894860689352196086,481.649658092772603273242802490) (499.775468099116178528439127268,390.525700776362713675101904630)(137.482167998551285434155871308,283.432619135759543741007488966) (462.517832001448714565844128692,516.567380864240456258992511034)(100.224531900883821471560872732,409.474299223637286324898095370) (418.542739441042961225234296967,561.082646259033743019600284479)(112.929783953928248371043658046,470.744146531540253930607365488) (487.070216046071751628956341954,329.255853468459746069392634512)(181.457260558957038774765703033,238.917353740966256980399715521) (461.082646259033743019600284479,518.542739441042961225234296967)(370.744146531540253930607365488,212.929783953928248371043658046) (229.255853468459746069392634512,587.070216046071751628956341955)(138.917353740966256980399715521,281.457260558957038774765703033) (416.567380864240456258992511033,562.517832001448714565844128692)(309.474299223637286324898095370,200.224531900883821471560872732) (290.525700776362713675101904630,599.775468099116178528439127268)(183.432619135759543741007488967,237.482167998551285434155871308) (489.624053747842418146039029213,336.417626340035324437851577191)(236.417626340035324437851577191,589.624053747842418146039029213) (489.624053747842418146039029213,463.582373659964675562148422808)(236.417626340035324437851577191,210.375946252157581853960970788) (500.000000000000000000000000000,400.000000000000000000000000000)(100.000000000000000000000000000,400.000000000000000000000000000) (395.531250102563154769060803914,575.709363022695882833807827128)(204.468749897436845230939196086,224.290636977304117166192172871) (363.582373659964675562148422808,210.375946252157581853960970788)(110.375946252157581853960970788,463.582373659964675562148422808) (482.574185835055371152323260934,318.350341907227396726757197510)(117.425814164944628847676739066,318.350341907227396726757197510) (482.574185835055371152323260934,481.649658092772603273242802490)(117.425814164944628847676739066,481.649658092772603273242802490) (300.000000000000000000000000000,600.000000000000000000000000000)(300.000000000000000000000000000,200.000000000000000000000000000) (402.944119480984316310573587501,571.471596086013106714295632487)(128.528403913986893285704367513,297.055880519015683689426412499) (471.471596086013106714295632487,297.055880519015683689426412499)(197.055880519015683689426412499,571.471596086013106714295632487) (471.471596086013106714295632487,502.944119480984316310573587501)(197.055880519015683689426412499,228.528403913986893285704367513) (402.944119480984316310573587501,228.528403913986893285704367513)(128.528403913986893285704367513,502.944119480984316310573587501) (363.582373659964675562148422808,589.624053747842418146039029213)(110.375946252157581853960970788,336.417626340035324437851577191) (381.649658092772603273242802490,582.574185835055371152323260934)(381.649658092772603273242802490,217.425814164944628847676739066) (218.350341907227396726757197510,582.574185835055371152323260934)(218.350341907227396726757197510,217.425814164944628847676739066) (475.709363022695882833807827128,495.531250102563154769060803914)(124.290636977304117166192172871,304.468749897436845230939196086) (475.709363022695882833807827128,304.468749897436845230939196086)(124.290636977304117166192172871,495.531250102563154769060803914) (395.531250102563154769060803914,224.290636977304117166192172871)(204.468749897436845230939196086,575.709363022695882833807827128)
$(n_k)$ configurations of lines
===============================
A configuration of lines and points is an *$(n_k)$ configuration* if it consists of $n$ lines and $n$ points, each of which is incident to exactly $k$ of the other type. It is called *geometric* if these are points and lines in the real projective plane.
There are many results concerning geometric $(n_4)$ configurations:
1. There exist geometric $(n_4)$ configurations of lines if and only if $n\ge 18$ except possibly for $n\in\{19,22,23,26,37,43\}$.
2. There is no geometric $(19_4)$ configuration.
3. There exist geometric $(37_4)$ and $(43_4)$ configurations.
Thus for the existence of geometric $(n_4)$ configurations, only the cases $n\in \{22,23,26\}$ were open. Using the above algorithm we can produce examples when $n$ is $22$ and $26$.
We will denote both projective lines and points with coordinates $(a:b:c)$ since points and lines are dual to each other in the plane.
$(22_4)$
--------
The key idea to obtain $(n_4)$ configurations with the above algorithm is to choose an arrangement $\Ac_0$ which already has some points of multiplicity $4$. This way, the orbit $\Ac$ is likely to have a large number of quadruple points as well. Indeed, starting with an arrangement in $\FF_{19}\PP^2$ with two quadruple points and a group $H$ of order $4$, we find the following arrangement of lines (see Figure \[fig\_22\_4\] and \[fig\_22\_4\_e\]): $$\begin{aligned}
\Ac_{22_4}&=&\{
(1:0:0),(0:1:0),(0:0:1),(1:1:1),(24:-5 w - 13:0), \\
&& (24:5 w + 13:24 w),(1:0:w),(2:0:w),\\
&& (24:-5 w - 13:-4 w + 52),(24:5 w + 13:28 w - 52),\\
&& (6:-w + 13:-w + 13),(24:-5 w - 13:16 w + 104),\\
&& (48:w + 65:24 w),(24:5 w + 13:-32 w + 104),\\
&& (18:-w + 13:4 w + 26),(12:-w + 13:0),\\
&& (96:w + 65:56 w - 104),(48:w + 65:-8 w + 104),\\
&& (48:w + 65:20 w + 52),(39:-w + 52:-w + 52),\\
&& (4:w + 13:4 w),(24:w + 26:12 w)
\}\end{aligned}$$ where $w$ is a root of $x^2+7x-26$. Each of the $22$ lines has $13$ intersection points, $4$ quadruple and $9$ double points. The dual configuration (in which the $22$ quadruple points are the lines) has $12$ lines with $4$ quadruple, one triple, and $7$ double points, and $10$ lines with $4$ quadruple and $9$ double points (see Figure \[fig\_22\_4\] and \[fig\_22\_4\_e\]).
1. Since there are two roots $w$ of $x^2+7x-26$, we obtain two arrangements $\Ac_{22_4}$ up to projectivities. The corresponding matroids are isomorphic, but the CW complexes are different. This is why we find four arrangements including the duals.
2. The corresponding matroid has a group of symmetries isomorphic to $\ZZ/2\ZZ\times\ZZ/2\ZZ$. This rather small group is probably the reason why this example did not appear in an earlier publication.
3. The above search finds these examples within a few seconds. The difficulty in finding such a configuration with the above algorithm is thus not about optimizing code.
$(26_4)$
--------
The same technique yields the following $(26_4)$ configuration (and its dual), see Figure \[fig\_26\_4\]:
$$\begin{aligned}
\Ac_{26_4}&=&\{
(1:0:0),(0:1:0),(0:0:1),(1:1:1), \\
&& (1:-z^2 - 2z:z),(z:-z^2 - 2z:z),(-1:-z:-z), \\
&& (2z^2 + 2z:-6z^2 - 14z:-5z^2 - 4z + 21), \\
&& (10z^2 + 12z - 14:-6z^2 - 56z - 98:16z^2 + 24z - 28), \\
&& (6z^2 - 14:-2z^2 - 12z + 14:-16z^2 - 20z + 56), \\
&& (68z^2 + 56z - 196:20z^2 + 72z - 84:84z^2 + 100z - 280), \\
&& (-24z^2 - 60z - 28:0:-26z^2 - 40z + 14), \\
&& (-256z^2 - 112z + 784:-352z^2 - 624z + 336:-264z^2 - 224z + 392), \\
&& (0:-16z^2 + 4z + 28:-2z^2 + 12z - 14), \\
&& (68z^2 + 56z - 196:20z^2 + 72z - 84:20z^2 + 72z - 84), \\
&& (-1136z^2 - 256z + 3696:-3152z^2 - 2560z + 7952:-1840z^2 - 528z + 4928), \\
&& (-608z^2 - 1760z + 1792:1120z^2 - 1824z - 8064:-2624z^2 - 4064z + 6048), \\
&& (0:-1120z^2 + 1824z + 8064:-1872z^2 - 1120z + 7056), \\
&& (-12864z^2 - 14976z + 48832:-23616z^2 - 35968z + 90048:-27584z^2 - 19200z + 88256), \\
&& (4288z^2 + 37888z + 61376:-44736z^2 - 170752z - 157248:-2656z^2 + 19712z + 48608), \\
&& (8z^2 + 136z - 224:-304z^2 - 392z + 1176:-412z^2 - 632z + 1652), \\
&& (-784z^2 - 608z + 2800:-272z^2 - 288z - 1232:-1136z^2 - 256z + 3696), \\
&& (11264z^2 + 30464z - 75264:-193536z^2 - 190208z + 637952:-123776z^2 - 57344z + 307328), \\
&& (-65984z^2 - 13056z + 231616:-55360z^2 - 91904z + 37184:448z^2 + 55808z - 20160), \\
&& (8192z^2 - 31232z - 155904:55808z^2 + 147712z - 57344:-54912z^2 - 36096z + 17024), \\
&& (627968z^2 + 367104z - 1732864:2495232z^2 + 3188224z - 9105152:\\
&& 1453568z^2 + 1928704z - 5465600)
\}\end{aligned}$$
where $z$ is the real root of $x^3 + 3x^2 - x - 7$.
All the matroids presented in this note have realizations which are unique up to projectivities and Galois automorphisms. For $\Ac_{26_4}$ there is a complex realization which may not be transformed into a real arrangement by a projectivity, namely when $z$ is a complex root of $x^3 + 3x^2 - x - 7$.
$(23_4)$
--------
The arrangement of lines $$\begin{aligned}
\Ac_{23_4}&=&\{
(0:0:1),(0:1:0),(1:0:0),(2:0:1),(1:0:1), \\
&& (1:-1:1),(1:1:1),(2:2:i + 1),(1:1:i),(1:-i:0), \\
&& (2:-2i:i + 1),(1:-i:i + 1),(1:-i + 2:i), \\
&& (5:-3i + 4:i + 2),(2:-i + 1:i + 1),(5:-2i + 1:i + 2), \\
&& (5:-i - 2:i + 2),(5:-i + 2:-i + 2),(5:-i + 2:i + 3), \\
&& (5:-i + 2:3i + 4),(1:i:0),(1:i:-i),(1:i:i)
\}\end{aligned}$$ where $i=\sqrt{-1}$ has $25$ intersection points of multiplicity $4$. The right choice of $23$ points yields a $(23_4)$ configuration in the complex projective plane.
Notice that the above algorithm produces many more non isomorphic examples over finite fields and even (at least) three more examples over the complex numbers. Thus these results give no hint concerning the existence of geometric $(23_4)$ configurations.
**Acknowledgement.** I would like to thank J. Bokowski and V. Pilaud for calling my attention to the subject of $(n_k)$ configurations.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[1]{}
Jürgen Bokowski and Vincent Pilaud, *On topological and geometric [$(19_4)$]{} configurations*, European J. Combin. **50** (2015), 4–17.
, *Quasi-configurations: building blocks for point-line configurations*, Ars Math. Contemp. **10** (2016), no. 1, 99–112.
Jürgen Bokowski and Lars Schewe, *On the finite set of missing geometric configurations [$(n_4)$]{}*, Comput. Geom. **46** (2013), no. 5, 532–540.
M. Cuntz, *Minimal fields of definition for simplicial arrangements in the real projective plane*, Innov. Incidence Geom. **12** (2011), 49–60.
Branko Grünbaum, *Connected [$(n_4)$]{} configurations exist for almost all [$n$]{}—second update*, Geombinatorics **16** (2006), no. 2, 254–261.
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abstract: 'The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair $\frak g=\mathfrak{gl}(n,2m), \frak k=\mathfrak{osp}(n,2m)$ we establish a natural bijection between projective covers of spherically typical irreducible $\frak g$-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals $\frak{D}_{n,m}$ acting on the corresponding Laurent quasi-invariants $\frak A_{n,m}$.'
address:
- 'Department of Mathematics, Saratov State University Astrakhanskaya 83, Saratov 410012, Russia and National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia '
- 'Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK and Moscow State University, Moscow 119899, Russia'
author:
- 'A.N. Sergeev'
- 'A.P. Veselov'
title: 'Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems'
---
Introduction
============
In 1964 Berezin et al [@BPF] made an important remark that the radial part of the Laplace–Beltrami operator on the symmetric space $X=SL(n)/SO(n)$ $$L=\Delta+\sum_{i<j}^n\coth(x_i-x_j) (\partial_i-\partial_j)$$ is conjugated to the quantum Hamiltonian $$H=\Delta+ \sum_{i<j}^n\frac{1}{2\sinh^2(x_i-x_j)}$$ describing the pairwise interacting particle on the line.
This was probably the first recorded observation of the connection between the theory of symmetric spaces and the theory of what later became known as Calogero-Moser, or Calogero-Moser-Sutherland (CMS), integrable models [@CMS]. Olshanetsky and Perelomov suggested a class of generalisations of CMS systems related to any root system and showed that the radial parts of all irreducible symmetric spaces are conjugated to some particular operators from this class [@OP3]. The joint eigenfunctions of the corresponding commutative algebras of quantum integrals are zonal spherical functions. In the $A_n$ case this leads to an important notion of the Jack polynomials introduced by H. Jack independently around the same time [@Jack].
The discovery of the Dunkl operator technique led to an important link of the CMS systems with the representation theory of Cherednik algebras, see Etingof’s lectures [@Etingof].
It turned out that there are other integrable generalisations, which have only partial symmetry and called deformed CMS systems [@CFV]. Their relation with symmetric superspaces was first discovered by one of the authors in [@Ser] and led to a class of such operators related to the basic classical Lie superalgebras, which was introduced in [@SV].
In this paper we develop this link further to study the representation theory of symmetric Lie superalgebras and the related spherical functions. Such Lie superalgebra is a pair $(\frak g,\theta),$ where $\frak g$ is a Lie superalgebra and $\theta$ is an involutive automorphism of $\frak g$. It corresponds to the symmetric pair $X=(\frak g, \frak k),$ where $\frak k$ is $\theta$-invariant part of $\frak g$ and can be considered as an algebraic version of the symmetric superspace $G/K$.
In the particular case of $X=(\frak {gl}(n,2m), \frak {osp}(n,2m))$ the radial part of the corresponding Laplace-Beltrami operator in the exponential coordinates is a particular case of the deformed CMS operator related to Lie superalgebra $\mathfrak{gl}(n,m)$ [@SV] $${\mathcal L}=\sum_{i=1}^n
\left(x_{i}\frac{\partial}{\partial
x_{i}}\right)^2+k\sum_{j=1}^m
\left(y_{j}\frac{\partial}{\partial
y_{j}}\right)^2-k \sum_{i < j}^n
\frac{x_{i}+x_{j}}{x_{i}-x_{j}}\left(
x_{i}\frac{\partial}{\partial x_{i}}-
x_{j}\frac{\partial}{\partial x_{j}}\right)$$ $$\label{Lrad}
-\sum_{i <
j}^m \frac{y_{i}+y_{j}}{y_{i}-y_{j}}\left(
y_{i}\frac{\partial}{\partial y_{i}}-
y_{j}\frac{\partial}{\partial y_{j}}\right)-\sum_{i=1 }^n\sum_{
j=1}^m \frac{x_{i}+y_{j}}{x_{i}-y_{j}}\left(
x_{i}\frac{\partial}{\partial x_{i}}-k
y_{j}\frac{\partial}{\partial y_{j}}\right)$$ corresponding to the special value of parameter $k=-\frac12.$ According to [@SV] it has infinitely many commuting differential operators generating the algebra of quantum deformed CMS integrals $\frak{D}_{n,m}$.
We study the action of $\frak{D}_{n,m}$ on the algebra $\frak A_{n,m}$ of $S_n\times S_m$-invariant Laurent polynomials $f\in\Bbb C[x_1^{\pm1},\dots, x_n^{\pm1},y_1^{\pm1}\dots,y_m^{\pm1}]^{S_n\times S_m}$ satisfying the quasi-invariance condition $$\label{quasi}
x_i\frac{\partial f}{\partial x_i}-ky_j\frac{\partial f}{\partial y_j}\equiv 0$$ on the hyperplane $x_i=y_j$ for all $i=1,\dots,n$, $j=1,\dots,m$ with $k=-\frac12$. It turns out that the generalised eigenspaces $\frak A_{n,m}(\chi)$ in the corresponding spectral decomposition $$\frak A_{n,m}=\oplus_{\chi} \frak A_{n,m}(\chi),$$ where $\chi$ are certain homomorphisms $\chi: \frak{D}_{n,m} \rightarrow \mathbb C$, are in general not one-dimensional, similarly to the case of Jack–Laurent symmetric functions considered in our recent paper [@SV3]. We have shown there that the corresponding generalised eigenspaces have dimension $2^r$ and the image of the algebra of CMS integrals in the endomorphisms of such space is isomorphic to the tensor product of dual numbers $$\mathfrak A_r=\Bbb C[\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r]/(\varepsilon_1^2,\,\varepsilon_2^2,\dots,\varepsilon_r^2).$$ It is known that the algebra $\frak A_r$ also appears as the algebra of the endomorphisms of the projective indecomposable modules over general linear supergroup (see Brundan-Stroppel [@Brund]). It is natural therefore to think about possible links between our generalised eigenspaces and projective modules.
The main result of this paper is a one-to-one correspondence between the finite-dimensional generalised eigenspaces of $\frak{D}_{n,m}$ and projective covers of certain irreducible finite-dimensional modules of $\mathfrak{gl}(n,2m)$. More precisely, we prove the following main theorem.
Let $Z(\frak g)$ be the centre of the universal enveloping algebra of $\frak g = \mathfrak{gl}(n,2m).$ For a $\frak g$-module $U$ we denote by $U^\frak k$ its part invariant under $\frak k=\frak{osp}(n,2m).$
For any finite dimensional generalised eigenspace $\frak A_{n,m}(\chi)$ there exists a unique projective indecomposable module $P$ over $\frak{gl}(n,2m)$ and a natural map $$\Psi : (P^*)^{\frak k}\longrightarrow \frak A_{n,m}(\chi),$$ which is an isomorphism of $Z(\frak g)$-modules.
This establishes the bijection between the projective covers of spherically typical irreducible $\frak g$-modules and the finite dimensional generalised eigenspaces of the algebra of deformed CMS integrals $\frak{D}_{n,m}$ acting in $\frak A_{n,m}$.
The corresponding projective modules can be described explicitly in terms of the highest weights of $\mathfrak{gl}(n,2m)$ under certain typicality conditions, which are natural generalisation of Kac’s typicality conditions [@Kac].
As a corollary we have an algorithm for calculating the composition quotients in Kac flag of the corresponding projective covers in the spherically typical case (which may have any degree of atypicality in the sense of [@Brun]). The number of the quotients is equal to the number of elements in the corresponding equivalence class, which can be described combinatorially, and equals $2^s$, where $s$ is the degree of atypicality (see sections 6 and 7 below). Our algorithm is equivalent to Brundan-Stroppel algorithm [@Brund] in this particular case, but our technique is different and uses the theory of the deformed CMS systems.
The plan of the paper is following. In the next section we introduce the algebra $\frak{D}_{n,m}$ of quantum integrals of the deformed CMS system (mainly following [@SV4]) and study the corresponding spectral decomposition of its action on the algebra $\frak A_{n,m}.$
In section 3 we introduce symmetric Lie superalgebras and derive the formula for the radial part of the corresponding Laplace-Beltrami operators. In particular, we show that for the four classical series of symmetric Lie superalgebras this radial part is conjugated to the deformed CMS operators introduced in [@SV].
The rest of the paper is dealing mainly with the particular case corresponding to the symmetric pairs $(\frak g, \frak k)$ with $\frak g=\mathfrak{gl}(n,2m), \frak k=\mathfrak{osp}(n,2m).$ We call a finite dimensional $\frak g$-module $U$ spherical if the space of $\frak k$-invariant vectors $U^{\frak k}$ is non-zero. We describe the admissibility conditions on highest weight $\lambda,$ for which the corresponding Kac module $K(\lambda)$ is spherical. Under certain assumptions of typicality we describe the conditions on admissible highest weights for irreducible modules to be spherical (see sections 5 and 6) and study the equivalence relation on the admissible weights defined by the equality of central characters.
These results are used in section 7 to prove the main theorem, which implies in particular that any finite-dimensional generalised eigenspace contains at least one zonal spherical function, corresponding to an irreducible spherically typical $\frak g$-module. In the last section we illustrate all this, including explicit formulas for the zonal spherical functions, in the simplest example of symmetric pair $X=(\mathfrak{gl}(1,2), \mathfrak{osp}(1,2)).$
Algebra of deformed CMS integrals and spectral decomposition
============================================================
In this section we assume that the parameter $k$ is arbitrary nonzero, so everything is true for the special case $k=-\frac12$ as well.
To define the algebra of the corresponding CMS integrals $\mathfrak D_{n,m}$ it will be convenient to denote $x_{n+j}:=y_j,\,\, j=1,\dots,m$ and to introduce parity function $p(i)=0,\, i=1,\dots,n, \,\, p(i)=1,\,\, i=n+1,\dots, n+m.$ We also introduce the notation $$\partial_j=x_j\frac{\partial}{\partial x_j}, \,\,\, j=1,\dots, n+m.$$
By definition the algebra $\mathfrak D_{n,m}$ is generated by the deformed CMS integrals defined recursively in [@SV]. It will be convenient for us to use the following, slightly different choice of generators.
Define recursively the differential operators $\partial_{i}^{(p)}, 1\le i \le n+m, \, p \in \mathbb N$ as follows: for $p=1$ $$\partial_{i}^{(1)}=k^{p(i)}\partial_i$$ and for $p > 1$ $$\label{dif1}
\partial_{i}^{(p)}=\partial_{i}^{(1)}\partial_{i}^{(p-1)}-\sum_{j\ne i}k^{1-p(j)}\frac{x_{i}}{x_{i}-x_{j}}\left(\partial_{i}^{(p-1)}-\partial_{j}^{(p-1)}\right).$$
Then the higher CMS integrals ${\mathcal{L}}_{p}$ are defined as the sums $$\label{dif2} {\mathcal{L}}_{p}=\sum_{i\in I}k^{-p(i)}\partial_{i}^{(p)}.$$ In particular, for $p=2$ we have $$\label{L2}
\mathcal L_2=\sum_{i=1}^{n+m}k^{-p(i)} \partial_i^2-\sum_{i<j}^{n+m}\frac{x_{i}+x_{j}}{x_{i}-x_{j}}(k^{1-p(j)} \partial_i-k^{1-p(i)}\partial_j),$$ which coincides with the deformed CMS operator (\[Lrad\]).
The operators ${\mathcal{L}}_{p}$ are quantum integrals of the deformed CMS system: $$[ {\mathcal{L}}_{p}, {\mathcal{L}}_{2}]=0.$$
Following the idea of our recent work [@SV4] introduce a version of quantum Moser $(n+m)\times (n+m)$-matrices $L,\, M$ by $$L_{ii}=k^{p(i)}\partial_i-\sum_{j\ne i}k^{1-p(j)}\frac{x_{i}}{x_{i}-x_{j}},\,\,\, L_{ij}=k^{1-p(j)}\frac{x_i}{x_i-x_j},\, i\ne j$$ $$M_{ii}=-\sum_{j\ne i}\frac{2k^{1-p(j)}x_ix_j}{(x_{i}-x_j)^2},\quad M_{ij}=\frac{2k^{1-p(j)}x_ix_j}{(x_{i}-x_j)^2},\, i\ne j.$$ Note that matrix $M$ satisfies the relations $$Me=e^*M=0,\,\,e=(\underbrace{1,\dots,1}_{n+m})^t,\,\, e^*=(\underbrace{1\dots,1}_{n},\underbrace{1/k,\dots,1/k}_{m}).$$ Define also matrix Hamiltonian $H$ by $$H_{ii}=\mathcal L_2,\,\,\,H_{ij}=0,\,i\ne j.$$ Then it is easy to check that these matrices satisfy Lax relation $$[L,H]=[L,M].$$ Indeed, matrix $L$ is different from the Moser matrix (43) from [@SV4] by rank one matrix $e\otimes e^*$, which does not affect the commutator $[L,M]$ because of the relations $Me=e^*M=0.$ This implies as in [@SV4; @UHW] that the “deformed total trace” $${\mathcal{L}}_{p}=\sum_{i,j}k^{-p(i)}(L^p)_{ij}$$ commute with ${\mathcal{L}}_{2}.$
Define now the [*Harish-Chandra homomorphism*]{} $$\varphi: \mathfrak D_{n,m} \rightarrow \mathbb C[\xi_1, \dots, \xi_{n+m}]$$ by the conditions (cf. [@SV]): $$\varphi (\partial_i)=\xi_i, \quad \varphi \left(\frac{x_{i}}{x_{i}-x_{j}}\right)=1, \, \,\, {\text {if}} \,\, i<j.$$ In particular, $d_i^{(p)}(\xi):=\varphi(\partial_i^{(p)})$ satisfy the following recurrence relations $$\label{recu}
d_i^{(p)}=d_i^{(1)}d_{i}^{(p-1)}-\sum_{j> i}k^{1-p(j)}(d_i^{(p-1)}-d_j^{(p-1)}),$$ which determine them uniquely with $d_i^{(1)}=k^{p(i)}\xi_i, \, i=1,\dots, n+m.$
Let $\rho(k) \in \mathbb C^{n+m}$ be the following deformed analogue of the Weyl vector $$\label{rhok}
\rho(k)=\frac12\sum_{i=1}^n(k(2i-n-1)-m)e_i+\frac12\sum_{j=1}^{m}(k^{-1}(2j-m-1)+n)e_{j+n}$$ and consider the bilinear form $( , )$ on $\mathbb C^{n+m}$ defined in the basis $e_1,\dots, e_{n+m}$ by $$(e_i, e_i)=1, \,\, i=1,\dots, n, \quad (e_j, e_j)=k, \,\, j=n+1,\dots, n+m.$$
\[hc\] [@SV] Harish-Chandra homomorphism is injective and its image is the subalgebra $\Lambda_{n,m}(k) \subset \mathbb C[\xi_1, \dots, \xi_{n+m}]$ consisting of polynomials with the following properties: $$f(w(\xi+\rho(k)))=f(\xi+\rho(k)), \quad w \in S_n\times S_m$$ and for every $i \in \{1,\dots, n\},\,\,j \in \{n+1,\dots, n+m\}$ $$\label{hc1}
f(\xi-e_i+e_j)=f(\xi)$$ on the hyperplane $(\xi+\rho(k), e_i-e_j)=\frac12(1+k).$
Operators $\mathcal L_p$ commute with each other.
From the results of [@SV5] it follows that $\mathcal L_p$ generate the same algebra $\mathfrak D_{n,m}$ as commuting CMS integrals from [@SV], which gives another proof of their commutativity.
Let now $\frak A_{n,m}$ be the algebra consisting of $S_n\times S_m$-invariant Laurent polynomials $f\in\Bbb C[x_1^{\pm1},\dots, x_n^{\pm1},y_1^{\pm1}\dots,y_m^{\pm1}]^{S_n\times S_m}$ satisfying the quasi-invariance condition $$\label{quasix}
x_i\frac{\partial f}{\partial x_i}-ky_j\frac{\partial f}{\partial y_j}\equiv 0$$ on the hyperplane $x_i=y_j$ for all $i=1,\dots,n$, $j=1,\dots,m$ with $k$ being arbitrary and the same as in the definition of $\mathfrak D_{n,m}$. We claim that the algebra $\mathfrak D_{n,m}$ preserves it.
For any Laurent polynomial $$f=\sum_{\mu \in X_{n,m}}c_{\mu}x^{\mu}, \quad X_{n,m}=\mathbb Z^n\oplus\mathbb Z^m$$ consider the set $M(f)$ consisting of $\mu$ such that $c_{\mu}\ne0$ and define the [*support*]{} $S(f)$ as the intersection of the convex hull of $M(f)$ with $X_{n,m}.$
\[in\] The operators ${\mathcal{L}}_{p}$ for all $p=1,2,\dots$ map the algebra $\frak A_{n,m}$ to itself and preserve the support: for any $ D\in{\mathfrak D}_{n,m}$ and $f\in \frak{A}_{n,m}$ $$S(Df)\subseteq S(f).$$
The first part follows from the fact that if $f\in \frak A_{n,m}$ then $\partial_{i}^{(p)}f$ is a polynomial. The proof is essentially repeating the arguments from [@SV1] (see theorem 5 and lemmas 5 and 6), so we will omit it.
To prove the second part it is enough to show that $S(\partial_{i}^{(p)} f)\subseteq S(f)$ for any $f\in \frak A_{n,m}.$ From the recursion (\[dif1\]) we see that it is enough to prove that $$g=\frac{x_{i}}{x_{i}-x_{j}}\left(\partial_{i}^{(p-1)}-\partial_{j}^{(p-1)}\right)(f)$$ is a polynomial and $S(g)\subseteq S(f).$ Denote $\left(\partial_{i}^{(p-1)}-\partial_{j}^{(p-1)}\right)(f)$ as $h(x),$ which is known to be a Laurent polynomial. By induction assumption $S(h) \subseteq S(f)$. Since $$h(x)=\left(1-\frac{x_j}{x_i}\right)g(x)$$ and the support of a product of two Laurent polynomials is the Minkowski sum of the supports of the factors this implies that $S(g)\subseteq S(h) \subseteq S(f)$.
Now we are going to investigate the spectral decomposition of the action of the algebra of CMS integrals $\mathfrak D_{n,m}$ on $\frak{A}_{n,m}$.
We will need the following [*partial order*]{} on the set of integral weights $\lambda \in X_{n,m} = \mathbb Z^{n+m}$: we say that $\mu \preceq \lambda$ if and only if $$\label{partial}
\mu_1 \le \lambda_1, \, \mu_1+\mu_2 \le \lambda_1+\lambda_2, \dots, \mu_1+\dots +\mu_{n+m} \le \lambda_1+\dots +\lambda_{n+m}.$$
\[sup\] Let $f \in \mathfrak A_{n,m}$ and $\lambda$ be a maximal element of $M(f)$ with respect to partial order. Then for any $D \in \mathfrak D_{n,m}$ there is no $\mu$ from $M(D(f)), \, \mu \ne \lambda$ such that $\lambda \preceq \mu.$ The coefficient at $x^{\lambda}$ in $D(f)$ is $\varphi(D)(\lambda)c_\lambda$, where $c_\lambda$ is the coefficient at $x^{\lambda}$ in $f.$
If $\lambda$ is the only maximal element of $M(f)$ then $\mu \preceq \lambda$ for any $\mu$ from $M(D(f)).$
It is enough to prove this only for $D=\partial_{i}^{(p)}$. We will do it by induction on $p$. In the notations of the proof of Theorem \[in\] let us assume that there is $\mu \in M(g)$ such that $\lambda \preceq \mu, \, \mu \ne \lambda.$ Without loss of generality we can assume that $\mu$ is maximal in $M(g).$
From $h(x)=(1-x_j/x_i)g(x)$ with $i<j$ it follows that $\mu$ is also maximal in $M(h)$, which contradicts the inductive assumption. This implies that the coefficient at $x^\lambda$ in $\partial_{i}^{(p)}(f)$ satisfies the same recurrence relations (\[recu\]) with the initial conditions multiplied by $c_\lambda.$ This proves the first part. The proof of the second part is similar.
Let $\chi: \mathfrak D_{n,m} \rightarrow \mathbb C$ be a homomorphism and define the corresponding [*generalised eigenspace*]{} $\frak A_{n,m}(\chi)$ as the set of all $f\in \frak A_{n,m}$ such that for every $D\in \mathfrak D_{n,m}$ there exists $N\in \mathbb N$ such that $(D-\chi(D))^N(f)=0.$ If the dimension of $\frak A_{n,m}(\chi)$ is finite then such $N$ can be chosen independent on $f.$
\[dec\] Algebra $\frak A_{n,m}$ as a module over the algebra $\mathfrak D_{n,m}$ can be decomposed in a direct sum of generalised eigenspaces $$\label{sum}
\frak A_{n,m}=\oplus_{\chi}\frak A_{n,m}(\chi),$$ where the sum is taken over the set of some homomorphisms $\chi$ (explicitly described below).
Let $f\in \frak A_{n,m}$ and define a vector space $$V(f)=\{g\in \frak A_{n,m}\mid S(g)\subseteq S(f)\}.$$ By Theorem \[in\] $V(f)$ is a finite dimensional module over $\mathfrak D_{n,m}.$ Since the proposition is true for every finite-dimensional modules the claim now follows.
Now we describe all homomorphisms $\chi$ such that $\frak A_{n,m}(\chi)\ne0$. We say that the integral weight $\lambda \in X_{n,m}\in\mathbb Z^{n+m}$ [*dominant*]{} if $$\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n, \quad \lambda_{n+1} \ge \lambda_{n+2} \ge \dots \ge \lambda_{n+m}.$$ The set of dominant weights is denoted $X_{n,m}^+.$
For every $\lambda\in X_{n,m}^{+}$ we define the homomorphism $\chi_\lambda: \mathfrak D_{n,m} \rightarrow \mathbb C$ by $$\chi_{\lambda}(D)=\varphi(D)(\lambda), \,\, D\in \mathfrak D_{n,m}$$ where $\varphi$ is the Harish-Chandra homomorphism.
\[max\] $1)$ For any $\lambda\in X_{n,m}^+$ there exists $\chi$ and $f\in \frak A_{n,m}(\chi)$, which has the only maximal term $x^{\lambda}$.
$2)$ $\frak A_{n,m}(\chi)\ne0$ if and only if there exists $\lambda\in X_{n,m}^+$ such that $\chi=\chi_{\lambda}$.
$3)$ If $\frak A_{n,m}(\chi)$ is finite dimensional then its dimension is equal to the number of $\lambda \in X_{n,m}^+$ such that $\chi_{\lambda}=\chi$.
Let $\mu_1=\lambda_1, \dots, \mu_n=\lambda_n, \,\, \, \nu_1=\lambda_{n+1}, \dots, \nu_{m}=\lambda_{n+m}$. Consider the Laurent polynomial $$g(x,y)=s_{\mu}(x)s_{\nu}(y)\prod_{i,j}(1-y_j/x_i)^2$$ where $s_{\mu}(x), \, s_{\nu}(y)$ are the Schur polynomials [@Ma]. It is easy to check that $g$ belongs to the algebra $\frak A_{n,m}$ and has the only maximal weight $\lambda$. By Proposition \[dec\] we can write $g=g_1+\dots+g_N$, where $g_i$ belong to different generalised eigenspaces. Therefore there exists $i$ such that $\lambda \in M(g_i)$. Since $g_i$ can be obtained from $g$ by some element from the algebra $\mathfrak A_{n,m}$ (which is a projector to the corresponding generalised eigenspace in some finite-dimensional subspace containing $g$), then $\lambda$ is the only maximal element of $M(g_i)$ by Proposition \[sup\]. This proves the first part.
Let $ \frak A_{n,m}(\chi)\ne0$. Pick up a nonzero element $f$ from this subspace and choose some maximal element $\lambda^{(1)}$ from $M(f)$ and an operator $D \in \mathfrak D_{n,m}$. Then according to Proposition \[sup\] element $x^{\lambda^{(1)}}$ does not enter in $f_1=(D-\chi_{\lambda^{(1)}}(D))(f)$ and $S(f_1) \subset S(f)$. Repeating this procedure we get the sequence of nonzero elements $f_0=f,\,f_1,\,\dots,f_N$ and the numbers $a_1=\chi_{\lambda^{(1)}}(D),\dots,a_N=\chi_{\lambda^{(N)}}(D)$ such that $$f_i=(D-a_i)f_{i-1},\,i=1,\dots,N,\quad (D-a_N)f_{N-1}=0.$$ Therefore $$P(t)=\prod_{i=1}^N(t-a_i)$$ is a minimal polynomial for $D$ in the subspace $<f_0,\dots,f_{N-1}>$. But this subspace is in $\frak A_{n,m}(\chi)$. Therefore this polynomial should be some power of $t-\chi(D)$ and hence $a_1=a_2=\dots=a_N=\chi(D)$. In particular, this implies that $\chi(D)=a_1=\chi_{\lambda^{(1)}}(D)$ for some $\lambda^{(1)}\in X^+_{n,r}$ as required.
Conversely, let $\lambda\in X_{n,m}^+$. According to the first part there exists $\chi$ and $f\in \frak A_{n,m}(\chi)$ such that $\lambda$ is its maximal weight. Therefore the previous considerations show that $\chi=\chi_{\lambda}$ and thus $ \frak A_{n,m}(\chi_\lambda)\ne0.$
To prove the third part suppose that $\frak A_{n,m}(\chi)$ is finite dimensional and that $\lambda^{(1)},\dots,\lambda^{(N)}$ are all different elements from $X_{n,m}^+$ such that $\chi_{\lambda^{(i)}}=\chi,\, i=1,\dots, N$. According to the first two parts there exists $f_i \in \frak A_{n,m}(\chi)$ with the only maximal weight $\lambda^{(i)}$. It is easy to see that $f_1,\dots,f_N$ are linearly independent. To show that they form a basis consider any $f \in \frak A_{n,m}(\chi)$ and take a maximal weight $\mu$ from $M(f).$ According to Proposition \[sup\] $\chi_\mu=\chi$ and thus $\mu$ must coincide with one of $\lambda^{(i)}.$ By subtracting from $f$ a suitable multiple of $f_i$ and using induction we get the result.
The set of homomorphisms in Proposition \[dec\] consists of $\chi=\chi_\lambda, \,\, \lambda \in X^+_{n,m}.$
Symmetric Lie superalgebras and Laplace-Beltrami operators
==========================================================
We will be using an algebraic approach to the theory of symmetric superspaces based on the notion of symmetric Lie superalgebras going back to Dixmier [@Dix]. More geometric approach with relation to physics and random-matrix theory can be found in Zirnbauer [@Zirn]. For the classification of real simple Lie superalgebras and symmetric superspaces see Serganova [@Se].
[*Symmetric Lie superalgebra*]{} is a pair $(\frak g,\theta),$ where $\frak g$ is a complex Lie superalgebra, which will be assumed to be basic classical [@Kac0],[^1] and $\theta$ is an involutive automorphism of $\frak g$. We have the decomposition $ \mathfrak g = \mathfrak k \oplus \frak p,$ where $\mathfrak k$ and $\frak p$ are $+1$ and $-1$ eigenspaces of $\theta:$ $$[\frak k, \frak k]\subset \frak k, \, [\frak k, \frak p]\subset \frak p, \, [\frak p, \frak p]\subset \frak k.$$ Alternatively, one can talk about [*symmetric pair*]{} $X=(\frak g, \frak k).$
In this paper we restrict ourselves by the following 4 classical series of symmetric pairs (in Cartan’s notations [@Hel0; @Zirn]): $$AI/AII=(\frak {gl}(n,2m), \frak {osp}(n,2m)),\,\,\, DIII/CI=(\frak{osp} (2l, 2m), \frak{gl}(l,m)),$$ $$\label{4series}
AIII=(\frak{gl} (n_1+n_2, m_1+m_2), \frak{gl}(n_1,m_1)\oplus\frak{gl}(n_2,m_2)),$$ $$BDI/CII=(\frak{osp} (n_1+n_2, 2m_1+2m_2), \frak{osp}(n_1,2m_1)\oplus\frak{osp}(n_2,2m_2)).$$ In fact, we will give all the details only for the first series, which will be our main case (see next Section). For a more general approach we refer to the work by Alldridge et al [@Ald].
Commutative subalgebra $\frak a \subset\frak p$ is called [*Cartan subspace*]{} if it is reductive in $\frak g$ and the centraliser of $\frak a$ in $\frak p$ coincides with $\frak a$ [@Dix]. We will consider only the cases when Cartan subspace can be chosen to be even (“even type” in the terminology of [@Ald]).
The Lie superalgebra $\frak g$ has an even invariant supersymmetric bilinear form with restriction on $\frak a$ being non-degenerate. The corresponding quadratic form on $\frak a$ we denote $Q.$
We have the decomposition of $\frak g$ with respect to $\frak a$ into nonzero eigenspaces $$\frak g=\frak g^\frak a_0\oplus \bigoplus_{\alpha\in R(X)}\frak g^\frak a_{\alpha}.$$ The corresponding set $R(X) \subset \frak a^*$ is called [*restricted root system*]{} of $X$ and $\mu_{\alpha}=sdim \, \frak g^\frak a_{\alpha}$ are called [*multiplicities*]{}, where $sdim \, \frak g^\frak a_{\alpha}$ is the super dimension: $sdim \, g^\frak a_{\alpha}=\dim g^\frak a_{\alpha}$ for even roots and $sdim \, g^\frak a_{\alpha}=-\dim g^\frak a_{\alpha}$ for odd roots.
For the symmetric pairs $X=(\frak {gl}(n,2m), \frak {osp}(n,2m))$ of type $AI/AII$ we have the following root system consisting of the even roots $\pm (x_i-x_j), \, 1\le i < j \le n$ with multiplicity $\mu=1$, $\pm (y_i-y_j), \, 1\le i < j \le m$ with multiplicity $\mu=4$ and odd roots $\pm (x_i-y_j), \, 1\le i\le n, \, 1\le j\le m$ with multiplicity $\mu=-2$. The corresponding invariant quadratic form is $$\label{Q}
Q= x_1^2+\dots+x_n^2+k^{-1}(y_1^2+\dots+y_m^2)$$ with $k=-1/2$ (see the next section).
For the remaining 3 classical series we have the following restricted root systems of $BC(n,m)$ type, see [@OP; @Ald].
For $X=(\frak{osp} (2l, 2m), \frak{gl}(l,m))$ of type $DIII/CI$ the restricted root system depends on the parity of $l.$ For odd $l=2n+1$ the restricted even roots are $\pm x_i$ with $\mu=4$, $\pm 2x_i$ with $\mu=1$ for $i=1,\dots, n,$ $\pm x_i \pm x_j$ with $\mu=4$ for $1 \le i < j \le n,$ $\pm 2y_i$ with $\mu=1$ for $i=1,\dots, m,$ $\pm y_i \pm y_j$ with $\mu=1$ for $1 \le i < j \le m$ and odd roots $\pm x_i \pm y_j$ and $\pm y_j$ with $\mu=-2$ with for $1 \le i \le n,\, 1 \le j \le m.$ The quadratic form $Q$ by (\[Q\]) with $k=-2.$
For even $l=2n$ the restricted even roots are $\pm 2x_i$ with $\mu=1$ for $i=1,\dots, n,$ $\pm x_i \pm x_j$ with $\mu= 4$ for $1 \le i < j \le n,$ $\pm 2y_i$ with $\mu=1$ for $i=1,\dots, m,$ $\pm y_i \pm y_j$ with $\mu=1$ for $1 \le i < j \le m$ and odd roots $\pm x_i \pm y_j$ with $\mu=-2$ with for $1 \le i \le n,\, 1 \le j \le m.$ The quadratic form $Q$ is given by (\[Q\]) with $k=-2.$
For the symmetric pairs $(\frak{gl} (n_1+n_2, m_1+m_2), \frak{gl}(n_1,m_1)\oplus\frak{gl}(n_2,m_2))$ of type $AIII$ the even type means that $(n_1-m_1)(n_2-m_2)\geq 0$ (see [@Ald]). We have then $n=\min(n_1,n_2), \, m=\min(m_1,m_2)$ and the even roots $\pm x_i$ with $\mu=2|n_1-n_2|$, $\pm 2x_i$ with $\mu=1$ for $i=1,\dots, n,$ $\pm x_i \pm x_j$ with $\mu=2$ for $1 \le i < j \le n,$ $\pm y_i$ with $\mu=2|m_1-m_2|,$ $\pm 2y_i$ with $\mu=1$ for $i=1,\dots, m,$ $\pm y_i \pm y_j$ with $\mu=2$ for $1 \le i < j \le m$ and odd roots $\pm x_i \pm y_j$ with $\mu=-2$, $\pm x_i$ with $\mu=-2|m_1-m_2|$, $\pm y_j$ with $\mu=-2|n_1-n_2|$ for $1 \le i \le n,\, 1 \le j \le m.$ The form $Q$ is given by (\[Q\]) with $k=-1.$
For the even type $BDI/CII$ pairs $(\frak{osp} (n_1+n_2, 2m_1+2m_2), \frak{osp}(n_1,2m_1)\oplus\frak{osp}(n_2,2m_2))$ with $(n_1-m_1)(n_2-m_2)\geq 0$ we have again $n=\min(n_1,n_2), \,\, m=\min(m_1,m_2)$ and the even roots $\pm x_i$ with $\mu=|n_1-n_2|$ for $i=1,\dots, n,$ $\pm x_i \pm x_j$ with $\mu=1$ for $1 \le i < j \le n,$ $\pm y_i$ with $\mu=4|m_1-m_2|,$ $\pm 2y_i$ with $\mu=3$ for $i=1,\dots, m,$ $\pm y_i \pm y_j$ with $\mu=4$ for $1 \le i < j \le m$ and odd roots $\pm x_i \pm y_j$ with $\mu=-2$, $\pm x_i$ with $\mu=-2|m_1-m_2|$, $\pm y_j$ with $\mu=-2|n_1-n_2|$ for $1 \le i \le n,\, 1 \le j \le m.$ The form $Q$ is given by (\[Q\]) with $k=-1/2.$
Let $U(\frak g)$ be the universal enveloping algebra of $\frak g.$ Let $e_1, \dots, e_N$ be a basis in $\frak g.$ The dual space $U(\frak g)^*$ is known to be the algebra isomorphic to the algebra of formal series $\mathbb C[[X_1, \dots, X_N]],$ where $X_1, \dots, X_N \in \frak g^*$ is a dual basis (see Dixmier [@Dix], Chapter 2).
By a [*zonal function*]{} for the symmetric pair $X=(\mathfrak{g}, \mathfrak k)$ we mean a linear functional $f \in U(\frak g)^*$, which is two-sided $\mathfrak{k}$-invariant: $$f(xu)=f(ux)=0,\,\,x\in\frak k,\,\, u\in U(\frak{g}).$$ The space of such functions we denote $\mathcal Z(X) \subset U(\frak g)^*.$
Let $Y=\frak{k}U(\frak{g})+ U(\frak{g})\frak{k}$ be a subspace in $U(\frak{g}),$ on which the zonal functions vanish. Let also $U(\frak a)=S(\frak a)$ be the symmetric algebra of $\frak a.$
\[ress1\] $$U(\frak{g})=S(\frak{a})+Y.$$
For any $x\in\frak g$ set $$x^+=\frac12(x+\theta(x)), \quad x^-=\frac12(x-\theta(x)).$$ Let $\alpha \in \frak h^*$ be a root of $\mathfrak g$ and $T_{\alpha}: U(\frak a) \to U(\frak a)$ be the automorphism defined by $x \to x + \alpha(x), \, x \in \frak a.$ Define $R_{\alpha}^{\pm}: U(\frak a) \to U(\frak a)$ by $$R_{\alpha}^{+}=\frac12(T_{\alpha}+T_{-\alpha}), \quad R_{\alpha}^{-}=\frac12(T_{\alpha}-T_{-\alpha}).$$ Let $X_{\alpha}\in \frak g_{\alpha}, X_{-\alpha}\in\frak g_{-\alpha}$ be corresponding root vectors and define $$h_{\alpha}=[X_{\alpha},X_{-\alpha}] / (X_{\alpha}, X_{-\alpha}) \in \frak h.$$
We will need the following lemma, which can be checked directly.
\[comm\] For any $u \in U(\frak a)$ the following equalities hold true:
[*i)*]{} $X_{\alpha}^+u=R_{\alpha}^+uX_{\alpha}^+-R_{\alpha}^-uX_{\alpha}^-$
[*ii)*]{} $R_{\alpha}^- uX_{\alpha}X_{-\alpha}-T_{\alpha} u
[X_{\alpha}^+, X_{-\alpha}^-]\in Y$
[*iii)*]{} $[X_{\alpha}^+,X_{\alpha}^-]=\frac12h_{\alpha}^-(X_{\alpha}, X_{-\alpha}). $
To prove the proposition it is enough to show that for $q>0$ $$v=uX^-_{\alpha_1} \ldots X^-_{\alpha_q}\in Y$$ for any roots ${\alpha_1},\ldots, \alpha_q$, where $u\in S(\frak{a})$. We prove this by induction in $q$.
If $q=1$ and $w\in S(\frak{a})$, then by the first part of Lemma \[comm\] we have $R_{\alpha}^-wX_{\alpha}^-=R_{\alpha}^+wX_{\alpha}^+-X_{\alpha}^+w$, which clearly belongs to $Y.$ But any $u\in S(\frak a)$ can be represented in the form $u=R_{\alpha}^-w$ for some $ w\in S(\frak a)$, therefore $uX_{\alpha}^-\in Y$. Let now $q>1$. Then modulo $Y$ we have using Lemma \[comm\] $$R^-_{{\alpha_1} }uX^-_{\alpha_1}
\ldots X^-_{\alpha_q} = R^+_{\alpha_1} uX^+_{\alpha_1} X^-_{\alpha_2}\ldots X^-_{\alpha_q}-X^+_{\alpha_1}uX^-_{\alpha_2}\ldots X^-_{\alpha_q}$$ $${\equiv}R^+_{\alpha_1} uX^+_{\alpha_1} X^-_{\alpha_2}
\ldots X^-_{\alpha_q}
\equiv R^+_{\alpha_1} u\cdot [X^+_{\alpha_1},
X^-_{\alpha_2} \ldots X^-_{\alpha_q}]$$ $$= R^+_{\alpha_1}
u\cdot[X^+_{\alpha_1}, X^-_{\alpha_2}] X^-_{\alpha_3}\ldots X^-_{\alpha_q}
+R^+_{\alpha_1} u X^-_{\alpha_2}[X^+_{\alpha_1}, X^-_{\alpha_3}]\ldots
X^-_{\alpha_q}+\ldots\in Y$$ by inductive assumption.
Let $\alpha\in R$ be a root of $\frak g$ such that the restriction of $\alpha$ on $\frak{a}$ is not zero. Let also $f \in \mathcal Z(X)$ be a two sided $\frak k$-invariant functional on $U(\frak{g})$. By proposition \[ress1\] $f$ is uniquely determined by its restriction to $U(\frak a) = S(\frak a),$ and thus we can consider $\mathcal Z(X)$ as a subalgebra $S(\frak a)^*.$
Identify $S(\frak a)^*$ with the algebra of formal power series as follows (see [@Dix]). Let $e_1, \dots, e_N$ be a basis in $\frak a$ and $x_1, \dots, x_N \in \frak a^*$ be the dual basis. Then we can define for any $f \in S(\frak a)^*$ the formal power series $\hat f \in \mathbb C[[x_1,\dots,x_N]]$ by $$\hat f = \sum_{M \in \mathbb Z_+^N} f(e_M)x^M,$$ where $$e_M=\frac{1}{m_1!\dots m_N!}e_1^{m_1}\dots e_N^{m_N} \in U(\frak a), \,\, x^M=x_1^{m_1}\dots x_N^{m_N}.$$ It is easy to see that the operator of multiplication by, say, $e_1$ corresponds to the partial derivative $\frac{\partial}{\partial x_1}$ in this realisation: $$\hat f(u e_1)=\frac{\partial}{\partial x_1} \hat f(u).$$ Similarly, the shift operator $T_\lambda, \, \lambda \in \frak a^*$ corresponds to multiplication by $e^{\lambda}$: $$\hat f(T_\lambda u)=e^\lambda \hat f(u), \,\, u \in S(\frak a).$$
Let $S \subset S(\mathfrak a)^*$ be the multiplicative set generated by $e^{2\alpha}-1, \, \alpha \in R(X)$ and $S(\mathfrak a)^*_{loc}=S^{-1}S(\mathfrak a)^*$ be the corresponding localisation.
Let now $\frak h$ be a Cartan subalgebra of $\frak g$, $R$ be the root system of $\frak g$ and $X_\alpha$ are the corresponding root vectors with respect to $\frak h.$
Choose an orthogonal basis $h_i \in \mathfrak h, \, i=1,\dots, r$ and define the [*quadratic Casimir element*]{} $\mathcal C_2$ from the centre $Z(\frak g)$ of the universal enveloping algebra $U(\frak g)$ by $$\label{casimir}
\mathcal C_2= \sum_{i=1}^r \frac{h_i^2}{(h_i,h_i)} + \sum_{\alpha \in R} \frac{X_\alpha X_{-\alpha}}{(X_{-\alpha}, X_{\alpha})},$$ where the brackets denote the invariant bilinear form on $\frak g.$
It can be defined invariantly as an image of the element of $\frak g \otimes \frak g$ representing the invariant form itself and determines the corresponding [*Laplace-Beltrami operator*]{} $\frak L$ on $X$ acting on left $\frak k$-invariant functions $f \in \frak F(X)=U(\frak g)^{*\frak k}$ (which are algebraic analogues of the functions on the symmetric superspace $X=G/K$) by $$\frak L f(x)=f(x \mathcal C_2), \, x \in U(\frak g).$$
The restriction of the invariant bilinear form on $\frak g$ to $\frak a$ is a non-degenerate form, which we also denote by $( , ).$ Let $\Delta$ be the corresponding Laplace operator on $\frak a$ and $\partial_\alpha, \, \alpha \in \frak a^*$ be the differential operator on $\frak a$ defined by $$\label{defini}
\partial_\alpha e^{\lambda}=(\alpha, \lambda) e^{\lambda}.$$
Consider the following operator $\frak L_{rad}: S(\mathfrak a)^* \to S(\mathfrak a)^*$ defined by $$\label{radpart}
\frak L_{rad}=\Delta+\sum_{\alpha \in R_+(X)}\mu_\alpha \frac{e^{2\alpha}+1}{e^{2\alpha}-1} \,\, \partial_\alpha,$$ where the sum is taken over positive restricted roots considered with multiplicities $\mu_\alpha$. This operator is the [*radial part*]{} of the Laplace-Beltrami operator $\frak L$ in the following sense.
The following diagram is commutative $$\label{commutdun1}
\begin{array}{ccc}
\mathcal Z(X)&\stackrel{\frak L}{\longrightarrow}&\mathcal Z(X)\\ \downarrow
\lefteqn{i^*}& &\downarrow \lefteqn{i^*}\\
S(\mathfrak a)^*_{loc}&\stackrel{\frak L_{rad}}{\longrightarrow}&
S(\mathfrak a)^*_{loc}.\\
\end{array}$$
For any root $\alpha$ of $\frak g$ define the operators $D_{\alpha}, \, \partial_{\alpha}: \mathcal Z(X) \to S(\frak a)^*$ by $$D_{\alpha}(f)(u)=f\left(\frac{uX_{\alpha}X_{-\alpha}}{(X_{-\alpha}, X_{\alpha})}\right), \,\,
\partial_{\alpha}(f)(u)= f(u h_{\alpha}^-),\,\, u\in S(\frak{a}),$$ where we consider $\mathcal Z(X)$ as a subset of $S(\frak a)^*.$ One can check that the definition of the operator $\partial_\alpha$ agrees with (\[defini\]).
We claim that the operators $D_{\alpha}, \, \partial_{\alpha}$ in the formal power series realisation satisfy the relation $$\label{Dalpha}
(e^{\alpha}-e^{-\alpha})D_{\alpha}=(-1)^{p(\alpha)}e^{\alpha}\partial_{\alpha},$$ where $p(\alpha)$ is parity function: $p(\alpha)=0$ for even roots and $p(\alpha)=1$ for odd roots.
Indeed, since the restriction of $f \in \frak F(X)$ on $Y$ vanishes, from parts $ii)$ and $iii)$ of Lemma \[comm\] it follows that $$\hat f\left(\frac{R_{\alpha}^- uX_{\alpha}X_{-\alpha}}{(X_{-\alpha}, X_{\alpha})}\right)=\hat f\left(\frac{T_{\alpha} u
[X_{\alpha}^+, X_{-\alpha}^-]}{(X_{-\alpha}, X_{\alpha}) }\right) = \frac{(-1)^{p(\alpha)}}{2}e^\alpha \hat f(u
h_{\alpha}^-),$$ since $(X_{-\alpha},X_{\alpha})=(-1)^{p(\alpha)}(X_{\alpha},X_{-\alpha}).$ Since $R_{\alpha}^- =\frac 12(T_{\alpha}-T_{-\alpha})$ we have $\hat f(R_{\alpha}^- uX_{\alpha}X_{-\alpha})=\frac 12(e^{\alpha}-e^{-\alpha})\hat f(uX_{\alpha}X_{-\alpha})$ and thus the claim.
In the localisation $S(\mathfrak a)^*_{loc}$ we can write the operator $D_\alpha$ as $$\label{dalpha}
D_\alpha=\frac{(-1)^{p(\alpha)}e^\alpha}{e^\alpha-e^{-\alpha}}\partial_\alpha=\frac{(-1)^{p(\alpha)}e^{2\alpha}}{e^{2\alpha}-1}\partial_\alpha$$ and extend it to the whole $S(\mathfrak a)^*_{loc}$.
Summing over all $\alpha \in R$ and taking into account that the multiplicities $\mu_\alpha$ are defined with the sign $(-1)^{p(\alpha)}$ after the restriction to $\frak a$ we have the second term in formula (\[radpart\]). One can check that the first part of the Casimir operator (\[casimir\]) gives the Laplace operator $\Delta.$
For 4 classical series of symmetric pairs (\[4series\]) of even type the radial parts of Laplace-Beltrami operators are conjugated to the deformed CMS operators of classical type.
More precisely, for the classical series $X=(\frak{gl} (n, 2m), \frak{osp} (n,2m))$ the corresponding radial part (\[radpart\]) is conjugated to the deformed CMS operator related to generalised root system of type $A(n-1,m-1)$ from [@SV] with parameter $k=-1/2$ (as it was already pointed out in [@Ser]).
For three other classical series the corresponding radial part is conjugated to the following deformed CMS operator of type $BC(n,m)$ introduced in [@SV] $$\begin{aligned}
\label{bcnm} L& =& -\Delta_n
-k \Delta_m +\sum_{i<j}^{n}\left(\frac{2k(k+1)}{\sinh^2(x_{i}-x_{j})}+\frac{2k(k+1)}{\sinh^2(x_{i}+x_{j})}\right)\nonumber \\& &
+\sum_{i<j}^{m}\left(\frac{2(k^{-1}+1)}{\sinh^2(y_{i}-y_{j})}+\frac{2(k^{-1}+1)}{\sinh^2(y_{i}+y_{j})}\right)
\nonumber
\\& & +\sum_{i=1}^{n}\sum_{j=1}^{m}\left(\frac{2(k+1)}{\sin^2(x_{i}-y_{j})}+
\frac{2(k+1)}{\sinh^2(x_{i}+y_{j})}\right) +\sum_{i=1}^n
\frac{p(p+2q+1)}{\sinh^2x_{i}} \nonumber \\& & +\sum_{i=1}^n
\frac{4q(q+1)}{\sinh^22x_{i}} +\sum_{j=1}^m \frac{k
r(r+2s+1)}{\sinh^2y_{j}}+\sum_{j=1}^m \frac{4k
s(s+1)}{\sinh^22y_{j}},\end{aligned}$$ where the parameters $k,p,q,r,s$ must satisfy the relation $$\label{rel}
p=kr,\quad 2q+1=k(2s+1).$$ Indeed, using the description of the restricted roots given above and the definition of the deformed root system of $BC(n,m)$ type from [@SV], one can check that $n=\min(n_1,n_2),\, m=\min (m_1,m_2)$ and the parameters $$k=-1, \,\, p=|m_1-m_2|-|n_1-n_2|=-r, \,\, q=s=-1/2$$ for the symmetric pairs $X=(\frak{gl} (n_1+n_2, m_1+m_2), \frak{gl}(n_1,m_1)\oplus\frak{gl}(n_2,m_2)),$ $$k=-\frac{1}{2}, \,\, p=|m_1-m_2|-\frac{1}{2}|n_1-n_2|=-\frac{1}{2}r, \,\, q=0, \,\, s=-\frac{3}{2}$$ for the pairs $X=(\frak{osp} (n_1+n_2, 2m_1+2m_2), \frak{osp}(n_1,2m_1)\oplus\frak{osp}(n_2,2m_2)).$ For the symmetric pairs $X=(\frak{osp} (2l, 2m), \frak{gl}(l,m))$ we have two different cases depending on the parity of $l$: when $l=2n$ then $$k=-2, \,\, p=0=r, \,\, q=s=-\frac{1}{2},$$ and when $l=2n+1$ then $$k=-2, \,\, p=-2, \,\, r=1, \,\, q=s=-\frac{1}{2}.$$
In the rest of the paper we will restrict ourselves to the case of symmetric pairs $X=(\frak{gl} (n, 2m), \frak{osp} (n,2m)).$ In particular, we will show that the radial part homomorphism maps the centre of the universal enveloping algebra of $\frak{gl} (n, 2m)$ to the algebra of the deformed CMS integrals $\mathcal D_{n,m}.$
Symmetric pairs $X=(\frak{gl} (n, 2m), \frak{osp} (n,2m))$
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Recall that the Lie superalgebra $\frak g=\mathfrak{gl}(n,2m)$ is the sum $\frak g = \frak g_0 \oplus \frak g_1$, where $\mathfrak{g}_{0}=\mathfrak{gl}(n)\oplus
\mathfrak{gl}(2m)$ and $\mathfrak{g}_{1}=V_{1}\otimes V_{2}^*\oplus
V_{1}^*\otimes V_{2},$ where $V_{1}$ and $V_{2}$ are the identical representations of $\mathfrak{gl}(n)$ and $\mathfrak{gl}(2m))$ respectively. As a Cartan subalgebra $\frak h \subset \frak g_0$ we choose the diagonal matrices.
A bilinear form $(\, ,\, )$ on a $\mathbb Z_2$-graded vector space $V = V_0 \oplus V_1$ is said to be [*even*]{}, if $V_0$ and $V_1$ are orthogonal with respect to this form and it is called to be [*supersymmetric*]{} if $$(v,w)=(-1)^{p(v)p(w)}(w,v)$$ for all homogeneous elements $v$, $w$ in $V$. If $V$ is endowed with an even non-degenerate supersymmetric form $(\, ,\, )$, then the involution $\theta$ is defined by the relation $$\label{theta} (\theta(x)v,w)+(-1)^{p(x) p(v)}(v,xw)=0, \quad x \in \mathfrak{gl}(V).$$ When $\dim V_0=n, \, \dim V_1=2m$ and the form $(\, ,\, )$ coincides with Euclidean structure on $V_0$ and symplectic structure on $V_1$ we can define the [*orthosymplectic Lie superalagebra*]{} $\mathfrak{osp}(n,2m) \subset \mathfrak{gl}(n,2m)$ as $$\mathfrak{osp}(n,2m)=\{x\in \mathfrak{gl}(n,2m): \, \theta(x)=x\}.$$
Let $\varepsilon_{1},\dots,\varepsilon_{n+2m} \in \frak h^*$ be the weights of the identical representation of $\mathfrak{gl}(n,2m)$. It will be convenient also to introduce $\delta_{p}:=\varepsilon_{p+n},\: 1\le p\le 2m.$
The root system of $\mathfrak{g}$ is $R=R_0 \cup R_1$, where $$R_{0}=\{\varepsilon_{i}-\varepsilon_{j}, \delta_{p}-\delta_{q} :
\: i\ne j\,:\: 1\le i,j\le n\ , p\ne q,\, 1\le p,q\le 2m\},$$ $$R_{1}=\{\pm(\varepsilon_{i}- \delta_{p}), \quad 1\le i\le n, \,
1\le p\le 2m\}$$ are even and odd (isotropic) roots respectively. We will use the following distinguished system of simple roots $$B=\{\varepsilon_{1}-\varepsilon_{2},\dots,\varepsilon_{n-1}-\varepsilon_{n},
\varepsilon_{n}- \delta_{1},\delta_{1}-
\delta_{2},\dots,\delta_{2m-1}-\delta_{2m}\}.$$
The invariant bilinear form is determined by the relations $$(\varepsilon_{i},\varepsilon_{i})=1,\:
(\delta_{p},\delta_{p})=-1$$ with all other products to be zero. The integral weights are $$\label{pogl}
P_{0}=\{\lambda\in\mathfrak{h}^{*}\mid
\lambda=\sum_{i=1}^{n+2m}
\lambda_{i}\varepsilon_{i} =\sum_{i=1}^{n}
\lambda_{i}\varepsilon_{i}+\sum_{p=1}^{2m}\mu_{p}\delta_{p},\:
\lambda_{i}, \mu_j \in \mathbb
Z\} .$$ The Weyl group $W_0=
S_{n}\times S_{2m}$ acts on the weights by separately permuting $\varepsilon_{i},\; i=1,\dots, n$ and $\delta_{p},\; p=1,\dots, 2m.$
The involution $\theta$ is acting on $\frak h^*$ by mapping $\delta_{2j-1} \to -\delta_{2j}, \, \delta_{2j} \to -\delta_{2j-1},$ $ j=1,\dots, m$ and $\varepsilon_i \to -\varepsilon_i, \, i=1,\dots, n.$ The dual $\frak a^*$ of Cartan subspace $\frak a$ can be described as the $\theta$ anti-invariant subspace of $\frak h^*$ $$\frak a^*=\{\theta(x)-x, \,\,\, x \in \frak h^*\}$$ and is generated by $$\label{tildee}
\tilde\varepsilon_i=\varepsilon_i, \, i=1,\dots, n, \quad \tilde \delta_j=\frac12(\delta_{2j-1}+\delta_{2j}),\, \, j=1,\dots, m.$$ The induced bilinear form in this basis is diagonal with $$(\tilde \varepsilon_{i},\tilde \varepsilon_{i})=1,\,\,\,
(\tilde \delta_{p},\tilde \delta_{p})=-\frac12.$$
Let us introduce the following superanalogue of Gelfand invariants [@Molev] $$\label{Gel}
Z_s=\sum_{i_1, \dots, i_s}^{n+2m} (-1)^{p(i_2)+\dots +p(i_{s})}E_{i_1 i_2}E_{i_2 i_3}\dots E_{i_{s-1} i_{s}}E_{i_{s} i_{1}}, \,\, s \in \mathbb N,$$ where $E_{ij}, i,j =1, \dots, n+2m$ is the standard basis in $\frak{gl} (n,2m).$ One can define them also as $Z_s=\sum_{i=1}^{n+2m}E_{ii}^{(s)}$, where elements $E_{ij}^{(s)}$ are defined recursively by $$\label{recc}
E_{ij}^{(s)}=\sum_{l=1}^{n+2m}(-1)^{p(l)}E_{il}E_{lj}^{(s-1)}$$ with $
E_{ij}^{(1)}=E_{ij}.
$ One can check that these elements satisfy the following commutation relations $$[E_{ij},E_{st}^{(l)}]=\delta_{js}E^{(l=1)}_{it}-(-1)^{(p(i)+p(j))(p(s)+p(t))}\delta_{it}E^{(l)}_{sj},$$ which imply that the elements $Z_l$ are central.
Let as before $\mathcal Z(X) \subset U(\frak g)^*$ be the subspace of zonal (two-sided $\frak k$-invariant) functions and $S(\frak a)^*, \, S(\frak a)^*_{loc}$ be as in the previous section.
Let $i^*: \mathcal Z(X) \to S(\frak a)^*_{loc}$ be the restriction homomorphism induced by the embedding $i: \frak a \to \frak g.$
\[radial\] The restriction homomorphism $i^*$ is injective and there exists a unique homomorphism $\psi: Z(\frak g)\to \mathfrak D_{n,m}$ such that the following diagram is commutative $$\label{commutdun1}
\begin{array}{ccc}
{\mathcal Z(X)}&\stackrel{L_z}{\longrightarrow}&{\mathcal Z(X)}\\ \downarrow
\lefteqn{i^*}& &\downarrow \lefteqn{i^*}\\
S(\frak a)^*_{loc}&\stackrel{\psi(z)}{\longrightarrow}&
S(\frak a)^*_{loc}.\\
\end{array}$$ where $L_z$ is the multiplication operator by $z \in Z(\frak g).$ The image of Gelfand invariants (\[Gel\]) are the deformed CMS integrals (\[dif2\]): $$\psi( Z_s)= 2^s \mathcal L_s.$$
We will call $\psi$ the [*radial part homomorphism*]{}. For the Casimir element $\frak C$ the operator $\psi(\frak C)$ is the deformed CMS operator (\[Lrad\]).
Let us prove first that for any $z\in Z(\frak g)$ there exists not more than one element $\psi(z)\in \mathfrak D_{n,m}$ which makes the diagram commutative. It is enough to prove this only when $z=0$. Therefore we need to prove the following statement: if $D\in \frak D_{n,m}$ and $D(i^*(f))=0$ for any $f\in \mathcal Z(X)$ then $D=0$.
Let $\frak A_{n,m} \subset C(\frak a)=\Bbb C[x_1^{\pm1},\dots, x_n^{\pm1},y_1^{\pm1}\dots,y_m^{\pm1}]$ be the subalgebra consisting of $S_n\times S_m$-invariant Laurent polynomials $f\in C(\frak a),$ satisfying the quasi-invariance conditions (\[quasi\]).
Let us take $f=\phi_\lambda(x) \in \mathcal Z(X)$ from Proposition \[restweyl\], where $\lambda \in P^+(X)$ satisfies Kac condition (\[Kac2\]). By Proposition \[restweyl\] (which proof is independent from the results of this section) $i^*(f) \in \frak A_{n,m}$ and by Proposition \[sup\] $$D(i^*(f))=\varphi(D)(\lambda)e^{\lambda}+\dots,$$ where $\dots$ mean lower order terms in partial order (\[partial\]) and $\varphi$ is the Harish-Chandra homomorphism. If $D(i^*(f))=0$ then $\varphi(D)(\lambda)=0$ for all $\lambda$ which are admissible and $K(\lambda)$ is irreducible. By Proposition \[restweyl\] the set of such $\lambda$ is dense in Zarisski topology in $\frak a^*$. Therefore $\varphi(D)=0$ and since the Harish-Chandra homomorphism is injective we have $D=0$.
Now let us prove that for every $z \in Z(\frak g)$ element $\psi(z)$ indeed exists. It is enough to prove this only for the Gelfand generators $Z_s$. Actually we prove now that $\psi(Z_s)=2^s\mathcal L_s$, where $\mathcal L_s$ are the deformed CMS integrals defined by (\[dif2\]).
Let $Y= \frak k U(\frak g)+ U(\frak g) \frak k$ and $T_{\alpha}, R_{\alpha}: U(\frak a) \to U(\frak a)$ be defined as in the previous section for any root $\alpha \in \frak h^*$ of $\mathfrak g$. For $\alpha= \varepsilon_i-\varepsilon_j$ choose $X_{\alpha}=E_{ij}\in \frak g_{\alpha}$ and define $$X_{\alpha}^{(s)}=E_{ij}^{(s)},\,\,h_{\alpha}^{(s)}=[X_{\alpha},X_{-\alpha}^{(s)}], \,\, s \in \mathbb N,$$ where $E_{ij}^{(s)}$ are given by (\[recc\]).
\[comm1\] For any $u \in U(\frak a)$ the following equalities hold true:
[*i)*]{} $R_{\alpha}^- uX_{\alpha}X^{(s)}_{-\alpha}-T_{\alpha} u
[X_{\alpha}^+, (X^{(s)}_{-\alpha})^-]\in Y$
[*ii)*]{} $[X_{\alpha}^+,(X^{(s)}_{\alpha})^-]=\frac12(h_{\alpha}^{(s)})^-$, where $$h_{\alpha}^{(s)}=[X_{\alpha},X^{(s)}_{-\alpha}]=E_{ii}^{(s)}-(-1)^{p(i)+p(j)}E_{jj}^{(s)}.$$
Proof is by direct calculation.
Let us denote the image $\tilde {\mathcal Z}(X) = i^*(\mathcal Z(X)) \subset S(\frak a)^*$ and define the operators $D^{(s)}_{\alpha}, \, \partial^{(s)}_{\alpha}, \partial^{(s)}_{i}: \tilde {\mathcal Z}(X) \to S(\frak a)^*$ by the relations $$D^{(s)}_{\alpha}(i^*(f))(u)=f\left(uX_{\alpha}X^{(s)}_{-\alpha}\right), \,
\partial^{(s)}_{\alpha}(i^*(f))(u)= f\left(u h^{(s)}_{\alpha}\right),$$ $$\partial_i^{(s)}(i^*(f))(u)=f(E_{ii}^{(s)}u)$$ for any $f \in \mathcal Z(X),$ $u\in S(\frak{a}).$ Since $i^*$ is injective these operators are well-defined.
\[rad\] For any root $\alpha=\varepsilon_i-\varepsilon_j$ of $\frak g$ the operators $D^{(s)}_{\alpha}, \, \partial^{(s)}_{\alpha}$ in the formal power series realisation satisfy the relation $$(e^{\alpha}-e^{-\alpha})D^{(s)}_{\alpha}=e^{\alpha}\partial^{(s)}_{\alpha},\,\,\partial_{\alpha}^{(s)}=\partial_i^{(s)}-(-1)^{p(i)+p(j)}\partial_{j}^{(s)}.$$
Since the restriction of $f \in \mathcal Z(X)$ on $Y$ vanishes, from Lemma \[comm1\] it follows that $$\hat f(R_{\alpha}^- uX_{\alpha}X_{-\alpha}^{(s)})=\hat f(T_{\alpha} u
[X_{\alpha}^+, (X_{-\alpha}^{(s)})^-]) =\frac12 \hat f(T_{\alpha} u
(h_{\alpha}^{(s)})^-)$$ $$= \frac12e^\alpha \hat f(u (h_{\alpha}^{(s)})^-)= \frac12 e^\alpha \partial^{(s)}_{\alpha} \hat f(u).$$ Since $R_{\alpha}^- =\frac 12(T_{\alpha}-T_{-\alpha})$ we have $$\hat f(R_{\alpha}^- uX_{\alpha}X_{-\alpha}^{(s)})=\frac 12(e^{\alpha}-e^{-\alpha})\hat f(uX_{\alpha}X_{-\alpha}^{(s)})=\frac 12(e^{\alpha}-e^{-\alpha})D^{(s)}_{\alpha}\hat f(u),$$ which implies the claim.
Now from the recurrence relation (\[recc\]) and lemma we have $$\partial_i^{(s)}=(-1)^{p(i)}\partial_i^{(1)}\partial_i^{(s-1)}+\sum_{j\ne i}(-1)^{p(j)}\frac{e^{2\alpha}}{e^{2\alpha}-1}(\partial_i^{(s-1)}-(-1)^{p(i)+p(j)}\partial_{j}^{(s-1)}),$$ $i=1, \dots, n+2m,$ where $\alpha=\varepsilon_i-\varepsilon_j.$
Define the operators $\hat \partial_i^{(s)}=(-1)^{p(i)}\partial_i^{(s)}$, then the new operators satisfy the recurrence relation $$\hat \partial_i^{(s)}=\hat \partial_i^{(1)}\hat \partial_i^{(s-1)}-\sum_{j\ne i}(-1)^{1-p(j)}\frac{e^{2\alpha}}{e^{2\alpha}-1}(\hat \partial_i^{(s-1)}-\hat \partial_{j}^{(s-1)}), \,\, i=1,\dots, n+2m.$$
After the restriction to $\frak a$ we have $$\delta_{2j-1}=\delta_{2j}=\tilde\delta_j,\,\varepsilon_i=\tilde\varepsilon_i,\,\partial_{n+2j-1}=\partial_{n+2j}.$$ Let us introduce $$x_i=e^{2\tilde \varepsilon_i}, i=1,\dots, n, \quad y_j=e^{2\tilde \delta_j}, j=1,\dots, m.$$ From the above recurrence relations we have $\partial_{n+2j-1}^{(s)}=\partial_{n+2j}^{(s)}$. We also have $$\partial_i(x_i)=\partial_{i}(e^{2\tilde{\varepsilon_i}})=2\tilde\varepsilon_{i}(E_{ii})e^{2\tilde{\varepsilon_i}}=2x_i,\,i=1,\dots,n,$$ $$\partial_{n+2j}(y_{j})=\partial_{n+2j}(e^{2\tilde\delta_j})=2\tilde\delta_{j}(E_{n+2j})e^{2\tilde{\delta_j}}=y_j,\,j=1,\dots, m.$$
Therefore if we set $x_{n+j}=y_{j}, \, j=1,\dots,m$ we will have $$\hat \partial_i^{(s)}=\hat \partial_i^{(1)}\hat \partial_i^{(s-1)}-\sum_{j\ne i}^{n+m}(-1)^{1-p(j)}\frac{2^{p(j)}x_i}{x_{i}-x_j}(\hat \partial_i^{(s-1)}-\hat \partial_{j}^{(s-1)}),$$ and $\partial^{(1)}_i=2k^{p(i)}x_i\frac{\partial}{\partial x_i}, \, i=1,\dots n+m$ with $k=-1/2.$ So, if we define $\tilde\partial^{(s)}_i=2^{-s}\hat\partial^{(s)}_i$ we will have $$\tilde \partial_i^{(s)}=\tilde \partial_i^{(1)}\tilde \partial_i^{(s-1)}-\sum_{j\ne i}^{n+m}k^{1-p(j)}\frac{x_i}{x_i-x_j}(\tilde \partial_i^{(s-1)}-\tilde \partial_j^{(s-1)}),\, \tilde\partial^{(1)}_i=k^{p(i)}x_i\frac{\partial}{\partial x_i},$$ where $k=-1/2.$ This last recurrence relation and initial conditions coincide with (\[dif1\]) for this $k.$ So we have $$\psi(Z_s)=\sum_{i=1}^{n+2m}\partial_{i}^{(s)}=\sum_{i=1}^{n+2m}(-1)^{p(i)}\hat\partial_{i}^{(s)}=2^s\sum_{i=1}^{n+m}(-\frac12)^{-p(i)}\tilde\partial_{i}^{(s)},$$ which coincides with formula (\[dif2\]) with $k=-1/2$. Therefore we proved that $\psi(Z_s)=2^s \mathcal L_s \in\mathcal D_{n,m}$.
So it is only left to prove that $\psi$ is a homomorphism. Let $\psi(z_1)=D_1,\,\psi(z_2)=D_2$ then $D_1D_2$ makes the diagram commutative for $z_1z_2$. From the unicity it follows that $\psi(z_1z_2)=D_1D_2$. Theorem is proved.
Spherical modules and zonal spherical functions
===============================================
Let $X=(\frak g, \frak k)$ be symmetric pair and $\mathcal Z(X) \subset U(\frak g)^*$ be the set of the corresponding zonal functions.
We will call such function $f \in \mathcal Z(X)$ [*zonal spherical function*]{} if it is an eigenfunction of the action of the centre $Z(\frak g)$ on $U(\frak g)^*$ (cf. [@Hel]). Recall that the action of $U(\frak g)$ on $U(\frak g)^*$ is defined by the formula $$\label{leftaction}
(y l)(x)=(-1)^{p(y)p(l)}l(y^t x), \,\, x,\, y \in U(\frak g),$$ where $y^t$ is the principal anti-automorphism of $U(\frak g)$ uniquely defined by the condition that $y^t=-y, \, y \in \frak g.$
If $f$ is a generalised eigenfunction of $\mathcal Z(X)$, we will call it [*generalised zonal spherical function*]{} on $X.$ The appearance of such functions is a crucial difference of the super case from the classical one.
Let now $U$ be a finite dimensional $\frak g$-module and $U^{\frak k}$ be the space of all $\frak k$-invariant vectors $u \in U$ such that $x u =0$ for all $x \in \frak k$ and additionally that $g u=u$ for all $g \in O(n) \subset OSP(n, 2m).$ The last assumption is not essential but will be convenient. This allows us to exclude the possibility of tensor multiplication by one-dimensional representation given by Berezinian. Consider also similar space $U^{*\frak k}$ for the dual module $U^*.$
For any $u \in U^{\frak k}$ and $l \in U^{*\frak k}$ we can consider the corresponding zonal function $\phi_{u,l}(x) \in \mathcal Z(X) \subset U(\frak g)^*$ defined by $$\label{fi}
\phi_{u,l}(x):=l(x u), \quad x \in U(\frak g).$$ We denote the linear space of such functions for given $U$ as $\mathcal Z(U).$
A finite dimensional $\frak g$-module $U$ is called [*spherical*]{} if the space $U^{\frak k}$ is non-zero.
The following result should be true in general, but we will prove it only in the special case for the symmetric pair $X=(\frak{gl} (n, 2m), \frak{osp} (n,2m)).$
So from now on we assume that $\frak g = \frak{gl} (n, 2m), \, \frak k = \frak{osp} (n,2m).$
\[irrz\] Let $U$ be irreducible spherical module. Then $\dim \mathcal Z(U)=1$ and the corresponding function (\[fi\]) is zonal spherical.
Let $\theta$ be the automorphism of $\frak g= \frak{gl}(n, 2m)$ defined in the previous section. It is easy to check that the correspondence $$F(U)=U^{\theta},\,\, x\circ u=\theta(x)u$$ defines a functor on the category of finite dimensional $\frak g$-modules. From the definition (\[theta\]) it follows that for the standard representation we have $U^{\theta}=U^*$. It turns out that this is true for any irreducible module.
\[dual\] For any finite dimensional irreducible $\frak g$-module $U$ $$U^{\theta}=U^*.$$
In the case of Lie algebras the proof uses the fact that the longest element of the corresponding Weyl group $W$ maps the Borel subalgebra to the opposite one. In our case we do not have proper Weyl group, so we have to choose a special Borel subalgebra to prove this.
Let $\theta$ and $V$ be the same as in the previous section. Choose a basis in $V$: $$V_0=<f_1,\dots,f_n>,\,\, V_1=<f_{n+1},\dots,f_{n+2m}>$$ such that $$(f_i,f_{n-i+1})=1, i=1,\dots, n,\, (f_{n+j},f_{2m+n-j+1})=1=-(f_{2m+n-j+1},f_{n+j}),$$ where $j=1,\dots, m$ and other products are $0$.
Let $\frak h\subset\frak g$ be the Cartan subalgebra consisting of the matrices, which are diagonal in this basis, and the subalgebras $\frak n_{+},\, \frak n_{-} \subset\frak g$ be the set of upper triangular and low triangular matrices respectively in the basis $(f_{n+1}, \dots, f_{n+m}, f_1,\dots, f_n, f_{n+m+1}, \dots, f_{n+2m}).$
Consider the automorphism $\omega$ acting by conjugation by the matrix $C$ with $$C_{i \, n-i+1}=1, \, i=1,\dots, n, \,\,\,C_{n+j \, n+2m-j+1}=1, \, j=1,\dots, 2m$$ and all other entries being zero, corresponding to the product of two longest elements of groups $S_n$ and $S_{2m}.$
Then one can check that $\theta(\frak{h})=\frak h,\,\,\theta( \frak{n}_{+})=\frak{n}_{+}, \, \omega(\frak{h})=\frak h,\,\,\omega( \frak{n}_{+})=\frak{n}_{-},$ and $\theta(h)+\omega(h)=0$ for any $h\in\mathfrak h.$
Let $v\in U$ be highest weight vector with respect to Borel subalgebra $\frak h \oplus \frak n_+$ and let $\lambda\in\frak h^*$ be its weight. Then for module $U^{\theta}$ we also have $\frak{n}_{+}\circ v=\theta(\frak{n}_{+}) v=\frak{n}_{+} v=0$ and $\theta(\lambda)\in\frak h^*$ is the weight $v$ in $U^{\theta}$.
Let $u \in U$ be the highest vector with respect to Borel subalgebra $\frak h \oplus \frak n_-,$ then the corresponding weight is $\omega(\lambda)$ (cf. [@Bourbaki], Ch. 7, prop. 11).
Let $u^{*}\in U^*$ be the linear functional such that $u^{*}(u)=1$ and $u^{*}(u')=0$ for any other eigenvector $u'$ of $\frak h$ in $\in U$. Then it is easy to see that $n_{+}u^{*}=0$ and its weight with respect to Cartan subalgebra $\frak h$ is $-\omega(\lambda)$.
So we see that both irreducible modules $U^{\theta}$ and $U^*$ have the same highest weights $\theta(\lambda)=-\omega(\lambda)$ with respect to the same Borel subalgebra. Therefore they are isomorphic.
\[form\] Every finite dimensional irreducible $\frak g$-module $U$ has even non-degenerate bilinear form $( \, , \, )$ such that $$(\theta(x)u,v)+(-1)^{p(x)p(u)}(u,xv)=0,\,\, u,v\in U, \, x \in \frak g.$$ The $\frak g$-modules $U$ and $U^{*}$ are isomorphic as $\frak k$-modules.
Let us choose now the standard Borel subalgebra $\frak b$ consisting of upper triangular matrices in the basis $e_1,\dots, e_{n+2m}$ in $V$ such that $$V_0=<e_1,\dots,e_n>, \,\, V_1=<e_{n+1},\dots,e_{n+2m}>,$$ $(e_i,e_i)=1, \, i=1,\dots, n,\,\, \,\, (e_{n+2j-1},e_{n+2j})=-(e_{n+2j},e_{n+2j-1})=1,\, \, j=1,\dots, m$ (with all other products being zero).
For every $\frak g$ module $U$ we will denote by $U^{\frak k}$ the subspace of $\frak k$-invariant vectors.
\[dim\] For every irreducible finite dimensional $\frak g$-module $U$ we have $$\dim U^{\frak k}\le1.$$
[^2] Prove first that the map $$\frak k\times\frak b\longrightarrow \frak{g},\, (x,y)\rightarrow x+y$$ is surjective. The kernel of this map coincides with the set of the pairs $(x,-x),x\in\frak k\cap\frak b,$ which is the linear span of vectors $$E_{n+2j-1,n+2j-1}-E_{n+2j,n+2j}, \,\,\,\, E_{n+2j-1,n+2j}, \,\, j=1,\dots, m$$ and has the dimension $2m$. Since $$\dim \frak k+\dim\frak b-2m=\frac12n(n-1)+m(2m+1)+2nm$$ $$+\frac12(n+2m)(n+2m+1)-2m=(n+2m)^2=\dim \frak g,$$ which implies the claim.
Let $v \in U$ be the highest weight vector with respect to Borel subalgebra $\frak b$ and $w$ be vector invariant with respect to $\frak k$: $y w=0$ for all $y \in \frak k$, and such that $(w,v)=0.$ We claim that this implies that $w =0.$
To show this we prove by induction in $N$ that $
(w, x_1x_2\dots x_N v)=0$ for all $x_1, \dots, x_N \in \frak g.$ This is obviously true when $N=0$. Suppose that this is true for $N$. Take any $x\in\frak g$ and represent it in the form $x=y+z,\,y\in \frak{k},\, z\in \frak b,$ so that $$(w, xx_1x_2\dots x_N v)=(w, yx_1x_2\dots x_N v)+(w, zx_1x_2\dots x_N v).$$ By definition we have $(w, yx_1x_2\dots x_N v)=\pm(yw, x_1x_2\dots x_N)=0.$ We have $$(w, zx_1x_2\dots x_N v)=(w, [z,x_1x_2\dots x_N] v)\pm (w, x_1x_2\dots x_N z v)=0.$$ By inductive assumption $$(w, [z,x_1x_2\dots x_N] v)=(w, [z,x_1]x_2\dots x_N v)\dots\pm(w, x_1x_2\dots [z,x_N] v)=0$$ and since $zv = cv$ for any $z \in \frak b$ we also have $(w, x_1x_2\dots x_N z v)=0.$ Thus we have $(w, u)=0$ for any $u\in U$, so $\omega =0$ since the form is non-degenerate.
Let now $w_1, w_2$ be two $\frak k$-invariant non-zero vectors. Then we have $$(w_1,v)=c_1\ne0,\, (w_2,v)=c_2\ne0,$$ so $(c_1w_1-c_2w_2,v)=0$ and thus $c_1w_1=c_2w_2$. Thus the dimension of the space $U^{\frak k}$ of $\frak k$-invariant vectors in $U$ can not be greater than 1.
Now let us deduce the Theorem. Let $u \in U^{\frak k}$ be a non-zero vector, then by Corollary \[form\] there exists a non-zero $l \in U^{*\frak k}.$ By Proposition \[dim\] they are unique up to a multiple, therefore $\dim \mathcal Z(U)\leq1.$ So we only need to show that $\mathcal Z(U) \neq 0.$
From the proof of Proposition \[dim\] it follows that $l(v)\neq 0$ for a highest vector $v \in U$. Since module $U$ is irreducible $v = xu$ for some $x \in U(\frak g)$, so $\phi_{u,l} \neq 0$ and thus $\mathcal Z(U) \neq 0.$
This proves that if $U^{\frak k} \neq 0$ then $U$ is spherical. The converse is trivial.
Since the space $\mathcal Z(U)$ is one-dimensional and centre $Z(\frak g)$ preserves it, it follows that the function $\phi_{u,l}$ is zonal spherical.
Now we would like to describe the conditions on the highest weights for irreducible modules to be spherical.
Let $\varepsilon_i$ be the basis in $\frak h^*$ dual to the basis $E_{ii},\,i=1,\dots,n+2m$. Let us call the weight $$\lambda=\sum_{i=1}^{n+2m}\lambda_i\varepsilon_i \in \frak h^*$$ [*admissible*]{} for symmetric pair $X$ if $$\lambda_i\in2\Bbb Z,\,i=1,\dots,n,\,\,\,\,\lambda_{n+2j-1}=\lambda_{n+2j}\in\Bbb Z,\,j=1,\dots,m.$$ Denote the set of all such weights as $P(X)$. Let also $P^+(X) \subset P(X)$ be the subset of highest admissible weights: $$\lambda_1\ge\dots\ge\lambda_{n},\,\,\lambda_{n+1}\ge\dots\ge\lambda_{n+2m}.$$
Let $U=L(\lambda)$ be a finite-dimensional irreducible module with highest weight $\lambda$ and $U^*=L(\mu)$. Proposition \[restweyl\] implies that both $\lambda$ and $\mu$ are admissible. We conjecture that this condition is also sufficient.
[**Conjecture.**]{} [*If the highest weights $\lambda$ and $\mu$ of both $U$ and $U^*$ are admissible then $U$ is spherical.*]{}
We will prove this only under additional assumption of typicality, which is a natural generalisation of Kac’s typicality conditions for Kac modules (see Corollary \[corcon\] below).
Let us remind the notion of Kac module [@Kac2]. Let $\frak g_0$ be the even part of the Lie superalgebra $\frak g$ and $$\frak p=\frak p_0\oplus \frak p_1,$$ where $\frak p_0=\frak g_0$ and $\frak p_1 \subset \frak g_1$ be the linear span of positive odd root subspaces.
Let $V^{(0)}$ be irreducible finite dimensional $\frak g_0$-module. Define the structure of $\frak p$-module on it by setting $\frak p_1 V^{(0)}=0.$ The Kac module is defined as an induced module by $$K(V^{(0)})= U(\frak g)\otimes_{U(\frak p)}V^{(0)}$$ If $V^{(0)}=L^{(0)}(\lambda)$ is the highest weight $\frak g_0$-module with weight $\lambda,$ then the corresponding Kac module is denoted by $K(\lambda)$.
The following theorem describes the main properties of the Kac modules.
Recall that $\frak k = \frak{osp}(n, 2m) \subset \frak g=\frak{gl}(n, 2m).$
\[Kac\]
$1)$ We have the isomorphism of $\frak k$-modules $$K(\lambda)=U(\frak k)\otimes_{U(\frak k_0)} L^{(0)}(\lambda).$$
$2)$ $K(\lambda)$ is projective as $\frak k$ module.
$3)$ As $\frak g$ modules $$K(\lambda)^*=K(2\rho_1-w_0(\lambda)),$$ where $2\rho_1$ is the sum of odd positive roots and $w_0$ is the longest element of the Weyl group $S_n\times S_{2m}$.
$4)$ $K(\lambda)$ is spherical if and only if $\lambda\in P^+(X)$, in which case $$\dim K(\lambda)^{\frak k}=1.$$
We start with the following important fact, which can be easily checked: $$(1+\theta)\frak p_{1}=\frak{k}_{1},$$ where $\frak k_1$ is the odd part of $\frak k.$ Let $$\varphi : U(\frak k)\otimes_{U(\frak k_0)} L^{(0)}(\lambda) \rightarrow K(\lambda)$$ be the homomorphism of $\frak k$-modules induced by natural inclusion $L^{(0)}(\lambda) \subset K(\lambda)$. Let $\frak p_{-1}$ be the the linear span of negative odd root subspaces and consider the filtration on $K(\lambda)$ such that $$L^{(0)}(\lambda)=K_{0}\subset K_{1}\subset\dots\subset K_{N-1}\subset K_{N}=K(L^{(0)}(\lambda)),$$ where $N=\dim \frak p_{-1}$ and $K_{r}$ is the linear span of $x_{1}\dots x_{s}v, \, v \in L^{(0)}(\lambda)$, where $x_{1},\dots, x_{s}\in \frak p_{-1},\,\,s\le r$.
Now let us prove by induction in $r$ that $K_{r}\subset Im \,\varphi$. Case when $r=0$ is obvious. Let $K_{r}\subset Im \,
\varphi $, then $(x+\theta (x))K_{r}\subset
Im \, \varphi $ for all $x \in \frak p_{-1}$ since $x+\theta(x) \in \frak k.$ Since $\theta (\frak p_{-1})= \frak p_1$ we have $\theta (x)K_{r}\subset K_{r-1}$.
Therefore $xK_{r}\subset Im \, \varphi $ and thus $K(\lambda)=K_N\subset Im \, \varphi $. This means that the homomorphism $\varphi$ is surjective. Since both modules have the same dimension $\varphi$ is an isomorphism. This proves the first part of the theorem.
Part $2)$ now follows since every induced module from $\frak p_0$ to $\frak p$ is projective, see [@Zou]. Part $3)$ can be found in Brundan [@Bru1], see formula (7.7).
So we only need to prove part $4)$. From the first part we have the isomorphism of the vector spaces $$(K(\lambda)^*)^{\frak k}=(L^{(0)}(\lambda)^*)^{\frak k_0}.$$ Thus we reduced the problem to the known case of Lie algebras. In particular, according to [@GW] $$\dim (L^{(0)}(\lambda)^*)^{\frak k_0} = 1$$ if $\lambda\in P^{+}(X)$ and $0$ otherwise. From part $3)$ it follows that the same is true for the module $K(\lambda).$
We need some formula for the zonal spherical functions related to irreducible modules.
Let $W(X)=S_n\times S_m$ be the [*restricted Weyl group*]{}. It acts naturally on $\frak a^*$ permuting $\tilde \varepsilon_i, \tilde \delta_j$ given by (\[tildee\]). Let $$C(\frak a)=\Bbb C[x_1^{\pm1},\dots, x_n^{\pm1},y_1^{\pm1}\dots,y_m^{\pm1}]$$ be the subalgebra of $U(\frak a)^*$, where $x_i=e^{2\tilde \varepsilon_i}, \, y_j=e^{2\tilde \delta_j}.$ The subalgebra $\frak A_{n,m} \subset C(\frak a)$ consists of $S_n\times S_m$-invariant Laurent polynomials $f\in C(\frak a)$ satisfying the quasi-invariance conditions (\[quasi\]).
\[restweyl\] Let $L(\lambda)$ be an irreducible spherical finite-dimensional module and $\phi_\lambda(x)$ be the corresponding zonal spherical function (\[fi\]). Then its restriction to $U(\frak a)$ belongs to $\frak A_{n,m} $ and has a form $$\label{formx}
i^*(\phi_\lambda(x))=\sum_{\mu\preceq\lambda}c_{\lambda,\mu}e^{\mu}(x), \,\, x \in U(\frak a),$$ where $\lambda \in P^+(X),\,\mu\in P(X)$ and $c_{\lambda,\lambda}=1.$
The restrictions to $\frak a$ of the weights $\lambda,$ for which $L(\lambda)$ is an irreducible spherical finite-dimensional module, are dense in Zarisski topology in $\frak a^*.$
The proof of the form (\[formx\]) and of the invariance under $W(X)$ can be reduced to the case of Lie algebras by considering $L(\lambda)$ as a module over $\frak g_0$ (see e.g. Goodman-Wallach [@GW]). Note that Goodman and Wallach consider the representations of Lie groups (rather than Lie algebras), so in order to use their result we need the invariance of vector under $O(n),$ which is assumed in the definition of spherical modules.
To prove the quasi-invariance conditions we use the fact that $i^*(\phi_\lambda)$ is an eigenfunction of the Laplace-Beltrami operator on $X$. Since the radial part of this operator has the form (\[radpart\]) we see that the derivative $\partial_{\alpha}\phi_\lambda$ must vanish when $e^{2\alpha}-1=0$ for all roots $\alpha$, which implies the conditions (\[quasi\]) in the variables $x_i, y_j.$
According to [@Kac] Kac module $K(\lambda)$ is irreducible if and only if [*Kac’s typicality conditions*]{} $$\label{Kac2}
\prod_{i=1}^n \prod_{j=1}^{2m}(\lambda + \rho, \varepsilon_i -\delta_j) \ne 0$$ are satisfied, where $\rho$ is given by $$\label{rho2}
\rho=\frac12\sum_{i=1}^n(n-2m-2i+1)\varepsilon_i+\frac12\sum_{j=1}^{2m}(2m+n-2j+1)\varepsilon_{n+j}.$$ Let $\lambda\in P^+(X)$ be an admissible weight, satisfying this condition. Then the corresponding Kac module $K(\lambda)$ is irreducible and spherical by Theorem \[Kac\]. Since such $\lambda$ are dense in $\frak a^*$ the proposition follows.
Spherical typicality
====================
Let $\frak g= \frak {gl}(n, 2m)$ and $V$ be finite-dimensional irreducible $\frak g$-module. By Schur lemma any element from the centre $Z(\frak g)$ of the universal enveloping algebra $U(\frak g)$ acts as a scalar in $V$ and therefore we have the homomorphism $$\label{central}
\chi_{V} : Z(\frak g)\longrightarrow \Bbb C,$$ which is called the [*central character*]{} of $V$.
The following notion is an analogue of Kac’s typicality conditions [@Kac].
Irreducible finite-dimensional spherical module $L(\lambda)$ is called ***spherically typical*** if it is uniquely defined by its central character among the spherical irreducible $\frak g$-modules.
In other words, if $L(\mu)$ is another irreducible finite-dimensional spherical $\frak g$-module and $\chi_{\lambda}=\chi_{\mu},$ then $\lambda=\mu$.
In order to formulate the conditions on the highest admissible weight to be typical we need several results from the representation theory of Lie superalgebras [@Kac; @PS; @Serge1].
Let $
\varphi: Z(g)\longrightarrow S(\frak h)
$ be the Harish-Chandra homomorphism [@Kac]. Let $\varepsilon_i \in \frak h^*, \, i=1,\dots, n+2m$ be the same as in the previous section and $\rho$ is given by (\[rho2\]).
\[HCh\] [@Serge1] The image of the Harish-Chandra homomorphism $\varphi$ is isomorphic to the algebra of polynomials from $P(\frak h^*)=S(\frak h)$, which have the following properties:
$1)$ $
f(w(\lambda+\rho))=f(\lambda+\rho),\, w\in S_n\times S_{2m}, \lambda \in \frak h^*,
$
$2)$ for all odd roots $\alpha \in \frak h^*$ and $\lambda \in \frak h^*$ such that $(\lambda+\rho,\alpha)=0$ $$f(\lambda+\alpha)=f(\lambda).$$
It is generated by the following polynomials: $$\label{gener}
P_{r}(\lambda)=\sum_{i=1}^n(\lambda+\rho,\varepsilon_i)^r-\sum_{j=1}^{2m}(\lambda+\rho,\varepsilon_{n+j})^r,\,r \in \mathbb N.$$
We are going to apply this theorem in the case of admissible highest weights $\lambda \in P^+(X)$. For any such weight define two sets $$A=\{a_1,\dots,a_n\},\quad B=\{b_1,\dots,b_m\},$$ where $$a_i=(\lambda+\rho,\varepsilon_i)-\frac12(n-2m-1)=\lambda_i+1-i,\, i=1,\dots,n,$$ $$b_j=(\lambda+\rho,\varepsilon_{n+2j})-\frac12(n-2m-1)=-\lambda_{n+2j}-n+2j,\, j=1,\dots, m.$$ It is easy to check that this establishes a bijection between the set $P^+(X)$ of the highest admissible weights and the set $\mathcal T$ of pairs $(A,B)$, where $A, B \subset \mathbb Z$ are finite subsets of $n$ and $m$ elements respectively, which satisfy the following conditions:
$i)$ If $a,\tilde a\in A$ and there is no any other element in $A$ between them (in the natural order in $\mathbb Z$), then $a-\tilde a$ is odd integer.
$ii)$ If we denote by $B-1$ the shift of the set $B$ by $-1$ then $B\cap(B-1)=\emptyset$.
The dominance partial order on the set of admissible highest weights $P^+(X)$ induces some partial order on $\mathcal T,$ which we will be denote by the same symbol $\prec$.
The Harish-Chandra homomorphism defines an equivalence relation on the set $P^+(X):$ $\lambda \sim \mu$ if and only if $\chi_{\lambda}=\chi_{\mu}$. The following lemma describes the corresponding equivalence relation on the set $\mathcal T$.
\[equival\] Let $\lambda,\tilde\lambda$ be admissible highest weights and $(A,B),\,(\tilde A,\tilde B)$ be the corresponding elements in $\mathcal T$. Then $\chi_{\lambda}=\chi_{\tilde\lambda}$ if and only if the following conditions are satisfied: $$A\setminus(B\cup B-1)=\tilde A\setminus(\tilde B\cup \tilde B-1),$$ $$(B\cup B-1)\setminus A=(\tilde B\cup \tilde B-1)\setminus\tilde A.$$
Let $C=B\cup(B-1), \, \tilde C=\tilde B\cup(\tilde B-1),$ then from (\[gener\]) it follows that $$\sum_{i=1}^n a_i^r - \sum_{i=1}^{2m} c_i^r = \sum_{i=1}^n \tilde a_i^r - \sum_{i=1}^{2m} \tilde c_i^r, \, \, r \in \mathbb N.$$ Therefore $$\sum_{i=1}^n a_i^r + \sum_{i=1}^{2m} \tilde c_i^r= \sum_{i=1}^n \tilde a_i^r +\sum_{i=1}^{2m} c_i^r , \, \, r \in \mathbb N.$$ Hence the sequences $(a_1,\dots, a_n, \tilde c_1, \dots, \tilde c_{2m})$ and $(\tilde a_1,\dots, \tilde a_n, c_1, \dots, c_{2m})$ coincide up to a permutation. Hence $A\setminus C=\tilde A\setminus \tilde C$ and $C\setminus A=\tilde C\setminus \tilde A.$
\[atip\]
If $A\cap B\ne\emptyset$, then there exists $(\tilde A,\tilde B) \in \mathcal T$, which is equivalent to $(A,B),$ such that $(\tilde A,\tilde B)\prec(A,B)$. In particular, every finite equivalence class contains a representative $(A, B)$ such that $A\cap B=\emptyset.$
If $A\cap B=\emptyset$, then the equivalence class of $(A,B)$ in $\mathcal T$ is finite and contains $2^s$ elements, where $s=|A\cap(B-1)|$ and $(A,B)\preceq (\tilde A,\tilde B)$ for any $(\tilde A,\tilde B)\in\mathcal T$.
Let $A\cap B\ne\emptyset$. Represent $B\cup B-1$ as the disjoint union of the segments of integers $$B\cup (B-1)=\cup_{i}\Delta_i,$$ where a segment is a finite set of integers $\Delta$ such that $a,b\in \Delta$ and $a\le c\le b$ imply $c\in\Delta$. Since $B$ and $B-1$ do not intersect then any $\Delta_i$ consists of even number of integers and if we set $C_i = B\cap \Delta_i$ then $C_i\cap (C_i-1)=\emptyset$ and $\Delta_i=C_i\cup (C_i-1)$.
Consider two cases. In the first case suppose that there exist $i$ and $a\in A \cap C_i$ such that if $a'\in \Delta_i$ and $a'<a$ then $a'\notin A$. Let $\Delta_i=[c,d]$ and set $$\tilde A=(A\setminus\{a\})\cup\{c-1\},\quad \tilde B=B\setminus \{a\}\cup\{c-1\}.$$ We need to prove that $(\tilde A,\tilde B)$ is equivalent to $(A,B)$ and that $(\tilde A,\tilde B)\in\mathcal T$.
Indeed, it is clear that $c-1\notin B\cup (B-1)$ and since $a-(c-1)$ is an even number we have $c-1\notin A$. Therefore $(\tilde A,\tilde B)$ is equivalent to $(A,B)$.
It is easy to see that if $a'\in A$ is a neighbour of $a$ then the difference $a'-(c-1)$ is an odd integer. Further we have $$\tilde B\cup (\tilde B-1)=\cup_{j\ne i}\Delta_j\cup[c-1,a-1]\cup[a+1,d],$$ which proves that $(\tilde A,\tilde B)\in\mathcal T$. Since $c-1<a$ we have $(\tilde A,\tilde B)\prec( A, B)$ (see [@Brund]).
Now suppose that the conditions of the first case are not fulfilled. From the assumption $A\cap B \ne\emptyset$ we see that there exist $i$ and $a\in C_i\cap A$ such that there exists $a'\in \Delta_i \cap A$ and $a'<a$. Since $[a',a]\subset[c,d]$ we can assume that $a',a$ are neighbours. Since the difference $a-a'$ must be odd, we have $a'\in (C_i-1)$.
Let $a_{min}$ be the minimal element from $A$. Choose a set $E=\{ e-1, e\}$ such that $a_{min}-e$ is positive odd, $E\cap (B\cup(B-1))=\emptyset$ and define $$\tilde A=(A\setminus\{a,a'\})\cup E,\,\,\tilde B=(B\setminus\{a\})\cup \{e\}.$$ It is easy to verify that the $(\tilde A,\tilde B) \sim (A,B)$, $(\tilde A,\tilde B)\in \mathcal T$ and $(\tilde A,\tilde B)\prec(A,B)$. Note that in this case we have always an infinite equivalence class.
Now let us prove the second part.
Assume that $A\cap B=\emptyset$. In that case every $\Delta_i$ contains not more than one element of $A$, which must belong to $(C_i-1)$ since $A\cap C_i=\emptyset.$ Indeed, if $\Delta_i \cap A$ contains two elements $a$ and $a'$ then $[a, a'] \subset \Delta_i$, so we can assume without loss of generality that $a$ and $a'$ are neighbours. Since both of them belong to $(C_i-1)$ the difference $a'-a$ is even, which is a contradiction.
Let $a \in A \cap \Delta_i$ and $\Delta_i=[c,d].$ Let $(\tilde A, \tilde B) \sim (A,B), \, (\tilde A, \tilde B) \in \mathcal T.$ Then there are two possibilities: $(\tilde B \cup (\tilde B-1)$ contains $\Delta_i$ or not. In the last case the only possibility for $\Delta_i$ is to be replaced by the union $[c,a-1] \cup [a+1, d+1]$, which leads to $2^s$ possibilities.
We now need the following Serganova’s lemma [@Serga; @PS], which connects two highest weight vectors in an irreducible module with respect to an odd reflection.
Let $\frak g=\frak{gl}(n,l)$ and $\frak b,\,\frak b'$ be two Borel subalgebras such that $$\frak b'=(\frak b\setminus\{\gamma\})\cup\{-\gamma\}$$ where $\gamma=\varepsilon_p-\varepsilon_{n+q}$ is a simple odd root, and $\rho$ and $\rho'=\rho-\gamma$ be the corresponding Weyl vectors.
Let $V$ be a simple finite-dimensional $\frak g$-module and $v$ and $v'$ be the highest weight vectors with respect to $\frak b$ and $\frak b'$ respectively. Let $\lambda$ and $\lambda'$ be the corresponding weights.
Define the sequences $A=\{a_1,\dots,a_n\}, B=\{b_1,\dots,b_{l}\}, A'=\{a'_1,\dots,a'_n\}, $ $B'=\{b'_1,\dots,b'_{l}\},$ where $$\label{sets}
a_i=(\lambda+\rho,\varepsilon_i),\,i=1,\dots,n,\,\,\, b_j=(\lambda+\rho,\varepsilon_{n+j}),\,j=1,\dots, l,$$ $$a'_i=(\lambda'+\rho',\varepsilon_i),\,i=1,\dots,n,\,\,\, b'_j=(\lambda'+\rho',\varepsilon_{n+j}),\,j=1,\dots, l.$$
[@Serga; @PS] If $a_p\ne b_q $ then $$A'=A,\,\, B'=B.$$ If $a_p=b_q$ then $$a'_p=a_p+1, \,\, b'_q=b_q +1,
$$ and $ \, a'_i=a_i, i\ne p, \, \, b'_j=b_j, j \ne q.$
Define the following operation on the pairs of sequences $F: (A, B) \rightarrow (\tilde B, \tilde A)$ recursively. If $A$ and $B$ consist of one element $a$ and $b$ respectively then $$\label{rule}
F(a,b)=
\begin{cases}
(b,a), \, b \ne a\\
(b+1, a+1), b=a.
\end{cases}$$ If $A=\{a_1,\dots,a_n\}$, $B=\{b_1,\dots,b_{l}\}$, then we repeat this procedure for all elements of $A$ starting with $a_n$ and moving them to the right of $B$ using the rule (\[rule\]).
If $A=(3,2,5), B=(3,1,2,4)$ then $$F(A,B)=(\tilde B, \tilde A)=((4,1,3,5), (5, 3, 5)).$$
Let $\frak b$ be the standard Borel subalgebra of $\frak {gl}(n,l)$ and $ \tilde {\frak b}$ be its “odd opposite” with the same even part and odd part replaced by the linear span of negative odd root vectors.
\[lowhighest\] [@Serga; @PS] Let $(A,B)$ be the sequences (\[sets\]) corresponding to the highest weight of $\frak g$-module $V$ with respect to standard Borel subalgebra $\frak b$, then the highest weight of $V$ with respect to the odd opposite Borel subalgebra $\tilde {\frak b}$ is $(\tilde A,\,\tilde B)$, where $(\tilde B,\,\tilde A)=F(A,B).$
There is a natural bijection $
\sharp: P^+(X)\longrightarrow X_{n,m}^{+}
$ mapping the admissible weight $\lambda=(2\lambda_1,\dots, 2\lambda_n,\mu_1,\mu_1,\dots,\mu_m,\mu_m)$ to $$\label{sharp}
\lambda^\sharp=(\lambda_1,\dots,\lambda_n,\mu_1,\dots,\mu_m).$$
\[typical\] A finite-dimensional irreducible $\frak g$-module $L(\lambda)$ is spherically typical if and only if $\lambda \in P^+(X)$ and $$\label{star}
\prod_{1\le i\le n,\, 1\le j\le m}(\lambda+\rho,\varepsilon_i-\delta_{2j})\ne0,$$ where $\rho$ is given by (\[rho2\]). In terms of the restricted roots $\alpha \in R(X) \subset \frak a^*$ this condition can be written in the invariant form $$\label{starinv}
\prod_{\alpha \in R_+(X)}[(\lambda^\sharp+\rho(k),\alpha)-\frac12(\alpha, \alpha)]\ne 0,$$ where $\rho(k)$ is given by (\[rhok\]) with $k=-\frac12$ and the form on $\frak a^*$ is induced from the restricted form on $\frak a.$
Let us prove first that the conditions are necessary.
Let $L(\lambda)$ be spherically typical irreducible finite dimensional $\frak g$-module. Then the dual module $L(\lambda)^*$ is also spherical by corollary \[form\]. Since $L(\lambda)$ is the homomorphic image of $K(\lambda)$ the dual Kac module $K(\lambda)^*\supset L(\lambda)^*$ is spherical. But by part $3$ of Theorem \[Kac\] we have $K(\lambda)^*=K(2\rho_1-w_0(\lambda))$. Therefore by part $4$ of the same theorem we have $2\rho_1-w_0(\lambda)\in P^+(X),$ which implies that $\lambda\in P^+(X)$.
Now let us prove that $\lambda$ satisfies the condition (\[star\]). Suppose that this is not the case. This means that $A\cap B\ne\emptyset$ for the corresponding sets in $\mathcal T$. By Proposition \[atip\] there exists $(\tilde A,\tilde B)\in \mathcal T$ such that $(\tilde A,\tilde B)\prec( A, B)$ and $(\tilde A,\tilde B)\sim( A, B)$. Therefore there exists $\mu\in P^+(X)$ such that $\mu\prec\lambda$ and $\chi_{\mu}=\chi_{\lambda}$. Then by Theorem \[Kac\], part $4$ module $K(\mu)$ contains an invariant vector $\omega$. Consider the Jordan–Hölder series of $K(\mu):$ $$K(\mu)=K_0\supset K_1\supset\dots\supset K_N=0.$$ There exists $0\le i\le N-1$ such that $\omega\in K_i,\,\omega\notin K_{i+1}$. Therefore sub-quotient $L(\nu)=K_i/K_{i+1}$ is an irreducible spherical module and $\chi_{\nu}=\chi_{\mu}=\chi_{\lambda}$. Since $L(\lambda)$ is spherically typical then $\lambda=\nu\preceq\mu$. Contradiction means that (\[star\]) is satisfied.
To prove that the conditions are sufficient assume that $\lambda\in P^+(X)$ and (\[star\]) is fulfilled. Then $A\cap B=\emptyset$. We claim that $L(\lambda)$ contains a $\frak k$-invariant vector. Indeed, since $\lambda\in P^+(X)$ Kac module $K(\lambda)$ contains a $\frak k$-invariant vector $\omega$. Let $L(\mu)$ be an irreducible spherical sub-quotient of $K(\lambda)$, such that the image of $\omega$ in $L(\mu)$ is non-zero. As before, $\mu\in P^+(X)$, $\mu\preceq\lambda$ and $\chi_{\lambda}=\chi_{\mu}$. But since $A\cap B=\emptyset$ then by part 2 of Proposition \[atip\] $\lambda\preceq\mu$. Therefore $\lambda =\mu$ and $L(\lambda)$ is spherical.
So we only need to prove that $L(\lambda)$ is spherically typical module.
Let $L(\nu)$ be an irreducible spherical module such that $\chi_{\lambda}=\chi_{\nu}$. As we have already shown $\nu\in P^+(X)$. Suppose that $\nu\ne\lambda$. By Corollary \[form\] the dual module $L(\nu)^*$ also spherical. It is known that $L(\nu)^*=L(-w_0(\mu)),$ where $w_0$ is the longest element of the Weyl group $S_n\times S_{2m}$ and $\mu$ is the weight of the highest weight vector in $L(\nu)$ with respect to the odd opposite Borel subalgebra $\tilde {\frak b}.$ Therefore $\mu\in P^+(X)$.
Let $(\hat A, \hat B),\,(\tilde A,\tilde B)\in \mathcal T$ be the pairs corresponding to $\nu$ and $\mu$ respectively. By Proposition \[lowhighest\] we have $
F(\hat A, \hat B)=(\tilde B,\tilde A).
$ Let us represent $ \hat B\cup( \hat B-1)$ as the disjoint union of integer segments $$\hat B\cup( \hat B-1)=\bigcup_{i}\Delta_i.$$ We assume that the labelling is done in the increasing order: if $i'<i$ and $x'\in\Delta_{i'},\,x\in\Delta_{i}$ then $x'<x$. Note that since $(\hat A, \hat B)\in \mathcal T$ every segment consists of even number of points. From the description of the equivalence class containing $(A, B)$ (see Proposition \[atip\]) it follows that for any $i$ there are the following possibilities:
$1)$ $\Delta_i \cap \hat A = \emptyset$,
$2)$ $\Delta_i$ contains one element from $\hat A\cap (\hat B-1)$,
$3)$ $\Delta_i=[c_i,d_i]$ contains one element $ a\in \hat A\cap \hat B$ and $\hat A\cap[c_i,a-1]=\emptyset$.
Let us choose $i$ such that $\Delta_i$ has property $3)$ and $i$ is minimal. Since $\hat A\cap \hat B\ne\emptyset$ such $i$ indeed does exist. Then one can check using Proposition \[lowhighest\] that $a-1\in \tilde B\cup(\tilde B-1)$ and the segment containing this element consists of odd number of points. Therefore $(\tilde A,\tilde B)\notin\mathcal T$, which is a contradiction. Theorem is proved.
The usual typicality (\[Kac2\]) for admissible weights implies the spherical typicality (\[star\]), but the converse is not true. As it follows from the proof of the theorem the degree of atypicality [@Brun] of a spherically typical module $L(\lambda)$ is equal to $s$, where $2^s$ is the number of elements in the equivalence class of $\lambda,$ and thus can be any number (see Proposition \[atip\]).
\[corcon\] If the highest weight $\lambda$ of the irreducible module $U=L(\lambda)$ is admissible and satisfies (\[star\]) then the highest weight $\mu$ of $U^*$ is also admissible and $U$ is spherical.
We should mention that a different proof of the sphericity of $U$ under the assumption that the weight $\lambda$ is large enough was found in [@AldSch].
Proof of the main theorem
=========================
Let $\mathfrak D_{n,m}$ be the algebra of quantum integrals of the deformed CMS system with parameter $k=-\frac12.$ It acts naturally on the algebra $\frak A_{n,m}$ of $S_n\times S_m$-invariant Laurent polynomials $f\in\Bbb C[x_1^{\pm1},\dots, x_n^{\pm1},y_1^{\pm1}\dots,y_m^{\pm1}]^{S_n\times S_m}$ satisfying the quasi-invariance condition (\[quasi\]) with the parameter $k=-\frac12.$ Let $$\frak A_{n,m}=\bigoplus_{\chi}\frak A_{n,m}(\chi)$$ be the corresponding decomposition into the direct sum of the generalised eigenspaces (\[sum\]).
On the set of highest admissible weights $P^+(X)$ there is a natural equivalence relation defined by the equality of the corresponding central characters (\[central\]). Under bijection (\[sharp\]) it goes to the equivalence relation on $X^+_{n,m}$ when $\lambda \sim \mu$ if $\chi_\lambda = \chi_\mu.$
Consider $\frak A_{n,m}$ as $Z(\frak g)$-module with respect to the radial part homomorphism $\psi.$
For any finite dimensional generalised eigenspace $\frak A_{n,m}(\chi)$ there exists a unique projective indecomposable module $P$ over $\frak{gl}(n,2m)$ and a natural map from $\frak k$-invariant part of $P^*$ $$\Psi : (P^*)^{\frak k}\longrightarrow \frak A_{n,m}(\chi)$$ which is an isomorphism of $Z(\frak g)$-modules.
Let $\chi :\mathfrak D_{n,m}\rightarrow \Bbb C$ be a homomorphism such that $\frak A_{n,m}(\chi)$ is a finite dimensional vector space. By Proposition \[max\] there exists $\nu \in X^+_{n,m}$ such that $\chi=\chi_{\nu}$.
Let $E$ be the equivalence class in $P^+(X)$ corresponding to the equivalence class of $\nu$ via $\sharp$ bijection. Since the corresponding equivalence class is finite by Proposition \[atip\] and the definition of the sets $A$ and $B$ there exists $\lambda\in E,$ which satisfies condition (\[star\]).
By Theorem \[typical\] the corresponding irreducible module $L(\lambda)$ is spherically typical. Let $K(\lambda)$ is the corresponding Kac module. Consider the Jordan–Hölder series of $K(\lambda):$ $$K(\lambda)=K_0\supset K_1\supset\dots\supset K_N=0.$$ Since $\lambda\in P^+(X)$ Kac module $K(\lambda)$ contains a non-zero $\frak k$-invariant vector $v$. Let $L(\mu)$ be an irreducible spherical sub-quotient of $K(\lambda)$, such that the image of $v$ in $L(\mu)$ is non-zero. From the proof of Theorem \[typical\] it follows that $\mu\in P^+(X)$, $\mu\preceq\lambda$ and $\chi_{\lambda}=\chi_{\mu}$. Since $L(\lambda)$ is spherically typical this implies that $\lambda =\mu$. Thus the image of $v$ under natural homomorphism $\varphi : K(\lambda)\longrightarrow L(\lambda)$ is not zero: $\varphi(v) \ne 0.$
Let us consider the projective cover $P(\lambda)$ of $L(\lambda)$ (see e.g. [@Zou]) and prove that it is generated by a $\frak k$-invariant vector.
Since $K(\lambda)$ is projective as $\frak k$-module (see Theorem \[Kac\]), there exists a $\frak k$-invariant vector $\omega \in P(\lambda)$ such that $\psi(\omega)\ne 0$ under natural homomorphism $\psi: P(\lambda)\longrightarrow L(\lambda).$
Let $N\subset P(\lambda)$ be the $\frak g$-submodule generated by $\omega$. Since $\psi(\omega) \ne 0$ we have that $N$ is not contained in $Ker (\psi)$, which is known to be the only maximal submodule of $P(\lambda)$ [@Zou]. Therefore $N=P(\lambda)$ and $\omega$ generates $P(\lambda)$ as $\frak g$-module.
Now let us construct the map $$\Psi : (P(\lambda)^*)^{\frak k}\longrightarrow \frak A_{n,m}(\chi).$$ Let $l\in P(\lambda)^*$ be a $\frak k$-invariant linear functional on $P(\lambda)$ and define $\Psi(l)$ as the restriction of $\phi_{\omega,l}=l(x\omega)$ on to $S(\frak a).$
Let us show that $\Psi(l)\in \frak A_{n,m}(\chi).$ Similarly to the proof of Proposition \[restweyl\] we can claim that $\Psi(l)$ has a form $$\Psi(l)=\sum_{\mu \in M}c_{\mu}e^{\mu}(x), \,\, x \in U(\frak a),$$ where the sum is taken over some finite subset of $M \subset P(X)$. Let $\alpha$ be a root of $\frak g$ and $X_\alpha, X_{-\alpha} \in \frak g$ be the corresponding root vectors. The product $X_\alpha X_{-\alpha} \in U(\frak g)$ commutes with $\frak a$, so the restriction of $X_\alpha X_{-\alpha}\phi_{\omega,l}$ has a similar form with the same set $M.$ From (\[dalpha\]) it follows that $\partial_\alpha \Psi(l)$ is divisible by $e^{2\alpha}-1$, which implies the claim.
From Theorem \[radial\] it follows that $\Psi: (P(\lambda)^*)^{\frak k}\longrightarrow \frak A_{n,m}(\chi)$ is a homomorphism of $Z(\frak g)$-modules.
Now we are going to prove that $\Psi$ is an isomorphism. Let us prove first that $\Psi$ is injective. Suppose that $\Psi(l)=0$. This means that the restriction of $\phi_{\omega,l}$ on $S(\frak a)$ is zero. Therefore by Theorem \[radial\] $\phi_{\omega,l}(x)=l(x\omega)=0$ for all $x\in U(\frak g)$. Since $\omega$ generates $P(\lambda)$ as $\frak g$-module we have that $l=0$ and thus $\Psi$ is injective.
In order to prove that $\Psi$ is surjective it is enough to show that the dimension of $(P(\lambda)^*)^{\frak k}$ is not less than the dimension of $\frak A_{n,m}(\chi)$. It is known that $P(\lambda)$ has the Kac flag (see [@Zou]). Let $n_{\lambda,\mu}$ be the multiplicity of $K(\mu)$ in the Kac flag of $P(\lambda)$. Since Kac modules are projective as $\frak k$-modules we have $$P(\lambda)=\bigoplus_{\mu\in Y}n_{\lambda,\mu}K(\mu),$$ where $Y$ is a finite subset of highest weights $\mu$ of $\frak g$ such that $n_{\lambda,\mu}>0.$ Therefore by part 3 of Theorem \[Kac\] $$P(\lambda)^*=\bigoplus_{\mu\in Y}n_{\lambda,\mu}K(2\rho_1-w_0\mu)$$ as $\frak k$-modules. Hence by the same Theorem we have $$\dim (P(\lambda)^*)^{\frak k}=\sum_{\mu\in Y\cap P^+(X)}n_{\lambda,\mu}.$$ We claim that $Y\subset P^+(X)$. Indeed by BGG duality for classical Lie superalgebras of type I (see [@Zou]) we have $n_{\lambda,\mu}=m_{\mu,\lambda},$ where $m_{\mu,\lambda}$ is the multiplicity of $L(\lambda)$ in the Jordan–Hölder series of $K(\mu)$. So if $n_{\lambda,\mu}>0$ then $L(\lambda)$ is a subquotient of $K(\mu)$. This means that there are submodules $M\supset N$ in $K(\mu)$ such that $L(\lambda)=M/N$. Therefore we have natural homomorphism $
\varphi: K(\lambda)\longrightarrow M/N.
$ As we have just seen the image $\varphi(v)$ of $\frak k$-invariant vector $v \in K(\lambda)$ is not zero. Since $K(\lambda)$ is projective as $\frak k$-module we can lift previous homomorphism to a homomorphism $K(\lambda)\longrightarrow M$ and the image of vector $v$ is a non-zero $\frak k$-invariant vector in $M\subset K(\mu)$. Therefore as before $\mu\in P^+(X)$, so $Y\subset P^+(X)$. Thus we have $$\dim (P(\lambda)^*)^{\frak k}=\sum_{\mu\in Y}n_{\lambda,\mu}\ge |Y|.$$
By Proposition \[max\] the dimension of $\frak A_{n,m}(\chi)$ is equal to the number of $\tau \in X_{n,m}^+$ such that $\chi_{\tau}=\chi_{\nu},$ or equivalently, to the number $|E|$ of the elements in the equivalence class $E.$ Let us show that $E=Y.$
It is obvious that $Y \subseteq E.$ To prove that $E \subseteq Y$ consider $\mu \in E$ and corresponding Kac module $K(\mu).$ By Theorem \[Kac\] it is spherical. Hence there exists its sub-quotient, which is spherical and irreducible. Since $\lambda$ is spherically typical this sub-quotient must be isomorphic to $L(\lambda).$ By BGG duality $\lambda \in Y.$ Thus $\Psi$ is an isomorphism.
Now let us prove the uniqueness of $P.$ If $P(\lambda)$ and $P(\mu)$ satisfy the conditions of the theorem then they have the same central characters and from spherical typicality it follows that $\lambda=\mu$. Theorem is proved.
Any finite-dimensional generalised eigenspace of $\frak D_{n,m}$ in $\frak A_{n,m}$ contains at least one zonal spherical function on $X$, corresponding to an irreducible spherically typical $\mathfrak g$-module.
Since $Y=E$ and all $n_{\lambda,\mu}=1$ as another corollary we have an effective description of the Kac flag of the projective cover $P(\lambda)$ in the spherically typical case (which may have any degree of atypicality in the sense of [@Brun], see above). Our description is equivalent to Brundan-Stroppel algorithm [@Brun; @Brund] in this particular case.
Zonal spherical functions for $X=(\mathfrak{gl}(1,2), \mathfrak{osp}(1,2))$.
============================================================================
Let us illustrate this in the simplest example $m=1,n=1$, corresponding to the symmetric pair $X=(\mathfrak{gl}(1,2), \mathfrak{osp}(1,2)).$
The corresponding algebra $\frak A_{1,1}$ consists of the Laurent polynomials $f\in\Bbb C[x, x^{-1}, y, y^{-1}]$, satisfying the quasi-invariance condition $$\label{quasi11}
(\partial_x+\frac12 \partial_y) f\equiv 0$$ on the line $x=y$, where $
\partial_x =x \frac{\partial}{\partial x}, \, \, \partial_y =y \frac{\partial}{\partial y}.
$ Writing $f=\sum_{i,j \in \mathbb Z}a_{i,j}x^iy^j,$ where only finite number of coefficients are non-zero, we can write the quasi-invariance conditions as an infinite set of linear relations $$\sum_{i+j=l}(2i+ j)a_{i,j}=0, \quad l \in \mathbb Z.$$ Note that the algebra $\frak A_{1,1}$ is naturally $\mathbb Z$-graded by the degree defined for the Laurent monomial $x^iy^j$ as $i+j,$ so we have one linear relation in each degree.
The radial part of the Laplace-Beltrami operator (\[radpart\]) in these coordinates has the form $$\label{rad11}
\mathcal L_2=\partial_x^2-\frac12 \partial_y^2 - \frac{x+y}{x-y}(\partial_x+\frac12 \partial_y).$$ It commutes with the grading (momentum) operator $
\mathcal L_1=\partial_x+ \partial_y,
$ but in contrast with the usual symmetric spaces these two do not generate the whole algebra of the deformed CMS integrals $\frak D_{1,1}:$ one has to add the third order quantum integral $$\label{radord3}
\mathcal L_3=\partial_x^3 + \frac{1}{4} \partial_y^3 -\frac{3}{2}\frac{x+y}{x-y} (\partial_x^2-\frac{1}{4} \partial_y^2)
+\frac{3}{4}\frac{x^2+4xy+y^2}{(x-y)^2}(\partial_x + \frac{1}{2} \partial_y).$$ To describe the corresponding spectral decomposition let us introduce the functions $$\label{fun11}
\varphi_{ij}=x^iy^j-\frac{2i+j}{2i+j-1}x^{i-1}y^{j+1}, \quad 2i+j\neq 1,$$ $$\label{fun112}
\psi_{i}=x^{i+1}y^{-1-2i}+x^{i-1}y^{1-2i}, \quad i\in \mathbb Z.$$ One can easily check that they satisfy the quasi-invariance conditions and form a basis in the algebra $\frak A_{1,1}$. Denote also $\varphi_{ij}$ with $2i+j=0$ as $\varphi_i$: $$\varphi_i=x^i y^{-2i}, \, i\in \mathbb Z.$$
\[jord11\] The generators $\mathcal L_i, \, i=1,2,3$ of algebra $\frak D_{1,1}$ act in the basis (\[fun11\], \[fun112\]) as follows $$\mathcal L_1 \varphi_{ij}=(i+j)\varphi_{ij}, \quad
\mathcal L_2\varphi_{ij}=\lambda_{ij}\varphi_{ij}, \quad \lambda_{ij}=i(i-1)-\frac12 j(j+1),$$ $$\mathcal L_3 \varphi_{ij}=\mu_{ij}\varphi_{ij}, \quad \mu_{ij}=i^3+\frac{1}{4} j^3 -\frac{3}{2} (i^2-\frac{1}{4} j^2)+\frac{3}{4} (i+\frac{1}{2}j), \quad 2i+j\neq 1,$$ $$\mathcal L_1 \psi_i= -i \psi_i, \quad \mathcal L_1 \varphi_i= -i \varphi_i, \quad \mathcal L_2\psi_{i}=-i^2 \psi_i-\varphi_i, \quad
\mathcal L_2\varphi_{i}=-i^2 \varphi_i,$$ $$\mathcal L_3\psi_{i}=-i^3 \psi_i-3 \varphi_i, \quad
\mathcal L_3\varphi_{i}=-i^3 \varphi_i, \quad i \in \mathbb Z.$$
Let $W_i=< \varphi_i, \psi_i>$ be the linear span of $\varphi_i$ and $\psi_i.$ We see that $W_i$ is two-dimensional generalised eigenspace (Jordan block) for the whole algebra $\frak D_{1,1},$ while $V_{ij}=<\varphi_{ij}>$ with $2i+j \neq 0,1$ are its eigenspaces. In our terminology $\varphi_{ij}$ with $i+2j\neq 1$ are the zonal spherical functions, while $\psi_i$ are the generalised zonal spherical functions of $X=(\mathfrak{gl}(1,2), \mathfrak{osp}(1,2)).$
\[irrz\] The spectral decomposition of $\frak A_{1,1}$ with respect to the action of $\frak D_{1,1}$ has the form $$\label{spec11}
\frak A_{1,1}=\bigoplus_{2i+j\neq 0,1}<\varphi_{ij}>\oplus \bigoplus_{i\in \mathbb Z}<\varphi_{i}, \psi_i>.$$
This is in a good agreement with the equivalence relation $\sim$ on $$\mathcal T=\{(a,b), \, a \in 2\mathbb Z, b \in \mathbb Z\},$$ where $a=2i, \, b =-j+1.$ Indeed, it is easy to check using Lemma \[equival\] that the corresponding equivalence classes consist of one-element classes $(a,b)$ with $a\neq b, b-1$ and of two-element classes $(a,a) \sim (a-2, a-1).$ In terms of $i,j$ we have one element classes $(i,j)$ with $2i+j\neq 0,1$, corresponding to $V_{ij}=<\varphi_{ij}>$ and two-element classes $(i, -2i+1)\sim (i, -2i)$, corresponding to $W_i=< \varphi_i, \psi_i>.$
This also agrees with the representation theory. Let $L(\lambda)$ and $P(\lambda)$ be the irreducible spherical $\mathfrak g$-module with highest weight $\lambda$ and its projective cover respectively. In our case $\lambda=(2i, j, j), \, i,j \in \mathbb Z.$ The spherical typicality condition (\[star\]) means that $2i+j\neq 1.$ In fact, one can show that if $2i+j=1$ the module $L(\lambda)$ is not spherical, in agreement with the spectral decomposition (\[spec11\]). If $2i+j\neq 0,1$ the projective cover $P(\lambda)=K(\lambda)$ coincides with the corresponding Kac module. One can check that it contains only one (up to a multiple) $\mathfrak {osp}(1,2)$-invariant vector with the corresponding spherical function $\varphi_{ij}.$ A bit more involved calculations show that if $2i+j=0$ the projective cover $P(\lambda)$ contains two dimensional $\mathfrak{osp}(1,2)$-invariant subspace, corresponding to the generalised eigenspace $W_i.$
Concluding remarks
==================
Although we have considered only one series of the classical symmetric Lie superalgebras we believe that a similar relation of spectral decomposition of the algebra of the deformed CMS integrals and projective covers holds also at least for the remaining classical series (\[4series\]). Note that the notion of spherically typical modules can be easily generalised to all these cases. Since the corresponding deformed root system is of $BC(n,m)$ type, to describe the corresponding zonal spherical functions one can use the super Jacobi polynomials [@SV2].
The type $AI/AII$ we have considered in that sense is different since the corresponding deformed root system is of type $A(n-1,m-1).$ To describe the zonal spherical functions in this case we can use the theory of Jack–Laurent symmetric functions developed in [@SV5; @SV3].
Recall that such functions $P^{(k,p_0)}_{\alpha}$ are certain elements of $\Lambda^{\pm}$ labelled by bipartitions $\alpha=(\lambda,\mu)$, where $\Lambda^{\pm}$ is freely generated by $p_a$ with $a \in \mathbb Z \setminus \{0\}$ and variable $p_0$ is considered as an additional parameter [@SV5].
In [@SV3] we considered the case of special parameters $p_0=n+k^{-1}m$ with natural $m,n.$ In that case the spectrum of the algebra of quantum CMS integrals acting on $\Lambda^{\pm}$ is not simple. For generic $k$ we showed that any generalised eigenspace has dimension $2^r$, which coincides with the number of elements in the corresponding equivalence class of bipartitions. This equivalence can be described explicitly in terms of geometry of the corresponding Young diagrams $\lambda, \mu$ (see [@SV3]). In each equivalence class $E$ there is only one bipartition $\alpha$ such that the corresponding Jack-Laurent symmetric function $P^{(k,p_0)}_{\alpha}$ is regular at $p_0=n+k^{-1}m.$ At such $p_0$ there is a natural homomorphism $$\varphi_{n,m}: \Lambda^{\pm} \rightarrow \frak A_{n,m}(k),$$ sending $p_a$ to the deformed power sum $$p_a(x,y,k)= x_1^a+\dots +x_n^a+k^{-1} (y_{1}^a+\dots + y_{m}^a), \,\, a \in \mathbb Z.$$ The image of the corresponding function $\varphi_{n,m}(P^{(k,n+k^{-1}m)}_{\alpha}) \in \frak A_{n,m}(k)$ is an eigenfunction of the algebra $\frak{D}_{n,m}(k)$ of the deformed CMS integrals, so its specialisation at $k=-1/2$ (provided it exists) determines a zonal spherical function for $X=(\mathfrak{gl}(n,2m), \mathfrak{osp}(n,2m)).$ A natural question is whether this relation can be extended to an isomorphism of the corresponding generalised eigenspaces.
We would like to mention also that Brundan and Stroppel [@Brund] showed that the algebra of the endomorphisms of a projective indecomposable module over general linear supergroup is isomorphic to $$\mathfrak A_r=\Bbb C[\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r]/(\varepsilon_1^2,\,\varepsilon_2^2,\dots,\varepsilon_r^2).$$ We believe that using this and our main theorem it is possible to describe in a similar way the action of the algebra $\frak{D}_{n,m}$ in its generalised eigenspace, which in particular would imply that it contains only one zonal spherical function. This would be in a good agreement with the results of [@SV3].
Acknowledgements
================
We are grateful to Professor A. Alldridge for attracting our attention to the preprint [@AldSch] and helpful comments on the first version of this paper.
This work was partially supported by the EPSRC (grant EP/J00488X/1). ANS is grateful to the Department of Mathematical Sciences of Loughborough University for the hospitality during the autumn semesters in 2012-14.
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[^1]: Strictly speaking, the Lie superalgebra $\frak {gl}(n,m)$ is not basic classical, but it is more convenient for us to consider than $\frak {sl}(n,m).$
[^2]: A different proof in the general even type case can be found in [@AldSch].
|
---
abstract: 'Data representation is crucial for the success of machine learning models. In the context of quantum machine learning with near-term quantum computers, equally important considerations of how to efficiently input (encode) data and effectively deal with noise arise. In this work, we study data encodings for binary quantum classification and investigate their properties both with and without noise. For the common classifier we consider, we show that encodings determine the classes of learnable decision boundaries as well as the set of points which retain the same classification in the presence of noise. After defining the notion of a robust data encoding, we prove several results on robustness for different channels, discuss the existence of robust encodings, and prove an upper bound on the number of robust points in terms of fidelities between noisy and noiseless states. Numerical results for several example implementations are provided to reinforce our findings.'
author:
- Ryan LaRose
- Brian Coyle
title: Robust data encodings for quantum classifiers
---
Introduction {#sec:intro}
============
Fault-tolerant quantum computers which can efficiently simulate physics [@feynman_simulating_1982] and factor prime numbers [@shor_polynomial-time_1997] lie on an unclear timeline. Current quantum processors in laboratories or in the cloud [@larose_overview_2019] contain fewer than one-hundred qubits with short lifetimes and noisy gate operations. Such devices cannot yet implement fault-tolerant procedures and are said to belong to the noisy, intermediate scale quantum (NISQ) era [@preskill_quantum_2018]. While the gap between NISQ capabilities and required resources for, e.g., factoring 2048-bit integers is large [@gidney_how_2019], recent hardware advancements have led to the first quantum computation which classical computers cannot emulate [@arute_quantum_2019]. This combination of improved hardware and unclear timeline for fault-tolerance makes the question “What (useful) applications can NISQ computers implement?” both interesting and important to consider. A leading candidate for NISQ applications is a class of algorithms known as variational quantum algorithms (VQAs) [@mcclean_theory_2016]. VQAs use a NISQ computer to evaluate an objective function and a classical computer to adjust input parameters to optimize the function. VQAs have been proposed or used for many applications including quantum chemistry [@peruzzo_variational_2014], approximate optimization [@farhi_quantum_2014], quantum state diagonalization [@larose_variational_2019], quantum compilation [@khatri_quantum-assisted_2019; @jones_quest_2019], quantum field theory simulation [@klco_quantum-classical_2018; @klco_digitization_2019], linear systems of equations [@bravo-prieto_variational_2019; @xu_variational_2019; @huang_near-term_2019], and even quantum foundations [@arrasmith_variational_2019]. More fundamental questions about the computational complexity [@mcclean_theory_2016; @biamonte_universal_2019], trainability [@mcclean_barren_2018; @grant_initialization_2019; @cerezo_cost-function-dependent_2020] and noise-resilience [@sharma_noise_2020] of VQAs have also been considered.
Variational quantum algorithms have strong overlap with machine learning algorithms, which seek to train a computer to recognize patterns by designing and minimizing a cost function defined over an input dataset [@goodfellow_deep_2016]. In this context, VQAs can be considered quantum neural networks (QNNs) [@farhi_classification_2018; @benedetti_parameterized_2019], and a multitude of applications from classical machine learning can be realized with (simulated) quantum computers. Such applications include generative modeling [@verdon_quantum_2017; @benedetti_generative_2019; @liu_differentiable_2018; @coyle_born_2019; @dallaire-demers_quantum_2018], transfer learning [@mari_transfer_2019], and classification [@farhi_classification_2018; @schuld_circuit-centric_2018; @schuld_supervised_2018; @grant_hierarchical_2018; @perez-salinas_data_2019; @blank_quantum_2019; @abbas_quantum_2020]. These QNN applications, along with additional techniques and applications based on quantum kernel methods [@schuld_quantum_2019; @havlicek_supervised_2019; @kubler_quantum_2019; @suzuki_analysis_2019], define the emerging field of quantum machine learning (QML) [@wittek_quantum_2014; @schuld_supervised_2018][^1].
Despite these applications, several fundamental questions lie at the forefront of QML. Perhaps the most pressing question is whether quantum models can provide any advantages over classical models. While some (generally) negative theoretical results have been shown for sampling complexity [@arunachalam_optimal_2017] and information capacity [@wright_capacity_2019], other (generally) positive results have been shown for expressive power [@du_expressive_2018; @coyle_born_2019] and problem-specific sampling complexity [@servedio_equivalences_2004; @arunachalam_quantum_2020], and the question is largely open. Any practical experiments toward demonstrating advantage give rise to additional crucial questions. One question, general to all NISQ applications, is how to deal with noise present in NISQ computers. A second question, specific to QML applications, is how to (efficiently) input or *encode* data into a QML model.
For the first question, several general-purpose strategies for dealing with noise such as dynamical decoupling [@viola_dynamical_1999], probabilistic error cancellation & zero-noise extrapolation [@temme_error_2017], and quantum subspace expansion [@mcclean_decoding_2020] have been proposed. However, the robustness (resilience) of particular VQAs in the presence of noise has not been thoroughly investigated, with the exception of preliminary studies in quantum compiling [@khatri_quantum-assisted_2019; @sharma_noise_2020] and approximate optimization [@xue_effects_2019; @alam_analysis_2019; @marshall_characterizing_2020]. Understanding robustness properties of VQAs is key to making progress towards practical NISQ implementations. For the second question on inputting data, most studies in QML focus on the design of the QNN while assuming a full wavefunction representation of arbitrary data [@schuld_circuit-centric_2018; @farhi_classification_2018; @harrow_quantum_2009; @kerenidis_quantum_2016; @kerenidis_quantum_2018; @dervovic_quantum_2018; @zhao_smooth_2018]. This assumption is not suitable for practical implementations as it is well-known that preparing an arbitrary quantum state requires a number of gates exponential in the number of qubits [@Knill_Laflamme_Milburn_2001].
In this paper, we study both of the above questions in the context of binary quantum classification. In particular, we define different methods of encoding data and analyze their properties both with and without noise. For the noiseless case, we demonstrate that different data encodings lead to different sets of learnable decision boundaries for the quantum classifier. For the noisy case, given a quantum channel, we define the notion of a robust point for the quantum classifier — a generalization of fixed points for quantum operations. We completely characterize the set of robust points for example quantum channels, and discuss how encoding data into robust points is a type of problem-specific error mitigation for quantum classifiers.
To these ends, the rest of the paper is organized as follows. Section \[sec:definitions\] presents definitions which are used in the remainder to prove our results, including a formal definition of a common binary quantum classifier we consider (Sec. \[subsec:intro/quantum-classifiers\]), definitions and examples of data encodings (Sec. \[sec:data-encodings\]), noise channels we consider (Sec. \[sec:noise-models\]), and definitions of robust points and robust encodings (Sec. \[subsec:robustness-definition\]). After this, we present analytic results and proofs for robustness in Section \[sec:noise\_robustness\]. We begin by showing that different encodings lead to different classes of learnable decision boundaries in Sec. \[ssec:classes\_learnable\_decision\_boundaries\], then characterize the set of robust points for example quantum channels in Sec. \[ssec:characterize\_robust\_points\]. In Sec. \[subsec:robustness-results\], we state and prove robustness results, and in Sec. \[subsec:existence-of-robust-encodings\] we discuss the existence of robust encodings. Finally, we prove an upper bound on the number of robust points in terms of fidelities between noisy and ideal states in Sec. \[ssec:fidelity\_bounds\]. Last, we include several numerical results in Sec. \[sec:numerical\_results\] that reinforce and extend our findings, and finally conclude in Sec. \[sec:conclusions\].
Preliminary Definitions {#sec:definitions}
=======================
Quantum Classifiers {#subsec:intro/quantum-classifiers}
-------------------
In classical machine learning, classification problems are a subclass of supervised learning problems in which the computer model is presented with labeled data and asked to learn some pattern. For binary classification, the input is a set of labeled feature vectors $$\label{eqn:labeled-data-for-classifier}
\{({\boldsymbol{x}}_i, y_i)\}_{i = 1}^{M}$$ where ${\boldsymbol{x}}_i \in \mathcal{X}$ is a feature vector, $\mathcal{X}$ is an arbitrary set[^2] , and ${y}_i \in \{0, 1\}$ is a binary label. Given this data, the goal of the learner is to output a rule $f: \mathcal{X} \rightarrow \{0, 1\}$ which accurately classifies the data and can be used to make predictions on new data. In practice, this is accomplished by defining a model (e.g., a neural network) and cost function, then minimizing this cost function by “training” the model over the input data.
In this work, we restrict to binary (quantum) classifiers, henceforth called simply (quantum) classifiers. We remark that multi-label classification problems can be reduced to binary classification by standard methods.
![A common architecture for a binary quantum classifier that we study in this work. The general circuit structure is shown in **(a)** and the structure for a single qubit is highlighted in **(b)**. In both, a feature vector ${\boldsymbol{x}}$ is encoded into a quantum state ${\rho_{{\boldsymbol{x}}}}$ via a “state preparation” unitary $S_{{\boldsymbol{x}}}$. The encoded state ${\rho_{{\boldsymbol{x}}}}$ then evolves to $U {\rho_{{\boldsymbol{x}}}}U^\dagger =: {\tilde{\rho}_{{\boldsymbol{x}}}}$ where $U({\boldsymbol{\alpha}})$ is a unitary ansatz with trainable parameters ${\boldsymbol{\alpha}}$. A single qubit of the evolved state ${\tilde{\rho}_{{\boldsymbol{x}}}}$ is measured to yield a predicted label ${\hat{y}}$ for the vector ${\boldsymbol{x}}$.[]{data-label="fig:classifier"}](images/classifier.pdf)
A quantum classifier is essentially the same as a classical one, the key difference being the model (ansatz) of the classifier. As mentioned in the Introduction, many QML architectures use a variational quantum algorithm, nominally consisting of parameterized one- and two-qubit gates, which is called a quantum neural network (QNN) in the context of machine learning. QNNs can be considered function approximators analogous to classical neural networks (e.g., feedforward neural networks [@fine_feedforward_1999]), and the procedure for training the QNN consists of adjusting gate parameters such that this function approximator outputs good predictions for the input data.
While the use of QNNs as machine learning models may present the possibility of advantage for particular problems [@du_expressive_2018; @coyle_born_2019], QNNs also present key challenges for machine learning. For nearly all QML problems, a pressing challenge is inputting (arbitrary) data to the model such that the QNN can process it. We refer to this input process as *data encoding*, and discuss it in detail in Sec. \[sec:data-encodings\]. Another potential challenge with QNNs is outputting information, since the data propagated through the QNN is a quantum state. For machine learning applications, this means that the output feature vector (amplitudes of the quantum state) cannot be accessed efficiently. Rather, as is usual in quantum mechanics, quantities of the form ${{\rm Tr}}[\rho \hat{O}]$ where $\rho$ is the quantum state and $\hat{O}$ is some Hermitian operator can be efficiently evaluated.
Fortunately for quantum classification, outputting information (predictions) can be done in a straightforward manner. As several authors have noted [@farhi_classification_2018; @schuld_circuit-centric_2018; @schuld_supervised_2018; @grant_hierarchical_2018; @perez-salinas_data_2019; @blank_quantum_2019], it is natural to use the measurement outcome of a single qubit as a class prediction as produces a binary outcome. We adopt this strategy in our work.
Informally, we define a (binary) quantum classifier as a procedure for encoding data into a quantum circuit, processing it through trainable QNN, and outputting a (binary) predicted label. Given a feature vector ${\boldsymbol{x}} \in \mathcal{X}$, a concise description of such a classifier can be written $$\begin{aligned}
{\boldsymbol{x}} &\mapsto {\rho_{{\boldsymbol{x}}}}&& \text{(encoding)} \label{eqn:encoding} \\
&\mapsto {\tilde{\rho}_{{\boldsymbol{x}}}}&& \text{(processing)} \label{eqn:processing}\\
&\mapsto {\hat{y}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] . && \text{(prediction)} \label{eqn:prediction}\end{aligned}$$ Several remarks are in order. First, a given data point ${\boldsymbol{x}}$ in the training set [(\[eqn:labeled-data-for-classifier\])]{} is encoded in a quantum state $\rho_{{\boldsymbol{x}}} \in \mathcal{D}_n$ via a state preparation unitary $S_{{\boldsymbol{x}}}$ (see Fig. \[fig:classifier\]. Throughout the paper, we use $\mathcal{D}_n \subset \mathbb{C}^{2^n \times 2^n}$ to denote the set of density operators (matrices) on $n$ qubits. We remark that each ${\boldsymbol{x}}$ in the training set leads to a (unique) $S_{{\boldsymbol{x}}}$, so the state preparation unitary can be considered a parameterized family of unitary ansätze. We discuss encodings in detail in Sec. \[sec:data-encodings\].
For the processing step [(\[eqn:processing\])]{}, there have been many proposed QNN architectures in recent literature, including quantum convolutional neural networks [@cong_quantum_2019; @henderson_quanvolutional_2019], strongly entangling ansätze [@schuld_circuit-centric_2018], and more [@stoudenmire_supervised_2016; @grant_hierarchical_2018]. In this work, we allow for a general unitary evolution $U({\boldsymbol{\alpha}})$ such that $$\label{eqn:unitary_evolution}
{\tilde{\rho}_{{\boldsymbol{x}}}}= U({\boldsymbol{\alpha}}) {\rho_{{\boldsymbol{x}}}}U^\dagger({\boldsymbol{\alpha}}) .$$ We remark that some QNN architectures involve intermediate measurements and conditional processing (notably [@cong_quantum_2019]) and so do not immediately fit into [(\[eqn:unitary\_evolution\])]{}. Our techniques for showing robustness could be naturally extended to such architectures, however, and so we consider [(\[eqn:unitary\_evolution\])]{} as a simple yet general model. We also note that training the classifier via minimization of a well-defined cost function is an important task with interesting questions, but we primarily focus on data encodings and their properties in this work. For this reason we often suppress the trainable parameters ${\boldsymbol{\alpha}}s$ and write $U$ for $U({\boldsymbol{\alpha}})$.
Finally, the remaining step is to extract information from the state ${\tilde{\rho}_{{\boldsymbol{x}}}}$ to obtain a predicted label. As mentioned, a natural method for doing this is to measure a single qubit which yields a binary outcome $0$ or $1$ taken as the predicted label ${\hat{y}}$. Since measurements are probabilistic, we measure $N_m$ times and take a “majority vote.” That is, if $0$ is measured $N_0$ times and $N_0 {\geqslant}N_m / 2$, we take $0$ as the class prediction, else $1$. Generalizing the finite statistics, this condition can be expressed analytically as $$\label{eqn:decision_rule}
{\hat{y}}[{\tilde{\rho}_{{\boldsymbol{x}}}}] = \begin{cases}
0 \qquad \text{if } {{\rm Tr}}[ \Pi_0^c {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 1 / 2 \\
1 \qquad \text{otherwise}
\end{cases}$$ where $$\label{eqn:pi0-projector-definition}
\Pi_0^c := |0{\rangle}{\langle}0|_c \equiv |0{\rangle}{\langle}0|_c \otimes I_{\bar{c}}$$ is the projector onto the ground state of the classification qubit, labeled $c$, and the remaining qubits are labeled $\bar{c}$. For brevity we often omit these labels when it is clear from context. Throughout the paper, we use ${\hat{y}}$ for predicted labels and $y$ for true labels, and we refer to [(\[eqn:decision\_rule\])]{} as the *decision rule* for the classifier. Equation [(\[eqn:decision\_rule\])]{} is not the only choice for such a decision rule. In particular, one could choose a different “weight” $\lambda$ such that ${\hat{y}}= 0$ if ${{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge \lambda$ as in Ref. [@perez-salinas_data_2019], add a bias to the classifier as in Ref. [@schuld_circuit-centric_2018], or measure the classification qubit in a different basis (e.g., the Hadamard basis instead of the computational basis). Our techniques for showing robustness (Sec. \[subsec:robustness-results\]) could be easily adapted for such alternate decision boundaries, and we consider [(\[eqn:decision\_rule\])]{} as a simple yet general rule for the remainder. The preceding discussion is summarized in the following formal definition of a quantum classifier.
\[def:binary-quantum-classifier\] A (binary) quantum classifier consists of three well-defined functions:\
(i) an encoding function $$\begin{aligned}
E: \mathcal{X} &\rightarrow \mathcal{D}_n \label{eqn:encoding-formal} \\
E({\boldsymbol{x}}) &= {\rho_{{\boldsymbol{x}}}},
\end{aligned}$$ (ii) a function which evolves the state $$\begin{aligned}
\mathcal{U}: \mathbb{C}^{2^n \times 2^n} &\rightarrow \mathbb{C}^{2^m \times 2^m} \label{eqn:evolution-formal} \\
\mathcal{U}({\rho_{{\boldsymbol{x}}}}) &= {\tilde{\rho}_{{\boldsymbol{x}}}},
\end{aligned}$$ and (iii) a decision rule $$\begin{aligned}
\label{eqn:decision-rule-formal}
\hat{y}: \mathbb{C}^{2^m \times 2^m} \rightarrow \{0, 1\} .
\end{aligned}$$ For training, a quantum classifier is provided with labeled data [(\[eqn:labeled-data-for-classifier\])]{}, a cost function $C$, and an optimization routine for minimizing the cost function.
Specification of the functions $E$, $\mathcal{U}$, and ${\hat{y}}$ — along with training data, a cost function, and an optimization routine — uniquely define a quantum classifier. In this work, we let $\mathcal{U}$ be a general unitary evolution [(\[eqn:unitary\_evolution\])]{} and always take the decision rule ${\hat{y}}$ to be [(\[eqn:decision\_rule\])]{}. In the remainder, we study the effects of different encoding functions [(\[eqn:encoding-formal\])]{}, which we now discuss in more detail.
Data Encodings {#sec:data-encodings}
--------------
![(Color online.) A visual representation of data encoding [(\[eqn:encoding-formal\])]{} for a single qubit. On the left is shown a set of randomly generated points $\{{\boldsymbol{x}}_i, y_i\}_{i = 1}^{M}$ normalized to lie within the unit square, separated by a true decision boundary shown by the dashed black line. A data encoding maps each ${\boldsymbol{x}}_i \in \mathbb{R}^2$ to a point on the Bloch sphere $\rho_{{\boldsymbol{x}}_i} \in \mathbb{C}^2$, here using the dense angle encoding [(\[eqn:dae-single-qubit\])]{}. The dashed black line on the Bloch sphere shows the initial decision boundary of the quantum classifier. During the training phase, unitary parameters are adjusted to rotate the dashed black line to correctly classify as many training points as possible. Different data encodings lead to different learnable decision boundaries and different robustness properties, as discussed in the main text.[]{data-label="fig:data-encoding-schmatic-single-qubit"}](images/classical-to-quantum-data.pdf){width="\columnwidth"}
An encoding can be thought of as “loading” a data point ${\boldsymbol{x}} \in \mathcal{X}$ from memory into a quantum state so that it can be processed by a QNN. Unlike classical machine learning, this presents a unique challenge in QML. The “loading” is accomplished by an encoding [(\[eqn:encoding-formal\])]{} from the set $\mathcal{X}$ to $n$-qubit quantum states $\mathcal{D}_n$. As mentioned, many QML papers [@harrow_quantum_2009; @kerenidis_quantum_2016; @kerenidis_quantum_2018; @dervovic_quantum_2018; @zhao_smooth_2018] assume a full wavefunction encoding with $n = \log_2 N$. This provides an exponential saving in “space” at the cost of an exponential increase in “time.” That is, a quantum state of $n = \log_2 N$ qubits can represent a data point with $N$ features, but in general such a quantum state takes time $O(2^n)$ to prepare [@Knill_Laflamme_Milburn_2001].
In practice, data is encoded via a state preparation circuit (unitary) $S_{{\boldsymbol{x}}}$ — written in terms of one- and two-qubit gates — which acts on an initial state $|\phi{\rangle}$, nominally the all zero state $|\phi{\rangle}= |0{\rangle}^{\otimes n}$. This realizes the encoding $$\label{eqn:encoding_unitary}
{\boldsymbol{x}} \mapsto E({\boldsymbol{x}}) = S_{{\boldsymbol{x}}}{| \hspace{1pt} \phi \rangle \langle \phi \hspace{1pt} |}S_{{\boldsymbol{x}}}^\dagger = |{\boldsymbol{x}} {\rangle}{\langle}{\boldsymbol{x}} | =: {\rho_{{\boldsymbol{x}}}}.$$ For $S_{{\boldsymbol{x}}}$ to be useful for a data encoding, it should have several desirable properties. First, $S_{{\boldsymbol{x}}}$ should have a number of gates which is at most polynomial in the number of qubits. For machine learning applications, we want the family of state preparation unitaries to have enough free parameters such that there is a unique quantum state ${\rho_{{\boldsymbol{x}}}}$ for each feature vector ${\boldsymbol{x}}$ — i.e., such that the encoding function $E$ is bijective. Additionally, for NISQ applications, sub-polynomial depth is even more desirable, and we want $S_{{\boldsymbol{x}}}$ to be “hardware efficient” — meaning that the one- and two-qubit gates comprising $S_{{\boldsymbol{x}}}$ can be realized without too much overhead due to, e.g., compiling into the computer’s native gate set and implementing swap gates to connect disjoint qubits.
Motivated by such NISQ limitations, some recent authors [@schuld_supervised_2018; @grant_hierarchical_2018; @stoudenmire_supervised_2016; @schuld_supervised_2018; @cao_cost_2019] have considered a “qubit encoding” $$\label{eqn:qubit_encoding_grant}
|{\boldsymbol{x}}{\rangle}= \bigotimes_{i=1}^{N} \cos (x_i){| 0 \rangle} + \sin (x_i){| 1 \rangle}$$ for the feature vector ${\boldsymbol{x}} = [x_1, ..., x_N]^T \in \mathcal{X}^N$. (Note that for pure state encodings, we often write only the wavefunction $|{\boldsymbol{x}}{\rangle}\in \mathbb{C}^{2^n}$, from which the density matrix ${\rho_{{\boldsymbol{x}}}}= |{\boldsymbol{x}}{\rangle}{\langle}{\boldsymbol{x}}| \in \mathcal{D}_n$ is implicit.) We will also refer to [(\[eqn:qubit\_encoding\_grant\])]{} as an “angle encoding.” The angle encoding uses $N$ qubits with a constant depth quantum circuit and is thus amenable to NISQ computers. The state preparation unitary is $S_{{\boldsymbol{x}}_j} = \bigotimes_{i = 1}^{N} U_i$ where $$U_i := \left[
\begin{matrix}
\cos (x_j^{(i)}) & -\sin(x_j^{(i)}) \\
\sin(x_j^{(i)}) & \cos(x_j^{(i)})
\end{matrix}
\right] ,$$ a strategy which encodes one feature per qubit.
This encoding can be slightly generalized to encode two features per qubit by exploiting the relative phase degree of freedom. We refer to this as the “dense angle encoding” and include a definition below.
\[def:dae\] Given a feature vector ${\boldsymbol{x}} = [x_1, ..., x_N]^T \in \mathbb{R}^N$, the dense angle encoding maps ${\boldsymbol{x}} \mapsto E({\boldsymbol{x}})$ given by $$\label{eqn:dae-general}
|{\boldsymbol{x}}{\rangle}= \bigotimes_{i=1}^{\ceil*{N / 2}} \cos (\pi x_{2i -1}){| 0 \rangle} + e^{2 \pi i x_{2i}} \sin (\pi x_{2i - 1}){| 1 \rangle} .$$
For some of our analytic and numerical results, we highlight the dense angle encoding for two-dimensional data ${\boldsymbol{x}} \in \mathbb{R}^2$ with a single qubit given by $$\label{eqn:dae-single-qubit}
|{\boldsymbol{x}}{\rangle}= \cos (\pi x_{1}) {| 0 \rangle} + e^{2 \pi i x_{2}} \sin (\pi x_{1}){| 1 \rangle} ,$$ which has density matrix $$\label{eqn:dae-density-matrix}
{\rho_{{\boldsymbol{x}}}}=
\left[ \begin{matrix}
\cos^2 \pi x_1 & e^{ - 2 \pi i x_2} \cos \pi x_1 \sin \pi x_1 \\
e^{ 2 \pi i x_2} \cos \pi x_1\sin \pi x_1& \sin^2 \pi x_1 \\
\end{matrix} \right] .$$
Although the angle encoding [(\[eqn:qubit\_encoding\_grant\])]{} and dense angle encoding [(\[eqn:dae-general\])]{} use sinuosoids and exponentials, there is nothing special about these functions (other than, perhaps, they appear in common parameterizations of qubits and unitary matrices [@nielsen_quantum_2010]). We can easily abstract these to a general class of qubit encodings which use arbitrary functions.
Given a feature vector ${\boldsymbol{x}} = [x_1, ..., x_N]^T \in \mathbb{R}^N$, the general qubit encoding maps ${\boldsymbol{x}} \mapsto E({\boldsymbol{x}})$ given by $$\label{eqn:qubit-encoding-general}
|{\boldsymbol{x}}{\rangle}= \bigotimes_{i=1}^{\ceil*{N / 2}} f_i(x_{2i - 1}, x_{2i}) {| 0 \rangle} + g_i(x_{2i - 1}, x_{2i}) {| 1 \rangle} .$$ where $f, g : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{C}$ are such that $|f_i|^2 + |g_i|^2 = 1 \ \forall i$.
We remark that a similar type of generalization was used in [@perez-salinas_data_2019] with a single qubit classifier that allowed for repeated application of an arbitrary state preparation unitary. While [(\[eqn:qubit-encoding-general\])]{} is the most general description of a qubit encoding — and is the encoding we primarily focus on in this work — it is of course not the most general encoding [(\[eqn:encoding-formal\])]{}. The previously mentioned wavefunction encoding maps $N$ features into $n = \log_2 N$ qubits as follows.
The wavefunction encoding of a vector ${\boldsymbol{x}} \in \mathbb{R}^N$ is $$\label{eqn:wavefunction_encoding_amplitude}
|{\boldsymbol{x}}{\rangle}:= \frac{1}{||{\boldsymbol{x}}||_2^2} \sum_{i = 1}^{N} x_i |i{\rangle}$$ where $x_i$ is the $i$th feature of ${\boldsymbol{x}}$.
As with the dense angle encoding, we highlight the wavefunction encoding for ${\boldsymbol{x}} = [x_1 \ x_2]^T$ with $||{\boldsymbol{x}}||_2 = 1$: $$\label{eqn:wavefunction-encoding-density-matrix-onequbit}
{\rho_{{\boldsymbol{x}}}}= \left[ \begin{matrix}
x_1^2 & x_1 x_2 \\
x_1 x_2 & x_2^2 \\
\end{matrix} \right] .$$
While we do not consider it in this work, we note that the wavefunction encoding can be slightly generalized to allow for parameterizations of features (amplitudes).
\[def:amplitude-encoding\] For ${\boldsymbol{x}} \in \mathbb{R}^N$, the amplitude encoding maps ${\boldsymbol{x}} \mapsto E({\boldsymbol{x}})$ given by $$\label{eqn:amplitude_encoding_grant} {| {\boldsymbol{x}} \rangle} = \sum\limits_{i=1}^{N} f_i({\boldsymbol{x}}){| i \rangle}$$ where $\sum_i |f_i|^2 = 1$.
The functions $f_i$ could only act on the $i$th feature, e.g. $f_i({\boldsymbol{x}}) = \sin x_i$, or could be more complicated functions of several (or all) features.
Thus far, we have formally defined a data encoding [(\[eqn:qubit-encoding-general\])]{} and its role in a quantum classifier (Def. \[def:binary-quantum-classifier\]), and we have given several examples. While we have discussed different properties of state preparation circuits which implement data encodings (depth, overhead, etc.), we have not yet discussed the two main properties of data encodings we consider in this work: *learnability* and *robustness*. For the first property, we show in Sec. \[ssec:classes\_learnable\_decision\_boundaries\] that different data encodings lead to different classes of learnable decision boundaries. For the second property, we show that different data encodings lead to different sets of robust points (to be defined) in Sec. \[ssec:characterize\_robust\_points\] — Sec. \[subsec:existence-of-robust-encodings\]. For the latter results which constitute the bulk of our work, we first need to introduce the noise channels we consider and define the notion of a robust point, which we do in the following two sections.
Noise in Quantum Systems {#sec:noise-models}
------------------------
In this section, we introduce our notation for the common quantum channels we use in this work. While we provide brief exposition on quantum noise, we refer the reader desiring more background to the standard references [@nielsen_quantum_2010; @john_preskill_quantum_1998; @Watrous_2018].
Noise occurs in quantum systems due to interactions with the environment. Letting $\rho$ denote the quantum state of interest and $\rho_{\text{env}}$ the environment, noise can be characterized physically by the process $$\rho \mapsto {{\rm Tr}}_{\text{env}} \left[ U \left( \rho \otimes \rho_{\text{env}} \right) U^\dagger \right]$$ where $U$ is a unitary on the composite Hilbert space. This can be written in the equivalent, often more convenient, operator-sum representation $$\label{eqn:quantum-operation}
\rho \mapsto \sum_{k = 1}^{K} E_k \rho E_k^\dagger$$ where the Kraus operators $E_k$ satisfy the completeness relation $$\sum_{k = 1}^{K} E_k^\dagger E_k = I .$$ Equation [(\[eqn:quantum-operation\])]{} is known as a quantum operation or quantum channel. Physically, it can be interpreted as randomly replacing the state $\rho$ by the (properly normalized) state $E_k \rho E_k^\dagger$ with probability ${{\rm Tr}}[ E_k \rho E_k ^\dagger ]$.
The quantum channels we study here are standard and often used in theoretical work as reasonable noise models [@nielsen_quantum_2010]. For readers familiar with these channels, the following definitions are solely to introduce our notation.
A widely used noise model is the Pauli channel.
\[def:pauli-channel\] The Pauli channel maps a single qubit state $\rho$ to ${{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}(\rho)$ defined by $$\label{eqn:pauli-channel}
{{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}(\rho) := p_I \rho + p_X X \rho X + p_Y Y \rho Y + p_Z Z \rho Z$$ where $p_I + p_X + p_Y + p_Z = 1$.
While the Pauli channel acts on a single qubit, it can be generalized to a $d$-dimensional Hilbert space via the *Weyl channel* $$\label{eqn:weyl-channel-def}
\mathcal{E}_{p}^{\text{W}} (\rho) := \sum_{k, l = 0}^{d - 1} p_{kl} W_{kl} \rho W_{kl}^\dagger$$ where $p_{kl}$ are probabilities and the Weyl operators are $$W_{kl} := \sum_{m = 0}^{d - 1} e^{2 \pi i m k / d} |m{\rangle}{\langle}m + 1| .$$ For $d = 2$, Eqn. [(\[eqn:weyl-channel-def\])]{} reduces to Eqn. [(\[eqn:pauli-channel\])]{}.
Two special cases of the Pauli channel are the bit-flip and phase-flip (dephasing) channel.
\[def:bit-flip-channel\] The bit-flip channel maps a single qubit state $\rho$ to ${{\mathcal{E}}_p^{\text{deph}}}(\rho)$ defined by $$\label{eqn:bitflip-channel}
{{\mathcal{E}}_p^{\text{BF}}}(\rho) := (1 - p) \rho + p X \rho X$$ where $0 \le p \le 1$.
While a bit-flip channel flips the computational basis state with probability $p$, the phase-flip channel introduces a relative phase with probability $p$.
\[def:dephasing-channel\] The phase-flip (dephasing) channel maps a single qubit state $\rho$ to ${{\mathcal{E}}_p^{\text{deph}}}(\rho)$ defined by $$\label{eqn:dephasing-channel}
{{\mathcal{E}}_p^{\text{deph}}}(\rho) := ( 1 - p) \rho + p Z \rho Z$$ where $0 \le p \le 1$.
Another special case of the Pauli channel is the depolarizing channel which occurs when each Pauli is equiprobable $p_X = p_Y = p_Z = p$ and $p_I = 1-3p$. This channel can be equivalently thought of as replacing the state $\rho$ by the maximally mixed state $I / 2$ with probability $p$.
\[def:depolarizing-channel\] The depolarizing channel maps a single qubit state $\rho$ to ${{\mathcal{E}}_p^{\text{depo}}}(\rho)$ defined by $$\label{eqn:depolarizing-channel}
{{\mathcal{E}}_p^{\text{depo}}}(\rho) := (1 - p) \rho + p I / 2$$ where $0 \le p \le 1$.
The $d = 2^n$-dimensional generalization of Def. \[def:depolarizing-channel\] is straightforward.
\[def:global-depolarizing-channel\] The global depolarizing channel maps an $n$-qubit state $\rho$ to ${{\mathcal{E}}_p^{\text{GD}}}(\rho)$ defined by $$\label{eqn:global-depolarizing-channel}
{{\mathcal{E}}_p^{\text{GD}}}(\rho) := (1 - p) \rho + p I / d$$ where $0 \le p \le 1$, $d = 2^n$, and $I \equiv I_d$ is the $d$-dimensional identity.
Finally, we consider amplitude damping noise which models decay from the excited state to the ground state via spontaneous emission of a photon.
\[def:amp-damp-channel\] The amplitude damping channel maps a single qubit state $\rho$ to ${{\mathcal{E}}_p^{\text{AD}}}(\rho)$ defined by $$\label{eqn:amp-damp-channel}
{{\mathcal{E}}_p^{\text{AD}}}(\rho) := \left[ \begin{matrix}
\rho_{00} + p \rho_{11} & \sqrt{1 - p} \rho_{01} \\
\sqrt{1 - p} \rho_{10} & (1 - p) \rho_{11} \\
\end{matrix} \right]$$ where $0 \le p \le 1$.
Now that we have introduced quantum noise and several common channels, we can define robust data encodings for quantum classifiers. We remark that we also consider measurement noise in Appendix \[app:further\_proofs\] but omit the definition and results from the main text for brevity.
Robust Data Encodings {#subsec:robustness-definition}
---------------------
In this section, we define robust points and robust data encodings of quantum classifiers. Informally, the intuition is as follows: the quantum classifier with decision rule [(\[eqn:decision\_rule\])]{} requires only a “coarse-grained” measurement to extract a predicted label. For example, with a single qubit classifier, all points in the “top” hemisphere of the Bloch sphere are predicted to have label $0$, while all points in the “bottom” hemisphere are predicted to have label $1$. The effect of noise is to shift points on the Bloch sphere, but certain points can get shifted such that they get assigned the same labels they would without noise. This is the idea of robustness, represented schematically in Fig. \[fig:single\_qubit\_binary\_classifier\_with\_noise\]. For classification purposes, we do not require completely precise measurements, only that the point remain “in the same hemisphere” in order to get the same predicted label.
Formally, we define a robust point as follows.
\[def:robust-point\] Let $\mathcal{E}$ be a quantum channel, and consider a (binary) quantum classifier with decision rule ${\hat{y}}$ as defined in [(\[eqn:decision\_rule\])]{}. We say that the state ${\rho_{{\boldsymbol{x}}}}\in \mathcal{D}_n$ encoding a data point ${\boldsymbol{x}} \in \mathcal{X}$ is a *robust point* of the quantum classifier if and only if $$\label{eqn:robust-point-definition}
{\hat{y}}[ \mathcal{E} ( {\tilde{\rho}_{{\boldsymbol{x}}}})] = {\hat{y}}[{\tilde{\rho}_{{\boldsymbol{x}}}}]$$ where ${\tilde{\rho}_{{\boldsymbol{x}}}}$ is the processed state via [(\[eqn:evolution-formal\])]{}.
As mentioned, for the purpose of classification, Eqn. [(\[eqn:robust-point-definition\])]{} is a well-motivated and reasonable definition of robustness. We remark that [(\[eqn:robust-point-definition\])]{} is expressed in terms of probability; in practice, additional measurements may be required to reliably determine robustness[^3]. Further, we note that [(\[eqn:robust-point-definition\])]{} assumes that noise occurs only after the evolution ${\rho_{{\boldsymbol{x}}}}\mapsto {\tilde{\rho}_{{\boldsymbol{x}}}}$. While this may be a useful theoretical assumption, in practice noise happens throughout a quantum circuit. We can therefore consider robustness for an *ideal data encoding* as in Def. \[def:robust-point\], or for a *noisy data encoding* in which some noise process $\mathcal{E}_1$ occurs after encoding and another noise process $\mathcal{E}_2$ occurs after evolution: $$\label{eqn:robust-point-noisy-encoding}
{\hat{y}}[ \mathcal{E}_2 ( \mathcal{U} ( \mathcal{E}_1 ( {\rho_{{\boldsymbol{x}}}}) ) ) ] =
{\hat{y}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] .$$ For our results, we primarily consider [(\[eqn:robust-point-definition\])]{}, although we show robustness for [(\[eqn:robust-point-noisy-encoding\])]{} in some cases.
Robust points [(\[eqn:robust-point-definition\])]{} are related but not equivalent to (density operator) fixed points of a quantum channel, and can be considered an application-specific generalization of fixed points. In Sec. \[ssec:characterize\_robust\_points\], we characterize the set of robust points for example channels, and in Sec. \[subsec:existence-of-robust-encodings\] we use this connection to prove the existence of robust data encodings. For classification, we are concerned with not just one data point, but rather a set of points (e.g., the set $\mathcal{X}$ or training set [(\[eqn:labeled-data-for-classifier\])]{}). We therefore define the set of robust points, or robust set, in the following natural way.
\[def:robust-set\] Consider a (binary) quantum classifier with encoding $E : \mathcal{X} \rightarrow \mathcal{D}_n$ and decision rule ${\hat{y}}$ as defined in [(\[eqn:decision\_rule\])]{}. Let $\mathcal{E}$ be a quantum channel. The set of robust points, or simply *robust set*, is $$\label{eqn:robust-point-set-definition}
\mathcal{R} (\mathcal{E}, E, {\hat{y}}) := \left\{ {\boldsymbol{x}} \in \mathcal{X}: {\hat{y}}[ \mathcal{E} ( {\tilde{\rho}_{{\boldsymbol{x}}}})] = {\hat{y}}[{\tilde{\rho}_{{\boldsymbol{x}}}}] \right\}$$ where ${\tilde{\rho}_{{\boldsymbol{x}}}}$ is the processed state via [(\[eqn:evolution-formal\])]{} and $\rho_{{\boldsymbol{x}}} = E({\boldsymbol{x}})$.
While the robust set generally depends on the encoding $E$, there are cases in which $\mathcal{R}$ is independent of $E$. In this scenario, we say all encodings are robust to this channel. Otherwise, the size of the robust set (i.e., number of robust points) can vary based on the encoding, and we distinguish between two cases. If the robust set is the set of all possible points, we say that the encoding is *completely robust* to the given noise channel.
\[def:complete\_noise\_robustness\] Consider a (binary) quantum classifier with encoding $E$ and decision rule ${\hat{y}}$ as defined in [(\[eqn:decision\_rule\])]{}. Let ${\boldsymbol{x}} \in \mathcal{X}$ and let $\mathcal{E}$ be a quantum channel. We say that $E$ is a *completely robust data encoding* for the quantum classifier if and only if $$\label{eqn:completely-robust-encoding-condition}
\mathcal{R}(\mathcal{E}, E, {\hat{y}}) = \mathcal{X} .$$
We note that in practice (e.g., for numerical results), complete robustness is determined relative to the training set [(\[eqn:labeled-data-for-classifier\])]{}. That is, we empirically observe that $E$ is a completely robust data encoding if and only if $$\mathcal{R}(\mathcal{E}, E, {\hat{y}}) = \{{\boldsymbol{x}}_i\}_{i = 1}^{M} .$$
Complete robustness can be a strong condition, so we also consider a partially robust data encoding, defined as follows.
\[def:partial\_noise\_robustness\] Consider a (binary) quantum classifier with encoding $E$ and decision rule ${\hat{y}}$ as defined in [(\[eqn:decision\_rule\])]{}. Let ${\boldsymbol{x}} \in \mathcal{X}$ and let $\mathcal{E}$ be a quantum channel. We say that $E$ is a *partially robust data encoding* for the quantum classifier if and only if $$\label{eqn:partially-robust-encoding-condition}
\mathcal{R}(\mathcal{E}, E, \hat{y}) \subsetneq \mathcal{X} .$$
Similar to complete robustness, partial robustness is determined in practice relative to the training set. For $0 \le \delta \le 1$, we say that $E$ is a $\delta$-robust data encoding if and only if $$\label{eqn:delta_robust_encoding_condition}
| \mathcal{R}(\mathcal{E}, E, \hat{y}) | = \delta M$$ where $|\cdot|$ denotes cardinality so that $ | \mathcal{R}(\mathcal{E}, E, \hat{y}) | \in [M]$.
Analytic Results {#sec:noise_robustness}
================
Using the definitions from Sec. \[sec:definitions\], we now state and prove results about data encodings. First, we show that different encodings lead to different classes of decision boundaries in Sec. \[ssec:classes\_learnable\_decision\_boundaries\]. Next, we characterize the set of robust points for example quantum channels in Sec. \[ssec:characterize\_robust\_points\]. In Sec. \[subsec:robustness-results\], we prove several robustness results for different quantum channels, and in Sec. \[subsec:existence-of-robust-encodings\] we discuss the existence of robust encodings as well as an observed tradeoff between learnability and robustness. Finally, in Sec. \[ssec:fidelity\_bounds\], we prove an upper bound on the number of robust points in terms of fidelities between noisy and noiseless states.
Classes of Learnable Decision Boundaries {#ssec:classes_learnable_decision_boundaries}
----------------------------------------
We defined several different encodings in Sec. \[sec:data-encodings\] and discussed differences in the state preparation circuits which realize the encodings. Here, we show that different encodings lead to different sets of decision boundaries for the quantum classifier, thereby demonstrating that the success of the quantum classifier in Def. \[def:binary-quantum-classifier\] depends crucially on the data encoding [(\[eqn:qubit-encoding-general\])]{}.
The decision boundary according to the decision rule [(\[eqn:decision\_rule\])]{} is implicitly defined by $$\label{eqn:decision_boundary_defining rule}
{{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] = 1 / 2 .$$ Consider a single qubit encoding [(\[eqn:qubit-encoding-general\])]{} so that $${\rho_{{\boldsymbol{x}}}}= \left[ \begin{matrix}
f(x_1, x_2)^2 & f(x_1, x_2) g(x_1, x_2)^* \\
f(x_1, x_2) g(x_1, x_2) & |g(x_1, x_2)|^2
\end{matrix} \right]$$ where we assumed without loss of generality that $f$ is real valued. Let the unitary $U$ such that ${\tilde{\rho}_{{\boldsymbol{x}}}}= U {\rho_{{\boldsymbol{x}}}}U^\dagger$ have matrix elements $U_{ij}$. Then, one can write the decision boundary [(\[eqn:decision\_boundary\_defining rule\])]{} as (see [(\[eqn:useful-00mtx-elt\])]{}) $$\label{eqn:projector_expanded}
|U_{00}|^2 f^2 + 2 {\text{Re}}[ U_{00}^* U_{01} f g] + |U_{01}|^2 |g|^2 = 1 / 2$$ where we have let $f = f(x_1, x_2)$ and $g = g(x_1, x_2)$ for brevity. Eqn. [(\[eqn:projector\_expanded\])]{} implicitly defines the decision boundary in terms of the data encoding $f$ and $g$. The unitary matrix elements $U_{ij}$ act as hyperparameters to define a class of learnable decision boundaries.
Eqn. [(\[eqn:projector\_expanded\])]{} can be solved numerically for different encodings, and we do so in Sec. \[ssec:classes-decision-boundaries-numerical\] (Fig. \[fig:random\_decision\_boundaries\_encodings\]) to visualize decision boundaries for single qubit classifiers. At present, we can proceed further analytically with a few inconsequential assumptions to simplify the equations.
For the wavefunction encoding, we have $f(x_1, x_2) = x_1$ and $g(x_1, x_2) = x_2$. Suppose for simplicity that matrix elements $U_{00} \equiv a$ and $U_{01} \equiv b$ are real. Then, Eqn. [(\[eqn:projector\_expanded\])]{} can be written $$\label{eqn:learnable-decision-boundary-wavefunction}
(a x_1 + b x_2)^2 = 1/2 ,$$ which defines a line $x_2 = x_2(x_1)$ with slope $-a / b$ and intercept $1 / \sqrt{2} b$. Thus, a single qubit classifier in Def. \[def:binary-quantum-classifier\] which uses the wavefunction encoding [(\[eqn:wavefunction\_encoding\_amplitude\])]{} can learn decision boundaries that are straight lines.
Now consider the dense angle encoding [(\[eqn:dae-general\])]{} on a single qubit, for which $f(x_1, x_2) = \cos(\pi x_1)$ and $g(x_1, x_2) = e^{2 \pi i x_2} \sin (\pi x_1)$. Supposing again that matrix elements $U_{00} \equiv a$ and $U_{01} \equiv b$ are real, we can write [(\[eqn:projector\_expanded\])]{} as $$\begin{gathered}
a^2 \cos^2 \pi x_1 + 2 a b \cos \pi x_1 \sin \pi x_1 \cos 2 \pi x_2 \\+ b^2 \sin^2 \pi x_1 = 1/2 .\end{gathered}$$ This can be rearranged to $$\label{eqn:learable_decision_boundary_dae}
\cos 2 \pi x_2 = \frac{1 - 2 a^2 + (2a^2 - 2b^2) \sin^2 \pi x_1}{a b \sin 2 \pi x_1} ,$$ which defines a class of sinusoidal functions $x_2 = x_2(x_1)$. (See Sec. [(\[ssec:classes-decision-boundaries-numerical\])]{} and Fig. \[fig:random\_decision\_boundaries\_encodings\].)
The different decision boundaries defined by [(\[eqn:learnable-decision-boundary-wavefunction\])]{} and [(\[eqn:learable\_decision\_boundary\_dae\])]{} emphasize the effect that encoding has on learnability. A classifier may have poor performance due to its encoding, and switching the encoding may lead to better results. We note that a similar phenomenon occurs in classical machine learning — a standard example being that a dot product kernel cannot separate data on a spiral, but a Gaussian kernel can. It may not be clear *a priori* what encoding to use (similarly in classical machine learning with kernels), but different properties of the data may lead to educated guesses. We note that Lloyd *et al.* [@lloyd_quantum_2020] consider training over hyperparameters to find good encodings, and we introduce a similar idea in [Section \[ssec:encoding\_learn\_alg\]]{} to find good *robust* encodings.
In Sec. \[ssec:classes-decision-boundaries-numerical\], we numerically evaluate decision boundaries for additional single-qubit encodings, as well as two-qubit encodings, to further illustrate the differences that arise from different encodings.
Characterization of Robust Points {#ssec:characterize_robust_points}
---------------------------------
For a given quantum channel $\mathcal{E}$, it is a standard exercise to characterize the set of density operator fixed points, i.e., states $\rho \in \mathcal{D}_n$ such that $$\label{eqn:fixed-point-definition}
\mathcal{E}(\rho) = \rho .$$ In this section, we characterize the set of robust points for example quantum channels. This demonstrates the relationship between robust points and fixed points which we further elaborate on in Sec. \[subsec:existence-of-robust-encodings\]. We remark that the characterizations similar to the ones in this Section may be of independent interest from a purely theoretical perspective, as robust points can be considered a type of generalized fixed point, or symmetry, of quantum channels.
The pure states which are fixed points of the dephasing channel [(\[eqn:dephasing-channel\])]{} are $\Pi_0 := |0{\rangle}{\langle}0|$ and $\Pi_1 := |1{\rangle}{\langle}1|$, and $$\label{eqn:fixed-points-of-dephasing-channel}
\rho = a \Pi_0 + b \Pi_1$$ with $a + b = 1$ is the general mixed-state density operator fixed point. In contrast, let us now consider the robust points of the same dephasing channel, which satisfy $$\label{eqn:robust-condition-for-dephasing}
{\hat{y}}[ {{\mathcal{E}}_p^{\text{deph}}}(\rho) ] = {\hat{y}}[\rho]$$ instead of [(\[eqn:fixed-point-definition\])]{}. Certainly the state in Eqn. [(\[eqn:fixed-points-of-dephasing-channel\])]{} will satisfy [(\[eqn:robust-condition-for-dephasing\])]{} — i.e., any fixed point is a robust point — but the set of robust points may contain more elements. To completely characterize the robust set, we seek the set of $\rho \in \mathcal{D}_2$ such that $$\label{eqn:robust-condition-for-dephasing-inequality1}
{{\rm Tr}}[ \Pi_0 \rho ] \ge 1/2 \implies {{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{deph}}}(\rho)] \ge 1 / 2$$ and $$\label{eqn:robust-condition-for-dephasing-inequality2}
{{\rm Tr}}[ \Pi_0 \rho ] < 1/2 \implies {{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{deph}}}(\rho)] < 1 / 2 .$$ Using simple properties of the trace and Pauli matrices (see Appendix \[app:useful-formulae\] if desired), we can write $${{\rm Tr}}[\Pi_0 {{\mathcal{E}}_p^{\text{deph}}}(\rho)] = (1 - p) {{\rm Tr}}[\Pi_0 \rho] + p {{\rm Tr}}[\Pi_0 Z \rho Z] = {{\rm Tr}}[ \Pi_0 \rho ].$$ Thus [(\[eqn:robust-condition-for-dephasing-inequality1\])]{} and [(\[eqn:robust-condition-for-dephasing-inequality2\])]{} are satisfied for all density operators $\rho \in \mathcal{D}_{2}$. That is, every data point ${\boldsymbol{x}} \in \mathcal{X}$ is a robust point of the dephasing channel (independent of the encoding) for the quantum classifier in Def. \[def:binary-quantum-classifier\].
Consider now an amplitude damping channel [(\[eqn:amp-damp-channel\])]{} with $p = 1$, for which the only fixed point is the pure state $\Pi_0$. By evaluating $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{AD}}}(\rho) ] = (1 - p) {{\rm Tr}}[ \Pi_0 \rho ] + p ,$$ we see that a robust point $\sigma$ must satisfy ${{\rm Tr}}[\Pi_0 \sigma] = 1$. That is, the only robust point is $\Pi_0$, and in this case the set of robust points is identical to the set of fixed points. The previous two examples illustrate how to find the robust points of a quantum channel, and the relationship between robust points and fixed points for the given channels. As expected from [(\[eqn:robust-point-definition\])]{} and [(\[eqn:fixed-point-definition\])]{}, these examples confirm that $$\label{eqn:fixed-points-are-subset-of-robust-points}
\mathcal{F}(\mathcal{E}) \subseteq \mathcal{R}(E, \mathcal{E}, \hat{y})$$ where $\mathcal{F}(\mathcal{E})$ denotes the set of fixed points of $\mathcal{E}$. In Sec. \[subsec:existence-of-robust-encodings\], we use this connection to generalize the above discussion and prove the existence of robust data encodings.
Robustness Results {#subsec:robustness-results}
------------------
In this section, we state and prove results on robust encodings. In particular, we prove robustness results for Pauli, depolarizing and amplitude damping channels, given certain conditions on the noise parameters in each channel.
First, we consider when robustness can be achieved for a Pauli channel.
\[thm:robustness\_pauli\_noise\_xy\] Let ${{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}$ be a Pauli channel [(\[eqn:pauli-channel\])]{} and consider a quantum classifier on data from the set $\mathcal{X}$. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_2$, we have complete robustness $$\mathcal{R} ({{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}, E, {\hat{y}}) = \mathcal{X}$$ if $p_X + p_Y \le 1/2$. (Recall that ${\boldsymbol{p}} = [p_I, p_X, p_Y, p_Z]$.)
The predicted label in the noisy case is identical to [(\[eqn:decision\_rule\])]{} with ${\tilde{\rho}_{{\boldsymbol{x}}}}$ replaced by ${{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})$. That is, $$\label{eqn:classification-scheme-pauli-noise}
{\hat{y}}[ {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}({\tilde{\rho}_{{\boldsymbol{x}}}}) ] = \begin{cases}
0 \qquad \text{if } {{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] \ge 1 / 2 \\
1 \qquad \text{otherwise}
\end{cases} .$$ By definition [(\[eqn:pauli-channel\])]{}, we have $$\label{eqn:pauli-proof1}
\begin{split}
{{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] = p_I & {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + p_X {{\rm Tr}}[ \Pi_0 X {\tilde{\rho}_{{\boldsymbol{x}}}}X] \\
+ \ p_Y {{\rm Tr}}& [ \Pi_0 Y {\tilde{\rho}_{{\boldsymbol{x}}}}Y] + p_Z {{\rm Tr}}[ \Pi_0 Z {\tilde{\rho}_{{\boldsymbol{x}}}}Z] .
\end{split}$$ Using straightforward substitutions (Appendix \[app:useful-formulae\]), we may write [(\[eqn:pauli-proof1\])]{} as $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] = (p_I + p_Z) {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + (p_X + p_Y) {{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] .$$ By resolution of the identity $$1 = {{\rm Tr}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] = {{\rm Tr}}[\Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + {{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] ,$$ we come to the simplified expression $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] = \left[ 1 - 2 \nu \right] {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + \nu .$$ where $\nu := p_X + p_Y$.
Suppose the noiseless classification is ${\hat{y}}= 0$ so that ${{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 1 / 2$. Since $\nu \le 1/2$, we have $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] \ge \left[ 1 - 2 \nu \right] \frac{1}{2} + \nu = \frac{1}{2}$$ Hence, classification of data points with label ${\hat{y}}= 0$ is robust for any encoding. Suppose the noiseless classification is ${\hat{y}}= 1$ so that ${{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] < 1 / 2$. Since $\nu \le 1/2$, we have $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] < \left[ 1 - 2 \nu \right] \frac{1}{2} + \nu = \frac{1}{2}.$$ Hence, classification of data points with label ${\hat{y}}= 1$ is also robust for any encoding.
Returning to the condition, $p_X + p_Y \le 1/2$, one can imagine a NISQ computer in which either $p_X$ or $p_Y$ were large enough such that this condition is not satisfied. In this regard, we note two things. The first is that if this condition is *not* satisfied, then not every encoding strategy will be robust to the Pauli channel in this model. In particular, the set of robust points will now be *dependent* on the encoding strategy. This is similar to the behavior of the amplitude damping channel (which we demonstrate shortly), and we illustrate in [Section \[sec:numerical\_results\]]{}. Secondly, the requirement $p_X + p_Y \le 1/2$ appears because the decision rule uses a measurement in the computational basis. In this case, we can still achieve robustness by using a modified decision rule which measures in a different basis.
\[corr:pauli\_x\_robustness\] Consider a quantum classifier on data from the set $\mathcal{X}$ with modified decision rule $$\label{eqn:classification-scheme-measure-hadamard-basis}
\hat{z} [ {\tilde{\rho}_{{\boldsymbol{x}}}}] = \begin{cases}
0 \qquad \text{if } {{\rm Tr}}[ \Pi_+ {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 1 / 2 \\
1 \qquad \text{otherwise}
\end{cases} .$$ Here, $\Pi_+ := |+{\rangle}{\langle}+ |$ is the projector onto the $+1$ eigenstate $|+{\rangle}$ of Pauli $X$. Then, for any $E: \mathcal{X} \rightarrow \mathcal{D}_2$, $$\mathcal{R} ({{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}, E, \hat{z}) = \mathcal{X}$$ for a Pauli channel ${{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}$ such that $p_Y + p_Z \le 1/2$ .
The proof mimics that of Theorem \[thm:robustness\_pauli\_noise\_xy\]. We note that the analogous statement for measurements in the $Y$-basis also holds. These results suggest that device-specific encoding strategies may be important for achieving robustness in practice on NISQ computers. Theorem \[thm:robustness\_pauli\_noise\_xy\] also implies the following result for dephasing noise, which is a Pauli channel with $p_X = p_Y = 0$.
\[thm:robustness-single-qubit-dephasing-noise\] Let ${{\mathcal{E}}_p^{\text{deph}}}$ be a dephasing channel [(\[eqn:dephasing-channel\])]{}, and consider a quantum classifier on data from the set $\mathcal{X}$. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_2$, $$\mathcal{R} ({{\mathcal{E}}_p^{\text{deph}}}, E, {\hat{y}}) = \mathcal{X} .$$
This result follows immediately from the discussion of the dephasing channel in the above Section \[ssec:characterize\_robust\_points\].
Similar to Corollary \[corr:pauli\_x\_robustness\], we can consider a modified decision rule to achieve robustness for a bit-flip channel.
\[corr:bit-flip-robustness\] Consider a quantum classifier on data from the set $\mathcal{X}$ with modified decision rule $\hat{z}$ defined in Eqn. [(\[eqn:classification-scheme-measure-hadamard-basis\])]{}. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_2$, $$\mathcal{R} ({{\mathcal{E}}_p^{\text{BF}}}, E, \hat{z}) = \mathcal{X} .$$
We note that a decision rule which measures in the $Y$-basis yields robustness to combined bit/phase-flip errors. (That is, the error channel $\mathcal{E}(\rho) = (1 - p) \rho + p Y \rho Y$.) v
We now consider robustness for depolarizing noise [(\[eqn:depolarizing-channel\])]{}. A simple calculation shows that $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{depo}}}({\tilde{\rho}_{{\boldsymbol{x}}}}) ] = p / 2 + (1 - p) {{\rm Tr}}[ \Pi_0 \rho ] .$$ If ${\hat{y}}= 0$ so that ${{\rm Tr}}[ \Pi_0 \rho ] \ge 1/2$, then we have that ${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{depo}}}({\tilde{\rho}_{{\boldsymbol{x}}}}) ] \ge 1/2$. Similarly for the case ${\hat{y}}= 1$. Thus, we have shown the following.
\[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_single\_qubit\] Let ${{\mathcal{E}}_p^{\text{depo}}}$ be a depolarizing channel [(\[eqn:depolarizing-channel\])]{}, and consider a quantum classifier on data from the set $\mathcal{X}$. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_2$, $$\mathcal{R} ({{\mathcal{E}}_p^{\text{depo}}}, E, {\hat{y}}) = \mathcal{X} .$$
We remark that Theorem \[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_single\_qubit\] holds with measurements in any basis, not just the computational basis. Further, we will soon generalize this result to (i) multi-qubit classifiers and (ii) noisy data encoding [(\[eqn:robust-point-noisy-encoding\])]{}.
We now consider amplitude damping noise, for which the robust set $\mathcal{R}$ depends on the encoding $E$. From the channel definition [(\[eqn:amp-damp-channel\])]{}, it is straightforward to see that $$\label{eqn:amp-damp-simple-expansion}
{{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{AD}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] = {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + p {{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] .$$ Suppose first that the noiseless prediction is ${\hat{y}}= 0$ so that ${{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 1/2$. Then, certainly ${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{AD}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] \ge 1/2$ because $p \ge 0$ and ${{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 0$. Thus, the noisy prediction is always identical to the noiseless prediction when the noiseless prediction is ${\hat{y}}= 0$. This can be understood intuitively because an amplitude damping channel models the $|1\rangle \mapsto |0\rangle$ transition [@john_preskill_quantum_1998] which only increases the probability of the ground state.
Suppose now that the noiseless prediction is ${\hat{y}}= 1$. From [(\[eqn:amp-damp-simple-expansion\])]{}, we require that $${{\rm Tr}}[ \Pi_0 {{\mathcal{E}}_p^{\text{AD}}}( {\tilde{\rho}_{{\boldsymbol{x}}}})] = {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + p {{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] < 1/2$$ to achieve robustness. We use resolution of the identity $${{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] = 1 - {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}]$$ to arrive at the condition $$\label{eqn:amp-damp-robustness-condition}
{{\rm Tr}}[ \Pi_1 {\tilde{\rho}_{{\boldsymbol{x}}}}] > \frac{1}{2(1 - p)} .$$ Let ${\rho_{{\boldsymbol{x}}}}$ be given by the general qubit encoding [(\[eqn:qubit-encoding-general\])]{} so that [(\[eqn:amp-damp-robustness-condition\])]{} can be written (see [(\[eqn:useful-11mtx-elt\])]{}) $$|U_{10}|^2 f^2 + 2 {\text{Re}}[ U_{11}^* U_{10} f g^*] + |U_{11}|^2 |g|^2 > \frac{1}{2(1 - p)}$$ where $U_{ij}$ denote the optimal unitary matrix elements.
We have thus shown the following.
\[thm:robustness-amp-damp\] Consider a quantum classifier on data from the set $\mathcal{X}$, and let ${{\mathcal{E}}_p^{\text{AD}}}$ denote the amplitude damping channel [(\[eqn:amp-damp-channel\])]{}. Then, for any qubit encoding $E$ defined in [(\[eqn:qubit-encoding-general\])]{} which satisfies $$\label{eqn:amp_damp_robustness_condition_expanded_unitary}
|U_{10}|^2 f^2 + 2 {\text{Re}}[ U_{11}^* U_{10} f g^*] + |U_{11}|^2 |g|^2 > \frac{1}{2(1 - p)} ,$$ we have $$\mathcal{R} ({{\mathcal{E}}_{\mathbf{p}}^{\text{P}}}, E, {\hat{y}}) = \mathcal{X} .$$ If $E$ is not completely robust, the set of points ${\boldsymbol{x}}$ such that that [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{} holds define the partially robust set.
We note that [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{} depends on the optimal unitary $U$ as well as the encoding $E$. This is expected as the final state ${\tilde{\rho}_{{\boldsymbol{x}}}}$ has been processed by the QNN. In practice, since we do not know the optimal unitary parameters *a priori*, it remains a question of how large the (partially) robust set will for a given an encoding. To address this point, we discuss in Sec. \[ssec:encoding\_learn\_alg\] how training over hyperparameters in the encoding function can help find the robust region even after application of the *a priori* unknown optimal unitary. Additionally, in the next Section we discuss whether we can find an encoding which satisfies [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{}, or more generally whether a robust encoding exists for a given channel.
Given the robustness condition [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{} for the amplitude damping channel, it is natural to ask whether such an encoding exists. In Sec. \[subsec:existence-of-robust-encodings\], we show the answer is yes by demonstrating there always exists a robust encoding for any trace preserving quantum operation. This encoding may be trivial, which leads to the idea of a tradeoff between learnability and robustness. (See Sec. \[subsec:existence-of-robust-encodings\].)
We now consider global depolarizing noise on a multi-qubit classifier. It turns out that any encoding is completely robust to this channel applied at any point throughout the circuit. To clearly state the theorem, we introduce the following notation. First, let $$\label{eqn:short-notation-depo-channel}
{\mathcal{E}}_{p_i} (\rho) = p_i \rho + (1 - p_i) I_d / d$$ be shorthand for a global depolarizing channel with probability $p_i$. (Note $p_i$ and $1 - p_i$ are intentionally reversed compared to Def. \[def:global-depolarizing-channel\] to simplify the proof.) Then, let $$\label{eqn:depo-robustness-notation-of-unitary-noise-application}
\tilde{\rho}_{{\boldsymbol{x}}}^{(m)} \equiv \left[ \prod_{i = 1}^{m} U_i \circ {\mathcal{E}}_{p_i} \right] \circ {\rho_{{\boldsymbol{x}}}}$$ denote the state of the encoded point ${\rho_{{\boldsymbol{x}}}}$ after $m$ applications of a global depolarizing channel and unitary channel. For instance, $m = 1$ corresponds to $$U_1 \circ {\mathcal{E}}_{p_1} \circ {\rho_{{\boldsymbol{x}}}}\equiv U_1( {\mathcal{E}}_{p_1} ( {\rho_{{\boldsymbol{x}}}}) )$$ and $m = 2$ corresponds to $$U_2 \circ {\mathcal{E}}_{p_2} \circ U_1 \circ {\mathcal{E}}_{p_1} \circ {\rho_{{\boldsymbol{x}}}}\equiv U_2 ( {\mathcal{E}}_{p_2} ( U_1( {\mathcal{E}}_{p_1} ( {\rho_{{\boldsymbol{x}}}}) ) ) ) .$$ We remark that $U_i$ can denote any unitary in the circuit.
With this notation, we state the theorem as follows.
\[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_multiple\_qubits\] Consider a quantum classifier on data from the set $\mathcal{X}$ with decision rule $\hat{y}$ defined in Eqn. [(\[eqn:classification-scheme-measure-hadamard-basis\])]{}. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_n$, $$\mathcal{R} \left({{\mathcal{E}}_p^{\text{GD}}}, E, \hat{y} \right) = \mathcal{X} .$$ where ${{\mathcal{E}}_p^{\text{GD}}}$ denotes the composition of global depolarizing noise acting at any point in the circuit — i.e., such that the final state of the classifier is given by [(\[eqn:depo-robustness-notation-of-unitary-noise-application\])]{}.
To prove Theorem \[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_multiple\_qubits\], we use the following lemma.
\[lem:lemma-for-global-depo-proof\] The state in Eqn. [(\[eqn:depo-robustness-notation-of-unitary-noise-application\])]{} can be written as (adapted from [@sharma_noise_2020]) $$\label{eqn:lemma-for-global-depo-proof}
\tilde{\rho}_{{\boldsymbol{x}}}^{(m)} =
\prod_{i = 1}^{m} p_i U_m \cdots U_1 {\rho_{{\boldsymbol{x}}}}U_1^\dagger \cdots U_m^\dagger + \left( 1 - \prod_{i = 1}^{m} p_i \right) \frac{I_d}{d}$$ where $d = 2^n$ is the dimension of the Hilbert space.
Using the definition of the global depolarizing channel [(\[eqn:short-notation-depo-channel\])]{}, it is straightforward to evaluate $$\tilde{\rho}_{{\boldsymbol{x}}}^{(1)} = U_1 \circ {\mathcal{E}}_{p_1} \circ {\rho_{{\boldsymbol{x}}}}= p_1 U_1 {\rho_{{\boldsymbol{x}}}}U_1^\dagger + (1 - p_1) I_d / d .$$ Thus [(\[eqn:lemma-for-global-depo-proof\])]{} is true for $m = 1$. Assume [(\[eqn:lemma-for-global-depo-proof\])]{} holds for $m = k$. Then, for $k + 1$ we have $$\begin{aligned}
\tilde{\rho}_{{\boldsymbol{x}}}^{(k + 1)} &= U_{k + 1} \circ {\mathcal{E}}_{p_{k + 1}} \circ \tilde{\rho}_{{\boldsymbol{x}}}^{(k)} \\
&= p_{k + 1} U_{k + 1} \tilde{\rho}_{{\boldsymbol{x}}}^{(k)} U_{k + 1}^\dagger + (1 - p_{k + 1}) I_d / d .
\end{aligned}$$ The last line can be simplified to arrive at $$\begin{aligned}
\tilde{\rho}_{{\boldsymbol{x}}}^{(k + 1)} =
\prod_{i = 1}^{k + 1} p_i U_{k + 1} \cdots U_1 {\rho_{{\boldsymbol{x}}}}U_1^\dagger \cdots U_{k + 1}^\dagger \nonumber \\
+ \left( 1 - \prod_{i = 1}^{k + 1} p_i \right) I / d , \nonumber
\end{aligned}$$ which completes the proof.
We can now prove Theorem \[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_multiple\_qubits\] as follows. Let $l$ denote the total number of alternating unitary gates with depolarizing noise in the classifier circuit so that [(\[eqn:lemma-for-global-depo-proof\])]{} can be written $$\label{eqn:final-state-global-depo-circuit}
\tilde{\rho}_{{\boldsymbol{x}}}^{(l)} = \bar{p} {\tilde{\rho}_{{\boldsymbol{x}}}}+ (1 - \bar{p}) I / d.$$ Here, we have let $\bar{p} := \prod_{i = 1}^{l} p_i$ and noted that $U_l \cdots U_1 {\rho_{{\boldsymbol{x}}}}U_1^\dagger \cdots U_l^\dagger = {\tilde{\rho}_{{\boldsymbol{x}}}}$ is the final state of the noiseless circuit before measuring. Eqn. [(\[eqn:final-state-global-depo-circuit\])]{} is thus the final state of the noisy circuit before measuring. We can now evaluate $$\begin{aligned}
{{\rm Tr}}[ \Pi_0 \tilde{\rho}_{{\boldsymbol{x}}}^{(l)} ] = \bar{p} {{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] + (1 - \bar{p}) / 2\end{aligned}$$ where we have used ${{\rm Tr}}[ \Pi_0 I_d ] = 2^{d - 1}$. To prove robustness, suppose that ${\hat{y}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] = 0$ so that ${{\rm Tr}}[ \Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] \ge 1/2$. Then, $${{\rm Tr}}[ \Pi_0 \tilde{\rho}_{{\boldsymbol{x}}}^{(l)} ] \ge \bar{p} / 2 + (1 - \bar{p}) / 2 = 1 / 2$$ so that ${\hat{y}}[ \tilde{\rho}_{{\boldsymbol{x}}}^{(l)} ] = 0$. Similarly for the case ${\hat{y}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] = 1$, which completes the proof of Theorem \[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_multiple\_qubits\].
Thus, any encoding strategy exhibits complete robustness to global depolarizing noise. We remark again (see footnote on Page ) that our definition of robustness (Def. \[def:robust-point\]) is in terms of probability, meaning that more measurements for sampling may be required to reliably evaluate robustness. With this remark, we note an interesting connection to explain a phenomenon observed in recent literature: In Ref. [@grant_hierarchical_2018], the authors found that classification accuracy decreased under the presence of depolarizing noise. Theorem \[thm:robustness\_to\_depolarizing\_on\_unitary\_ansatz\_multiple\_qubits\] implies this was a feature exclusively of the finite shot noise used to obtain the predicted label.
While global depolarizing noise admits a clean robustness result for an arbitrary $d$-dimensional circuit, general channels can lead to complicated equations which are best handled numerically. We include several numerical results in Sec. \[sec:numerical\_results\], and we discuss avenues for proving more analytical results with certain classes of channels in future work in Sec. \[sec:conclusions\]. To close the present discussion, we highlight the special case of multi-qubit classifiers with “factorizable noise,” for which it is straightforward to apply previous results proved in this section.
In particular, suppose that ${\mathcal{E}}: \mathcal{D}_n \rightarrow \mathcal{D}_n$ is a noise channel which factorizes into single qubit channels, e.g. $$\label{eqn:factorizable-noise-channel-into-single-qubits-definition}
{\mathcal{E}}= {\mathcal{E}}_1 \otimes \cdots \otimes {\mathcal{E}}_n$$ where ${\mathcal{E}}_i : \mathcal{D}_2 \rightarrow \mathcal{D}_2$ for $i \in [n]$. Without loss of generality, let the classification qubit be the $n$th qubit. Then, if the processed state of the classification qubit is robust to the channel ${\mathcal{E}}_n$, the encoded state will be robust to the entire channel ${\mathcal{E}}$ in [(\[eqn:factorizable-noise-channel-into-single-qubits-definition\])]{}. This result, which is precisely stated and proved in Appendix \[app:factorizable-noise\], also holds for general $n - 1$ qubit channels which act on every qubit except the classification qubit. Although this is relatively straightforward, the result could be used as a building block to better understand more intricate robustness properties of quantum classifiers.
Existence of Robust Encodings {#subsec:existence-of-robust-encodings}
-----------------------------
In Sec. \[ssec:characterize\_robust\_points\], we considered example channels and characterized their robust points and fixed points. We found that the set of fixed points $\mathcal{F}(\mathcal{E})$ is always a subset of the robust set $\mathcal{R}(\mathcal{E}, E, {\hat{y}})$ in [(\[eqn:fixed-points-are-subset-of-robust-points\])]{}. Here, we use this connection to show that there always exists a robust encoding for a trace-preserving channel $\mathcal{E}$ (regardless of optimal unitary parameters which may appear in the robustness condition, e.g. [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{}).
(Existence of Fixed Points [@schauder_fixpunktsatz_1930; @nielsen_quantum_2010]) \[thm:schauder\_fixed\_point\] Any trace-preserving quantum operation has at least one density operator fixed point [(\[eqn:fixed-point-definition\])]{}.
Using this and the observation that $\mathcal{F}(\mathcal{E}) \subset \mathcal{R}(\mathcal{E}, E, {\hat{y}})$, we have the following existence theorem for robust encodings.
\[thm:robust\_encodings\_existence\] Given a data point ${\boldsymbol{x}} \in \mathcal{X}$, a trace-preserving quantum channel $\mathcal{E}$, and decision rule ${\hat{y}}$ defined in [(\[eqn:decision\_rule\])]{}, there exists an encoding $E$ such that $$\label{eqn:existenstence-robust-encoding}
{\hat{y}}[ \mathcal{E} ( E({\boldsymbol{x}}) ) ] = {\hat{y}}[ E({\boldsymbol{x}}) ] .$$
We note that the optimal unitary of the QNN affects the “location” of the robust set, but not the existence. We emphasize that Theorem \[thm:robust\_encodings\_existence\] is with respect to a single data point ${\boldsymbol{x}} \in \mathcal{X}$. As mentioned in Sec. \[subsec:robustness-definition\], it is more relevant for applications to consider the training set [(\[eqn:labeled-data-for-classifier\])]{} or entire set $\mathcal{X}$. Appropriately, one can ask whether a completely robust encoding (Def. \[def:complete\_noise\_robustness\]) exists for a given channel ${\mathcal{E}}$. This answer also turns out to be yes, but in a potentially trivial way.
In particular, suppose that there is a unique fixed point $\sigma$ of the channel ${\mathcal{E}}$, e.g. depolarizing noise or amplitude damping noise with $p = 1$. Then, consider the encoding $${\mathcal{E}}({\boldsymbol{x}}) = \sigma$$ for all ${\boldsymbol{x}} \in \mathcal{X}$. From a robustness perspective, this has the desirable property of complete robustness. From a machine learning perspective, however, this has very few desirable properties: all training data is mapped to the same point so that it is impossible to successfully train a classifier[^4] .
The previous example, while extreme, serves to illustrate the tradeoff between learnability and robustness. By “learnability,” we mean the ability of the classifier to predict correct labels (without regard to noise), and by robustness we mean that the prediction is the same with or without noise (without regard to correctness). The two links are schematically connected below: $$\label{eqn:learnability-robustness-links}
y[{\boldsymbol{x}}] \ \ \xleftrightarrow{\text{Learnability}} \ \ {\hat{y}}[ {\tilde{\rho}_{{\boldsymbol{x}}}}] \ \
\xleftrightarrow{\text{Robustness}} \ \ {\hat{y}}[ {\mathcal{E}}( {\tilde{\rho}_{{\boldsymbol{x}}}}) ]$$ The tradeoff we observe is that more learnability comes at the price of less robustness, and *vice versa*. See Sec. \[ssec:encoding\_learn\_alg\] for a discussion.
Upper Bounds on Partial Robustness {#ssec:fidelity_bounds}
----------------------------------
In this section, we consider a slightly modified binary quantum classifier which embeds the cost function in the circuit and computes the cost by measuring expectation values. In contrast to the classifier in Def. \[def:binary-quantum-classifier\], the output of this circuit is thus the cost $C$ instead of an individual predicted label ${\hat{y}}$. Correspondingly, the input to the circuit is all data points in the training set [(\[eqn:labeled-data-for-classifier\])]{} (using a “mixed state encoding” discussed below) instead of a single data point ${\boldsymbol{x}}$. Such a classifier was recently introduced by Cao *et al.* in Ref. [@cao_cost_2019] and presents an interesting framework to analyze in the context of noise, which we do in this Section.
Since the output of the circuit is the cost $C$ for all points instead of a predicted label ${\hat{y}}$ for an individual point, the definition of a single robust point does not immediately apply to this classifier. However, it is still natural to compare the noisy and noiseless outcomes — in the same spirit as robustness — by comparing the difference between the output cost $C_{\mathcal{E}}$ when some noise channel ${\mathcal{E}}$ occurs in the circuit to the output cost $C$ from an ideal (noiseless) circuit. In fact, we show this quantity provides an upper bound on the size of the partially robust set and therefore can be used as a proxy to assess robustness of different encodings.
To do so, we consider the indicator cost function $$\label{eqn:indicator_cost_over_dataset}
C := \frac{1}{M} \sum_{i = 1}^{M} \mathcal{I} (\hat{y}_i({\tilde{\rho}_{{\boldsymbol{x}}_i}}) \neq y_i) .$$ Here, the indicator $\mathcal{I}$ evaluates to the truth value of its argument — i.e., $\mathcal{I}({\hat{y}}_i \neq y_i) = 0$ if $y_i = {\hat{y}}_i$, else $1$. We note again that $C = C({\boldsymbol{\alpha}})$ is parameterized by some angles ${\boldsymbol{\alpha}}$ but we omit ${\boldsymbol{\alpha}}$ for brevity.
The indicator cost function [(\[eqn:indicator\_cost\_over\_dataset\])]{} relates naturally to the robust set in Def. \[def:robust-set\]. Even though we cannot say individually which points are robust, a decrease in the cost due to some noise channel implies that some points were misclassified (assuming we had perfect classification, in the absence of the channel). Hence, how much the noisy cost function decreases is a useful proxy of robustness. We quantify this as $$\label{eqn:change-in-cost}
\Delta_{\mathcal{E}}C := | C_{\mathcal{E}}- C |.$$ In the encoding strategy of Cao *et al* [@cao_cost_2019], each feature vector ${\boldsymbol{x}}$ is encoded along with its true label $y$ on an ancilla qubit as per $$\label{eqn:pure-state-of-mixed-state-encoding-for-embedded-cost-classifier}
{\sigma_{{\boldsymbol{x}}}}= E({\boldsymbol{x}}) \otimes |y{\rangle}{\langle}y| = {\rho_{{\boldsymbol{x}}}}\otimes |y{\rangle}{\langle}y| .$$ where $E$ is an encoding function. Then, the entire dataset [(\[eqn:labeled-data-for-classifier\])]{} is prepared in the mixed state $$\sigma = \frac{1}{M} \sum_{i = 1}^{M} {\rho_{{\boldsymbol{x}}_i}}\otimes |y_i{\rangle}{\langle}y_i| .$$ Such a mixed state encoding may not be reliably preparable on NISQ computers, but in principle could be prepared using purification or by probabilistically preparing one of the pure states [(\[eqn:pure-state-of-mixed-state-encoding-for-embedded-cost-classifier\])]{} — which could be amenable to NISQ computers depending on the encoding $E$. The QNN acts only on the “data subsystem” so that the evolved state before measurement is $$\label{eqn:mixed_state_over_data}
\tilde{\sigma} = \frac{1}{M} \sum_{i = 1}^{M} {\tilde{\rho}_{{\boldsymbol{x}}_i}}\otimes |y_i{\rangle}{\langle}y_i|$$ We now consider the application of a noisy channel ${\mathcal{E}}$ so that $$\label{eqn:mixed_state_over_data_noisy_with_label}
{\mathcal{E}}(\tilde{\sigma} ) = \frac{1}{M} \sum_{i=i}^M {\mathcal{E}}\left( {\tilde{\rho}_{{\boldsymbol{x}}_i}}\otimes {| \hspace{1pt} y_i \rangle \langle y_i \hspace{1pt} |} \right) .$$ While ${\mathcal{E}}$ could most generally act on the entire system, to match with previous analyses we assume that $$\label{eqn:mixed_state_over_data_noisy}
{\mathcal{E}}(\tilde{\sigma} ) = \frac{1}{M} \sum_{i=i}^M {\mathcal{E}}\left( {\tilde{\rho}_{{\boldsymbol{x}}_i}}\right) \otimes {| \hspace{1pt} y_i \rangle \langle y_i \hspace{1pt} |}$$ for simplicity. That is, we assume the true labels are invariant with respect to the noise channel[^5].
The cost[^6] $C$ can be evaluated by measuring the expectation of $$\label{eqn:observable_cost}
D := I^{\otimes n-1} \otimes Z_c \otimes Z_l$$ where $c$ and $l$ denote classification and label qubits, respectively. (See Ref. [@cao_cost_2019] for more details.) That is, the (noiseless) cost is given by $$\label{eqn:quantum_cost}
C = {{\rm Tr}}(D \tilde{\sigma}) ,$$ and the noisy cost is identical with $\tilde{\sigma}$ replaced by ${\mathcal{E}}(\tilde{\sigma})$.
We can now evaluate the change in cost due to noise [(\[eqn:change-in-cost\])]{} as (following [@gentini_noise-assisted_2019]) $$\begin{aligned}
\Delta_{\mathcal{E}}C &:= \left| C_{\mathcal{E}}- C \right| \nonumber \\[1.0ex]
&= \left| {{\rm Tr}}[ D ( {\mathcal{E}}(\tilde{\sigma} ) - \tilde{\sigma}) ] \right| \nonumber \\[1.0ex]
&\le ||D||_\infty || {\mathcal{E}}(\tilde{\sigma}) - \tilde{\sigma} ||_1 \nonumber \\[1.0ex]
&\le 2 \sqrt{ 1 - F({\mathcal{E}}(\tilde{\sigma}), \tilde{\sigma}) } . \label{eqn:fidelity_bound_mixed_state}\end{aligned}$$ Here, $F$ is the fidelity of states $\tau, \omega \in \mathcal{D}_n$ defined by $$\label{eqn:fidelity_defintion}
F(\tau, \omega) := {{\rm Tr}}\left[ \sqrt{ \sqrt{ \tau } \omega \sqrt {\tau } } \right] ^2 .$$ The third line in this derivation follows from Hölders inequality and the last line from the Fuchs-van de Graaf inequality [@gentini_noise-assisted_2019; @fuchs_cryptographic_1999]. We also used the fact that $||D||_\infty := \max_j | \lambda_j(D) | = 1$.
We can also derive an alternative inequality based on the average trace distance between the individual encoded states, namely $$\label{eqn:fidelity_bound_average}
\Delta_{\mathcal{E}}C {\leqslant}\frac{2}{M} \sum_{i=1}^M \sqrt{1-F({\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}_i}}), {\tilde{\rho}_{{\boldsymbol{x}}_i}})} .$$ A proof is included in Appendix \[app:further\_proofs\].
Due to our choice of cost function [(\[eqn:indicator\_cost\_over\_dataset\])]{}, the quantity $\Delta_{\mathcal{E}}C$ corresponds exactly to the $\delta$-robustness of the model in [Definition \[def:partial\_noise\_robustness\]]{} since a classification difference in a single point due to noise causes the error to increase by $1/M$. In particular, the quantity $\Delta_{\mathcal{E}}C$ is exactly the fraction of robust points in the dataset [(\[eqn:delta\_robust\_encoding\_condition\])]{}. Specifically, we have $$|\mathcal{R}(\mathcal{E}, E, \hat{y})| = M \Delta_{\mathcal{E}}C$$
Thus, Eqn. [(\[eqn:fidelity\_bound\_mixed\_state\])]{} provides an upper bound on how large the (partially) robust set can be, namely $$\label{eqn:partially_robust_set_bound}
|\mathcal{R}(\mathcal{E}, E, \hat{y})| {\leqslant}2 \sum_{i=1}^M \sqrt{1-F({\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}_i}}), {\tilde{\rho}_{{\boldsymbol{x}}_i}})} .$$ In Sec. \[ssec:fidelity\_analysis\_exp\], we use these inequalities to bound the size of the robust set for several different encodings on an example implementation.
Numerical Results {#sec:numerical_results}
=================
In this Section, we present numerical evidence to reinforce the theoretical results proved in Sec. \[sec:noise\_robustness\] and build on the discussions. In Sec. \[ssec:classes-decision-boundaries-numerical\], we show classes of learnable decision boundaries for example encodings, building on the previous discussion in Sec. \[ssec:classes\_learnable\_decision\_boundaries\]. We then plot the robust sets for partially robust encodings in Sec. \[subsec:robust-sets-for-partially-robust-encodings\] to visualize the differences that arise from different encodings. We also generalize some encodings defined in Sec. \[sec:data-encodings\] to include hyperparameters and study the effects. This leads us to attempt to train over these hyperparameters, and we present an “encoding learning algorithm” in Sec. \[ssec:encoding\_learn\_alg\] to perform this task. Finally, in Sec. \[ssec:fidelity\_analysis\_exp\] we compute upper bounds on the size of robust sets based on [Section \[ssec:fidelity\_bounds\]]{}. We note that we include code to reproduce all results in this Section at Ref. [@coyle_noiserobustclassifier_2020]. For all numerical results in the following sections related to single qubit classifier, we use three simple datasets; the first is the “moons” dataset from [scikit-learn]{}, [@pedregosa_scikit-learn_2011], and two we denote “vertical” and “diagonal”. Representative examples can be found in [Appendix \[app:numerical\_results\]]{}.
Decision Boundaries and Implementations {#ssec:classes-decision-boundaries-numerical}
---------------------------------------
In Sec. \[sec:data-encodings\], we defined an encoding [(\[eqn:encoding-formal\])]{} and gave several examples. In Sec. \[ssec:classes\_learnable\_decision\_boundaries\], we showed that a classifier with the wavefunction encoding [(\[eqn:wavefunction-encoding-density-matrix-onequbit\])]{} can learn decision boundaries that are straight lines, while the same classifier with the dense angle encoding [(\[eqn:dae-single-qubit\])]{} can learn sinusoidal decision boundaries. We show this in Fig. \[fig:random\_decision\_boundaries\_encodings\], and we build on this discussion in the remainder of this section.
![(Color online.) Examples of learnable decision boundaries for a single qubit classifier with the (a) dense angle encoding, (b) wavefunction encoding, and (c) superdense angle encoding where $\theta = \pi$ and $\phi = 2 \pi$. Colors denote class labels. The QNN used here consisted of an arbitrary single qubit rotation (see Fig. \[fig:specific\_classifier\_circuits\]) with random parameters.[]{data-label="fig:random_decision_boundaries_encodings"}](images/boundaries_from_encodings.pdf){width="\columnwidth"}
Figure \[fig:random\_decision\_boundaries\_encodings\](c) shows a “striped” decision boundary which was learned by a “superdense” angle encoding, defined below. The superdense encoding introduces a linear combination of features into the qubit (angle) encoding [(\[eqn:qubit\_encoding\_grant\])]{}.
\[def:sdae\] Let ${\boldsymbol{x}} = [x_1, ..., x_N]^T \in \mathbb{R}^N$ be a feature vector and ${\boldsymbol{\theta}}, {\boldsymbol{\phi}} \in \mathbb{R}^N$ be parameters. Then, the superdense angle encoding maps ${\boldsymbol{x}} \mapsto E({\boldsymbol{x}})$ given by $$\label{eqn:sdae-general}
|{\boldsymbol{x}}{\rangle}= \bigotimes_{i=1}^{\ceil*{N / 2}}\cos (\theta_i x_{2i-1} + \phi_i x_{2i}) {| 0 \rangle} + \cos (\theta_i x_{2i-1} + \phi_i x_{2i}) {| 1 \rangle} .$$
For a single qubit, the SDAE is $$\label{eqn:superdae_encoding_single_qubit}
|{\boldsymbol{x}}{\rangle}:= \cos \left(\theta x_1+ \phi x_2\right) |0{\rangle}+ \sin \left(\theta x_1+ \phi x_2\right)|1{\rangle}.$$ We observe that $\phi = 0$ recovers the qubit (angle) encoding [(\[eqn:qubit\_encoding\_grant\])]{} considered by [@stoudenmire_supervised_2016; @schuld_supervised_2018; @cao_cost_2019] and [(\[eqn:sdae-general\])]{} encodes two features per qubit.
We note that Def. \[def:sdae\] includes hyperparameters $\mathbf{\theta}$ and $\mathbf{\phi}$. The reason for this will become clear in Sec. \[ssec:encoding\_learn\_alg\] when we consider optimizing over encoding hyperparameters to increase robustness. As previously mentioned, a similar idea was investigated by Lloyd *et al.* in Ref. [@lloyd_quantum_2020] for the purpose of (in our notation) learnability.
As a final example to explore the importance of encodings, we consider an example implementation on a standard dataset using different encodings. The dataset we consider is the Iris flower dataset [@fisher_use_1936] in which each flower is described by four features so that ${\boldsymbol{x}} \in \mathbb{R}^4$. The original dataset includes three classes (species of flower) but we only consider two for binary classification. A quantum classifier using the qubit angle encoding [(\[eqn:qubit\_encoding\_grant\])]{} and a tree tensor network (TTN) ansatz was considered in Ref. [@grant_hierarchical_2018]. Using this encoding and QNN, the authors were able to successfully classify all points in the dataset.
Since the angle encoding maps one feature into one qubit, a total of four qubits was used for the example in [@grant_hierarchical_2018]. Here, we consider encodings which map two features into one qubit and thus require only two qubits. Descriptions of the encodings, QNN ansatze, and overall classification accuracy are shown in Table \[table:two\_qubit\_iris\_class\_accuracy\].
------------------ --------- -------------------- ------------------- --------------
**Encoding** **QNN** $\boldsymbol{N_P}$ $\boldsymbol{n} $ **Accuracy**
\[0.5ex\] Angle TTN 7 4 100%
Dense Angle $U(4)$ 12 2 100%
Wavefunction $U(4)$ 12 2 100%
Superdense Angle $U(4)$ 12 2 77.6%
------------------ --------- -------------------- ------------------- --------------
: Classification accuracy achieved on the Iris dataset using different encodings and QNNs in the quantum classifier. The top row is from Ref. [@grant_hierarchical_2018] and the remaining rows are from this work. The heading $N_p$ indicates number of parameters in the QNN and $n$ is the number of qubits in the classifier. The accuracy is the overall performance using a train-test ratio of $80\%$ on classes $0$ and $2$. (See Ref. [@coyle_noiserobustclassifier_2020] for full implementation details.)[]{data-label="table:two_qubit_iris_class_accuracy"}
As can be seen, we are able to achieve 100% accuracy using the wavefunction and dense angle encoding. For the SDAE, the accuracy drops. Because the SDAE performs worse than other encodings, this implementation again highlights the importance of encoding on learnability. Additionally, the fact that we can use two qubits instead of four highlights the importance of encodings from a resource perspective. Specifically, NISQ applications with fewer qubits are less error prone due to fewer two-qubit gates, less crosstalk between qubits, and reduced readout errors. The reduction in the number of qubits here due to data encoding parallels, e.g., the reduction in the number of qubits in quantum chemistry applications due to qubit tapering [@bravyi_tapering_2017]. For QML, such a reduction is not always beneficial as the encoding may require a significantly higher depth. For this implementation, however, the dense angle encoding has the same depth as the angle encoding, so the reduction in number of qubits is meaningful.
Robust Sets for Partially Robust Encodings {#subsec:robust-sets-for-partially-robust-encodings}
------------------------------------------
In Sec. \[subsec:robustness-results\], we proved conditions under which an encoding is robust to a given error channel. Typically in practice, encodings may not completely satisfy such robustness criteria, but will exhibit partial robustness — i.e., some number of training points will be robust, but not all. In this section, we characterize such robust sets for different partially robust encodings. We emphasize two points that (i) the number of robust points is different for different encodings, and (ii) the “location” of robust points is different for different encodings.
To illustrate the first point, we consider amplitude damping noise — which has robustness condition [(\[eqn:amp\_damp\_robustness\_condition\_expanded\_unitary\])]{} — for two different encodings: the dense angle encoding and the wavefunction encoding. For each, we use a dataset which consists of 500 points in the unit square separated by a vertical decision boundary at $x_1 = 0.5$.
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The results for the dense angle encoding are shown in Fig. \[fig:dae\_encoding\_linear\_boundary\]. Without noise, the classifier is able to reach an accuracy of $\sim 99\%$ on the training set. When the amplitude damping channel with strength $p = 0.2$ is added, the test accuracy reduces to $\sim 78\%$. This encoding is thus partially robust, and the set of robust points is shown explicitly in Fig. \[fig:dae\_encoding\_linear\_boundary\](c).
The results for the wavefunction encoding are shown in Fig. \[fig:wf-encoding-linear-boundary\]. Here, the classifier is only able to reach $\sim 82\%$ test accuracy without noise. When the same amplitude damping channel with strength $p = 0.4$ is added, the test accuracy drops to $\sim 43\%$. We consider also the effect of amplitude damping noise with strength $p = 0.2$ in Fig. \[fig:wf-encoding-linear-boundary\], for which the classifier achieves test accuracy $\sim 61\%$. The robust set for both channels is also shown in Fig. \[fig:wf-encoding-linear-boundary\].
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Encoding Learning Algorithm {#ssec:encoding_learn_alg}
---------------------------
For this purpose, we introduce an “encoding learning algorithm” to try and search for good encodings. The goal is crudely illustrated in Fig. \[fig:single\_qubit\_binary\_classifier\_encoding\_learning\]. As mentioned above, Ref. [@lloyd_quantum_2020] trains over hyperparameters using the re-uploading structure of Ref. [@perez-salinas_data_2019] to increase learnability. Here, the encoding learning algorithm adapts to noise to increase robustness. We note the distinction that in our implementations we train the unitary in a noiseless environment and do not alter its parameters during the encoding learning algorithm.
![(Color online.) Cartoon illustration of the encoding learning algorithm with a single qubit classifier. In (a), a preset encoding with no knowledge of the noise misclassifies a large number of points. In (b), the encoding learning algorithm detects misclassifications and tries to adjust points to achieve more robustness, attempting to encode into the robust set for the channel.[]{data-label="fig:single_qubit_binary_classifier_encoding_learning"}](images/robust_encoding_learning_image_white_v3.pdf){width="\columnwidth"}
The encoding learning algorithm is similar to the Rotoselect algorithm [@ostaszewski_quantum_2019] which is used to learn circuit structure. For each function pair $(f_j, g_j)$ from a discrete set of parameterized functions $\{f_i({\boldsymbol{\theta}}_i), g_i({\boldsymbol{\theta}}_i)\}_{i = 1}^{K}$ we train the unitary $U({\boldsymbol{\alpha}})$ to minimize the cost while keeping the encoding (hyper)parameters ${\boldsymbol{\theta}}_j$ fixed. Next, we add a noise channel ${\mathcal{E}}$ which causes some points to be misclassified. Now, we optimize the encoding parameters ${\boldsymbol{\theta}}_j$ in the noisy environment. For this optimization, the same cost function is used, and the goal is to further decrease the cost (and hence increase the set of robust points) by varying the encoding hyperparameters. Pseudocode for the algorithm is shown in Appendix \[app:numerical\_results\].
We test the algorithm on linearly separable and non-linearly separable datasets in Fig. \[fig:encoding\_learning\_algorithm\]. In particular, we use three different encodings on three datasets. The encodings used are the dense angle encoding, superdense angle encoding, and “generalized wavefunction encoding (GWFE)” given by
$$\label{eqn:gwf_single_qubit}
| {\boldsymbol{x}} {\rangle}:= \frac{\sqrt{1+\theta x_2^2}}{||{\boldsymbol{x}}||_2} x_1 |0{\rangle}+ \frac{\sqrt{1-\theta x_1^2}}{||{\boldsymbol{x}}||_2} x_2 |1{\rangle}.$$
for a single qubit.
Using these encodings and the datasets shown in Appendix \[app:numerical\_results\], we study performance for the noiseless case, noisy case, and the effect of the encoding learning algorithm. We observe that the algorithm is not only capable of recovering the noiseless classification accuracy achieved, but is actually able to outperform it in some cases, as can be seen in Fig. \[fig:encoding\_learning\_algorithm\].
Finally, we consider the discussion in Sec. \[subsec:existence-of-robust-encodings\] about the tradeoff between learnability and robustness. We make this quantitative in Fig. \[fig:dae\_encoding\_learnability\_versus\_robustness\] by plotting accuracy (percent learned correctly) and robustness against hyperparameters $\theta$ and $\phi$ in a generalized dense angle encoding $$|{\boldsymbol{x}}{\rangle}= \cos (\theta x_{1}) {| 0 \rangle} + e^{i \phi x_{2}} \sin (\theta x_{1}){| 1 \rangle} .$$ More specifically, in Fig. \[fig:dae\_encoding\_learnability\_versus\_robustness\], we illustrate how the noise affects the hyperparameters, $\theta^*$ and $\phi^*$, which give maximal classification *accuracy* in both the noiseless and noisy environments, and also those which give maximal *robustness* (in the sense of [Definition \[def:partial\_noise\_robustness\]]{}). Fig. \[fig:dae\_encoding\_learnability\_versus\_robustness\](a), shows the percentage misclassified in the noiseless environment, where red indicates the lowest accuracy on the test set, and blue indicates the highest accuracy. We then repeat this in Fig. \[fig:dae\_encoding\_learnability\_versus\_robustness\](b) and Fig. \[fig:dae\_encoding\_learnability\_versus\_robustness\](c) to find the parameters which maximize accuracy in the presence of noise, and the maximize robustness. As expected, for the amplitude damping channel, the best parameters (with noise) are closer to the fixed point of the channel (i.e. $\theta^* \rightarrow 0$ implies encoding in the ${| 0 \rangle}$ state), thereby demonstrating the tradeoff between learnability and robustness.
(img) ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
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Fidelity Bounds on Partial Robustness {#ssec:fidelity_analysis_exp}
-------------------------------------
As a final numerical implementation, we compute the upper bounds on partial robustness proved in Sec. \[ssec:fidelity\_bounds\] for several different encodings and error channels. The implementation we consider is the previously-discussed Iris datasest classification problem using two qubits. The results are shown in Fig. \[fig:fidelity\_analysis\_iris\]. In this Figure, each plot corresponds to a different error channel with strength varied across the horizontal axis. Each curve in the top row corresponds to the fidelity of noisy and noiseless states using different encodings. Each curve in the bottom row shows the upper bounds on partial robustness proved in Sec. \[ssec:fidelity\_bounds\].
As can be seen in the bottom row of Fig. \[fig:fidelity\_analysis\_iris\], upper bounds on partial robustness are different for different encodings, particularly at small noise values. (Recall that a trivial upper bound on the size of partial robustness is one so that curves at large channel strengths above one are mostly uninformative.) For such low values of noise, they give us some information about the maximum cost function deviation we can expect. Based on the average fidelity over the datasets, in Figs. (\[subfig:fidelity\_compare\_iris\_bit\_flip\] - \[subfig:fidelity\_compare\_iris\_depo\]) all three encodings behave qualitatively the same. However, the cost function error for the three encodings is significantly different, especially for bit flip and dephasing errors, Figures (\[subfig:fidelity\_bound\_iris\_bit\_flip\] - \[subfig:fidelity\_bound\_iris\_dephase\]). As expected, a depolarizing channel causes no misclassification, as seen in [Figure \[subfig:fidelity\_bound\_iris\_depo\]]{}, despite the decrease in fidelity of the states. The apparent erratic behavior of the cost function error is largely due to the low number of samples in the Iris dataset. (Recall that the superdense angle encoding was not able to achieve perfect classification accuracy on the Iris dataset, so under amplitude damping noise, e.g., the cost function error can only decrease by about $25\%$ ($\sim 77\% \rightarrow 50 \%$).) We can also observe that the dense angle encoding is less susceptible to bit flip and phase errors than the wavefunction encoding in Fig. \[subfig:fidelity\_bound\_iris\_bit\_flip\] and Fig. \[subfig:fidelity\_bound\_iris\_dephase\].
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Discussion and Conclusions {#sec:conclusions}
==========================
In this paper, we have formally defined a model for a binary quantum classifier common in recent literature and studied encoding functions in detail. In particular, we showed that different encodings lead to different learnable decision boundaries and thus have an important effect on the overall success of the classifier. We introduced and formally defined the concept of robust points as well as robust sets and (completely) robust encodings. In addition to studying the robustness criteria for several common noise models, we have characterized robust sets for example channels and discussed their relationship to the fixed points of the channels. We used this connection to provide an existence proof of robust encodings and discussed an empirically observed tradeoff between learnability and robustness. Finally, we considered an embedded cost function classifier using an indicator cost function and provided upper bounds on the robustness of an encoding in terms of fidelities between ideal and noisy evolution.
In addition to the above theoretical results, we performed several numerical implementations to confirm and extend our findings. Specifically, we numerically evaluated decision boundaries for different encodings and performed example implementations on standard datasets in machine learning. Additionally, we used numerics to show that different encodings lead to different robust sets, and quantified the size and location of such sets for different encodings. Finally, we presented an “encoding learning algorithm” which optimizes over hyperparameters in the encoding to attempt to find robust sets. We provided proof of principle implementations on three datasets to show the performance of this algorithm. Finally, we showed upper bounds on partial robustness by computing the fidelities between ideal and noisy final states for an example implementation. We provide code to reproduce all numerical results at Ref. [@coyle_noiserobustclassifier_2020].
Within quantum machine learning, this work constitutes one of a relatively small number of papers to study encoding functions in detail. Our concept of robust points and robust encodings is novel to this work, and some of our analytical results help explain phenomena observed in recent literature — namely, that misclassifications due to depolarizing noise found in Ref. [@grant_hierarchical_2018] are solely due to finite shot statistics.
As this work introduces new concepts and discusses a relatively under-studied area in quantum machine learning, there are several avenues for future research. While we have provided multiple analytic results, the framework we introduced for proving these results is perhaps more impactful. In future work, these ideas can be applied to prove more robustness results for different channels and different decision boundaries than the one we considered in this work. Specific tasks we leave to future work include generalizing results to classes of quantum channels (e.g., unital channels), quantifying the tradeoff between learnability and robustness, and characterizing the conditions under which a *completely* robust encoding exists. It is also interesting from a purely theoretical perspective to extend the notion of a robust point to a “generalized fixed point” of a channel, i.e. points which satisfy $f({\mathcal{E}}(\rho)) = f(\rho)$ for some function $f$. (When $f = {\hat{y}}$, the “generalized fixed point” is a robust point, but other arbitrary functions $f$ could be considered.)
From an applications perspective, a clear task for future work is to test the ideas introduced here on a NISQ computer. While the channels we considered are standard theoretical tools for analyzing noise, more complicated effects such as crosstalk occur in real devices. Implementations on NISQ computers would assess our results *in situ* and potentially give additional insight into how robust encodings can be designed. Also, for future numerical work one could consider additional datasets (e.g., the MNIST dataset [@lecun_gradient-based_1998]) and test the performance of different encodings both in terms of learnability and robustness. Last, to further incorporate with recent literature, one could consider robustness in the context of “data re-uploading” [@perez-salinas_data_2019] or in the adversarial setting of Refs. [@lu_quantum_2019; @kechrimparis_channel_2019].
More broadly for NISQ applications, our work introduces a problem-specific strategy for dealing with errors. In contrast to error mitigation or even error correction techniques which allow errors to occur then attempt to mitigate (correct) them, our strategy of robust data encoding attempts to set up the problem such that any errors which do occur have no effect on the final result. We exploit the natural machine learning concept of data representation to achieve this effect, but in principle a similar idea could be used in other settings. This work defines fundamental concepts and proves several results for important questions that must be addressed on the path to practical applications with near-term quantum computers.
Acknowledgments {#sec:acknowledgements .unnumbered}
===============
We thank Nana Liu and Arkin Tikku for useful discussions and feedback. RL thanks Filip Wudarski, Stuart Hadfield, and Tad Yoder for helpful comments. BC thanks Niraj Kumar, Matty Hoban, and Atul Mantri for helpful comments. BC was supported by the Engineering and Physical Sciences Research Council (grant EP/L01503X/1), EPSRC Centre for Doctoral Training in Pervasive Parallelism at the University of Edinburgh, School of Informatics and Entrapping Machines, (grant FA9550-17-1-0055). RL acknowledges partial support from the [Unitary Fund](https://unitary.fund).
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Preliminaries and useful formulae {#app:useful-formulae}
=================================
The single qubit Pauli operators are $$\label{eqn:paulis}
I := \left[ \begin{matrix}
1 & 0\\
0 & 1 \\
\end{matrix} \right], \
X := \left[ \begin{matrix}
0 & 1\\
1 & 0 \\
\end{matrix} \right], \
Y := \left[ \begin{matrix}
0 & -i \\
i & 0 \\
\end{matrix} \right], \
Z := \left[ \begin{matrix}
1 & 0 \\
0 & -1 \\
\end{matrix} \right]$$
Let $\rho$ be a single qubit state with matrix elements $\rho_{ij}$, i.e. $$\rho := \left[ \begin{matrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11} \\
\end{matrix} \right] .$$ Then, $$\begin{aligned}
X \rho X &= \left[ \begin{matrix}
\rho_{11} & \rho_{10} \\
\rho_{01} & \rho_{00} \\
\end{matrix} \right] \\
Y \rho Y &= \left[ \begin{matrix}
\rho_{11} & - \rho_{10} \\
- \rho_{01} & \rho_{00} \\
\end{matrix} \right] \\
Z \rho Z &= \left[ \begin{matrix}
\rho_{00} & - \rho_{01} \\
- \rho_{10} & \rho_{11} \\
\end{matrix} \right] \end{aligned}$$
Defining the projectors $\Pi_0 = |0{\rangle}{\langle}0|$ and $\Pi_1= |1{\rangle}{\langle}1|$, one can show $$\begin{aligned}
{{\rm Tr}}[ \Pi_0 X \rho X] &= {{\rm Tr}}[ \Pi_1 \rho] \\
{{\rm Tr}}[ \Pi_0 Y \rho Y] &= {{\rm Tr}}[ \Pi_1 \rho] \\
{{\rm Tr}}[ \Pi_0 Z \rho Z] &= {{\rm Tr}}[ \Pi_0 \rho] ,\end{aligned}$$ For any hermitian matrix $A = [A_{ij}]$ and any unitary matrix $U = [U_{ij}]$, we have $$\begin{gathered}
\label{eqn:useful-00mtx-elt}
{{\rm Tr}}[ \Pi_0 U A U^\dagger] =\\
|U_{00}|^2 A_{00} + 2 {\text{Re}}[ U_{00}^* U_{01} A_{10}] + |U_{01}|^2 A_{11} .\end{gathered}$$ Similarly, one can show that $$\begin{gathered}
\label{eqn:useful-11mtx-elt}
{{\rm Tr}}[ \Pi_{1} U A U^\dagger ] =\\
|U_{10}|^{2} A_{00} + 2 {\text{Re}}[ U_{11}^* U_{10} A_{01}] + |U_{11}|^2 A_{11} .\end{gathered}$$ If we further assume the single qubit parameterized unitary, $U(\boldsymbol \alpha)$, has the following decomposition: $R_z(2\alpha_1)R_y(2\alpha_2)R_z(2\alpha_3)$ (up to a global phase) [@nielsen_quantum_2010], we get: $$\label{eqn:single_qubit_unitary_decompostion}
U(\boldsymbol\alpha) = \left[ \begin{matrix}
e^{i ( - \alpha_1 - \alpha_3)} \cos\alpha_2&
-e^{i ( - \alpha_1+ \alpha_3 )} \sin \alpha_2\\
e^{i ( \alpha_1 - \alpha_3 )} \sin \alpha_2 &
e^{i ( \alpha_1 + \alpha_3)} \cos \alpha_2 \\
\end{matrix} \right]$$ Therefore, we get the various terms to be: $$\begin{aligned}
&|U_{00}|^2 = \cos^2(\alpha_2) \\
&|U_{01}|^2 = |U_{10}|^2 = \sin^2(\alpha_2) \\
&|U_{11}|^2 = \cos^2(\alpha_2)\\
&U_{00}^*U_{01} = -e^{i2\alpha_3}\cos(\alpha_2)\sin(\alpha_2) = -\frac{1}{2}e^{i2\alpha_2}\sin(2\alpha_2) \\
&U_{11}^*U_{10}= e^{-i2\alpha_3}\cos(\alpha_2)\sin(\alpha_2) = \frac{1}{2}e^{-2i\alpha_3}\sin(2\alpha_2)\end{aligned}$$ So the conditions, (\[eqn:useful-00mtx-elt\], \[eqn:useful-11mtx-elt\]) become: $$\begin{gathered}
{{\rm Tr}}[ \Pi_0 U A U^\dagger] \\
=|U_{00}|^2 A_{00} + 2 {\text{Re}}[ U_{00}^* U_{01} A_{10}] + |U_{01}|^2 A_{11} \\
= \cos^2\left(\alpha_2\right)A_{00} + \sin^2\left(\alpha_2\right)A_{11} - {\text{Re}}[e^{2i\alpha_3}\sin\left(\alpha_2\right)A_{10}] \label{eqn:useful-00mtx-elt-unitary-filled} \end{gathered}$$ $$\begin{gathered}
{{\rm Tr}}[ \Pi_{1} U A U^\dagger ]
=|U_{10}|^{2} A_{00} + 2 {\text{Re}}[ U_{11}^* U_{10} A_{01}] + |U_{11}|^2 A_{11} \\
=\sin^2\left(\alpha_2\right)A_{00} +\cos^2\left(\alpha_2\right)A_{11}
+{\text{Re}}[e^{-2i\alpha_3}\sin\left(2\alpha_2\right)A_{01}] \label{eqn:useful-11mtx-elt-unitary-filled}\end{gathered}$$
Proofs and Additional Results {#app:further_proofs}
=============================
Here we give the explicit proofs for the remaining theorems (which we also repeat here for completeness) in the main text, and some others introduced here.
Robustness to measurement noise {#app_ssec:meas_noise}
-------------------------------
Just as the case of quantum compilation [@sharma_noise_2020], we can deal with measurement noise in the classifier:
\[defn:measurement\_noise\] *Measurement noise* (MN) is defined as a modification of the standard POVM basis elements, $\{\Pi_0 = {| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}, \Pi_1 = {| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |}\}$ by the channel $\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}$ with assignment matrix ${\boldsymbol{p}}$ for a single noiseless qubit: $$\begin{aligned}
\Pi_0 &= {| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} \overset{\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}}{\rightarrow} \tilde{\Pi}_0 = p_{00}{| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} + p_{01}{| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |}\nonumber\\
\Pi_1 &= {| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |} \overset{\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}}{\rightarrow} \tilde{\Pi}_1 = p_{10}{| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} + p_{11}{| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |} \label{eqn:noisy_povm_elements}\\
{\boldsymbol{p}}& := \left(\begin{array}{cc}
p_{00} & p_{01} \\
p_{10} & p_{11}
\end{array}\right)\nonumber\end{aligned}$$ where $p_{00} + p_{10} = 1, p_{10} +p_{11} = 1$, and hence $p_{kl}$ is the probability of getting the $k$ outcome given the $l$ input. Furthermore, we assume that $p_{kk} > p_{kl}$ for $k \neq l$.
The definition for the general case of $n$ qubit measurements can be found in [@sharma_noise_2020], but we shall not need it here, since we only require measuring a single qubit to determine the decision function. More general classifiers which measure multiple qubits (e.g. and then take a majority vote for the classification) could also be considered, but these are outside the scope of this work. Now, we can show the following result in a similar fashion to the above proofs:
\[thm:measurement\_noise\_robustness\] Let $\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}$ define measurement noise acting on the classification qubit and consider a quantum classifier on data from the set $\mathcal{X}$. Then, for any encoding $E: \mathcal{X} \rightarrow \mathcal{D}_2$, we have complete robustness $$\mathcal{R} (\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}, E, {\hat{y}}) = \mathcal{X}$$ if the measurement assignment probabilities satisfy $p_{00} > p_{01}, p_{11} > p_{10}$.
We can write the measurement noise channel acting on the POVM elements as: $$\begin{aligned}
\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}(\Pi^{(c)}_0\otimes I^{\otimes n-1}) = (p_{00}{| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} + p_{01}{| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |}) \otimes I^{\otimes n-1} \nonumber\end{aligned}$$ Again, if we had correct classification before the noise, ${{\rm Tr}}({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}) {\geqslant}1/2$, then: $$\begin{aligned}
&{{\rm Tr}}[\mathcal{E}_{{\boldsymbol{p}}}^{\textnormal{meas}}(\Pi^{(c)}_0\otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}]\\
&= {{\rm Tr}}[(\{p_{00}{| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} + p_{01}{| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |}\} \otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}]\\
&= p_{00}{{\rm Tr}}[({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}\otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}] + p_{01}{{\rm Tr}}[({| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |} \otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}]\\
&= p_{00}{{\rm Tr}}[({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}\otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}] \\
&+ p_{01}(1- {{\rm Tr}}[({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}\otimes I^{\otimes n-1}) {\tilde{\rho}_{{\boldsymbol{x}}}}])\\
&= (p_{00} - p_{01}){{\rm Tr}}[({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}\otimes I^{\otimes n-1}){\tilde{\rho}_{{\boldsymbol{x}}}}] + p_{01}\\
&{\geqslant}(p_{00} - p_{01})1/2 + p_{01} = 1/2(p_{00} + p_{01}) = 1/2\end{aligned}$$ where in the last line, we used the fact that $p_{00}+p_{01}=1$ and our assumption that $p_{00} > p_{01}$. The same result holds if the vector was classified as $1$, and hence the classifier is robust to measurement noise.
Just as above, we can replace the ideal state, ${\tilde{\rho}_{{\boldsymbol{x}}_i}}$ with a noisy state, $\mathcal{E}({\tilde{\rho}_{{\boldsymbol{x}}_i}})$, where the operator accounts for other forms of noise, not including measurement noise. We can see this allows us to take a model which is robust without measurement noise, and ‘upgrade’ it to one which is. However, we may be able to find looser restrictions by considering different types of noise together, rather than in this modular fashion.
To illustrate the results of [Theorem \[thm:measurement\_noise\_robustness\]]{} we focus on the dense angle encoding, which can achieve nearly 100% accuracy on the “vertical” dataset. We then compute the percentage which would be misclassified as a function the assignment probabilities in the noisy projectors in [(\[eqn:noisy\_povm\_elements\])]{}. The results are seen in [Figure \[fig:measurement\_noise\_results\]]{}.
(img) ; ; ;
Robustness for Factorizable Noise {#app:factorizable-noise}
---------------------------------
\[thm:factorizable-noise-before-meas\] If ${\mathcal{E}}$ is any noise channel which factorizes into a single qubit channel, and a multiqubit channel as follows: $$\begin{aligned}
{\mathcal{E}}(\rho) = {\mathcal{E}}_{\Bar{c}}({\tilde{\rho}_{{\boldsymbol{x}}}}^{\Bar{c}}) \otimes {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}}^c)\end{aligned}$$ where WLOG ${\mathcal{E}}_c$ acts only on the classification qubit $({\tilde{\rho}_{{\boldsymbol{x}}}}^c = {{\rm Tr}}_{\Bar{c}}({\tilde{\rho}_{{\boldsymbol{x}}}}))$ after encoding and unitary evolution, and ${\mathcal{E}}_{\Bar{c}}$ acts on all other qubits arbitrarily, $({\tilde{\rho}_{{\boldsymbol{x}}}}^{\Bar{c}} = {{\rm Tr}}_c({\tilde{\rho}_{{\boldsymbol{x}}}}))$. Further assume the state meets the robust classification requirements for the single qubit error channel ${\mathcal{E}}_c$. Then the classifier will be robust to ${\mathcal{E}}$.
The correct classification again depends on the classification qubit measurement probabilities: ${{\rm Tr}}({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |}^{(c)}\otimes I^{\otimes n-1}{\tilde{\rho}_{{\boldsymbol{x}}}}), {{\rm Tr}}({| \hspace{1pt} 1 \rangle \langle 1 \hspace{1pt} |}^{(c)}\otimes I^{\otimes n-1} {\tilde{\rho}_{{\boldsymbol{x}}}})$. If ${\tilde{\rho}_{{\boldsymbol{x}}}}$ is robust to the single qubit error channel ${\mathcal{E}}_c$, this means $$\begin{aligned}
{{\rm Tr}}(\Pi_0^c{\tilde{\rho}_{{\boldsymbol{x}}}}) {\geqslant}1/2 &\implies {{\rm Tr}}(\Pi_0^c {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}})) {\geqslant}1/2 \\
{{\rm Tr}}(\Pi_1^c {\tilde{\rho}_{{\boldsymbol{x}}}}) < 1/2 &\implies {{\rm Tr}}(\Pi_1^c {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}})) < 1/2\end{aligned}$$ Then WLOG, assume the point ${\boldsymbol{x}}$ classified as $y({\tilde{\rho}_{{\boldsymbol{x}}}}) = 0$ before the noise, then: $$\begin{aligned}
{{\rm Tr}}\left(\Pi_0^c{\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}}})\right)&= {{\rm Tr}}\left(\Pi_0^c \left[{\mathcal{E}}_{\Bar{c}}({\tilde{\rho}_{{\boldsymbol{x}}}}^{\Bar{c}} \otimes {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}}^c)\right]\right) \\
&={{\rm Tr}}_{\Bar{c}}\left({\mathcal{E}}_{\Bar{c}}({\tilde{\rho}_{{\boldsymbol{x}}}}^{\Bar{c}})\right) {{\rm Tr}}_{c}\left({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}}^c)\right) \\
&={{\rm Tr}}\left({| \hspace{1pt} 0 \rangle \langle 0 \hspace{1pt} |} {\mathcal{E}}_c({\tilde{\rho}_{{\boldsymbol{x}}}}^c)\right) {\geqslant}1/2 \qedhere\end{aligned}$$
The above theorem is a simple consequence of causality in the circuit, only errors which have to happen before the measurement can corrupt the outcome. As such, outside of single qubit errors, we only need to consider errors before the measurement which specifically involve the classification qubit.
Fidelity Bound {#app_ssec:fidelity_bound}
--------------
Here we derive the fidelity bound, [(\[eqn:fidelity\_bound\_average\])]{} in a similar fashion to [(\[eqn:fidelity\_bound\_mixed\_state\])]{}: $$\begin{aligned}
\Delta_{\mathcal{E}}C &:= \left| C_{\mathcal{E}}- C \right| \nonumber \\[1.0ex]
&= \left| {{\rm Tr}}[ D ( {\mathcal{E}}(\tilde{\sigma} ) - \tilde{\sigma}) ] \right| \nonumber\\
&{\leqslant}\frac{1}{M}\sum\limits_{i=1}^M| {{\rm Tr}}(D\left[ {\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}_i}})\otimes{| \hspace{1pt} y_i \rangle \langle y_i \hspace{1pt} |} - {\tilde{\rho}_{{\boldsymbol{x}}_i}}\otimes{| \hspace{1pt} y_i \rangle \langle y_i \hspace{1pt} |}\right])| \nonumber\\
&{\leqslant}\frac{1}{M}\sum\limits_{i=1}^M||D||_{\infty} ||\left[{\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}_i}})- {\tilde{\rho}_{{\boldsymbol{x}}_i}}\right]\otimes{| \hspace{1pt} y_i \rangle \langle y_i \hspace{1pt} |}||_1 \nonumber\\
&{\leqslant}\frac{2}{M}\sum\limits_{i=1}^M \sqrt{1-F({\mathcal{E}}({\tilde{\rho}_{{\boldsymbol{x}}_i}}), {\tilde{\rho}_{{\boldsymbol{x}}_i}})} \label{eqn:fidelity_bound_average_appendix}\end{aligned}$$ Again, we use Hölders, the Fuchs-van de Graaf and the triangle inequalities, with $||D||_\infty := \max_j | \lambda_j(D) | = 1$.
More Details on Numerical Results {#app:numerical_results}
=================================
In this section, we present supplementary numerical results to those in the main text. Firstly, in [Figure \[fig:single\_qubit\_datasets\]]{}, we illustrate the three single qubit datasets we employ here, namely the “vertical”, “diagonal” and “moons”. The former two are linearly separable, whereas the “moons” dataset is nonlinear.
-1.5em -1.5em
Secondly, to complement the results of [Figure \[fig:dae\_encoding\_learnability\_versus\_robustness\]]{} in the main text, in Table \[table:vertical\_boundary\_encodin\_params\_plot\] we provide the best parameters found in the procedure. Each set of parameters (each row, measured in radians) performs optimally in one of three areas. The first is the noiseless environment, in which a $\theta\approx 2.9$ parameter performs optimally. The second is the amplitude-damped environment, in which $\theta\approx 1.6$ achieves the best accuracy, and finally, $\theta=0$ is the most robust point to encode in, for the whole dataset. For each of these parameter sets, we also test them in the other scenarios, for example, the best parameters found in the noisy environment ($[\theta, \phi] = [1.6, 3.9]$) have a higher $\delta$-Robustness ($81\%$) than those in the noiseless environment ($70\%$), since these parameters force points to be encoded closer to the ${| 0 \rangle}$ state, i.e., the fixed point of the channel in question.
----------------------------------------- ----------- ---------- ---------------------
Parameters Accuracy Accuracy $\delta$-Robustness
w/o Noise w/ Noise
\[0.5ex\] $[\theta, \phi] = [2.9, 2.9]$ $100\%$ $84\%$ $70\%$
$[\theta, \phi] = [1.6, 3.9]$ $49\%$ $100\%$ $81\%$
$[\theta, \phi] = [0, 0]$ $43\%$ $43\%$ $100\%$
----------------------------------------- ----------- ---------- ---------------------
: Optimal parameters $[\theta, \phi]$ for dense angle encoding (with parameters in $U({\boldsymbol{\alpha}})$ trained in noiseless environment) in (a) noiseless environment, (b) noisy environment (i.e. amplitude damping channel is added) and (c) for maximal robustness. Optimal parameters in noisy environment are closer to fixed point of amplitude damping channel (${| 0 \rangle}$, i.e. $\theta \equiv 0$) and give a higher value of $\delta$-robustness. []{data-label="table:vertical_boundary_encodin_params_plot"}
[Figure \[fig:specific\_classifier\_circuits\]]{} illustrates the specific decompositions for the single and two qubit classifiers we utilize for the numerical results in the text. For the matrix representation of the circuit shown in [Figure \[fig:specific\_classifier\_circuits\]]{}(a), see [(\[eqn:single\_qubit\_unitary\_decompostion\])]{}.
To illustrate the results of [Theorem \[thm:robustness\_pauli\_noise\_xy\]]{}, we focus on the dense angle encoding, which can achieve nearly 100% accuracy on the “vertical” dataset. We then compute the percentage which would be misclassified as a function of and Pauli noise parameters. The results are seen in [Figure \[fig:pauli\_noise\_results\]]{}, similar to that observed in [Figure \[fig:measurement\_noise\_results\]]{}. We note here, that for values of $p_X + p_Y > 1/2$, one has two strategies to achieve robustness. The first is to adjust the measurement basis as per [Corollary \[corr:pauli\_x\_robustness\]]{} and requires changing the model itself. Alternatively, one can apply an extra step of post processing and relabel every output ‘${\hat{y}}= 0$’ to ‘${\hat{y}}= 1$’, and vice versa.
(img) ; ; ;
Algorithm \[alg:robust-algorithm\] contains pseudocode for the encoding learning algorithm discussed in the main text.
\[alg:robust-algorithm\] Initialize encoding, $(f^*, g^*) \leftarrow \{f_l, g_l\}_{l=1}^K$ and parameters, $\{{\boldsymbol{\theta}}_j\} \leftarrow (0, 2\pi ]_j ~\forall j$ heuristically or at random. Initialize $C^*= M$
[^1]: Although quantum machine learning is generally considered an emerging field, foundational ideas have been published more than two decades ago [@behrman_quantum_1996].
[^2]: In practice we typically have $\mathcal{X} = \mathbb{R}^N$, but other sets — e.g., $\mathcal{X} = \mathbb{Z}^N$ or $\mathcal{X} \in \mathbb{Z}_2^N$ — are possible, so we write $\mathcal{X}$ for generality.
[^3]: \[footnote:shots\]Recall that expectation values evaluated with $N_m$ measurements have variance $1 / \sqrt{N_m}$ [@mcclean_theory_2016]. If ${{\rm Tr}}[\Pi_0 {\tilde{\rho}_{{\boldsymbol{x}}}}] \le 1/2$ and ${{\rm Tr}}[\Pi_0 \mathcal{E} ( {\tilde{\rho}_{{\boldsymbol{x}}}})] = 1/2 - \epsilon$ where $\epsilon > 0$, then at least $N_m = 1 / \epsilon^2$ measurements are required to determine robustness. Intuitively, this means that points “on the border” between classes require more measurements to distinguish.
[^4]: In principle, one can achieve an encoding which is completely robust and able to correctly classify all data if there are at least two orthogonal fixed points in $\mathcal{F}({\mathcal{E}})$. For example, if ${\mathcal{E}}$ the bit flip channel, the encoding ${\boldsymbol{x}}_i \mapsto {| 0 \rangle} + (-1)^{y_i} {| 1 \rangle}$ is both completely robust and completely learnable (the optimal unitary is a Hadamard gate), but assumes the true labels $y_i$ are known.
[^5]: Interestingly, the case where the true label is corrupted by noise can be linked to a commonly studied case in classical supervised learning — i.e., “learning from noisy examples” [@angluin_learning_1988].
[^6]: Note this actually gives a slightly more general cost function than the indicator we use here [(\[eqn:indicator\_cost\_over\_dataset\])]{}, but can be related by a simple transformation. See Ref. [@cao_cost_2019].
|
---
abstract: 'Recent high-resolution observations of the central region of Galactic globular clusters have shown the presence of a large variety of exotic stellar objects whose formation and evolution may be strongly affected by dynamical interactions. In this paper I review the main properties of two classes of exotic objects: the so-called Blue Stragglers stars and the recently identified optical companions to Millisecond pulsar. Both these class of objects are invaluable tools to investigate the binary evolution in very dense environments and are powerful tracers of the dynamical history of the parent cluster.'
author:
- 'Francesco R. Ferraro'
title: Exotic populations in Galactic Globular Clusters
---
Introduction
============
Ultra-dense cores of Galactic Globular Clusters (GCs) are very efficient “furnaces” for generating exotic objects, such as low-mass X-ray binaries, cataclysmic variables, millisecond pulsars (MSPs), blue stragglers (BSS), etc. Most of these objects are thought to result from the evolution of various kinds of binary systems originated and/or hardened by stellar interactions. The nature and even the existence of binary by-products can be strongly affected by the cluster core dynamics, thus serving as a diagnostic of the dynamical evolution of GCs. This topic has received strong impulse in the recent years and many studies have been devoted to investigate the possible link between the dynamical evolution of clusters and the evolution of their stellar population.
In particular two aspects can be investigated: (1) the environment effects on canonical evolutionary sequences (as for example the possible effect of different environments on the blue tail extension of the Horizontal Branch - HB, see Fusi Pecci et al 1992, Buonanno et al 1997); (2) the creation of artificial sequences as BSS and other exotic objects (see Bailyn 1995 and reference therein).
In this paper I will review the main properties of two anomalous sequences in the color-magnitude diagram (CMD):
- the most known [*anomalous sequence*]{} in GCs: the so-called BSS sequence, indeed the very first sequence of exotic objects discovered in the CMD of GCs;
- the most recently discovered [*anomalous sequence*]{}: the one defined by MSP companions.
Blue Straggler Stars
====================
BSS, first discovered by Sandage (1953) in M3, are commonly defined as stars brighter and bluer (hotter) than the main sequence (MS) turnoff (TO), lying along an apparent extension of the MS, and thus mimicking a rejuvenated stellar population. The existence of such a population has been a puzzle for many years, and even now its formation mechanism is not completely understood, yet. At present, the leading explanations involve mass transfer between binary companions, the merger of a binary star system or the collision of stars (whether or not in a binary system). Direct measurements (Shara et al. 1997) and indirect evidence show that BSS are more massive than the normal MS stars, pointing again towards collision or merger of stars. Thus, BSS represent the link between classical stellar evolution and dynamical processes (see Bailyn 1995). The realization that BSS are the ideal diagnostic tool for a quantitative evaluation of the dynamical interaction effects inside star clusters has led to a remarkable burst of searches and systematic studies, using UV and optical broad-band photometry.
The UV approach to the study of BSS
-----------------------------------
The observational and interpretative scenario of BSS has significantly changed in the last 20 years. In fact, since their discovery and for almost 40 years, BSS have been detected only in the outer regions of GCs or in relatively loose clusters, thus forming the idea that a low-density environment is the [*natural habitat*]{} for BSS. Of course, it was an observational bias: starting from the early ’90 high resolution studies allowed to properly image the central region of high density clusters (see the case of NGC6397 by Auriere & Ortolani 1990). Moreover, with the advent of the Hubble Space Telescope ([HST]{}) it became possible for the first time to search dense cluster cores for BSS. This was a really turning point in BSS studies since HST, thanks to its unprecedented spatial resolution and imaging/spectroscopic capabilities in the UV, has given a new impulse to the study of BSS (see Paresce et al 1992, Paresce & Ferraro (1993), Guhatahakurtha et al 1994, etc).
Based on these observations, the first catalogs of BSS have been published (Fusi Pecci et al 1992; Sarajedini 1992; Ferraro, Bellazzini & Fusi Pecci 1995, hereafter FFB95) until the most recent collection of BSS which counts nearly 3000 candidates (Piotto et al 2004). These works have significantly contributed to form the nowadays, commonly accepted idea that BSS are a normal stellar population in clusters, since they are present in all of the properly observed GCs. However, according to Fusi Pecci et al. (1992) BSS in different environments could have different origin. In particular, BSS in loose GCs might be produced from coalescence of primordial binaries, while in high density GCs (depending on survival-destruction rates for primordial binaries) BSS might arise mostly from stellar interactions, particularly those which involve binaries. Thus, while the suggested mechanisms for BSS formation could be at work in clusters with different environments (FFB95; Ferraro et al. 1999), there are evidence that they could also act simultaneously within the same cluster (as in the case of M3, see Ferraro et al. 1993 - hereafter F93; Ferraro et al. 1997 - hereafter F97). Moreover, as shown by Ferraro et al. (2003 - hereafter F03), both the BSS formation channels (primordial binary coalescence and stellar interactions) seem to be equally efficient in producing BSS in different environments, since the two clusters that show the largest known BSS specific frequency, i.e. NGC 288 (Bellazzini et al. 2002) and M 80 (Ferraro et al. 1999), represent two extreme cases of central density concentration among the GCs ($Log
\rho_0=2.1$ and $5.8$, respectively).
However, a major problem in the systematic study of the BSS still persists, especially in the central region of high density clusters by using the classical CMD even with HST. In fact, the CMD of an old stellar population (as a GC) in the [*classical*]{} $(V,B-V)$ plane is dominated by the cool stellar component, hence the observations and the construction of complete sample of hot stars (as extreme blue HB, BSS, various by-products of binary system evolution etc.) is “intrinsically” difficult in this plane. Moreover, in visible CMDs the BSS region could be severely affected by photometric blends which mimic BSS.
In the ultraviolet (UV) plane, where the sub-giants and red giant stars which cause BSS-like blends are faint and the hot stellar populations are relatively bright, problems are much less severe, allowing to obtain complete BSS samples even in the densest cluster core regions The advantage of studying BSS in the mid-UV CMD is shown in Figure 1, where the shapes of the main evolutionary sequences in the traditional (V,B-V) plane ([*panel (a)*]{}) and in the UV plane ([*panel (b)*]{}) are compared. As can be seen, in the UV plane the main branches display very different morphologies with respect to those in the optical CMD (i.e. $V,~ V-I$). As can be seen, the red giant branch (RGB) is very faint in the UV, while the HB, excluding the hottest section, which bends downward because of the increasing bolometric correction, appears diagonal. Since red giants are faint in UV, the photometric blends, which mimic BSS in visible CMDs, are less problematic. The BSS define a narrow, nearly vertical sequence spanning $\sim 3$ mag in this plane, thus, a complete BSS sample can be obtained even in the densest cores: indeed, the ($m_{255}$, $m_{255} - m_{336}$) plane is an ideal tool for selecting BSS.
M3: a new approach to study GC stellar populations
--------------------------------------------------
M3 has played a fundamental role in the BSS history, since it is the GC where BSS have been identified for the first time, but also the first GC in which the BSS radial distribution has been studied over the entire cluster extension. In fact by combining UV HST observations in the central region of the cluster (F97, see Figure 2) and extensive wide field ground-based observations (F93 - Buonanno et al 1994), F97 presented the radial distribution of BSS over the entire radial extension ($r\sim 6 '$).
The UV search for BSS in the central region of M3 led to the discovery of a large population of [*centrally segregated*]{} BSS (see Figure 2), contrary to the previous claim by Bolte, Hesser & Stetson (1993) who suggested a possible depletion of BSS in the central region of the cluster with respect to the external regions. As expected, the BSS candidates occupy a narrow, nearly vertical, sequence spanning $\sim 3$ mag in $m_{255}$. Two limits (one in color, and one in magnitude) have been assumed to properly select the BSS sample in the UV-CMD shown in Figure 2. The BSS sequence blends smoothly into the MS near the cluster TO. In order to select ‘safe’ BSS, only stars brighter than $m_{255} \sim
19.4$ (0.3 mag brighter than the cluster TO) have been considered.
Beside this, a much more unexpected result was found from the analysis of BSS in M3. In fact, in order to extend the BSS analysis to the whole radial extension of the cluster and to compare the HST sample with the external (ground-based) one, F97 limited the BSS analysis to the brighter portion ($m_{255}<19$) of BSS population. The radial distribution of the BSS candidates was compared to that of a sample of RGB stars assumed as “reference” population. The cumulative radial distributions of the entire sample split into two sub-sets (at $r=150\arcsec$) are reported in Figure 3: as can be seen the BSS (solid line) are more centrally concentrated than RGB stars (dotted line) in the central regions (out to r$<150\arcsec$), while are less concentrated in the outer ones.
In order to further investigate this surprising result, F97 computed the doubly normalized ratios for the BSS and the RGB stars, following the definitions by F93:
$$R_{\rm BSS} = {{(N_{\rm BSS}/N_{\rm BSS}^{\rm tot})} \over
{(L^{sample}/L_{tot}^{sample})}}$$
and
$$R_{\rm RGB} = {{(N_{\rm RGB}/N_{\rm RGB}^{\rm tot})} \over
{(L^{\rm sample}/L_{\rm tot}^{\rm sample})}}$$
respectively.
The surveyed cluster region has been divided in a number of concentric annuli and the numbers of BSS and RGB counted in each annulus has been normalized to the sampled luminosity accordingly to the above relations. The [*relative frequency*]{} of BSS is compared with that computed for the RGB “reference” stars as a function of the distance from the cluster center, as shown in Figure 4. As can be seen, the radial distribution of BSS is clearly bimodal: it reaches its maximum at the center of the cluster (showing no evidence of a BSS depletion in the core); it has a clear-cut dip in the intermediate region (at $100\arcsec<r<200\arcsec$) and a rising trend in the outer region (out to $r\sim360\arcsec$), as first noted by F93.
47 Tuc: another surprise!
-------------------------
While the bimodality detected in M3 was considered for years to be [*peculiar*]{}, the most recent results demonstrated that this is not the case. In fact, in the last years the same observational strategy adopted by F97 in M3 has been applied to a number of clusters with the aim of determining the BSS frequency over the entire cluster extent. To do this two data-set are generally combined:
[*(i) High resolution set—*]{} consisting of a series of high-resolution WFPC2-HST images (typically in the UV) of the cluster center. In this data set the planetary camera (PC, which has the highest resolution $\sim 0\farcs{046}/{\rm
pixel}$) is roughly centered on the cluster center while the Wide Field (WF) cameras (at lower resolution $\sim 0\farcs{1}/{\rm pixel}$) sample the surrounding outer regions;
[*(ii) Wide Field set—*]{} consisting of a series of multi-filter ($B$, $V$, $I$) wide field images obtained by using the last generation of wide field imagers (as for example the Wide Field Imager (WFI) mounted at the 2.2m ESO-MPI telescope at ESO (La Silla)). The WFI is a mosaic of 8 CCD chips (each one with a field of view of $8'\times
16'$) giving a global coverage of $33'\times 34'$.
Figure 5 shows the typical cluster coverage adopting this observing strategy, also used for 47 Tuc (see Ferraro et al 2001a and Ferraro et al 2004a, hereafter F04). Figure 6 shows the corresponding CMD. As can be seen a well defined sequence of BSS has been obtained in both samples. To study the BSS radial distribution, F04 applied the procedure described in F93 (used also in F97) where the surveyed area has been divided into a set of concentric annuli. 11 concentric annuli, each one containing roughly $\sim 10$% of the reference population) have been defined in the case of 47 Tuc. The BSS specific frequency has been computed in two different ways: (1) the ratio $F_{\rm BSS}^{HB}=N_{\rm BSS}/N_{\rm HB}$ and (2) the double-normalized ratio $R$ (see above) considering the fraction of luminosity sampled in each annulus.
Figure 7 shows the distribution of both ratios as a function of the effective radius of each annulus. The distribution is clearly bimodal, with the highest value in the innermost annulus where the $F_{\rm BSS}^{HB}$ ratio reaches $\sim 0.4$, it significantly decreases to less than 0.1 as $r$ increases and then slowly rises up to $\sim 0.3$ in the outer region.
This trend is fully confirmed by using the [*relative BSS frequency*]{} $R_{BSS}$ (see F97 and above). The behavior of this ratio as a function of the distance from the cluster center is shown in Figure 7 [*(panel (b))*]{} and compared with the corresponding one for the HB “reference” stars. As can be seen, the HB specific frequency remains essentially constant over the surveyed area since the fraction of HB stars (as any post-main sequence stage) in each annulus strictly depends on the fraction of luminosity sampled in that annulus (see the relation by Renzini & Buzzoni 1986, eq 2 in F03). In contrast, the BSS specific frequency reaches its maximum at the center of the cluster, then decreases to an approximately constant value in the range 100–500 from the cluster center and then rises again. [*The trend found in 47 Tuc closely resembles that discovered in M3 by F97*]{}.
To further demonstrate the similarity with M3 and to study possible differences we show the BSS specific frequencies for the two clusters on the same figure (see Figure 8). The radial coordinate is given in units of the core radius $r_c$, adopting $r_c=21\arcsec$ and $r_c=24\arcsec$ for 47 Tuc and M3, respectively (see F04 and F03). A few major characteristics about Figure 8 are worth noticing: (1) the central values are similar; (2) while the BSS specific frequency decreases in both clusters as $r$ increases from 0 to $\sim 4r_c$, the decrease is much larger in M3. In 47 Tuc it is a factor of 5.5, dropping from 2.64 down to 0.48; in M3 the drop is a factor of 15 (from 2.76 to 0.2). (3) the specific frequency minimum in 47 Tuc appears to be much broader than that observed in M3. In 47 Tuc the depletion zone extends from $\sim 4 r_c$ to 20–$22~r_c$ with the upturn of the BSS density occurring at $\sim 25~r_c$, while in M3 the BSS specific frequency is already rising at $\sim 8 r_c$.
F97 argued that the bimodal distribution of BSS in M3 was a signature that two formation scenarios were active in the same cluster, the [*external*]{} BSS arising from mass transfer in primordial binaries and the [*central*]{} BSS arising from stellar interactions which lead to mergers. As earlier noted by Bailyn & Pinsonneault (1995), the luminosity functions of the two BSS samples differ as theoretically expected for the two different mechanisms.
Sigurdsson et al. (1994) offered another explanation for the bimodal BSS distribution in M3. They suggested that the [*external*]{} BSS were formed in the core and then ejected into the outer regions by the recoil from the interactions. Those binaries which get kicked out to a few core radii ($r_c$) rapidly drift back to the center of the cluster due to mass segregation, leading to a concentration of BSS near the center and a paucity of BSS in the outer parts of this region. More energetic kicks will take the BSS to larger distances; these stars require much more time to drift back toward the core and may account for the overabundance of BSS at large distances. In order to discern between different possibilities accurate simulations with suitable dynamical codes are necessary. Mapelli et al (2004) modeled the evolution of BSS in 47 Tuc, mimicking their dynamics in a multimass King model, by using a new version of the dynamical code described by Sigurdsson & Phinney (1995). Their results demonstrate that the observed spatial distribution cannot be explained within a purely collisional scenario in which BSS are generated exclusively in the core through stellar interactions. In fact, an accurate reproduction of the BSS radial distribution can be obtained only requiring that a sizable fraction of BSS is generated in the peripheral regions of the cluster inside primordial binaries that evolve in isolation and experiencing mass transfer.
A BSS specific upturn similar to that observed in 47 Tuc and M3 has also been detected in M55 by Zaggia et al. (1997). This result is based on ground-based observations sampling only a quadrant of the cluster. Since ground-based observations tend to hide BSS in the central region of the cluster, the bimodality in M55 could be even stronger than that found by Zaggia et al (1997). This is of particular significance because M55 has a central density significantly lower than M3 and 47 Tuc and could suggest that the bimodal distribution is not related to the central density of the parent cluster. Moreover a BSS bimodal radial distribution has been also recently detected in NGC6752 by Sabbi et al (2004). At present, there are at least 4 GCs in which the BSS radial distribution seems to be bimodal: M3, 47 Tuc, M55 and NGC6752 (see Figure 9).
Though the number of the surveyed clusters is low, these discoveries suggest that the [*peculiar*]{} radial distribution first found in M3 is much more [*common*]{} that was thought. [*Indeed, it could be the “natural” BSS radial distribution*]{}. Clearly, generalizations cannot be made from a sample of a few clusters. Hence we need to characterize the BSS radial distribution on a much more solid statistical base. First results from simulations indicate that bimodality is a signature that both collisions and primordial binaries play an important role.
However, there is recent evidence that BSS do not follow this rule in at least one cluster. Using combined (HST+ground-based) observations Ferraro et al (2005) selected the largest population of BSS ever observed in a stellar system in the giant cluster $\omega$ Cen. Conversely to any other GC surveyed up to now, they found that the BSS frequency in this stellar system [*does not peak in the center and does not vary with the distance from the cluster center*]{}. As can be seen from Figure 10, the double normalized BSS ratio turns out to be nicely constant over the entire cluster extension and it is fully consistent with the reference population. This is the very first time that such a trend has been found for BSS. This result is surprising since the relaxation time for the core of the cluster ($\tau \sim 7$ Gyr) is a factor two lower than the estimated cluster age ($t \sim 13$ Gyr). Clearly this evidence could be related to the complex history of this peculiar stellar system (see recent results in Lee et al 1999; Pancino et al 2000, 2002; Ferraro et al 2002, 2004b). [*However, our observations give the cleanest evidence that this cluster has not yet reached the energy equipartition even in the central core*]{} and further support the use of BSS as probe of the dynamical cluster evolution.
BSS and the parent GC properties
--------------------------------
### Specific BSS frequencies and structural parameters
The collection of an homogeneous dataset of BSS in different clusters allows a direct cluster-to-cluster comparison. However, it should be emphasized that the interpretation of BSS specific frequency in terms of structural cluster parameters can be risky (see FFB95). In particular, one should keep in mind the intrinsic dependence of any BSS specific frequency on the cluster luminosity.
Recently, Piotto et al. (2004) noted a correlation between BSS specific frequency and cluster absolute magnitude. The same correlation had been discussed by FFB95 (see Fig. 11 and 12), who showed that much of this effect arose from the normalization of the population. Indeed such a trend could be generated by the correlation between the sampled and the total luminosity as shown in Fig. 4 of FFB95. A similar analysis should be performed on the Piotto et al (2004) sample in order to establish the role of the normalization factor in defining the observed trend.
Based on the results of Piotto et al., Davies et al. (2004) developed a model for the production of BSS in GGCs. In the low mass systems ($M_V > -8$) BSS arise mostly from mass exchange in primordial binaries. In more massive systems collisions produce mergers of the primordial binaries early in the cluster history. BSS resulting from these mergers long ago evolved away. Once the primordial binaries were used up, BSS produced via this channel disappeared. In the cores of the most massive systems ($M_V < -9$) collisional BSS are produced (see Fig. 6 of Davies et al.). The working hypothesis proposed by Davies et al (2004) is interesting, however detailed cluster-to-cluster comparison has shown that the scenario is much more complex than that, since the dynamical history of each cluster apparently plays a significant role in determining the origin and radial distribution BSS content (see next Section).
### Cluster to cluster comparison
In the contest of a direct comparison of the BSS content in different clusters, particularly interesting is the case of M80, which shows an exceptionally high BSS content: more than 300 BSS have been discovered (Ferraro et al 1999). This is among the largest and most concentrated BSS population ever found in a GGC (see Figure 13). Indeed only the largest stellar system in the Halo, $\omega$ Centauri, has been found to harbor a BSS population (Ferraro et al 2005) larger than that discovered in M80. M80 is the GGC which has the largest central density among those not core-collapsed yet. However, the stellar density cannot explain such a large population, since other clusters with similar central density harbor much fewer BSS (see the case of 47 Tuc, Ferraro et al 2001a, NGC6388 Piotto et al 2003). Ferraro et al (1999) suggested that M80 is in a transient dynamical state during which stellar interactions are delaying the core-collapse process, leading to an exceptionally large population of collisional BSS. If this hypothesis would be further confirmed, this discovery could be the first direct evidence that stellar collisions could indeed be effective in delaying the core collapse.
Interesting enough, clusters that have already experienced (or are experiencing) the collapse of the core show a small BSS population. Indeed this is the case of NGC6752. This cluster has been recently found (see Ferraro et al 2003d) to be dynamically evolved probably undergoing a Post Core Collapse bounce (see Section 3.4). A recent search for BSS in the central region of this cluster indicated a surprisingly low BSS content: the specific number of BSS is among the lowest ever measured in a GC (Sabbi et al 2004, see Figure 14).
F03 presented a direct comparison of the BSS content for 6 clusters observed in the UV with HST (namely M3, M13, M80, NGC288, M92 and M10). Figure 15 shows the ($m_{255},~m_{255}-m_{336}$) CMDs for these clusters. More than 50,000 stars are plotted in the six panels of Figure 15. The CMD of each cluster has been shifted to match that of M3 using the brightest portion of the HB as the normalization region. The solid horizontal line (at $m_{255}=19$) in the figure marks the threshold magnitude for the selection of the bright (hereafter bBSS) sample. Such a dataset allows a direct cluster-to-cluster comparison. A number of interesting results have been obtained and in the following we just briefly discuss the two major ones:
- the specific frequency of BSS largely varies from cluster to cluster. The specific frequency of BSS compared to the number of HB stars varies from 0.07 to 0.92 for these six clusters, and does not seem to be correlated with central density, total mass, velocity dispersion, or any other obvious cluster property. Twins clusters as M3 and M13 harbor a quite different BSS population: the specific frequency in M13 is the lowest ever measured in a GC (0.07), and it turns out to be 4 times lower than that measured in M3 (0.28). Which is the origin of this difference? The paucity of BSS in M13 suggests that either the primordial population of binaries in M13 was poor or that most of them were destroyed. Alternatively, as suggested by F97, the mechanism producing BSS in the central region of M3 is more efficient than M13 because M3 and M13 are in different dynamical evolutionary phases.
In this respect, the most surprising result is that the two clusters with the largest BSS specific frequency are at the central density extremes of our sample: NGC 288 (lowest central density) and M80 (highest). The BSS specific frequency measured in these clusters (Bellazzini et al 2002, and Ferraro et al 1999) suggest that both of them have almost as many BSS as HB stars in the central region. F03 have shown that the collision channel is more than one order of magnitude more efficient in a dense cluster like M80 than in a low density cluster like NGC 288. Therefore, the high specific frequency of BSS in NGC 288 suggests that the binary fraction in this cluster is much higher than the one in M80. Only then would one expect a similar encounter frequency in the two clusters. Note that a cluster like M80 may have originally had a higher binary fraction but because of the efficiency of encounters, those primordial binaries were “used up” early in the history of the cluster, producing some collisional BSS which have evolved away from the MS (see for example the evolved BSS population found in M3 and M80 by F97 and Ferraro et al. (1999, see also Section 2.6).
However, without invoking [*ad hoc*]{} binary content, [*a more natural explanation*]{} for the origin of BSS in NGC 288 (as discussed in Bellazzini et al (2002)) is the mass transfer process in primordial binary systems (Carney et al 2001). We also know that the binary fraction in the core is $\sim 10$ to 38% (Bellazzini et al 2002), so BSS formed in this low density cluster should be the result of binary evolution rather than stellar collisions. Here we probably have another confirmation of the scenario suggested by Fusi Pecci et al (1992, and references therein): BSS living in different environments have different origins. In this case the above result demonstrated that both channels are quite efficient in producing BSS.
- In Figure 16 the magnitude distributions (equivalent to a Luminosity Function) of bBSS for the six clusters are compared. In doing this we use the parameter $\delta m_{255}$ defined as the magnitude of each bBSS with respect to the magnitude threshold (assumed at $m_{255}=19$ - see Figure 15): then $\delta m_{255}= m^{bBSS}_{255}-19.0$. From the comparison shown in Figure 16 ([*panel(a)*]{}) the bBSS magnitude distributions for M3 and M92 appear to be quite similar and both are significantly different from those obtained in the other clusters. This is essentially because in both clusters the bBSS magnitude distribution seems to have a tail extending to brighter magnitudes (the bBSS magnitude tip reaches $\delta m_{255}\sim -2.5$). A KS test applied to these two distributions yields a probability of $93$% that they are extracted from the same distribution. In [*panel(b)*]{} we see that the bBSS magnitude distribution of M13, M10 and M80 are essentially indistinguishable from each other and significantly different from M3 and M92. A KS test applied to the three LFs confirms that they are extracted from the same parent distribution. Moreover, a KS test applied to the total LFs obtained by combining the data for the two groups: M3 and M92 ([*group(a)*]{}), and M13, M80 and M10 ([*group(b)*]{}) shows that the the bBSS-LFs of [*group(a)*]{} and [*group(b)*]{} are not compatible (at $3\sigma$ level).
It is interesting to note that the clusters grouped on the basis of bBSS-LFs have some similarities in their HB morphology. The three clusters of [*group(b)*]{} have an extended HB blue tail; the two clusters of [*group(a)*]{} have no HB extension. Could there be a connection between the bBSS photometric properties and the HB morphology? This possibility needs to be further investigated.
Comparison with collisional models
----------------------------------
A preliminary comparison between the observed BSS distribution in the CMD (and their LF) with those obtained from collisional models have been presented in F03. The models used there are described in detail in Sills & Bailyn (1999). The main assumption is that all the BSS in the central regions are [*formed through stellar collisions during an encounter between a single star and a binary system*]{}. Unfortunately, while binary-binary collisions may well be important, we do not currently have the capability of modeling the BSS they might produce.
The other ingredients of the collisional models are discussed in F03, in the following we list the major assumptions:
- The trajectories of the stars during the collision are modeled using the STARLAB software package (McMillan & Hut 1996).
- The adopted mass function has an index $x=-2$, and the mass distribution within the binary systems are drawn from a Salpeter mass function ($x=1.35$).
- The adopted binary fraction is 20% with a binary period distribution which is flat in $\log P$.
- The total stellar density is derived from the central density of each cluster.
In order to explore the effects of non-constant BSS formation rates, we considered a series of truncated rates, namely constant for some portions of the cluster lifetime, and zero otherwise. This assumption is obviously unphysical—the relevant encounter rates would presumably change smoothly on timescales comparable to the relaxation time. However these models do demonstrate how the distribution of BSS in the CMD depends on when the BSS were created, and thus provide a basis for understanding more complicated and realistic formation rates.
Figure 17 shows an example of the distribution of BSS in the $m_{255},~m_{255}-m_{336}$ CMD as predicted by collisional models in the case of M80. In each panel, a different assumption for the BSS formation rate has been adopted. The differences between the models can be understood in terms of lifetimes of the individual collision products which make up the distributions. For example, if BSS production stopped 5 Gyr ago (central panel), we predict that there should be no observed bBSS at present because all the massive BSS had time to evolve off to the RGB. At the other extreme, if all the BSS were formed in the last Gyr (panel marked “1 Gyr to now”), only a few (if any) of the brightest (i.e. most massive) BSS started moving towards the subgiant branch.
As a general result, F03 concluded that the BSS formation occurred in all clusters over last 5 Gyr at least, and more were formed 2–5 Gyr ago than in the recent past. Of course this result needs to be tested using models of GC dynamical evolution in which the feedback between stellar collisions and cluster evolution is modeled explicitly. Our assumption of a BSS formation rate which is either constant or zero is unphysical, and more complicated models are clearly required.
Evolved BSS on the HB
---------------------
Renzini & Fusi Pecci (1988) suggested to search for Evolved BSS (hereafter E-BSS) during their core helium burning phase since they should appear to be redder and brighter than [*normal*]{} HB stars. Following this prescription, Fusi Pecci et al. (1992) identified a few E-BSS candidates in several clusters with predominantly blue HBs, where the likelihood of confusing E-BSS stars with true HB or evolved HB stars was minimized. Because of the small numbers there is always the possibility that some or even most of these candidate E-BSS are due to field contamination. However, near the cluster centers, field contamination should be less of a problem. Following Fusi Pecci et al (1992), F97 identified a sample of 19 E-BSS candidates in M3 (see Figure 18) and argued that the radial distribution of E-BSS was similar to that of the BSS.
The large population of BSS discovered by Ferraro et al (1999) in M80 allowed a further possibility to search for E-BSS, in fact, M80 offers some advantages over M3 in selecting E-BSS: (1) it has a very blue HB, so there should be less confusion between red HB stars and E-BSS; (2) it has a larger number of BSS. In Figure 19 [*Panel (a)*]{} a zoomed $(U,~U-V)$ CMD of the HB region is shown and the expected location for E-BSS has been indicated as a box; 19 E-BSS (plotted as large filled circles) lie in the box. The cumulative radial distribution of the E-BSS stars shown in [*Panel (b)*]{} of Figure 19 is quite similar to the BSS distribution and significantly different from that of the HB-RGB. A Kolmogorov-Smirnov test shows that the probability that the E-BSS and BSS populations have been extracted from the same distribution is $\sim 67$%, while the probability that the E-BSS and the RGB-HB population have the same distribution is only $\sim 1.6 $%. This result confirms the expectation that the E-BSS share the same distribution of the BSS and they are both a more massive population than the bulk of the cluster stars.
Under the assumption that the connection between the E-BSS and the BSS population is real, one can relate the population ratios and the lifetimes of these evolutionary stages. Earlier studies (Fusi Pecci et al 1992) have suggested that the ratio of bright BSS (b-BSS) to E-BSS is $N_{\rm b-BSS}/N_{\rm E-BSS} \approx 6.5$. Interesting enough, the population ratios are quite similar in both clusters: in M3 F97 counted 19 E-BSS over a population of 122 bBSS; the same number of E-BSS candidates have been selected by Ferraro et al (1999) in M80 compared with a population of 129 b-BSS. The population ratio turns out to be $N_{\rm b-BSS}/N_{\rm
E-BSS} = 6.6$, fully consistent with earlier studies. Moreover because both BSS and E-BSS samples are so cleanly defined in the case of M80, the ratio of the total number of BSS to E-BSS, $N_{\rm BSS} /N_{\rm
E-BSS} \sim 16$, should be useful in testing lifetimes of BSS models.
MSP companions in GCs
=====================
Among the possible collisional by-product zoo, MSPs are invaluable probes to study cluster dynamics: they form in binary systems containing a neutron star (NS) which is eventually spun up through mass accretion from the evolving companion. Despite the large difference in total mass between the disk of the Galaxy and the GC system, about $50$% of the entire MSP population has been found in the latter. This is not surprising since in the Galactic disk MSPs can only form through the evolution of primordial binaries, while in GC cores dynamical interactions can lead to the formation of several different binary systems, suitable for recycling NS.
The search for optical counterpart to MSP companion in GCs is a relatively recent branch of this research, since the first identification has been done only a few years ago in the core of 47 Tuc. Edmonds et al (2001) identified $U_{opt}$, the companion to PSR J0024$-$7203U: this object turned out to be a faint blue variable whose position in the CMD is consistent a cooling helium WD. This is fully in agreement with the MSP recycling scenario, where the usual companion to a binary MSP is an exhausted star.
The surprising companion to the MSP in NGC6397
-----------------------------------------------
A major surprise came from the optical identification of the companion to the binary MSP PSR J1740-5340 in NGC6397.
PSR J1740-5340 was identified by D’Amico et al. (2001a) during a systematic search for MSP in GCs carried out with the Parkes radio telescope. The pulsar displays eclipses at a frequency of 1.4 GHz for more than 40% of the 32.5 hr orbital period and exhibits striking irregularities of the radio signal at all orbital phases (D’Amico et al 2001b). This suggests that the MSP is orbiting within a large envelope of matter released from the companion, whose interaction with the pulsar wind could be responsible for the modulated and probably extended X-ray emission detected with CHANDRA (Grindlay et al. 2001, 2002). By using high resolution multiband HST observations and the position of the MSP inferred from radio timing, Ferraro et al (2001b) identified a bright variable star (hereafter COM J1740-5340) as the optical counterpart to the MSP companion, whose optical modulation nicely agrees with the orbital period of the MSP itself. The optical counterpart shows a quite anomalous position in the CMD since it is located at the luminosity of the TO point but it has an anomalous red color (see Figure 20).
As quoted above, in the classical framework of the MSP recycling scenario, the usual companion to a binary MSP should be either a WD or a very light ($0.01-0.03~M_{\odot}$), almost exhausted star. None of these scenarios can be applied to COM J1740-5340: it is too luminous to be a WD ($V\sim16.5$, comparable to the TO stars of NGC6397); moreover its mass ($M\sim
0.2 M_{\odot}$), recently constrained by radial velocity observations (see Figure 21), is too high for a very light stellar companion. As a consequence, a wealth of intriguing scenarios have flourished in order to explain the nature of this binary (see Orosz & van Kerkwijk 2003, Grindlay et al. 2002 for a review). In particular, Burderi et al. (2002) suggested that the position of COM J1740-5340 in the CMD (CMD) is consistent with the evolution of an (slightly) evolved Sub Giant Branch (SGB) star orbiting the NS and loosing mass. The future evolution of this system will generate a He-WD/MSP pair (see [*panel (d)*]{} in Figure 20). COM J1740-5340 could be a star acquired by exchange interaction in the cluster core or alternatively the same star that spun up the MSP and still overflowing its Roche lobe. The latter case suggests the fascinating possibility that PSR J1740-5340 is a new-born MSP, the very first observed just after the end of the recycling process.
This is the first example ever observed of a MSP companion whose light curve is dominated by ellipsoidal variations, suggestive of a tidally distorted star, which almost completely fills (and is still overflowing) its Roche lobe. Thanks to the unusual brightness of the companion ($V\sim
16.5$), this system represents an unique laboratory to study the formation mechanism of binary MSP in GCs, allowing unprecedented detailed spectroscopic observations. In this contest, our group is coordinating a spectro-photometric programme at ESO-Telescopes. In particular a first set of high resolution spectra have been acquired at the [*Very Large Telescope*]{} (VLT) with UVES. From these data we obtained a number of interesting results:
- The determination of the radial velocity curve (see Figure 21) allowed an accurate measure of the mass ratio of the system ($M_{PRS}/M_{COM}=5.83\pm0.13$) which suggests a mass of $M_{COM}\sim 0.25 M_{\odot}$ by assuming $M_{PSR}\sim 1.4 M_{\odot}$. (Ferraro et al 2003b).
- The $H_{\alpha}$ emission from the system was already noted by Ferraro et al (2001b, see also [*Panel (c)*]{} in Figure 20) and fully confirmed by the high-resolution spectra (Sabbi et al 2003a). In particular, the complex structure of the $H_{\alpha}$ line suggests the presence of a matter stream exaping from the companion towards the NS. Note that because of the radiation flux from the pulsar, the material would never reach the NS surface, creating a cometary-like gaseous tail which feeds the presence of (optically thin) hydrogen gas outside the Roche lobe.
- The unexpected detection of strong He I absorption lines (see Figure 22) implies the existence of a region at $T>10,000K$, significantly hotter than the rest of the star (Ferraro et al 2003b, Sabbi et al 2003a,b). The intensity of the He I line correlates with the orbital phase, suggesting the presence of a region on the companion surface, heated by the millisecond pulsar flux.
- COM J1740-5340 has been found to show a large rotation velocity ($V sin i = 50\pm 1 Km s^{-1}$). The derived abundances are found fully consistent with those of normal unperturbed stars in NGC 6397, with the exception of a few elements (Li, Ca, and C). In particular, the lack of C suggests that the star has been peeled down to regions where incomplete CNO burning occurs (Sabbi et al. 2003b), favoring a scenario where the companion is a SGB star which has lost most of its mass (see also Ergma & Sarna 2003).
The case of NGC6752
-------------------
Another interesting object has captured our attention: the binary MSP PSR J1911-5958A (hereafter PSR-A), recently discovered in the outskirts of the nearby GC NGC 6752 (D’Amico et al. 2001a). It is located quite far away (at about $6'$) from the cluster optical center. Indeed PSR-A is the more off-centered pulsar among the sample of 41 MSPs with known positions in the parent cluster.
By using deep high-resolution multiband images taken at the ESO VLT we recently identified the optical binary companion to PSR-A (COM-PSR-A, Ferraro et al 2003c). The object turns out to be a blue star whose position in the CMD is consistent with the cooling sequence of a low mass ($M\sim 0.17-0.20M_{\odot}$), low metallicity Helium WD (He-WD) (see Figure 23, see also Bassa et al 2003). The anomalous position of PSR-A with respect to the GC center suggested that this system has been recently ($< 1$ Gyr) ejected from the cluster core as the result of a strong dynamical interaction.
Photometric properties of MSP companion in GCs
----------------------------------------------
The companion to PSR-A is the second He-WD which has been found to orbit a MSP in GCs. Curiously, both objects lie on the same mass He-WD cooling sequence.
Since the first discovery of $U_{opt}$, in 47 Tuc (Edmonds et al 2001), the zoo of the optical MSP counterparts in GCs is rapidly enlarging. Figure 24 shows a comparison of the photometric properties of the available optical identifications of MSP companions hosted in GCs. Note that we also include an additional potential MSP companion (W34 in 47 Tuc) discussed by Edmonds et al. (2003). 2 out of the 5 sources are really peculiar: the bright object in NGC6397 (which is as luminous as the turn off stars and shows quite red colors) and the faint W29 in 47 Tuc, which is also too red to be a He-WD (Edmonds et al 2002). Indeed, $U_{opt}$ and COM-PSR-A lie nearly on the same mass He-WD cooling sequence and W34 in 47 Tuc curiously shares the same photometric properties of COM-PSR-A. Indeed, if confirmed as a MSP companion, W34 would be the third He-WD companion orbiting a MSP in GCs roughly located on the same-mass cooling sequence. If further supported by additional cases, this evidence could confirm that a low mass $\sim
0.15-0.2~M_{\odot}$ He-WD orbiting a MSP is the favored system generated by the recycling process of MSPs in GCs (Rasio et al 2000).
MSPs as probes of the cluster dynamics
--------------------------------------
Among the list of the GCs harboring MSP, NGC6752 presents a number of surprising features:
- NGC 6752 hosts 5 known MSP (hereafter PSR A, B, C, D, E, D’Amico et al. 2001; D’Amico et al. 2002).
- As discussed in the previous section, NGC6752 harbors the two most off-centered pulsars (PSR-A and PSR-C) among the sample of MSPs whose position in the corresponding cluster is known.
- In the plane of the sky, the positions of the other 3 known MSPs in the cluster (PSR-B, D, E, all isolated pulsars, see D’Amico et al. 2001a, 2002) are close to the cluster center, as expected from mass segregation in the cluster. D’Amico et al. (2002) found two of them (PSR-B and E) showing large [*negative*]{} values of $\dot{P}$, implying that the pulsars are experiencing an acceleration with a line-of-sight component $a_l$ directed toward the observer and a magnitude significantly larger than the positive component of $\dot{P}$ due to the intrinsic pulsar spin-down.
- By combining HST and wide field observations, Ferraro et al (2003d) constructed the most extended ($0'-27'$) and complete radial density profile ever obtained for this cluster (see Figure 25). The observed radial density profile shows a significant deviation from a canonical King model in the innermost region, indicating, beyond any doubt, that the cluster core is experiencing the collapse phase. On the basis of this data set Ferraro et al (2003d) also re-determined the center of gravity $C_{\rm grav}$, which turns out to be $\sim
10\arcsec$ S and $\sim 2\arcsec$ E of the $C_{\rm lum}$ reported by Djorgovski (1993). Interestingly, the barycenter of the 9 innermost X-ray sources detected by [*Chandra*]{} (Pooley et al 2002) is located only $1\farcs 9$ from the new $C_{\rm grav}.$
On the basis of these new results, Ferraro et al (2003d) suggested two viable explanations of the observed negative $\dot{P}$: [*(i)*]{} the accelerating effect of the cluster gravitational potential well or [*(ii)*]{} the presence of some close perturbator(s) exerting a gravitational pull on the pulsars.
The hypothesis that the line-of-sight acceleration of the MSPs with negative $\dot{P}$ is dominated by the cluster gravitational potential has been routinely applied to many globulars. In the case of NGC6752, under the hypothesis that the line-of-sight acceleration of PSR-B and PSR-E are entirely due to the cluster gravitational potential, a ${\cal{M}}/{\cal L}_V {\sim} 6$–7 is inferred (see Figure 26). Collapsed globulars typically show values of ${\cal{M}}/{\cal L}_V {\sim} 2$–3.5 (Pryor & Meylan 1993). Assuming such a value, the expected total mass located within the inner $r_{\perp,B}=0.08$ pc of NGC 6752 (equivalent to the projected displacement of PSR-B from $C_{\rm grav}$) would be $\sim
1200$–$2000\,{M_\odot}$. On the other hand, the observed ${\cal{M}}/{\cal{L}}_V \sim 6$–7 implies the existence of an additional $\sim 1500$–$2000\,{M_\odot}$ of low-luminosity matter segregated inside the projected radius $r_{\perp,B}.$ This extra amount of mass cannot be a relatively massive ($> 10^3\,{M_\odot}$) black hole (BH), since it would produce a central power-law cusp in the radial density profile, which is not observed (see Figure 25). The high ${\cal{M}}/{\cal{L}}_V$ ratio could be more likely due to a very high central concentration of dark remnants of stellar evolution, like NS and heavy ($\sim 1.0\,{M_\odot}$) WD, which sank into the NGC 6752 core during the cluster dynamical evolution (as also proposed for M15 by Gerssen et al. 2003 and Baumgardt et al. 2003).
One can imagine as an alternative possibility that the acceleration imparted to PSR-B and PSR-E is due to some local perturbators. Could a [*single*]{} object, significantly more massive than a typical star in the cluster simultaneously produce the accelerations detected both in PSR-B and PSR-E? Recently, Colpi, Possenti & Gualandris (2002) suggested the presence of a binary BH of moderate mass ($M_{\rm
bh+bh}\sim 100$–$200\,{ M_\odot}$) in the center of NGC 6752, in order to explain the unexpected position of PSR-A in the outskirts of the cluster. The projected separation of PSR-B and PSR-E is only $d_{\perp}=0.03$ pc. A binary BH of total mass $M_{\rm bh+bh}$, approximately located in front of them within a distance of the same order of $d_{\perp}$, could accelerate both pulsars without leaving any observable signature on the cluster density profile. As the BH binarity ensures a large cross section to interaction with other stars, the recoil velocity $v_{\rm rec}$ due to a recent dynamical encounter could explain the offset position (with respect to $C_{\rm grav}$) of the BH. However this scenario has allow probability to occur: indeed, placing the BH at random within the core gives roughly a 1% chance that it would land in the right location to produce the observed pulsar accelerations.
It is a pleasure to thank Bob Rood, Alison Sills, Michela Mapelli, Monica Colpi, Giacomo Beccari, Elena Sabbi, Andrea Possenti, Nichi D’Amico and the many other friends who are collaborating to this project. This research was supported by the [*Ministero della Istruzione dell’Università e della Ricerca*]{} (MIUR). A special thanks to Livia for her lovely support.
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---
abstract: 'In this paper, we obtain the uniqueness of the 2D MHD equations, which fills the gap of recent work [@1] by Chemin et al.'
address: 'Department of Mathematics, Zhejiang University, Hangzhou 310027, China'
author:
- Renhui Wan
title: On the uniqueness for the 2D MHD equations without magnetic diffusion
---
MHD equations, Uniqueness, Magnetic diffusion
Introduction
============
This paper considers the 2D MHD equations given by $$\label{1.1}
\left\{
\begin{array}{l}
\partial_t u + u\cdot\nabla u + \nabla p-\nu\Delta u = B\cdot\nabla B, \\
\partial_t B + u\cdot\nabla B -B\cdot\nabla u=0,\\
{\rm div} u=0,\quad {\rm div}B=0, \\
u(x,0) =u_0(x), \quad B(x,0) =B_0(x),
\end{array}
\right.$$ here $t\ge0$, $x\in \mathbb{R}^2$, $u=u(x,t)$ and $B=B(x,t)$ are vector fields representing the velocity and the magnetic field, respectively, $p=p(x,t)$ denotes the pressure and $\nu$ is a positive viscosity constant. .1in (\[1.1\]) has been investigated by many mathematicians. In 2014, by establishing a generalized Kato-Ponce estimate (see [@Kato] for the well-known result): $$<u\cdot\nabla B\mid B>_{\dot{H}^s}\le C\|\nabla u\|_{H^s}\|B\|_{H^s}^2,\ \ s>\frac{d}{2},\ d=2,3,$$ Fefferman et al. [@Fef2] obtained the local existence and uniqueness for (\[1.1\]) and related models with the initial data $(u_{0},B_{0})\in H^s(\mathbb{R}^d),\ s>\frac{d}{2}.$ For other results concerning regularity criterions, we refer to [@Fan1] and [@Zhou1].
Very recently, Chemin et al. in [@1] obtain the local existence for (\[1.1\]) in 2D and 3D. But for the 2D case, the uniqueness was not obtained. Our main result is filling the gap of their works. The details can be described as follows
\[t1.1\] For $u_{0}\in B_{2,1}^0(\mathbb{R}^2)$ and $B_{0}\in B_{2,1}^1(\mathbb{R}^2)$ with ${\rm div}u_{0}={\rm div}B_{0}=0$, there exists a time $T=T(\nu,\|u_{0}\|_{B_{2,1}^0},\|B\|_{B_{2,1}^1})>0$ such that the system (\[1.1\]) has a unique solution $(u,B)$ with $$u\in C([0,T];B_{2,1}^0(\mathbb{R}^2))\cap L^1([0,T];B_{2,1}^2)$$ and $$B\in C([0,T];B_{2,1}^1(\mathbb{R}^2)).$$
.3in
Preliminaries
=============
Let $\mathfrak{B}=\{\xi\in\mathbb{R}^d,\ |\xi|\le\frac{4}{3}\}$ and $\mathfrak{C}=\{\xi\in\mathbb{R}^d,\ \frac{3}{4}\le|\xi|\le\frac{8}{3}\}$. Choose two nonnegative smooth radial function $\chi,\ \varphi$ supported, respectively, in $\mathfrak{B}$ and $\mathfrak{C}$ such that $$\chi(\xi)+\sum_{j\ge0}\varphi(2^{-j}\xi)=1,\ \ \xi\in\mathbb{R}^d,$$ $$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\ \ \xi\in\mathbb{R}^d\setminus\{0\}.$$ We denote $\varphi_{j}=\varphi(2^{-j}\xi),$ $h=\mathfrak{F}^{-1}\varphi$ and $\tilde{h}=\mathfrak{F}^{-1}\chi,$ where $\mathfrak{F}^{-1}$ stands for the inverse Fourier transform. Then the dyadic blocks $\Delta_{j}$ and $S_{j}$ can be defined as follows $$\Delta_{j}f=\varphi(2^{-j}D)f=2^{jd}\int_{\mathbb{R}^d}h(2^jy)f(x-y)dy,$$ $$S_{j}f=\sum_{k\le j-1}\Delta_{k}f=\chi(2^{-j}D)f=2^{jd}\int_{\mathbb{R}^d}\tilde{h}(2^jy)f(x-y)dy.$$ Formally, $\Delta_{j}=S_{j}-S_{j-1}$ is a frequency projection to annulus $\{\xi:\ C_{1}2^j\le|\xi|\le C_{2}2^j\}$, and $S_{j}$ is a frequency projection to the ball $\{\xi:\ |\xi|\le C2^j\}$. One can easily verifies that with our choice of $\varphi$ $$\Delta_{j}\Delta_{k}f=0\ if \ |j-k|\ge2\ \ {\rm and}\ \ \Delta_{j}(S_{k-1}f\Delta_{k}f)=0\ if |j-k|\ge5.$$ With the introduction of $\Delta_{j}$ and $S_{j}$, let us recall the definition of the Besov space.
Let $s\in \mathbb{R}$, $(p,q)\in[1,\infty]^2,$ the homogeneous space $\dot{B}_{p,q}^s$ is defined by $$\dot{B}_{p,q}^{s}=\{f\in \mathfrak{S}';\ \|f\|_{\dot{B}_{p,q}^{s}}<\infty\},$$ where $$\|f\|_{\dot{B}_{p,q}^s}=\left\{\begin{aligned}
&\displaystyle (\sum_{j\in \mathbb{Z}}2^{sqj}\|\Delta_{j}f\|_{L^p}^{q})^\frac{1}{q},\ \ \ \ {\rm for} \ \ 1\le q<\infty,\\
&\displaystyle \sup_{j\in\mathbb{Z}}2^{sj}\|\Delta_{j}f\|_{L^p},\ \ \ \ \ \ \ \ \ {\rm for}\ \ q=\infty,\\
\end{aligned}
\right.$$ and $\mathfrak{S}'$ denotes the dual space of $\mathfrak{S}=\{f\in\mathcal{S}(\mathbb{R}^d);\ \partial^{\alpha}\hat{f}(0)=0;\ \forall\ \alpha\in \ \mathbb{N}^d $ [multi-index]{}} and can be identified by the quotient space of $\mathcal{S'}/\mathcal{P}$ with the polynomials space $\mathcal{P}$.
Let $s>0,$ and $(p,q)\in [1,\infty]^2$, the inhomogeneous Besov space $B_{p,q}^s$ is defined by $${B}_{p,q}^{s}=\{f\in \mathcal{S'}(\mathbb {R}^d);\ \|f\|_{{B}_{p,q}^{s}}<\infty\},$$ where $$\|f\|_{B_{p,q}^s}=\|f\|_{L^p}+\|f\|_{\dot{B}_{p,q}^s}.$$ Let’s recall space-time space.
\[d1\] Let $s\in\mathbb{R}.$ $1\le p,q,r\le \infty$, $I\subset\mathbb{R}$ is an interval. The homogeneous mixed time-space Besov space $\tilde{L}^r(I;\dot{B}_{p,q}^s)$ is defined as the set of all the distributions $f$ satisfying $$\|f\|_{\tilde{L}^r(I;\dot{B}_{p,q}^s)}=\left\|2^{sj}\left(\int_{I}\|\Delta_{j}f(\tau)\|_{L^p}^rd\tau\right)^\frac{1}{r}\right\|_{l^q(\mathbb{Z})}<\infty.$$
For convenience, we sometimes use $\tilde{L}^r_{t}\dot{B}_{p,q}^s$ and $ L^r_{t}\dot{B}_{p,q}^s$ to denote $\tilde{L}^r(0,t;\dot{B}_{p,q}^s)$ and $L^r(0,t;\dot{B}_{p,q}^s)$.
.1in Bernstein’s inequalities are useful tools in dealing with Fourier localized functions and these inequalities trade integrability for derivatives. The following proposition provides Bernstein type inequalities for fractional derivatives.
\[p2.1\] Let $\alpha\ge0$. Let $1\le p\le q\le \infty$.
1. If $f$ satisfies $$\mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, |\xi|
\le K 2^j \},$$ for some integer $j$ and a constant $K>0$, then $$\|(-\Delta)^\alpha f\|_{L^q(\mathbb{R}^d)} \le C_1\, 2^{2\alpha j +
j d(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p(\mathbb{R}^d)}.$$
2. If $f$ satisfies $$\label{spp}
\mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, K_12^j
\le |\xi| \le K_2 2^j \}$$ for some integer $j$ and constants $0<K_1\le K_2$, then $$C_1\, 2^{2\alpha j} \|f\|_{L^q(\mathbb{R}^d)} \le \|(-\Delta)^\alpha
f\|_{L^q(\mathbb{R}^d)} \le C_2\, 2^{2\alpha j +
j d(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p(\mathbb{R}^d)},$$ where $C_1$ and $C_2$ are constants depending on $\alpha,p$ and $q$ only.
For more details about Besov space such as some useful embedding relations and the equivalency $$\|f\|_{\dot{B}_{2,2}^s}\approx\|f\|_{\dot{H}^s},\qquad \|f\|_{B_{2,2}^s}\approx\|f\|_{H^s},$$ see [@2],[@3] and [@4]. .3in
Proof of The main result
========================
Before the proof of Theorem \[t1.1\], we need the following lemma.
\[l3.1\] $\forall\ t>0,$ $$\label{3.1}
\int_{0}^{t}\|f(\tau)\|_{\dot{B}_{2,1}^1}d\tau\le C(\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}+\|f\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}})\log\left(e+\frac{\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^0}+\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^2} }{\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}+\|f\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}}\right).$$
Using the definition of homogeneous Besov space, we have $$\begin{aligned}
\|f\|_{L^1_{t}\dot{B}_{2,1}^1}=&\sum_{j\in\mathbb{Z}}2^j\|\Delta_{j}f\|_{L^1_{t}L^2}\\
=&\sum_{j<-N}2^j\|\Delta_{j}f\|_{L^1_{t}L^2}
+\sum_{-N\le j\le N}2^j\|\Delta_{j}f\|_{L^1_{t}L^2}+\sum_{j>N}2^j\|\Delta_{j}f\|_{L^1_{t}L^2}\\
\le& 2^{-N}\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^0}+2N\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}+2^{-N}\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^2}.
\end{aligned}$$ Choosing $$N=\log\left(e+\frac{\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^0}+\|f\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^2} }{\|f\|_{\tilde{L}^1\dot{B}_{2,\infty}^1}+\|f\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}}\right),$$ we can get the inequality (\[3.1\])
Now, we begin the proof of Theorem \[t1.1\]. The existence of the solution to (\[1.1\]) was obtained in [@1], while the continuity in time can be obtained by the definition of Besov space. So here we only deal with the uniqueness. Let $(u_{j},B_{j}),\ j=1,2,$ be two solution of (\[1.1\]), denote $\delta u=u_{1}-u_{2}$, $\delta B=B_{1}-B_{2}$ and $\delta p=p_{1}-p_{2}$, then we obtain $$\label{3.2}
\partial_{t}\delta u+(u_{1}\cdot\nabla)\delta u+(\delta u\cdot\nabla)u_{2}-\nu\Delta\delta u+\nabla\delta p=(B_{1}\cdot\nabla)\delta B+(\delta B\cdot\nabla)B_{2}$$ and $$\label{3.3}
\partial_{t}\delta B+(u_{1}\cdot\nabla)\delta B+(\delta u\cdot\nabla)B_{2}=(B_{1}\cdot\nabla)\delta u+(\delta B\cdot\nabla)u_{2}.$$ First, we consider (\[3.2\]). By a standard argument, we have $$\begin{aligned}
\frac{d}{dt}\|\Delta_{j}\delta u\|_{L^2}&+\nu2^{2j}\|\Delta_{j}\delta u\|_{L^2}\le \|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^2}\\
&+\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^2}+\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^2}+\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^2},
\end{aligned}$$ which with Gronwall’s inequality yields that $$\begin{aligned}
\|\Delta_{j}\delta u\|_{L^2}\le& \int_{0}^{t} e^{\nu2^{2j}(\tau-t)}(\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^2}\\
&+\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^2}+\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^2}+\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^2})d\tau.
\end{aligned}$$ Taking the $L^r(0,t)$ norm, and using Young’s inequality to obtain $$\begin{aligned}
\|\Delta_{j}\delta u\|_{L^r_{t}L^2}\le& \|e^{-\nu2^{2j}\tau}\|_{L^r_{t}} (\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^1_{t}L^2}\\
&+\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^1_{t}L^2}+\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^1_{t}L^2}+\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^1_{t}L^2})\\
\le&(\nu2^{2j}r)^{-\frac{1}{r}}(\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^1_{t}L^2}\\
&+\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^1_{t}L^2}+\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^1_{t}L^2}+\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^1_{t}L^2}).
\end{aligned}$$ Multiplying $2^{-j}$, and taking the $l^\infty$ norm, we obtain $$\begin{aligned}
\nu^\frac{1}{r}\|\delta u\|_{\tilde{L}^r_{t}\dot{B}_{2,\infty}^{-1+\frac{2}{r}}}\le& \sup_{j\in\mathbb{Z}}2^{-j}(\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^1_{t}L^2}\\
&+\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^1_{t}L^2}+\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^1_{t}L^2}+\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^1_{t}L^2})\\
=&K_{1}+K_{2}+K_{3}+K_{4},
\end{aligned}$$ where $$\begin{aligned}
&K_1 =\sup_{j\in\mathbb{Z}}2^{-j}\|[\Delta_{j},u_{1}\cdot\nabla]\delta u\|_{L^1_{t}L^2}, \qquad K_2 = \sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(\delta u\cdot\nabla u_{2})\|_{L^1_{t}L^2},\\
&K_3 = \sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(B_{1}\cdot\nabla\delta B)\|_{L^1_{t}L^2}, \qquad K_4 =\sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}(\delta B\cdot\nabla B_{2})\|_{L^1_{t}L^2}.\end{aligned}$$ In the following, we will bound $K_{i}$, $i=1,2,3,4.$ By homogeneous Bony decomposition, we can split $K_{1}$ into four parts, $$\label{3.4}
\begin{aligned}
K_{1}\le& \sup_{j\in\mathbb{Z}}2^{-j}\sum_{|k-j|\le4}\|[\Delta_{j},S_{k-1}u_{1}\cdot\nabla]\Delta_{k}\delta u\|_{L^1_t L^2}
+\sup_{j\in\mathbb{Z}}2^{-j}\sum_{|k-j|\le4}\|\Delta_{j}(\Delta_{k}u_{1}\cdot\nabla S_{k-1}\delta u)\|_{L^1_t L^2}\\
&+\sup_{j\in\mathbb{Z}}2^{-j}\sum_{k\ge j-3}\|\Delta_{k}u_{1}\cdot\nabla\Delta_{j}S_{k+1}\delta u\|_{L^1_t L^2}
+\sup_{j\in\mathbb{Z}}2^{-j}\sum_{k\ge j-3}\|\Delta_{j}(\Delta_{k}u_1 \cdot\nabla\tilde{\Delta}_k\delta u)\|_{L^1_t L^2}\\
=&K_{11}+K_{12}+K_{13}+K_{14},
\end{aligned}$$ where $\tilde{\Delta}_{k}=\Delta_{k-1}+\Delta_{k}+\Delta_{k+1}.$\
By Hölder’s inequality, standard commutator estimate and Bernstein’s inequality, $$K_{11}\le C\sup_{j\in\mathbb{Z}}2^{-j}\|\nabla u_{1}\|_{L^{1}_{t}L^\infty}\|\Delta_{j}\delta u\|_{L^\infty_{t}L^2}\le C\|u_{1}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}},$$ $$K_{12}\le C\sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j} u_{1}\|_{L^{1}_{t}L^\infty}\|\nabla S_{j-1}\delta u\|_{L^\infty_{t}L^2}\le C\|u_{1}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}},$$ $$\begin{aligned}
K_{13}\le& C\sup_{j\in\mathbb{Z}}2^{-j}\sum_{k\ge j-3}2^j\|\Delta_{j}\delta u\|_{L^\infty_{t}L^2}\|\Delta_{k}u_{1}\|_{L^1_{t}L^\infty}\\
\le& C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^{j-k}2^{-j}\|\Delta_{j}\delta u\|_{L^\infty_{t}L^2}2^k\|\Delta_{k}u_{1}\|_{L^1_{t}L^\infty}\\
\le& C\|u_{1}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}\end{aligned}$$ and $$K_{14}\le C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^j\|\Delta_{k}u_{1}\|_{L^1_{t}L^2}\|\tilde{\Delta}_{k}\delta u\|_{L^\infty_{t}L^2}\le C\|u_{1}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}.$$ Collecting the estimates above in (\[3.4\]), we obtain $$K_{1}\le C\|u_{1}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}.$$ By homogeneous Bony decomposition again, $$\begin{aligned}
K_{2}\le& \sup_{j\in\mathbb{Z}}2^{-j}\sum_{|k-j|\le4}\|\Delta_{j}(\Delta_{k}\delta u\cdot\nabla S_{k-1}u_{2})\|_{L^1_{t}L^2}\\
&+\sup_{j\in\mathbb{Z}}2^{-j}\sum_{|k-j|\le4}\|\Delta_{j}(S_{k-1}\delta u\cdot\nabla \Delta_{k}u_{2})\|_{L^1_{t}L^2}\\
&+\sup_{j\in\mathbb{Z}}2^{-j}\sum_{k\ge j-3}\|\Delta_{j}(\Delta_{k}\delta u\cdot\nabla \tilde{\Delta}_{k}u_{2})\|_{L^1_{t}L^2}\\
=&K_{21}+K_{22}+K_{23}.\end{aligned}$$ By Hölder’s inequality and Bernstein’s inequality, $$K_{21}\le C\|\nabla u_{2}\|_{L^1_{t}L^\infty}\sup_{j\in\mathbb{Z}}2^{-j}\|\Delta_{j}\delta u\|_{L^\infty_{t}L^2}\le C\|u_{2}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}},$$ $$\begin{aligned}
K_{22}\le& C\sup_{j\in\mathbb{Z}}2^{-j}\|S_{j-1}\delta u\|_{L^\infty_{t}L^\infty}\|\nabla \Delta_{j}u_{2}\|_{L^1_{t}L^2}\\
\le& C\sup_{j\in\mathbb{Z}}2^j\|\nabla\Delta_{j}u\|_{L^1_{t}L^2}2^{-j}\|S_{j-1}\delta u\|_{L^\infty_{t}L^2}\\
\le& C\|u_{2}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}},\end{aligned}$$ $$\begin{aligned}
K_{23}\le& C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^j\|\Delta_{k}\delta u\|_{L^\infty_{t}L^2}\|\tilde{\Delta}_{k} u_{2}\|_{L^1_{t}L^2}\\
\le& C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^{j-k}2^{2k}\|\tilde{\Delta}_{k}u_{2}\|_{L^1_{t}L^2}2^{-k}\|\Delta_{k}\delta u\|_{L^\infty_{t}L^2}\\
\le& C\|u_{2}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}.\end{aligned}$$ Thus we have $$K_{2}\le C\|u_{2}\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}.$$ Similarly, we can bound $K_{3}$ and $K_{4}$ as follows: $$K_{3}\le \|B_{1}\cdot\nabla \delta B\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^{-1}}\le \int_{0}^{t}\|B_{1}\cdot\nabla\delta B\|_{\dot{B}_{2,\infty}^{-1}}d\tau\le
\int_{0}^{t}\|B_{1}\|_{\dot{B}_{2,1}^1}\|\delta B\|_{\dot{B}_{2,\infty}^0}d\tau$$ and $$K_{4}\le \|\delta B\cdot\nabla B_{2}\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^{-1}}\le \int_{0}^{t}\|\delta B\cdot\nabla B_{2}\|_{\dot{B}_{2,\infty}^{-1}}d\tau\le
\int_{0}^{t}\|B_{2}\|_{\dot{B}_{2,1}^1}\|\delta B\|_{\dot{B}_{2,\infty}^0}d\tau.$$ Therefore, $$\label{3.5}
\begin{aligned}
\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}&+\nu\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}\\
\le& C\|(u_{1},u_{2})\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+C\int_{0}^{t}\|(B_{1},B_{2})\|_{\dot{B}_{2,1}^1}\|\delta B\|_{\dot{B}_{2,\infty}^0}d\tau.
\end{aligned}$$ Next, we consider (\[3.3\]), we have the following estimate, $$\begin{aligned}
\frac{d}{dt}\|\delta B\|_{\dot{B}_{2,\infty}^0}\le& \sup_{j\in\mathbb{Z}}\|[\Delta_{j},u_{1}\cdot\nabla]\delta B\|_{L^2}\\
&+\sup_{j\in\mathbb{Z}}\|\Delta_{j}(\delta u\cdot\nabla B_{2})\|_{L^2}+\sup_{j\in\mathbb{Z}}\|\Delta_{j}(B_{1}\cdot\nabla \delta u)\|_{L^2}+\sup_{j\in\mathbb{Z}}\|\Delta_{j}(\delta B\cdot\nabla u_{2})\|_{L^2}\\
=&J_{1}+J_{2}+J_{3}+J_{4}.
\end{aligned}$$ By homogeneous Bony decomposition, $$\begin{aligned}
J_{1}\le& \sup_{j\in\mathbb{Z}}\sum_{|k-j|\le 4}\|[\Delta_{j},S_{k-1}u_{1}\cdot\nabla]\Delta_{k}\delta B\|_{L^2}+\sup_{j\in\mathbb{Z}}\sum_{|k-j|\le4}\|\Delta_{j}(\Delta_{k}u_{1}\cdot\nabla S_{k-1}\delta B)\|_{L^2}\\
&+\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}\|\Delta_{k}u_{1}\cdot\nabla S_{k+1}\Delta_{j}\delta B\|_{L^2}+
\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}\|\Delta_{j}(\Delta_{k}u_{1}\cdot\nabla\tilde{\Delta}_{k}\delta B)\|_{L^2}\\
=&J_{11}+J_{12}+J_{13}+J_{14}.
\end{aligned}$$ By Hölder’s inequality and Bernstein’s inequality, $$J_{11}\le C\sup_{j\in\mathbb{Z}}\|\nabla u_{1}\|_{L^\infty}\|\Delta_{j}\delta B\|_{L^2}\le C\|u_{1}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0},$$ $$J_{12}\le C\sup_{j\in\mathbb{Z}}2^{2j}\|\Delta_{j}u_{1}\|_{L^2}2^{-2j}\|\nabla S_{j-1}\delta B\|_{L^\infty}\le C\|u_{1}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0},$$ $$\begin{aligned}
J_{13}\le& C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^j\|\Delta_{k}u\|_{L^\infty}\|\Delta_{j}\delta B\|_{L^2}\\
\le& C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^{j-k}2^{2k}\|\Delta_{k}u\|_{L^2}\|\Delta_{j}\delta B\|_{L^2}
\le C\|u_{1}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0},\end{aligned}$$ $$J_{14}\le C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^{2j}\|\Delta_{k}u\|_{L^2}\|\tilde{\Delta}_{k}\delta B\|_{L^2}\le C\|u_{1}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}.$$ Hence we have $$J_{1}\le C\|u_{1}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}.$$ By the inequality $$\|fg\|_{\dot{B}_{2,1}^{1}}\le C\|f\|_{\dot{B}_{2,1}^{1}}\|g\|_{\dot{B}_{2,1}^{1}},$$ (see, e.g., [@5]), we have $$J_{2}+J_{3}\le \|\delta u\cdot \nabla B_{2}\|_{\dot{B}_{2,\infty}^0}+\|B_{1}\cdot \nabla\delta u\|_{\dot{B}_{2,\infty}^0}\le C\|\delta u\|_{\dot{B}_{2,1}^1}\|(B_{1},B_{2})\|_{\dot{B}_{2,1}^1}.$$ Finally, we bound $J_{4}$. By homogeneous Bony decomposition, $$\begin{aligned}
J_{4}\le& \sup_{j\in \mathbb{Z}}\sum_{|k-j|\le4}\|\Delta_{j}(\Delta_{k}\delta B\cdot\nabla S_{k-1}u_{2})\|_{L^2}+
\sup_{j\in \mathbb{Z}}\sum_{|k-j|\le4}\|\Delta_{j}(S_{k-1}\delta B\cdot\nabla \Delta_{k}u_{2})\|_{L^2}\\
&+\sup_{j\in \mathbb{Z}}\sum_{k\ge j-3}\|\Delta_{j}(\Delta_{k}\delta B\cdot\nabla\tilde{\Delta}_{k}u_{2})\|_{L^2}\\
=&J_{41}+J_{42}+J_{43},\end{aligned}$$ where $$J_{41}\le C\sup_{j\in\mathbb{Z}}\|\nabla u_{2}\|_{L^\infty}\|\Delta_{j}\delta B\|_{L^2}\le C\|u_{2}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0},$$ $$J_{42}\le C\sup_{j\in\mathbb{Z}}2^j\|\nabla\Delta_{j}u_{2}\|_{L^2}2^{-j}\|S_{j-1}\delta B\|_{L^\infty}\le C\|u_{2}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}$$ and $$J_{43}\le C\sup_{j\in\mathbb{Z}}\sum_{k\ge j-3}2^{2j}\|\tilde{\Delta}_{k}u_{2}\|_{L^2}\|\Delta_{k}\delta B\|_{L^2}\le C\|u_{2}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}.$$ So $$J_{4}\le C\|u_{2}\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}.$$ Therefore, $$\frac{d}{dt}\|\delta B\|_{\dot{B}_{2,\infty}^0}\le C\|(u_{1},u_{2})\|_{\dot{B}_{2,1}^2}\|\delta B\|_{\dot{B}_{2,\infty}^0}+C\|(B_{1},B_{2})\|_{\dot{B}_{2,1}^1}\|\delta u\|_{\dot{B}_{2,1}^1},$$ which implies that $\forall\ 0\le t\le T,$ $T$ is the lifespan of the solution, $$\label{3.6}
\|\delta B\|_{L^\infty_{t}\dot{B}_{2,\infty}^0}\le C\|(u_{1},u_{2})\|_{L^1_{t}\dot{B}_{2,1}^2}\|\delta B\|_{L^\infty_{t}\dot{B}_{2,\infty}^0}+
C\|(B_{1},B_{2})\|_{L^\infty_{t}\dot{B}_{2,1}^1}\|\delta u\|_{L^1_{t}\dot{B}_{2,1}^1}.$$ Set $0<\bar{T}<T$ such that $$\int_{0}^{\bar{T}}\|(u_{1},u_{2})\|_{\dot{B}_{2,1}^2}dt<\frac{1}{4C},$$ then $\forall 0\le t\le \bar{T}$, (\[3.5\]) and (\[3.6\]) reduce to $$\label{3.7}
\frac{3}{4}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+\nu\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}\le
\int_{0}^{t}\|(B_{1},B_{2})\|_{\dot{B}_{2,1}^1}\|\delta B\|_{\dot{B}_{2,\infty}^0}d\tau$$ and $$\label{3.8}
\frac{3}{4}\|\delta B\|_{L^\infty_{t}\dot{B}_{2,\infty}^{0}}\le C\|(B_{1},B_{2})\|_{L^\infty_{t}\dot{B}_{2,1}^1}\|\delta u\|_{L^1_{t}\dot{B}_{2,1}^1}.$$ Plugging (\[3.8\]) into (\[3.7\]) yields that $$\begin{aligned}
\frac{3}{4}\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+\nu\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1}\le&
C\int_{0}^{t}\|(B_{1},B_{2})(\tau)\|_{\dot{B}_{2,1}^1}\|(B_{1},B_{2})\|_{L^\infty_{\tau}\dot{B}_{2,1}^1}\|\delta u\|_{L^1_{\tau}\dot{B}_{2,1}^1}d\tau\\
\le&C_{T}\int_{0}^{t}\|\delta u\|_{L^1_{\tau}\dot{B}_{2,1}^1}d\tau.\end{aligned}$$ Thanks to the Log-type inequality (\[3.1\]), denote $$X(t)=\|\delta u\|_{L^\infty_{t}\dot{B}_{2,\infty}^{-1}}+\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^1},$$ we have $$\begin{aligned}
X(t)\le
C_{T,\nu}\int_{0}^{t} X(\tau)\log\left(e+\frac{V(\tau)}{X(\tau)}\right)d\tau,\end{aligned}$$ where $V(t)=\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^0}+\|\delta u\|_{\tilde{L}^1_{t}\dot{B}_{2,\infty}^2}$, is bounded in $[0,T]$. Applying the Osgood’s Lemma (see,[@3] p.125) and combining with (\[3.6\]) can yields $\delta u=\delta B=0$ in $[0,\bar{T}].$ By a standard continuous argument, we can show that $\delta u=\delta B=0$ in $[0,T]\times \mathbb{R}^2.$ This completes the proof of Theorem \[1.1\].
.4in
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank Prof. J-Y. Chemin for his comments.
.4in
[99]{} H. Bahouri, J-Y. Chemin and R. Danchin, *Fourier Analysis and Nonliear Partial Differential Equations.* Springer, 2011. J-Y. Chemin, D. McCormick, J. Robinson and J. Rodrigo, Local existence for the non-resistive MHD equations in Besov space, arXiv:1503.01651v1 \[math.AP\] 5 Mar 2015. R. Danchin, Local theory in critical spaces for the compressible viscous and heat-conductive gases, [*Comm. Partial Differential Equations **26(7-8)***]{} (2001), 1183-1233. C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, [*J. Funct. Anal. **267(4)***]{} (2014), 1035-1056.
J. Fan and T. Ozawa, Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model, [*Kinet. Relat. Models **2(2)***]{}, (2009), 293-305. L. Grafakos, *Modern Fourier Analysis.* 2nd Edition., Grad. Text in Math., **250**, Springer-Verlag, 2008.
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier- Stokes equations, [*Comm. Pure Appl. Math. **41(7)***]{}, (1988), 891-907.
E. Stein, *Singular Integrals and Differentiability Properties of Functions*, Princeton University Press, Princeton, 1970. Y. Zhou and J, Fan, A regularity criterion for the 2D MHD system with zero magnetic diffusivity, [*J. Math. Anal. Appl. **378(1)***]{}, (2011), 169-172.
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---
author:
- |
Lerrel Pinto and Abhinav Gupta\
Carnegie Mellon University
bibliography:
- 'references.bib'
title: '**Learning to Push by Grasping: Using multiple tasks for effective learning**'
---
|
---
abstract: 'We present an explicit proof that a minimal model of rank-2 antisymmetric field with spontaneous Lorentz violation and a classically equivalent vector field model are also quantum equivalent, by calculating quantum effective actions of both theories. We comment on the issues encountered while checking quantum equivalence in curved spacetime.'
author:
- Sandeep Aashish
- Sukanta Panda
bibliography:
- 'ref.bib'
title: On the Quantum equivalence of an antisymmetric tensor field with spontaneous Lorentz violation
---
\[intro\]Introduction
=====================
Antisymmetric tensor fields appear in all superstring theories and are especially relevant for studies in the low-energy limit [@rohm1986; @ghezelbash2009]. They have been studied in the past in several contexts, including strong-weak coupling duality and phase transitions [@quevedo1996; @olive1995; @polchinski1995; @siegel1980; @hata1981; @buchbinder1988; @duff1980; @bastianelli2005a; @*bastianelli2005b].
A study relevant to the present work was carried out by Altschul [*et al*.]{}[@altschul2010], where spontaneous Lorentz violation with various rank-2 antisymmetric field models minimally and non minimally coupled to gravity was investigated. A remarkable feature of that study is the presence of distinctive physical features with phenomenological implications for tests of Lorentz violation, even with relatively simple antisymmetric field models with a gauge invariant kinetic term. More recently, quantisation and propagator for such theories have been studied in Refs. [@maluf2019; @aashish2019b]. Lorentz violation is also a strong candidate signal for quantum gravity, and is part of the Standard Model Extension research program [@bonder2015]. Such interesting phenomenological possibilities have been a strong motivation for various works on spontaneous Lorentz violation (SLV) [@hernaski2016; @azatov2006; @kostelecky1989b; @*kostelecky1998; @*kostelecky2004; @bluhm2005; @carroll1990; @jackiw1999; @coleman1999; @bertolami1999].
Antisymmetric tensors, and $n-$forms in general, display interesting properties with regard to their equivalence with scalar and vector fields. For instance, in four dimensions theory of a massless 2-form field (with a gauge-invariant kinetic term) is classically equivalent to a massive nonconformal scalar field, while a massless 3-form theory does not have any physical degrees of freedom (see [@buchbinder2008] and references therein). Likewise, a massive rank-2 antisymmetric field is clasically equivalent to a massive vector field, and a rank-3 antisymmetric field is equivalent to massive scalar field [@buchbinder2008]. Such properties are useful in the analysis of degrees of freedom of these theories [@altschul2010]. Classical equivalence implies that the actions of two theories are equivalent. However, quantum equivalence is established at the level of effective actions, and it is in general not straightforward to check especially in curved spacetime. Moreover, classical equivalence between two theories does not necessarily carry over to the quantum level, particularly in the case of spontaneously broken Lorentz symmetry [@seifert2010a; @aashish2019b], and thus makes for an interesting study.
Quantum equivalence in the context of massive rank-2 and rank-3 antisymmetric fields in curved spacetime, without SLV, was first studied by Buchbinder [*et al*.]{}[@buchbinder2008] and later confirmed in Ref. [@shapiro2016]. The proof of quantum equivalence in Ref. [@buchbinder2008] was based on the zeta-function representation of functional determinants of $p$-form Laplacians appearing in the 1-loop effective action, and identities satisfied by zeta-functions for massless case [@rosenberg1997; @elizalde1994; @hawking1977]. Quantum equivalence results from these identities generalized to the massive case. In flat spacetime though, the proof is trivial as operators appearing in the effective action reduce to d’Alembertian operators due to vanishing commutators of covariant derivatives and equivalence follows by taking into account the independent components of each field.
We consider a particularly simple but interesting model of a rank-2 antisymmetric field minimally coupled to gravity, with the simplest choice of spontaneously Lorentz violating potential [@altschul2010]. Its classical equivalence was studied in Ref. [@altschul2010] in terms of an equivalent Lagrangian consisting of a vector field $A_{\mu}$ coupled to auxiliary field $B_{\mu\nu}$ in Minkowski spacetime. However, checking the quantum equivalence of such classically equivalent theories is not straightforward, in flat as well as curved spacetime. We find that the simple structure of operators breaks down due to the presence of SLV terms. As a result, the difference of their effective actions does not vanish in Minkowski spacetime, contrary to the case without SLV. However, this does not threaten quantum equivalence due to a lack of field dependence in the effective actions, which will therefore cancel after normalization.
In curved spacetime, making a conclusive statement about quantum equivalence is a nontrivial task for the following reasons. First, directly comparing effective actions using known proper time methods as in Ref. [@shapiro2016] is a difficult mathematical problem. Unlike the minimal operators (of the form $g_{\mu\nu}\nabla^{\mu}\nabla^{\nu} + Q$, where $Q$ is a functional without any derivative terms) found in [@buchbinder2008] for instance, we encounter nonminimal operators in functional determinants of the effective action, for which finding heat kernel coefficients to evaluate the determinants is a highly nontrivial task. Second, the formal arguments made in Ref. [@buchbinder2008] do not apply to the present case due to the non-trivial structure of operators appearing in effective actions. We point out that a resolution to this problem lies in doing a perturbative analysis of effective actions in nearly flat spacetimes, as in Ref. [@aashish2019b]. Nevertheless, in this work we present a derivation of one-loop effective actions of concerned theories in operator form and demonstrate the above difficulties by writing down operator structures explicitly. We use the quantization method developed in Ref. [@aashish2018a] to calculate the effective actions in curved spacetime using the DeWitt-Vilkovisky’s covariant effective action approach (see [@parker2009] for review) and St[ü]{}ckelberg procedure [@stuckelberg1957; @buchbinder2007].
The organization of this paper is as follows. Section \[sec2\] contains a review of the antisymmetric field Lagrangian in consideration, and a derivation of classically equivalent Lagrangian. In section \[sec3\], we calculate the effective action for the two classically equivalent theories. Section \[sec4\] deals with checking their quantum equivalence and problems therein.
\[sec2\]Classical action
========================
We consider the minimal model of a rank-2 antisymmetric tensor field, $B_{\mu\nu}$, with the simplest choice of spontaneously Lorentz violating potential [@altschul2010], $$\begin{aligned}
\label{amslv0}
\mathcal{L} = -\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda} - \frac{1}{2}\lambda\Big(B_{\mu\nu}B^{\mu\nu} - b_{\mu\nu}b^{\mu\nu}\Big)^{2}.\end{aligned}$$ $\lambda$ here is a massless coefficient. The first term in Eq. (\[amslv0\]) is the gauge invariant kinetic term, where, $$\label{amslv1}
H_{\mu\nu\lambda} \equiv \nabla_{\mu}B_{\nu\lambda} + \nabla_{\lambda}B_{\mu\nu} + \nabla_{\nu}B_{\lambda\mu},$$ and the second term is responsible for spontaneous Lorentz violation, giving rise to a non-zero vacuum expectation value, $$\begin{aligned}
\label{amslv2}
\langle B_{\mu\nu}\rangle = b_{\mu\nu}. \end{aligned}$$ $b_{\mu\nu}$ is also an antisymmetric tensor, which in general may not have a simple structure, but it is possible to transform to a special observer frame in which $b_{\mu\nu}$ has a block-diagonal form with its components being real numbers, provided that $b_{\mu\nu}b^{\mu\nu}$ is nonzero [@altschul2010].
It is clear from Eq. (\[amslv0\]) that the potential contains self-interaction terms for $B_{\mu\nu}$. Although it would be interesting to investigate quantum corrections in such a theory, it is out of scope of the current work. For the present study we are interested in the quantum properties of this theory with upto quadratic order terms in $B_{\mu\nu}$, and thus it is relevant to consider fluctuations of $B_{\mu\nu}$ around its vacuum expectation value $b_{\mu\nu}$ so that all higher order terms, including self-interaction terms can be ignored. We define the fluctuations $\crtilde{B}_{\mu\nu}$ as, $$\begin{aligned}
\label{amslv3}
\crtilde{B}_{\mu\nu} = B_{\mu\nu} - b_{\mu\nu}.\end{aligned}$$ Substituting Eq. (\[amslv3\]) in Eq. (\[amslv0\]) and neglecting higher order terms and constants, yields, $$\begin{aligned}
\label{amslv4}
\mathcal{L} = -\frac{1}{12}\crtilde{H}_{\mu\nu\lambda}\crtilde{H}^{\mu\nu\lambda} - 2\lambda\Big(b_{\mu\nu}\crtilde{B}^{\mu\nu}\Big)^{2}.\end{aligned}$$ where $\crtilde{H}_{\mu\nu\lambda}$ is now defined in terms of fluctuations $\crtilde{B}_{\mu\nu}$. For convenience, we define $$\begin{aligned}
\label{amslv5}
b_{\mu\nu} = b n_{\mu\nu},\end{aligned}$$ where $n_{\mu\nu}$ is an antisymmetric tensor satisfying $n_{\mu\nu}n^{\mu\nu} = 1$ so that, $$\begin{aligned}
\label{amslv6}
b_{\mu\nu}b^{\mu\nu} = b^{2}.\end{aligned}$$ Using Eq. (\[amslv5\]), Lagrangian (\[amslv4\]) can be written in a convenient form, $$\begin{aligned}
\label{amslv7}
\mathcal{L} = -\frac{1}{12}\crtilde{H}_{\mu\nu\lambda}\crtilde{H}^{\mu\nu\lambda} - \dfrac{1}{4}\alpha^{2} \Big(n_{\mu\nu}\crtilde{B}^{\mu\nu}\Big)^{2}.\end{aligned}$$ where $\alpha\equiv 8\lambda b^{2}$ is now a massive coefficient.
Our intention is to check the quantum equivalence of theory (\[amslv7\]) with a classically equivalent vector theory. Classical equivalence here means equivalence at the level of Lagrangian, that is, one Lagrangian can be obtained from other and vice versa, after manipulations. Knowledge of equivalence is quite useful in analysing the degrees of freedom and propagating modes of theories, as documented in Ref. [@altschul2010]. In the case $\alpha=0$ in Eq. (\[amslv7\]), $B_{\mu\nu}$ is known to be equivalent to a scalar field $\phi$ with Lagrangian $L = -\partial_{\mu}\phi\partial^{\mu}\phi/2$ [@altschul2010]. For $\alpha\neq 0$, an equivalent Lagrangian can be obtained by introducing a vector field $A_{\mu}$ along with the field strength and its dual defined as, $$\begin{aligned}
\label{aceq0}
F_{\mu\nu} = \nabla_{\mu}A_{\nu} - \nabla_{\nu}A_{\mu}, \nonumber \\
\mathcal{F}_{\mu\nu} = \dfrac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}, \end{aligned}$$ such that, Lagrangian (\[amslv7\]) is equivalent to [@altschul2010], $$\begin{aligned}
\label{aceq1}
\mathcal{L} = \dfrac{1}{2}\crtilde{B}_{\mu\nu}\mathcal{F}^{\mu\nu} - \frac{1}{2}A^{\mu}A_{\mu} - \dfrac{1}{4}\alpha^{2} \Big(n_{\mu\nu}\crtilde{B}^{\mu\nu}\Big)^{2}.\end{aligned}$$ For present purposes, we want to get rid of $\crtilde{B}_{\mu\nu}$ entirely in favour of a Lagrangian described solely by a vector field, and thus it is handy to make use of projections of a tensor along and transverse to $n_{\mu\nu}$, $$\begin{aligned}
\label{aceq2}
T_{||\mu\nu} = n_{\rho\sigma}T^{\rho\sigma} n_{\mu\nu}, \nonumber \\
T_{\perp\mu\nu} = T_{\mu\nu} - T_{||\mu\nu},\end{aligned}$$ Substituting Eq. (\[aceq2\]) in Eq. (\[aceq1\]), the Lagrangian density becomes, $$\begin{aligned}
\label{aceq3}
\mathcal{L} = \dfrac{1}{2}\crtilde{B}_{\perp\mu\nu}\mathcal{F}_{\perp}^{\mu\nu} + \dfrac{1}{2}\crtilde{B}_{||\mu\nu}\mathcal{F}_{||}^{\mu\nu} - \frac{1}{2}A^{\mu}A_{\mu} - \dfrac{1}{4}\alpha^{2} \crtilde{B}_{||\mu\nu}\crtilde{B}_{||}^{\mu\nu}.\end{aligned}$$ Using the equations of motion of $\crtilde{B}_{||\mu\nu}$ and $\crtilde{B}_{\perp\mu\nu}$ in (\[aceq3\]) allows us to write, $$\begin{aligned}
\label{aceq4}
\alpha^{2}\mathcal{L} &=& \dfrac{1}{4}\mathcal{F}_{||\mu\nu}\mathcal{F}_{||}^{\mu\nu} - \frac{1}{2}\alpha^{2} A^{\mu}A_{\mu} \nonumber \\
&=& \dfrac{1}{4}\left(n_{\mu\nu}\mathcal{F}^{\mu\nu}\right)^{2} - \frac{1}{2}\alpha^{2} A^{\mu}A_{\mu}.\end{aligned}$$ Note that Eq. (\[aceq4\]) incorporates the condition $\mathcal{F}_{\perp}^{\mu\nu} \approx 0$, i.e. all modes orthogonal to $n_{\mu\nu}$ are non-propagating; at the same time, the term proportional to $A_{\mu}A^{\mu}$ generates the mass term for modes along $n_{\mu\nu}$ [@altschul2010]. Introducing the dual of $n_{\mu\nu}$, given by $\tilde{n}_{\mu\nu} = \dfrac{1}{2}\epsilon_{\mu\nu\rho\sigma}n^{\rho\sigma}$, the classically equivalent Lagrangian in terms of $F_{\mu\nu}$ reads, $$\begin{aligned}
\label{aceq5}
\alpha^{2}\mathcal{L} &=& \dfrac{1}{4}\left(\tilde{n}_{\mu\nu}F^{\mu\nu}\right)^{2} - \frac{1}{2}\alpha^{2} A^{\mu}A_{\mu}\end{aligned}$$ A distinctive feature of Lagrangian (\[aceq5\]) when compared to a generic massive vector field Lagrangian like the Proca model, is its peculiar kinetic term. Infact, kinetic terms with a Lorentz violating factor have been explored in past literatures in the context of Chern-Simons modification to Maxwell theory and alternatives to Higgs mechanism [@chung1999; @jean2011]. Moreover, the sign of kinetic term in (\[aceq5\]) is opposite to that in Proca model. In the context of SLV, another noteworthy feature of Lagrangian (\[aceq5\]) is that the potential term is not affected by $n_{\mu\nu}$ unlike other vector models with SLV, for instance the Bumblebee model. It will be observed in later sections that these features lead to an effective action that has a structure different from the corresponding effective action for Lagrangian (\[amslv7\]).
\[sec3\]The Effective Action
============================
The classical analysis of the previous section did not take into account the gauge symmetries of equivalent Lagrangians (\[amslv7\]) and (\[aceq5\]). While these Lagrangians are technically not gauge invariant, they belong to a class of theories having a softly broken gauge symmetry: the kinetic terms of Lagrangians (\[amslv7\]) and (\[aceq5\]) are invariant under the transformations $\crtilde{B}^{\mu\nu} \longrightarrow \crtilde{B}^{\mu\nu} + \nabla_{\mu}\xi_{\nu} - \nabla_{\nu}\xi_{\mu}$ and $A_{\mu}\longrightarrow A_{\mu} + \nabla_{\mu}\Lambda$, respectively, but the potential terms are not. A standard approach for quantization of these theories is to employ the St[ü]{}ckelberg procedure [@stuckelberg1957; @buchbinder2007].
We first consider the Lagrangian (\[amslv7\]). The first step is to restore the softly broken gauge symmetry through the introduction of a St[ü]{}ckelberg field [@stuckelberg1957] $C_{\mu}$ such that the Lagrangian, $$\begin{aligned}
\label{aeaa0}
\mathcal{L} = -\frac{1}{12}\crtilde{H}_{\mu\nu\lambda}\crtilde{H}^{\mu\nu\lambda} - \frac{1}{4}\alpha^{2}\Big[n_{\mu\nu}\Big(\crtilde{B}^{\mu\nu} + \frac{1}{\alpha}F^{\mu\nu}[C]\Big)\Big]^{2},\end{aligned}$$ becomes gauge invariant (here, $F_{\mu\nu}\equiv\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}$), and reduces to original Lagrangian (\[amslv7\]) in the gauge $C_{\mu}=0$. The new Lagrangian (\[aeaa0\]) is invariant under the symmetries, $$\begin{aligned}
\label{aeaa1}
\crtilde{B}^{\mu\nu} &\longrightarrow & \crtilde{B}^{\mu\nu} + \nabla_{\mu}\xi_{\nu} - \nabla_{\nu}\xi_{\mu}, \nonumber \\
C_{\mu} &\longrightarrow & C_{\mu} - \alpha\xi_{\mu},\end{aligned}$$ and, $$\begin{aligned}
\label{aeaa2}
C_{\mu} &\longrightarrow & C_{\mu} + \nabla_{\mu}\Lambda , \nonumber \\
\crtilde{B}^{\mu\nu} &\longrightarrow & \crtilde{B}^{\mu\nu} .\end{aligned}$$ In addition to the above symmetries of fields, there exists a set of transformation of gauge parameters $\Lambda$ and $\xi_{\mu}$ that leaves the fields $B_{\mu\nu}$ and $C_{\mu}$ invariant, $$\begin{aligned}
\label{aeaa3}
\xi_{\mu} &\longrightarrow & \xi_{\mu} + \nabla_{\mu}\psi , \nonumber \\
\Lambda &\longrightarrow & \Lambda + \alpha\psi .\end{aligned}$$
Now, the gauge fixing procedure requires that a gauge condition be chosen for each of the fields $B_{\mu\nu}$ and $C_{\mu}$ as well as for the parameter $\xi_{\mu}$, so that the redundant degrees of freedom due to symmetries (\[aeaa1\]), (\[aeaa2\]) and (\[aeaa3\]) are taken care of. An important consideration while choosing a gauge condition is to ensure that all cross terms of fields in the Lagrangian cancel out or lead to a total derivative term, so that path integral can be computed with ease. Keeping this in mind, we choose the gauge condition for $B_{\mu\nu}$ to be (a similar choice for gauge condition in the context of Bumblebee model was considered in Ref. [@escobar2017]) $$\begin{aligned}
\label{aeaa6}
\chi_{\xi_{\nu}} = n_{\mu\nu}n_{\rho\sigma}\nabla^{\mu}\crtilde{B}^{\rho\sigma} + \alpha C_{\nu}.\end{aligned}$$ It turns out that the gauge fixing action term corresponding to Eq. (\[aeaa6\]) introduces yet another soft symmetry breaking in $C_{\mu}$ [@aashish2018a], so one has to introduce another St[ü]{}ckelberg field $\Phi$ so that, $$\begin{aligned}
\label{aeaa7}
C_{\mu} \longrightarrow C_{\mu} + \dfrac{1}{\alpha}\nabla_{\mu}\Phi.\end{aligned}$$ This modifies the symmetry in Eq. (\[aeaa2\]) by an additional shift transformation, $$\begin{aligned}
\label{aeaa8}
\Phi \longrightarrow \Phi - \alpha\Lambda.\end{aligned}$$ From Eqs. (\[aeaa2\]) and (\[aeaa8\]), the gauge condition for $C_{\mu}$ can be chosen to be, $$\begin{aligned}
\label{aeaa9}
\chi_{\Lambda} = \nabla^{\mu}C_{\mu} + \alpha\Phi.\end{aligned}$$ Similarly, for the symmetry of parameters, Eq. (\[aeaa3\]), we choose $$\begin{aligned}
\label{beaa0}
\check{\chi}_{\psi} = \nabla^{\mu}\xi_{\mu} - \alpha\Lambda.\end{aligned}$$ The gauge conditions chosen above are incorporated in the action for (\[aeaa0\]) through “gauge-fixing Lagrangian" terms of the form $-\frac{1}{2}\chi_{(\cdot)}^{2}$ for each of the conditions (\[aeaa6\]), (\[aeaa9\]) and (\[beaa0\]). The total gauge fixed Lagrangian is given by $$\begin{aligned}
\label{beaa1}
\mathcal{L}_{2}^{GF} = -\frac{1}{12}\crtilde{H}_{\mu\nu\lambda}\crtilde{H}^{\mu\nu\lambda} - \frac{1}{4}\alpha^{2}\Big(n_{\mu\nu}\crtilde{B}^{\mu\nu}\Big)^{2} - \dfrac{1}{4}\Big(n_{\mu\nu}F^{\mu\nu}\Big)^{2} - \frac{1}{2}\Big(n_{\mu\nu}n_{\rho\sigma}\nabla^{\mu}\crtilde{B}^{\rho\sigma}\Big)^{2} \nonumber \\ - \frac{1}{2}\alpha^{2}C_{\nu}C^{\nu} - \frac{1}{2}(\nabla_{\mu}\Phi)^{2} - \frac{1}{2}(\nabla^{\mu}C_{\mu})^{2} - \frac{1}{2}\alpha^{2}\Phi^{2}.\end{aligned}$$ Following the method developed in Ref. [@aashish2018a], the calculation of ghost determinants proceeds as follows. We rewrite $\chi_{\xi_{\nu}}$ as, $$\begin{aligned}
\label{beaa4}
\chi_{\xi_{\nu}}[\crtilde{B}^{\mu\nu}_{\xi_{\nu}}, C^{\mu}_{\xi_{\nu}}] = \chi_{\xi_{\nu}}[\crtilde{B}^{\mu\nu},C^{\mu},{\xi_{\nu}},\Lambda,\check{\chi}_{\psi}],\end{aligned}$$ which yields, $$\begin{aligned}
\label{beaa5}
\chi_{\xi_{\nu}} = n_{\mu\nu}n_{\rho\sigma}\nabla^{\mu}\crtilde{B}^{\rho\sigma} + \alpha C_{\nu}
+ 2n_{\mu\nu}n_{\rho\sigma}\nabla^{\mu}\nabla^{\rho}\xi^{\sigma} + \nabla_{\nu}\nabla_{\mu}\xi^{\mu} - \alpha^{2}\xi_{\nu} - \nabla_{\nu}\check{\chi}_{\psi}.\end{aligned}$$ Then, using the definition of $Q'^{\xi_{\mu}}_{\xi_{\nu}}$, we get $$\begin{aligned}
\label{beaa6}
Q'^{\xi_{\nu}}_{\xi_{\alpha}} = \left(\dfrac{\delta\chi_{\xi_{\nu}}}{\delta\xi_{\alpha}}\right)_{\xi_{\mu} = 0} &=& 2n_{\mu\nu}n_{\rho\alpha}\nabla^{\mu}\nabla^{\rho} + \nabla_{\nu}\nabla_{\alpha} - \alpha^{2}\delta_{\nu\alpha}.\end{aligned}$$ A straightforward calculation leads to other non-zero components of ghost determinant, $$\begin{aligned}
\label{beaa7}
Q'^{\Lambda}_{\Lambda} = \dfrac{\delta\chi_{\Lambda}}{\delta\Lambda} = \Box_{x} - \alpha^{2} \\
\label{beaa8}
\check{Q}^{\psi}_{\psi} \equiv \dfrac{\delta\check{\chi}_{\psi}}{\delta\psi} = \Box_{x} - \alpha^{2}\end{aligned}$$ Using the definition of effective action obtained in [@aashish2018a], $$\begin{aligned}
\label{beaa9}
\exp(i\Gamma[\bar{B},\bar{C}]) = \int\prod_{\mu}dC_{\mu}\prod_{\rho\sigma}dB_{\rho\sigma}\prod_{x}d\Phi
\det(Q'^{\Lambda}_{\Lambda})\det(Q'^{\xi_{\nu}}_{\xi_{\alpha}})(\det\check{Q}^{\psi}_{\psi})^{-1}\times \nonumber \\
\exp\left\{i \Bigg(\int d v_{x} \mathcal{L}_{2}^{GF}\Bigg) + (\bar{B}_{\mu\nu}-B_{\mu\nu})\dfrac{\delta}{\delta \bar{B}_{\mu\nu}}\Gamma[\bar{B},\bar{C}] \right. \nonumber \\ \left. + (\bar{C}_{\mu}-C_{\mu})\dfrac{\delta}{\delta \bar{C}_{\mu}}\Gamma[\bar{B},\bar{C}] \right\},\end{aligned}$$ The 1-loop effective action is obtained as, $$\begin{aligned}
\label{ceaa0}
\Gamma_{2}^{(1)} = \frac{i\hbar}{2}\Big[\ln\det(\crtilde{D}_{2} - \alpha^{2}n^{\mu\nu}n_{\rho\sigma}) - \ln\det(\crtilde{D}_{1}-\alpha^{2}) + \ln\det(\Box_{x} - \alpha^{2})\Big]\end{aligned}$$ where, $$\begin{aligned}
\label{cea1}
\crtilde{D}_{2}{}^{\mu\nu}_{\ \ \rho\sigma}B^{\rho\sigma} &\equiv & \nabla_{\alpha}\nabla^{\alpha}B^{\mu\nu} + \nabla_{\alpha}\nabla^{\mu}B^{\nu\alpha} + \nabla_{\alpha}\nabla^{\nu}B^{\alpha\mu} + 2 n^{\mu\nu}n_{\rho\sigma}n^{\alpha\sigma}n_{\beta\gamma}\nabla^{\rho}\nabla_{\alpha}B^{\beta\gamma}, \nonumber \\
\crtilde{D}_{1}{}^{\mu}_{\ \nu}C^{\nu} &\equiv & 2 n^{\nu\mu}n_{\rho\sigma}\nabla_{\nu}\nabla^{\rho}C^{\sigma} + \nabla^{\mu}\nabla_{\nu}C^{\nu}.\end{aligned}$$ It is to be noted that the coefficient of $\alpha^{2}$ in the first term in Eq. (\[ceaa0\]) ensures that massive modes correspond to field components along vacuum expectation tensor $n_{\mu\nu}$ and massless modes correspond to transverse components. An interesting observation here is the last term, which is unaffected by $n_{\mu\nu}$. In case of no SLV, the last term causes the quantum discontinuity when going from massive to massless case [@shapiro2016]. To compare Eq. (\[ceaa0\]) with the effective action of classically equivalent Lagrangian, the Lagrangian in (\[aceq5\]) is treated with the St[ü]{}ckelberg procedure to obtain, $$\begin{aligned}
\label{ceaa1}
\tilde{\mathcal{L}}_{1} &=& \dfrac{1}{4}\Big(\tilde{n}_{\mu\nu}F^{\mu\nu}\Big)^{2} - \dfrac{1}{2}\alpha^{2}(C_{\mu} + \dfrac{1}{\alpha}\nabla_{\mu}\Phi)^{2}\end{aligned}$$ The above Lagrangian is invariant under a transformation identical to Eqs. (\[aeaa2\]) and (\[aeaa8\]), $$\begin{aligned}
\label{ceaa2}
C_{\mu}\longrightarrow C_{\mu} + \nabla_{\mu}\Lambda, \quad \Phi\longrightarrow \Phi - \alpha\Lambda.\end{aligned}$$ With the gauge condition Eq. (\[aeaa9\]), the gauge fixed Lagrangian reads, $$\begin{aligned}
\label{ceaa3}
\tilde{\mathcal{L}}_{1}^{GF} = \dfrac{1}{2}C_{\mu}D_{1}C^{\mu} - \dfrac{1}{2}\alpha^{2}C_{\mu}C^{\mu} + \dfrac{1}{2}\Phi (\Box_{x} - \alpha^{2})\Phi.\end{aligned}$$ where, $$\begin{aligned}
\label{deaa0}
D_{1}C_{\mu} = -2\tilde{n}_{\nu\mu}\tilde{n}_{\rho\sigma}\nabla^{\nu}\nabla^{\rho}C^{\sigma} + \nabla_{\mu}\nabla_{\nu}C^{\nu}.\end{aligned}$$ It is straightforward to check that the 1-loop effective action is, $$\begin{aligned}
\label{ceaa4}
\Gamma_{1}^{(1)} = \frac{i\hbar}{2}\Big[\ln\det(D_{1}-\alpha^{2}) - \ln\det(\Box_{x} - \alpha^{2})\Big].\end{aligned}$$ Similar to Eq. (\[ceaa0\]), the scalar term is unaffected by $n_{\mu\nu}$ and the operator $D_{1}$ possesses a non-trivial structure. The expression for $D_{1}$ has a striking resemblance to that of $\crtilde{D}_{1}$, which has opposite sign in the first term and $n_{\mu\nu}$ instead of $\tilde{n}_{\mu\nu}$. Particularly interesting is the fact that this difference is, by design, built into the equivalent Lagrangian (\[aceq5\]) and is apparent even before, in Eq. (\[beaa1\]), where the kinetic part of St[ü]{}ckelberg field $-\frac{1}{4}(n_{\mu\nu}F^{\mu\nu})^{2}$ has a sign opposite to that of Eq. (\[aceq5\]).
\[sec4\]Quantum equivalence in flat spacetime
=============================================
To compare Eqs. (\[ceaa0\]) and (\[ceaa4\]), we define the difference in 1-loop effective actions given by, $$\begin{aligned}
\label{aqeq0}
\Delta\Gamma &=& \Gamma_{2}^{(1)} - \Gamma_{1}^{(1)} \nonumber \\
&=& \frac{i\hbar}{2}\Big[\ln\det(\crtilde{D}_{2} - \alpha^{2}n^{\mu\nu}n_{\rho\sigma}) - \ln\det(\crtilde{D}_{1}-\alpha^{2}) - \ln\det(D_{1}-\alpha^{2})\nonumber \\
&& + 2\ln\det(\Box_{x} - \alpha^{2})\Big].\end{aligned}$$ In contrast, the corresponding difference in 1-loop effective action in the case of massive antisymmetric and vector fields, with mass $m$, with no spontaneous Lorentz violation is given by [@buchbinder2008], $$\begin{aligned}
\label{aqeq1}
\Delta\Gamma' = \frac{i\hbar}{2}\Big[\ln\det(\Box_{2} - m^{2}) - 2\ln\det(\Box_{1}-m^{2}) + 2\ln\det(\Box_{x} - m^{2})\Big],\end{aligned}$$ where, $$\begin{aligned}
\label{aqeq2}
\Box_{2}B_{\mu\nu} &=& \Box_{x}B_{\mu\nu} - [\nabla^{\rho},\nabla_{\nu}]B_{\mu\rho} - [\nabla^{\rho},\nabla_{\mu}]B_{\rho\nu}, \nonumber \\
\Box_{1}C^{\mu} &=& \Box_{x}C^{\nu} - [\nabla^{\nu},\nabla_{\mu}]C^{\mu}.\end{aligned}$$ This comparison between cases with and without SLV is quite insightful, because it helps in understanding how the functional operators change due to the presence of Lorentz violating terms. In the later case, the operator for St[ü]{}ckelberg vector field and that for vector field of equivalent Lagrangian are equal, while in the former case they are not, as was noted earlier. Moreover, operators in Eq. (\[aqeq0\]) do not contain the commutator terms due to presence of $n_{\mu\nu}$, and hence do not simplify in flat spacetime unlike their counterparts in Eq. (\[aqeq1\]).
In flat spacetime, it can be explicitly checked that Eq. (\[aqeq1\]) vanishes, taking into account the number of independent components of respective fields (eight, four and one for antisymmetric, vector and scalar fields respectively), because the commutators in Eq. (\[aqeq2\]) vanish and hence the operators $\Box_{2}$, $\Box_{1}$, and $\Box_{x}$ are identical. Inferring quantum equivalence is thus trivial. However, this is clearly not the case in Eq. (\[aqeq0\]) due to the non-trivial structure of operators $\crtilde{D}_{2}$ and $\crtilde{D}_{1}$. This can be demonstrated in a rather simple example when a special choice of tensor $n_{\mu\nu}$ is considered. It can be shown that in Minkowski spacetime, $n_{\mu\nu}$ can be chosen to have a special form $$\begin{aligned}
\label{afse0}
n_{\mu\nu} =
\left(\begin{matrix}
0 & -a & 0 & 0\\
a & 0 & 0 & 0\\
0 & 0 & 0 & b\\
0 & 0 & -b & 0
\end{matrix}\right),\end{aligned}$$ where $a$ and $b$ are real numbers, provided atleast one of the quantities $x_{1}\equiv -2(a^{2}-b^{2})$ and $x_{2}\equiv 4ab$ are non-zero [@altschul2010]. For simplicity, and dictated by the requirements for non-trivial monopole solutions [@seifert2010b], we may choose $b=0$. Further, the constraint $n_{\mu\nu}n^{\mu\nu}=1$ implies that $a=1/\sqrt{2}$. Therefore, the only non-zero components of $n_{\mu\nu}$ are $n_{10}=1/\sqrt{2}$ and $n_{01}=-1/\sqrt{2}$. For the dual tensor $\tilde{n}_{\mu\nu}$, the non-zero components are $\tilde{n}_{32}=-1/\sqrt{2}$ and $\tilde{n}_{23}=1/\sqrt{2}$. Substituting in Eqs. (\[cea1\]) and (\[deaa0\]), one obtains, for the non-zero components of $n_{\mu\nu}$ and $\tilde{n}_{\mu\nu}$, $$\begin{aligned}
\label{afse1}
D_{1}C^{2} &=& \partial_{2}^{2}C^{2} - \partial_{3}^{2}C^{2} + 2\partial_{3}\partial^{2}C^{3} + \partial^{2}\partial_{i}C^{i}, \nonumber \\
D_{1}C^{3} &=& - \partial_{2}^{2}C^{3} + \partial_{3}^{2}C^{3} + 2\partial^{3}\partial_{2}C^{2} + \partial^{3}\partial_{i}C^{i}, \nonumber \\
\crtilde{D}_{1}C^{0} &=& \partial_{0}^{2}C^{0} + \partial_{1}^{2}C^{0} + \partial_{0}\partial_{j}C^{j}, \\
\crtilde{D}_{1}C^{1} &=& \partial_{0}^{2}C^{1} + \partial_{1}^{2}C^{1} + \partial_{1}\partial_{j}C^{j} , \nonumber \\
\crtilde{D}_{2}\crtilde{B}^{10} &=& \Box_{x}\crtilde{B}^{10} + \partial_{j}\left(\partial_{1}\crtilde{B}^{0j} + \partial_{0}\crtilde{B}^{j1}\right) = - \crtilde{D}_{2}\crtilde{B}^{01}, \nonumber\end{aligned}$$ where, $j=2,3$ and $i=0,1$. The remaining components of operators $\crtilde{D}_{2}$, $\crtilde{D}_{1}$ and $D_{1}$ are given by, $$\begin{aligned}
\label{afse2}
\crtilde{D}_{2}\crtilde{B}^{jk} &=& \Box_{x}\crtilde{B}^{jk} + \partial_{\mu}\partial^{j}\crtilde{B}^{k\mu} + \partial_{\mu}\partial^{k}\crtilde{B}^{\mu j}, \quad \crtilde{B}^{jk}\neq \crtilde{B}^{10} \nonumber \\
\crtilde{D}_{1}C^{l} &=& \partial^{l}\partial_{\nu}C^{\nu}, \quad l=2,3 \\
D_{1}C^{k} &=& \partial^{k}\partial_{\nu}C^{\nu}, \quad k=0,1. \nonumber\end{aligned}$$ An interesting feature here, compared to the case of Eq. (\[aqeq1\]), is that Eqs. (\[afse1\]) and (\[afse2\]) substituted in Eq. (\[aqeq0\]) show explicitly that $\Delta\Gamma$ does not vanish. However, functional determinants in Eq. (\[aqeq0\]) do not have field dependence and can only contribute as infinite (regularization-dependent) constants [@simon1977; @dunne2008]. Hence, each determinant in Eq. (\[aqeq0\]) can be normalized to identity and will thus be equal to each other. And once again, taking into account the degrees of freedom of corresponding tensor, vector and scalar fields, similar to Eq. (\[aqeq1\]), they will cancel for all physical processes. This proves the quantum equivalence of theories (\[ceaa0\]) and (\[ceaa4\]) in flat spacetime.
A check of quantum equivalence in curved spacetime is out of the scope of present work, because of the lack of mathematical tools to compute functional determinants in Eq. (\[aqeq0\]). More specifically, to evaluate the expansions these determinants one needs the correct heat kernels for operators $\crtilde{D}_{2}$, $\crtilde{D}_{1}$ and $D_{1}$. However, a perturbative approach can be undertaken to address this issue in a nearly flat spacetime as implemented in Ref. [@aashish2019b].
Summary
=======
We derived the Lagrangian for a vector field $C_{\mu}$ which is classically equivalent to a rank-2 antisymmetric tensor field with a spontaneously Lorentz violating potential, by extending the calculations carried out in Ref. [@altschul2010]. The 1-loop effective action in terms of functional operators was obtained for both theories, and it was found that the operators have complicated structures due to the presence of vacuum expectation tensor $n_{\mu\nu}$. In flat spacetime, we explicitly checked for a simple choice of $n_{\mu\nu}$ that although the difference of effective actions, $\Delta\Gamma$, does not vanish, their quantum equivalence still holds for physical processes once normalization of functional determinants are taken into account. This confirms, in flat spacetime, the fact that two free field theories which are classically equivalent, must also be quantum equivalent.
In curved spacetime, however, it is difficult to make a precise statement because an explicit comparison of operators is not possible unless one uses a regularization scheme to find an appropriate expression for operators in $\Delta\Gamma$, as done in Refs. [@shapiro2016] and [@buchbinder2008]. The question of quantum equivalence is important, not only due to its usefulness in analysing the degrees of freedom, as documented in Ref. [@altschul2010], but also due to the consequences to formal properties of theories. For example, Seifert [@seifert2010a] showed that interaction of vector and tensor theories with gravity are different when topologically non-trivial monopole-like solutions of the spontaneous symmetry breaking equations exist. Although perturbative methods can be employed in a nearly flat spacetime [@aashish2019b], a good starting point for addressing this issue in a general spacetime would be to explicitly write the heat kernel for these operators.
This work was partially funded by DST (Govt. of India), Grant No. SERB/PHY/2017041. The authors are grateful to Prof. Alan Kostelecky for useful comments on an earlier version of this paper.
|
---
abstract: 'We propose a set of photonic crystals that realize a nonlinear quantum Rabi model equivalent to a two-level system driven by the phase of a quantized electromagnetic field. The crystals are exactly solvable in the weak-coupling regime; their dispersion relation is discrete and the system is diagonalized by normal modes similar to a dressed state basis. In the strong-coupling regime, we use perturbation theory and find that the dispersion relation is continuous. We give the normal modes of the crystal in terms of continued fractions that are valid for any given parameter set. We show that these photonic crystals allow state reconstruction in the form of coherent oscillations in the weak-coupling regime. In the strong-coupling regime, the general case allows at most partial reconstruction of single waveguide input states, and non-symmetric coherent oscillations that show partial state reconstruction of particular phase-controlled states.'
address: 'Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, México'
author:
- 'B. M. Rodríguez-Lara'
- 'Alejandro Zárate Cárdenas, Francisco Soto-Eguibar, Héctor Manuel Moya-Cessa'
title: 'A photonic crystal realization of a phase driven two-level atom'
---
Photonic crystals ,Classical and quantum physics ,Classical simulation of quantum optics
Introduction
============
Photonic crystals as classical simulators of quantum processes have been the focus of attention in recent years [@Longhi2006p110402; @Perets2008p170506; @Bromberg2009p253904; @Dreisow2009p076802; @Lahini2010p163905; @Dreisow2010p143902; @Longhi2010p075102; @Longhi2011p3248; @Keil2011p103601; @RodriguezLara2011p053845; @Longhi2011p453; @Longhi2012p435601; @Longhi2012p012112; @Garanovich2012]. In particular, it has been shown theoretically and experimentally that the so-called quantum Rabi model describing the interaction of a two-level system with a quantum field may be realized by photonic superlattices [@Longhi2011; @Crespi2012p163601]. The quantum Rabi model in the weak-coupling regime, i.e. the Jaynes-Cummings model [@Jaynes1963p89], describes a variety of quantum mechanical systems that have been experimentally implemented; e.g. cavity-quantum electrodynamics (cavity-QED) [@Walther2006p1325], ion traps [@MoyaCessa2012p229] and circuit-QED [@Blais2004p062320]. Strong-coupling is not feasible in a majority of simple quantum optical systems but photonic crystals provide a classical realization of the quantum model for all coupling regimes [@Longhi2011; @Crespi2012p163601].
In quantum optics, diverse non-linear models describing the interaction between a two-level system and a quantum field have been proposed as deformations of the Jaynes-Cummings model [@Kundu2004p281; @delosSantosSanchez2012p015502]. One example of these nonlinear models is the Buck-Sukumar (BS) model where the atom–field coupling depends on the intensity of the quantum field [@Buck1981p132]. The BS model, which is exactly solvable and does not have a feasible experimental representation, unless it is classically realized in a couple of binary photonic crystals where the coupling depends linearly on the position of the waveguide, helps in understanding the apparition of collapses and revivals of the two-level inversion in the radiation–matter interaction systems.
In the following, we propose a semi-infinite photonic crystal that classically simulates a novel non-linear quantum optics model describing an atom driven by just the phase of a quantum field. Up to our knowledge both the non-linear radiation-matter interaction model and its photonic realization are missing in the literature. Then, we find the exact dispersion curves and normal modes of the waveguide lattice in the weak-coupling regime. In the strong-coupling regime, the dispersion relation is continuous and we find the normal modes as continued fractions. The transition from discrete to continuous spectrum, appearing in our photonic crystal, does not show in the spectra of Rabi [@Tur2000p574; @Tur2001p899; @Casanova2010p263603] and BS [@Buck1981p132] models which are discrete in both regimes, weak and strong coupling. Thus, parameter sets delivering coherent oscillations in Rabi or BS models only produce coherent oscillations in the weak-coupling regime of our model.
The model and its photonic crystal analogue
===========================================
Let us consider the Hamiltonian describing a two-level system driven by just the phase of a quantum field, $$\begin{aligned}
\hat{H} = \omega_{f} \hat{a}^{\dagger} \hat{a} + \frac{\omega_{0}}{2} \hat{\sigma_{z}} + \lambda \left( e^{\imath \hat{\phi}} + e^{-\imath \hat{\phi}} \right) \hat{\sigma}_{x}, \label{eq:Hamiltonian}\end{aligned}$$ where the exponential of the quantum phase operator is given by the Susskind-Glogower operator [@Susskind1964p49] $$\begin{aligned}
e^{\imath \hat{\phi}} \equiv \hat{V} = \frac{1}{\sqrt{\hat{a} \hat{a}^{\dagger}}} \hat{a}.\end{aligned}$$ The field mode of frequency $\omega_{f}$ is described by the annihilation (creation) operators $\hat{a}$ ($\hat{a}^{\dagger}$), the two-level system of transition frequency $\omega_{0}$ by Pauli matrices $\sigma_{x,y,z}$, and their interaction by the real coupling $\lambda$. It is possible to separate this system in two uncoupled Hamiltonians, $$\begin{aligned}
\hat{H}_{\pm} = \omega_{f} \hat{n} \mp \frac{\omega_{0}}{2} (-1)^{\hat{n}} + \lambda \left( \hat{B} + \hat{B}^{\dagger} \right),\end{aligned}$$ belonging to one of two parity chain basis, $$\begin{aligned}
|+,n \rangle &=& \hat{B}^{\dagger n} | 0, g\rangle, \\
|-,n \rangle &=& \hat{B}^{\dagger n} | 0, e\rangle,\end{aligned}$$ defined such that parity, $$\begin{aligned}
\hat{\Pi} = -\sigma_{z} (-1)^{\hat{n}},\end{aligned}$$ is conserved, $\langle \pm ,n \vert \hat{\Pi} \vert \pm, n \rangle = \pm$; the bases annihilation (creation) operator is given by $\hat{B} = \hat{V} \hat{\sigma}_{x}$ ($\hat{B}^{\dagger} = \hat{V}^{\dagger} \hat{\sigma}_{x}$) and the number operator is defined as $\hat{n} \vert \pm, m \rangle = m \vert \pm, m \rangle $. By defining the general state, $$\begin{aligned}
\vert \psi_{\pm} \rangle = \sum_{j=0}^{\infty} \mathcal{E}_{j}^{(\pm)} \vert \pm, j \rangle,\end{aligned}$$ the equations of motion for any given initial state under the dynamics given by Hamiltonian (\[eq:Hamiltonian\]) are reduced to the differential set $$\begin{aligned}
i \partial_{t} \mathcal{E}^{(\pm)}_{j} = \left[ \omega_{f} j \mp \frac{\omega_{0}}{2} (-1)^{j} \right] \mathcal{E}^{(\pm)}_{j} + \lambda \left( \mathcal{E}^{(\pm)}_{j-1} + \mathcal{E}^{(\pm)}_{j+1} \right), \label{eq:DiffSet}\end{aligned}$$ where the shorthand notation $\partial_{t}$ has been used for the partial derivative with respect to $t$. This differential set is equivalent, up to a phase and substituting $t \rightarrow z$, to that describing the propagation equation of a classical field through a photonic waveguide lattice. In this equivalent photonic waveguide lattice, $\mathcal{E}_{j}$ is the amplitude of the field at the $j$th waveguide, the waveguides are homogeneously coupled, and the refraction indices grow proportional to $\omega_{f}$ and to their position on the lattice plus a position depending bias proportional to $\omega_{0}/2$.
For the sake of simplicity, hereby we will refer to the photonic crystals as $H_{+}$ or $H_{-}$, depending on the sign of Eq.(\[eq:DiffSet\]). In order to construct these photonic crystals, one can choose to either implement a static, Fig. \[fig:Fig1\](a), or dynamic, Fig. \[fig:Fig1\](b), relation between parameters $\omega_{f}$ and $\omega_{0}$. The parameter $\omega_{0}$ will be fixed in both cases. Straight waveguides produce a fixed parameter $\omega_{f}$ independent of the wavelength of the impinging light. Bending the waveguides along a circle introduces an index gradient inversely proportional to the wavelength of the impinging light [@Longhi2011; @Crespi2012p163601]; thus, in this case the parameter $\omega_{f}$ depends on the wavelength of the impinging light and one can vary the ratio between $\omega_{f}$ and $\omega_{0}$ by choosing the color of the impinging light.
![(Color online) Two different schemes to produce the set of photonic crystals realizing the phase driven two-level atom. A semi-infinite set of homogeneously coupled waveguides where the refraction index behaves as the function $ n^{(\pm)}_{j} \propto \omega_{f} j \mp \frac{\omega_{0}}{2} (-1)^{j} $. (a) Straight waveguides deliver fixed $\omega_{f}$ and $\omega_{0}$ parameters. (b) Circularly bent waveguides produce a parameter $\omega_{f}$ that is proportional to the frequency of impinging light and a fixed parameter $\omega_0$.[]{data-label="fig:Fig1"}](Fig1.pdf)
It is interesting to notice that in the case $\omega_{0}=0$ our differential set in Eq.(\[eq:DiffSet\]) reduces to the one describing a semi-infinite waveguide lattice with linearly increasing refractive indexes that presents Bloch oscillations [@Peschel1998p1701; @Pertsch1999p4752]. It is also known that an infinite lattice, considering $\omega_{0}=0$ with a two-waveguides input with a phase difference between them, i.e. $\mathcal{E}(t=0) = \mathcal{E}_{0}(0) + e^{i \phi} \mathcal{E}_{1}(0) $ with $\phi \neq 0$ and $\mathcal{E}_{0,1}(0) \in \mathbb{R}$, shows a ratchet-like behavior controlled by the phase $\phi$ [@Thompson2011p214302]. In our idea of classical simulation of a radiation-matter interaction system, it may be possible to argue that the case $\omega_{0}=0$ corresponds to emulating a hot trapped ion coupled to just the phase of a bosonic mode; such an argument has been used in the quantum Rabi problem [@Casanova2010p263603].
Dispersion relation
===================
Up to our knowledge and means, it is not possible to find an exact dispersion relation for the photonic crystals described above, but it is possible to separate the relevant coupling parameter in two regimes, weak and strong, in order to obtain some results. In the first of these regimes, we can borrow techniques from quantum optics and find an exact dispersion relation. While on the last, we can only deal with the problem through perturbation theory.
Weak-coupling regime: $\lambda \ll \omega_{f}, \omega_{0}$.
-----------------------------------------------------------
In the weak coupling regime one can find the exact spectrum and normal modes of each one of the photonic crystals described by the differential set (\[eq:DiffSet\]) by taking a step back and implementing the rotating wave approximation in the Hamiltonian of the system, $$\begin{aligned}
\hat{H}_{RWA} = \omega_{f} \hat{a}^{\dagger} \hat{a} + \frac{\omega_{0}}{2} \hat{\sigma_{z}} + \lambda \left( e^{\imath \hat{\phi}} \hat{\sigma}_{+} + e^{-\imath \hat{\phi}} \hat{\sigma}_{-} \right), \label{eq:HRWA}\end{aligned}$$ before establishing the classical analogue. Then, it is simpler to find the spectrum and normal modes in this representation by using the basis set $\left\{ \vert n, e \rangle, \vert n+1, g \rangle \right\}$ belonging to the manifold with $n+1$ excitations. This leads to the discrete spectrum, $$\begin{aligned}
E_{\pm,n} = \omega_{f} \left( n + \frac{1}{2} \right) \pm \frac{\Omega}{2}, \qquad \Omega = \sqrt{ \delta^2 + 4 \lambda^{2}}, \label{eq:EigValRWA}\end{aligned}$$ where the detuning is defined as $\delta = \omega_{f} - \omega_{0}$. The proper states are given by $$\begin{aligned}
\vert n, \pm \rangle = \alpha_{\pm} \vert n, e \rangle + \beta_{\pm} \vert n+1, g \rangle, \label{eq:DressedStates}\end{aligned}$$ with $$\begin{aligned}
\frac{\alpha_{\pm}}{\beta_{\pm}} = \frac{- \delta \pm \Omega }{2 \lambda}.\end{aligned}$$ Note that $\vert 0,g \rangle$ is an eigenstate of the Hamiltonian with energy $E_{-,0}=-\omega_{0}/2$.
In our photonic crystals, Eq. (\[eq:DiffSet\]), we can approximate the dispersion relation, equivalent to the discrete spectrum found above, by proposing a collective proper mode and realizing that the three-term recurrence relations can be summarized by the tridiagonal matrix, $$\begin{aligned}
H_{\pm,W} &=& H_{0}^{(\pm)} + P, \\
\left(H_{0}^{(\pm)}\right)_{i,j} &=& \left[ \omega_{f} j \mp \frac{\omega_{0}}{2} (-1)^{j} \right] \delta_{i,j}, \\
\left(P\right)_{i,j} &=& \lambda \left( \delta_{i,j+1} + \delta_{i+1,j} \right),\end{aligned}$$ where the notation $\left( M \right)_{i,j}$ stands for the $(i,j)$th term of Matrix $M$ and the symbol $\delta_{a,b}$ is Kronecker’s delta. As $\lambda \ll \omega_{f}, \omega_{0}$, we can treat matrix $P$ as a perturbation on matrix $H_{0}$ and find the eigenvalues of $H_{\pm}$ up to second order corrections as the first order correction is equal to zero. Thus, we obtain the approximated dispersion relation, $$\begin{aligned}
\omega(q)^{(\pm)} &\approx& \omega_{f} q \mp \frac{ \omega_{0}}{2} (-1)^{q} \left(1 + \frac{ 4 \lambda^{2}}{ \omega_{0}^2 - \omega_{f}^{2}} \right). \label{eq:DispRelWeak}\end{aligned}$$ Figure \[fig:Fig2\] shows good agreement between the dispersion relation given by the exact eigenvalues in the rotating wave approximation, Eq. (\[eq:EigValRWA\]), and the perturbation approach, Eq. (\[eq:DispRelWeak\]), at zero and second order.
![(Color online) A segment of the dispersion relation in the weak-coupling regime. Exact closed form from rotating wave approximation (solid black), $0$th order perturbation (dashed blue) and second order perturbation (dotted red) are shown. We have used a detuning given by $\omega_{0}=1.1 ~\omega_{f}$.[]{data-label="fig:Fig2"}](Fig2.pdf)
Strong-coupling regime: $\lambda \gg \omega_{f}, \omega_{0}$.
-------------------------------------------------------------
In the case when the coupling parameter is larger than the field and transition frequencies, sometimes also called deep-coupling regime, it is possible to write the three term recurrence as $$\begin{aligned}
H_{\pm,S} &=& H_{0} + P_{\pm}, \\
\left(H_{0}\right)_{i,j} &=& \lambda \left( \delta_{i,j+1} + \delta_{i+1,j} \right), \\
\left(P_{\pm}\right)_{i,j} &=& \left[ \omega_{f} j \mp \frac{\omega_{0}}{2} (-1)^{j} \right] \delta_{i,j}.\end{aligned}$$ Notice that lead order matrix is diagonalized via $H_{0} = V \Lambda V^{-1}$, where the diagonal matrix $\Lambda$ contains the values of the dispersion relation in its diagonal and the matrix $V$ has as columns the coefficients of the normal modes given by $$\begin{aligned}
\left( V \right)_{j,q} = U_{j} \left( \frac{\mu(q)}{2 \lambda} \right), \quad \mu(q) \in \mathbb{R},\end{aligned}$$ where the function $U_{n}(x)$ is the $n$th Chebyshev polynomial of the second kind evaluated at $x$; i.e., the dispersion relation for this case is continuous.\
The first order correction for the dispersion relation delivers, $$\begin{aligned}
\omega(q)^{\pm} \approx \mu(q) + \int_{0}^{\infty} d\mu(q) \frac{\sum_{k=0}^{\infty} \left[U_{k}\left( \frac{\mu(q)}{2\lambda} \right) \right]^2 \left[ \omega_{f} k \mp \frac{\omega_{0}}{2} (-1)^{k} \right]}{ \sum_{j=0}^{\infty} \left[U_{j}\left( \frac{\mu(q)}{2\lambda} \right) \right]^2}.\end{aligned}$$
Collective modes
================
For any given set of parameters, decomposition in normal modes delivers the three term recurrence mentioned before, $$\begin{aligned}
\left[a^{(\pm)}_{0} - \omega(q)\right] c^{(\pm,q)}_{0} + \lambda c^{(\pm,q)}_{1} &=&0,\label{eq:RecRel1} \\
\left[a^{(\pm)}_{j} - \omega(q)\right] c^{(\pm,q)}_{j} + \lambda ( c^{(\pm,q)}_{j-1} + c^{(\pm,q)}_{j+1}) &=&0, \label{eq:RecRel2}\end{aligned}$$ with $$\begin{aligned}
a^{(\pm)}_{j} &=& \omega_{f} j \mp \frac{\omega_{0}}{2}(-1)^{j},\end{aligned}$$ where the coefficients $c^{(\pm,q)}_{k}$ are the $k$th coefficients of the $q$th collective mode corresponding to the proper value $\omega(q)$. These coefficients are given by, $$\begin{aligned}
c^{(\pm,q)}_{j} &=& \Pi_{k=0}^{j-1} s^{(\pm,q)}_{k} c^{(\pm,q)}_{0},\end{aligned}$$ where we have used the continued fraction $$\begin{aligned}
s^{(\pm,q)}_{j} &=& \frac{c^{(\pm,q)}_{j+1}}{c^{(\pm,q)}_{j}}, \\
&=& \frac{\lambda}{\omega(q) - a^{(\pm)}_{j+1} - \lambda s^{(\pm,q)}_{j+1}}, \\
&=& \frac{\lambda}{\omega(q) - a^{(\pm)}_{j+1} - \frac{\lambda^2}{\omega(q) - a^{(\pm)}_{j+2} - \ldots} },\end{aligned}$$ where one can always set $c_{0}=1$ and normalize the semi-infinite set later. In the weak-coupling case, the continued fraction is cut at the second term as the parameter $\lambda^2$ is negligible with respect to the field frequency. This delivers a normal mode equivalent to that found in the rotating wave approximation treatment.
In any given regime, we can take the continued fraction result and use it to write the eigenvectors of the quantum Hamiltonian in the reduced form, $$\begin{aligned}
| e_{\pm,m} \rangle &=& \sum_{j=0}^{\infty} c_{j}^{(\pm,m)} |\pm,j\rangle, \\
&=& \tilde{c}_{0} \Pi_{k=0}^{\hat{n}-1} s_{k}^{(m)} \hat{n}! e^{\hat{B}^{\dagger} - \hat{B}} | \pm, 0 \rangle,\end{aligned}$$ where $\tilde{c}_{0}$ is chosen such that $\langle e_{\pm,m} \vert e_{\pm,n} \rangle = \delta_{m,n}$.
For the case $\omega_{0}=0$, the amplitudes of the eigenvectors are well known and calculated by moving the differential set to Fourier domain and solving it there [@Peschel1998p1701]. We want to mention here that it is trivial to obtain an equivalent form from the recurrence relations in Eqs.(\[eq:RecRel1\]-\[eq:RecRel2\]) by setting $\omega_{0}=0$, $c_{j}^{(+,m)}=c_{j}^{(-,m)}= c_{j}(\omega_{q})\equiv c_{j}$, $\omega(q)\equiv \omega$, and $c_{0}=1$, $$\begin{aligned}
c_{j}&=& \frac{\pi}{\omega_{f}} \left\{ Y_{j-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right) \left[ (\omega - j \omega_{f}) J_{-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right) - \lambda J_{1-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right) \right] \right. + \nonumber \\
&& \left. - J_{j-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right) \left[ (\omega - j \omega_{f}) Y_{-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right) - \lambda Y_{1-\frac{\omega}{\omega_{f}}}\left( -\frac{2 \lambda}{\omega_{f}}\right)\right] \right\},\end{aligned}$$ where the symbols $J_{\alpha}(x)$ and $Y_{\alpha}(x)$ stand for the modified Bessel functions of the first and second kind, and the coefficients are given up to a normalization factor.
Propagation examples
====================
The dressed state basis that diagonalize our model in the weak-coupling regime, Eq.(\[eq:DressedStates\]), implies that starting in a state of the kind $\vert j, e \rangle$ or $\vert j+1, g \rangle$, with $j\ge0$, will produce coherent oscillations. Such an initial state translates to laser light impinging the $j$th or $(j+1)$th waveguide of one of the lattices; the case $j$ even (odd) corresponds to the crystal $H_-$ ($H_+$ ). Figure \[fig:Fig3\](a) shows the propagation of light impinging at the $0$th waveguide of our photononic crystal $H_{-}$ in the weak-coupling regime which is equivalent to an initial state $\vert 0,e \rangle$ in the quantum optics model. Accordingly, we observe the intensity oscillate between the $0$th and first wave corresponding to an oscillation between the $\vert 0,e \rangle$ and $\vert 1,g \rangle$ states in the quantum optics model. Figure \[fig:Fig3\](c) presents a detail of the intensity at the $0$th and first waveguide. This normalized intensity at the 0th waveguide is proportional to the probability of finding the time evolution of the quantum system back in the initial state, $I_{0} \leftrightarrow P_{-,0}$, with $$\begin{aligned}
P_{-,0} = \vert \langle -,0 \vert \psi(t) \rangle \vert^2, \quad \vert \psi(0) \rangle = \vert -,0 \rangle. \end{aligned}$$
In quantum optics literature, it is known that the quantum Rabi model produces coherent oscillations in the two-level system inversion for $\omega_{0}=0$ [@Casanova2010p263603]. In our model, due to the continuous spectra, all proper states are scattering states and, in the general case, at most we observe partial recovery of the original state when the field starts localized at a given waveguide. Figure \[fig:Fig3\](b) shows the propagation of light impinging at the $0$th waveguide of the photonic lattice $H_{+}$ in the strong-coupling regime, which is equivalent to an initial state $\vert 0,g \rangle$ in the quantum optics model; Figure \[fig:Fig3\](d) focus on the intensity at the first two waveguides. Again, the normalized intensity at the 0th waveguide is proportional to the probability of finding the time evolution of the quantum system back in the initial state, $I_{0} \leftrightarrow P_{+,0} = \vert \langle +,0 \vert \psi(t) \rangle \vert^2$, with $\vert \psi(0) \rangle = \vert +,0 \rangle$, for this case. Thus, our model presents both similar and different behaviors from the full quantum Rabi model in the weak- and strong-coupling regimes, in that order.
![(Color online) Examples of propagation in our negative parity photonic crystal. The left column (a,c) shows the case of weak-coupling, $\lambda = 0.1 ~\omega_{f}$, and the right (b,d) the case of strong-coupling, $\lambda = 2 ~ \omega_{f}$. The first row (a,b) depicts intensity propagation on the first ten waveguides of a total of five thousand when light impinges the zeroth waveguide. The second row (c,d) shows the normalized intensity at the $0$th (solid black) and first (dashed blue) waveguides. The dimensionless time parameter $\omega_{f} t$ is equivalent to the typical dimensionless propagation parameter.[]{data-label="fig:Fig3"}](Fig3.pdf)
We can go further than comparing with the full quantum Rabi problem. For $\omega_{0}=0$, it is known that single waveguide initial states far from the edge will produce a breather mode evolution and will reconstruct periodically, while multiple-waveguide initial states with Gaussian weight distributions will present Bloch oscillations [@Peschel1998p1701; @Pertsch1999p4752]. Also, if we consider a double contiguous waveguide input with a phase difference between the weights of the components, $$\begin{aligned}
\vert \psi_{k}(0) \rangle = \frac{1}{\sqrt{2}} \left( \vert k \rangle + e^{i \phi} \vert k+1 \rangle \right), \quad k \gg 0, \label{eq:Superposition}\end{aligned}$$ it is possible to obtain a periodical reconstruction [@Thompson2011p214302]. This is witnessed by the fidelity, $$\begin{aligned}
\mathcal{F} = \vert \langle \psi(0) \vert \psi(t) \rangle\vert^2. \label{eq:Fidelity}\end{aligned}$$ Such an initial state produces a break in the symmetry of the breather mode of a single input for $\phi \ne 0$ [@Thompson2011p214302]. This breaking in the propagation symmetry is accompanied by a variable center of mass of the beam, $$\begin{aligned}
x_{cm} &=& \langle \psi(t) \vert \hat{n} \vert \psi(t) \rangle, \label{eq:CenterMass} \\
&=& \sum_{j}^{\infty} j \vert c_{j} \vert ^{2}, \end{aligned}$$ where the amplitudes $c_{j}$ are normalized.
Figure 4 shows the propagation of double waveguide initial state, $\vert \psi_{k}(0)\rangle$ in Eq.(\[eq:Superposition\]) with $k=20$ and $\phi = \pi/6$. The left column shows the case where $\omega_{0}=0$; it is possible to see that a nonsymmetric breather mode is formed upon propagation, Fig.\[fig:Fig4\](a), with periodic reconstruction, Fig.\[fig:Fig4\](c), and a so-called ratchet like behavior of the center of mass of the beam, Fig.\[fig:Fig4\](e). The introduction of the binary modification, $\omega_{0}=\omega_{f}$, destroys the nonsymmetric propagation breather mode, Fig.\[fig:Fig4\](b), with partial reconstruction of the input, Fig.\[fig:Fig4\](d), and a quasi-periodical behavior of the center of mass of the beam, Fig.\[fig:Fig4\](f).
![(Color online) Examples of propagation in our positive parity photonic crystal in the strong-coupling regime with $\lambda = 2 ~\omega_{f}$. The initial state is the balanced superposition of light impinging the 20th and 21th waveguide with a phase difference $\phi = \pi/6$. The left column (a,c,e) shows the case, $\omega_{0} = 0$, and the right column (b,d,f) the case, $\omega_{0} = \omega_{f}$. The first row (a,b) depicts intensity propagation on the waveguides from 10th to 30th of a total of five thousand. The second row (c,d) shows the evolution of the Fidelity, Eq.(\[eq:Fidelity\]). The third row (e,f) shows the evolution of the center of mass of the beam, Eq.(\[eq:CenterMass\]). The dimensionless time parameter $\omega_{f} t$ is equivalent to the typical dimensionless propagation parameter.[]{data-label="fig:Fig4"}](Fig4.pdf)
All numerical simulations correspond to a finite photonic lattice of size 5000.
Conclusion
==========
We have proposed a set of two photonic crystals that classically simulates a new radiation-matter interaction where a two-level system is driven by just the phase of a quantum field. Up to our knowledge such a radiation–matter interaction systems does not occur in nature and has not been proposed before.
We show that it is possible to determine exactly the dispersion relation of the photonic waveguide lattices in the so-called weak-coupling regime and that in the strong-coupling regime we can use perturbation theory to approximate the dispersion relation up to second order perturbation. In the first case, the dispersion relation is discrete and, in the latter, continuous. Accordingly, the spectra of the radiation–matter interaction model is equivalent to that of the classical simulator.
The normal modes of the crystals are easily expressed in terms of continued fractions as a function of the dispersion relation. In a simplified version, they can be expressed in terms of modified Bessel functions of the first and second kind.
In the simplified version, an initial state consisting of light impinging two contiguous waveguides with a phase difference between them produces a phase controlled, so-called ratchet-like behavior of the center of mass of the beam upon propagation. This behavior is attenuated and becomes noisy in the most general model as the extra parameter increases in value.
Our optical realization of a phase driven two-level system provides a scheme to explore an interesting process that is not accessible by usual means; i.e. cavity-QED or trapped ions. And describes a phase controlled phenomenon that may be realized in other radiation–matter interaction systems.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors are grateful to an anonymous reviewer for his valuable comments and bringing forward important references.
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|
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abstract: 'We study the task of learning from non-data. In particular, we aim at learning predictors that minimize the *conditional risk* for a stochastic process, the expected loss of the predictor on the next point conditioned on the set of training samples observed so far. For non-data, the training set contains information about the upcoming samples, so learning with respect to the conditional distribution can be expected to yield better predictors than one obtains from the classical setting of minimizing the *marginal risk*. Our main contribution is a practical estimator for the conditional risk based on the theory of non-parametric time-series prediction, and a finite sample concentration bound that establishes uniform convergence of the estimator to the true conditional risk under certain regularity assumptions on the process.'
author:
- |
Alexander Zimin\
IST Austria\
3400 Klosterneuburg, Austria\
`azimin@ist.ac.at`\
Christoph H. Lampert\
IST Austria\
3400 Klosterneuburg, Austria\
`chl@ist.ac.at`\
bibliography:
- 'biblio.bib'
title: |
Conditional Risk Minimization\
for Stochastic Processes
---
|
---
address: 'Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WB, U.K.'
author:
- Ivan Smith
date: January 2020
title: 'Floer theory of higher rank quiver 3-folds'
---
Introduction
============
Fix a pair of positive integers $(g,d)$ and consider a closed surface ${\mathbb{S}}$ of genus $g>0$ with a non-empty collection of $d> 0$ marked points $\bP \subset {\mathbb{S}}$. Fix in addition a positive integer $m> 0$, called the ‘rank’. We work over the characteristic zero field $\bK = \Lambda_{\bC}$ which is the one-variable Novikov field over $\bC$. Associated to this data, there is
1. a $\bK$-linear CY$_3$ $A_{\infty}$-category $({\EuScript{C}}, W)$, obtained from a choice of potential $W$ on a quiver $Q(\Delta_m)$ associated to a choice of ideal triangulation $\Delta$ of ${\mathbb{S}}$ with vertices at $\bP$ and with no self-folded triangles, see [@Ginzburg; @Goncharov] and Section \[Sec:Quivers\];
2. a non-compact Kähler Calabi-Yau threefold $(Y,\omega)$, which is the total space of a fibration by $A_m$-surfaces over ${\mathbb{S}}$ with fibres over $\bP$ being disjoint unions of $m+1$ planes $\bC^2$, and a further collection of Lefschetz singular fibres, see [@Abrikosov; @DDP] and Section \[Sec:quiver\_3folds\].
Both the above depend on choices: the potential $W$ on the quiver (up to gauge equivalence) in the first case, and the (cohomology class of) Kähler form $\omega$ in the second. We will make a further choice, which is an ordering of the $m+1$ connected components of the reducible singular fibre of $Y$ over each point of $\bP$. The sum of the even-indexed components in this ordering, summed over each such reducible fibre, defines a class $b\in H^2(Y;\bZ/2)$. Relative to this background class, there is a sign-twisted Fukaya category ${\EuScript{F}}((Y,\omega); b)$, which is an $A_{\infty}$-category over $\bK$ whose objects are $b$-relatively spin graded $\omega$-Lagrangian submanifolds equipped with suitable brane data. (The choice of cycle representative for $b$ is natural in a particular setting encountered later, but monodromy considerations show that up to quasi-isomorphism the category only depends on the number of components at each $p \in \bP$.)
\[thm:main\] There is a non-empty open subset $U\subset H^2(Y;\bR)$ of the Kähler cone, and a map $U \to \{\mathrm{potentials}\}/\{\mathrm{gauge}\}$, $[\omega] \mapsto W_{[\omega]}$, such that for $[\omega] \in U$ there is a fully faithful embedding $({\EuScript{C}}, W_{[\omega]}) \hookrightarrow {\EuScript{F}}((Y,\omega);b)$.
The hypothesis $g({\mathbb{S}})>0$ simplifies the holomorphic curve theory (it implies the Fukaya category can be constructed using classical transversality theory). After passing to twisted complexes on both sides, the image of the embedding of Theorem \[thm:main\] is a split-closed triangulated subcategory. We conjecture that image co-incides with the full subcategory of $\mathrm{Tw}\,{\EuScript{F}}(Y,\omega;b)$ generated by Lagrangian spheres, and is therefore intrinsic to the symplectic topology of $(Y,\omega)$. One could view the algebraic model $({\EuScript{C}}, W_{[\omega]})$ as a ‘non-commutative mirror’ to $(Y,\omega)$, and Theorem \[thm:main\] as a statement of homological mirror symmetry in this setting[^1].
Goncharov conjectured in [@Goncharov Conjecture 6.2] that the $CY_3$-category associated to $Q(\Delta_m)$ and the ‘canonical’ potential $W = W(\Delta_m)$ on the underlying bipartite graph (as introduced in [@Franco-Hanany-etal]) should be realised as a subcategory of a Fukaya category. Goncharov’s conjecture, stemming from general expectations around ‘categorifications’ of cluster varieties, was futher elaborated by Abrikosov [@Abrikosov Conjecture 1.4]; Theorem \[thm:main\] proves the formulation given there. The result also relates to questions of Shende, Treumann and Williams [@STW Problems 1.15 & 1.16] on the existence of potentials governing local Calabi-Yau 3-folds associated to surfaces. It should be possible to relate the canonical potential, which defines a $\bC$-linear category, to the one coming from symplectic topology when the surface ${\mathbb{S}}$ is punctured and the associated threefold is exact as a symplectic manifold (this is true in the simplest case when there are punctures and no marked points / reducible fibres, in which case the canonical potential has only cubic and quartic terms; one can more generally work in a ‘relative Fukaya category’, cf. Remark \[rmk:relative\]). In the non-exact case, holomorphic curves are weighted by their areas encoded in the Novikov variable; the resulting potential never has trivial Novikov valuation. One can still recover the cohomology class $[\omega]$ from the potential, cf. Remark \[rmk:class\_of\_omega\].
The theorem is proved, following [@Smith:quiver] for $m=1$, by finding a collection of Lagrangian 3-spheres $\{L_v \, | \, v\in \mathrm{Vert}(Q(\Delta_m))\}$ in $Y$ whose Floer cohomology algebra $\oplus_{v,v'} HF^*(L_v,L_{v'})$ agrees with the Koszul dual to the Ginzburg algebra associated to $Q(\Delta_m)$. (The open subset $U = U(\Delta) \subset H^2(Y;\bR)$ of the Kähler cone, which in principle depends on $\Delta$, is any for which the relevant configuration of Lagrangian spheres exists. It is not clear if $\cup_{\Delta} U(\Delta)$ covers the Kähler cone.) The general theory of cyclic $A_{\infty}$-structures then implies that the subcategory ${\EuScript{F}}(\scrL) \subset {\EuScript{F}}(Y;b)$ generated by $\{L_v\}$ is governed by some potential $W_{[\omega]}$ on $Q(\Delta_m)$. Further study of non-vanishing holomorphic polygon counts shows that $W_{[\omega]} = W_{\bf c}(\Delta_m) + W'$ for a $\bK$-coefficient vector ${\bf c}$ recording areas of polygons associated to certain distinguished ‘primitive’ (chordless) cycles, and some ‘nonlocal’ terms $W'$, which cannot *a priori* be controlled. (The ‘canonical’ potential $W(\Delta_m)$ is exactly $W_{\bf c}(\Delta_m)$ for a vector of coefficients each of which is $\pm 1$; in the non-exact case we record information on $[\omega]$ in ${\bf c}$.) We conclude that some $A_{\infty}$-deformation of $({\EuScript{C}}, W_{\bf c}(\Delta_m))$ embeds into the Fukaya category, without specifying exactly which; resolving this ambiguity is a version of fixing a mirror map.
On the geometric side, the crucial new ingredient when passing from $m=1$ to $m>1$ is the presence of ‘tripod’ Lagrangian spheres in $A_m$-Milnor fibres, see Section \[Sec:Tripods\], and their appearance in the sphere configurations associated to $\Delta_m$.
Theorem \[thm:main\] relates to work of Gaiotto, Moore and Neitzke [@GMN:spectral; @GMN:snakes] on ‘theories of class $\mathcal{S}$’, certain four-dimensional $\mathcal{N}=2$ field theories. They relate the BPS degeneracies of solitons in such theories to ‘spectral networks’ on a Riemann surface equipped with a tuple of meromorphic differentials. In the rank $m=1$ case, this relates BPS states and saddle connections of meromorphic quadratic differentials [@GMN; @BridgelandSmith]. Long-standing expectations in both mathematics and physics suggest that the counting of BPS states should be formalised by counts of stable objects in triangulated categories such as Fukaya categories. The tripod spheres which enter into the proof of Theorem \[thm:main\] exactly correspond to the simplest spectral networks after saddle connections, see [@GMN:spectral Figure 3]. The possible *embedded* graded Lagrangians in $Y_{\Phi}$ are constrained by results of [@GanatraPomerleano], and one only obtains connect sums of copies of $S^1\times S^2$ and $3$-tori. It would be interesting to construct unobstructed immersed special Lagrangian representatives for more general spectral networks.
One can have pairs of tripods which meet at all three feet, and give rise to Lagrangian 3-spheres $L_0, L_1$ which meet at 3 transverse intersection points of equal Maslov grading, and bounding no holomorphic discs. The subcategory $\langle L_0, L_1 \rangle \subset {\EuScript{F}}(Y_{\Phi};b)$ is quasi-isomorphic to the Ginzburg category of the three-arrow Kronecker quiver. It then follows from [@Reineke], see also [@Mainiero], that there are classes $\eta \in K({\EuScript{F}}(Y_{\Phi};b))$ for which the DT-invariant of $d\cdot \eta$ grows exponentially with $d$, a phenomenon that does not occur for the 3-folds in rank one [@BridgelandSmith]. (The associated field theories of class $\mathcal{S}$ have ‘wild BPS spectra’ and ‘BPS giants’.) The 3-folds $Y_{\Phi}$ for rank $m>1$ contain graded Lagrangian submanifolds diffeomorphic to $(S^1\times S^2) \# (S^1 \times S^2)$, obtained from Lagrange surgery $L_0 \# L_1$ on $L_0$ and $L_1$. Irreducible modules over the based loop space $\Omega(L_0\# L_1)$ give rise to candidate stable objects to realise wild BPS states in the Fukaya category.
#### Acknowledgements.
Dmitry Tonkonog contributed several ideas at an early stage. Thanks to Mohammed Abouzaid, Efim Abrikosov, Tom Bridgeland, Andy Neitzke and Pietro Longhi for helpful conversations, and to Sasha Goncharov for his interest. The author is partially funded by a Fellowship from the Engineering and Physical Sciences Research Council, U.K.
Quivers and potentials from ideal triangulations\[Sec:Quivers\]
===============================================================
Categories from quivers with potential
--------------------------------------
A well-known construction due to Ginzburg [@Ginzburg] associates to a quiver with potential $(Q,W)$ a 3-dimensional Calabi-Yau cyclic $A_{\infty}$-category ${\EuScript{C}}(Q,W)$. The category ${\EuScript{C}}(Q,W)$ is the total $A_{\infty}$-endomorphism algebra of a collection of spherical objects $S_v$ indexed by the vertices $v\in Q_0$ of $Q$; it is concentrated in degrees $0 \leq * \leq 3$, the degree one morphism spaces are based by the arrows $Q_1$ of $Q$, and the potential gives a cyclic encoding of the non-trivial $A_{\infty}$-products, see [@Smith:quiver Section 2] for a summary of the construction. We denote by $\scrD(Q,W)$ the corresponding derived category. Mutations of $(Q,W)$ induce equivalences of the derived categories by [@Keller-Yang].
\[lem:lagrangians\_to\_quiver\] Let $(X,\omega)$ be a symplectic manifold with a well-defined Fukaya category ${\EuScript{F}}(X)$. Suppose we have objects $\{L_v \in {\EuScript{F}}(X) \, | \, v \in Q_0\}$ for which the total (cohomological) endomorphism algebras $$\oplus_{v, v' \in Q_0} \, HF^*(L_v, L_{v'}) \cong \oplus_{v,v' \in Q_0} \, \Hom_{{\EuScript{C}}}(S_v, S_{v'})$$ are isomorphic as graded algebras over the semisimple ring $\oplus_{v\in Q_0} \bK_v$ (with idempotents the units of the objects $L_i$ respectively $S_i$). Then the full $A_{\infty}$-subcategory $\scrL \subset {\EuScript{F}}(X)$ generated by the $\{L_v\}$ is encoded up to quasi-isomorphism by a potential $W_{\scrL}$ on $Q$.
This is almost tautological. The $A_{\infty}$-structure on Floer cochains $\oplus_{i.j} CF^*(L_i,L_j)$ can always be taken to be strictly unital, since the reduced Hochschild complex is quasi-isomorphic to the full Hochschild complex over a field, and that strictly unital structure can be pushed to cohomology by homological perturbation. The fact that $\bR \subset \bK$ means that the $A_{\infty}$-structure can also be taken to be strictly cyclic (this holds by abstract theory over any characteristic zero field [@KontSoi], but can be achieved for geometric reasons for fields $\bK \supset \bR$ [@Fukaya:cyclic]). The book-keeping in the Ginzburg construction then shows that the subcategory $\scrL$ is encoded by a cyclic potential on $Q$.
In the setting of Lemma \[lem:lagrangians\_to\_quiver\], the cubic terms in the potential encode the Floer product, which is well-defined; the higher order terms determine the higher $A_{\infty}$-products which depend on choices of almost complex structure and perturbation data. Lemma \[lem:lagrangians\_to\_quiver\] implies that, given a finite collection $\{L_v\, | \, v \in Q_0\}$ of Lagrangian rational homology spheres in a CY 3-fold for which the morphism spaces $HF^*(L_v, L_{v'})$ are concentrated in degrees $1,2$ for all $v\neq v'$, then the $A_{\infty}$-structure on $\oplus_{v,v'} HF^*(L_v,L_{v'})$ is encoded by a cyclic potential on the quiver with vertices $Q_0$ and arrow spaces $Q_1$ indexed by bases for $HF^1(L_v,L_{v'})$.
Gauge transformations
---------------------
Potentials are called *right-equivalent* if they are related by an automorphism of the completed path algebra; right equivalent potentials $W$ and $W'$ on $Q$ yield quasi-isomorphic $A_{\infty}$-categories ${\EuScript{C}}(Q,W) \simeq {\EuScript{C}}(Q,W')$, cf. [@Ginzburg; @KontSoi]. There are $A_{\infty}$-equivalences which do not arise from right equivalences, for instance ones acting non-trivially on cohomology, and ones arising from the canonical $\bK^*$-action on $A_{\infty}$-structures which rescales $m^k$ by $\lambda^{k-2}$.
The group $\scrG$ of right equivalences of the completed path algebra decomposes as a semidirect product $$\scrG = \scrG^{un} \rtimes \scrG^{diag}$$ where the second factor of ‘diagonal’ automorphisms are those which arise from automorphisms of the vector space of arrows (as bimodules over the semisimple ring given by the idempotent lazy paths at the vertices), and the first factor of ‘unitriangular’ automorphisms are those induced by maps from the arrow space into the subspace of paths of length $\geq 2$. When the arrow space between any two vertices is at most one-dimensional, then $\scrG^{diag} \cong (\bK^*)^{|\mathrm{Vert}(Q)|}$ acts just by diagonally rescaling the arrows.
A quiver has a finite distinguished set of ‘chordless’ cycles, see [@DWZ]. A potential is ‘primitive’ if it is a combination of chordless cycles, and every chordless cycle appears with non-zero coefficient. A potential is ‘generic’ if its projection to the span of chordless cycles is primitive. Using the fact that the set of chordless cycles is intrinsic to the quiver, [@Abrikosov Section 5.1] asserts that projection to the primitive part of a potential yields a $\scrG^{diag}$-equivariant projection $$\label{eqn:project_potential}
\{\mathrm{Generic \ potentials}\} / \scrG \longrightarrow \{\mathrm{Primitive \ potentials}\}/\scrG^{diag}.$$ Return to the situation of Lemma \[lem:lagrangians\_to\_quiver\]. Lagrangian 3-spheres will persist as Lagrangians under any sufficiently small deformation of the symplectic form $[\omega] \in U \subset H^2(X;\bR)$ on $X$, and (appealing to sufficient technology in the non-weakly-exact case) will remain unobstructed. Suppose furthermore that the Floer cohomologies $\oplus_{v,v'} HF^*(L_v,L_{v'})$ do not change, as graded $\bK$-vector spaces, as one varies the symplectic form in $U$. Then one obtains maps $$\label{eq:mirror}
H^2(X;\bR) \supset U \longrightarrow \{\mathrm{Potentials}\} / \scrG \longrightarrow \{\mathrm{Primitive \ potentials}\}/\scrG^{diag}.$$ The RH group above is *a priori* finite dimensional, whilst the set $\{\mathrm{Potentials}\}/\scrG$ of all cyclic $A_{\infty}$-structures need not be. Our aim is to determine the composite map ; giving its lift to $\{\mathrm{Potentials}\} / \scrG$ is somewhat like finding a ‘mirror map’, which we leave undetermined.
The map is not a local isomorphism (the domain and codomain have different dimensions). For one thing, the coefficients of the potential – which record areas of holomorphic polygons determined by areas of polygonal regions in the dual cellulation $\Delta_m^{\vee}$ – are governed by $[\omega] \in H^2(Y_{\Phi}, \sqcup_v L_{v};\bR)$, whilst the Fukaya category ${\EuScript{F}}(Y_{\Phi})$ only depends on $[\omega] \in H^2(Y_{\Phi};\bR)$.
Quivers with potential from triangulations
------------------------------------------
We summarise some results from [@Abrikosov]. Take an ideal triangulation $\Delta$ of ${\mathbb{S}}$ with vertices at $\bP \subset {\mathbb{S}}$ and with no self-folded triangles (i.e. all triangles have three distinct edges). We place $m$ vertices on each edge of the ideal triangulation, and then subdivide the triangulation (cf. Figure \[Fig:inscribed\], showing the case $m=4$) to obtain a new triangulation $\Delta_m$. We view this as bicoloured as in Figure \[Fig:inscribed\], so each triangle of $\Delta$ now has inscribed within it $m(m+1)/2$ black triangles. We then orient the edges of these inscribed black triangles as in Figure \[Fig:inscribed\]; doing this for each ideal triangle in $\Delta$ yields a quiver drawn on the surface ${\mathbb{S}}$, each vertex of which is one of those originally placed on $\Delta$. We denote the resulting quiver by $Q(\Delta_m)$; it depends on $\Delta$ and the choice of rank $m \geq 1$.
\[rmk:numerics\] An ideal triangulation of a surface of genus $g$ with $d$ marked points (vertices) has $6g-6+3d$ edges and $4g-4+2d$ faces.
(-3,0) – (3,0); (-3,0) – (0,5); (0,5) – (3,0);
(-1.5,0) – (-3 + 0.75, 1.25); (0,0) – (-3+1.5, 2.5); (1.5,0) – (-3+2.25, 3.75);
(-1.5,0) – (3-2.25,3.75); (0,0) – (3-1.5,2.5); (1.5,0) – (3-0.75,1.25);
(-2.25,1.25) – (2.25,1.25); (-1.5,2.5) – (1.5,2.5); (-0.75,3.75) – (0.75,3.75);
(-0.75,3.75) – (0.75,3.75) – (0, 2.5); (-1.5,2.5) – (0,2.5) – (-0.75, 1.25); (0,2.5)–(1.5,2.5) – (0.75,1.25); (-2.25,1.25) – (-0.75,1.25) – (-1.5,0); (-0.75,1.25) – (0.75,1.25) – (0,0); (0.75,1.25) – (2.25,1.25) – (1.5,0);
(0.75,3.75)– node [[(0,0) – +(.1,0);]{}]{} (-0.75,3.75); (-.75,3.75)– node [[(0,0) – +(.1,0);]{}]{} (0,2.5); (0,2.5) – node [[(0,0) – +(.1,0);]{}]{} (0.75,3.75);
(0,2.5)– node [[(0,0) – +(.1,0);]{}]{} (-1.5,2.5); (-1.5,2.5)– node [[(0,0) – +(.1,0);]{}]{} (-0.75,1.25); (-.75,1.25) – node [[(0,0) – +(.1,0);]{}]{} (0,2.5);
(0,2.5)– node [[(0,0) – +(.1,0);]{}]{} (.75,1.25); (.75,1.25)– node [[(0,0) – +(.1,0);]{}]{} (1.5,2.5); (1.5,2.5) – node [[(0,0) – +(.1,0);]{}]{} (0,2.5);
(-1.5,0)– node [[(0,0) – +(.1,0);]{}]{} (-0.75,1.25); (-.75,1.25)– node [[(0,0) – +(.1,0);]{}]{} (-2.25,1.25); (-2.25,1.25) – node [[(0,0) – +(.1,0);]{}]{} (-1.5,0);
(1.5,0)– node [[(0,0) – +(.1,0);]{}]{} (2.25,1.25); (2.25,1.25)– node [[(0,0) – +(.1,0);]{}]{} (.75,1.25); (.75,1.25) – node [[(0,0) – +(.1,0);]{}]{} (1.5,0);
(-.75,1.25)– node [[(0,0) – +(.1,0);]{}]{} (0,0); (0,0)– node [[(0,0) – +(.1,0);]{}]{} (0.75,1.25); (.75,1.25)– node [[(0,0) – +(.1,0);]{}]{} (-.75,1.25);
There are three visible collections of closed cycles on the inscribed quiver $Q(\Delta_m)$ (see Figures \[Fig:inscribed\], \[Fig:cycle\_types\_both\] and Figure \[Fig:cycle\_types\_punctures\]). We will call these ‘primitive cycles’. Namely, one has
(0,0) – (3,0); (0,0) – (-3,0); (0,0) – ([3\*cos(60)]{}, [3\*sin(60)]{}); (0,0) – ([-3\*cos 60]{}, [3\*sin 60]{}); (0,0) – ([3\*cos 60]{}, [-3\*sin 60]{}); (0,0) – ([-3\*cos 60]{}, [-3\*sin 60]{});
([-3\*cos 60]{}, [3\*sin 60]{}) – ([3\*cos 60]{}, [3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – ([3\*cos 60]{}, [-3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – (-3,0); ([3\*cos 60]{}, [-3\*sin 60]{}) – (3,0); ([3\*cos 60]{}, [3\*sin 60]{}) – (3,0); ([-3\*cos 60]{}, [3\*sin 60]{}) – (-3,0);
(1,0) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [sin 60]{}); ([cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (1,0);
(2,0) – node [[(0,0) – +(.1,0);]{}]{} ([2+cos 60]{}, [sin 60]{}); ([2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (2,0);
([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+2\*cos 60]{}, [2\*sin 60]{}); ([1+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2\*cos 60]{}, [2\*sin 60]{}); ([2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{},[sin 60]{});
(-1,0) – node [[(0,0) – +(.1,0);]{}]{} ([-1+cos 60]{}, [sin 60]{}); ([-1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-1,0);
(-2,0) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-3+cos 60]{}, [sin 60]{}); ([-3+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-2,0);
([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+2\*cos 60]{}, [2\*sin 60]{}); ([-2+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-3+2\*cos 60]{}, [2\*sin 60]{}); ([-3+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{},[sin 60]{});
([-cos 60, sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [sin 60]{}); ([cos 60, sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [sin 60]{});
(0,[2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (1, [2\*sin 60]{}); (1, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{},[3\*sin 60]{}); ([cos 60, 3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{});
(-1,[2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{},[3\*sin 60]{}); ([-cos 60, 3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (-1, [2\*sin 60]{});
([cos 60, -sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [-sin 60]{}); ([-cos 60, -sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [-sin 60]{});
(0,[-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (-1, [-2\*sin 60]{}); (-1, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{},[-3\*sin 60]{}); ([-cos 60, -3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{});
(1,[-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{},[-3\*sin 60]{}); ([cos 60, -3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (1, [-2\*sin 60]{});
(1,0) – node [[(0,0) – +(.1,0);]{}]{} ([1-cos 60]{}, [-sin 60]{}); ([1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (1,0);
(2,0) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([3-cos 60]{}, [-sin 60]{}); ([3-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (2,0);
([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-2\*cos 60]{}, [-2\*sin 60]{}); ([2-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([3-2\*cos 60]{}, [-2\*sin 60]{}); ([3-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{},[-sin 60]{});
(-1,0) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [-sin 60]{}); ([-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-1,0);
(-2,0) – node [[(0,0) – +(.1,0);]{}]{} ([-2-cos 60]{}, [-sin 60]{}); ([-2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-2,0);
([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-2\*cos 60]{}, [-2\*sin 60]{}); ([-1-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2\*cos 60]{}, [-2\*sin 60]{}); ([-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{},[-sin 60]{});
(0,0) – (3,0); (0,0) – (-3,0); (0,0) – ([3\*cos(60)]{}, [3\*sin(60)]{}); (0,0) – ([-3\*cos 60]{}, [3\*sin 60]{}); (0,0) – ([3\*cos 60]{}, [-3\*sin 60]{}); (0,0) – ([-3\*cos 60]{}, [-3\*sin 60]{});
([-3\*cos 60]{}, [3\*sin 60]{}) – ([3\*cos 60]{}, [3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – ([3\*cos 60]{}, [-3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – (-3,0); ([3\*cos 60]{}, [-3\*sin 60]{}) – (3,0); ([3\*cos 60]{}, [3\*sin 60]{}) – (3,0); ([-3\*cos 60]{}, [3\*sin 60]{}) – (-3,0);
(1,0) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [sin 60]{}); ([cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (1,0);
(2,0) – node [[(0,0) – +(.1,0);]{}]{} ([2+cos 60]{}, [sin 60]{}); ([2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (2,0);
([1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+2\*cos 60]{}, [2\*sin 60]{}); ([1+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2\*cos 60]{}, [2\*sin 60]{}); ([2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([1+cos 60]{},[sin 60]{});
(-1,0) – node [[(0,0) – +(.1,0);]{}]{} ([-1+cos 60]{}, [sin 60]{}); ([-1+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-1,0);
(-2,0) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-3+cos 60]{}, [sin 60]{}); ([-3+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-2,0);
([-2+cos 60]{}, [sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+2\*cos 60]{}, [2\*sin 60]{}); ([-2+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-3+2\*cos 60]{}, [2\*sin 60]{}); ([-3+2\*cos 60]{}, [2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2+cos 60]{},[sin 60]{});
([-cos 60, sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [sin 60]{}); ([cos 60, sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [sin 60]{});
(0,[2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (1, [2\*sin 60]{}); (1, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{},[3\*sin 60]{}); ([cos 60, 3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{});
(-1,[2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{},[3\*sin 60]{}); ([-cos 60, 3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (-1, [2\*sin 60]{});
([cos 60, -sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [-sin 60]{}); ([-cos 60, -sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{}, [-sin 60]{});
(0,[-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (-1, [-2\*sin 60]{}); (-1, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([-cos 60]{},[-3\*sin 60]{}); ([-cos 60, -3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{});
(1,[-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} ([cos 60]{},[-3\*sin 60]{}); ([cos 60, -3\*sin 60]{}) – node[[(0,0) – +(.1,0);]{}]{} (1, [-2\*sin 60]{});
(1,0) – node [[(0,0) – +(.1,0);]{}]{} ([1-cos 60]{}, [-sin 60]{}); ([1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (1,0);
(2,0) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([3-cos 60]{}, [-sin 60]{}); ([3-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (2,0);
([2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-2\*cos 60]{}, [-2\*sin 60]{}); ([2-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([3-2\*cos 60]{}, [-2\*sin 60]{}); ([3-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([2-cos 60]{},[-sin 60]{});
(-1,0) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-cos 60]{}, [-sin 60]{}); ([-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-1,0);
(-2,0) – node [[(0,0) – +(.1,0);]{}]{} ([-2-cos 60]{}, [-sin 60]{}); ([-2-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} (-2,0);
([-1-cos 60]{}, [-sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-2\*cos 60]{}, [-2\*sin 60]{}); ([-1-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-2\*cos 60]{}, [-2\*sin 60]{}); ([-2\*cos 60]{}, [-2\*sin 60]{}) – node [[(0,0) – +(.1,0);]{}]{} ([-1-cos 60]{},[-sin 60]{});
- anticlockwise-oriented 3-cycles $\{t_b\}$, each the boundary of a single inscribed black triangle $b$;
- clockwise-oriented 3- and 4-cycles $\{q_w\}$, boundaries of the white regions $w$ which are those complementary regions on ${\mathbb{S}}$ to the black triangles which do not contain a point $p\in \bP$; see Figure \[Fig:cycle\_types\_both\] for a four-cycle $q_w$;
- for each point $p\in \bP$, which has valence $k$ as a vertex of $\Delta$, larger clockwise-oriented $k$-cycles $L_p^{(j)}$ for $1\leq j \leq m$; see Figure \[Fig:cycle\_types\_punctures\].
When $m=1$, the middle class is not present; when $m=2$, there are only quadrilaterals in the middle class, and no 3-cycles.
The primitive cycles are exactly the chordless cycles for $Q(\Delta_m)$.
The decomposition of ${\mathbb{S}}$ into the black and white regions of the quiver and its complement amounts to giving a bipartite graph on ${\mathbb{S}}$, and leads to a ‘canonical’ potential $Q(\Delta_m)$, originating in the string theory community [@Franco-Hanany-etal] and emphasised in this setting by Goncharov in [@Goncharov]. We will write $N$ for the total number of primitive cycles, so $N = [(4g-4+2d)m(m+1)/2] +[(6g-6+3d)(m-2) + (4g-4+2d)(m-2)(m-1)/2]+ [dm]$ for the numbers of $t_b, q_w, L_p^{(j)}$ respectively.
For a vector ${\bf c} \in (\bK^*)^N$ of pointwise non-zero coefficients, we will write $W_{\bf c}(\Delta_m) = \sum c_b \cdot t_b + \sum c_w \cdot q_w + \sum_{p,j} c_p^{(j)} L_p^{(j)}$.
In the $m=1$ case, and assuming $|\bP| > 1$, [@GLFS] show that every generic potential is right-equivalent to a generic primitive potential, i.e. one with zero non-primitive part. This fails when $m>1$: then the moduli space of $A_{\infty}$-structures on the cohomological category underlying ${\EuScript{C}}(Q(\Delta_m))$ has positive dimension.
\[rmk:abrikosov\] By diagonal automorphisms, any generic potential can be related to a ‘normalised’ one in which the coefficients of all primitive cycles other than the $L_p^{(j)}$ are equal to $1$. If $m=2$, a normalised generic potential $W$ is *strongly generic* if, for each of the points $p\in \bP$, the (necessarily non-zero) coefficients $c_p^{(1)}$ and $c_p^{(2)}$ of $L_p^{(1)}$ and $L_p^{(2)}$ in $W$ satisfy the non-degeneracy condition $$\label{eqn:strongly_generic}
c_p^{(1)} + (-1)^{\mathrm{valence}(p)} c_p^{(2)} \neq 0.$$ Right-equivalence preserves generic potentials, and the subset of those generic potentials which when normalised satisfy strong genericity. The space of strongly generic potentials with fixed primitive part, up to right equivalence, is isomorphic to $\bA^1_{\bK}$, cf. [@Abrikosov Theorem 2] and [@Abrikosov Proposition 5.14]. When $m>2$, there is no explicit description of the generic fibre of .
Any two ideal triangulations $\Delta$ and $\Delta'$ of $({\mathbb{S}},\bP)$ (without self-folded triangles) can be related by a sequence of flips. Goncharov [@Goncharov] showed that the effect of a flip on the pair $(Q(\Delta_m), W(\Delta_m))$ could itself be effected by a sequence of $m(m+1)(m+2)/6$ mutations, and that the mutation of the canonical potential is right-equivalent to the canonical potential on the mutated quiver. Mutations induce auto-equivalences of the associated $CY_3$-categories [@Keller-Yang]. It follows that there is a well-defined $CY_3$-category $\scrD({\mathbb{S}},\bP,m)$, quasi-isomorphic to the derived category of ${\EuScript{C}}(Q(\Delta_m), W(\Delta_m))$ for any choice of ideal triangulation $\Delta$ of $({\mathbb{S}},\bP)$. More generally, the family of $CY_3$-categories associated to all possible generic potentials can be realised by generic potentials on a fixed quiver $Q(\Delta_m)$.
Quiver 3-folds and Lagrangian sphere configurations\[Sec:quiver\_3folds\]
=========================================================================
$A_m$-fibred 3-folds
--------------------
In this section we discuss the symplectic topology of the threefolds $Y({\mathbb{S}},\bP,m)$. These threefolds were introduced in [@Abrikosov], and are associated to tuples of meromorphic differentials $(\phi_2,\ldots,\phi_{m+1})$ on a Riemann surface $S$ underlying ${\mathbb{S}}$ with poles at a subset $D\subset S$ of cardinality $|\bP|$; thus $\phi_j \in H^0(K_S(D)^{\otimes j})$. The threefolds associated to tuples of holomorphic differentials were previously introduced in [@DDP], and those associated to meromorphic quadratic differentials in [@Smith:quiver].
Fix a Riemann surface $S$ of genus $g$, and a section $\delta \in H^0(\mathcal{O}_S(D))$ which vanishes to order 1 at a divisor $D$ of degree $d$ (we think of $D$ as lying at the points of $\bP \subset {\mathbb{S}}$, where ${\mathbb{S}}$ is the topological surface underlying the Riemann surface $S$). Note that $\delta$ is unique up to scale. We also fix a decomposition of the log canonical bundle $$\label{eqn:log_canonical}
K_S(D) = \calL_1 \otimes \calL_2.$$ We consider the rank 3 vector bundle $$\label{eqn:ambient}
\mathcal{W} = \calL_1^{\otimes(m+1)}(-D) \oplus \calL_1 \calL_2 \oplus \calL_2^{\otimes(m+1)}$$ over $S$. Given a tuple $$\Phi = (\phi_2,\ldots,\phi_{m+1}) \quad \mathrm{with} \ \phi_j \in H^0(K_S(D)^{\otimes j})$$ we consider the hypersurface $$Y_{\Phi} = \left\{ (a,b,c) \in \mathcal{W} \, \big| \, (\delta\cdot a)\cdot c = b^{m+1} - \sum_j b^{m+1-j} \cdot \phi_j \right\}$$ Here $(a,b,c)$ are written with respect to the decomposition of the rank 3 bundle $\mathcal{W}$. The terms $(\delta\cdot a)c$ and $b^{m+1} - \sum_j b^{m+1-j} \cdot \phi_j $ both belong to $K_S(D)^{\otimes(m+1)}$, so the defining equation makes sense. (We ask that the sum of the roots of $\Phi(b) = 0$ vanishes for compatibility and comparison with [@GMN:snakes].)
$Y_{\Phi}$ has vanishing canonical class, so is a quasi-projective Calabi-Yau variety.
See [@Abrikosov Section 6].
The *spectral curve* $\Sigma \subset \mathrm{Tot}(K_S(D))$ is the vanishing locus $\{b \, | \, \Phi(b) = 0\}$. We say that $\Phi$ is generic when $\Sigma$ is smooth and projection $\Sigma \to S$ is a simple branched covering; it then has covering degree $m+1$ and $m(m+1)(2g-2+d)$ simple branch points, arising from the zeroes of $\det(\Phi)$. The threefold is almost a conic $\bC^*$-bundle over $\mathrm{Tot}(K_S(D))$ with singular fibres $\bC\vee\bC$ along the spectral curve $\Sigma$: this is precisely true after a finite number of affine modifications of the bundle $K_S(D)$ at the fibres over points of $D$, see [@Abrikosov Section 6.3].
We will work with Kähler forms on $Y_{\Phi}$ which are small perturbations of those induced from a choice of Kähler form on $S$ and on the total space of $\mathcal{W} \to S$; in particular our Kähler forms tame integrable complex structures for which the projection $Y_{\Phi} \to S$ is holomorphic. Because the defining equation for $Y_{\Phi}$ is weighted homogeneous, parallel transport vector fields have polynomial growth on the fibres with respect to a Kähler metric on $Y_{\Phi}$ induced from a metric on the vector bundle $\mathcal{W}$, and there are globally defined parallel transport maps of the fibres of $Y_{\Phi}$ over paths in $S$. Furthermore, there are parallel transport maps defined on compact subsets of $Y_{\Phi}$ over compact subsets in the universal family of threefolds obtained by varying $\Phi$. (Neither the monodromy of $Y_{\Phi}\to S$, nor of the universal family, is naturally compactly supported; in the former case this is because if one compactifies the fibration vertically, the divisor at infinity is not locally trivial but degenerates over $D$.)
Given an ideal triangulation of ${\mathbb{S}}$ with vertices at $\bP$, and its inscribed quiver, we pass to the dual graph of the quiver, as in Figures \[Fig:a3\_lag\_cellulation\] and \[Fig:dual\_lagrangians\].
(-3,0) – (3,0); (-3,0) – (0,5); (0,5) – (3,0);
(-1.5,0) – (-3 + 0.75, 1.25); (0,0) – (-3+1.5, 2.5); (1.5,0) – (-3+2.25, 3.75);
(-1.5,0) – (3-2.25,3.75); (0,0) – (3-1.5,2.5); (1.5,0) – (3-0.75,1.25);
(-2.25,1.25) – (2.25,1.25); (-1.5,2.5) – (1.5,2.5); (-0.75,3.75) – (0.75,3.75);
(-0.75,3.75) – (0.75,3.75) – (0, 2.5); (-1.5,2.5) – (0,2.5) – (-0.75, 1.25); (0,2.5)–(1.5,2.5) – (0.75,1.25); (-2.25,1.25) – (-0.75,1.25) – (-1.5,0); (-0.75,1.25) – (0.75,1.25) – (0,0); (0.75,1.25) – (2.25,1.25) – (1.5,0);
(0,3/4) circle (0.05); (0, 13/4) circle (0.05); (-1.5, 3/4) circle (0.05); (1.5,3/4) circle (0.05); (-.75, 2) circle (0.05); (.75,2) circle (0.05);
(0,3/4)–(-0.75,1.25); (0,3/4) – (0.75,1.25); (0,3/4) – (0, -1/4); (-1.5,3/4) – (-1.5,-1/4); (-1.5,3/4) – (-0.75,1.25); (-1.5,3/4) – (-2.25-1/5,1.25+1/10); (1.5,3/4) – (1.5,-1/4); (1.5,3/4) – (0.75,1.25); (1.5,3/4) – (2.25+1/5,1.25+1/10); (-0.75,2) – (0,2.5); (-0.75,2) – (-0.75,1.25); (-0.75,2) – (-1.5-1/5,2.5+1/10); (0.75,2) – (0,2.5); (0.75,2) – (0.75,1.25); (0.75,2) – (1.5+1/5,2.5+1/10); (0,13/4) – (0,2.5); (0,13/4) – (-0.75-1/5,3.75+1/10); (0,13/4) – (0.75+1/5,3.75+1/10);
(0,0) – (3,0); (0,0) – (-3,0); (0,0) – ([3\*cos(60)]{}, [3\*sin(60)]{}); (0,0) – ([-3\*cos 60]{}, [3\*sin 60]{}); (0,0) – ([3\*cos 60]{}, [-3\*sin 60]{}); (0,0) – ([-3\*cos 60]{}, [-3\*sin 60]{});
([-3\*cos 60]{}, [3\*sin 60]{}) – ([3\*cos 60]{}, [3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – ([3\*cos 60]{}, [-3\*sin 60]{}); ([-3\*cos 60]{}, [-3\*sin 60]{}) – (-3,0); ([3\*cos 60]{}, [-3\*sin 60]{}) – (3,0); ([3\*cos 60]{}, [3\*sin 60]{}) – (3,0); ([-3\*cos 60]{}, [3\*sin 60]{}) – (-3,0);
(1,0) – node ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node ([cos 60]{}, [sin 60]{}); ([cos 60]{}, [sin 60]{}) – node (1,0);
(2,0) – node ([2+cos 60]{}, [sin 60]{}); ([2+cos 60]{}, [sin 60]{}) – node ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node (2,0);
([1+cos 60]{}, [sin 60]{}) – node ([1+2\*cos 60]{}, [2\*sin 60]{}); ([1+2\*cos 60]{}, [2\*sin 60]{}) – node ([2\*cos 60]{}, [2\*sin 60]{}); ([2\*cos 60]{}, [2\*sin 60]{}) – node ([1+cos 60]{},[sin 60]{});
(-1,0) – node ([-1+cos 60]{}, [sin 60]{}); ([-1+cos 60]{}, [sin 60]{}) – node ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node (-1,0);
(-2,0) – node ([-2+cos 60]{}, [sin 60]{}); ([-2+cos 60]{}, [sin 60]{}) – node ([-3+cos 60]{}, [sin 60]{}); ([-3+cos 60]{}, [sin 60]{}) – node (-2,0);
([-2+cos 60]{}, [sin 60]{}) – node ([-2+2\*cos 60]{}, [2\*sin 60]{}); ([-2+2\*cos 60]{}, [2\*sin 60]{}) – node ([-3+2\*cos 60]{}, [2\*sin 60]{}); ([-3+2\*cos 60]{}, [2\*sin 60]{}) – node ([-2+cos 60]{},[sin 60]{});
([-cos 60, sin 60]{}) – node ([cos 60]{}, [sin 60]{}); ([cos 60, sin 60]{}) – node (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node ([-cos 60]{}, [sin 60]{});
(0,[2\*sin 60]{}) – node (1, [2\*sin 60]{}); (1, [2\*sin 60]{}) – node ([cos 60]{},[3\*sin 60]{}); ([cos 60, 3\*sin 60]{}) – node (0, [2\*sin 60]{});
(-1,[2\*sin 60]{}) – node (0, [2\*sin 60]{}); (0, [2\*sin 60]{}) – node ([-cos 60]{},[3\*sin 60]{}); ([-cos 60, 3\*sin 60]{}) – node (-1, [2\*sin 60]{});
([cos 60, -sin 60]{}) – node ([-cos 60]{}, [-sin 60]{}); ([-cos 60, -sin 60]{}) – node (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node ([cos 60]{}, [-sin 60]{});
(0,[-2\*sin 60]{}) – node (-1, [-2\*sin 60]{}); (-1, [-2\*sin 60]{}) – node ([-cos 60]{},[-3\*sin 60]{}); ([-cos 60, -3\*sin 60]{}) – node (0, [-2\*sin 60]{});
(1,[-2\*sin 60]{}) – node (0, [-2\*sin 60]{}); (0, [-2\*sin 60]{}) – node ([cos 60]{},[-3\*sin 60]{}); ([cos 60, -3\*sin 60]{}) – node (1, [-2\*sin 60]{});
(1,0) – node ([1-cos 60]{}, [-sin 60]{}); ([1-cos 60]{}, [-sin 60]{}) – node ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node (1,0);
(2,0) – node ([2-cos 60]{}, [-sin 60]{}); ([2-cos 60]{}, [-sin 60]{}) – node ([3-cos 60]{}, [-sin 60]{}); ([3-cos 60]{}, [-sin 60]{}) – node (2,0);
([2-cos 60]{}, [-sin 60]{}) – node ([2-2\*cos 60]{}, [-2\*sin 60]{}); ([2-2\*cos 60]{}, [-2\*sin 60]{}) – node ([3-2\*cos 60]{}, [-2\*sin 60]{}); ([3-2\*cos 60]{}, [-2\*sin 60]{}) – node ([2-cos 60]{},[-sin 60]{});
(-1,0) – node ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node ([-cos 60]{}, [-sin 60]{}); ([-cos 60]{}, [-sin 60]{}) – node (-1,0);
(-2,0) – node ([-2-cos 60]{}, [-sin 60]{}); ([-2-cos 60]{}, [-sin 60]{}) – node ([-1-cos 60]{}, [-sin 60]{}); ([-1-cos 60]{}, [-sin 60]{}) – node (-2,0);
([-1-cos 60]{}, [-sin 60]{}) – node ([-1-2\*cos 60]{}, [-2\*sin 60]{}); ([-1-2\*cos 60]{}, [-2\*sin 60]{}) – node ([-2\*cos 60]{}, [-2\*sin 60]{}); ([-2\*cos 60]{}, [-2\*sin 60]{}) – node ([-1-cos 60]{},[-sin 60]{});
(0,[1.5\*sin 60]{}) circle (0.05); ([cos 60, 2.5\*sin 60]{}) circle (0.05); ([-cos 60, 2.5\*sin 60]{}) circle (0.05); (0,[1.5\*sin 60]{}) – (0,[2\*sin 60]{}); ([cos 60,2.5\*sin 60]{}) – (0,[2\*sin 60]{}); ([-cos 60,2.5\*sin 60]{}) – (0,[2\*sin 60]{});
(1,[0.5\*sin 60]{}) circle (0.05); (2,[0.5\*sin 60]{}) circle (0.05); (1.5,[1.5\*sin 60]{}) circle (0.05);
(1,[0.5\*sin 60]{}) – (1.5,[sin 60]{}); ([2, 0.5\*sin 60]{}) – (1.5,[sin 60]{}); ([1.5,1.5\*sin 60]{}) – (1.5,[sin 60]{});
(-1,[0.5\*sin 60]{}) circle (0.05); (-2,[0.5\*sin 60]{}) circle (0.05); (-1.5,[1.5\*sin 60]{}) circle (0.05);
(-1,[0.5\*sin 60]{}) – (-1.5,[sin 60]{}); ([-2, 0.5\*sin 60]{}) – (-1.5,[sin 60]{}); ([-1.5,1.5\*sin 60]{}) – (-1.5,[sin 60]{});
(0,[-1.5\*sin 60]{}) circle (0.05); ([cos 60, -2.5\*sin 60]{}) circle (0.05); ([-cos 60, -2.5\*sin 60]{}) circle (0.05); (0,[-1.5\*sin 60]{}) – (0,[-2\*sin 60]{}); ([cos 60,-2.5\*sin 60]{}) – (0,[-2\*sin 60]{}); ([-cos 60,-2.5\*sin 60]{}) – (0,[-2\*sin 60]{});
(-1,[-0.5\*sin 60]{}) circle (0.05); (-2,[-0.5\*sin 60]{}) circle (0.05); (-1.5,[-1.5\*sin 60]{}) circle (0.05);
(-1,[-0.5\*sin 60]{}) – (-1.5,[-sin 60]{}); ([-2, -0.5\*sin 60]{}) – (-1.5,[-sin 60]{}); ([-1.5,-1.5\*sin 60]{}) – (-1.5,[-sin 60]{});
(1,[-0.5\*sin 60]{}) circle (0.05); (2,[-0.5\*sin 60]{}) circle (0.05); (1.5,[-1.5\*sin 60]{}) circle (0.05);
(1,[-0.5\*sin 60]{}) – (1.5,[-sin 60]{}); ([2, -0.5\*sin 60]{}) – (1.5,[-sin 60]{}); ([1.5,-1.5\*sin 60]{}) – (1.5,[-sin 60]{});
(0,[1.5\*sin 60]{}) – (1, [0.5\*sin 60]{}); (1,[0.5\*sin 60]{}) – (1, [-0.5\*sin 60]{}); (1,[-0.5\*sin 60]{}) – (0, [-1.5\*sin 60]{}); (0,[-1.5\*sin 60]{}) – (-1, [-.5\*sin 60]{}); (-1,[-0.5\*sin 60]{}) – (-1, [.5\*sin 60]{}); (-1,[0.5\*sin 60]{}) – (0, [1.5\*sin 60]{});
(0.5, [2.5\*sin 60]{}) – (1.5, [1.5\*sin 60]{}); (2, [.5\*sin 60]{}) – (2, [-.5\*sin 60]{}); (1.5, [-1.5\*sin 60]{}) – (0.5, [-2.5\*sin 60]{});
(-0.5, [2.5\*sin 60]{}) – (-1.5, [1.5\*sin 60]{}); (-2, [.5\*sin 60]{}) – (-2, [-.5\*sin 60]{}); (-1.5, [-1.5\*sin 60]{}) – (-0.5, [-2.5\*sin 60]{});
(0.5,[2.5\*sin 60]{}) – (0.5,[3.25\*sin 60]{}); (-0.5,[2.5\*sin 60]{}) – (-0.5,[3.25\*sin 60]{}); (0.5,[-2.5\*sin 60]{}) – (0.5,[-3.25\*sin 60]{}); (-0.5,[-2.5\*sin 60]{}) – (-0.5,[-3.25\*sin 60]{}); (1.5,[1.5\*sin 60]{}) – ([2.25, 2.25\*sin 60]{}); (-1.5,[1.5\*sin 60]{}) – ([-2.25, 2.25\*sin 60]{}); (2,[0.5\*sin 60]{}) – ([2.75, 1.25\*sin 60]{}); (-2,[0.5\*sin 60]{}) – ([-2.75, 1.25\*sin 60]{});
(1.5,[-1.5\*sin 60]{}) – ([2.25, -2.25\*sin 60]{}); (-1.5,[-1.5\*sin 60]{}) – ([-2.25, -2.25\*sin 60]{}); (2,[-0.5\*sin 60]{}) – ([2.75, -1.25\*sin 60]{}); (-2,[-0.5\*sin 60]{}) – ([-2.75, -1.25\*sin 60]{});
The vertices of this dual ‘Lagrangian’ cellulation $\Delta_m^{\vee}$ are at the centres of the inscribed black triangles of $\Delta_m$; there are $m(m+1)/2$ vertices of $\Delta_m^{\vee}$ in each ideal triangle of $\Delta$. Thus, the total number of vertices of the Lagrangian cellulation is $$m(m+1)/2 \cdot (4g-4+2d) = m(m+1)(2g-2+d)$$ which co-incides with the number of branch points of $\Sigma \to S$, cf. Remark \[rmk:numerics\].
\[lem:delta\_m\_controls\_singular\_fibres\] Given an ideal triangulation $\Delta$ of $({\mathbb{S}},\bP)$, there is a tuple $\Phi$ such that, up to isotopy, the associated fibration $Y_{\Phi} \to S$ has reducible fibres at the points of $\bP$, and has Lefschetz singular fibres over the vertices of $\Delta_m^{\vee}$.
Recall from [@GMN; @BridgelandSmith] that a generic choice of quadratic differential $\phi_2$ on $S$ with double poles at $D\subset S$ defines an ideal triangulation $\Delta$, with a (simple) zero of $\phi_2$ at the centre of each triangle. We consider a point of the higher rank Hitchin base $(\phi_2,\ldots, \phi_m)$, with $\phi_j \in H^0(K_S(D)^{\otimes j})$, which is a small perturbation of the degenerate tuple $(\phi_2,0,\beta_1\cdot \phi_2^2,0,\beta_2\cdot \phi_2^4,\ldots)$ for constants $\beta_j \in \bC^*$ chosen so that the associated spectral curve factorizes: $$\label{eqn:reducible_spectral_curve}
\begin{aligned}
\Sigma_0 & = & \{(b^2 - \phi_2)(b^2-\alpha_1\phi_2)\cdots(b^2-\alpha_k\phi_2) = 0\} \subset \mathrm{Tot}(K_S(D)) \qquad m=2k+1; \\
\Sigma_0 & = & \{b \,(b^2-\phi_2)(b^2-\alpha_1\phi_2)\cdots(b^2-\alpha_k\phi_2) = 0\} \subset \mathrm{Tot}(K_S(D)) \qquad m=2k+2;
\end{aligned}$$ where $b$ denotes a co-ordinate on $K_S(D)$ and the $\alpha_j$ are pairwise distinct and not equal to $0$ or $1$. These curves are reducible, and the 3-fold conic fibration over $K_S(D)$ with discriminant $\Sigma_0$ has (for sufficiently general $\alpha_j$) an isolated singularity at each zero of $\phi_2$. In local co-ordinates near a zero of $\phi_2 \approx z$ on $S$ (and recalling the zeroes of $\delta$ and of $\phi_2$ are different so locally $\delta \approx 1$), the local model for the 3-fold is $$\{ac = (b^2-z)\cdots(b^2-\alpha_k z)\} \subset \bC^4 \qquad \mathrm{or} \qquad \{ac = b(b^2-z)\cdots(b^2-\alpha_k z)\} \subset \bC^4,$$ which is the stabilisation of a weighted homogeneous plane curve singularity with Milnor number $m(m-1)/2$. A small perturbation of the degenerate tuple to a generic tuple $\Phi$ of differentials will both smooth the threefold, and yield a smooth spectral curve which is generically branched over the $z$-plane, the branch points encoding the positions of the Lefschetz singularities of $Y_\Phi \to S$. (The affine modifications which relate $Y_\Phi$ to a conic bundle singular along the spectral curve do not affect the current local discussion, since they take place in fibres over $D$, which lie far from the singularities of the total space for the degenerate tuple, and from the locations of the Lefschetz singularities of the fibration after small perturbation of that tuple.)
The topology of the threefold is thus encoded, up to birational modifications far from the Lefschetz singularities, by the braid monodromy of the smoothed spectral curve $\Sigma \subset K_S(D)$. Away from points of $D \subset S$, the initial 3-fold has fibre locally modelled on $\{ac = b^{m+1}\}$ near an isolated simple zero of the differential $\phi_2$; the singularity in the total space is an $A_m$-singularity in the fibre. A small generic perturbation of the tuple gives rise to a smooth 3-fold locally cut out by $\{ac=b^{m+1} + P(b,z)\}$ in which the corresponding map from the $z$-plane to configurations of roots is transverse to the discriminant locus of repeated roots; the Lefschetz singular fibres of the projection $Y\to S$ local to the given zero of $\phi_2$ then arise from values $z$ where $b^{m+1} + P(b,z)$ has a double root. The discriminant of has degree $(2k+2)(2k+1)/2$ respectively $(2k+3)(2k+2)/2$ (as a function of $z$) in the two cases, which in both cases yields the value $m(m+1)/2$. After a smooth area-preserving isotopy of the base, which can be lifted to a symplectic isotopy of the total space, one can arrange that these Lefschetz critical points lie at the vertices of $\Delta_m^{\vee}$. Compare to [@GMN:snakes Figures 1–3], and Proposition \[prop:sphere\_configuration\] below.
Over points of $\bP$, i.e. points of the divisor $D$ where $\delta$ vanishes (simply), a local model for the 3-fold is given by $\{ \delta \,ac = b^{m+1}-1\} \subset \bC^4$, and the $\delta=0$ fibre is given by $m+1$ pairwise disjoint copies of $\bC^2$ with co-ordinates $(u,v)$.
\[rmk:exceptional\] The case $g=1$ and $|\bP| = 1$ of an elliptic curve $S=E$ with one marked point $D=\{p\}$ is exceptional; in that case $\delta$ and $b$ both belong to the same one-dimensional space $H^0(\mathcal{O}_E(p))$, and the equation for the threefold associated to a reducible spectral curve becomes degenerate. However, after including the perturbation terms the corresponding threefold is still smooth.
We do not assert that there is a tuple $\Phi$ for which the holomorphic projection has the described structure, only that it is symplectically isotopic to such.
\[lem:H2Y\] The rational cohomology $H^2(Y_{\Phi};\bQ)$ has rank $dm+1$, and is spanned by the components of the reducible fibres over points of $\bP \subset {\mathbb{S}}$, modulo the relation that their sum is independent of $p\in \bP$.
The rank computation is given in [@Abrikosov Section 6]. The generators can be extracted from his argument. Note that the total fibre class, which co-incides with the sum of the classes of the components of a fixed reducible fibre, agrees with the pullback of an area form on ${\mathbb{S}}$ of total area $1$.
When $m=2$, the space of Kähler forms on $Y_{\Phi}$ is an open cone of dimension $2d+1$, whilst the space of right equivalence classes of potentials has dimension $d+2$ by Remark \[rmk:abrikosov\]. This underscores the fact that one cannot expect the ‘mirror’ map in to be a local isomorphism.
\[rmk:kahler\_monodromy\] There is a family of 3-folds $Y_{\Phi}$ over the space of generic tuples $\Phi$, and it is natural to consider Kähler forms which vary locally trivially over the family. The monodromy permutes components of the reducible fibres, and the monodromy-invariant subspace of $H^2(Y_{\Phi};\bR)$ has rank 2. Up to global rescaling, there is thus just a one-parameter family of invariant Kähler forms.
Tripod spheres\[Sec:Tripods\]
-----------------------------
We recall Donaldson’s ‘matching sphere’ construction. Consider a symplectic Lefschetz fibration $X^{2n} \to \bC$ with two singular fibres lying over $\pm 1$, and a path $\gamma: [-1,1] \to \bC$ with $\gamma(\pm 1) = \pm 1$ and $\gamma(t) \not \in \{-1,+1\}$ for $t\in (-1,1)$. Parallel transport along $\gamma$ gives rise to two Lagrangian $S^{n-1}$ vanishing cycles in the fibre $X_{\gamma(0)}$. If these are Hamiltonian isotopic, then after a deformation of the symplectic connexion on $X$ in a neighbourhood of the preimage of $\gamma$, one can arrange that the vanishing cycles agree exactly, and glue to form a Lagrangian $S^n \subset X$, which is well-defined up to Hamiltonian isotopy. In this case, $\gamma$ is called a ‘matching path’; see [@AMP Lemma 8.4] and [@Seidel:FCPLT Section 16g] for the details of the construction. Because of the deformation of symplectic connexions, the matching sphere will in general only approximately lie over $\gamma$ for the original symplectic form $\omega$.
If $X \to B$ is a symplectic Lefschetz fibration with fibre $T^*S^2 = A_1$, then since the fibre contains a unique Lagrangian sphere up to Hamiltonian isotopy [@Hind], any path between critical values in the base (disjoint from critical values in its interior) is a matching path. For $A_m$ fibred 3-folds with $m>1$, the fibre contains infinitely many Hamiltonian isotopy classes of Lagrangian sphere. It will be useful to have a minor generalisation of the matching path construction, namely a ‘matching tripod’ construction giving rise to ‘tripod’ Lagrangian spheres.
The $A_m$-surface $\{x^2+y^2+\prod_{j=1}^{m+1} (z-j) = 0\} \subset \bC^3$ inherits an exact Kähler structure from $(\bC^3, \omega_{{\mathrm{st}}})$. It deformation retracts to a compact core (or skeleton) comprising an $A_m$-chain of Lagrangian spheres, which arise as matching spheres for the paths $[j.j+1] \subset\bR\subset\bC$ for $1 \leq j \leq m$. Let $a$ and $b$ denote a pair of Lagrangian 2-spheres in $A_m$ which meet transversely at a single point, for instance (but not necessarily) a consecutive pair of spheres in the compact core, see Figure \[Fig:matching\_paths\_in\_A2\].
(-2,0) – (0,0) – (2,0); (-2,0) circle (0.1); (0,0) circle (0.1); (2,0) circle (0.1); (-2,0) arc (130:50:3.15); (-2,0) arc (230:310:3.15); (-1,-0.2) node [$a$]{}; (1,-0.2) node [$b$]{}; (0,1.2) node [$\tau_a(b)$]{}; (0,-1.2) node [$\tau_a^{-1}(b)$]{};
\[lem:tripod\_vs\_matching\_sphere\] Let $p: X \to \bC$ be a symplectic Lefschetz fibration with three singular fibres and fibre $A_m$ with $m>1$. Suppose that the vanishing cycles are as shown in the first image of Figure \[Fig:tripod\_paths\], i.e. $a, \tau_a^{-1}(b), b$ for paths as drawn. Then there is a Lagrangian sphere which maps under $p$ to a small neighbourhood of the tripod spanning the three critical points.
(0,-1.7) circle (0.1); (1.5,1.5) circle (0.1); (-1.5,1.5) circle (0.1);
(0,-1.7) – (0,-0.5) – (0.3,0.3) – (1.5,1.5); (0,-1.7) – (0,-0.5) – (-0.3,0.3) – (-1.5,1.5); (-1.5,1.5) – (-0.3,0.3) – (0.3,0.3) –(1.5,1.5); (0.8,0) node [$L$]{};
(-6+0,-1.7) circle (0.1); (-6+1.5,1.5) circle (0.1); (-6+-1.5,1.5) circle (0.1); (-6+1.5,-3) circle (0.1); (-4.5,-3) – (-6,-1.7); (-4.5,-3) arc (10:56:6.8); (-4.5,-3) arc (-22:22:6);
(-4.4,-3.5) node [$q$]{}; (-5.5,-2.6) node [$a$]{}; (-7,0) node [$\tau_a^{-1}(b)$]{}; (-3.6,0) node [$b$]{};
(+6+0,-1.7) circle (0.1); (+6+1.5,1.5) circle (0.1); (+6+-1.5,1.5) circle (0.1); (+6+1.5,-3) circle (0.1); (12-4.5,-3) – (12-6,-1.7); (12-4.5,-3) arc (10:56:6.8); (12-4.5,-3) arc (-22:22:6); (12-4.5,-3) arc (280:150:3);
(12-4.4,-3.5) node [$q$]{}; (12-5.5,-2.6) node [$a$]{}; (12-3.6,0) node [$b$]{}; (12-7.5, -2.6) node [$b$]{};
The existence of a Lagrangian 3-sphere $L$ mapping to the tripod neighbourhood follows from the construction of a Lagrangian cobordism from a Lagrange surgery. We apply this to the Polterovich surgery of the core spheres $a$ and $b$ in the $A_2$-Milnor fibre; this yields a cobordism in the total space of the product of an $A_2$-surface and a disc, whose three ends carry the three Lagrangians $a,b,\tau_a^{-1}(b)$. The cobordism completes to a smooth closed Lagrangian submanifold of a Lefschetz fibration with three critical points as in the left picture, compare to [@BiranCornea:lefschetz]. It is straightforward to check that the resulting closed Lagrangian is a smooth sphere.
On the right side of Figure \[Fig:tripod\_paths\], the two outer paths both have vanishing cycle $b$, so their concatenation defines a matching path $\gamma$ and Lagrangian 3-sphere $L_{\gamma}$ in the total space of the Lefschetz fibration. The fact that $L_{\gamma}$ is Hamiltonian isotopic to the tripod sphere $L$ constructed previously is proved by isotoping the matching path $\gamma$ through the ‘lowest’ critical value, see [@AbouzaidSmith19 Lemma A.25] for details. (Cancelling one critical point and one handle in the fibre by a Weinstein surgery as in [@Cieliebak-Eliashberg], the total space of the Lefschetz fibration $X$ is symplectomorphic to $T^*S^3$, with the matching sphere for $\gamma$ and hence the tripod sphere $L$ Hamiltonian isotopic to the zero-section.)
The tripod sphere does not map exactly to a tripod of arcs, but to a neighbourhood of that which is fattened near the vertex; we will call such spheres ‘essentially fibred’.
If we had taken the matching paths $\langle b,\tau_a(b),a \rangle$ as the input ordered triple (for the same vanishing paths), rather than $\langle b, \tau_a^{-1}(b), a\rangle$, then we would still obtain a Lagrangian sphere in the total space, but the corresponding description as a matching sphere would break, since the leftmost vanishing path on the right image of Figure \[Fig:tripod\_paths\] would have associated vanishing cycle $\tau_a^2(b) \not\simeq b$. (More precisely, the matching sphere would now lie over a path in a different homotopy class on the right hand picture.)
The right hand picture of Figure \[Fig:tripod\_paths\] is symmetric in a way which is not manifest. See Figure \[Fig:more\_tripod\_paths\]. There are matching spheres over both non-dashed paths (or the analogous 3rd path, not shown), using the fact that $\tau_a^{-1}(b) \simeq \tau_b(a)$ in the $A_2$-fibre, compare to Figure \[Fig:matching\_paths\_in\_A2\].
(+0,-1.7) circle (0.1); (+1.5,1.5) circle (0.1); (+-1.5,1.5) circle (0.1); (+1.5,-3) circle (0.1); (-1.5,-2.5) circle (0.1); (-1.5,-2.5) arc (270:330:1.7); (-1.5,-2.5) arc (240:45:2.55);
(-3,1) node [$ a$]{}; (-1,-2) node [$a$]{};
(6-4.5,-3) – (6-6,-1.7); (6-4.5,-3) arc (10:56:6.8); (6-5.5,-2.6) node [$a$]{}; (6-5.7,0.7) node [$\tau_b(a)$]{}; (6-4.5,-3) arc (-22:22:6); (6-3.6,0) node [$b$]{};
(+6+0,-1.7) circle (0.1); (+6+1.5,1.5) circle (0.1); (+6+-1.5,1.5) circle (0.1); (+6+1.5,-3) circle (0.1); (12-4.5,-3) – (12-6,-1.7); (12-4.5,-3) arc (10:56:6.8); (12-4.5,-3) arc (-22:22:6); (12-4.5,-3) arc (280:150:3);
(12-4.4,-3.5) node [$q$]{}; (12-5.5,-2.6) node [$a$]{}; (12-5.5,0.7) node [$\tau_a^{-1}(b)$]{}; (12-3.6,0) node [$b$]{}; (12-7.5, -2.6) node [$b$]{};
(6-4.4,-3.5) node [$q$]{}; (-1.5,-3) node [$q'$]{};
The essential point of the symmetry of the ‘good’ tripod on the right of Figure \[Fig:tripod\_paths\] is that one can arrange a collection of tripods around a polygon so that the corresponding vanishing cycles agree and the configuration ‘closes up’, cf. Figure \[Fig:tripod\_cycle\].
(-1.25,0) circle (0.05); (1.25,0) circle (0.05); (-2,-2) circle (0.05); (2,-2) circle (0.05); (0,-3.5) circle (0.05); (2,0.5) circle (0.05); (-2,0.5) circle (0.05); (2.75,0.5) circle (0.05); (-2.75,0.5) circle (0.05); (2,1.25) circle (0.05); (-2,1.25) circle (0.05);
(2.75,-2) circle (0.05); (3.25,-2.5) circle (0.05); (3.25,-1.5) circle (0.05); (-2.75,-2) circle (0.05); (-3.25,-2.5) circle (0.05); (-3.25,-1.5) circle (0.05);
(0,-4.25) circle (0.05); (-.5,-4.75) circle (0.05); (0.5,-4.75) circle (0.05);
(0,-1.75) circle (0.05);
(-1.25,0) – (1.25,0) – (2,-2) – (0,-3.5) – (-2,-2) – cycle; (1.25,0) – (2, 0.5); (2,0.5) – (2.75, 0.5); (2,0.5) – (2,1.25); (-1.25,0) – (-2, 0.5); (-2,0.5) – (-2.75, 0.5); (-2,0.5) – (-2,1.25);
(2,-2) – (2.75,-2); (2.75,-2) – (3.25,-1.5); (2.75,-2) – (3.25,-2.5); (-2,-2) – (-2.75,-2); (-2.75,-2) – (-3.25,-1.5); (-2.75,-2) – (-3.25,-2.5);
(0,-4.25) – (0,-3.5); (0,-4.25) – (-.5,-4.75); (0,-4.25) – (.5,-4.75);
(2.75,0.5) – (3.25,-1.5); (-2.75,0.5) – (-3.25,-1.5); (.5,-4.75) – (3.25,-2.5); (-.5,-4.75) – (-3.25,-2.5); (2,1.25) – (-2,1.25);
(0,-.25) node [$a$]{}; (1.25,-1) node [$a$]{}; (-1.25,-1) node [$a$]{}; (-1,-2.5) node [$a$]{}; (1,-2.5) node [$a$]{}; (-1.5,0.35) node [$a$]{}; (-1.85,0.9) node [$b$]{}; (0,1) node [$b$]{}; (1.75,-3.3) node [$b$]{}; (2.4,-1.8) node [$a$]{};
(0,-1.75) – (1,0.75) – (2,1.25); (0,-1.75) – (2,-1) – (2.75,0.5); (0,-1.75) – (2,-2.5) – (3.25,-2.5); (0,-1.75) – (1.2,0.6) – (2.2,0.8) – (2.75,0.5);
(1, 0.75) node [$b$]{}; (2.4,-0.9) node [$b$]{}; (2,-2.5) node [$b$]{}; (2.6,0.95) node [$\tau_a^{-1}(b)$]{};
The arrangement displayed in Figure \[Fig:tripod\_cycle\] is local and topological: given two Lagrangian spheres $L_a, L_b \subset A_m$ which meet transversely once, one can construct a Lefschetz fibration over a disc with the given critical fibres and vanishing cycles. We need a globalisation of this local picture.
Spectral networks and sphere configurations
-------------------------------------------
Recall that the threefold $Y_{\Phi}$ is defined by an equation $\delta a c = b^{m+1} - \sum_{j=2}^{m+1} \phi_j b^{m+1-j} = 0$ in the total space of a vector bundle $\mathcal{W}$ over $S$. Because we have well-defined parallel transport over paths in $S$, we can consider matching and tripod spheres in the total space.
\[prop:sphere\_configuration\] Fix an ideal triangulation $\Delta$ and the dual Lagrangian cellulation $\Delta_m^{\vee}$ of its subdivision $\Delta_m$. There is a configuration of Lagrangian spheres essentially fibred over $\Delta_m^{\vee}$, in which each ideal triangle in $\Delta$ contains $m(m-1)/2$ tripod Lagrangians, and these clusters are joined by $m$ matching spheres for each edge of $\Delta$.
This is an extension of Lemma \[lem:delta\_m\_controls\_singular\_fibres\], and is again implicit in the work of Gaiotto-Moore-Neitzke relating their spectral networks to ideal triangulations [@GMN:snakes]. Focus first on the geometry inside a single ideal triangle. In the $A_1$ situation, the spectral cover is a double cover and the monodromy at a simple zero of $\phi_2$ swaps the sheets. We are taking a perturbation of a degenerate case in which we replace $b^2-\phi$ by , for which the monodromy around a zero of $\phi_2$ completely reverses the order of the sheets, giving the longest element $(1, m) (2, m-1) (3, m-2) \cdots$ of the symmetric group. The braid monodromy for such a reducible curve was computed in [@Cohen-Suciu Section 5] [@VietDung Lemma 4.1], and yields the Garside element of the braid group. This is the lift of the longest element of the symmetric group, which admits a factorization as the canonical sequence of $m(m+1)/2$ half-twists lifting the permutations $$\begin{gathered}
[(m, m-1) (m-1, m-2)\ldots (4, 3) (3, 2) (2, 1)] \cdot [(m, m-1) (m-1, m-2) \ldots(4, 3) (3, 2)] \cdot \\ \cdot [(m, m-1)\ldots (4, 3)]\cdot \ldots \cdot [(m, m-1) (m-1, m-2)] \cdot [(m, m-1)],\end{gathered}$$ compare to [@Brieskorn-Saito], [@Looijenga Section 2] and the labellings of the pairs of sheets in Figure \[Fig:branching\_data\]. The above factorisation *defines* a local smooth symplectic surface in the four-ball simply branched over the $z$-plane, compare to [@Loi-Piergallini; @Orevkov], and gives a local model for the smoothing of the spectral curve in the proof of Lemma \[lem:delta\_m\_controls\_singular\_fibres\]. In any ideal triangle for $\Delta_2^{\vee}$, three sheets of the spectral cover interact. The vertices of $\Delta_m^{\vee}$ can be grouped into (overlapping) triples governed by the same local geometry, see Figure \[Fig:branching\_data\] (and compare to the corresponding discussion around [@GMN:snakes Figure 1]). Each such triple then bounds a Lagrangian tripod sphere, and the previous discussion implies these tripods fit together as in (the perhaps higher rank analogue of) Figure \[Fig:a3\_lag\_cellulation\].
Before perturbation, the only branching of the reducible spectral curve happens at the zeroes of $\phi_2$ or equivalently the centres of the ideal triangles. These have fibrewise $A_m$-singularities in which the whole compact core of the fibre degenerates into the critical point. For any given edge between two ideal triangles, one can place the branch cuts away from that edge, so the labelling of sheets is consistent along the paths of the cellulation which cross the edge of the given ideal triangle; indeed, for a given point $p \in D$ and associated vertex of the ideal triangulation, one can place all branch cuts across the edges of triangles which are not adjacent to $p$. (This labelling of sheets is incorporated into the data of the ‘eigen-ordering’ at $p$ introduced below in Definition \[defn:eigen\].) This yields a system of matching paths across all edges of triangles adjacent to $p$ after perturbation; compare to Figure \[Fig:tripod\_cycle\], cf. also the discussion in [@GMN:snakes Section 4] of the ‘asymptotic behaviour of the $\mathcal{S}$-walls for lifted theories’.
The local model near the reducible fibres from the end of Lemma \[lem:delta\_m\_controls\_singular\_fibres\] is $\bC^*$-equivariant for a $\bC^*$-action of weight one in $\delta$. The local Kähler form can be deformed to be $S^1$-invariant, and the monodromy around the reducible fibre over a point $p \in D$ is then symplectically trivial on the $A_m$-surface. The monodromy around the outer boundary of the configuration of ideal triangles adjacent to $p$ is a power of the Garside element, and acts either trivially on the compact core of the $A_m$-surface or preserving the core but reversing the chosen order of its components, depending on the parity of the valence of $p$ in the ideal triangulation. In either case, the configuration of matching spheres constructed from the viewpoint of $p$ is compatible with that one would construct around another point $q\in D$.
Consider the case $m=2$, $g=1$ and $|\bP|=1$, see Figure \[Fig:globalising\_configurations\]. As indicated by the labelled vanishing cycles, the monodromy around the boundary of a fundamental domain of the torus is the square $\Delta^2 = (\tau_b\tau_a\tau_b)^2$ of the Garside element, which is central in the braid group. There is non-trivial monodromy around both generating loops[^2] for $\pi_1(T^2)$, i.e. the meridian and longitude depicted as the black boundaries of the fundamental domain; indeed, the underlying $m=1$ theory in this case has a single branch cut along each side, and the monodromy on both edges of the fundamental domain induces the permutation $(1,3)(2)$ of the three sheets of the spectral curve, compare to [@Hollands-Neitzke Figure 10]. For a global Kähler form, the monodromy around the reducible fibre is non-trivial but centralises the braid group. The $\bC^*$-invariant model in Proposition \[prop:sphere\_configuration\] trivialises this monodromy by an isotopy which is not compactly supported at infinity.
(0,0) – (3,0); (0,0) – (0,3); (0,3) – (3,3); (3,0) – (3,3);
(0,0) – (-3,0); (-3,0) – (-3,3); (0,3) – (-3,3);
(0,0) – (0,-3); (0,-3) – (3,-3); (3,0) – (3,-3);
(0,-3) – (-3,-3); (-3,0) – (-3,-3);
(-3,-3) – (3,3); (-3,-2) – (2,3); (-3,-1) – (1,3); (-3,0) – (3,0); (-3,1) – (-1,3); (-3,2) – (-2,3); (-1,-3) – (3,1); (1,-3) – (3,-1); (0,-3) – (3,0); (2,-3) – (3,-2); (-3,0) – (0,3); (-2,-3) – (3,2);
(-3,1) – (3,1); (-3,2) – (3,2); (-3,-1) – (3,-1); (-3,-2) – (3,-2); (-2,3) – (-2,-3); (-1,3) – (-1,-3); (1,3) – (1,-3); (2,3) – (2,-3);
(3,3) circle (0.07); (-3,3) circle (0.07); (3,-3) circle (0.07); (-3,-3) circle (0.07); (0,0) circle (0.07); (3,0) circle (0.07); (-3,0) circle (0.07); (0,3) circle (0.07); (0,-3) circle (0.07);
(0.75,1.25) circle (0.05); (0.75,2.25) circle (0.05); (1.75,2.25) circle (0.05); (0.75,1.25) – (1,2); (0.75,2.25) – (1,2); (1.75,2.25) – (1,2);
(1.25,0.75) circle (0.05); (2.25,0.75) circle (0.05); (2.25,1.75) circle (0.05); (1.25,0.75) – (2,1); (2.25,0.75) – (2,1); (2.25,1.75) – (2,1);
(-3+0.75,1.25) circle (0.05); (-3+0.75,2.25) circle (0.05); (-3+1.75,2.25) circle (0.05); (-3+0.75,1.25) – (-3+1,2); (-3+0.75,2.25) – (-3+1,2); (-3+1.75,2.25) – (-3+1,2);
(-3+1.25,0.75) circle (0.05); (-3+2.25,0.75) circle (0.05); (-3+2.25,1.75) circle (0.05); (-3+1.25,0.75) – (-3+2,1); (-3+2.25,0.75) – (-3+2,1); (-3+2.25,1.75) – (-3+2,1);
(0.75,-3+1.25) circle (0.05); (0.75,-3+2.25) circle (0.05); (1.75,-3+2.25) circle (0.05); (0.75,-3+1.25) – (1,-3+2); (0.75,-3+2.25) – (1,-3+2); (1.75,-3+2.25) – (1,-3+2);
(1.25,-3+0.75) circle (0.05); (2.25,-3+0.75) circle (0.05); (2.25,-3+1.75) circle (0.05); (1.25,-3+0.75) – (2,1-3); (2.25,-3+0.75) – (2,1-3); (2.25,-3+1.75) – (2,1-3);
(0.75-3,-3+1.25) circle (0.05); (0.75-3,-3+2.25) circle (0.05); (1.75-3,-3+2.25) circle (0.05); (0.75-3,-3+1.25) – (1-3,-3+2); (0.75-3,-3+2.25) – (1-3,-3+2); (1.75-3,-3+2.25) – (1-3,-3+2);
(1.25-3,-3+0.75) circle (0.05); (2.25-3,-3+0.75) circle (0.05); (2.25-3,-3+1.75) circle (0.05); (1.25-3,-3+0.75) – (2-3,1-3); (2.25-3,-3+0.75) – (2-3,1-3); (2.25-3,-3+1.75) – (2-3,1-3);
(1.25,0.75) – (0.75,1.25); (0.75,1.25) – (-0.75,.75); (-0.75,0.75) – (-1.25,-0.75); (-1.25,-0.75) – (-0.75,-1.25); (-.75,-1.25) – (0.75,-.75); (0.75,-0.75) – (1.25,0.75);
(1+1.25,1.75) – (1+0.75,2.25); (0.75,2.25) – (-0.75,1.75); (-1-0.75,0.75) – (-2.25,-0.75); (-2.25,-1.75) – (-1.75,-2.25); (-.75,-2.25) – (0.75,-1.75); (1.75,-0.75) – (2.25,0.75);
(-2+1.25,1.75) – (-2+0.75,2.25); (-3+1.25,0.75) – (-3+0.75,1.25); (1.75,-0.75) – (2.25,-1.25); (0.75,-1.75) – (1.25,-2.25);
(0.75,2.25) – (1,3); (1.75,2.25) – (2,3); (-1.25,2.25) – (-1,3); (-2.25,2.25) – (-2,3); (-2.25,2.25) – (-3,2); (-2.25,1.25) – (-3,1); (-2.25,-0.75) – (-3,-1); (-2.25,-1.75) – (-3,-2); (-1.75,-2.25) – (-2,-3); (-.75,-2.25) – (-1,-3); (1.25,-2.25) – (1,-3); (2.25,-2.25) – (2,-3); (2.25,-2.25) – (3,-2); (2.25,-1.25) – (3,-1); (2.25,0.75) – (3,1); (2.25,1.75) – (3,2);
(0.75,1.25) – (0,0); (1.25,0.75) – (0,0); (-0.75,0.75) – (0,0); (-1.25,-0.75) – (0,0); (-0.75,-1.25) – (0,0); (0.75,-0.75) – (0,0); (2.25,0.75) – (0,0); (0.75,2.25) – (0,0); (1.75,2.25) – (0,0); (2.25,1.75) – (0,0);
(3.5,0.5) .. controls (2.5,0.2) and (1.75,0.2) .. (1.5,0.45); (3.7,0.5) node [$b$]{}; (3.5,-0.5) .. controls (2.5,-0.75) and (1,-0.25) .. (0.75,0.4); (3.7,-0.5) node [$a$]{};
Recall from Lemma \[lem:delta\_m\_controls\_singular\_fibres\] that the 3-fold associated to the reducible spectral curve has isolated singularities of Milnor number $m(m-1)/2$ at the zeroes of $\phi_2$; the set of $m(m-1)/2$ tripod spheres in the corresponding ideal triangle presumably gives a distinguished basis of vanishing cycles of the singularity (this should follow from [@ACampo], but we will not need it).
The total number of Lagrangian spheres in the configuration of tripods and matching spheres is then $$(6g-6+3d)\cdot m + (2g-2+d)\cdot m(m-1).$$ Let $\Gamma(\Delta_m^{\vee})$ denote this set of Lagrangian spheres in $Y_{\Phi}$.
(0,0) node\[cross,red\] ; (-1,0) node\[cross,red\] ; (1,0) node\[cross,red\] ; (.5,.5) node\[cross,red\] ; (-.5,.5) node\[cross,red\] ; (0,1) node\[cross,red\] ;
(0,0) – ([0+5\*cos 100]{},[0+5\*sin 100]{}); (1,0) – ([1+5\*cos 100]{},[0+5\*sin 100]{}); (-1,0) – ([-1+5\*cos 100]{},[0+5\*sin 100]{}); (0.5,0.5) – ([0.5+5\*cos 100]{},[0.5+5\*sin 100]{}); (-0.5,0.5) – ([-0.5+5\*cos 100]{},[0.5+5\*sin 100]{}); (0,1) – ([0+5\*cos 100]{},[1+5\*sin 100]{});
(0,0) – ([0+5\*cos 190]{},[0+5\*sin 190]{}); (1,0) – ([1+5\*cos 190]{},[0+5\*sin 190]{}); (-1,0) – ([-1+5\*cos 190]{},[0+5\*sin 190]{}); (0.5,0.5) – ([0.5+5\*cos 190]{},[0.5+5\*sin 190]{}); (-0.5,0.5) – ([-0.5+5\*cos 190]{},[0.5+5\*sin 190]{}); (0,1) – ([0+5\*cos 190]{},[1+5\*sin 190]{});
(0,0) – ([0+5\*cos 350]{},[0+5\*sin 350]{}); (1,0) – ([1+5\*cos 350]{},[0+5\*sin 350]{}); (-1,0) – ([-1+5\*cos 350]{},[0+5\*sin 350]{}); (0.5,0.5) – ([0.5+5\*cos 350]{},[0.5+5\*sin 350]{}); (-0.5,0.5) – ([-0.5+5\*cos 350]{},[0.5+5\*sin 350]{}); (0,1) – ([0+5\*cos 350]{},[1+5\*sin 350]{});
([0+3\*cos 100]{},[0+3\*sin 100]{}) node [${12}$]{}; ([-1+3\*cos 100]{},[0+3\*sin 100]{}) node [${12}$]{}; ([1+3\*cos 100]{},[0+3\*sin 100]{}) node [${12}$]{}; ([0.5+4\*cos 100]{},[0.5+4\*sin 100]{}) node [${23}$]{}; ([-0.5+4\*cos 100]{},[0.5+4\*sin 100]{}) node [${23}$]{}; ([0+5.25\*cos 100]{},[1+5.25\*sin 100]{}) node [${34}$]{};
([0+3.5\*cos 190]{},[0+3.5\*sin 190]{}) node [${32}$]{}; ([-1+5.25\*cos 190]{},[0+5.25\*sin 190]{}) node [${21}$]{}; ([1+3\*cos 190]{},[0+3\*sin 190]{}) node [${43}$]{}; ([0.5+3\*cos 190]{},[0.5+3\*sin 190]{}) node [${43}$]{}; ([-0.5+3.5\*cos 190]{},[0.5+3.5\*sin 190]{}) node [${32}$]{}; ([0+3\*cos 190]{},[1+3\*sin 190]{}) node [${43}$]{};
([0+3.5\*cos 350]{},[0+3.5\*sin 350]{}) node [${32}$]{}; ([-1+3\*cos 350]{},[0+3\*sin 350]{}) node [${43}$]{}; ([1+5.25\*cos 350]{},[0+5.25\*sin 350]{}) node [${21}$]{}; ([0.5+3.5\*cos 350]{},[0.5+3.5\*sin 350]{}) node [${32}$]{}; ([-0.5+3\*cos 350]{},[0.5+3\*sin 350]{}) node [${43}$]{}; ([0+3\*cos 350]{},[1+3\*sin 350]{}) node [${43}$]{};
(0,0) –(0.2,-3); (1,0) –(1.2,-3); (-1,0) –(-0.8,-3); (0.5,0.5) –(0.7,-2.5); (-0.5,0.5) –(-0.3,-2.5); (0,1) –(0.2,-2);
The cyclic potential from holomorphic polygons
==============================================
Floer theory background
-----------------------
By construction, $Y_{\Phi}$ is equipped with an integrable complex structure $I$, arising as an algebraic subvariety of an algebraic $\bC^3$-bundle over the Riemann surface $S$. We can take its closure in the fibrewise completion, a $\bC\bP^3$-bundle over $S$, to obtain a projective compactification. This is in general singular, but resolving singularities yields a projective compactification $\bar{Y}$ which comes with a map $p: \bar{Y} \to S$ to $S$ and has normal crossing boundary.
We are assuming $g({\mathbb{S}})>0$, so any rational curve in $\bar{Y}$ maps by a constant map to $S$ so lies in a fibre of $p$, hence meets the boundary. Since the fibre $p^{-1}(x) \backslash \{p^{-1}(x) \cap Bd(Y)\}$ is affine, the boundary is relatively ample on the singular compactification, hence relatively nef on the resolution.
For Floer theory, it will be useful to perturb the complex structure $I$. We work with the class $\scrJ_{\pi}$ of almost complex structures on $Y$ which tame an $I$-Kähler form on $Y$ and which make projection $\pi: Y_{\Phi} \to S$ holomorphic, and which agree with $I$ outside a compact set. In this case, polygons with boundary conditions on the Lagrangians $\Gamma(\Delta_m^{\vee})$ map to holomorphic discs with boundary on the edges of the cellulation $\Delta_m^{\vee}$, to which one can apply the open mapping theorem. Since the fibres of $\pi$ are exact, and contain no rational curves, it is standard that one can achieve transversality in the class $\scrJ_{\pi}$. Although $Y_{\Phi}$ is non-compact and not manifestly of contact type at infinity, we have:
\[lem:compact\] The moduli space of holomorphic polygons in $Y_{\Phi}$ with Lagrangian boundary conditions belonging to a compact subset (e.g. to a given finite set of closed Lagrangian submanifolds) is compact.
This follows by considering intersections with the singular divisor $Bd(Y) \subset \bar{Y}_{\Phi}$ at infinity. More precisely, given a sequence $u_j$ of holomorphic discs with boundary conditions in a compact subspace and converging to a stable map $u_{\infty}$, if $u_{\infty}$ does not have image contained in $Y_\Phi$ it must have either a disc component or a rational curve component which meets $Bd(Y)$ but is not wholly contained in the boundary. Such components meet $Bd(Y)$ strictly positively by positivity of intersection, and relative nefness of $Bd(Y)$ on rational curves (in particular on components contained in the boundary) shows that $u_{\infty} \cdot Bd(Y) > 0$. This contradicts $u_j \cdot Bd(Y) = 0$ for finite $j$.
The Lagrangians we consider are tautologically unobstructed in $Y_{\Phi}$ for almost complex structures making projection $Y_{\Phi} \to S$ holomorphic. Given this, and with compactness from Lemma \[lem:compact\], a version of the Fukaya category ${\EuScript{F}}(Y)$ containing Lagrangian matching and tripod spheres can be constructed following the methods of [@Seidel:FCPLT], but working over a Novikov field to take account of convergence issues for holomorphic polygons. Since all the Lagrangians we consider are spin, and indeed relatively spin for any background class $b \in H^2(Y;\bZ/2)$ supported on the reducible fibres and hence disjoint from $\Gamma(\Delta_m^{\vee})$, we may define ${\EuScript{F}}(Y;b)$ over $\Lambda_{\bC}$.
Holomorphic triangles
---------------------
Once $m>2$ there are pairs of Lagrangian spheres in the $A_m$-Milnor fibre which are disjoint. Nonetheless:
\[lem:constant\_triangle\] At any vertex $b$ of $\Delta_m^{\vee}$, the three adjacent Lagrangian spheres $L_u, L_v, L_w$ meet pairwise transversely at a single point $\hat{b}$ of the corresponding fibre of $Y_{\Phi}$. The constant holomorphic triangle to $\hat{b}$ is regular and contributes to the product $HF(L_v, L_w) \otimes HF(L_u, L_v) \to HF(L_u, L_w)$ (where $L_u, L_v, L_w$ project to arcs ordered clockwise locally at $b$; all three Floer groups are $\bK$).
There is a unique Lefschetz singularity in the $A_m$-fibre lying over a vertex of $\Delta_m^{\vee}$, and locally the three Lagrangians are given by different Lefschetz thimbles near that point. (The isotopy in the construction of the matching spheres which makes them only essentially fibred can be taken to be supported away from the common critical end-point.) It follows that they meet pairwise transversely, and indeed can be locally described by three linear Lagrangian subspaces in $\bC^n$. Such a triple of Lagrangian subspaces meeting at a point can be modelled on a product of copies of three real lines in $\bC$ meeting at the origin. Regularity of the constant map for the correct cyclic order is standard, see [@Smith:quiver Lemma 4.9].
For $m>2$ there are internal ‘white’ triangles in the quiver $Q(\Delta_m)$, i.e. primitive 3-cycles of the form $q_w$, which contribute regions to the Lagrangian cellulation $\Delta_m^{\vee}$ which have three tripod Lagrangian boundary components (cf. the internal ‘hexagons’ in Figure \[Fig:a3\_lag\_cellulation\], note these have only three geometrically distinct Lagrangian boundaries).
\[lem:more\] Suppose $m>2$. Consider a primitive 3-cycle $q_w$ with tripod Lagrangian boundaries $L_u, L_v, L_w$ in cyclic order. The Floer product $HF^1(L_v, L_w) \otimes HF^1(L_u,L_v) \to HF^2(L_u,L_w)$ is non-zero.
Since only four sheets of the spectral cover are involved in the geometry of Figure \[Fig:a3\_lag\_cellulation\], it suffices to consider the case $m=3$. By Lemma \[lem:tripod\_vs\_matching\_sphere\], the three tripod spheres can be replaced by matching spheres for the local structure of $Y_{\Phi}$ as an $A_3$-fibred Lefschetz fibration. The matching paths can moreover be taken to meet at a unique point, the centre of the white triangle, see Figure \[Fig:force\_constant\_triangle\]. Then the corresponding 3-spheres meet only in the fibre over that point (more precisely, this is true after a symplectomorphism of an exact subdomain containing the given triple of spheres, after which they can be taken to be exactly fibred over arcs in the base). The Hamiltonian isotopies from tripods to matching spheres induce quasi-isomorphisms of the corresponding objects in the Fukaya category, and one can arrange that there is no wall-crossing since the isotopies are through weakly exact Lagrangians. We need to determine the vanishing 2-spheres in the smooth $A_3$-fibre $F$ over the central point in Figure \[Fig:force\_constant\_triangle\].
From the original configuration of the tripods, these 2-spheres meet pairwise with rank one Floer cohomology. By [@Khovanov-Seidel] this is only possible for matching spheres in $A_k$ if the spheres are fibred over paths which pairwise share a single end-point (so meet geometrically once). This reduces us to the geometry which entered into the discussion of the constant triangle in Lemma \[lem:constant\_triangle\].
(0,0) – (4,0); (0,0) – ([4\*cos(60)]{}, [4\*sin(60)]{}); ([4\*cos 60]{}, [4\*sin 60]{}) – (4,0);
(1,0) – node ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node ([cos 60]{}, [sin 60]{}); ([cos 60]{}, [sin 60]{}) – node (1,0);
(2,0) – node ([2+cos 60]{}, [sin 60]{}); ([2+cos 60]{}, [sin 60]{}) – node ([1+cos 60]{}, [sin 60]{}); ([1+cos 60]{}, [sin 60]{}) – node (2,0);
(3,0) – node ([3+cos 60]{}, [sin 60]{}); ([3+cos 60]{}, [sin 60]{}) – node ([2+cos 60]{}, [sin 60]{}); ([2+cos 60]{}, [sin 60]{}) – node (3,0);
([1+cos 60]{}, [sin 60]{}) – node ([1+2\*cos 60]{}, [2\*sin 60]{}); ([1+2\*cos 60]{}, [2\*sin 60]{}) – node ([2\*cos 60]{}, [2\*sin 60]{}); ([2\*cos 60]{}, [2\*sin 60]{}) – node ([1+cos 60]{},[sin 60]{});
([2+cos 60]{}, [sin 60]{}) – node ([2+2\*cos 60]{}, [2\*sin 60]{}); ([2+2\*cos 60]{}, [2\*sin 60]{}) – node ([1+2\*cos 60]{}, [2\*sin 60]{}); ([1+2\*cos 60]{}, [2\*sin 60]{}) – node ([2+cos 60]{},[sin 60]{});
(2, [2\*sin 60]{}) – node ([2+cos 60]{}, [3\*sin 60]{}); ([2+cos 60]{}, [3\*sin 60]{}) – node ([1+cos 60]{}, [3\*sin 60]{}); ([1+cos 60]{}, [3\*sin 60]{}) – node ([2]{},[2\*sin 60]{});
(1,[0.5\*sin 60]{}) circle (0.05); (2,[0.5\*sin 60]{}) circle (0.05); (3,[0.5\*sin 60]{}) circle (0.05); (1.5,[1.5\*sin 60]{}) circle (0.05); (2.5,[1.5\*sin 60]{}) circle (0.05); (2,[2.5\*sin 60]{}) circle (0.05);
(2,[1.25\*sin 60]{}) circle (0.03);
(1,[0.5\*sin 60]{}) – (1.5,[sin 60]{}); ([2, 0.5\*sin 60]{}) – (1.5,[sin 60]{}); ([1.5,1.5\*sin 60]{}) – (1.5,[sin 60]{});
(2,[0.5\*sin 60]{}) – (2.5,[sin 60]{}); ([3, 0.5\*sin 60]{}) – (2.5,[sin 60]{}); ([2.5,1.5\*sin 60]{}) – (2.5,[sin 60]{});
(1.5,[1.5\*sin 60]{}) – (2,[2\*sin 60]{}); ([2.5, 1.5\*sin 60]{}) – (2,[2\*sin 60]{}); ([2,2.5\*sin 60]{}) – (2,[2\*sin 60]{});
(1, [0.5\*sin 60]{}) – (1,1.25) – (1.5,1.65) – (2,1.25) – (2,[0.5\*sin 60]{});
(1.5,[1.5\*sin 60]{}) – (2, [1.25\*sin 60]{}) – (3,[1.25\*sin 60]{}) – (2.5,[2.25\*sin 60]{}) – (2,[2.5\*sin 60]{});
(2.5, [1.5\*sin 60]{}) – (2,[1.25\*sin 60]{}) – (1.5,[0.25\*sin 60]{}) – (2.5, [0.25\*sin 60]{}) – (3,[0.5\*sin 60]{});
Geometry near the reducible fibre
---------------------------------
We briefly recall the geometry near the reducible fibre. After deformation, there is a local model for $Y$ $$\label{eqn:reducible_local_model}
\{\delta(a^2 + c^2) + \prod_{j=1}^m (b-j) = 0\} \subset \bC^3\times \bC_{\delta}$$ in which the projection $Y\to S$ is modelled on projection to the $\delta$-plane. The general fibres are type $A_m$-surfaces $b^m + O(b^{m-1}) + (\mathrm{const.})(a^2+c^2) = 0$, whilst the fibre over $\delta =0$ is given by the $m$-tuple of planes $\{b=j\} \times \bC^2_{a,c}$, for $j \in \{1,\ldots, m\}$. There is a Lagrangian boundary condition $$\delta = e^{i\theta}, \ a,c \in e^{-i\theta/2}\cdot (-1)^m i \cdot \bR, \ b \in [j,j+1]$$ for each $1\leq j\leq m-1$, defining a totally real $S^1\times S^2$ lying over the unit circle in the $\delta$-plane. One can deform the standard symplectic structure in a neighbourhood of this totally real submanifold to make it Lagrangian, and fixed by an antiholomorphic involution, cf. [@Smith:quiver Section 4.6]. The only holomorphic discs with boundary on this Lagrangian cylinder are given by (multiple covers of) the constant sections over the unit disc in the $\delta$-plane: $$\label{eqn:two_discs}
u: (D, \partial D) \to (Y, (S^1\times S^2)_j), \quad u(z) = (0,\ast,0, z), \ \ast \in \{j,j+1\}.$$
There is another viewpoint which can be helpful. There is a unitary local change of co-ordinates which transforms the local model to the form $$\{\delta\cdot uv + \prod_j (b-j) = 0\} \subset \bC^4$$ and one can consider projection to the $b$-plane. The generic fibre is now $\{\delta u v = \mathrm{const}\} \subset \bC^3$, which is a copy of $(\bC^*)^2$; the fibres over $b=j$ are isomorphic to the union of the three co-ordinate planes $\delta u v = 0 \subset \bC^3$. The local model of the map $xyz: \bC^3\to \bC$ has been studied extensively in [@AAK]. Again consider the matching path $[j,j+1] \subset \bR$ between two critical values of the projection. One can parallel transport the Lagrangian $T^2 \subset T^*T^2 = (\bC^*)^2$ along this path, to obtain a Lagrangian $S^1\times S^2$ which is another model for that considered above. The two holomorphic discs with boundary on the Lagrangian now lie entirely over the end-points: the Lagrangian meets the fibre over $j$ in the unit circle in the $\delta$-plane, and bounds the obvious disc lying entirely in the singular locus of the $j$-fibre. Note that from the second viewpoint, there are three Lagrangian $(S^1\times S^2)$’s associated to $[j,j+1]$, given the symmetry in the co-ordinates $\delta,u,v$; only one of these is fibred with respect to the $\delta$-plane projection.
The second viewpoint makes it especially clear that the holomorphic discs with boundary on $S^1\times S^2$ have vanishing Maslov class, by comparing to the toric model $xyz: \bC^3 \to \bC$.
Given a choice of spin structure on $L_j \cong S^1\times S^2$ and hence orientation of the moduli space of rigid discs with boundary on $L_j$, the two holomorphic discs from have opposite sign.
The geometry is local near the given $S^1\times S^2$, and the argument from [@Smith:quiver Section 4.6] applies.
\[defn:eigen\] An *eigen-ordering* of a generic tuple $\Phi$ is a choice of ordering of the roots of $\Phi(b) = 0$ near each point $p\in D$. One can equivalently think of an eigen-ordering as giving an ordering of the irreducible components of the fibre $(Y_{\Phi})_p$ over each point $p\in D \subset S$.
The space of eigen-ordered generic tuples is an unramified ${\mathrm{Sym}}_m^{\times d}$ cover of an open subset of the Hitchin base. In analogy with [@BridgelandSmith], one expects eigen-ordered generic tuples to define stability conditions on the category ${\EuScript{C}}$.
Note that each one of the discs meets exactly one component of the reducible fibre of $Y$ over $0 \in \bC_{\delta}$. Moreover, consideration of the branching behaviour in Figure \[Fig:branching\_data\] shows that as one varies the value $j$ when considering sections over $L_p^{(j)}$, different pairs of irreducible components of the fibre over $p \equiv 0 \in \bC_{\delta}$ meet the corresponding holomorphic discs; compare to the final co-ordinate in . Given an ordering of the components of the fibres over $\bP$, there is an associated choice of cycle $Z_b$ comprising $\lceil (m+1)/2 \rceil$ of the irreducible components for which the total signed intersection number of the discs of with $Z_b$ is necessarily non-zero (because only one of each pair of discs hits one of the components included in $Z_b$). Up to monodromy in the space of eigen-ordered tuples, which induces a permutation representation of the components of the reducible fibres, we can assume that this choice of cycle is just the even-indexed components as specified in the Introduction (which is a suitable cycle choice if the local geometry agrees with the labelling of sheets from Figure \[Fig:branching\_data\]).
Holomorphic discs for other primitive cycles
--------------------------------------------
Fix an ideal triangulation $\Delta$ and the collection $\{L_v \, | \, v \in \Gamma(\Delta_m^{\vee})\}$ of Lagrangian spheres associated to the dual Lagrangian cellulation. We wish to understand the holomorphic discs which contribute to the $A_{\infty}$-structure on $\scrA_{\Gamma}$. There are constant holomorphic triangles indexed by the primitive cycles $c_b$ of the quiver, which we have already encountered. The other two classes of primitive cycle also give rise to holomorphic discs.
\[prop:rigid\_discs\] Fix a vertex $p \in \bP$ of $\Delta$. For each $1 \leq j \leq m$, the moduli space of rigid holomorphic discs with boundary $L_p^{(j)}$ is non-empty. Moreover, there is a choice of cycle representative for the background class $b \in H^2(Y_{\Phi};\bZ/2)$ of for which the algebraic count of such discs is non-zero.
(-1.25,0) circle (0.05); (1.25,0) circle (0.05); (-2,-2) circle (0.05); (2,-2) circle (0.05); (0,-3.5) circle (0.05); (2,0.5) circle (0.05); (-2,0.5) circle (0.05); (2.75,0.5) circle (0.05); (-2.75,0.5) circle (0.05); (2,1.25) circle (0.05); (-2,1.25) circle (0.05);
(2.75,-2) circle (0.05); (3.25,-2.5) circle (0.05); (3.25,-1.5) circle (0.05); (-2.75,-2) circle (0.05); (-3.25,-2.5) circle (0.05); (-3.25,-1.5) circle (0.05);
(0,-4.25) circle (0.05); (-.5,-4.75) circle (0.05); (0.5,-4.75) circle (0.05);
(0,-1.75) circle (0.05);
(-1.25,0) – (1.25,0) – (2,-2) – (0,-3.5) – (-2,-2) – cycle; (1.25,0) – (2, 0.5); (2,0.5) – (2.75, 0.5); (2,0.5) – (2,1.25); (-1.25,0) – (-2, 0.5); (-2,0.5) – (-2.75, 0.5); (-2,0.5) – (-2,1.25);
(2,-2) – (2.75,-2); (2.75,-2) – (3.25,-1.5); (2.75,-2) – (3.25,-2.5); (-2,-2) – (-2.75,-2); (-2.75,-2) – (-3.25,-1.5); (-2.75,-2) – (-3.25,-2.5);
(0,-4.25) – (0,-3.5); (0,-4.25) – (-.5,-4.75); (0,-4.25) – (.5,-4.75);
(2.75,0.5) – (3.25,-1.5); (-2.75,0.5) – (-3.25,-1.5); (.5,-4.75) – (3.25,-2.5); (-.5,-4.75) – (-3.25,-2.5); (2,1.25) – (-2,1.25);
(2,1.25) – (0.25,.25) – (1,-0.8) – (2.75,0.5); (-2,1.25) – (-0.25,.25) – (-1,-.8) – (-2.75,0.5);
(3.25,-1.5) – (1,-1.5) – (1,-2.5) – (3.25,-2.5); (-3.25,-1.5) – (-1,-1.5) – (-1,-2.5) – (-3.25,-2.5); (0.5,-4.75) – (0.5,-2.75) – (-0.5,-2.75) – (-.5,-4.75);
The argument for counting discs over $L_p^{(1)}$ is almost the same as in [@Smith:quiver], relying on a degeneration technique to reduce to the count of discs on a Lagrangian $S^1\times S^2$, as found in , and the behaviour of holomorphic discs under Lagrange surgery from [@FO3 Chapter 10]. For the higher $L_p^{(j)}$, there is a trick to reduce to the computation to the previously studied case, indicated schematically in Figure \[Fig:second\_tier\_after\_isotopy\] in the case $j=2$. Namely, if one replaces the tripod spheres by their Hamiltonian deformations as shown in red in the figure, then one reduces to the case of a region of the base $S$ containing a single point of $\bP$ and with boundary a polygon of matching paths, each labelled by the same Lagrangian vanishing cycle in the fibre. This is exactly the situation of the disc count over $L_p^{(1)}$, except the particular components of the reducible fibre which the holomorphic sections intersect will depend on $j$, compare to the discussion at the end of the previous section. Note that when $j>2$ (which arises only when $m>2$), the boundary configuration of $L_p^{(j)}$ involves adjacent tripod spheres, and not only alternating tripod and matching spheres. However, this doesn’t affect the argument.
It would be reasonable to expect that there is a holomorphic 3-form on $Y_{\Phi}$ with respect to which all the Lagrangian spheres in the configuration $\Gamma(\Delta_m^{\vee})$ admit gradings making them special of phase zero. At least the topological analogue of this holds:
One can grade the Lagrangians in the configuration $\Gamma(\Delta_m^{\vee})$ consistently so that all polygons in the cellulation have Maslov index zero, and the Floer algebra is concentrated in degrees $0\leq \ast \leq 3$.
Fix an element $p\in \bP$ and grade all but one of the Lagrangians encircling $p$ – the boundaries of a polygon projecting to $L_p^{(1)}$ – so that their intersections have degree $1$ as Floer inputs. The existence of the rigid disc of Proposition \[prop:rigid\_discs\] implies that the Floer output has degree 2, so the gradings are in fact cyclically symmetric. The existence of the rigid polygons over $L_p^{(j)}$ with $j \geq 1$, together with the fact that every matching sphere belongs to the boundary of a unique $L_p^{(j)}$, imply that the gradings propagate consistently to yield a grading satisfying the required conditions. (It follows by additivity of Maslov index that the gradings are consistent with the existence of the rigid quadrilaterals constructed in Proposition \[prop:rigid\_quadrilateral\] below.)
We now fix the background class $b\in H^2(Y_{\Phi};\bZ/2)$ which is Poincaré dual to a four-cycle $Z_b$ defined by ‘half’ the irreducible components at all the singular fibres. Precisely, $$\label{eqn:background}
b = PD[Z_b], \quad Z_b = \sum_{p \in \bP} \bC^2_{(ev)}$$ where the fibre $\pi^{-1}(p) \subset Y_{\Phi}$ of $p: Y_{\Phi} \to S$ is a disjoint union of $(m+1)$ ordered copies of $\bC^2$, the union of the even-indexed components of which we have labelled $\bC^2_{(ev)}$. If there are no 3-valent vertices in the ideal triangulation, one can compute the endomorphism algebra $\scrA_{\Gamma}$ equivalently by working either in ${\EuScript{F}}(Y)$ or in ${\EuScript{F}}(Y;b)$; but in the presence of 3-valent vertices and $L_p^{(1)}$ triangles, twisting by $b$ potentially affects the cohomological algebra.
\[prop:algebra\_ok\] The algebra $\scrA_{\Gamma} := \oplus_{v, v' \in \Gamma(\Delta_m^{\vee})} \, HF^*(L_v, L_{v'})$ is isomorphic to the total endomorphism algebra of the category ${\EuScript{C}}(Q(\Delta_m), W_{\bf c}(\Delta_m))$ for a vector ${\bf c}$ of non-zero coefficients.
There are three types of holomorphic triangles which contribute to the Floer product in the algebra:
1. constant triangles with image a vertex of $\Delta_m^{\vee}$;
2. the triangles of Lemma \[lem:more\];
3. triangles which map to a cycle $L_p^{(1)}$ for a vertex $p$ of $\Delta$ of valence 3 (if any such exist).
Each of these three triangle types has a non-zero count, and all the corresponding terms appear in the potential $W_{\bf c}(\Delta_m)$. Working over $\Lambda_{\bC}$, we take the coefficients of the first set of triangles to be $+1$, the second set of triangles to be of lowest order valuation (since the triangles become constant only after a Hamiltonian isotopy of the spheres in the configuration). The coefficients in ${\bf c}$ for a triangular $L_p^{(1)}$-region $R$ will be ‘large’, in the sense that it will be counted by $q^{\langle [\omega], R\rangle}$.
\[prop:rigid\_quadrilateral\] For a primitive cycle associated to a white quadrilateral $b_w$ the corresponding count of holomorphic discs is non-trivial.
(0-3,-1.7) circle (0.1); (1.5-3,1.5) circle (0.1); (-1.5-3,1.5) circle (0.1); (0-3,-1.7) – (0-3,0); (1.5-3,1.5) – (0-3,0); (-1.5-3,1.5) –(0-3,0);
(0-9,-1.7) circle (0.1); (1.5-9,1.5) circle (0.1); (-1.5-9,1.5) circle (0.1); (0-9,-1.7) – (0-9,0); (1.5-9,1.5) – (0-9,0); (-1.5-9,1.5) –(0-9,0);
(1.5-9,1.5) – (-1.5-3,1.5); (-9,-1.7) – (-3,-1.7); (-6,2) node [$b$]{}; (-6,-2.1) node [$a$]{}; (-1.5-9,1.5) – (-1.5-9,-2.5) – (1.5-9,-2.5) – (1.5-9,1.5); (-1.5-3,1.5) – (-1.5-3,-2.5) – (1.5-3,-2.5) – (1.5-3,1.5); (-11,-1) node [$b$]{}; (-1,-1) node [$b$]{};
(-0.3,0) – (1.2,0); (0+3,-1.7) circle (0.1); (1.5+3,1.5) circle (0.1); (-1.5+3,1.5) circle (0.1); (0+3,-1.7) – (0+3,0); (1.5+3,1.5) – (0+3,0); (-1.5+3,1.5) –(0+3,0);
(0+9,-1.7) circle (0.1); (1.5+9,1.5) circle (0.1); (-1.5+9,1.5) circle (0.1); (0+9,-1.7) – (0+9,0); (1.5+9,1.5) – (0+9,0); (-1.5+9,1.5) –(0+9,0);
(6,-2.1) node [$a$]{}; (1.5,1.5) – (1.5,-2.5) – (4.5,-2.5) – (4.5,0) – (7.5,0) – (7.5,-2.5) – (10.5,-2.5) – (10.5,1.5); (11,-1) node [$b$]{};
(4.5,-1.7) rectangle ++(3,3); (4.5,-2) – (4.5,0) – (7.5,0) – (7.5,-2) – (4.5,-2);
(9,-1.7) – (3,-1.7);
See Figure \[Fig:quadrilateral\_discs\]. The count of holomorphic discs is obtained by an invertible continuation isomorphism from the count in which the tripod boundary conditions are replaced by their red Hamiltonian images. The relation between holomorphic discs and polygons before and after Lagrange surgery [@FO3 Chapter 10] relates this to the count of holomorphic strips on the right hand side of the Figure. In this picture, the two Lagrangian 3-sphere boundary conditions are Hamiltonian disjoinable; the Floer complex has total rank two, with exactly one intersection point lying over each intersection of the black and red curves in the image. The Floer differential must be non-trivial (and hence an isomorphism). Putting one marked point in the interior of the strip $\bR\times [0,1]$ to stabilise the domain, since we are counting sections of the fibration $Y_{\Phi} \to S$ over the quadrilateral, it follows that the count of holomorphic quadrilaterals over the original domain $b_w$ is algebraically $\pm 1$.
The $A_{\infty}$-structure on $\scrA_{\Gamma}$ is encoded by a generic potential, i.e. one of the form $W = W_{\bf c}(\Delta_b) + W'$ for some ${\bf c} \in (\bK^*)^N$ and $W'$ concentrated on non-primitive cycles.
The previous lemmata show that the coefficients of all the primitive cycles are non-zero; in the case of the $L_p^{(j)}$ this relies on twisting by the background class $b$ to ensure that, for each $j$, the two contributing holomorphic discs (which have the same area) cannot cancel.
\[rmk:class\_of\_omega\] Lemma \[lem:H2Y\] implies that the cohomology class of a Kähler form $[\omega] \in H^2(Y_{\Phi};\bR)$ is determined by the total area of the base $S$ and its evaluation on a collection of $m$ closed surfaces at each reducible fibre whose intersection matrix with the irreducible components has rank $m$. Such a collection of surfaces can be obtained as follows: at a point of $D$, fix some $1 \leq j \leq m$, and consider the two holomorphic discs lying over $L^{(j)}_p$. Interpolating their boundaries fibrewise inside a component of the core $A_m$-chain of spheres in the fibres gives a 2-sphere meeting exactly two of the irreducible components. The symplectic area of such a sphere is just twice the coefficient in the potential of the primitive cycle $L_p^{(j)}$. It follows that the map $H^2(Y_{\Phi};\bR) \supset U \to \{\mathrm{primitive \ potentials}\}$ is locally injective. On the other hand, if we divide out by gauge equivalence, then one can normalise the coefficient of $L_p^{(1)}$ to be $1$, so the ‘mirror map’ $U \to \{\mathrm{potentials}\}/\{\mathrm{gauge}\}$ is not injective even after factoring out global rescaling of $[\omega]$.
\[rmk:relative\] One could consider the subcategory $\scrA_{\Gamma}$ inside the ‘relative Fukaya category’ ${\EuScript{F}}(Y_{\Phi},\scrD;b)$, see [@Sheridan:CDM], for $\scrD\subset Y_{\Phi}$ the divisor given by the union of fibres over $D \subset S$. The coefficients of holomorphic polygons in the relative Fukaya category record intersection numbers with $\scrD$; one can then take all the coefficients for $L_p^{(j)}$ equal. Similarly, if one works with a monodromy-invariant Kähler form as in Remark \[rmk:kahler\_monodromy\], then the coefficients of all the $L_p^{(j)}$ in the potential will be some fixed power $q^a$ of the Novikov variable, which brings one closer to the ‘canonical’ potential.
[^1]: A related result for $A_m$-fibrations over ${\mathbb{S}}= \bC$, but concerning derived categories rather than Fukaya categories, appears in [@Velez-Boer].
[^2]: These loops do not have canonical lifts to elements of the fundamental group of the smooth locus.
|
---
author:
- 'Tian-Shu Yang'
- 'Zong-Quan Zhou$\footnote{email:zq\_{}zhou@ustc.edu.cn}$'
- 'Yi-Lin Hua'
- Xiao Liu
- 'Zong-Feng Li'
- 'Pei-Yun Li'
- Yu Ma
- Chao Liu
- 'Peng-Jun Liang'
- Xue Li
- 'Yi-Xin Xiao'
- Jun Hu
- 'Chuan-Feng Li$\footnote{email:cfli@ustc.edu.cn}$'
- 'Guang-Can Guo'
title: 'Multiplexed storage and real-time manipulation based on a multiple-degree-of-freedom quantum memory'
---
Introduction {#introduction .unnumbered}
============
Large-scale quantum networks enable the long-distance quantum communication and optical quantum computing [@Internet; @repeater1; @computer1]. Due to the exponential photon loss in optical fibers [@limit], quantum communication via ground based optical fibers is currently limited to distances on hundred of kilometers. To overcome this problem, the idea of quantum repeater [@BDCZ; @DLCZ] has been proposed to establish quantum entanglement over long distances based on quantum memories and entanglement swapping. It has been shown that to reach practical data rates using this approach, the most significant improvements can be achieved through the use of multiplexed quantum memories [@repeater2; @repeater3; @multiplex].
The multiplexing of quantum memory can be implemented in any degree-of-freedom (DOF) of photons, such as those in the temporal [@TM1], spectral [@FM] and spatial [@SM] domains. Rare-earth-ion doped crystals (REIC) offer interesting possibilities as multiple-DOF quantum memories for photons by virtue of their large inhomogeneous bandwidths [@FM; @inhom; @sequencer2] and long coherence time [@sixhour] at cryogenic temperatures. Recently, there have been several important demonstrations using REIC, such as the simultaneous storage of 100 temporal modes [@TM100] by atomic frequency comb (AFC) [@AFC1] featuring pre-programmed delays, the storage of tens of temporal modes by spin-wave AFC with on-demand readout [@on-d2; @on-d15; @on-d5; @NC-14; @afc50] and the storage of 26 frequency modes with feed-forward controlled readout [@FM]. The orbital-angular-momentum (OAM) of a photon receives much attention because of the high capacity of OAM states for information transmission and spatial multimode operations [@OAMreviw]. Tremendous developments have recently been achieved in quantum memories for OAM states [@OAMbits; @OAMding; @OAMzhou], paving the way to quantum networks and scalable communication architectures based on this DOF.
{width="100.00000%"}
To date, most previous experiments with quantum memories have been confined to the storage of multiple modes using only one DOF, e.g. temporal, spectral or spatial. To significantly improve the communication capacity of quantum memory and quantum channel, we consider a quantum memory using more than one DOF simultaneously [@FM; @TF1; @MD; @SZ]. Here, we report on the experimental realization of an on-demand quantum memory storing single photons encoded with three-dimensional OAM states in a REIC. We present the results of a multiplexed spin-wave memory operating simultaneously in temporal, spectral and spatial DOF. In addition to expanding the number of modes in the memory through parallel multiplexing, a quantum mode converter (QMC) [@Convertera] can also be realized that can perform mode conversion in the temporal and spectral domains simultaneously and independently. Indeed, our quantum memory enables arbitrary temporal and spectral manipulations of spatial-qutrit-encoded photonic pulses and thus can serve as a real-time sequencer [@sequencer2], a real-time multiplexer/demultiplexer [@sequencer1], a real-time beam splitter [@BS1], a real-time frequency shifter [@S2], a real-time temporal/spectral filter [@sequencer1], among other functionalities.
Results {#results .unnumbered}
=======
Experimental Setup {#experimental-setup .unnumbered}
------------------
A schematic drawn of our experimental set-up and relevant atomic level structure of Pr$^{3+}$ ions are presented in Fig. 1. The memory crystal (MC) and filter crystal (FC) used in this set-up are 3 $\times$ 6 $\times$ 3 mm crystals of 0.05$\%$ doped Pr$^3$$^+$:Y$_2$SiO$_5$, which are cooled to 3.2 K using a cryogen-free cryostat (Montana Instruments Cryostation). In order to maximize absorption, the polarization of input light is close to the D$_2$ axis of Y$_2$SiO$_5$ crystal. To realize reliable quantum storage with high multimode capacity, we created a high-contrast AFC in MC (the AFC structure is shown in Supplementary Note 1). Spin-wave storage is employed to enable on-demand retrieval and extend storage time [@AFC1]. The control and input light are steered towards the MC in opposite directions with an angular offset $\sim$ $4^{\circ}$ to reject the strong control field and avoid the detection of free induction decay noise [@OD]. To achieve a low noise floor, we increase the absorption depth of the FC by employing a double-pass configuration. Fig. 2a presents the time histograms of the input photons (blue) and the photons retrieved at 12.68 $\mu$s (green) with a spin-wave storage efficiency $\eta$$_\textit{{SW}}$ = 5.51%. For an input with a mean photon number $\mu$ = 1.12 per pulse, we have measured a signal-to-noise ratio (SNR) $\sim$ 39.7 $\pm$ 6.7 with the input photons in the Gaussian mode.
Quantum process tomography {#quantum-process-tomography .unnumbered}
--------------------------
The ability to realize the on-demand storage of photonic OAM superposition states in solid-state systems is crucial for the construction of OAM based high dimensional quantum networks [@OAMbits]. The quantum process tomography for qutrit operations [@OAMtomography] is benchmark the storage performance for OAM qutrit in our solid-state quantum memory. The qutrit states are prepared in the following basis of OAM states: $\mid$LG$^{l=-1}_{p=0}$$\rangle$, $\mid$LG$^{l=0}_{p=0}$$\rangle$, $\mid$LG$^{l=1}_{p=0}$$\rangle$. Here, $\mid$LG$^{l}_{p}$$\rangle$ corresponds to OAM states defined as Laguerre-Gaussian (LG) modes, where $l$ and $p$ are the azimuthal and radial indices, respectively. In the following, we use the kets $\mid$L$\rangle$, $\mid$G$\rangle$ and $\mid$R$\rangle$ to denote the OAM states $\mid$LG$^{l=1}_{p=0}$$\rangle$, $\mid$LG$^{l=0}_{p=0}$$\rangle$ and $\mid$LG$^{l=-1}_{p=0}$$\rangle$, respectively. For $\mu$ = 1.12, we first characterized the input states before the quantum memory using quantum state tomography (see Methods section for details). The reconstructed density matrices of input are not ideal because of the preparation and measurements based on spatial light modulators (SLM) and single-mode fiber (SMF) are not perfect [@OAMzhou]. We then characterized the memory process using quantum process tomography. Fig. 2b presents the real part of the experimentally reconstructed process matrix $\chi$. It is found to have a fidelity of 0.909 $\pm$ 0.010 to Identity operation. This fidelity exceeds the classical bound of 0.831 (see Supplementary Note 4 for details), thereby confirming the quantum nature of the memory operation. The non-unity value of the memory fidelity may be caused by the limited beam waist of the pump/control light, which may result in imperfect overlap with different OAM modes. Moreover, we noted that the memory performance for superposition states of $\mid$L$\rangle$ and $\mid$R$\rangle$ is much better than that achieved here (as detailed in Supplementary Note 2). The visibility of such two-dimensional superposition states is higher than the fidelity of the memory process in all three dimensions. This result indicates that the storage efficiency is balanced for the symmetrical LG modes but is not balanced for all three considered spatial modes.
![\[Fig:2\] Time histograms and the reconstructed process matrix of quantum storage process in three dimensional OAM space. **a**, Time histograms of the input photons (blue), the photons retrieved at 12.68 $\mu$s for $\mu$ = 1.12 (green) and the unconditional noise for $\mu$ = 0 (black). **b**, Graphical representation of the process matrix $\chi$ of memory process as estimated via quantum process tomography. Details of operators $\lambda_i$ are shown in the Methods. Only the real part of the experimentally reconstructed process matrix is shown. All values are in the imaginary part are smaller than 0.090.](Tomography.pdf){width="50.00000%"}
Multiplexing storage in multiple DOF {#multiplexing-storage-in-multiple-dof .unnumbered}
------------------------------------
Carrying information in multiple DOF on photons can expand the channel capacity of quantum communication protocols [@FM; @MDOF]. Here, we show that our solid-state memory can be simultaneously multiplexed in temporal, spectral and spatial DOF. As shown in Fig. 3a, two AFC are created in the MC with an interval of 80 MHz between them to achieve spectral multiplexing. The two AFC have the same peak spacing of $\Delta$ = 200 kHz and the same bandwidth of $\Gamma_\textit{AFC}$ = 2 MHz. The spin-wave storage efficiencies are 5.05% and 5.13%, for the first AFC and the second AFC, respectively. The temporal multimode capacity of an AFC is limited by $\Gamma_\textit{AFC}$/$\Delta$ [@AFC1]. However, increasing the number of modes, the time interval between the last control pulse and the first output signal pulse will be reduced. Therefore, we employed only two temporal modes to reduce the noise caused by the last control pulse. The spatial multiplexing is realized by using three independent paths as input as shown in Fig. 3b. These paths, s$_1$, s$_2$ and s$_3$, correspond to the OAM states as $\mid$L$\rangle$, $\mid$R$\rangle$, and $\mid$G$\rangle$ defined above. By combining all three DOF together, we obtain 2 $\times$ 2 $\times$ 3 = 12 modes in total. The FC is employed to select out the desired spectral modes. Fig. 3c illustrates the results of multimode storage over these three DOF for $\mu$ = 1.04. The minimum crosstalk as obtained from mode crosstalk for each mode is 19.7 $\pm$ 3.41, which is calculated as one takes the counts in the diagonal term as the signal and then locates the large peaks over the range of output modes as the noise.
{width="80.00000%"}
Here, the temporal, spectral and spatial DOF are employed as classical DOF for multiplexing. One can choose any DOF to carry quantum information. As a typical example, now we use the temporal and spectral DOF for multiplexing and each channel is encoded with spatial qutrit state of $\mid$$\psi_1$$\rangle$ = ($\mid$L$\rangle$ + $\mid$G$\rangle$ + $\mid$R$\rangle$)/$\sqrt{3}$. Each channel is labeled as *$f_it_j$*, where $f_i$ represent spectral modes $i$ and $t_j$ similarly represent temporal modes $j$. Fig. 4a shows the experimental results for $\mu$ = 1.04. The minimum crosstalk as obtained from the mode crosstalk is approximately 15.2. We measured the fidelities of the spatial qutrit state for each channel as shown in Fig. 4b.
A quantum mode converter (QMC) can transfer photonic pulses to a target temporal or spectral mode without distorting the photonic quantum states. A real-time QMC that can operate on many DOF is essential for linking the components of a quantum network [@Convertera; @conveter1]. By adjusting the timing of the control pulse, one can specify the recall time in an on-demand manner to realize the temporal mode conversion. The two-AOM gate in our system can be used as a high-speed frequency shifter by tuning its driving frequency. Therefore, spectral and temporal mode conversion can be realized independently and simultaneously. Fig. 4c presents the results of QMC operation for $\mu$ = 1.04. We can convert from $f_i$ to $f_j$ and from $t_i$ to $t_j$, where these notations represent all different spectral modes and temporal modes. The noise level is significantly weaker than the strength of the converted signal, which indicates that the QMC operates quietly enough to avoid introducing any mode crosstalk. All these modes are encoded with OAM spatial qutrit of $\mid$$\psi_1$$\rangle$. To demonstrate that the qutrit state coherence is well preserved after QMC operation, we measured the fidelities (see Methods for details) between the input and converted states. The results are presented in Fig. 4d, which indicate the QMC can convert arbitrary temporal and spectral modes in real-time while preserving their quantum properties. Our device is expected to find applications in quantum networks comprising two quantum memories, in which mismatched spectral or temporal photon modes may need to be converted [@Convertera]. This device can ensure the indistinguishability of the photons which are retrieved from any quantum memories. This device is required for many photonic information processing protocols, e.g., a Bell-state measurement [@FM], and quantum-memory-assisted multi-photon generation [@multiplex].
{width="80.00000%"}
Arbitrary manipulations in real time {#arbitrary-manipulations-in-real-time .unnumbered}
------------------------------------
The precise and arbitrary manipulation of photonic pulses while preserving photonic coherence is an important requirement for many proposed photonic technologies [@sequencer1]. In addition to the QMC functionality demonstrated above, the developed quantum memory can enable arbitrary manipulations of photonic pulses in the temporal and spectral domains in real-time. As an example, we prepared the OAM qutrit state $\mid$$\psi_1$$\rangle$ in the $f_1t_1$ and $f_2t_2$ modes (Fig. 5a) as the input. Four typical operations were demonstrated, i.e., exchange of the readout times for the $f_1$ and $f_2$ photons, the simultaneous retrieval of the $f_1$ and $f_2$ photons at $t_1$, shifting the frequency of $f_1$ photons to $f_2$ but keeping the frequency of $f_2$ photons unchanged and temporal beam splitting the $f_1$ photons but filtering out the $f_2$ photons. These operations correspond to output of $\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{2}}$, $_{f_{2}}$$_{t_{1}}$, $\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{1}}$, $\mid$$\psi_1$$\rangle$$_{f_{2}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{2}}$ and $\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{1}}$$_{t_{2}}$, respectively. Another example was implemented with the OAM qutrit state $\mid$$\psi_2$$\rangle$ = ($\mid$L$\rangle$ + $\mid$G$\rangle$ - i$\mid$R$\rangle$)/$\sqrt{3}$ encoded in the $f_1t_2$ and $f_2t_2$ modes as the input, as shown in Fig. 5b with same output. The retrieved states were then characterized via quantum state tomography as usual (see Methods). Table 1 shows the fidelities between output states and input states.
![\[Fig:4\] Arbitrary temporal and spectral manipulations in real time. Four typical operations are presented for two different input states. **a**, The OAM qutrit state $\mid$$\psi_1$$\rangle$ is encoded on the $f_1t_1$ and $f_2t_2$ modes. The $f_1$ photons are marked as red color and the $f_2$ photons are marked as blue color. These operations, from up to down, correspond to a pulses sequencer, a multiplexer, a selective spectral shifter and a configurable beam splitter (the $f_2$ photons are filtered out), respectively. The little “$\times$2" indicates that the integration time of temporal beam splitting is two times of the other operations. **b**, The OAM qutrit state $\mid$$\psi_2$$\rangle$ is encoded on the $f_1t_2$ and $f_2t_2$ modes. These operations, from up to down, correspond to a demultiplexer, a pulses sequencer, a selective spectral shifter (the $f_1$ photons is frequency shifted to $f_2$ and retrieved at $t_1$ while the $f_2$ photons are retrieved at $t_2$) and a configurable beam splitter (the $f_2$ photons are filtered out), respectively. All of these operations can be determined after the photons have been absorbed into the quantum memory. The inputs are shifted earlier by 3 $\mu$s in the histograms for visual effects.](manipulate-final.pdf){width="50.00000%"}
Discussion {#discussion .unnumbered}
==========
In conclusion, we have experimentally demonstrated a multiplexed solid-state quantum memory that operates simultaneously in three DOF. The currently achieved multimode capacity is certainly not the fundamental limit for the physical system. Pr$^3$$^+$:Y$_2$SiO$_5$ has an inhomogeneous linewidth of 5 GHz, which can support more than 60 independent spectral modes. The number of temporal modes that can be achieved using the AFC protocol [@AFC1] is proportional to the number of absorption in the comb, which has already been improved to 50 in Eu$^3$$^+$:Y$_2$SiO$_5$ [@afc50]. There is no fundamental limit on the multimode capacity in the OAM DOF since it is independent on the AFC bandwidth. The capacity in this DOF is simply determined by the useful size of the memory in practice. We have recently demonstrated the faithful storage of 51 OAM spatial modes in a Nd$^3$$^+$:YVO$_4$ crystal [@OAMzhou]. The combination of these start-of-the-art technologies could result in a multimode capacity of 60 $\times$ 50 $\times$ 51 = 153000 modes. This large capacity could greatly enhance the data rate in memory-based quantum networks and in portable quantum hard drives with extremely long lifetimes [@sixhour].
The developed multiple-DOF quantum memory can serve as a QMC, which is the fundamental requirement for the construction of scalable networks based on multiplexed quantum repeaters. Although it is not demonstrated in the current work, mode conversion in the spatial domain should also be feasible using a high-speed digital micro-mirror device [@SLM]. QMC can also find applications in linear optical quantum computations. One typical example is to solve the mode mismatch caused by fiber-loop length effects and the time jitter of the photon sources in a boson sampling protocol [@BosonS1; @BosonS2].
Quantum communication and quantum computation in a large-scale quantum network rely on the ability of faithful storage and manipulation of photonic pulses carrying quantum information. The presented quantum memory can apply arbitrary temporal and spectral manipulations to photonic pulses in real-time, which indicates that this single device can serve as a variable temporal beam splitter [@BS1; @beamspliter] and a relative phase shifter [@AFCprep] that enables arbitrary control of splitting ratio and phase for each output. Therefore, this device can perform arbitrary single-qubit operations [@LOC]. Combining with the recent achievements on generation of heralded single photons [@source; @source1], this device should provide the sufficient set of operations to allow for universal quantum computing in the KLM scheme [@KLM]. Our results are expected to find applications in large-scale memory-based quantum networks and advanced photonic information processing architectures.
Input Mode Output Mode Fidelity
------------------------------------------------------------------------- ------------------------------------------------------------------- -----------------
\*[$\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{2}}$]{} $\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{2}}$, $_{f_{2}}$$_{t_{1}}$ 0.881$\pm$0.022
$\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{1}}$ 0.876$\pm$0.022
$\mid$$\psi_1$$\rangle$$_{f_{2}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{2}}$ 0.897$\pm$0.020
$\mid$$\psi_1$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{1}}$$_{t_{2}}$ 0.828$\pm$0.019
\*[$\mid$$\psi_2$$\rangle$$_{f_{1}}$$_{t_{2}}$, $_{f_{2}}$$_{t_{2}}$]{} $\mid$$\psi_2$$\rangle$$_{f_{1}}$$_{t_{2}}$, $_{f_{2}}$$_{t_{1}}$ 0.896$\pm$0.017
$\mid$$\psi_2$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{1}}$ 0.898$\pm$0.013
$\mid$$\psi_2$$\rangle$$_{f_{2}}$$_{t_{1}}$, $_{f_{2}}$$_{t_{2}}$ 0.898$\pm$0.016
$\mid$$\psi_2$$\rangle$$_{f_{1}}$$_{t_{1}}$, $_{f_{1}}$$_{t_{2}}$ 0.829$\pm$0.025
Methods {#methods .unnumbered}
=======
AFC preparation. {#afc-preparation. .unnumbered}
----------------
We tailored the absorption spectrum of Pr$^{3+}$ ions to prepare the AFC using spectral hole burning [@AFCprep]. The frequency of the pump light was first scanned over 16 MHz to create a wide transparent window in the Pr$^3$$^+$ absorption line. Then, a 1.6 MHz sweep was performed outside the pit to prepare the atoms into the 1/2g state. The burn-back procedure created an absorbing feature of 2 MHz in width resonant with the 1/2g - 3/2e transition but simultaneously populated the 3/2g state, which, in principle, must be empty for spin-wave storage. Thus, a clean pulse was applied at the 3/2g - 3/2e transition to empty this ground state. After the successful preparation of absorbing band in the 1/2g state, a stream of hole-burning pulses was applied on the 1/2g-3/2e transition. An AFC structure with a periodicity of $\Delta$ = 200 kHz is prepared in this step. These pulses burned the desired spectral comb of ions on the 1/2g - 3/2e transition and anti-holes at the 3/2g - 3/2e transition; thus, a short burst of clean pulses was applied to maintain the emptiness of the 3/2g state. For AFC preparation, the remaining 5/2g ground state is used as an auxiliary state which stores those atoms which don’t contribute to the AFC components. To reduce the noise generated by the control pulses during spin-wave storage, we applied 100 control pulses separated by 25 $\mu$s and another 50 control pulses with a separation of 100 $\mu$s after the preparation of the comb [@NC-14]. An example of the AFC with a periodicity $\Delta$ = 200 kHz is illustrated in Fig. 3a. A detailed estimation of the structure and storage efficiency of the AFC memory is presented in Supplementary Note 1. The signal photons are mapped onto the AFC, leading to an AFC echo after a time 1/$\Delta$. Spin-wave storage is achieved by applying two on-resonance control pulses to induce reversible transfer between the 3/2e state and 3/2g state before the AFC echo emission. The complete storage time is 12.68 $\mu$s in our experiment which includes an AFC storage time of 5 $\mu$s and a spin-wave storage time of 7.68 $\mu$s.
Filtering the noise. {#filtering-the-noise. .unnumbered}
--------------------
In order to achieve a low noise floor, temporal, spectral and spatial filter methods are employed. The input and control beams are sent to the MC in opposite directions with a small angular offset for spatial filtering. Temporal filtering is achieved by means of a temporal gate implemented with two AOM. This AOM gate temporally blocked the strong control pulses. This is important to avoid burning a spectral hole in the FC and to avoid blinding the single-photon detector. We used two 2-nm bandpass filters at 606 nm to filter out incoherent fluorescence noise. The spectral of the filter mode was achieved by narrow-band spectral filter in the FC (shown by the dashed black line in Fig. 3a), which is created by 0.8 MHz sweep around the input light frequency, leading to a transparent window of approximately 1.84 MHz due to the power broadening effect. Furthermore, the FC is implemented in a double-pass configuration to achieve high absorption.
Quantum tomography. {#quantum-tomography. .unnumbered}
-------------------
To characterize the memory performance for three dimensional OAM states, a quantum process tomography for the quantum memory operation is performed. Reconstructing the process matrix $\chi$ of any three-dimensional state requires nine linearly independent measurements. We chose three OAM eigenstates and six OAM superposition states as our nine input states, which are listed as follows: $\mid$L$\rangle$, $\mid$G$\rangle$, $\mid$R$\rangle$, ($\mid$L$\rangle$ + $\mid$G$\rangle$)/$\sqrt{2}$, ($\mid$R$\rangle$ + $\mid$G$\rangle$)/$\sqrt{2}$, (i$\mid$L$\rangle$ + $\mid$G$\rangle$)/$\sqrt{2}$, (-i$\mid$R$\rangle$ + $\mid$G$\rangle$)/$\sqrt{2}$, ($\mid$L$\rangle$ + $\mid$R$\rangle$)/$\sqrt{2}$, and ($\mid$L$\rangle$ - i$\mid$R$\rangle$)/$\sqrt{2}$ [@OAMzhou]. The complete operators for the reconstruction of the matrix $\chi$ are as follows [@Quditstate; @OAMzhou]:
$$\lambda_1 ={
\left[ \begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{array}
\right ]}$$
$$\lambda_2 ={
\left[ \begin{array}{ccc}
0& 1 & 0\\
1& 0 & 0\\
0 & 0 & 0
\end{array}
\right ]}$$
$$\lambda_3 ={
\left[ \begin{array}{ccc}
0& -i & 0\\
i& 0 & 0\\
0& 0 & 0
\end{array}
\right ]}$$
$$\lambda_4 ={
\left[ \begin{array}{ccc}
1& 0 & 0\\
0& -1 & 0\\
0 & 0 & 0
\end{array}
\right ]}$$
$$\lambda_5 ={
\left[ \begin{array}{ccc}
0& 0 & 1\\
0& 0 & 0\\
1 & 0 & 0
\end{array}
\right ]}$$
$$\lambda_6 ={
\left[ \begin{array}{ccc}
0& 0 & -i\\
0& 0 & 0\\
i & 0 & 0
\end{array}
\right ]}$$
$$\lambda_7 ={
\left[ \begin{array}{ccc}
0& 0 & 0\\
0& 0 & 1\\
0 & 1 & 0
\end{array}
\right ]}$$
$$\lambda_8 ={
\left[ \begin{array}{ccc}
0& 0 & 0\\
0& 0 & -i\\
0 & i & 0
\end{array}
\right ]}$$
$$\lambda_9 =\frac{1}{\sqrt3}{
\left[ \begin{array}{ccc}
1& 0 & 0\\
0& 1 & 0\\
0 & 0 & -2
\end{array}
\right ]}$$
Here, $\lambda_1$ is the identity operation. The process matrix $\chi$ can be expressed on the basis of $\lambda_i$ and maps an input matrix $\rho_{\textsl{in}}$ onto the output matrix $\rho_{\textit{out}}$ [@OAMstate; @Quditstate]. Density matrices of the input states $\rho_{\textit{in}}$ and density matrices of the output states $\rho_{\textit{out}}$ are reconstructed using quantum state tomography [@OAMstate; @Quditstate]. For a given qutrit state ($\mid$$\psi$$\rangle$), we used $\rho$=$\mid$$\psi$$\rangle$$\langle$$\psi$$\mid$ to reconstruct the density matrix $\rho$ from the measurement results. As examples, we presented graphical representations of the reconstructed density matrices for the OAM qutrit states $\mid$$\psi_1$$\rangle$ and $\mid$$\psi_2$$\rangle$ in Supplementary Note 3.The fidelities of the output states with respect to the input states can be calculated based on the reconstructed density matrices using the formula $Tr$($\sqrt{\sqrt{\rho_{\textit{output}}}\rho_{\textit{input}}\sqrt{\rho_{\textit{output}}}})$$^2$.
Data availability {#data-availability .unnumbered}
=================
The data that support the findings of this study are available from the corresponding authors on request.
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Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the National Key R&D Program of China (No. 2017YFA0304100,2016YFA0302700), the National Natural Science Foundation of China (Nos. 61327901,11774331,11774335,61490711,11504362,11654002), Anhui Initiative in Quantum Information Technologies (No. AHY020100),Key Research Program of Frontier Sciences, CAS (No. QYZDY-SSW-SLH003), the Fundamental Research Funds for the Central Universities (Nos. WK2470000023, WK2470000026).
Author contributions {#author-contributions .unnumbered}
====================
Z.Q.Z. and C.F.L. designed experiment. T.S.Y. and Z.Q.Z. carried out the experiment assisted by Y.L.H., X.L., Z.F.L., P.Y.L., Y.M., C.L., P.J.L., Y.X.X., J.H. and X.L. T.S.Y. and Z.Q.Z. wrote the paper with input from other authors. C.F.L. and G.C.G supervised the project. All authors discussed the experimental procedures and results.
Additional information {#additional-information .unnumbered}
======================
**Competing interests:** The authors declare no competing interests.
|
---
abstract: 'We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan’s summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. We also show the $q \to 1 - 0$ limit of our summation formulae.'
address:
- 'Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
- 'Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Hironori Mori
- Takeshi Morita
title: 'Summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1 ( a; b; q, z )$'
---
OU-HET 892
Introduction {#Intro}
============
In this paper, we give the following summation formulae for the bilateral series: $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha\beta; q )_\infty}
\sum_{n \in \mathbb{Z}}
\frac{( \alpha \beta; q )_n}{( \frac{\gamma}{\alpha}; q )_n}
\left( \frac{\gamma w}{\alpha q^{1/2}} \right)^n
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
\sum_{n \in \mathbb{Z}}
\frac{( \frac{1}{\beta}; q )_n}{( \beta^2 \gamma; q )_n}
\left( \frac{\gamma w}{\alpha q^{1/2}} \right)^n
\right\} \notag \\ &=
\frac
{\left( q^2, \frac{{\gamma}}{\alpha^2 \beta}, \frac{\alpha q^{3/2}}{{\gamma}w}, \alpha \beta w q^{1/2}; q^2 \right)_\infty}
{\left( q, \alpha^2 \beta^2, \frac{{\gamma}w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{main1}\end{aligned}$$ and $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha\beta; q )_\infty}
\sum_{n \in \mathbb{Z}}
\frac{( \alpha \beta; q )_n}{( \frac{\gamma}{\alpha}; q )_n}
\left( \frac{\gamma w}{\alpha q^{1/2}} \right)^n
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
\sum_{n \in \mathbb{Z}}
\frac{( \frac{1}{\beta}; q )_n}{( \beta^2 \gamma; q )_n}
\left( \frac{\gamma w}{\alpha q^{1/2}} \right)^n
\right\} \notag \\ &=
\alpha \beta
\frac
{\left( q^2, \frac{{\gamma}}{\alpha^2 \beta}, \frac{\alpha q^{5/2}}{{\gamma}w}, \alpha \beta w q^{3/2}; q^2 \right)_\infty}
{\left( q, \alpha^2 \beta^2, \frac{{\gamma}w q^{1/2}}{\alpha}, \frac{q^{3/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{main2}\end{aligned}$$ where $w \in \mathbb{C}^*$, and the parameters are constrained by $\alpha \beta^2 = - 1$ and $\beta \gamma = q$. Here, $( a; q )_n$ is the $q$-shifted factorial defined by $$\begin{aligned}
( a; q )_n := {\left\{}
\newcommand{\rc}{\right\}}\begin{aligned}
& 1, && n = 0, \\ & ( 1 - a ) ( 1 - a q ) \dots ( 1 - a q^{n - 1} ), && n \ge 1, \\ & {\left[}( 1 - a q^{-1} ) ( 1 - a q^{-2} ) \dots ( 1 - a q^n ) {\right]}^{-1}, && n \le - 1. \end{aligned}
\right.\end{aligned}$$ Moreover, $( a; q )_\infty := \lim_{n \to \infty} ( a; q )_n$ and we use the shorthand notation $$\begin{aligned}
( a_1, a_2, \dots, a_m; q )_\infty := ( a_1; q )_\infty ( a_2; q )_\infty \dots ( a_m; q )_\infty.\end{aligned}$$ Surprisingly, the special case of the formulae and are originally shown in the context of physics [@MMT]. These are realized as the equality of a physical quantity called the superconformal index, which is calculated to get strong evidence for so-called Abelian mirror symmetry in three-dimensional supersymmetric quantum field theories.
In Section \[Main\], we provide the detailed proofs for the formulae and , which are the generalized versions of the formulae found in [@MMT], based on the theta function of Jacobi and Ramanujan’s summation formula $$\begin{aligned}
\label{rama}
\sum_{n \in \mathbb{Z}}
\frac{( a; q )_n}{( b; q )_n}
z^n
=
\frac{( q, b/a, a z, q/a z; q )_\infty}{( b, q/a, z, b/a z; q )_\infty}, \quad |b/a| < |z| < 1,\end{aligned}$$ where the left-hand side is the bilateral basic hypergeometric series ${}_1\psi_1 ( a; b; q, z )$ (see Section \[Notation\] for the explicit definition). This was given by S. Ramanujan [@Ramanujan]. Further contributions to summation formulae and transformations for the bilateral basic hypergeometric series with the base $q$ were given by Bailey [@Bailey; @B1], Slater [@Slater], Jackson [@J1], Jackson [@J2], K. R. Vasuki and K. R. Rajanna [@VR], and D. D. Somashekara, K. N. Murthy and S. L. Shalini [@SNS]. The specific point we should mention is that our summation formulae and connect the geometric series with two different bases $q$ and $q^2$. As an application, we also show that our formulae with taking the specific parameter combination reproduce the results obtained by the physical study of Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$ [@MMT] where $\mathbf{R}P^2$ is a real projective plane. This means that our generic formulae can be powerful tools for checking exactly the nontrivial statement in physics.
In Section \[Limit\], we show the $q \to 1 - 0$ limit of our new formulae and from the viewpoint of connection problems on $q$-difference equations [@Birkhoff; @Z0; @M0]. The limit results in the relation among the gamma function, power functions and the bilateral hypergeometric series ${}_1H_1 ( a; b; z )$ with a suitable condition. Other summation formulae for some bilateral hypergeometric series were studied by J. Dougall [@Dougall] and W. N. Bailey [@Bailey]. For instance, Dougall derived the bilateral hypergeometric identity $$\begin{aligned}
{}_2H_2 ( a, b; c, d; 1 )
=
\frac{\Gamma ( 1 - a )\Gamma ( 1 - b ) \Gamma ( c ) \Gamma ( d ) \Gamma ( c + d - a - b - 1 )}{\Gamma ( c - a ) \Gamma ( c - b ) \Gamma ( d - a ) \Gamma ( d - b )},\end{aligned}$$ where $\Re ( c + d - a - b ) > 1$. Dougall also proved that a *well-poised* series ${}_5H_5$ [@GR] could be evaluated at $w = 1$, and then we can obtain another summation formula for a well-poised series ${}_5H_5$ with $w = 1$ [@Slater]. In the $q \to 1 - 0$ limit of the formulae and , we reach to the formula for ${}_1H_1$ under a suitable condition shown in Theorem \[mainlim2\].
Notation {#Notation}
========
In this section, we review basic notation. The bilateral basic hypergeometric series with the base $q$ is given by $$\begin{aligned}
\label{bbh} {}_r\psi_s ( a_1, \dots, a_r; b_1, \dots, b_s; q, z )
:=
\sum_{n \in \mathbb{Z}}
\frac{( a_1, \dots, a_r; q )_n}{( b_1, \dots, b_s; q )_n}
\left\{ ( - 1 )^n q^{\frac{n ( n - 1 )}{2}} \right\}^{s - r} z^n.\end{aligned}$$ The series diverges for $z \not = 0$ if $s < r$ and converges for $|b_1 \dots b_s/a_1 \dots a_r| < |z| < 1$ if $r = s$ (see [@GR] for more details). We remark that the $q$-shifted factorial $( a; q )_n$ is the $q$-analogue of the shifted factorial $$\begin{aligned}
( \alpha )_n
=
\alpha \{ \alpha + 1 \} \cdots \{ \alpha + ( n - 1 ) \},\end{aligned}$$ and the series is the $q$-analogue of the bilateral hypergeometric function $$\begin{aligned}
{}_rH_s ( \alpha_1, \dots, \alpha_r; \beta_1, \dots, \beta_s; z )
:=
\sum_{n \in \mathbb{Z}}
\frac{( \alpha_1, \dots, \alpha_r )_n}{( \beta_1, \dots, \beta_s )_n}
z^n.\end{aligned}$$ By D’Alembert’s ratio test, it can be checked that ${}_rH_r$ converges only for $|z| = 1$ [@Slater], provided that $\Re ( \beta_1 + \dots + \beta_r - \alpha_1 - \dots - \alpha_r ) > 1$.
The $q$-gamma function $\Gamma_q ( z )$ is defined by $$\begin{aligned}
\Gamma_q ( z )
:=
\frac{( q; q )_\infty}{( q^z; q )_\infty}
( 1 - q )^{1 - z}, \qquad 0 < q < 1.\end{aligned}$$ The $q \to 1 - 0$ limit of $\Gamma_q ( z )$ gives the gamma function [@GR] $$\begin{aligned}
\lim_{q \to 1 - 0}
\Gamma_q ( z ) = \Gamma ( z ).
\label{limgamma}\end{aligned}$$ The theta function of Jacobi with the base $q$ is given by $$\begin{aligned}
\label{thetaJ} \theta_q ( z )
:=
\sum_{n \in \mathbb{Z}}
q^{\frac{n^2}{2}}
( - z )^n, \qquad \forall z \in \mathbb{C}^*.\end{aligned}$$ Jacobi’s triple product identity is $$\begin{aligned}
\theta_q ( z )
=
( q, q^{1/2} z, q^{1/2}/z; q )_\infty.
\label{jtpi}\end{aligned}$$ The theta function has the inversion formula $$\begin{aligned}
\label{inv} \theta_q ( z )
=
\theta_q \left( 1/z \right),\end{aligned}$$ and satisfies the $q$-difference equation $$\begin{aligned}
\theta_q ( z q^k )
=
( - z )^{- k} q^{- \frac{k^2}{2}}
\theta_q ( z ).
\label{thetaperi}\end{aligned}$$
In our study, the following proposition about the theta function [@Z1] is useful to consider the $q \to 1 - 0$ limit of our formulae in Section \[Limit\].
For any $z \in \mathbb{C}^* ( - \pi < \arg z < \pi )$, we have $$\begin{aligned}
\lim_{q \to 1 - 0}
\frac{\theta_q ( q^\beta z )}{\theta_q ( q^\alpha z )}
=
( - z )^{\alpha - \beta}.
\label{limt1}\end{aligned}$$
We also use the following limiting formula [@GR]: $$\begin{aligned}
\lim_{q \to 1 - 0}
\frac{( z q^\alpha; q )_\infty}{( z; q )_\infty}
=
( 1 - z )^{- \alpha}, \qquad |z| < 1.
\label{limbin}\end{aligned}$$
Main theorem {#Main}
============
In this section, we show the formulae and as the summation formulae of the bilateral basic hypergeometric series ${}_1\psi_1 ( a; b; q, z )$ [@GR] by utilizing Ramanujan’s summation formula.
\[mainth1\] For any $w \in \mathbb{C}^*$, we have $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{m1}\end{aligned}$$ where $\alpha \beta^2 = - 1$ and $\beta \gamma = q$.
\[mainth2\] For any $w \in \mathbb{C}^*$, we have $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\alpha \beta
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty}
{\left( \frac{\gamma w q^{1/2}}{\alpha}, \frac{q^{3/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{m2}\end{aligned}$$ where $\alpha \beta^2 = - 1$ and $\beta \gamma = q$.
We obtain the following corollaries immediately from Theorem \[mainth1\] and \[mainth2\].
For any $w \in \mathbb{C}^*$, we have $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac{( q^2; q^2 )_\infty}{( q; q^2 )_\infty}
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{mcor1}\end{aligned}$$ where $\alpha \beta^2 = - 1$ and $\beta \gamma = q$.
For any $w \in \mathbb{C}^*$, we have $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\alpha \beta
\frac{( q^2; q^2 )_\infty}{( q; q^2 )_\infty}
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty}
{\left( \frac{\gamma w q^{1/2}}{\alpha}, \frac{q^{3/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{mcor2}\end{aligned}$$ where $\alpha \beta^2 = - 1$ and $\beta \gamma = q$.
We consider the $q \to 1 - 0$ limit of our formulae and in Section \[Limit\].
Ramanujan’s summation formula
-----------------------------
We review Ramanujan’s sum for ${}_1\psi_1 ( a; b; q, z )$ to establish our main theorems.
\[Ram\] For any $z \in \mathbb{C}$, we have $$\begin{aligned}
\label{rsum11}
{}_1\psi_1 ( a; b; q, z )
=
\frac{( q, b/a, a z, q/a z; q )_\infty}{( b, q/a, z, b/a z; q )_\infty}\end{aligned}$$ with $|b/a| < |z| < |1|$.
There are a number of proofs of the summation formula in the literature. The first published proof of the summation formula was given by W. Hahn [@Hahn] and M. Jackson [@J2]. We can find other proofs by G. E. Andrews [@A1; @A2], Askey [@As1], and M. E. H. Ismail [@Ismail]. Ramanujan’s summation formula is considered as the “bilateral extension” of the $q$-binomial theorem [@GR] $$\begin{aligned}
\label{qbinomial} \sum_{n \ge 0}
\frac{( a; q )_n}{( q; q )_n}
z^n
=
\frac{( a z; q )_\infty}{( z; q )_\infty}\end{aligned}$$ for $|z| < 1$. The $q$-binomial theorem was derived by Cauchy [@C1], Heine [@H1], and many mathematicians. We remark that the identity is also thought of as the $q$-analogue of the “bilateral binomial theorem” [@Mbi] found by M. E. Horn [@Horn] $$\begin{aligned}
{}_1H_1 ( a; c; z )
=
\frac{( 1 - z )^{c - a - 1}}{( - z )^{c - 1}}
\frac{\Gamma ( 1 - a ) \Gamma ( c )}{\Gamma ( c - a )},\end{aligned}$$ where $a$ and $c$ are complex numbers such that $\Re ( c - a ) > 1$, and $z$ is a complex number with $|z| = 1$ and $z \neq 1$.
For later use, let us write the application of Ramanujan’s summation formula . We have the following relations by Theorem \[Ram\].
\[l1\] For any $w \in \mathbb{C}^*$ (provided that $|\gamma w/( \alpha q^{1/2} )| < 1$), we have $$\begin{aligned}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
&=
\frac
{\left( q, \frac{\gamma}{\alpha^2 \beta}, \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}; q \right)_\infty}
{\left( \frac{\gamma}{\alpha}, \frac{q}{\alpha \beta}, \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}, \\ {}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
&=
\frac
{\left( q, \beta^3 {\gamma}, \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}; q \right)_\infty}
{\left( \beta^2 \gamma, q \beta, \frac{\gamma w}{\alpha q^{1/2}}, \frac{\alpha \beta^{3} q^{1/2}}{w}; q \right)_\infty}
=
\frac
{\left( q, \frac{{\gamma}}{\alpha^2 \beta}, \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}; q \right)_\infty}
{\left( \beta^2 \gamma, q \beta, \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}. $$
Recall that there exist relations $\alpha \beta^2 = - 1$ and $\beta \gamma = q$.
Relations between the theta functions
-------------------------------------
Before showing the precise proofs of our main theorems, in order to make them simpler, we prepare several relations for the theta functions with the same or the different bases. First of all, we obtain the following relations by using the inversion formula of the theta function :
\[lem1\] For any $\xi, \eta \in \mathbb{C}^*$, we have $$\begin{aligned}
\theta_{q^2} ( \xi \eta )
&=
\frac{1}{2}
\left\{
\theta_{q^2} ( \xi \eta ) + \theta_{q^2} \left( \frac{1}{\xi \eta} \right)
\right\}, \\ \theta_{q^2} \left( \frac{\xi}{\eta} \right)
&=
\frac{1}{2}
\left\{
\theta_{q^2} \left( \frac{\xi}{\eta} \right) + \theta_{q^2} \left( \frac{\eta}{\xi} \right)
\right\}. $$
\[co3\] If we put $\xi = \alpha \beta/q^{1/2}$ and $\eta = 1/w$ in Lemma \[lem1\], we obtain $$\begin{aligned}
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)}
=
\frac
{4}
{\left\{
\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right) + \theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\right\}
\left\{
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right) + \theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)
\right\}}. \end{aligned}$$
\[corl3\] If we put $\xi = \alpha \beta q/q^{1/2}$ and $\eta = 1/w$ in Lemma \[lem1\], we obtain $$\begin{aligned}
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)}
=
\frac
{4}
{\left\{
\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
+
\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta q} \right)
\right\}
\left\{
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)
+
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta q w} \right)
\right\}}.\end{aligned}$$
We can derive the product formula of the theta functions with the different bases.
\[p1\] For any $\xi$, $\eta \in \mathbb{C}^*$, we have $$\begin{aligned}
\theta_{q^2} \left( \frac{\xi}{\eta} \right) \theta_{q^2} ( \xi \eta )
=
\theta_q ( \xi ) \theta_q ( - \eta ).
\label{ptp1}\end{aligned}$$
We put $\xi = e^{\pi i x}$ and $\eta = e^{\pi i y}$ in the definition of the theta function on the left-hand side of , $$\begin{aligned}
&
\theta_{q^2} \left( \frac{\xi}{\eta} \right)
\theta_{q^2} ( \xi \eta ) \\ &=
\left(
\sum_{n \in \mathbb{Z}}
\exp \left\{ \pi i n + \pi i n^2 \tau + \pi i n ( x - y ) \right\}
\right)
\left(
\sum_{m \in \mathbb{Z}}
\exp \left\{ \pi i m + \pi i m^2 \tau + \pi i m ( x + y ) \right\}
\right) \\ &=
\sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}}
\exp \left\{ \pi i ( n + m ) + \pi i ( n^2 + m^2 ) \tau + \pi i n ( x - y ) + \pi i m ( x + y ) \right\} \\ &=
\sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}}
\exp \left\{ \frac{\pi i}{2} ( n + m )^2 \tau + \frac{\pi i}{2} ( n - m )^2 \tau + \pi i ( n + m ) x - \pi i ( n - m ) y + \pi i ( n + m ) \right\} \\ &=
\sum_{N \in \mathbb{Z}}
\exp \left\{ \frac{\pi i}{2} N^2 \tau + \pi i N x + \pi i N \right\}
\sum_{ - M \in \mathbb{Z}}
\exp \left\{ \frac{\pi i}{2} M^2 \tau + \pi i M y \right\} \\ &=
\theta_q ( \xi )
\theta_q ( - \eta ),\end{aligned}$$ provided that $N := n + m$ and $M := n - m$. Therefore, we obtain the identity .
By Proposition \[p1\] and the inversion formula , we have the following relations:
\[co4\] For the specific parameter combinations of $\xi$ and $\eta$, the identity can be expressed as $$\begin{aligned}
\xi = \frac{\alpha \beta}{q^{1/2}},\
\eta = w
&{\Rightarrow}\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)
=
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q ( - w ), \\ \xi = \frac{1}{w},\
\eta = \frac{q^{1/2}}{\alpha \beta}
&{\Rightarrow}\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)
=
\theta_q \left( \frac{1}{w} \right)
\theta_q \left( - \frac{q^{1/2}}{\alpha \beta} \right)
=
\theta_q \left( w \right)
\theta_q \left( - \frac{q^{1/2}}{\alpha \beta} \right), \\ \xi = w,\
\eta = \frac{\alpha \beta}{q^{1/2}}
&{\Rightarrow}\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)
=
\theta_q ( w )
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} \right)
=
\theta_q \left( w \right)
\theta_q \left( - \frac{q^{1/2}}{\alpha \beta} \right), \\ \xi = \frac{q^{1/2}}{\alpha \beta},\
\eta = \frac{1}{w}
&{\Rightarrow}\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)
=
\theta_q \left( \frac{q^{1/2}}{\alpha \beta} \right)
\theta_q \left( - \frac{1}{w} \right)
=
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q ( - w ). $$
We apply the relations in Corollary \[co4\] to the equality in Corollary \[co3\] in order to change the base of the theta functions.
\[co5\] For any $\alpha$, $\beta$ and $w \in \mathbb{C}^*$, we have $$\begin{aligned}
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)}
=
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}.\end{aligned}$$
Bringing Corollary \[co3\] and \[co4\] together leads to $$\begin{aligned}
&
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)} \\ &=
\frac
{4}
{\left\{
\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
+
\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\right\}
\left\{
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)
+
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)
\right\}} \\ &=
\frac
{4}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)
+
\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)
+
\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)
+
\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta} \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta w} \right)} \\ &=
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( - w \right)
+
\theta_q \left( w \right)
\theta_q \left( - \frac{q^{1/2}}{\alpha \beta} \right)} \\ &=
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \beta q^{1/2} \right)} \\ &=
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}, $$ where we used the condition $\alpha \beta^2 = - 1$ and the inversion formula in the last two steps.
Moreover, we need other product relations connecting the theta functions with different bases derived from Proposition \[p1\] and the inversion formula .
\[corl4\] For the specific parameter combinations of $\xi$ and $\eta$, the identity can be expressed as $$\begin{aligned}
\xi = \frac{\alpha \beta}{q^{1/2}} q,\
\eta = w
&{\Rightarrow}\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)
=
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} q \right)
\theta_q ( - w ), \\ \xi = \frac{1}{w},\
\eta = \frac{q^{1/2}}{\alpha \beta q}
&{\Rightarrow}\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta q w} \right)
=
\theta_q \left( \frac{1}{w} \right)
\theta_q \left( - \frac{q^{1/2}}{\alpha \beta q} \right)
=
\theta_q \left( w \right)
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} q \right), \\ \xi = w,\
\eta = \frac{\alpha \beta}{q^{1/2}} q
&{\Rightarrow}\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta q} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)
=
\theta_q ( w )
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} q \right), \\ \xi = \frac{q^{1/2}}{\alpha \beta q},\
\eta = \frac{1}{w}
&{\Rightarrow}\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta q} \right)
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta q w} \right)
=
\theta_q \left( \frac{q^{1/2}}{\alpha \beta q} \right)
\theta_q \left( - \frac{1}{w} \right)
=
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} q \right)
\theta_q ( - w ). $$
Then, these relations in Corollary \[corl4\] can be used to show another product-to-sum identity of the theta function.
\[corol1\] For any $\alpha$, $\beta$ and $w \in \mathbb{C}^*$, we have $$\begin{aligned}
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)}
=
\frac
{2 \alpha \beta}
{-
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}.\end{aligned}$$
As done in Corollary \[co5\], bringing Corollary \[corl3\] and \[corl4\] together leads to $$\begin{aligned}
\frac
{1}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)}
&=
\frac
{4}
{\left\{
\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
+
\theta_{q^2} \left( \frac{q^{1/2} w}{\alpha \beta q} \right)
\right\}
\left\{
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)
+
\theta_{q^2} \left( \frac{q^{1/2}}{\alpha \beta q w} \right)
\right\}} \\ &=
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} q \right)
\theta_q \left( - w \right)
+
\theta_q \left( w \right)
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} q \right)}. $$ By applying the $q$-difference equation of the theta function , $$\begin{aligned}
&
\frac
{2}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} q \right)
\theta_q \left( - w \right)
+
\theta_q \left( w \right)
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} q \right)} \\ &=
\frac
{2}
{{\left(}- \frac{q^{1/2}}{\alpha \beta} q^{- 1/2} {\right)}\cdot
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( - w \right)
+
{\left(}\frac{q^{1/2}}{\alpha \beta} q^{- 1/2} {\right)}\cdot
\theta_q \left( w \right)
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} \right)} \\ &=
\frac
{2 \alpha \beta}
{-
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( - w \right)
+
\theta_q \left( w \right)
\theta_q \left( - \frac{\alpha \beta}{q^{1/2}} \right)} \\ &=
\frac
{2 \alpha \beta}
{-
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}, $$ where we used the condition $\alpha \beta^2 = - 1$ in the last equality.
Proof of Theorem \[mainth1\]
----------------------------
In this subsection, we give the exact proof of Theorem \[mainth1\] based on the relations for the theta functions we showed above. We aim to translate the left-hand side of the relation into its right-hand side. By Corollary \[l1\], we can rewrite the left-hand side of as follows: $$\begin{aligned}
\label{proof1} &
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{(\alpha \beta; q )_\infty}
\frac
{\left( q, \frac{\gamma}{\alpha^2 \beta}, \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w} \right)_\infty}
{\left( \frac{\gamma}{\alpha}, \frac{q}{\alpha \beta}, \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
\frac
{\left( q, \frac{\gamma}{\alpha^2 \beta}, \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}; q \right)_\infty}
{\left( \beta^2 \gamma, \beta q, \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\right\} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac{\left( q, \frac{\gamma}{\alpha^2 \beta}; q \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\left\{
\frac{\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}; q \right)_\infty}{\left( \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty}
+
\frac
{\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}; q \right)_\infty}
{\left( \frac{1}{\beta}, \beta q; q \right)_\infty}
\right\} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{ ( q^2; q^2 )_\infty}
\frac
{\left( q, \frac{\gamma}{\alpha^2\beta}; q \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac
{1}
{\left( \alpha \beta, \frac{q}{\alpha \beta}, \frac{1}{\beta}, \beta q; q \right)_\infty} \notag \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, \beta q; q \right)_\infty
+
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha\beta}; q \right)_\infty
\right\}. $$ Now, we focus on the overall factor in the first line of . By using the relations $( a^2; q^2 )_\infty = ( a, - a; q )_\infty$ and $( a; q )_\infty = ( a, a q; q^2 )_\infty$ in addition to $\alpha \beta^2 = - 1$, this part is rewritten as $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac{\left( q, \frac{\gamma}{\alpha^2 \beta}; q \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha \beta, \frac{q}{\alpha \beta}, \frac{1}{\beta}, \beta q; q \right)_\infty} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac{\left( q, \frac{\gamma}{\alpha^2 \beta}; q \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha \beta, \frac{q}{\alpha \beta}, - \alpha \beta, - \frac{q}{\alpha \beta}; q \right)_\infty} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac
{\left( q, q^2, \frac{\gamma}{\alpha^2 \beta}, \frac{\gamma}{\alpha^2 \beta} q; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha^2 \beta^2, \frac{q^2}{\alpha^2 \beta^2}; q^2 \right)_\infty}. $$ Furthermore, the condition $\beta {\gamma}= q$ can simplify this part, $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac
{\left( q, q^2, \frac{\gamma}{\alpha^2 \beta}, \frac{\gamma}{\alpha^2 \beta} q; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha^2 \beta^2, \frac{q^2}{\alpha^2 \beta^2}; q^2 \right)_\infty} \notag \\ &=
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\frac
{\left( q, q^2, \frac{\gamma}{\alpha^2 \beta}, \frac{q^2}{\alpha^2 \beta^2}; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha^2 \beta^2, \frac{q^2}{\alpha^2 \beta^2}; q^2 \right)_\infty} \notag \\ &=
\frac{1}{2}
( q; q^2 )_\infty
\frac{\left( q, \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q \right)_\infty}
\frac{1}{\left( \alpha^2 \beta^2; q^2 \right)_\infty} \notag \\ &=
\frac{1}{2}
( q; q^2 )_\infty
\frac
{\left( q, \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w}, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty}
\frac{1}{\left( \alpha^2 \beta^2; q^2 \right)_\infty} \notag \\ &=
\frac{1}{2}
\frac{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \alpha^2 \beta^2; q^2 \right)_\infty}
\frac{1}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}
\frac{\left( q, q; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty}. $$ Thus, $$\begin{aligned}
( \text{the left-hand side of \eqref{m1}} )
&=
\frac{1}{2}
\frac{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \alpha^2 \beta^2; q^2 \right)_\infty}
\frac{1}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}
\frac{\left( q, q; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty} \notag \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, \beta q; q \right)_\infty
+
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. \label{proof2}\end{aligned}$$ As the next step, we would like to show the following relation:
\[co6\] For any $w \in {\mathbb{C}}^*$, we obtain $$\begin{aligned}
1
&=
\frac{1}{2}
\frac
{\left( q, q; q^2 \right)_\infty}
{\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty
\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty} \notag \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, q \beta; q \right)_\infty
+
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. \label{part2} $$
The first line on the right-hand side of can be rewritten in terms of the theta function through Jacobi’s triple product identity as $$\begin{aligned}
\frac{1}{2}
\frac
{\left( q, q; q^2 \right)_\infty}
{\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty
\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty}
&=
\frac{1}{2}
( q, q; q^2 )_\infty
\frac{( q^2; q^2 )_\infty}{\theta_{q^2} \left( \frac{\alpha q^{1/2}}{\gamma w} \right)}
\frac{( q^2; q^2 )_\infty}{\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)} \notag \\ &=
\frac{1}{2}
\frac
{( q, q; q )_\infty}
{\theta_{q^2} \left( \frac{\alpha q^{1/2}}{\gamma w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)} \notag \\ &=
\frac{1}{2}
\frac
{( q, q; q )_\infty}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} \right)} \notag \\ &=
\frac
{( q, q; q )_\infty}
{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
+
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}, \label{part12}\end{aligned}$$ where we use $\beta {\gamma}= q$ and Corollary \[co5\]. Similarly, Jacobi’s triple product identity can also be applied to the second line of so that $$\begin{aligned}
&
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, \beta q; q \right)_\infty
+
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\} \notag \\ &=
\frac{\theta_q \left( \frac{\beta \gamma}{q} w \right)}{( q; q )_\infty}
\frac{\theta_q \left( \frac{1}{\beta q^{1/2}} \right)}{( q; q )_\infty}
+
\frac{\theta_q \left( \frac{\gamma w}{\alpha \beta q} \right)}{( q; q )_\infty}
\frac{\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)}{( q; q )_\infty} \notag \\ &=
\frac{1}{( q, q; q)_\infty}
\left\{
\theta_q \left( \frac{\beta \gamma}{q} w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)
+
\theta_q \left( \frac{\gamma w}{\alpha \beta q} \right)
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\right\} \notag \\ &=
\frac{1}{( q, q; q)_\infty}
\left\{
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)
+
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\right\}, \label{part22}\end{aligned}$$ where we again insert the condition $\beta {\gamma}= q$ in the last equality. Combining and together, we obtain the relation .
From Corollary \[co6\], we can immediately state the following identity:
\[co7\] For any $w \in {\mathbb{C}}^*$, we obtain $$\begin{aligned}
\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty
&=
\frac{1}{2}
\frac{\left( q, q; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty} \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, q \beta; q \right)_\infty
+
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w},\alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. $$
Finally, substituting Corollary \[co7\] into results in $$\begin{aligned}
( \text{the left-hand side of \eqref{m1}} )
&=
\frac{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \alpha^2 \beta^2; q^2 \right)_\infty}
\frac{1}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}
\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty \\ &=
( \text{the right-hand side of \eqref{m1}} ). $$ Therefore, Theorem \[mainth1\] which we would like to show in the paper is proven.
Proof of Theorem \[mainth2\]
----------------------------
Let us turn to implementing the proof of Theorem \[mainth2\]. The basic process to prove it is the same of which we made use in the case of Theorem \[mainth1\]. Hence, in the following, we omit some middle steps which are already written down in the previous subsection.
Firstly, as for the path from to , we take Ramanujan’s summation formula to re-xpress the left-hand side of as $$\begin{aligned}
\label{proof3} &
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac{1}{2}
\frac{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \alpha^2 \beta^2; q^2 \right)_\infty}
\frac{1}{\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty}
\frac{\left( q, q; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty} \notag \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, \beta q; q \right)_\infty
-
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. $$
Secondly, we would like to show the following relation:
\[coro1\] For any $w \in {\mathbb{C}}^*$, we obtain $$\begin{aligned}
\alpha \beta
&=
\frac{1}{2}
\frac
{\left( q, q; q^2 \right)_\infty}
{\left( \frac{\alpha q^{1/2}}{{\gamma}w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty
\left( \frac{{\gamma}w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty} \notag \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta {\gamma}w}{q^{1/2}}, \frac{q^{3/2}}{\beta {\gamma}w}, \frac{1}{\beta}, \beta q; q \right)_\infty
-
\left( \frac{{\gamma}w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. \label{part2b} $$
We again rely on Jacobi’s triple product identity to rewrite the right-hand side of . Its first line is given by means of the theta function with base $q$ as follows: $$\begin{aligned}
\frac{1}{2}
\frac
{\left( q, q; q^2 \right)_\infty}
{\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty
\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty}
&=
\frac{1}{2}
( q, q; q^2 )_\infty
\frac{( q^2; q^2 )_\infty}{\theta_{q^2} \left( \frac{\alpha q^{1/2}}{\gamma w} q \right)}
\frac{( q^2; q^2 )_\infty}{\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)} \notag \\ &=
\frac{1}{2}
\frac
{( q, q; q )_\infty}
{\theta_{q^2} \left( \frac{\alpha \beta}{q^{1/2} w} q \right)
\theta_{q^2} \left( \frac{\alpha \beta w}{q^{1/2}} q \right)} \notag \\ &=
\alpha \beta
\frac
{( q, q; q )_\infty}
{\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)
-
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)}, \label{part12b}\end{aligned}$$ where we use $\beta {\gamma}= q$ and Corollary \[corol1\]. Also, the second line of become the linear combination of the products of the theta function thanks to Jacobi’s triple product identity , $$\begin{aligned}
&
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, \beta q; q \right)_\infty
-
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w}, \alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\} \notag \\ &=
\frac{1}{( q, q; q)_\infty}
\left\{
\theta_q \left( w \right)
\theta_q \left( \frac{1}{\beta q^{1/2}} \right)
-
\theta_q \left( \frac{w}{\alpha \beta^2} \right)
\theta_q \left( \frac{\alpha \beta}{q^{1/2}} \right)
\right\}, \label{part22b}\end{aligned}$$ where we again put the condition $\beta {\gamma}= q$. Combining and gives us the relation .
In addition, we can claim the following identity from Corollary \[coro1\]:
\[coro2\] For any $w \in {\mathbb{C}}^*$, we have $$\begin{aligned}
\alpha \beta
\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty
&=
\frac{1}{2}
\frac{\left( q, q; q^2 \right)_\infty}{\left( \frac{\gamma w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty} \\ &\hspace{1em} \times
\left\{
\left( \frac{\beta \gamma w}{q^{1/2}}, \frac{q^{3/2}}{\beta \gamma w}, \frac{1}{\beta}, q \beta; q \right)_\infty
-
\left( \frac{\gamma w}{\alpha \beta q^{1/2}}, \frac{\alpha \beta q^{3/2}}{\gamma w},\alpha \beta, \frac{q}{\alpha \beta}; q \right)_\infty
\right\}. $$
Finally, we insert the relation of Corollary \[coro2\] into which is the left-hand side of , $$\begin{aligned}
( \text{the left-hand side of \eqref{m2}} )
&=
\alpha \beta
\frac{\left( \frac{{\gamma}}{\alpha^2 \beta}; q^2 \right)_\infty}{\left( \alpha^2 \beta^2; q^2 \right)_\infty}
\frac{1}{\left( \frac{{\gamma}w}{\alpha q^{1/2}} q, \frac{q^{1/2}}{\alpha \beta w} q; q^2 \right)_\infty}
\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty \\ &=
( \text{the right-hand side of \eqref{m2}} ). $$ Consequently, we can provide the complete proof of Theorem \[mainth2\].
Application to physics
----------------------
Let us consider the special cases of Theorem \[mainth1\] and \[mainth2\]. We obtain the following formulae by setting $\alpha = - a$, $\beta = - a^{- \frac{1}{2}}$ and $\gamma = - a^{\frac{1}{2}} q$:
\[cormain1\] For any $w \in \mathbb{C}^*$ such that $|q/a| < |q^{1/2} w/a^{1/2}| < 1$, we have $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty} \left\{
\frac{( a^{- \frac{1}{2}} q; q )_\infty}{( a^{\frac{1}{2}}; q )_\infty}
{}_1\psi_1 \left( a^{\frac{1}{2}}; a^{- \frac{1}{2}} q; q, q^{\frac{1}{2}} a^{- \frac{1}{2}} w \right)
+
\frac{( - a^{- \frac{1}{2}} q; q )_\infty}{( - a^{\frac{1}{2}}; q )_\infty}
{}_1\psi_1 \left( - a^{\frac{1}{2}}; - a^{- \frac{1}{2}} q; q, q^{\frac{1}{2}} a^{- \frac{1}{2}} w \right)
\right\} \\ &=
\frac
{( \tilde{a}^{-1} q, \tilde{a}^{\frac{1}{2}} \tilde{w}^{- 1} q^{\frac{1}{2}}, \tilde{a}^{\frac{1}{2}} \tilde{w} q^{\frac{1}{2}}; q^2 )_\infty}
{( \tilde{a}, \tilde{a}^{- \frac{1}{2}} \tilde{w} q^{\frac{1}{2}}, \tilde{a}^{- \frac{1}{2}} \tilde{w}^{- 1} q^{\frac{1}{2}}; q^2 )_\infty}, $$ where $a = \tilde{a}$ and $w = \tilde{w}$.
\[cormain2\] For any $w \in \mathbb{C}^*$ such that $|q/a| < |q^{3/2} w/a^{1/2}| < 1$, we have $$\begin{aligned}
&
\frac{1}{2}
\frac{( q; q^2 )_\infty}{( q^2; q^2 )_\infty} \left\{
\frac{( a^{- \frac{1}{2}} q; q )_\infty}{( a^{\frac{1}{2}}; q )_\infty}
{}_1\psi_1 \left( a^{\frac{1}{2}}; a^{- \frac{1}{2}} q; q, q^{\frac{1}{2}} a^{- \frac{1}{2}} w \right)
-
\frac{( - a^{- \frac{1}{2}} q; q )_\infty}{( - a^{\frac{1}{2}}; q )_\infty}
{}_1\psi_1 \left( - a^{\frac{1}{2}}; - a^{- \frac{1}{2}} q; q, q^{\frac{1}{2}} a^{- \frac{1}{2}} w \right)
\right\} \\ &=
\tilde{a}^{\frac{1}{2}}
\frac
{( \tilde{a}^{-1} q, \tilde{a}^{\frac{1}{2}} \tilde{w}^{- 1} q^{\frac{3}{2}}, \tilde{a}^{\frac{1}{2}} \tilde{w} q^{\frac{3}{2}}; q^2 )_\infty}
{( \tilde{a}, \tilde{a}^{- \frac{1}{2}} \tilde{w} q^{\frac{3}{2}}, \tilde{a}^{- \frac{1}{2}} \tilde{w}^{- 1} q^{\frac{3}{2}}; q^2 )_\infty}, $$ where $a = \tilde{a}$ and $w = \tilde{w}$.
Those are the identities firstly found from the physics viewpoint to explore the simplest version of Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$ [@MMT]. This symmetry states that two distinct supersymmetric gauge theories named the supersymmetric quantum electrodynamics (SQED) and the XYZ-model are physically equivalent. The left-hand and right-hand sides of Corollary \[cormain1\] and \[cormain2\] are the exact formulae of the superconformal indices of the SQED and the XYZ-model, respectively. $a$ and $w$ (resp. $\tilde{a}$ and $\tilde{w}$) are weight parameters called fugacities associated to U$(1)$ symmetries in the SQED (resp. the XYZ-model). The conditions $a = \tilde{a}$ and $w = \tilde{w}$ represents the correspondence of each U$(1)$ symmetry of two theories, which is expected from physical arguments. Therefore, Theorem \[mainth1\] and \[mainth2\] are the generalized formulae motivated by the physical results [@MMT], and those are interesting examples that purely physical discussions lead to mathematically nontrivial results.
The $q \to 1 - 0$ limit of the new formulae {#Limit}
===========================================
In this section, we would like to show the $q \to 1 - 0$ limit of our new formulae $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
+
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac{( q^2; q^2 )_\infty}{( q; q^2 )_\infty}
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q, \frac{\alpha \beta w}{q^{1/2}} q; q^2 \right)_\infty}
{\left( \frac{\gamma w}{\alpha q^{1/2}}, \frac{q^{1/2}}{\alpha \beta w}; q^2 \right)_\infty},
\label{col}\end{aligned}$$ and $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\alpha \beta
\frac{( q^2; q^2 )_\infty}{( q; q^2 )_\infty}
\frac
{\left( \frac{\gamma}{\alpha^2 \beta}; q^2 \right)_\infty}
{( \alpha^2 \beta^2; q^2 )}
\frac
{\left( \frac{\alpha q^{1/2}}{\gamma w} q^2, \frac{\alpha \beta w}{q^{1/2}} q^2; q^2 \right)_\infty}
{\left( \frac{\gamma w q^{1/2}}{\alpha}, \frac{q^{3/2}}{\alpha \beta w}; q^2 \right)_\infty}, \label{mcor22}\end{aligned}$$ where $\alpha \beta^2 = - 1$ and $\beta \gamma = q$. At first, we set $\alpha = q^a$, $\beta = q^b$ and $\gamma = q^c$, then we have $q^{a + b + 1 - c} = - 1$. We also introduce the weight function $$\begin{aligned}
W ( a, b, c; q )
:=
( 1 - q^2 )^{- \frac{4 a + 3 b - c - 1}{2}},
\label{weight}\end{aligned}$$ where $a$ is constrained by $a = - 2b$. This condition and relation $b + c = 1$ which is equivalent to $\beta {\gamma}= q$ imply $$\begin{aligned}
- \frac{4 a + 3 b - c - 1}{2} = 3 b + c.
\label{rellim}\end{aligned}$$ The aim of this section is to show the following theorem as the $q \to 1 - 0$ limit of the formulae and multiplied by the weight function $W ( a, b, c; q )$:
\[mainlim2\] For any $w \in \mathbb{C}^*$, we have $$\begin{aligned}
\frac{2^{2 b + 1}}{{\Gamma}( b + 1 )}
{}_1H_1 \left( - b; b + 1; w \right)
=
\frac
{{\Gamma}{\left(}\frac{1}{2} {\right)}}
{{\Gamma}{\left(}\frac{2 b + 1}{2} {\right)}}
( - w )^{- b}
( 1 - w )^{2 b}.\end{aligned}$$
Let us start with considering the left-hand side of multiplied by the weight function .
1. The first term on the left-hand side of can be expressed in terms of the $q$-gamma function, $$\begin{aligned}
&
( 1 - q^2 )^{3 b + c}
\frac{\left( q^{c - a}; q \right)_\infty}{( q^{a + b}; q )_\infty}
{}_1\psi_1 \left( q^{a + b}; q^{c - a}; q, q^{c - a - \frac{1}{2}} w \right) \notag \\ &=
(1 + q)^{3 b + c}
\cdot
( 1 - q )^{c - a - 1}
\frac{\left( q^{c - a}; q \right)_\infty}{( q; q )_\infty}
\cdot
( 1 - q )^{1 - ( a + b )}
\frac{( q; q )_\infty}{( q^{a + b}; q )_\infty}
\cdot
{}_1\psi_1 \left( q^{a + b}; q^{c - a}; q, q^{c - a - \frac{1}{2}} w \right) \notag \\ &=
(1 + q)^{3 b + c}
\frac{{\Gamma}_{q} ( a + b )}{{\Gamma}_{q} ( c - a )}
{}_1\psi_1 \left( q^{a + b}; q^{c - a}; q, q^{c - a - \frac{1}{2}} w \right) \notag \\ &=
(1 + q)^{2 b + 1}
\frac{{\Gamma}_{q} ( - b )}{{\Gamma}_{q} ( b + 1 )}
{}_1\psi_1 \left( q^{- b}; q^{b + 1}; q, q^{b + \frac{1}{2}} w \right). \notag $$ The $q \to 1 - 0$ limit of this part becomes $$\begin{aligned}
2^{2 b + 1}
\frac{{\Gamma}( - b )}{{\Gamma}( b + 1 )}
{}_1H_1 \left( - b; b + 1; w \right).
\label{limitp2}\end{aligned}$$
2. The second term on the left-hand side of is rewritten in the same manner as the first term, $$\begin{aligned}
&
( 1 - q^2 )^{3 b + c}
\frac{( q^{2 b + c}; q )_\infty}{( q^{- b}; q )_\infty}
{}_1\psi_1 \left( q^{- b}; q^{2 b + c}; q, q^{c - a - \frac{1}{2}} w \right) \notag \\ &=
(1 + q)^{3 b + c}
\frac{{\Gamma}_{q} ( - b )}{{\Gamma}_{q} ( 2 b + c )}
{}_1\psi_1 \left( q^{- b}; q^{2 b + c}; q, q^{c - a - \frac{1}{2}} w \right) \notag \\ &=
(1 + q)^{2 b + 1}
\frac{{\Gamma}_{q} ( - b )}{{\Gamma}_{q} ( b + 1 )}
{}_1\psi_1 \left( q^{- b}; q^{b + 1}; q, q^{c - a - \frac{1}{2}} w \right), \notag $$ namely, the $q \to 1 - 0$ limit of this part also produces the contribution .
As a result, we can take the $q \to 1 - 0$ limit of the left-hand side of as $$\begin{aligned}
( \text{the left-hand side of \eqref{col}} )
&\to
\frac{1}{2}
\cdot
2
\cdot
2^{2 b + 1}
\frac{{\Gamma}( - b )}{{\Gamma}( b + 1 )}
{}_1H_1 \left( - b; b + 1; w \right) \notag \\ &=
2^{2 b + 1}
\frac{{\Gamma}( - b )}{{\Gamma}( b + 1 )}
{}_1H_1 \left( - b; b + 1; w \right). \label{limitp1}\end{aligned}$$
On the other hand, the right-hand side multiplied by the weight function can be rewritten by using, in addition to $a = - 2b$ and $b + c = 1$, the $q$-gamma function and the theta function with base $q^2$ as follows: $$\begin{aligned}
&
W ( a, b, c; q )
\frac
{\left( q^2; q^2 \right)_\infty}
{{\left(}q; q^2 {\right)}}
\frac
{\left( q^{c - 2 a - b}; q^2 \right)_\infty}
{{\left(}q^{2 ( a + b )}; q^2 {\right)}}
\frac
{\left( q^{a - c + \frac{3}{2}} w^{- 1}, q^{a + b + \frac{1}{2}} w; q^2 \right)_\infty}
{\left( q^{c - a - \frac{1}{2}} w, q^{- a - b + \frac{1}{2}} w^{- 1}; q^2 \right)_\infty} \notag \\ &=
( 1 - q^2 )^{\frac{1}{2}}
\frac
{\left( q^2; q^2 \right)_\infty}
{{\left(}( q^2 )^{\frac{1}{2}}; q^2 {\right)}}
\cdot
( 1 - q^2 )^{\frac{1}{2} ( c - 2 a - b ) - 1}
\frac
{\left( ( q^2 )^{\frac{1}{2} ( c - 2 a - b )}; q^2 \right)_\infty}
{{\left(}q^2; q^2 {\right)}}
\cdot
( 1 - q^2 )^{1 - ( a + b )}
\frac
{\left( q^2; q^2 \right)_\infty}
{{\left(}( q^2 )^{a + b}; q^2 {\right)}}
\notag \\ &\hspace{1em} \times
{\left\{}
\newcommand{\rc}{\right\}}\frac
{\left( q^{a - c + \frac{3}{2}} w^{- 1}; q^2 \right)_\infty}
{\left( q^{- a - b + \frac{1}{2}} w^{- 1}; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{- a + c + \frac{1}{2}} w, q^2; q^2 {\right)}_\infty}
{{\left(}q^{a + b + \frac{3}{2}} w, q^2; q^2 {\right)}_\infty}
\rc
\cdot
\frac
{\left( q^{a + b + \frac{1}{2}} w; q^2 \right)_\infty}
{\left( q^{c - a - \frac{1}{2}} w; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{a + b + \frac{3}{2}} w, q^2; q^2 {\right)}_\infty}
{{\left(}q^{- a + c + \frac{1}{2}} w, q^2; q^2 {\right)}_\infty} \notag \\ &=
\frac
{{\Gamma}_{q^2} {\left(}\frac{1}{2} {\right)}{\Gamma}_{q^2} {\left(}a + b {\right)}}
{{\Gamma}_{q^2} {\left(}\frac{c - 2 a - b}{2} {\right)}}
{\left\{}
\newcommand{\rc}{\right\}}\frac
{\theta_{q^2} {\left(}q^{- a + c - \frac{1}{2}} w {\right)}}
{\theta_{q^2} {\left(}q^{a + b + \frac{1}{2}} w {\right)}}
\rc \notag \\ &\hspace{1em} \times
\frac
{\left( q^{a + b + \frac{1}{2}} w; q^2 \right)_\infty}
{\left( w; q^2 \right)_\infty}
\frac
{\left( w; q^2 \right)_\infty}
{\left( q^{c - a - \frac{1}{2}} w; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{a + b + \frac{3}{2}} w, q^2; q^2 {\right)}_\infty}
{{\left(}w; q^2 {\right)}_\infty}
\frac
{{\left(}w; q^2 {\right)}_\infty}
{{\left(}q^{- a + c + \frac{1}{2}} w, q^2; q^2 {\right)}_\infty} \notag \\ &=
\frac
{{\Gamma}_{q^2} {\left(}\frac{1}{2} {\right)}{\Gamma}_{q^2} {\left(}- b {\right)}}
{{\Gamma}_{q^2} {\left(}\frac{2 b + 1}{2} {\right)}}
\frac
{\theta_{q^2} {\left(}q^{b + \frac{1}{2}} w {\right)}}
{\theta_{q^2} {\left(}q^{- b + \frac{1}{2}} w {\right)}} \notag \\ &\hspace{1em} \times
\frac
{\left( q^{- b + \frac{1}{2}} w; q^2 \right)_\infty}
{\left( w; q^2 \right)_\infty}
\frac
{\left( w; q^2 \right)_\infty}
{\left( q^{b + \frac{1}{2}} w; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{- b + \frac{3}{2}} w; q^2 {\right)}_\infty}
{{\left(}w; q^2 {\right)}_\infty}
\frac
{{\left(}w; q^2 {\right)}_\infty}
{{\left(}q^{b + \frac{3}{2}} w; q^2 {\right)}_\infty}. \label{limitp4}\end{aligned}$$ As $q \to 1 - 0$, the limiting formulae , , and reduce the function to $$\begin{aligned}
( \text{the right-hand side of \eqref{col}} )
&\to
\frac
{{\Gamma}{\left(}\frac{1}{2} {\right)}{\Gamma}{\left(}- b {\right)}}
{{\Gamma}{\left(}\frac{2 b + 1}{2} {\right)}}
( - w )^{- b}
( 1 - w )^{\frac{1}{2} {\left\{}
\newcommand{\rc}{\right\}}{\left(}b - \frac{1}{2} {\right)}+ {\left(}b + \frac{1}{2} {\right)}+ {\left(}b - \frac{3}{2} {\right)}+ {\left(}b + \frac{3}{2} {\right)}\rc} \notag \\ &=
\frac
{{\Gamma}{\left(}\frac{1}{2} {\right)}{\Gamma}{\left(}- b {\right)}}
{{\Gamma}{\left(}\frac{2 b + 1}{2} {\right)}}
( - w )^{- b}
( 1 - w )^{2 b}. \label{limitp5}\end{aligned}$$ Combining the results and , we obtain Theorem \[mainlim2\] as a conclusion.
Further, we can conclude that in the $q \to 1 - 0$ limit leads to Theorem \[mainlim2\].
First of all, we rely on $\alpha \beta^2 = - 1$ on the left-hand side of such that $$\begin{aligned}
&
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
-
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\} \notag \\ &=
\frac{1}{2}
\left\{
\frac{\left( \frac{\gamma}{\alpha}; q \right)_\infty}{( \alpha \beta; q )_\infty}
{}_1\psi_1 \left( \alpha \beta; \frac{\gamma}{\alpha}; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
+
\alpha \beta^2
\frac{( \beta^2 \gamma; q )_\infty}{( \frac{1}{\beta}; q )_\infty}
{}_1\psi_1 \left( \frac{1}{\beta}; \beta^2 \gamma; q, \frac{\gamma w}{\alpha q^{1/2}} \right)
\right\}. \label{limitn1}\end{aligned}$$ Because the only difference of from the left-hand side of is the presence of factor $\alpha \beta^{2} = q^{a + 2 b}$ in the second term, we can deform multiplied by the weight function in the same way as to obtain in the $q \to 1 - 0$ limit, thus, $$\begin{aligned}
( \text{the left-hand side of \eqref{mcor22}} )
\to
2^{2 b + 1}
\frac{{\Gamma}( - b )}{{\Gamma}( b + 1 )}
{}_1H_1 \left( - b; b + 1; w \right).
\label{limitn2}\end{aligned}$$
Let us turn to the right-hand side of multiplied by the weight function , $$\begin{aligned}
&
W ( a, b, c; q )
q^{a + b}
\frac
{\left( q^2, q^{c - 2 a - b}; q^2 \right)_\infty}
{{\left(}q, q^{2 ( a + b )}; q^2 {\right)}}
\frac
{\left( q^{a - c + \frac{5}{2}} w^{- 1}, q^{a + b + \frac{3}{2}} w; q^2 \right)_\infty}
{\left( q^{c - a + \frac{1}{2}} w, q^{- a - b + \frac{3}{2}} w^{- 1}; q^2 \right)_\infty} \notag \\ &=
q^{a + b}
\frac
{{\Gamma}_{q^2} {\left(}\frac{1}{2} {\right)}{\Gamma}_{q^2} {\left(}a + b {\right)}}
{{\Gamma}_{q^2} {\left(}\frac{c - 2 a - b}{2} {\right)}}
\frac
{\theta_{q^2} {\left(}q^{- a + c - \frac{3}{2}} w {\right)}}
{\theta_{q^2} {\left(}q^{a + b - \frac{1}{2}} w {\right)}} \notag \\ &\hspace{1em} \times
\frac
{\left( q^{a + b + \frac{3}{2}} w; q^2 \right)_\infty}
{\left( w; q^2 \right)_\infty}
\frac
{\left( w; q^2 \right)_\infty}
{\left( q^{c - a + \frac{1}{2}} w; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{a + b + \frac{1}{2}} w, q^2; q^2 {\right)}_\infty}
{{\left(}w; q^2 {\right)}_\infty}
\frac
{{\left(}w; q^2 {\right)}_\infty}
{{\left(}q^{- a + c - \frac{1}{2}} w, q^2; q^2 {\right)}_\infty} \notag \\ &=
q^{a + b}
\frac
{{\Gamma}_{q^2} {\left(}\frac{1}{2} {\right)}{\Gamma}_{q^2} {\left(}- b {\right)}}
{{\Gamma}_{q^2} {\left(}\frac{2 b + 1}{2} {\right)}}
\frac
{\theta_{q^2} {\left(}q^{b - \frac{1}{2}} w {\right)}}
{\theta_{q^2} {\left(}q^{- b - \frac{1}{2}} w {\right)}} \notag \\ &\hspace{1em} \times
\frac
{\left( q^{- b + \frac{3}{2}} w; q^2 \right)_\infty}
{\left( w; q^2 \right)_\infty}
\frac
{\left( w; q^2 \right)_\infty}
{\left( q^{b + \frac{3}{2}} w; q^2 \right)_\infty}
\cdot
\frac
{{\left(}q^{- b + \frac{1}{2}} w; q^2 {\right)}_\infty}
{{\left(}w; q^2 {\right)}_\infty}
\frac
{{\left(}w; q^2 {\right)}_\infty}
{{\left(}q^{b + \frac{1}{2}} w; q^2 {\right)}_\infty}. \label{limitn4}\end{aligned}$$ Again, the limiting formulae , , and provide $$\begin{aligned}
( \text{the right-hand side of \eqref{mcor22}} )
\to
\frac
{{\Gamma}{\left(}\frac{1}{2} {\right)}{\Gamma}{\left(}- b {\right)}}
{{\Gamma}{\left(}\frac{2 b + 1}{2} {\right)}}
( - w )^{- b}
( 1 - w )^{2 b}.
\label{limitn5}\end{aligned}$$ The expressions and are completely identical with and , respectively. Therefore, the $q \to 1 - 0$ limit of also results in Theorem \[mainlim2\].
Finally, we would like to comment on the physical perspective of Theorem \[mainlim2\]. As we mentioned, the new formula shown in this paper strongly supports Abelian mirror symmetry on $\mathbf{R}P^2 \times S^1$. The $q \to 1 - 0$ limit corresponds physically to the limit where the radius of $S^1$ is taken to be zero, and thus we expect that Theorem \[mainlim2\] may lead to the certain relationship of partition functions on $\mathbf{R}P^2$. We hope to provide the physical meaning to this purely mathematical conclusion as a future problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of H.M. was supported in part by the JSPS Research Fellowship for Young Scientists.
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|
---
abstract: 'Starting from topological quantum field theory, we derive space-time uncertainty relation with respect to the time interval and the spatial length proposed by Yoneya through breakdown of topological symmetry in the large N matrix model. This work suggests that the topological symmetry might be an underlying higher symmetry behind the space-time uncertainty principle of string theory.'
---
=cmr5 \#1[\#1\^[\^]{}]{}
\#1
EDO-EP-15\
September, 1997\
[**Space-time Uncertainty Principle from\
Breakdown of Topological Symmetry** ]{}
Ichiro Oda [^1]\
Edogawa University, 474 Komaki, Nagareyama City, Chiba 270-01, JAPAN\
Introduction
============
In spite of recent remarkable progress in nonperturbative formulations of M-theory [@M] and IIB superstring [@IKKT; @FKKT], we have not yet reached a complete understanding of the fundamental principle and the underlying symmetry behind string theory. Since a string has an infinite number of states in the perturbative level in addition to various extended objects as solitonic excitations in the nonperturbative regime, it is expected that in string theory the fundamental principle might have peculiar properties and the gauge symmetry would be quite huge compared to the usual ones in particle theory.
In a quest of the fundamental principle of string theory, Yoneya has advocated a space-time uncertainty principle which is of the form [@Y1; @Y2] $$\begin{aligned}
\Delta T \Delta X
\ge l_s^2,
\label{1.1}\end{aligned}$$ where $l_s$ denotes the string minimum length which is related to the Regge slope $\alpha'$ by $l_s = \sqrt{\alpha'}$. Provided that the relation (1) holds literally, string theory would lead a physical picture that space-time in itself is quantized at the short distance and the concept of space-time as a continuum manifold cannot be extrapolated beyond the fundamental string scale $l_s$. In more recent work [@Y2], making use of the “conformal constraint” which stems from the Schild action [@Schild] and essentially expresses the space-time uncertainty priciple (1), Yoneya has constructed a IIB matrix model from which the IKKT model [@IKKT] can be interpreted as an effective theory for D-branes [@Pol].
In this article, we would like to consider the space-time uncertainty principle from a different perspective from Yoneya’s one. Namely, we attempt to understand at least some aspects of the underlying gauge symmetry of string theory through the relation (1). We will see that topological quantum field theory [@Witten1] provides us with a nice framework in understanding the space-time uncertainty principle (1). It is worthwhile to point out that it has been already stated that a topological symmetry might be of critical importance in both string theory and quantum gravity in connection with the background independent formulation of string theory and the unbroken phase of quantum gravity [@Witten2]. Our study at hand may shed a light on this idea to some extent.
The paper is organized as follows. In section 2 we review Yoneya’s works [@Y2] relating to the present study. In section 3, we derive the space-time uncertainty principle from the topological field theory where the classical action is trivially zero. The final section is devoted to discussions.
The space-time uncertainty principle and\
the conformal constraint
==========================================
In this section, we review only a part of Yoneya’s works relevant to later study (See [@Y1; @Y2] for more detail). Let us start with the Schild action [@Schild] of a bosonic string. Then the Schild action has a form $$\begin{aligned}
S_{Schild} = -\frac{1}{2} \int d^2 \xi \left[\ -\frac{1}{2 \lambda^2}
\frac{1}{e} \left( \varepsilon^{ab} \partial_a X^\mu \partial_b
X^\nu \right)^2 + e \ \right],
\label{2.1}\end{aligned}$$ where $X^\mu (\xi)$ $(\mu = 0, 1, \ldots , D-1)$ are space-time coordinates, $e(\xi)$ is a positive definite scalar density defined on the string world sheet parametrized by $\xi^1$ and $\xi^2$, and $\lambda = 4\pi \alpha'$.
Taking the variation with respect to the auxiliary field $e(\xi)$, one obtains $$\begin{aligned}
e(\xi) = \frac{1}{\lambda} \sqrt{-\frac{1}{2}
\left( \varepsilon^{ab} \partial_a X^\mu \partial_b
X^\nu \right)^2 },
\label{2.2}\end{aligned}$$ which is also rewritten to be $$\begin{aligned}
\lambda^2 = -\frac{1}{2}
\left\{ X^\mu, X^\nu \right\}^2,
\label{2.3}\end{aligned}$$ where one has introduced the diffeomorphism invariant Poisson bracket defined as $$\begin{aligned}
\left\{ X^\mu, X^\nu \right\} =
\frac{1}{e(\xi)} \varepsilon^{ab} \partial_a X^\mu \partial_b
X^\nu.
\label{2.4}\end{aligned}$$ Then eliminating the auxiliary field $e(\xi)$ through (3) from (2) and using the identity $$\begin{aligned}
- \det \partial_a X \cdot \partial_b X
= -\frac{1}{2} \left( \varepsilon^{ab} \partial_a X^\mu \partial_b
X^\nu \right)^2,
\label{2.5}\end{aligned}$$ the Schild action (2) becomes at least classically equivalent to the famous Nambu-Goto action $S_{NG}$ $$\begin{aligned}
S_{Schild} &=& -\frac{1}{\lambda} \int d^2 \xi \sqrt{- \det
\partial_a X \cdot \partial_b X},{\nonumber}\\
&=& S_{NG}.
\label{2.6}\end{aligned}$$ Let us note that the “conformal” constraint (4) describes half the classical Virasoro conditions [@Y2] and the well-known relation between the Poisson bracket and the commutation relation in the large $N$ matrix model $$\begin{aligned}
\left\{ A, B \right\} \longleftrightarrow
\left[ A, B \right],
\label{2.7}\end{aligned}$$ leads the “conformal” constraint (4) to $$\begin{aligned}
\lambda^2 = -\frac{1}{2}
\left[ X^\mu, X^\nu \right]^2.
\label{2.8}\end{aligned}$$ Then it turns out that the commutation relation (9) yields the space-time uncertainty principle (1) [@Y2]. Here notice that it is not the whole Schild action but the “conformal” constraint in the large $N$ matrix model that produces the space-time uncertainty principle so that it is a natural next step to seek the fundamental action yielding the relation (9). Actually, Yoneya has derived such an action which has a close connection with the IKKT model \[5\]. His construction of the action is in itself quite interesting but seem to be a bit ambiguous. In particular, a natural question arises whether or not we can derive the relation (9) in terms of the more field-theoretic framework where we usually start with a classical action with some local symmetry. In the following section, we shall challenge this problem of which we will see an interesting possibility that the breakdown of a topological symmetry gives a generation of the quantum action including the essential content of the space-time uncertainty principle.
A topological model
=====================
Let us start by considering a topological theory [@Witten1] where the classical action is trivially zero but dependent on the fields $X^\mu(\xi)$ and $e(\xi)$ as follows: $$\begin{aligned}
S_{c} = S_{c}(X^\mu(\xi), e(\xi)) = 0.
\label{3.1}\end{aligned}$$ The BRST transformations corresponding to the topological symmetry are given by $$\begin{aligned}
\delta_B X^\mu = \psi^\mu, \ \delta_B \psi^\mu = 0, {\nonumber}\\
\delta_B e = e \ \eta, \ \delta_B \eta = 0, {\nonumber}\\
\delta_B \bar{c} = b, \ \delta_B b = 0,
\label{3.2}\end{aligned}$$ where $\psi^\mu$ and $\eta$ are ghosts, and $\bar{c}$ and $b$ are respectively an antighost and an auxiliary field. Note that these BRST transformations are obviously nilpotent. Also notice that the BRST transformation $\delta_B e$ shows the character as a scalar density of $e$.
The idea, then, is to fix partially the topological symmetry corresponding to $\delta_B e$ by introducing an appropriate covariant gauge condition. A conventional covariant and nonsingular gauge condition would be $e = 1$ but this gauge choice is not suitable for the present purpose since it makes difficult to move to the large $N$ matrix theory. Then almost unique choice up to its polynomial forms is nothing but the “conformal” condition (4). Hence the quantum action defined as $S_q = \int d^2
\xi \ e L_q$ becomes $$\begin{aligned}
L_q &=& \frac{1}{e} \delta_B \left[ \bar{c} \left\{ e \left(
\frac{1}{2} \left\{ X^\mu, X^\nu \right\}^2 + \lambda^2 \right)
\right\} \right],{\nonumber}\\
&=& b \left( \frac{1}{2} \left\{ X^\mu,
X^\nu \right\}^2 + \lambda^2 \right)
- \bar{c} \left( \eta \left( -\frac{1}{2} \left\{ X^\mu,
X^\nu \right\}^2 + \lambda^2 \right)
+ 2 \left\{ X^\mu, X^\nu \right\} \left\{ X^\mu, \psi^\nu
\right\} \right),
\label{3.3}\end{aligned}$$ where the BRST transformations (11) were used.
What is necessary to obtain a stronger form of the space-time uncertainty relation (9) is to move to the large $N$ matrix theory where in addition to (8) we have the following correspondences $$\begin{aligned}
\int d^2 \xi \ e \longleftrightarrow Trace,{\nonumber}\\
\int {\it D} e \longleftrightarrow \sum_{n=1}^\infty,
\label{3.4}\end{aligned}$$ where the trace is taken over $SU(n)$ group. These correspondences can be justified by expanding the hermitian matices by $SU(n)$ generators in the large $N$ limit as is reviewed by the reference [@F]. Here it is worth commenting one important point. As in the IKKT model [@IKKT] the matrix size $n$ is now regarded as a dynamical variable so that the partition function includes the summation over $n$. Even if the direct proof is missing, the summation over $n$ is expected to recover the path integration over $e(\xi)$. In fact, the authors of the reference [@FKKT] have recently shown that the model of Fayyazuddin et al. [@F] where a positive definite hermitian matrix $Y$ is introduced as a dynamical variable instead of $n$, belongs to the same universality class as the IKKT model [@IKKT] owing to irrelevant deformations of the loop equation [@FKKT]. Thus we think that the correspondences (8) and (13) are legitimate even in the context at hand.
Now in the large $N$ limit, we have $$\begin{aligned}
S_q = Tr \left( b \left( \frac{1}{2} \left[ X^\mu,
X^\nu \right]^2 + \lambda^2 \right)
- \bar{c} \left\{ \eta \left( - \frac{1}{2} \left[ X^\mu,
X^\nu \right]^2 + \lambda^2 \right)
+ 2 \left[ X^\mu, X^\nu \right] \left[ X^\mu, \psi^\nu \right]
\right\} \right).
\label{3.5}\end{aligned}$$ Next by redefining the auxiliary field $b$ by $b + \bar{c} \ \eta$, $S_q$ can be cast into a simpler form $$\begin{aligned}
S_q = Tr \left( b \left( \frac{1}{2} \left[ X^\mu,
X^\nu \right]^2 + \lambda^2 \right)
- 2 \lambda^2 \bar{c} \ \eta - 2 \bar{c} \left[ X^\mu, X^\nu \right]
\left[ X^\mu, \psi^\nu \right] \right).
\label{3.6}\end{aligned}$$ Then the partition function is defined as $$\begin{aligned}
Z &=& \int {\it D}X^\mu {\it D}\psi^\mu {\it D}e {\it D}\eta
{\it D}\bar{c} {\it D}b \ e^{- S_q},{\nonumber}\\
&=& \sum_{n=1}^\infty \int {\it D}X^\mu {\it D}\psi^\mu {\it D}\eta
{\it D}\bar{c} {\it D}b \ e^{- S_q}.
\label{3.7}\end{aligned}$$ At this stage, it is straightforward to perform the path integration over $\eta$ and $\bar{c}$, as a result of which one obtains $$\begin{aligned}
Z = \sum_{n=1}^\infty \int {\it D}X^\mu {\it D}\psi^\mu {\it D}b \
e^{- Tr \ b \ \left( \frac{1}{2} \left[ X^\mu, X^\nu \right]^2
+ \lambda^2 \right)}.
\label{3.8}\end{aligned}$$ In (17) there remains the gauge symmetry $$\begin{aligned}
\delta \psi^\mu = \omega^\mu,
\label{3.9}\end{aligned}$$ which is of course the remaining topological symmetry. Now let us factor out this gauge volume or equivalently fix this gauge symmetry by the gauge condition $\psi^\mu = 0$, so that the partition function is finally given by $$\begin{aligned}
Z = \sum_{n=1}^\infty \int {\it D}X^\mu {\it D}b \
e^{- Tr \ b \ \left( \frac{1}{2} \left[ X^\mu, X^\nu \right]^2
+ \lambda^2 \right)}.
\label{3.10}\end{aligned}$$ It is remarkable that the variation with respect to the auxiliary variable $b$ in (19) gives a stronger form of the space-time uncertainty relation (9) and the theory is “dynamical” in the sense that the ghosts have completely decoupled from (19). In other words, we have shown how to derive the space-time uncertainty principle from a topological theory through the breakdown of a topological symmetry in the large $N$ matrix model. Why has the topological theory yielded the nontrivial “dynamical” theory? The reason is very much simple. In moving from the continuous theory (12) to the matrix theory (14), the dynamical degree of freedom associated with $e(\xi)$ was replaced by the discrete sum over $n$, on the other hand, the corresponding BRST partner $\eta$ remains the continuous variable. This distinct treatment of the BRST doublet leads to the breakdown of the topological symmetry giving rise to a “dynamical” matrix theory. In this respect, it is worthwhile to point out that while the topological symmetry is “spontaneously” broken, the other gauge symmetries never be violated in the matrix model (Of course, correctly speaking, these gauge symmetries reduce to the global symmetries in the matrix model but this is irrelevant to the present argument.) Moreover, notice that the above-examined phenomenon is a peculiar feature in the matrix model with the scalar density $e(\xi)$, which means that an existence of the gravitational degree of freedom is an essential ingredient.
Discussions
=============
In this short article, we have investigated a possibility of the space-time uncertainty principle advocated by Yoneya [@Y1; @Y2] to be derived from the topological field theory [@Witten1]. The study at hand suggests that the underlying symmetry behind this principle in string theory might be a topological symmetry as mentioned before in a different context [@Witten2]. This rather unexpected appearance of the topological field theory seems to be plausible from the following arguments. Suppose that we live in the world where the topological symmetry is exactly valid. Then we have no means of measuring the distance owing to lack of the metric tensor field so that there is neither concept of distance nor the space-time uncertainty principle. If the topological symmetry, in particular, that associated with the gravitational field, is spontaneously broken by some dynamical mechanism, an existence of the dynamical metric together with a string would give us both concept of distance and the space-time uncertainty principle.
So far we have not paid attention to the number of the space-time dimensions so much except the implicit assumption $D \ge 2$. An intriguing case is $D = 2$ even if this specification is not always necessary within the present formulation. In this special dimension, the Nambu-Goto action which is at least classically equivalent to the Schild action as shown in (7) becomes not only the topological field theory but also almost a surface term as follows: $$\begin{aligned}
\sqrt{- \det \partial_a X \cdot \partial_b X}
&=& \sqrt{- \left( \det \partial_a X^\mu \right)^2}, {\nonumber}\\
&=& \pm \det \partial_a X^\mu, {\nonumber}\\
&=& \mp \frac{1}{2} \varepsilon^{ab} \varepsilon_{\mu\nu}
\partial_a X^\mu \partial_b X^\nu,
\label{4.1}\end{aligned}$$ where we have assumed a smooth parametrization of $X^\mu$ over $\xi^a$ in order to take out the absolute value. Actually, this topological model has been investigated to some extent in the past [@Fuji; @Roberto; @Oda]. In this case, it is interesting that we can start with the nonvanishing surface term as a classical action.
One of the most important problems in future is to understand the symmetry breaking mechanism of a topological symmetry proposed in this paper more clearly by physical picture. Another interesting problem is to introduce the spinors and construct a supersymmetric matrix model from the topological field theory. These problems will be reported in a separate publication.
1
[**Acknowledgement**]{}
The author thanks Y.Kitazawa and A.Sugamoto for valuable discussions. He is also indebted to M.Tonin for stimulating discussions and a kind hospitality at Padova University where most of parts of this study have been done. This work was supported in part by Grant-Aid for Scientific Research from Ministry of Education, Science and Culture No.09740212.
1
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[^1]: E-mail address: ioda@edogawa-u.ac.jp
|
---
abstract: 'We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset\mathbb{R}^2$ with directed and uniquely colored edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we show that one can construct a monomorphism from the sandpile group $G_{\Gamma_1}=\mathbb{Z}^{\Gamma_1}/\Delta(\mathbb{Z}^{\Gamma_1})$ on the domain (graph) $\Gamma_1=\hat{P}_1\cap\mathbb{Z}^2$ to the respective group on $\Gamma_2=\hat{P}_2\cap\mathbb{Z}^2$. We provide several examples of infinite series of such tilings with polyforms converging to $\mathbb{R}^2$, and thus the first definition of scaling-limits for the sandpile group on the plane. Additional results include an exact sequence relating sandpile configurations to harmonic functions, an alternative formula for the order of the sandpile group based on a basis for the module of integer-valued harmonic functions, and three examples of how to prove the existence of (cyclic) subgroups for infinite families of sandpile groups by constructing appropriate integer-valued harmonic functions. The main open question concerns if the scaling-limits of the sandpile group for different sequences of polyforms converging to $\mathbb{R}^2$ are isomorphic.'
author:
- 'Moritz Lang[^1]'
- Mikhail Shkolnikov
bibliography:
- 'Lang2019.bib'
title: Sandpile monomorphisms and limits
---
Introduction
============
Background
----------
Let $\bar{\Gamma}=\Gamma\cup\{s\}$ be the vertices of a finite connected (multi-)graph with sink $s$. Denote by $\partial\Gamma$ the boundary of $\Gamma$ – the set of all vertices adjacent to the sink. The standard discrete graph Laplacian $\bar{\Delta}_{\bar\Gamma}$ is then defined as the difference between the adjacency matrix of $\bar\Gamma$ and its degree/valency matrix. When we delete, from $\bar{\Delta}_{\bar\Gamma}$, the row and column corresponding to the sink, we obtain the reduced graph Laplacian $\Delta_\Gamma$. The sandpile group $G_\Gamma$ is then defined as the cokernel of $\Delta_\Gamma$ acting on $\mathbb{Z}^\Gamma$ [@Dhar1990; @Biggs1999; @Raza2014; @Alar2017], i.e. $$\begin{aligned}
G_\Gamma=\mathbb{Z}^\Gamma/\Delta_\Gamma(\mathbb{Z}^\Gamma).\end{aligned}$$ Note that the sandpile groups corresponding to different choices of the sink for the same graph $\bar{\Gamma}$ are isomorphic [@Cori2000]. Also note that the sandpile group was rediscovered several times and that it is, as a consequence, sometimes referred to as the critical group–based on the work of Biggs [@Biggs1999; @Biggs1999b]–or as the Jacobian and (sometimes) as the Picard group [@Bacher1997; @Biggs1997; @Baker2007].
The study of the sandpile group originated in the physical literature, and there mainly focuses on sandpile groups defined on finite connected domains of the standard square lattice $\mathbb{Z}^2$, i.e. on graphs $\bar\Gamma$ obtained from $\mathbb{Z}^2$ by contracting all vertices $\mathbb{Z}^2\setminus(\mathbb{Z}^2\cap P)$ outside of some finite open set $P\subset\mathbb{R}^2$ to the sink. The group naturally arises in the study of the sandpile model, a cellular automaton introduced by Bak, Tang and Wiesenfeld in 1987 [@Bak1987] as the first and archetypical example of a system showing self-organized criticality (SOC), a phenomenom which subsequently became important in several areas of physics, biology, geology and other fields (see [@Aschwanden2016] for a recent review). Shortly after the introduction of this cellular automaton, Dhar showed that its recurrent configurations form a group isomorphic to $G_\Gamma$, and laid the foundation for its analysis [@Dhar1990; @Dhar1995]. Due to this isomorphism, both $G_\Gamma$ as well as the group formed by the recurrent configurations of the sandpile model are commonly referred to as the sandpile group. This may cause some confusion, since the elements of $G_\Gamma$ rather correspond to the equivalence classes of recurrent configurations. For readers used to the notation of the literature on the sandpile model, we thus note that the distinction between transient and recurrent configurations does not apply when directly working with $G_\Gamma$. We also note that we denote the group operation by $+:G_\Gamma\times G_\Gamma\rightarrow G_\Gamma$, and not by $(.+.)^\circ$, with $(.)^\circ:\mathbb{Z}_{\geq 0}^\Gamma\rightarrow\{0,\ldots,3\}^\Gamma$ the relaxation operator [@Dhar1990].
The sandpile group, specifically when defined on domains of $\mathbb{Z}^2$, provides connections between various mathematical fields, including fractal geometry, graph theory and algebraic geometry (see below), tropical geometry [@Caracciolo2010; @Kalinin2016; @Kalinin2017; @Kalinin2018b; @Kalinin2019], domino tilings [@Florescu2015], and others. Via the so called “burning algorithm”, Dhar constructed bijections (in the category of sets) between the sandpile group and spanning trees [@Dhar1990; @Majumdar1992], and thus showed that the former is a refinement of the latter. Creutz was the first to study the recurrent configuration of the sandpile model corresponding to the identity on domains of $\mathbb{Z}^2$, and provided an iterative algorithm for its construction [@Creutz1990]. He found that, on many domains, this identity is composed of self-similar fractal patterns [@Creutz1990; @LeBorgne2002; @Holroyd2008; @Caracciolo2008; @Caracciolo2008b]; since these patterns appear to be remarkably similar on rectangular domains with the same aspect ratio, scaling limits for the sandpile identity have been conjectured [@Holroyd2008]. Recently, we have extended these conjectures and suggested that several scaling limits for each recurrent configuration exist, forming piecewise smooth “fractal movies” referred to as harmonic sandpile dynamics [@Lang2019]. Finally, based on an analogy of graphs and (discrete) Riemann surfaces established by Baker and Norine (including a Riemann-Roch theorem for graphs) [@Baker2007; @Baker2009], several connections between algebraic geometry and the study of sandpiles were established (see [@Perkinson2011] and [@Corry2018], p.65ff. and 191ff., for expositions). The most interesting connection, in the context of this article, is provided by (non-constant) harmonic morphisms between graphs, corresponding to holomorphic maps between surfaces, as these morphisms directly induce epimorphisms between the respective sandpile groups [@Baker2009] (see also [@Reiner2014; @Alfaro2012]).
Only for few infinite families of graphs, the structure of the respective sandpile groups has been (partly) determined, including complete graphs [@Lorenzini1991], complete multipartite graphs [@Jacobson2003], cycles (equivalent to domains of $\mathbb{Z}^1$) [@Lorenzini1991], thick cycles [@Alar2017] (see also [@Dhar1995]), wheels [@Biggs1999; @Norine2011], modified wheels [@Raza2014], wired regular trees [@Levine2009], thick trees [@Chen2007], polygon flowers [@Chen2019], nearly complete graphs [@Norine2011], threshold graphs [@Norine2011], Möbius ladders [@Pingge2006; @Deryagina2014], prism graphs/graphs $\mathcal{D}_n$ of the dihedral group [@Dartois2003; @Deryagina2014], and $n$-cubes [@Bai2003; @Alfaro2012]. While this list is certainly not complete, the decomposition of the sandpile group on domains of $\mathbb{Z}^2$, for which the sandpile model was originally defined [@Bak1987], is–to our knowledge–yet unknown. Numeric calculations of the order [@Dhar1990] or decomposition [@Dhar1995] of the sandpile group on small enough domains however indicate that the groups are in general “incompatible”, even when the domains have the same shape, in the sense that no group monomorphisms can exist between them. For example, the order of the sandpile group on a $3\times 3$ square domain is $2^{11}7^2$, while the one on a $5\times 5$ domain is $2^{18}3^55^211^213^2$. Recently, we have shown that the sandpile group can be considered as a discretization of a $|\partial\Gamma|$-dimensional torus, to which we refer to as the extended sandpile group [@Lang2019]. We have then derived epimorphisms from the extended sandpile group on a given domain to the corresponding group on a subdomain. On the level of the (usual) sandpile group, due to the discretization, this renormalization is however defined in the category of sets and in general only “approximates” group homeomorphism for sufficiently large domains [@Lang2019]. Under which conditions these “approximations” can be lifted to true group homeomorphisms, if possible at all, is yet unknown. This lack of known relationships in terms of homeomorphism between the sandpile groups on different domains of $\mathbb{Z}^2$ is in stark contrast to the role of the sandpile model as the archetypical example for self-organized criticality, given that the concept of criticality itself is based on the notion of scaling. Progress in this subject might also provide means by which the existence of the conjectured scaling limits of the sandpile identity [@Holroyd2008] and of other recurrent configurations [@Lang2019] might be proven.
Sandpile monomorphisms (main result)
------------------------------------
In this paper, we analyze the relationships between sandpile groups defined on different domains of the standard square lattice $\mathbb{Z}^2$. Specifically, given two domains $\Gamma_1, \Gamma_2\subset\mathbb{Z}^2$, $\Gamma_1\subseteq \Gamma_2$, our goal is to understand under which conditions group monomorphisms from $G_{\Gamma_1}$ to $G_{\Gamma_2}$ exist.
![A) Dark-gray points represent the vertices of the standard square lattice $\mathbb{Z}^2$, while the gray isoscele triangles correspond to $M$. B) The isocele triangles belonging to the $M$-polyform $P_1$ are highlighted by a green background. The black points and lines represent the vertices and edges of the graph $\Gamma(P_1)=\mathbb{Z}^2\cap P_1$ defined by $P_1$. The sides of the $M$-polyform are directed and colored, exemplifying the definition of $P_1^{DC}$. C) DC-tiling of a $M$-polyform $P_2$ (all colored isoscele triangles) by four copies of the $M$-polyform $P_1^{DC}$ from (B). The background of the tiles are colored green if they can be obtained from $P_1^{DC}$ by only translations and rotations, and blue if (additionally) reflections are required. Note that the graph $\Gamma(P_2)$ consists not only of the vertices and edges corresponding to the four tiles (black points and lines), but also of additional vertices lying on the common edges of pairs of tiles, and the edges connecting these vertices to the rest of the graph (red points and lines). []{data-label="fig:initialIllus"}](Figure1.pdf){width="90.00000%"}
To state our main result, we first introduce some notation. Let $M$ be the unique tiling of $\mathbb{R}^2$ by isosceles triangles with base length $1$ and height $\frac{1}{2}$ such that each vertex of $(\mathbb{Z}+0.5)^2$ coincides with the apecies of four triangles (Figure \[fig:initialIllus\]A). An $M$-polyform $P\subset M$ then consists of a finite connected subset of triangles in $M$ (Figure \[fig:initialIllus\]B). Note that $M$ is *not* the usual triangular tiling of the plane, that the corners and edges of its triangles do not form a lattice, and that $M$-polyforms thus differ from the usual definition of polyiamonds.
By a slight abuse of notation, we interpret each $M$-polyform $P$ to directly correspond to the open subset of $\mathbb{R}^2$ enclosed by its isosceles triangles, i.e. to the interior of $\bigcup_{m\in P}m$. To each $M$-polyform $P$, we then associate the domain $\Gamma(P)=\mathbb{Z}^2\cap P$. This domain is obtained from the standard square lattice $\mathbb{Z}^2$ as described in the Introduction, i.e. by contracting all vertices $\mathbb{Z}^2\setminus(\mathbb{Z}^2\cap P)$ to the sink (Figure \[fig:initialIllus\]B). We interchangeably denote by $G_\Gamma$ and $G_P$ the sandpile groups defined on the domain $\Gamma=\Gamma(P)$.
Denote by $P^{DC}$ the result of assigning directions and colors to the edges of an $M$-polyform $P$ such that each edge has a different color (Figure \[fig:initialIllus\]B). Given two $M$-polyforms $P_1$ and $P_2$, we say that $P_1$ DC-tiles $P_2$ if there exists a tiling $T^{P_1\rightarrow P_2}$ of $P_2$ by copies of $P_1^{DC}$ (allowing all transformations which correspond to automorphisms of $M$), such that every common edge of two adjacent tiles in $T^{P_1\rightarrow P_2}$ has the same color and direction (Figure \[fig:initialIllus\]C).
We can now state our main result:
\[theorem:main\] Let $P_1$ and $P_2$ be two convex $M$-polyforms, and assume that $P_1$ DC-tiles $P_2$. Then, there exists a group monomorphism $G_{P_1}\rightarrowtail G_{P_2}$ from $G_{P_1}$ to $G_{P_2}$.
In the proof of this theorem, we construct an explicit mapping $\mu(T^{P_1\rightarrow P_2})=(G_{P_1}\rightarrowtail G_{P_2})$ from $DC$-tilings to the corresponding sandpile group monomorphisms. We refer to Section \[proof:main\] for the details on the construction of this map, and here only discuss some of its properties. We note that the graph morphisms $\Gamma(P_2)\rightarrow\Gamma(P_1)$ induced by DC-tilings are in general not harmonic at the sink $s_2$ of $\Gamma(P_2)$ (Figure \[fig:initialIllus\]B&C), and that thus Theorem \[theorem:main\] is distinct from the theory on harmonic graph morphisms [@Baker2009].
Trivially, for two $M$-polyforms $P_1$ and $P_2$, there can exist more than one distinct DC-tiling of $P_2$ by $P_1$. For example, let the polyform $P$ describe a square with width $w$ and sides parallel to the standard axes of $\mathbb{R}^2$. Since the dihedral group $D_4$ of a square has order eight, there also exist eight different DC-tiling of $P$ by itself. For $w>2$, $\mu$ maps each of these tilings to a different automorphism of $G^{P}$, which directly correspond to the action of the respective element of $D_4$ on $\Gamma(P)$ (see proof of Theorem \[theorem:main\]). For $w=2$, the domain $\Gamma(P)$ however consists of only a single vertex, and all eight tilings are mapped to the trivial automorphism. Now, denote by $\hat{P}$ the result of extending a polyform $P$ by one triangle in $M$ adjacent to $P$ such that $\Gamma(\hat{P})=\Gamma(P)$. Then, there exist no DC-tilings of $\hat{P}$ by $P$, or vice versa. However, since $G^P=G^{\hat{P}}$, the set of automorphisms is non-empty. We thus conclude that the mapping $\mu$ is in general neither injective nor surjective.
Scaling-limits of the sandpile group
------------------------------------
Let ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ denote the poset of bounded convex $M$-polyforms, with $P_1\subseteq_{DC} P_2$ if there exists a DC-tiling $T^{P_1\rightarrow P_2}$ of the $M$-polyform $P_2$ by the $M$-polyform $P_1$ such that the position and orientation of one tile in $T^{P_1\rightarrow P_2}$ directly corresponds to $P_1^{DC}$, i.e. $P_1^{DC}\in T^{P_1\rightarrow P_2}$. We naturally identify ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ with its corresponding (small) category, with the (faithful) forgetful functor $U:{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}\rightarrow{{\ensuremath{\mathbf{Set}}\xspace}}$ to the category of sets mapping each $M$-polyform to its corresponding open subset of $\mathbb{R}^2$ and $\subseteq_{DC}$ to set inclusions.
Since the position and orientation of one tile uniquely identifies a $DC$-tiling (if it exists), the definition of ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ allows us to associate a $DC$-tiling $\nu(P_1\subseteq_{DC} P_2)\in\{T^{P_1\rightarrow P_2}\}$ to each morphism $P_1\subseteq_{DC} P_2$, i.e. the unique DC-tiling satisfying $P_1^{DC}\in T^{P_1\rightarrow P_2}$. We can then define the functor $F:{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}\rightarrow{{\ensuremath{\mathbf{Ab}}\xspace}}$ from ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ to the category ${{\ensuremath{\mathbf{Ab}}\xspace}}$ of abelian groups, with $F(P)=G_P$ and $F(P_1\subseteq_{DC} P_2)=\mu(\nu(P_1\subseteq_{DC} P_2))$. To see that $F(\operatorname{id}_P)=\operatorname{id}_{G_P}$ and $F((P_2\subseteq_{DC} P_3)\circ (P_1\subseteq_{DC} P_2))=(G_{P_2}\rightarrowtail G_{P_3})\circ (G_{P_1}\rightarrowtail G_{P_2}$), we refer to the construction of the map $\mu$ in Section \[proof:main\].
![Depiction of a small finite part of the category [$\mathcal{P}\mkern-12mu\mathcal{P}$]{}of $M$-polyforms. Each shape represents a $M$-polyform $P$, while arrows represent morphisms $P_1\subseteq_{DC}P_2$ (identities and composed morphisms omitted). For a better orientation, the position of the initial triangular $M$-polyform (black) is depicted by a gray background in each $M$-polyform. The category [$\mathcal{P}\mkern-12mu\mathcal{P}$]{}is not filtered, since there exist no $DC$-tiling of diamond-shaped $M$-polyforms by rectangular-shaped ones, or vice versa. However, both classes of $M$-polyforms can be reached by triangular-shaped ones. Note that the composition of all non-bounded sequences $S$ with the forgetful functor $U$ in the depicted part of ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ has a direct limit of $\mathbb{R}^2$. []{data-label="fig:category"}](Figure2.pdf){width="80.00000%"}
Of specific interest are infinite sequences $S=S_0 \subseteq_{DC} S_1 \subseteq_{DC} S_2\ldots$ of $M$-polyforms in ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ (identity and composed morphisms omitted), i.e. functors $S\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ from the usual linear order ${{\ensuremath{\mathbf{\omega}}\xspace}}=\{0,1,\ldots\}$ on the ordinal numbers to ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$. Trivially, each of these sequences, composed with the forgetful functor $U$, defines a direct limit $\varinjlim U S=\bigcup_i U(S_i)\subseteq\mathbb{R}^2$ (in the category of sets, since [$\mathcal{P}\mkern-12mu\mathcal{P}$]{}does not admit all filtered colimits), which we denote by $\hat{S}_\infty$. Furthermore, each sequence, composed with $F$, also defines a direct limit $\varinjlim F S$, denoted either as $G_{\hat{S}_\infty}^S$ or, equivalently, by $G_{\Gamma(\hat{S}_\infty)}^S$. We interpret $G_{\Gamma(\hat{S}_\infty)}^S$ as the limit of the sandpile group for $\Gamma(S_i)\rightarrow\Gamma(\hat{S}_\infty)$ (with respect to the sequence $S$). In Figure \[fig:category\], we depict the morphisms between four families of polyforms in ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$. The direct limit of each infinite sequence $S\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ which only contains these polyforms and morphisms, with $S_{i+1}\neq S_i$ for all $i\in{{\ensuremath{\mathbf{\omega}}\xspace}}$, is given by $\hat{S}^\infty=\mathbb{R}^2$, and thus $\Gamma(\hat{S}^\infty)=\mathbb{Z}^2$. To our knowledge, the respective limits of the sandpile group $G_{\mathbb{Z}^2}^S$ are the first[^2] definitions of scaling limits for the sandpile group on $\mathbb{Z}^2$.
If a given sequence $S$ of $M$-polyforms is upper bounded, i.e. if there exists an $u\in{{\ensuremath{\mathbf{\omega}}\xspace}}$ such that $U S_j=U S_u=\hat{S}_\infty$ for all $j\geq u$, it directly follows that $G_{\hat{S}_\infty}^S\cong G_{S_u}$. Thus, for such upper bounded sequences, the limit of the sandpile group is completely determined (up to isomorphisms) by the upper bound, i.e. $F$ preserves all finite direct limits. In such cases, we can drop the dependency of $G_{\hat{S}_\infty}^S$ on $S$ and simply write $G_{\hat{S}_\infty}$. We may ask if the same also holds for unbounded sequences:
\[question:limitPreservation\] Let $S^A,S^B\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ be two (possibly unbounded) sequences of $M$-polyforms with common limit $\hat{S}_\infty=\varinjlim U S^A=\varinjlim U S^B$. Is $G_{\hat{S}_\infty}^{S^A}$ isomorphic to $G_{\hat{S}_\infty}^{S^B}$?
Let $\hat{{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}}$ be the category with objects corresponding to all limits $\hat{S}_\infty=\varinjlim U S$ of sequences $S\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ of polyforms, and morphisms $\hat{S}_\infty^A\subseteq \hat{S}_\infty^B$ if there exists a natural transformation $S^a{ \mathrel{\vbox{\offinterlineskip
\mathsurround=0pt
\ialign{\hfil##\hfil\cr
\normalfont\scalebox{1.2}{.}\cr
$\longrightarrow$\cr}
}}}S^b$ between two sequences $S^a,S^b\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ with $\hat{S}^A_\infty=\varinjlim U S^a$ and $\hat{S}^B_\infty=\varinjlim U S^b$, i.e. if $S^a_i\subseteq_{DC}S^b_i$ for all $i\in{{\ensuremath{\mathbf{\omega}}\xspace}}$. We interpret ${\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}$ to represent a full subcategory of $\hat{{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}}$, with the object function of the (fully faithful) inclusion functor $I:{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}\rightarrow\hat{{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}}$ given by $I(P)=\varinjlim U\delta P$, where $\delta:{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}\rightarrow{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ denotes the diagonal functor with $(\delta P)_i=P$ for all $i\in{{\ensuremath{\mathbf{\omega}}\xspace}}$. Question \[question:limitPreservation\] then asks if there exists a functor $\hat{F}:\hat{P}\rightarrow{{\ensuremath{\mathbf{Ab}}\xspace}}$ which preserves all direct limits, and for which $F$ factors as $\hat{F}\circ U$.
In case Question \[question:limitPreservation\] can be answered in the affirmative, the dependency of the direct limit of the sandpile group on the sequence could be always dropped. Specifically, this would mean that there exists a unique scaling limit $G_{\mathbb{Z}^2}$ (up to isomorphisms) of the sandpile group on $\mathbb{Z}^2$. In this case, we would however immediately arrive at the following result:
\[equivalenceLimits\] Assume that $G_{\hat{S}_\infty}^{S^A}\cong G_{\hat{S}_\infty}^{S^B}$ whenever $\hat{S}_\infty=\varinjlim U S^A=\varinjlim U S^B$. Then, the limit of the sandpile group on $\mathbb{Z}^2$ is isomorphic to its limit on the upper-right quadrant of $\mathbb{Z}^2$, i.e. $G_{\mathbb{Z}^2}\cong G_{\mathbb{Z}_{\geq 0}^2}$.
This corollary, as well as several similar ones relating the limits of the sandpile group on different unbounded domains, arises because the mapping $\nu$ between morphisms $P_1\subseteq_{DC}P_2$ and DC-tilings $T^{P_1\rightarrow P_2}$ is not injective. We can thus construct two sequences $S^A$ and $S^B$ such that there exists a natural isomorphism $F S^A\cong F S^B$, but for which $\hat{S}^A_\infty\neq\hat{S}^B_\infty$. Corollary \[equivalenceLimits\] then follows when choosing $S^A_0=S^B_0$ to be square-shaped $M$-polyforms with side length $w_0$, $S^A_{i+1}$ and $S^B_{i+1}$ to have side lengths $w_{i+1}=5w_i$, $S^A_{i+1}$ to be positioned such that $S^A_{i}$ is in its center, and $S^B_{i+1}$ such that $S^B_{i}$ is at its bottom-left.
An exact sequence and the order of the sandpile group
-----------------------------------------------------
Theorem \[theorem:main\] is based on a close relationship between sandpile groups and certain modules of harmonic functions. Since this relationship is of interest itself, we summarize some of its properties in this section. We say that a domain $\Gamma\subseteq\mathbb{Z}^2$ is convex if there exists a convex open set $P\subseteq\mathbb{R}^2$ such that $\Gamma=P\cap\mathbb{Z}^2$. Note that, different to before, we do not require $P$ to be an $M$-polyform anymore. We say that an $R$-valued function $H:\Gamma\rightarrow R$, $R\in\{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$, is harmonic (on $\Gamma$) if $\Delta_\Gamma H(v)=0$ for all vertices $v\in\Gamma_0$ in the interior $\Gamma_0=\Gamma\setminus\partial\Gamma$ of the domain. The $R$-valued harmonic functions on $\Gamma$ form the module $\mathcal{H}^\Gamma_R$.
\[lemma:exactSequence\] For every finite convex domain $\Gamma\subset\mathbb{Z}^2$, $-\Delta_\Gamma:\mathcal{H}_G^\Gamma\cong G^\Gamma$ is an isomorphism from $\mathcal{H}_G^\Gamma=\{H\in\mathcal{H}_\mathbb{Q}^\Gamma|\Delta_\Gamma H|_{\partial\Gamma}\in\mathbb{Z}^{\partial\Gamma}\}/\mathcal{H}_\mathbb{Z}^\Gamma$ to the sandpile group $G_\Gamma$, with $\mathcal{H}_G^\Gamma$ the subgroup of the rational-valued harmonic functions $\mathcal{H}_\mathbb{Q}^\Gamma$ with integer-valued Laplacians, modulo the integer-valued harmonic functions $\mathcal{H}_\mathbb{Z}^\Gamma$. This isomorphism corresponds to the exact sequence $$\begin{aligned}
\xymatrix{
0 \ar[r] & G_\Gamma \ar[r] & \mathcal{H}_\mathbb{Q}^\Gamma/\mathcal{H}_\mathbb{Z}^\Gamma \ar[r] & (\mathbb{Q}/\mathbb{Z})^{\partial\Gamma} \ar[r] & 0.
}\end{aligned}$$
We derive an explicit construction for this isomorphism in the proof of Lemma \[lemma:exactSequence\].
Denote by $\mathcal{B}^\Gamma_R=\{B_i\}_{i=1,\ldots,|\partial\Gamma|}$ a basis for the module $\mathcal{H}^\Gamma_R$ of $R$-valued harmonic functions on a finite convex domain $\Gamma\subset\mathbb{Z}^2$ (in Section \[section:basis\], we present an algorithm for the construction of $\mathcal{B}^\Gamma_\mathbb{Z}$, and thus also for $\mathcal{B}^\Gamma_\mathbb{Q}$ and $\mathcal{B}^\Gamma_\mathbb{R}$). By definition, the Laplacian $\Delta_\Gamma H$ of every harmonic function $H\in\mathcal{H}^\Gamma_R$, and thus also of every basis function in $\mathcal{B}^\Gamma_R$, only has support at the boundary $\partial\Gamma$ of the domain. The Laplacian of every basis function in $\mathcal{B}^\Gamma_R$ can thus be restricted to $\partial\Gamma$ without information loss, and we refer to $\Delta\mathcal{B}^\Gamma_R=(\Delta_\Gamma B_1|_{\partial\Gamma},\ldots,\Delta_\Gamma B_{|\partial\Gamma|}|_{\partial\Gamma})\in R^{|\partial\Gamma|\times |\partial\Gamma|}$ as the potential matrix of $\Gamma$ (with respect to $\mathcal{B}^\Gamma_R$).
\[lemma:order\] Let $\Gamma\subset\mathbb{Z}^2$ be a finite convex domain, and $\mathcal{B}^\Gamma_\mathbb{Z}$ be a basis for the module of integer-valued harmonic functions $\mathcal{H}^\Gamma_\mathbb{Z}$ on $\Gamma$. Then, the order of the sandpile group $G_\Gamma$ is given by $$\begin{aligned}
|G_\Gamma|=|\det(\Delta\mathcal{B}^\Gamma_\mathbb{Z})|.\end{aligned}$$
Integer-valued harmonic functions and cyclic subgroups
------------------------------------------------------
In this section, we present three examples of constructions which directly link integer-valued harmonic functions to cyclic subgroups of the sandpile group. Such constructions might help to answer Question \[question:limitPreservation\] (in the negative), since they represent structural restrictions on an (eventually existing) unique scaling-limit $G_{\mathbb{Z}^2}$ of the sandpile group.
![Integer-valued harmonic functions used to prove the existence of cyclic subgroups. A) The harmonic function $H=xy$ is coprime on $N\times N$ domains, $N\in2\mathbb{N}+1$, (here: $N=5$) while its Laplacian is divisible by $\frac{N+1}{2}$ (here: by $3$). B) All values of the harmonic function $H^{\pi}$ are divisible by three on vertices with a blue background, by five on vertices with a green background, and by seven on vertices with a red background, respectively. This sequence however ends at the vertices with a yellow background, since nine is not prime. C) The harmonic function $H^\diamond_i$ corresponds to the sum of the four harmonic basis functions depicted in blue, green, red, and yellow, such that the values of $H^\diamond_i$ on $\partial(\mathbb{Z}^2\setminus\Gamma)$ are all divisible by four. []{data-label="fig:primeHarmonic"}](Figure3.pdf){width="75.00000%"}
Let $H:\mathbb{Z}^2\rightarrow\mathbb{Z}$, $\Delta_{\mathbb{Z}^2}H=0$, be an integer-valued harmonic function on $\mathbb{Z}^2$. Assume that the values of the restriction of $H$ to some finite convex domain $\Gamma\subset\mathbb{Z}^2$ are coprime, and that the values of $H$ on the boundary $\partial(\mathbb{Z}^2\setminus\Gamma)$ of the complement of the domain are all divisible by some integer $n\geq 2$ (Figure \[fig:primeHarmonic\]A). Recall that $$\begin{aligned}
\Delta_\Gamma H|_\Gamma(v)=-\sum_{\stackrel{w\in\partial(\mathbb{Z}^2\setminus\Gamma)}{w\sim v}}H(w)\end{aligned}$$ for all $v\in\Gamma$, and that, thus, also $\Delta_\Gamma H|_\Gamma$ is divisible by $n$ (Figure \[fig:primeHarmonic\]A). From the isomorphism in Theorem \[lemma:exactSequence\], it then follows that $
S_\Gamma^H=\{0,C,\ldots(n-1)C\}\subseteq G_\Gamma
$ forms a cyclic subgroup of the sandpile group $G_\Gamma$ with generator $C=[-\frac{1}{n}\Delta_\Gamma H|_\Gamma]$ and order $|S_\Gamma^H|=n$.
For a given integer-valued harmonic function $\hat{H}\in\mathcal{H}_{\mathbb{Z}}^\Gamma$, we recently introduced the harmonic sandpile dynamics $D_{\hat{H}}:\mathbb{R}/\mathbb{Z}\rightarrow G_\Gamma$, $D_{\hat{H}}(t) = [\lfloor t\Delta_\Gamma\hat{H}\rfloor]$, with $\lfloor.\rfloor$ the element-wise floor function [@Lang2019]. For $\hat{H}=H|_\Gamma$, the discussion above implies that the elements of $S_\Gamma^H$ appear exactly at times $t\equiv0, \frac{1}{n},\ldots,\frac{n-1}{n} (\operatorname{mod} 1)$ in the harmonic sandpile dynamics $D_{\hat{H}}$, in the sense that $[\lfloor t\Delta_\Gamma H|_\Gamma\rfloor]=[t\Delta_\Gamma H|_\Gamma]\in S_\Gamma^H$ at these times.
As a first example of how such constructions can impose restrictions on the limits of the sandpile group, consider the family $\{\Gamma_N\}_{N\in2\mathbb{N}+1}$ consisting of all $N\times N$ square domains $\Gamma_N\subset\mathbb{Z}^2$ with odd domain sizes $N$. Assume that each domain $\Gamma_N$ is defined such that its center lies at the origin $(x,y)=(0,0)$ of $\mathbb{Z}^2$. It is then easy to see that the values of the harmonic function $H=x y$ on $\partial(\mathbb{Z}^2\setminus\Gamma_N)$ are divisible by $\frac{N+1}{2}$ (Figure \[fig:primeHarmonic\]A), which directly proves the following lemma:
\[lemma:divFirstExample\] For every $N\in 2\mathbb{N}+1$, the sandpile group $G_{\Gamma_N}$ on an $N\times N$ square domain $\Gamma_N\subset\mathbb{Z}^2$ has a cyclic subgroup $\mathbb{Z}/\frac{N+1}{2}\mathbb{Z}\subseteq G_{\Gamma_N}$ of order $\frac{N+1}{2}$.
For every $n\geq 1$, we can construct a sequence of domains $S^n\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ with $S^n_\infty=\varinjlim U S^n=\mathbb{R}^2$, which starts at an $M$-polyform $S^n_0$ with $\Gamma(S^n_0)$ corresponding to an $N\times N$ square domain with $N=2n-1$ (Figure \[fig:category\]). If Question \[question:limitPreservation\] can be answered in the affirmative, this would directly imply that the scaling limit $G_{\mathbb{Z}^2}$ of the sandpile group on the standard square lattice $\mathbb{Z}^2$ would contain cyclic subgroups of every order.
For $N\times N$ square domains with even domain sizes $N\in 2\mathbb{N}$, somewhat similar results can be obtained when considering the integer-valued harmonic function depicted in Figure \[fig:primeHarmonic\]B. This harmonic function takes values on $\partial(\mathbb{Z}^2\setminus\Gamma_N)$ which are divisible by three ($N=2$), five ($N=4$) and seven ($N=6$), respectively. This pattern however breaks down at $N=8$, since $N+1=9$ is not prime.
\[lemma:divPrime\] Let $\Gamma_N\subset\mathbb{Z}^2$ be an $N\times N$ square domain. Then, if $N+1$ is prime, the sandpile group $G_{\Gamma_N}$ possesses a cyclic subgroup $\mathbb{Z}/(N+1)\mathbb{Z}\subseteq G_{\Gamma_N}$ of order $N+1$.
If $N+1$ is not prime, Theorem \[theorem:main\] implies that there exist group monomorphisms from $G_{\Gamma_M}$ to $G_{\Gamma_N}$ whenever $M+1$ divides $N+1$. Thus, the sandpile group on every $N\times N$ domain (independently if $N+1$ is prime or not) possesses a cyclic subgroup with an order given by the product of all distinct factors of $N+1$ (each to the power of one). Furthermore, when we denote by $p_k$ the $k^{th}$ prime number, the limit $G_{\mathbb{Z}^2}^S$ of the sandpile group with respect to the sequence $S\in{\ensuremath{\mathcal{P}\mkern-12mu\mathcal{P}}\xspace}^{{\ensuremath{\mathbf{\omega}}\xspace}}$ with $\Gamma(S_i)=\Gamma_{\prod_{k\leq i}(p_k-1)}$ contains at least one cyclic subgroup $\mathbb{Z}/p_k\mathbb{Z}\subset G_{\mathbb{Z}^2}^S$ for every prime number $p_k$.
Our last lemma is based on the existence of integer-valued harmonic functions which are zero in the interior of some diamond-shaped region of $\mathbb{Z}^2$, take values $\pm 1$ at its edges, the value $0$ at its corners, and which are divisible by four everywhere else (Figure \[fig:primeHarmonic\]C). By the discussion above, every such harmonic function for which all four corners lie on the boundary $\partial(\mathbb{Z}^2\setminus\Gamma_N)$ of the complement of an $N\times N$ domain $\Gamma_N\subset\mathbb{Z}^2$ can be directly mapped to a cyclic subgroup $\mathbb{Z}/4\mathbb{Z}\subseteq G_{\Gamma_N}$ of order four. If we also allow for one “degenerated diamond” in case $N$ is odd, there exist $N$ such harmonic functions which are linearly independent.
\[lemma:div4\] Let $G_{\Gamma_N}$ be the sandpile group on an $N\times N$ square domain $\Gamma_N\subset\mathbb{Z}^2$, with $N\in\mathbb{N}$. Then, $G_{\Gamma_N}$ has a subgroup $S_{N}\subseteq G_{\Gamma_N}$ isomorphic to the direct sum $S_{N}\cong\bigoplus_{i=1}^N(\mathbb{Z}/4\mathbb{Z})$ of $N$ cyclic groups of order four.
We note that this result was derived before in [@Dhar1995], and formed the basis for the proof that the minimal number of generators for the sandpile group on $\Gamma_N$ is $N$. Our alternative proof arguably provides more insights by directly constructing the respective cyclic subgroups of $G_{\Gamma_N}$, and can be easily generalized to other domains.
### Acknowledgements {#acknowledgements .unnumbered}
The authors thank Nikita Kalinin for carefully reading the manuscript. ML is grateful to the members of the Guet group for valuable comments and support. MS is grateful to Tamas Hausel, Ludmil Katzarkov, Maxim Kontsevich, Ernesto Lupercio, Grigory Mikhalkin and Andras Szenes for inspiring communications.
Overview of the proofs
======================
We first derive the two related Lemmata \[lemma:exactSequence\] and \[lemma:order\]. Lemma \[lemma:exactSequence\] then allows us to restate the question on the existence of monomorphisms between sandpile groups (Theorem \[theorem:main\]) into a question on the existence of monomorphisms between groups of harmonic functions, which we solve by an explicit construction. We then continue to state an algorithm for the construction of a basis for the integer-valued harmonic functions on a given finite convex domain. Finally, we use this basis to prove Lemmata \[lemma:divPrime\]&\[lemma:div4\]. We note that Corollary \[equivalenceLimits\] and Lemma \[lemma:divFirstExample\] were directly proved in the Introduction.
The sandpile group and harmonic functions
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In this section, we derive the isomorphism between the harmonic functions $\mathcal{H}_G^\Gamma$ and the sandpile group $G_\Gamma$ (Lemma \[lemma:exactSequence\]), as well as the formula for the order of the sandpile group (Lemma \[lemma:order\]). We start with an observation made by Creutz about the sandpile model, namely that every recurrent configuration can be reached from the empty configuration (or any other configuration) by only adding particles to the boundary of the domain and “relaxing” the sandpile [@Creutz1990]. Recall that the elements of the sandpile group, as defined in this article, correspond to the equivalence classes of the recurrent configurations of the sandpile model (see Introduction). Creutz’s observation can thus be restated as follows: for every element $C\in G^\Gamma$ of the sandpile group, there exist (infinitely many) functions $X\in\mathbb{Z}^{\Gamma}$ which only have support at the boundary $\partial\Gamma$ of the domain, and which satisfy that $[X]=C$, with $[.]:\mathbb{Z}^\Gamma\rightarrow G_\Gamma$ the canonical projection map to the sandpile group. For an algorithm for the construction of $X$, we refer to [@Creutz1990].
By the existence and uniqueness of solutions to the discrete Dirichlet problem on convex domains [@Lawler2010], it follows that, for every such $X\in\mathbb{Z}^{\Gamma}$, there exists a unique rational-valued harmonic function $H_X\in\mathcal{H}_\mathbb{Q}^\Gamma$ with $\Delta_\Gamma H_X=-X$. The composition $[.]\circ-\Delta_\Gamma$ of the discrete Laplacian with the canonical projection map then maps two harmonic functions $H_{X,1},H_{X,2}\in\mathcal{H}_\mathbb{Q}^\Gamma$, $\Delta_\Gamma H_{X,1},\Delta_\Gamma H_{X,2}\in\mathbb{Z}^{\Gamma}$, to the same element of the sandpile group if and only if $-\Delta_\Gamma(H_{X,1}-H_{X,2})\in\Delta_\Gamma(\mathbb{Z}^\Gamma)$. Since both $\Delta_\Gamma H_{X,1}$ and $\Delta_\Gamma H_{X,2}$ only have support at the boundary $\partial\Gamma$ of the domain, $H_{X,1}-H_{X,2}$ is thus an integer-valued harmonic function, which concludes our proof of Lemma \[lemma:exactSequence\].
We construct the inverse of $-\Delta_\Gamma:\mathcal{H}_G^\Gamma\cong G_\Gamma$ in two steps. For every configuration $C\in G_\Gamma$, we first define the coordinates $\sigma_\Gamma:G_\Gamma\rightarrow(\mathbb{Q}/\mathbb{Z})^{\partial\Gamma}$, $$\begin{aligned}
\sigma_\Gamma(C)\equiv-(\Delta\mathcal{B}^\Gamma_\mathbb{Z})^{-1}X\ \ (\operatorname{mod} 1),\end{aligned}$$ with respect to the basis $\mathcal{B}^\Gamma_\mathbb{Z}$. Note that, for two different choices $X^\alpha,X^\beta\in\mathbb{Z}^{\Gamma}$, $-(\Delta\mathcal{B}^\Gamma_\mathbb{Z})^{-1}(X^\alpha-X^\beta)\in\mathbb{Z}^{\partial\Gamma}$, and that thus the coordinates $\sigma_\Gamma$ don’t depend on the specific choice for $X$. This also implies that $\sigma_\Gamma$ correspond to toppling invariants as defined in [@Dhar1995].
In the second step, we then define the function $\phi_\Gamma:(\mathbb{R}/\mathbb{Z})^{\partial\Gamma}\rightarrow\mathcal{H}_\mathbb{R}^\Gamma/\mathcal{H}_\mathbb{Z}^\Gamma$, $\phi_\Gamma(s)=\sum_{i=1}^{|\partial\Gamma|}s_iB_i$. It is easy to check that the composition $\phi_\Gamma\circ\sigma_\Gamma:G_\Gamma\rightarrow\mathcal{H}_\mathbb{R}^\Gamma/\mathcal{H}_\mathbb{Z}^\Gamma$ is independent of the choice of the basis $\mathcal{B}^\Gamma_\mathbb{Z}$, and that $-\Delta_\Gamma\phi_\Gamma(\sigma_\Gamma([X]))=[X]$. The latter implies that $\phi_\Gamma\circ\sigma_\Gamma$ is the inverse of $-\Delta_\Gamma$.
The isomorphism between the sandpile group $G_\Gamma$ and $\mathcal{H}_G^\Gamma$ proposes to consider the sandpile group as a discrete subgroup of a continuous Lie group isomorphic to $\mathcal{H}_\mathbb{R}^\Gamma/\mathcal{H}_\mathbb{Z}^\Gamma$, to which we refer to as the extended sandpile group $\tilde{G}_\Gamma$ [@Lang2019]. More precisely, the extended sandpile group is an extension of the torus $(\mathbb{R}\setminus\mathbb{Z})^{\partial\Gamma}$ by the usual sandpile group, and is defined by the exact sequence $$\begin{aligned}
\xymatrix{
0 \ar[r] & G_\Gamma \ar[r] & \tilde{G}_\Gamma \ar[r] & (\mathbb{R}/\mathbb{Z})^{\partial\Gamma} \ar[r] & 0.
}\end{aligned}$$ In terms of the sandpile model, this Lie group is obtained by allowing each vertex $b\in\partial\Gamma$ in the boundary $\partial\Gamma$ of the domain to carry a real value $\tilde{C}(b)\in[0,4)$ of particles, while each vertex $v\in\Gamma_0$ in the interior $\Gamma_0=\Gamma\setminus\partial\Gamma$ of the domain is still only allowed to carry an integer number of particles, i.e. $\tilde{C}(v)\in\{0,1,2,3\}$ (the toppling rules are kept unchanged) [@Lang2019]. This definition lifts $\phi_\Gamma:(\mathbb{R}/\mathbb{Z})^{\partial\Gamma}\cong \tilde{G}_\Gamma$ to a group isomorphism, and a left-inverse of the inclusion map $G_\Gamma\rightarrow\tilde{G}_\Gamma$ is given by the floor function $\lfloor.\rfloor:\tilde{G}_\Gamma\rightarrow G_\Gamma$. We thus naturally arrive at the function $f=\lfloor.\rfloor\circ-\Delta_\Gamma\circ\phi_\Gamma:(\mathbb{R}/\mathbb{Z})^{\partial\Gamma}\rightarrow G_\Gamma$, $f(s)
=-[\lfloor\sum_{i=1}^{|\partial\Gamma|} s_i \Delta_\Gamma B_i\rfloor]$, which justifies to interpret the usual sandpile group $G_\Gamma$ as the discretization of an $|\partial\Gamma|$-dimensional torus [@Lang2019].
Due to the properties of the floor function, the preimage $f^{-1}(C)$ of an element $C\in G_\Gamma$ of the sandpile group under $f$ is connected. Denote by $\operatorname{vol}(f^{-1}(C))$ the volume of this preimage, with $\operatorname{vol}((\mathbb{R}/\mathbb{Z})^{\partial\Gamma})=1$. Since, for every $C\in G_\Gamma$, there exists a coordinate transformation $s\mapsto\tilde{s}$ such that $C$ has coordinates $\tilde{s}=0$, we get that $\operatorname{vol}(f^{-1}(C))=\operatorname{vol}(f^{-1}(\mathbf{0}))=\frac{1}{|G_\Gamma|}$ for all $C\in G_\Gamma$, with $\mathbf{0}$ the identity of the sandpile group. The preimage $f^{-1}(\mathbf{0})$ of the identity under $f$ forms a $|\partial\Gamma|$-parallelotope with edges $g_i$ given by $(\Delta\mathcal{B}^\Gamma_\mathbb{Z})g_i=e_i$, with $(e_i)_j=\delta_{ij}$ the $i^{th}$ unit vector and $\delta_{ij}$ the Kronecker delta. The volume of this parallelotope is $\operatorname{vol}(f^{-1}(\mathbf{0}))=|\det(\Delta\mathcal{B}^\Gamma_\mathbb{Z})^{-1}|$, and thus $|G_\Gamma|=|\det(\Delta\mathcal{B}^\Gamma_\mathbb{Z})|$, which proves Lemma \[lemma:order\].
Construction of sandpile monomorphisms {#proof:main}
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![A) Example of an harmonic function $H_A\in\mathcal{H}_G^{\Gamma_A}$ (left) corresponding to the element $[-\Delta_{\Gamma_A}H_A]\in G_{P_A}$ of the sandpile group (right) on a given $M$-polyform $P_A$. The directed and colored edges of $P_A^{DC}$ are indicated by arrows, and the green squares corresponds to the vertices in $\Gamma_A=\Gamma(P_A)$. B) The $M$-polyform $P_A$ from (A) $DC$-tiles the depicted $M$-polyform $P_B$. To construct the harmonic function $H_B\in\mathcal{H}_G^{\Gamma_B}$ onto which $H_A$ is mapped under the monomorphism from $\mathcal{H}_G^{P_A}$ to $\mathcal{H}_G^{P_B}$ induced by this tiling, we first define a rational-valued function $\hat{H}_B$ which is harmonic everywhere, except for the vertices directly next to an internal-boundary (gray backgrounds). For each tile in $T^{P_A\rightarrow P_B}$, we then construct an integer-valued function $X_i$ which cures the non-harmoniticity of its respective vertices. C) The procedure depicted in (B) leads to the harmonic function $H_B\in\mathcal{H}_G^{\Gamma_B}$ (left) corresponding to the element $[-\Delta_{\Gamma_B}H_B]\in G_{P_B}$ of the sandpile group (right) on $P_B$, onto which $H_A$ is mapped by the monomorphism from $\mathcal{H}_G^{P_A}$ to $\mathcal{H}_G^{P_B}$. []{data-label="fig:monomorphismProof"}](Figure4.pdf){width="60.00000%"}
Let $P_A$ and $P_B$ be two convex $M$-polyforms, and assume that there exists a $DC$-tiling $T^{P_A\rightarrow P_B}$ of $P_B$ by $P_A$. In this section, we then construct a monomorphism from the sandpile group on the domain $\Gamma_A=\Gamma(P_A)=\mathbb{Z}^2\cap P_A$ to the sandpile group on $\Gamma_B=\Gamma(P_B)=\mathbb{Z}^2\cap P_B$, and thus prove Theorem \[theorem:main\]. Before starting this construction, we derive three properties of $DC$-tilings.
The domains of different tiles do not overlap, i.e. $\Gamma_i\cap\Gamma_j=\{\}$ for all $i\neq j$.
This corollary directly follows from each tile $P_i$ being treated as an open subset of $\mathbb{R}^2$ when determining its domain $\Gamma_i=\mathbb{Z}^2\cap P_i$.
The domains of the tiles in general don’t cover $\Gamma_B$. Specifically, all vertices of $\Gamma_B$ which lie directly on common edges (including their endpoints) of two tiles are not elements of any $\Gamma_i$ (red vertices in Figure \[fig:initialIllus\]). We refer to the set $\partial^T\Gamma_B=\Gamma_B\setminus\bigcup_i\Gamma_i$ of these vertices as the internal boundaries of the tiling. These internal boundaries separate the domains $\Gamma_i$ in the following sense:
\[corollary:separation\] The removal of all vertices in $\partial^T\Gamma_B$ splits $\Gamma_B$ into the disconnected components $\{\Gamma_1, \Gamma_2,\ldots,\Gamma_{|T^{P_A\rightarrow P_B}|}\}$.
Because the tiles are convex by assumption, each pair of adjacent tiles can be separated by exactly one line (the extension of their common edge). By the definition of $M$, this line is either horizontal, vertical or diagonal, and passes through infinitely many vertices of $\mathbb{Z}^2$ (Figure \[fig:initialIllus\]A). In each of the cases, it splits $\mathbb{Z}^2$ into two unconnected components, from which the corollary directly follows.
By definition, each tile $P_i\in T^{P_A\rightarrow P_B}$ can be obtained from $P_A^{DC}$ by a combination of translations, rotations and reflections. If this is possible by using only translations and rotations, we assign the sign $s(P_i)=+1$ to the tile, and otherwise the sign $s(P_i)=-1$. This definition also induces signs $s(v)=s(\Gamma_i)=s(P_i)$ for the domains $\Gamma_i$ and vertices $v\in\Gamma_i$ belonging to the tiles. To the internal boundaries $\partial^T\Gamma_B$ and their vertices $b\in\partial^T\Gamma_B$, we assign the sign $s(\partial^T\Gamma_B)=s(b)=0$. The relationship of each tile $P_i$ with the polyform $P_A$ (i.e. the translations, rotations and reflections mapping $P_A$ on $P_i$) corresponds to a function $\psi_i:\Gamma_A\rightarrow\Gamma_B$ which maps vertices $v_A\in \Gamma_A$ of the polyform onto their corresponding vertices $v_i$ of the tile. For two vertices $v,w\in\Gamma_B$, we then define the equivalence relation $\equiv_{DC}$ such that $v\equiv_{DC}w$ if there exists a $v_A\in \Gamma_A$ such that $v=\psi_i(v_A)$ and $w=\psi_j(v_A)$ for some tiles $P_i$ and $P_j$, or if both vertices are part of the internal boundaries, i.e. $v,w\in\partial^T\Gamma_B$. We denote by $[v]_{DC}$ the equivalence class of $v$ induced by $\equiv_{DC}$.
\[corollary:IBs\] Let $b\in\partial^T\Gamma_B$ be a vertex of the internal boundaries. Then, the number of neighbors of $b$ in every equivalence class $[v_B]_{DC}$, $v_B\in\Gamma_B$ carrying a positive sign is equal to the number of neighbors carrying a negative sign, i.e. $\sum_{\stackrel{v\in[v_B]_{DC}}{v\sim b}}s(v)=0$.
Assume that $b$ has at least one neighbor in $[v_B]_{DC}$; otherwise the corollary is trivially satisfied. Also, assume $v_B\notin\partial^T\Gamma_B$, since otherwise $s(v)=0$ for all $v\in[v_B]_{DC}$, from which the corollary also trivially follows. Denote by $N(b,v_B)=\{v\in[v_B]_{DC}|v\sim b\}$ the set of neighbors of $b$ in the equivalence class of $v_B$. Every vertex can have maximally four neighbors, thus $|N(b,v_B)|\leq 4$. Being part of the internal boundaries, $b$ must lie on at least one common edge (including endpoints) of two tiles $P_i\neq P_j$. These two tiles can be mapped onto one another by reflection on the common edge, and thus must have opposite signs. If a vertex $v\in\Gamma_i$ of $P_i$ is a neighbor of $b$, it follows that there must be a vertex $w\in\Gamma_j$ of $P_j$ which is also a neighbor of $b$, and which has opposite sign, i.e. $s(w)=-s(v)$. This excludes the case $|N(b,v_B)|=1$, and proves the corollary for $|N(b,v_B)|=2$. For $|N(b,v_B)|\in\{3,4\}$, the structure of $M$ directly implies that $b$ has to lie on a common corner of three, respectively four, tiles. The corresponding internal angles of the tiles have to be smaller or equal to $360^\circ/3=120^\circ$, respectively $360^\circ/4=90^\circ$. The definition of $M$ only admits internal angles which are multiples of $45^\circ$ (Figure \[fig:initialIllus\]A). Thus, in both cases, only angles of $45^\circ$ or $90^\circ$ are possible. An angle of $45^\circ$ is only possible if all $v\in[v_B]$ lie on the internal boundaries, which implies $s(v)=0$ (see above). If the angle is $90^\circ$, $|N(b,v_B)|=3$ would imply that $P_B$ is not convex, which can thus be excluded. Finally, if the angle is $90^\circ$ and $|N(b,v_B)|=4$, each of the four tiles to which these vertices belong must have exactly two adjacent tiles with opposite signs, from which the corollary follows.
With this preparatory work, we can now prove Theorem \[theorem:main\]. By Lemma \[lemma:exactSequence\], the sandpile group $G_\Gamma$ is isomorphic to $\mathcal{H}_G^\Gamma=\{H\in\mathcal{H}_\mathbb{Q}^\Gamma|\left.\Delta_\Gamma H\right|_{\partial_\Gamma}\in\mathbb{Z}^{\partial\Gamma}\}/\mathcal{H}_\mathbb{Z}^\Gamma$. It thus suffices to construct a monomorphism from $\mathcal{H}_G^{P_A}$ to $\mathcal{H}_G^{P_B}$ whenever the $M$-polyform $P_A$ $DC$-tiles $P_B$. We construct this monomorphism in two steps. For the first step, assume that $T^{P_A\rightarrow P_B}$ is a given $DC$-tiling of $P_B$ by $P_A$, and let $H_A$ be a harmonic function in $\mathcal{H}_G^{\Gamma_A}$ (Figure \[fig:monomorphismProof\]A). Then, define the rational-valued function $\hat{H}_B\in\mathbb{Q}^{\Gamma_B}$ in the following way (Figure \[fig:monomorphismProof\]B): for each vertex $v\in\Gamma_i$ belonging to tile $P_i\in T^{P_A\rightarrow P_B}$, set $\hat{H}_B(v)=s(P_i)H_A(v_A)$ with $v_A\in\Gamma_A$ the unique vertex satisfying $\psi_i(v_A)=v$. Otherwise, that is if $v$ belongs to the internal boundaries, set $\hat{H}_B(v)=0$.
Because $\hat{H}_B(b)=0$ for all vertices $b\in\partial^T\Gamma_B$ of the internal boundaries, Corollary \[corollary:separation\] implies that $\Delta_{\Gamma_B}\hat{H}_B(v)=s(\Gamma_i)\Delta_{\Gamma_A} H_A(v_A)$ for all vertices $v\in\Gamma_i$ belonging to the domain of a tile $P_i$, with $\psi_i(v_A)=v$. This implies that the Laplacian of $\hat{H}_B$ is zero in the interior of the sub-domains $\Gamma_i$ of $\Gamma_B$, and integer-valued at their boundaries. From Corollary \[corollary:IBs\], on the other hand, it follows that $\Delta_{\Gamma_B}\hat{H}_B(b)=0$ for every vertex $b\in\partial^T\Gamma_B$ of the internal boundaries. Thus, $\hat{H}_B$ is harmonic nearly everywhere, except at the vertices directly adjacent to (but not including) the internal boundaries, for which $\Delta_{\Gamma_B}\hat{H}_B$ is integer-valued.
The “harmonic deficit” of $\hat{H}_B$ can be cured, one tile at a time: for a given tile $P_i$, we can define an integer-valued function $X_i\in\mathbb{Z}^{\Gamma_B}$ whose Laplacian is zero everywhere in the interior of $\Gamma_B$, except for those vertices $\{v\in\partial\Gamma_i\setminus\partial\Gamma_B|\exists b\in\partial^T\Gamma_B:v\sim b\}$ at the boundary of $\Gamma_i$ which are adjacent to at least one vertex of the internal boundaries, for which we require that $\Delta_{\Gamma_B}X_i(v)=-\Delta_{\Gamma_B}\hat{H}_B(v)$. For example, if we set $X_i$ to zero in $\Gamma_i$, $X_i$ directly corresponds to a solution $\tilde{X}_i$ of a Dirichlet problem on $\mathbb{Z}^2\setminus\Gamma_i$ with boundary conditions chosen such that $\sum_{\stackrel{w\in\partial(\mathbb{Z}^2\setminus\Gamma_i)}{w\sim v}}\tilde{X}_i(w)=-s(P_i)\Delta_{\Gamma_B}\hat{H}_B(v)$ for each $v\in\partial\Gamma_i$. Since $\Gamma_i$ is convex, we can always choose these boundary conditions to be integer-valued, and then there exist (infinitely many) integer-valued solutions. As can easily be seen in the following, any such solution results in the same outcome, since the difference $X_i^\alpha-X_i^\beta$ of any two possible choices $X_i^\alpha$ and $X_i^\beta$ is integer-valued harmonic.
Given the functions $X_i$, we define $H_B\in\mathbb{Q}^\Gamma$ by $$\begin{aligned}
H_B=\hat{H}_B+\sum_i X_i.\end{aligned}$$ By construction, $H_B$ is harmonic everywhere and has an integer-valued Laplacian. We can thus reinterpret $H_B$ to be an element of $\mathcal{H}_\mathbb{G}^{\Gamma_B}$. It is then easy to see that the function $\xi:\mathcal{H}_G^{P_A}\rightarrow\mathcal{H}_G^{P_B}$, $\xi(H_A)=H_B$, is injective, and that it satisfies $\xi(H_B^1+H_B^2)=\xi(H_B^1)+\xi(H_B^2)$. The function $\xi$ is thus a group monomorphism, and with the isomorphism $-\Delta_{\Gamma}:\mathcal{H}_G^{P}\cong G_P$ from Lemma \[lemma:exactSequence\], we get that $\mu(T^{P_A\rightarrow P_B})=\Delta_{\Gamma_B}\circ\xi\circ(\Delta_{\Gamma_A})^{-1}:G_{P_A}\rightarrowtail G_{P_B}$ is the group monomorphism which we claimed to exist in Theorem \[theorem:main\].
A basis for integer-valued harmonic functions {#section:basis}
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![Harmonic functions used in the construction of a basis for $\mathcal{H}^\Gamma_\mathbb{Z}$ (A&B), and construction of such a basis on a $4\times 4$ square domain (C). A&B) Two examples of harmonic functions which are zero for all vertices above/below some diagonal (A: $d^+_i$, and B: $d^-_{i+1}$), and which take values in $\{+1,-1\}$ on the diagonal. Both harmonic functions are additionally chosen such that they are zero on the diagonal $d^-_0$ (A), respectively $d^+_0$ (B), which is orthogonal to their respective defining diagonal. The rest of the values of the harmonic functions are chosen such that they are anti-symmetric (A), respectively symmetric (B) with respect to this orthogonal diagonal, depending if the two diagonals intersect on a vertex (B) or not (A). C) Each square depicts a step in the construction of the basis by the algorithm in Figure \[fig:basisAlgo\]. Light-gray backgrounds denote those vertices already belonging to the growing domain at the beginning of the step, and dark-gray backgrounds those vertices added during the step. The numbers correspond to the harmonic basis function added during the respective step. []{data-label="fig:properBasis"}](Figure5.pdf){width="65.00000%"}
In this section, we present an algorithm for the construction of a basis for the module of integer-valued harmonic functions $\mathcal{H}^\Gamma_\mathbb{Z}$ on a finite convex domain $\Gamma\subset\mathbb{Z}^2$. We note that the resulting bases provided important intuition during the research which lead to this article.
We first define four families of integer-valued harmonic functions from which our algorithm will then select a subset for the construction of the basis. Denote by $d^+_i=\{(x,y)\in\mathbb{Z}^2|x+y=i+c^+\}$ and $d^-_i=\{(x,y)\in\mathbb{Z}^2|x-y=i+c^-\}$ the diagonals of $\mathbb{Z}^2$, with $c^+,c^-\in\mathbb{Z}$. For a given diagonal, say $d_i^+$, it is then possible to construct an integer-valued harmonic function ${B^+_{\geq i}}\in\mathcal{H}^\Gamma_\mathbb{Z}$ which is zero for all vertices below $d_i^+$ (i.e. for all $v\in d_j^+$, $j<i$), but non-zero for nearly all vertices on and above $d_i^+$ (i.e. for $v\in d_j^+$, $j\geq i$, see [@Buhovsky2017]). For one vertex on each of the non-zero diagonals $d_j^+$, $j\geq i$, the value of ${B^+_{\geq i}}$ can be freely assigned, which then uniquely determines the value of all other vertices [@Buhovsky2017]. Here, we only assume that the free value of ${B^+_{\geq i}}$ on the defining diagonal $d_i^+$ is chosen to be $\pm 1$, while we yet do not pose any restrictions on the choice of the free values on the other diagonals. This construction results in a harmonic function ${B^+_{\geq i}}$ which is zero below $d_i$ and which alternates between $+1$ and $-1$ on $d_i^+$ (Figure \[fig:properBasis\]A&B). Similarly, we denote by ${B^+_{\leq i}}\in\mathcal{H}^\Gamma_\mathbb{Z}$ an harmonic function which is zero *above* $d_i^+$ and takes the values $\pm 1$ on $d_i^+$, and by ${B^-_{\geq i}}$ and ${B^-_{\leq i}}$ the corresponding harmonic functions when replacing $d_i^+$ by $d_i^-$ in the definitions above.
We make the following definitions:
Let $\Gamma\subset\mathbb{Z}^2$ be a finite convex domain. Denote by $\operatorname{bond}(\Gamma)=\{v\in\mathbb{Z}^2|v\in\Gamma\vee\exists w\in\Gamma: w\sim v\wedge\sum_{u\in\Gamma}\delta_{u\sim w}=3\}$ the domain obtained by extending $\Gamma$ by all vertices in its complement $\mathbb{Z}^2\setminus\Gamma$ which are the direct neighbors of vertices in $\Gamma$ which already have three neighbors in $\Gamma$. With $\operatorname{bond}^k=\operatorname{bond}^{k-1}\circ\operatorname{bond}$ and $\operatorname{bond}^0(\Gamma)=\Gamma$, we then define the diamond hull of $\Gamma$ as the limit $\operatorname{diam}(\Gamma)=\lim_{k\rightarrow\infty}\operatorname{bond}^k(\Gamma)$.
For every finite convex domain $\Gamma\subset\mathbb{Z}^2$, $\operatorname{diam}(\Gamma)\subset\mathbb{Z}^2$ is a finite convex domain, too. Furthermore, $|\partial\operatorname{diam}(\Gamma)|=|\partial\Gamma|$.
\[ext2diam\] Every harmonic function $H\in\mathcal{H}^\Gamma_R$, $R\in\{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$, on a finite convex domain $\Gamma\subset\mathbb{Z}^2$ can be uniquely extended to the diamond hull of $\Gamma$, i.e. there exists a unique $\hat{H}\in\mathcal{H}^{\operatorname{diam}(\Gamma)}_R$ such that $\hat{H}|_\Gamma=H$. The domain $\operatorname{diam}(\Gamma)$ is maximal with respect to this property.
It directly follows that, if $\mathcal{B}^{\operatorname{diam}(\Gamma)}_R$ denotes the result of extending all basis functions in $\mathcal{B}^\Gamma_R$ to $\operatorname{diam}(\Gamma)$, then $\mathcal{B}^{\operatorname{diam}(\Gamma)}_R$ is a basis for $\mathcal{H}^{\operatorname{diam}(\Gamma)}_R$. At least for the bases constructed below, the reverse is also true.
A vertex $v\in\Gamma$ is a line-segment in $\Gamma\subset\mathbb{Z}^2$ if it has exactly two neighbors $w_1$ and $w_2$ in $\Gamma$, and if $v$, $w_1$ and $w_2$ lie on a line. We denote by $\operatorname{lines}(\Gamma)\subset\Gamma$ the set of all line-segments in $\Gamma$.
With these definitions, our algorithm for the construction of a basis for the module $\mathcal{H}^\Gamma_\mathbb{Z}$ of integer-valued harmonic functions is given in Figure \[fig:basisAlgo\], and exemplified in Figure \[fig:properBasis\]C.
\[algoBasisLemma\] For every finite convex domain $\Gamma\subset\mathbb{Z}^2$, the algorithm terminates and returns a basis $\mathcal{B}^\Gamma_\mathbb{Z}$ for the module $\mathcal{H}^\Gamma_\mathbb{Z}$.
It is easy to see that, in every step $s$, there always exist at least one vertex $v_s$ such that $\Gamma_{s-1}\cup\{v_s\}$ is convex and $\operatorname{lines}(\Gamma_{s-1}\cup\{v_s\})\subseteq\operatorname{lines}(\Gamma)$. To prove termination of the algorithm, we thus only have to show that, independent of the choice of $v_s$, at least one of the harmonic functions ${B^+_{\geq i}}, {B^+_{\leq i}}, {B^-_{\geq j}}, {B^-_{\leq j}}$ is zero on $\Gamma_{s-1}$. For $s=1$, this is trivially true. For $s>1$, $v_s$ has at least one neighbor in $\Gamma_{s-1}$. Denote this vertex by $v_N$, and, w.l.o.g., assume that it is to the bottom of $v_s$. In $\Gamma_{s-1}$, $v_N$ must have at most two neighbors, since otherwise $v_s\in\operatorname{diam}(\Gamma_{s-1})$. Furthermore, $v_N$ must not have both a neighbor to the right and to the left in $\Gamma_{s-1}$, since this would constitute a line segment in $\Gamma_{s-1}$ which is not a line-segment in $\Gamma$. W.l.o.g., assume that the vertex to the right of $v_N$ is not in $\Gamma_{s-1}$, and denote this vertex by $v_R$. Furthermore, let $d^+_i$ be the diagonal going through $v_s$ and $v_R$. No vertex on this diagonal (and thus also not to the right of this diagonal) can be an element of $\Gamma_{s-1}$, since otherwise either $\Gamma_{s-1}\cup\{v_s\}$ would not be convex, $\operatorname{lines}(\Gamma_{s-1})\not\subseteq\operatorname{lines}(\Gamma)$, or $v_s\in\operatorname{diam}(\Gamma_{s-1})$. Thus, $B_s={B^+_{\geq i}}$ is a valid choice at step $s$. To show that $\mathcal{B}^\Gamma_\mathbb{Z}$ is a basis for $\mathcal{H}^\Gamma_\mathbb{Z}$, note that $B_1$ is the only harmonic function in $\mathcal{B}^\Gamma_\mathbb{Z}$ which is non-zero at $v_1$. By definition, $B_1$ takes the value $\pm 1$ at $v_1$, and thus only linear combination of $\mathcal{B}^\Gamma_\mathbb{Z}$ can be integer-valued for which the coefficient corresponding to $B_1$ is integer-valued. Assume that, in step $s$, this is true for all harmonic functions in $\mathcal{B}_{s-1}$. Then, since $B_s$ takes the value $\pm 1$ at $v_s$, this is also true for $B_s$. Thus, the functions in $\mathcal{B}^\Gamma_\mathbb{Z}$ are linearly independent. Since $|\mathcal{B}_s|=|\partial\Gamma_s|=s$, $|\mathcal{B}^\Gamma_\mathbb{Z}|=|\partial\Gamma|$. Thus, $\mathcal{B}^\Gamma_\mathbb{Z}$ is a basis for $\mathcal{H}^\Gamma_\mathbb{R}$. We conclude our proof by noting that $\mathcal{H}^\Gamma_\mathbb{Z}\subset\mathcal{H}^\Gamma_\mathbb{R}$.
\[corollary:properSquareBasis\] Let $\Gamma_N\subset\mathbb{Z}^2$ be an $N\times N$ square domain. Assume that $d^+_0$ and $d^-_0$ correspond to the main diagonals of the domain. Then, (i) for $N=1$, $\{{B^+_{\geq 0}}\}$ is a basis for $\mathcal{H}^\Gamma_\mathbb{Z}$; (ii) for $N\in 2\mathbb{N}$, $\{{B^+_{\geq i}},\allowbreak{B^+_{\leq -i}},\allowbreak{B^-_{\geq i}},\allowbreak{B^-_{\leq -i}}\}_{i=1,\ldots,N-1}$ is a basis; and (iii) for $N\in 2\mathbb{N}+1$, $\{{B^+_{\geq 0}},\allowbreak{B^+_{\leq -1}},\allowbreak{B^-_{\geq 1}},\allowbreak{B^-_{\leq -1}}\} \cup\allowbreak\{{B^+_{\geq i}},\allowbreak{B^+_{\leq -i}},\allowbreak{B^-_{\geq i}},\allowbreak{B^-_{\leq -i}}\}_{i=2,\ldots,N-1}$ is a basis for $\mathcal{H}^\Gamma_\mathbb{Z}$.
See construction in Figure \[fig:properBasis\]C.
A harmonic function related to primes
=====================================
![In each quadrant of the domain, $H^{\pi}=H^++H^-$ is the sum of two harmonic functions $H^+$ and $H^-$. Our proof of Lemma \[lemma:divPrime\] is based on showing that $(x-y)H^+(x,y)=-(x+y)H^+(x,-y)$. []{data-label="fig:primeConstruction"}](Figure7.pdf){width="75.00000%"}
To prove Lemma \[lemma:divPrime\], we first observe that, for $N=1$, $G^\Gamma\cong\mathbb{Z}/4\mathbb{Z}\supseteq\mathbb{Z}/2\mathbb{Z}$, in agreement with the lemma. Since $N+1$ must be prime, we thus assume that $N\in 2\mathbb{N}$ in the following.
Define the diagonals $d^+_0$ and $d^-_0$ (see previous section) such that they do not intersect at a vertex, and choose the harmonic basis functions ${B^+_{\geq 2i-1}}$, ${B^+_{\leq -2i+1}}$,${B^-_{\geq 2i-1}}$, and ${B^-_{\leq -2i+1}}$, $i\in\mathbb{N}$, as described in Figure \[fig:properBasis\]B. The harmonic function depicted in Figure \[fig:primeHarmonic\]B is then given by $$\begin{aligned}
H^\pi=\sum_{i=0}^\infty(-1)^{i}({B^+_{\geq 2i+1}}+{B^+_{\leq -(2i+1)}}-{B^-_{\geq 2i+1}}-{B^-_{\leq -(2i+1)}}).\end{aligned}$$
By definition, $H^\pi$ is symmetric with respect to $d^+_0$ and $d^-_0$. Thus, we w.l.o.g. only consider the quadrant of the domain below $d^+_0$ and above $d^-_0$, and label the vertices in this quadrant by coordinates $(x,y)$, with $x=1,3,5,\ldots$ and $y=\pm 1,\pm 3,\ldots\pm x$, as shown in Figure \[fig:primeConstruction\]. In this quadrant, $H^\pi=H^++H^-$ with $H^+=\sum_{i=0}^\infty(-1)^{i}{B^+_{\geq 2i+1}}$, and $H^-=\sum_{i=0}^\infty(-1)^{i+1}{B^-_{\geq 2i+1}}$ (Figure \[fig:primeConstruction\]). We claim that, in this quadrant, $(x-y)H^+(x,y)=-(x+y)H^+(x,-y)$. Since $H^+(x,y)=-H^-(x,-y)$, this implies that $$\begin{aligned}
H^\pi=&H^+(x,y)+H^-(x,y)
=H^+(x,y)-H^+(x,-y)\\
=&H^+(x,y)+\frac{x-y}{x+y}H^+(x,y)
=\frac{2x}{x+y}H^+(x,y).\end{aligned}$$ For $|y|<x$, since $y\neq 0$ and $H^+(x,y)$ is integer-valued, $H^\pi(x,y)$ is thus divisible by $x$ given that $x$ is prime. Thus, if our claim holds, Lemma \[lemma:divPrime\] directly follows from the discussion in the Introduction when setting $x=N+1$.
For small enough values of $x$, it is possible to directly check our claim for all $|y|\leq x$ (Figure \[fig:primeHarmonic\]). Furthermore, it is easy to validate (by induction, in this order) that $H^+(x,x)=\pm 1$, $H^+(x,-x)=0$, $H^+(x,x-2)=\mp (x-1)$, $H^+(x,-x+2)=\pm 1$, $H^+(x,x-4)=\pm(x-2)^2$, and $H^+(x,-x+4)=\mp (2x-4)$. Thus, our claim also holds close to the diagonals $d_0^+$ and $d_0^-$.
Now, assume that our claim holds for all $\hat{x}\leq x$. Then, [$$\begin{aligned}
H^+(x+2,y)=&4H^+(x,y)-H^+(x,y-2)-H^+(x,y+2)-H^+(x-2,y)\\
=&-4\frac{x+y}{x-y}H^+(x,-y)+\frac{x+y-2}{x-y+2}H^+(x,-y+2)\\
&+\frac{x+y+2}{x-y-2}H^+(x,-y-2)+\frac{x+y-2}{x-y-2}H^+(x-2,-y)\\
=&-\frac{x+y+2}{x-y+2}H^+(x+2,-y)+\frac{4}{x-y+2}\epsilon(x,-y),\end{aligned}$$ ]{} with [$$\begin{aligned}
\epsilon(x,y) =& -\frac{4y}{x-y}H^+(x,-y)-H^+(x,-y+2)\\
&+\frac{x+y+2}{x-y-2}H^+(x,-y-2)+\frac{2y}{x-y-2}H^+(x-2,-y)\\
=& \frac{4y}{x+y}H^+(x,y)+\frac{x-y+2}{x+y-2}H^+(x,y-2)\\
&-H^+(x,y+2)-\frac{2y}{x+y-2}H^+(x-2,y).\end{aligned}$$ ]{} The following calculations, which show that $\epsilon(x,y)=0$, are “a bit tedious” to check by hand, and we thus recommend using a computer algebra system. We first utilize that $H^+$ is harmonic to replace $H^+(x,y)$ by $4H^+(x-2,y)-H^+(x-2,y+2)-H^+(x-2,y-2)-H^+(x-4,y)$, and similarly for $H^+(x,y-2)$ and $H^+(x,y-2)$. We then utilize that, by the inductive assumption, $\epsilon(\hat{x},y)=0$ for all $\hat{x}<x$ to replace $H^+(x-2,y+4)$ by $\frac{4y+8}{x+y}H^+(x-2,y+2)+\frac{x-y-2}{x+y-2}H^+(x-2,y)-\frac{2y+4}{x+y-2}H^+(x-4,y+2)$, and similarly for $H^+(x-2,y+2)$ and $H^+(x-2,y-4)$, which also eliminates $H^+(x-2,y-2)$. The only remaining term in column $x-2$ is then in $H^+(x-2,y)$, which we replace again by $4H^+(x-4,y)-H^+(x-4,y+2)-H^+(x-4,y-2)-H^+(x-6,y)$. We then get that $\epsilon(x,y)=-\frac{x+y-6}{x+y-2}\epsilon(x-4,y)=0$, which concludes the proof.
Diamond-shaped harmonic functions
=================================
From the discussion in the Introduction, it becomes clear that every harmonic function of the type depicted in Figure \[fig:primeHarmonic\]C directly corresponds to a cyclic subgroup $\mathbb{Z}\setminus 4\mathbb{Z}\subseteq G_\Gamma$ of the sandpile group $G_\Gamma$ with order four. To prove Lemma \[lemma:div4\], we thus only have to show that there exist $N$ such harmonic functions which are linearly independent. Since, for $N=1$, the lemma is trivially satisfied ($S_1=\mathbb{Z}/4\mathbb{Z}\cong G^\Gamma$), we assume $N>1$ in the following.
For $N\in 2\mathbb{N}$, define $d^+_0$ and $d^-_0$ such that they correspond to the main diagonals of the domain. Then, by Corollary \[corollary:properSquareBasis\], $\mathcal{B}^\Gamma_\mathbb{Z}=\{{B^+_{\geq i}}|_\Gamma,\allowbreak{B^+_{\leq -i}}|_\Gamma,\allowbreak{B^-_{\geq i}}|_\Gamma,\allowbreak{B^-_{\leq -i}}|_\Gamma\}_{i=1,\ldots,N-1}$ is a basis for the $\mathcal{H}_\mathbb{Z}^\Gamma$ on $\Gamma$. Recall, that ${B^+_{\geq i}}$ is defined on the whole of $\mathbb{Z}^2$, takes values in $\{-1,+1\}$ on its defining diagonal $d^+_i$, and is zero below it (and similar for ${B^+_{\leq -i}}$, ${B^-_{\geq i}}$, and ${B^-_{\leq -i}}$). Also recall that on every diagonal $d_j^+$, $j>i$, we can still choose the value of one vertex, which then determines the values ${B^+_{\geq i}}$ of all other vertices on the same diagonal [@Buhovsky2017]. If we always choose these “free values” to be zero, ${B^+_{\geq i}}$ becomes divisible by four everywhere except on its defining diagonal $d_i^+$ (Figure \[fig:primeHarmonic\]C).
On the boundary $\partial\Gamma^{op}$ of the complement $\Gamma^{op}=\mathbb{Z}^2\setminus\Gamma$ of the domain, this implies that ${B^+_{\geq i}}$ is divisible by four, except for two vertices for which ${B^+_{\geq i}}$ takes a value of $\pm 1$ (Figure \[fig:primeHarmonic\]A). Furthermore, for each vertex $v\in\partial\Gamma^{op}$ on this boundary, there exist exactly two basis functions $B_i, B_j\in\mathcal{B}^{\Gamma}_\mathbb{Z}$ with $B_i(v), B_j(v)\in\{-1, +1\}$. Thus, for each $i=1,\ldots,N$, we can define the harmonic function $H^\diamond_i={B^+_{\geq i}}\pm{B^-_{\geq N-i+1}}\pm{B^+_{\leq -i}}\pm{B^-_{\leq -N+i-1}}$, where the signs are chosen such that the values of the basis functions on the boundary which are $\pm 1$ cancel each other out (Figure \[fig:primeHarmonic\]C). It then directly follows that $\Delta_\Gamma (H^\diamond_i|_\Gamma)$ is divisible by four.
We can always replace one basis function in a basis for $\mathcal{H}_\mathbb{Z}^\Gamma$ by the sum of it and an integer-multiple of another function in the same basis. That is, if $\{B_1,\ldots,B_i,\ldots B_{|\partial\Gamma|}\}$ is a basis for $\mathcal{H}_\mathbb{Z}^\Gamma$, $\{B_1,\ldots,B_i+z B_j,\ldots B_{|\partial\Gamma|}\}$ is so, too, for every $z\in\mathbb{Z}$ and $i\neq j$. Trivially, since ${B^+_{\geq N}}$, ${B^+_{\leq -N}}$,${B^-_{\geq N}}$ and ${B^-_{\leq -N}}$ evaluate to zero in $\Gamma$, we can also add their restriction to $\Gamma$ to any basis functions. Together, this means that $\{H^\diamond_1|_\Gamma,\allowbreak{B^+_{\leq -1}}|_\Gamma,\allowbreak H^\diamond_N|_\Gamma,\allowbreak{B^-_{\leq -1}}|_\Gamma\}\cup\{H^\diamond_i|_\Gamma,\allowbreak{B^+_{\leq -i}}|_\Gamma,\allowbreak{B^-_{\geq i}}|_\Gamma,\allowbreak{B^-_{\leq -i}}|_\Gamma\}_{i=2,\ldots,N-1}$ is also a basis for $\mathcal{H}_\mathbb{Z}^\Gamma$, from which Lemma \[lemma:div4\] directly follows.
For $N\in 2\mathbb{N}+1$, by Corollary \[corollary:properSquareBasis\], $\{{B^+_{\geq 0}}|_\Gamma,\allowbreak{B^+_{\leq -1}}|_\Gamma,\allowbreak{B^-_{\geq 1}}|_\Gamma,\allowbreak{B^-_{\leq -1}}|_\Gamma\} \cup\allowbreak\{{B^+_{\geq i}}|_\Gamma,\allowbreak{B^+_{\leq -i}}|_\Gamma,\allowbreak{B^-_{\geq i}}|_\Gamma,\allowbreak{B^-_{\leq -i}}|_\Gamma\}_{i=2,\ldots,N-1}$ is a basis for $\mathcal{H}^\Gamma_\mathbb{Z}$. By a similar argument as before, we get that also $\{{B^+_{\geq 0}}|_\Gamma,\allowbreak{B^+_{\leq -1}}|_\Gamma,\allowbreak H^\diamond_N|_\Gamma,\allowbreak{B^-_{\leq -1}}|_\Gamma\} \cup\allowbreak\{H^\diamond_i|_\Gamma,\allowbreak{B^+_{\leq -i}}|_\Gamma,\allowbreak{B^-_{\geq i}}|_\Gamma,\allowbreak{B^-_{\leq -i}}|_\Gamma\}_{i=2,\ldots,N-1}$ is a basis. Note that, different to before, this basis only contains $N-1$ diamond shaped basis functions $H^\diamond_i$. However, the Laplacian of ${B^+_{\geq 0}}$ is divisible by four, too, which concludes our proof of Lemma \[lemma:div4\].
[^1]: To whom correspondence should be addressed. E-mail: moritz.lang@ist.ac.at
[^2]: This claim of priority might be controversial: there exist several approaches to define sandpile models directly on $\mathbb{Z}^2$, which cope with the occurrence of infinite avalanches in various ways. The (weak) limits of the sandpile measures for some of these models can be associated to/concentrate on certain abelian groups, see e.g. [@Maes2004; @Athreya2004; @Maes2005; @Jarai2015] and references therein. At least to us, it is however unclear if and how these groups exactly relate to the categorical notion of (scaling) limits for the sandpile group employed in this article.
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abstract: 'In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.'
address:
- |
Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629\
2, rue de la Houssinière\
44322 Nantes Cedex 03, France.
- |
Laboratoire de Mathématiques J. Leray, Université de Nantes, UMR CNRS 6629\
2, rue de la Houssinière\
44322 Nantes Cedex 03, France.
author:
- Benoît Grébert
- Laurent Thomann
title: KAM for the quantum harmonic oscillator
---
[^1]
Introduction
============
Let $\Psi : { \mathbb{N} }\longrightarrow [0,+\infty[$ so that $\Psi(j)\geq j$ for all $j\geq 1$. We consider the (complex) Hilbert space ${\ell}^{2}_{\Psi}$ defined by the norm $$\|w\|_{\Psi}^{2}=\sum_{j\geq 1}|w_{j}|^{2}\Psi^{2}(j).$$ We define the symplectic phase space $\mathcal{P}=\mathcal{P}^{\Psi}$ as $$\label{def.esp}
\mathcal{P}={ \mathbb{T} }^{n}\times { \mathbb{R} }^{n}\times {\ell}^{2}_{\Psi} \times {\ell}^{2}_{\Psi},$$ equipped with the canonic symplectic structure: $$\sum_{j=1}^n d\theta_j\wedge dy_j\ +\ \sum_{j\geq 1} du_j\wedge dv_j.$$ For $(\theta,y,u,v) \in \mathcal{P}$ we introduce the following Hamiltonian in normal form $$\label{Ham.N}
N=\sum_{j=1}^{n}{ \omega }_{j}(\xi)y_{j}+\frac12\sum_{j\geq 1}\Omega_{j}(\xi)(u_{j}^{2}+v_{j}^{2}),$$ where $\xi \in { \mathbb{R} }^{n}$ is an external parameter.\
In [@Kuk1], (see also [@Kuk2] and a slightly generalised version in [@Poschel]) S.B. Kuksin has shown the persistence of $n-$dimensional tori for the perturbed Hamiltonians $H=N+P$ with general conditions on the frequencies ${ \omega }_{j}, \Omega_{j}$ and perturbation $P$ which essentially are the following : Firstly the frequencies satisfy some Melnikov conditions and the external frequencies $\Omega_{j}$ have to be well separated in the sense that there exists $d\geq 1$ so that roughly speaking (see Assumption \[AS2\] below) $$\label{freq}
\Omega_{j}(\xi)\approx j^{d}.$$ Denote by $\mathcal{P}^{a,p}$ the phase space given by the weight $\Psi(j)= j^{p/2}{ \text{e} }^{aj}$ where $p\geq 0$ and $a\geq 0$. Secondly, the perturbation is real analytic and the corresponding Hamiltonian vector field is so that $$\label{XP}
X_{P}:\mathcal{P}^{a,p}\longrightarrow \mathcal{P}^{a,{ \overline }{p}} \quad
\text{with} \quad \left\{\begin{array}{ll}
\quad { \overline }{p}\geq p &\text{for} \quad d>1, \\[6pt]
\quad { \overline }{p}> p &\text{for} \quad d=1, \end{array} \right.$$ where $d$ is the constant which appears in . For instance, the Schrödinger and the wave equation on $[0,\pi]$ with Dirichlet boundary conditions satisfy the previous conditions, see respectively the KAM results of Kuksin-Pöschel [@KukPosch] and Pöschel [@Poschel2]. Indeed the result in [@KukPosch] is stronger because there is no external parameter $\xi$ in the equation.\
Now, if we consider the nonlinear harmonic oscillator $$\label{NLS}
i\partial_t u=-\partial^{2}_{x} u +x^{2}u +V(x)u+ |u|^{2m}u ,\quad
(t,x)\in{ \mathbb{R} }\times {{ \mathbb{R} }},$$ with real and bounded potential $V$, we have $\Omega_{j}\sim 2j+1$, hence $d=1$ but the Hamiltonian perturbation which is here $$\label{pert}
P=\int_{{ \mathbb{R} }}(u\bar u)^{m+1}\text{d}x,$$ does not satisfy the strict smoothing condition (see Section \[Sect.5\] for more details).\
The aim of this paper is to prove a KAM theorem (Theorem \[thmKAM\]) in the case $d=1$ and $p={ \overline }{p}$ in and . To compensate the lack of smoothing effect of $X_{P}$ we need some additional conditions (see Assumption \[AS4\]) on the decay of the $P$ derivatives (in the spirit of the so-called Töplitz-Lipschitz condition used by Eliasson & Kuksin in [@ElKuk1]) which will be satisfied by the perturbation . The general strategy is explained with more details in Section \[Strategy\].\
Notice that S.B. Kuksin has already considered in [@Kuk2] the harmonic oscillator with a smoothing nonlinearity of type ${\displaystyle}P=\int_{{ \mathbb{R} }}{ \varphi }(|u\star \xi|)\text{d}x$ where $\xi$ is a fixed smooth function.\
We present two applications of our abstract result concerning the harmonic oscillator $T=-\partial^{2}_{x}+x^{2}$. Let $p\geq 2$ and denote by $\ell^{2}_{p}$ the space $\ell^{2}_{\Psi}$ with $\Psi(j)=j^{p/2}$. The operator $T$ has eigenfunctions $(h_{j})_{j\geq 1}$ (the Hermite functions) which satisfy $Th_{j}=(2j-1)h_{j}, \;j\geq 1$ and form a Hilbertian basis of $L^{2}({ \mathbb{R} })$. Let $u=\sum_{j\geq1}u_{j}h_{j}$ be a typical element of $L^{2}({ \mathbb{R} })$. Then $(u_{j})_{j\geq 1}\in \ell^{2}_{p}$ if and only if $u\in { \mathcal{H} }^{p}:= D(T^{p/2})=\{u\in L^{2}({ \mathbb{R} })\mid T^{p/2}u\in L^{2}({ \mathbb{R} })\}$. Indeed ${ \mathcal{H} }^{p}$ is a Sobolev space based on $T$ and we can check that $${ \mathcal{H} }^{p}=D(T^{p/2})=\{u\in L^{2}({ \mathbb{R} })\mid x^\alpha\partial^\beta u \in L^2({ \mathbb{R} }) \text{ for } \alpha+\beta\leq p\}.$$ In this context, we are able to apply our KAM result to and we obtain (see Theorem \[main1\] for a more precise statement)
\[theo:NLS\] Let $m\geq 1$ be an integer. For typical potential $V$ and for $\epsilon>0$ small enough, the nonlinear Schrödinger equation $$\label{NLS:theo}
i\partial_t u=-\partial^{2}_{x} u +x^{2}u +{\varepsilon}V(x)u\pm \epsilon |u|^{2m}u$$ has many quasi-periodic solutions in ${ \mathcal{H} }^{\infty}$.
Here the notion of “typical potential” is vague. This means that there exists rather a large class of perturbations of the harmonic oscillator so that the result of Theorem \[theo:NLS\] holds true (unfortunately our result does not cover the case $V=0$). Since the definition of this class is technical, we postpone it to Section \[Sect.5\].\
The physical motivation for considering equation (for $V=0$) comes from the Gross-Pitaevski equation used in the study of Bose-Einstein condensation (see [@GP]). The harmonic potential $x^{2}$ arises from a Taylor expansion near the bottom of general smooth well. In our work, we have to add a small linear perturbation $V$ to the harmonic potential in order to avoid resonances (see the non resonance condition below).\
The generalisation of such a result in a multidimensional setting is not evident for a spectral reason: the spectrum of the linear part is no more well separated. We could expect to adapt the tools introduced in [@ElKuk1] but the arithmetic properties of the corresponding spectra are not the same: in [@ElKuk1] the free frequencies are $j_1^2+j_2^2+\cdots+j_D^2$ for all $j_1,\cdots, j_D\in { \mathbb{Z} }$, while in our case they are $2(j_1+j_2+\cdots+j_D)+D$ for all $j_1,\cdots, j_D\in { \mathbb{N} }$. Nevertheless we mention that it is still possible to obtain a Birkhoff normal form for as recently proved in [@GIP].\
A consequence of Theorem \[theo:NLS\] is the existence of periodic solutions to . There are other approaches to construct periodic solutions of this equation. For instance, the gain of compacity yielded by the confining potential $x^{2}$ allows the use of variational methods. We develop this point of view in the appendix.\
The second application concerns the reducibility of a linear harmonic oscillator, $T=-\partial^{2}_{x}+x^{2}$, on $L^2({ \mathbb{R} })$ perturbed by a quasi periodic in time potential. Such kind of reducibility result for PDE using KAM machinery was first obtained by Bambusi & Graffi [@BamGra] for Schrödinger equation with an $x^\beta$ potential, $\beta$ being strictly larger than 2 (notice that in that case the exponent $d>1$ in the asymptotic of the frequencies ). This result was recently extended by Liu and Yuan [@LiuYuan] to include the Duffing oscillator.\
Here we follow the more recent approach developed by Eliasson & Kuksin (see [@ElKuk2]) for the Schrödinger equation on the multidimensional torus. Namely we consider the linear equation $$i\partial_t u=-\partial^{2}_{x} u +x^2u +\epsilon V(t\omega,x)u, \quad u=u(t,x),\ x\in { \mathbb{R} },$$ where $\epsilon >0$ is a small parameter and the frequency vector $\omega$ of forced oscillations is regarded as a parameter in $\mathcal U \subset { \mathbb{R} }^n$. We assume that the potential $V: \ { \mathbb{T} }^n\times { \mathbb{R} }\ni(\theta, x)\mapsto { \mathbb{R} }$ is analytic in $\theta$ on $|\text{Im}\,\theta|<s$ for some $s>0$, and $\mathcal{C}^{2}$ in $x$, and we suppose that there exists $\delta>0$ and $C>0$ so that for all $\theta \in [0,2\pi)^{n}$ and $x\in { \mathbb{R} }$ $$\label{*}
|V(\theta,x)|\leq C(1+x^{2})^{-\delta},\quad\; |\partial_{x}V(\theta,x)|\leq C,\quad\; |\partial_{xx}V(\theta,x)|\leq C.$$ In Section \[Sect.6\] we consider the previous equation as a linear non-autonomous equation in the complex Hilbert space $L^2({ \mathbb{R} })$ and we prove (see Theorem \[thmKAM2\] for a more precise statement)
\[theo:LS\] Assume that $V$ satisfies . Then there exists $\epsilon_0$ such that for all $0\leq\epsilon<\epsilon_0$ there exists $\Lambda_{{\varepsilon}} \subset [0,2\pi)^n$ of positive measure and asymptotically full measure: $\mbox{Meas}(\Lambda_{{\varepsilon}} ) \to (2\pi)^n$ as $\epsilon \to 0$, such that for all $\omega\in \Lambda_{{\varepsilon}} $, the linear Schrödinger equation $$\label{LS}
i\partial_t u=-\partial^{2}_{x} u +x^2u +\epsilon V(t\omega,x)u$$ reduces, in $L^2({ \mathbb{R} })$, to a linear equation with constant coefficients (with respect to the time variable).
In particular, we prove the following result concerning the solutions of .
\[Coro1.3\] Assume that $V$ is $\mathcal{C}^{\infty}$ in $x$ with all its derivatives bounded and satisfying . Let $p\geq 0$ and $u_{0}\in { \mathcal{H} }^{p}$. Then there exists ${\varepsilon}_{0}>0$ so that for all $0<{\varepsilon}<{\varepsilon}_{0}$ and ${ \omega }\in \Lambda_{{\varepsilon}}$, there exists a unique solution $u \in \mathcal{C}\big({ \mathbb{R} }\,;\,{ \mathcal{H} }^{p}\big)$ of so that $u(0)=u_{0}$. Moreover, $u$ is almost-periodic in time and we have the bounds $$(1-{\varepsilon}C)\|u_{0}\|_{{ \mathcal{H} }^{p}}\leq \|u(t)\|_{{ \mathcal{H} }^{p}}\leq (1+{\varepsilon}C)\|u_{0}\|_{{ \mathcal{H} }^{p}}, \quad \forall \,t\in { \mathbb{R} },$$ for some $C=C(p,{ \omega })$.
In the very particular case where $V$ satisfies and is independent of $\theta$, the result of Corollary \[Coro1.3\] is easy to prove. In that case, the solution of reads $$u(t,x)=\sum_{n\geq 0}c_{n}{ \text{e} }^{-i\lambda^{2}_{n}t}{ \varphi }_{n}(x),$$ where $({ \varphi }_{n})_{n\geq 0}$ and $(\lambda_{n})_{n\geq 0}$ are the eigenfunctions and the eigenvalues of $-\partial_{x}^{2}+x^{2}+{\varepsilon}V(x)$, and some $(c_{n})_{n\geq 0}\in { \mathbb{C} }$. The result follows thanks to the asymptotics of ${ \varphi }_{j}$ when ${\varepsilon}\to 0$ (see Section \[Sect.5\] for similar considerations.)
The previous results show that all solutions to remain bounded in time, for a large set of parameters ${ \omega }\in [0,2\pi)^{n}$. A natural question is whether we can find a real valued potential $V$, quasi-periodic in time and a solution $u\in { \mathcal{H} }^{p}$ so that $\|u(t)\|_{{ \mathcal{H} }^{p}}$ does not remain bounded when $t\longrightarrow +\infty$. J.-M. Delort [@Delort] has recently shown that this is the case if $V$ is replaced by a pseudo differential operator : he proves that there exist smooth solutions so that for all $p\geq 0$ and $t\geq 0$, $\|u(t)\|_{{ \mathcal{H} }^{p}}\geq c t^{p/2}$, which is the optimal growth. We also refer to the introduction of [@Delort] for a survey on the problem of Sobolev growth for the linear Schrödinger equation.\
Another way to understand the result of Theorem \[theo:LS\] is in term of Floquet operator (see [@Eli] and [@Wang] for mathematical considerations, and [@EV; @KY] for the physical meaning). Consider on $L^2({ \mathbb{R} })\otimes L^2({ \mathbb{T} }^n)$ the Floquet Hamiltonian $$\label{Floq}
K:=i\sum_{k=1}^n\omega_k \frac{\partial}{\partial \theta_k} -\partial^{2}_{x} +x^2 +\epsilon V(\theta,x),$$ then we have
Assume that $V$ satisfies . There exists ${\varepsilon}_{0}>0$ so that for all $0<{\varepsilon}<{\varepsilon}_{0}$ and ${ \omega }\in \Lambda_{{\varepsilon}}$, the spectrum of the Floquet operator $K$ is pure point.
A similar result, using a different KAM strategy, was obtained by W.M. Wang in [@Wang] in the case where $$V(t\omega,x)=|h_1(x)|^2\sum_{k=1}^n \cos(\omega_k t +{ \varphi }_k)$$ where $h_1$ is the first Hermite function.\
At the end of Section \[Sect.6\] we make explicit computations in the case of a potential which is independent of the space variable. This example shows that one can not avoid to restrict the choice of parameters ${ \omega }$ to a Cantor type set in Theorem \[theo:LS\].
The first author thanks Hakan Eliasson and Serguei Kuksin for helpful suggestions at the principle of this work. Both authors thank Didier Robert for many clarifications in spectral theory.
Statement of the abstract result
================================
We give in this section our abstract KAM result.\
The assumptions on the Hamiltonian and its perturbation
-------------------------------------------------------
\
Let $\Pi\in { \mathbb{R} }^{n}$ be a bounded closed set so that $\text{Meas}(\Pi)>0$, where $\text{Meas}$ denote the Lebesgue measure in ${ \mathbb{R} }^{n}$. The set $\Pi$ is the space of the external parameters $\xi$. Denote by $\Delta_{\xi\eta}$ the difference operator in the variable $\xi$ : $$\Delta_{\xi\eta}f=f(\cdot,\xi)-f(\cdot,\eta).$$ For $\l =(l_{1},\dots, l_{k},\dots)\in { \mathbb{Z} }^{\infty}$ so that only a finite number of coordinates are non zero, we denote by ${\displaystyle}|l|=\sum_{j=1}^{\infty}|l_{j}|$ its length, and ${\displaystyle}{ \langle }l{ \rangle }=1+|\sum_{j=1}^{\infty}jl_{j}|$. We set ${\displaystyle}\mathcal{Z}=\{(k,l)\neq 0,\;|l|\leq 2\}\subset { \mathbb{Z} }^{n}\times { \mathbb{Z} }^{\infty}.$\
The first two assumptions we make, concern the frequencies of the Hamiltonian in normal form
\[AS1\] Denote by ${ \omega }=({ \omega }_{1},\dots,{ \omega }_{n})$ the internal frequencies. We assume that the map $\xi \mapsto { \omega }(\xi)$ is an homeomorphism from $\Pi$ to its image which is Lipschitz continuous and its inverse also.\
Moreover we assume that for all $(k,l)\in \mathcal{Z}$ $$\label{Non.dg}
\text{Meas}\Big(\big\{\,\xi\;:\:k\cdot { \omega }(\xi)+l\cdot \Omega(\xi)=0\,\big\}\Big) =0,$$ and for all $\xi \in \Pi$ $$l\cdot \Omega(\xi)\neq 0, \quad \forall\, 1\leq |l|\leq 2.$$
\[AS2\] Set $\Omega_{0}=0$. We assume that there exists $m>0$ so that for all $i,j\geq 0$ and uniformly on $\Pi$ $$| \Omega_{i}-\Omega_{j}|\geq m|i-j|.$$ Moreover we assume that there exists $\beta>0$ such that the functions $$\xi \longmapsto j^{2\beta} \Omega_{j}(\xi),$$ are uniformly Lipschitz on $\Pi$ for $j\geq 1$.
If the previous assumptions are satisfied (and actually without assuming ), J. Pöschel [@Poschel] proves that there exist a finite set $\mathcal{X}\subset \mathcal{Z}$ and $\widetilde{\Pi}_{{ \alpha }}\subset \Pi$ with $\text{Meas}(\Pi\backslash \widetilde{\Pi}_{{ \alpha }})\longrightarrow 0$ when ${ \alpha }\longrightarrow 0$, such that for all $\xi \in \widetilde{\Pi}_{{ \alpha }}$ $$\label{2.2bis}
\big|k\cdot { \omega }(\xi) +l\cdot \Omega(\xi) \big|\geq \alpha\frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}},\quad (k,l)\in \mathcal{Z}\backslash \mathcal{X},$$ for some large $\tau$ depending on $n$ and ${ \beta }$.\
Then assuming , J. Pöschel proves [@Poschel Corollary C and its proof] that the non resonance condition remains valid on all $\mathcal{Z}$, i.e $$\label{dio}
\big|k\cdot { \omega }(\xi) +l\cdot \Omega(\xi) \big|\geq \alpha\frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}},\quad (k,l)\in \mathcal{Z},\,\xi \in \widetilde{\Pi}_{{ \alpha }}.$$
In the sequel, we will use the distance $$|\Omega-\Omega'|_{2\beta,\Pi}=\sup_{\xi \in \Pi}\,\sup_{j\geq 1}j^{2\beta}\,|\Omega_{j}(\xi)-\Omega'_{j}(\xi)|$$ and the semi-norm $$ |\Omega|^{{ \mathcal{L} }}_{2\beta,\Pi}=\sup_{\substack{ \xi,\eta \in \Pi\\ \xi\neq \eta}}\,\sup_{j\geq 1}j^{2\beta}\,\frac{|\Delta_{\xi\eta}\,\Omega_{j}|}{|\xi-\eta|}.$$ Finally, we set $$|\omega|^{{ \mathcal{L} }}_{\Pi}+ |\Omega|^{{ \mathcal{L} }}_{2\beta,\Pi}=M,$$ where ${\displaystyle}|\omega|^{{ \mathcal{L} }}_{\Pi}=\sup_{\substack{ \xi,\eta \in \Pi\\ \xi\neq \eta}}\,\max_{1\leq k\leq n}\,\frac{|\Delta_{\xi\eta}\,\omega_{k}|}{|\xi-\eta|}.$
The proof of crucially uses the control of the Lipschitz semi-norm $|\Omega|^{{ \mathcal{L} }}_{2\beta,\Pi}$ (see [@Poschel Lemma 5]). For this reason in assumptions 3 and 4 below we have to control the Lipschitz version of each semi-norms introduced on $P$ or $X_P$.
Recall that the phase space $\mathcal{P}$ is defined by , with a weight $\Psi$ so that $\Psi(j)\geq j$, as in the beginning of the introduction. As in [@Poschel], for $s,r>0$ we define the (complex) neighbourhood of ${ \mathbb{T} }^{n}\times \big\{0,0,0\big\}$ in $\mathcal{P}$. $$\label{def.dsr}
{D}(s,r)=
\big\{(\theta,y,u,v)\in \mathcal{P}^{}\;s.t.\ |\text{Im}\, \theta|<s, |y|<r^{2}, \|u\|_{\Psi}+\|v\|_{\Psi}<r\big\}.$$ Let $r>0$. Then for $W=(X,Y,U,V)$ we define $$|W|_{r}=|X|+\frac{1}{r^{2}}|Y|+\frac{1}r\big(\,\|U\|_{\Psi}+\|V\|_{\Psi}\big).$$ The next assumption concerns the regularity of the vector field associated to $P$. Denote by $${\displaystyle}X_{P}=(\,\partial_{y}P,\, -\partial_{\theta}P,\,\partial_{v}P,\,-\partial_{u}P\,).$$ Then
\[AS3\] We assume that there exist $s,r>0$ so that $$X_{P}\,:D(s,r)\times \Pi \longrightarrow \mathcal{P}^{}.$$ Moreover we assume that for all $\xi \in \Pi$, $X_{P}(\cdot,\xi)$ is analytic in $D(s,r)$ and that for all $w\in D(s,r)$, $P(w,\cdot)$ and $X_{P}(w,\cdot)$ are Lipschitz continuous on $\Pi$.
We then define the norms $$\|P\|_{D(s,r)}:=\sup_{D(s,r)\times \Pi}|P|<+\infty,$$ and $$\|P\|^{{ \mathcal{L} }}_{D(s,r)}= \sup_{\substack{\xi,\eta\in\Pi\\ \xi\neq \eta}} \,\sup_{D(s,r)}\,\frac{|\Delta_{\xi\eta}\,P|}{|\xi-\eta|},$$ where $\Delta_{\xi\eta}\,P=P(\cdot,\xi)-P(\cdot,\eta)$ and we define the semi-norms $$\|X_{P}\|_{r,D(s,r)}:=\sup_{D(s,r)\times \Pi}|X_{P}|_{r}<+\infty,$$ and $$\|X_{P}\|_{r,D(s,r)}^{{ \mathcal{L} }}:=\sup_{\substack{\xi,\eta\in\Pi\\ \xi\neq \eta}} \,\sup_{D(s,r)}\,\frac{|\Delta_{\xi\eta}X_{P}|_{r}}{|\xi-\eta|}<+\infty.$$ where $\Delta_{\xi\eta}X_{P}=X_{P}(\cdot,\xi)-X_{P}(\cdot,\eta)$.\
In the sequel, we will often work in the complex coordinates $$z=\frac{1}{\sqrt{2}}(u-iv),\quad { \overline }{z}=\frac{1}{\sqrt{2}}(u+iv).$$ Notice that this is not a canonical change of variables and in the variables $(\theta,y,z,\bar z)\in \mathcal{P}$ the symplectic structure reads $$\sum_{j=1}^n d\theta_j\wedge dy_j\ +\ i\sum_{j\geq 1} dz_j\wedge d\bar z_j,$$ and the Hamiltonian in normal form is $$\label{Ham.Comp}
N=\sum_{j=1}^{n}{ \omega }_{j}(\xi)y_{j}+\sum_{j\geq 1}\Omega_{j}(\xi)z_{j}{ \overline }{z}_{j}.$$ As we mentioned previously we need some decay on the derivatives of $P$. We first introduce the space ${ \Gamma }^{{ \beta }}_{r,D(s,r)}$: Let ${ \beta }>0$, we say that $P\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$ if ${ \langle }P{ \rangle }_{r,D(s,r)}+{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)} <\infty $ where :\
$\bullet$ The norm ${ \langle }\,\cdot\,{ \rangle }_{r,D(s,r)}$ is defined by the conditions [^2] $$\begin{aligned}
\big\|P\big\|_{D(s,r)}&\leq & r^{2}{ \langle }P{ \rangle }_{r,D(s,r)},\\
\max_{1\leq j\leq n}\Big\|\frac{\partial P}{\partial y_{j}}\Big\|_{D(s,r)} &\leq &{ \langle }P{ \rangle }_{r,D(s,r)},
\end{aligned}$$ $$\begin{aligned}
\Big\|\frac{\partial P}{\partial w_{j}}\Big\|_{D(s,r)} &\leq &\frac{r}{j^{{ \beta }}}{ \langle }P{ \rangle }_{r,D(s,r)}, \quad \forall \,j\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j},\\
\Big\|\frac{\partial^{2} P}{\partial w_{j}\partial w_{l}}\Big\|_{D(s,r)} &\leq &\frac{1}{(jl)^{{ \beta }}}{ \langle }P{ \rangle }_{r,D(s,r)},\quad \forall \,j,l\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j}.
\end{aligned}$$ $\bullet$ The semi-norm ${ \langle }\,\cdot\,{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}$ is defined by the conditions $$\begin{aligned}
\big \|P\big\|^{{ \mathcal{L} }}_{D(s,r)}&\leq & r^{2}{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)},\\
\max_{1\leq j\leq n}\Big\|\frac{\partial P}{\partial y_{j}}\Big\|^{{ \mathcal{L} }}_{D(s,r)} &\leq &{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)},
\end{aligned}$$ $$\begin{aligned}
\Big\|\frac{\partial P}{\partial w_{j}}\Big\|^{{ \mathcal{L} }}_{D(s,r)} &\leq &\frac{r}{j^{{ \beta }}}{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}, \quad \forall \,j\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j},\\
\Big\|\frac{\partial^{2} P}{\partial w_{j}\partial w_{l}}\Big\|^{{ \mathcal{L} }}_{D(s,r)} &\leq &\frac{1}{(jl)^{{ \beta }}}{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)},\quad \forall \,j,l\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j}.
\end{aligned}$$
The last assumption is then the following
\[AS4\] $P\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$ for some $\beta >0$.
The control of the second derivative is the most important condition. The other ones are imposed so that we are able to recover the last one after the KAM iteration (see Lemma \[lem.2\]). Furthermore the assumptions on the first derivatives are already contained in Assumption \[AS3\] as soon as $p>0$.
Statement of the abstract KAM Theorem
-------------------------------------
\
\[5pt\] Recall that $M=|{ \omega }|^{{ \mathcal{L} }}_{\Pi}+|\Omega|^{{ \mathcal{L} }}_{2{ \beta },\Pi}$.
\[thmKAM\] Suppose that $N$ is a family of Hamiltonians of the form on the phase space $\mathcal{P}^{}$ depending on parameters $\xi \in \Pi$ so that Assumptions \[AS1\] and \[AS2\] are satisfied. Then there exist ${\varepsilon}_0>0$ and $s>0$ so that every perturbation $H=N+P$ of $N$ which satisfies Assumptions \[AS3\] and \[AS4\] and the smallness condition $${\varepsilon}=\big({\bf \|}X_{P}{\bf\|}_{r,D(s,r)}+{ \langle }P{ \rangle }_{r,D(s,r)}\big)+\frac{\alpha}{M}\big({\bf \|}X_{P}{\bf\|}^{{ \mathcal{L} }}_{r,D(s,r)}+{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}\big)\leq {\varepsilon}_0\alpha,$$ for some $r>0$ and $0<\alpha\leq 1$, the following holds. There exist
(i) a Cantor set $\Pi_\alpha\subset \Pi$ with $\text{Meas}(\Pi\backslash \Pi_\alpha)\rightarrow 0$ as $\alpha\rightarrow 0$ ;
(ii) a Lipschitz family of real analytic, symplectic coordinate transformations $\Phi: D(s/2,r/2)\times \Pi_\alpha\rightarrow D(s,r)$ ;
(iii) a Lipschitz family of new normal forms $$N^\star=\sum_{j=1}^n \omega_j^\star(\xi) y_j+ \sum_{j\geq 1}\Omega_j^\star(\xi) z_j\bar z_j$$ defined on $D(s/2,r/2)\times \Pi_\alpha $ ;
such that $$H\circ \Phi = N^\star +R^\star$$ where $R^\star$ is analytic on $D(s/2,r/2)$ and globally of order 3 at ${ \mathbb{T} }^n\times\{0,0,0\}$. That is the Taylor expansion of $R^\star$ only contains monomials $y^mz^q\bar z^{\bar q}$ with $2|m|+|q+\bar q|\geq 3$.\
Moreover each symplectic coordinate transformation is close to the identity $$\label{Est.Phi}
{\bf \|}\Phi-Id {\bf\|}_{r,D(s/2,r/2)}\leq c {\varepsilon},$$ the new frequencies are close to the original ones $$\label{Est.Freq}
|{ \omega }^{\star}-{ \omega }|_{\Pi_{\alpha}}+|\Omega^\star-\Omega|_{2\beta,\Pi_{\alpha}}\leq c{\varepsilon},$$ and the new frequencies satisfy a non resonance condition $$\label{NewNR}
\big|k\cdot { \omega }^\star(\xi)+l\cdot\Omega^\star(\xi) \big|\geq \frac {\alpha} 2\ \frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}},\quad (k,l)\in \mathcal{Z},\ \xi\in \Pi_\alpha .$$
As the consequence, for each $\xi\in\Pi_\alpha$ the torus $\Phi\big({ \mathbb{T} }^n\times\{0,0,0\}\big)$ is still invariant under the flow of the perturbed Hamiltonian $H=N+P$, the flow is linear ( in the new variables) on these tori and furthermore all these tori are linearly stable.
General strategy {#Strategy}
----------------
\
The general strategy is the classical one used for instance in [@Kuk1; @Kuk2; @Poschel]. For convenience of the reader we recall it. Let $H=N+P$ be a Hamiltonian, where $N$ is given by and $P$ a perturbation which satisfies the assumptions of the previous section. We then consider the second order Taylor approximation of $P$ which is $$\label{taylor}
R=\sum_{2|m|+|q+{ \overline }{q}|\leq 2}\,\sum_{k\in{ \mathbb{Z} }^{n}}R_{kmq{ \overline }{q}}\,{ \text{e} }^{ik\cdot\theta}y^{m}z^{q}{ \overline{z} }^{{ \overline }{q}},$$ with $R_{kmq{ \overline }{q}}=P_{kmq{ \overline }{q}}$ and we define its mean value by $$[R]=\sum_{|m|+|q|=1}R_{0mqq} y^{m}z^{q}{ \overline{z} }^{q}.$$ Recall that in this setting $z,{ \overline }{z}$ have homogeneity 1, whereas $y$ has homogeneity 2.\
Let $F$ be a function of the form and denote by $X^{t}_{F}$ the flow at time $t$ associated to the vector field of $F$. We can then define a new Hamiltonian by $H\circ X^{1}_{F}:=N_{+}+P_{+}$, and the Hamiltonian structure is preserved, because $X^{1}_{F}$ is a symplectic transformation. The idea of the KAM step is to find, iteratively, an adequate function $F$ so that the new error term has a small quadratic part. Namely, thanks to the Taylor formula we can write $$\begin{aligned}
H\circ X^{1}_{F}&=&N\circ X^{1}_{F}+(P-R)\circ X_{F}^{1}+R\circ X^{1}_{F} \\
&=&N+{ \big\{\,{ N,F }\, \big\} }+ \int_{0}^{1}(1-t){ \big\{\,{ { \big\{\,{ N,F }\, \big\} },F }\, \big\} }\circ X^{t}_{F}\,\text{d}t+ \\
&&+ (P-R)\circ X_{F}^{1}+R+ \int_{0}^{1}{ \big\{\,{ R,F }\, \big\} }\circ X^{t}_{F}\,\text{d}t.
\end{aligned}$$ In view of the previous equation, we define the new normal form by $N_{+}=N+{ \widehat }{N}$, where ${ \widehat }{N}$ satisfies the so-called homological equation (the unknown are $F$ and ${ \widehat }{N}$) $$\label{homol}
\big\{F,N\big\}+{ \widehat }{N}=R.$$ The new normal form $N_+$ has the form with new frequencies given by $${ \omega }^{+}(\xi)={ \omega }(\xi)+{ \widehat }{{ \omega }}(\xi) \mbox{ and }\Omega^{+}(\xi)=\Omega(\xi)+{ \widehat }{\Omega}(\xi)$$ where $$\label{newfreq} {\displaystyle}{ \widehat }{{ \omega }}_{j}(\xi)=\frac{\partial { \widehat }{N}}{\partial y_{j}}(0,0,0,0,\xi) ) \mbox{ and } {\displaystyle}{ \widehat }{\Omega}_{j}(\xi)=\frac{\partial^{2} { \widehat }{N}}{\partial z_{j} \partial { \overline }{z}_{j}}(0,0,0,0,\xi).$$ Once the homological equation is solved, we define the new perturbation term $P_{+}$ by $$\label{newP}
P_{+}=(P-R)\circ X_{F}^{1}+\int_{0}^{1}{ \big\{\,{ R(t),F }\, \big\} }\circ X^{t}_{F}\,\text{d}t,$$ where $R(t)=(1-t){ \widehat }{N}+tR$ in such a way that $$H\circ X^{1}_{F}=N_+ +P_+\ .$$ Notice that if $P$ was initially of size ${\varepsilon}$, then $R$ and $F$ are of size ${\varepsilon}$, and the quadratic part of $P_{+}$ is formally of size ${\varepsilon}^{2}$. That is, the formal iterative scheme is exponentially convergent.\
Without any smoothing effect on the regularity, there is no decreasing property in the correction term added to the external frequencies . In that case it would be impossible to control the small divisors (see ) at the next step. In this work the smoothing condition on $X_{P}$ is replaced by Assumption \[AS4\] (see also Remark \[Rq4\]). The difficulty is to verify the conservation of this assumption at each step.
Plan of the proof of Theorem \[thmKAM\] {#plan-of-the-proof-of-theorem-thmkam .unnumbered}
----------------------------------------
In Section \[Sect.Lin\] we solve the homological equation and give estimates on the solutions. Then we study precisely the flow map $X_{F}^{t}$ and the composition $H\circ X_{F}^{1}$. In Section \[Sect.KAM\] we estimate the new error term and the new frequencies after the KAM step, and Section \[Sect.Conv\] is devoted to the convergence of the KAM method and the proof of Theorem \[thmKAM\].
[Notations]{} In this paper $c$, $C$ denote constants the value of which may change from line to line. These constants will always be universal, or depend on the fixed quantities $n,\beta,\Pi,p
$.\
We denote by ${ \mathbb{N} }$ the set of the non negative integers, and ${ \mathbb{N} }^{*}={ \mathbb{N} }\backslash\{0\}$. For $\l =(l_{1},\dots, l_{k},\dots)\in { \mathbb{Z} }^{\infty}$, we denote by ${\displaystyle}|l|=\sum_{j=1}^{\infty}|l_{j}|$ its length (if it is finite), and ${\displaystyle}{ \langle }l{ \rangle }=1+|\sum_{j=1}^{\infty}jl_{j}|$. We define the space $\mathcal{Z}=\big\{ (k,l)\neq 0, \,k\in { \mathbb{Z} }^{n}, l\in { \mathbb{Z} }^{\infty},\,|l|\leq 2\,\big\}$. The notation $\text{Meas}$ stands for the Lebesgue measure in ${ \mathbb{R} }^{n}$.
In the sequel, we will state without proof some intermediate results of [@Poschel] which still hold under our conditions ; hence the reader should refer to [@Poschel] for the details. For the convenience of the reader we decided to remain as close as possible to the notations of J. Pöschel.
The linear step {#Sect.Lin}
===============
In this section, we solve equation and study the Lie transform $X^{t}_{F}$.\
Following [@Poschel], $\|\cdot\|^{*}$ (respectively ${ \langle }\,\cdot\,{ \rangle }^{*}$) stands either for $\|\cdot\|$ or $\|\cdot\|^{{ \mathcal{L} }}$ (respectively ${ \langle }\,\cdot\,{ \rangle }$ or ${ \langle }\,\cdot\,{ \rangle }^{{ \mathcal{L} }}$) and $\|\cdot\|^{\lambda}$ stands for $\|\cdot\|+\lambda \|\cdot\|^{{ \mathcal{L} }}$.\
The homological equation
------------------------
\
The following result shows that it is possible to solve equation under the Diophantine condition .
\[lem.Poschel1\] Assume that the frequencies satisfy, uniformly on $\widetilde{\Pi}_{{ \alpha }}$, for some ${ \alpha }>0$ the condition . Then the homological equation has a solution $F$, ${ \widehat }{N}$ which is normalised by $[F]=0$, $[{ \widehat }{N}]={ \widehat }{N}$, and satisfies for all $0<{ \sigma }<s$, and $0\leq \lambda\leq \alpha/M$ $${\bf \|} X_{{ \widehat }{N}}{\bf\|}^{*}_{r,D(s,r)}\leq {\bf \|}X_{R}{\bf\|}^{*}_{r,D(s,r)},\quad {\bf \|} X_{F}{\bf\|}^{\lambda}_{r,D(s-{ \sigma },r)}\leq \frac{C}{\alpha { \sigma }^{t}}{\bf \|}X_{R}{\bf\|}^{\lambda}_{r,D(s,r)},$$ where $t$ only depends on $n$ and $\tau$.
The space ${ \Gamma }^{{ \beta }}_{r,D(s,r)}$ is not stable under the Poisson bracket. Therefore we need to introduce the space ${ \Gamma }^{{ \beta },+}_{r,D(s,r)} \subset { \Gamma }^{{ \beta }}_{r,D(s,r)}$ endowed with the norm ${ \langle }\,\cdot\, { \rangle }^{+}_{r,D(s,r)}+{ \langle }\,\cdot\, { \rangle }^{+,{ \mathcal{L} }}_{r,D(s,r)}$ defined by the following conditions. $$\big\|F\big\|^{*}_{D(s,r)}\leq r^{2}{ \langle }F{ \rangle }^{+,*}_{r,D(s,r)},\quad \max_{1\leq j\leq n}\Big\|\frac{\partial F}{\partial y_{j}}\Big\|^{*}_{D(s,r)} \leq { \langle }F{ \rangle }^{+,*}_{r,D(s,r)},$$ $$\begin{aligned}
\Big\|\frac{\partial F}{\partial w_{j}}\Big\|^{*}_{D(s,r)} &\leq &\frac{r}{j^{{ \beta }+1}}{ \langle }F{ \rangle }^{+,*}_{r,D(s,r)}, \quad \forall \,j\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j},\\
\Big\|\frac{\partial^{2} F}{\partial w_{j}\partial w_{l}}\Big\|^{*}_{D(s,r)} &\leq &\frac{1}{(jl)^{{ \beta }}(1+|j-l|)}{ \langle }F{ \rangle }^{+,*}_{r,D(s,r)}\quad \forall \,j,l\geq1\quad \text{and}\quad w_{j}=z_{j}, \,{ \overline{z} }_{j}.
\end{aligned}$$ \
This definition is motivated by the following result, which can be understood as a smoothing property of the homological equation
\[lem.1\] Assume that the frequencies satisfy , uniformly on $\widetilde{\Pi}_{{ \alpha }}$. Let $F, { \widehat }{N}$ be given by Lemma \[lem.Poschel1\]. Assume moreover that $R\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$, then there exists $C>0$ so that for any $0<{ \sigma }<s$, we have $F\in { \Gamma }^{{ \beta },+}_{r,D(s-{ \sigma },r)}$, ${ \widehat }{N}\in { \Gamma }^{{ \beta }}_{r,D(s-{ \sigma },r)}$ and $${ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}\leq \frac{C}{{ \alpha }{ \sigma }^{t}} { \langle }R{ \rangle }_{r,D(s,r)},$$ $$\label{Delta+}
{ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}\leq \frac{C}{{ \alpha }{ \sigma }^{t}} \Big( { \langle }R{ \rangle }_{r,D(s,r)} + \frac{M}{\alpha}{ \langle }R{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}\Big),$$ and $${ \langle }{ \widehat }{N}{ \rangle }_{r,D(s-{ \sigma },r)}\leq { \langle }R{ \rangle }_{r,D(s,r)},\quad { \langle }{ \widehat }{N}{ \rangle }^{{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}\leq { \langle }R{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)},$$ where $t$ only depends on $n$ and $\tau$.
For the proof of this result, we need the classical lemma
\[lem.1.0\] Let $f\,:\,{ \mathbb{R} }\longrightarrow { \mathbb{C} }$ be a periodic function and assume that $f$ is holomorphic in the domain $|\text{Im}\, \theta|<s$, and continuous on $|\text{Im}\, \theta|\leq s$. Then there exists $C>0$ so that its Fourier coefficients satisfy $$|{ \widehat }{f}(k)|\leq C{ \text{e} }^{-|k|s}\sup_{{|\text{Im}\, \theta|<s}}|f(\theta)|.$$
In [@Poschel], the author looks for a solution $F$ of of the form of , i.e. $$\label{def.F}
F=\sum_{2|m|+|q+{ \overline }{q}|\leq 2}\,\sum_{k\in{ \mathbb{Z} }^{n}}F_{kmq{ \overline }{q}}\,{ \text{e} }^{ik\cdot\theta}y^{m}z^{q}{ \overline{z} }^{{ \overline }{q}}.$$ A direct computation then shows that the coefficients in are given by $$\label{solution}
iF_{kmq{ \overline }{q}}= \left\{
\begin{array}{ll}
{\displaystyle}\frac{R_{kmq{ \overline }{q}}}{k\cdot { \omega }+(q-{ \overline }{q})\cdot \Omega},\quad &\text{if} \quad |k|+|q-{ \overline }{q}|\neq 0, \\[9pt]
0 , \;\, &\text{otherwise},
\end{array}
\right.$$ and that we can set ${ \widehat }{N}=[R]$.\
In the following we will use the notation $q_{j}=(0,\cdots,0,1,0,\cdots)$, where the 1 is at the $j^{th}$ position, and $q_{jl}=q_{j}+q_{l}$.\
The variables $z$ and ${ \overline }{z}$ exactly play the same role, therefore it is enough to study the derivatives in the variable $z$.\
In the sequel we write $A_{k}=1+|k|^{\tau}$. Then it easy to check that for any $j\geq 1$ and ${ \sigma }>0$, $$\sum_{k\in { \mathbb{Z} }^{n}}A_{k}^{j}{ \text{e} }^{-|k|{ \sigma }}\leq \frac{C}{{ \sigma }^{t}},$$ for some $C>0$ and $t=2j\tau+n+1$. In the sequel, $t$ may vary from line to line, but will remain independent of ${ \sigma }$.\
$\spadesuit$ We first prove that ${\displaystyle}{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}\leq \frac{C }{{ \alpha }{ \sigma }^{t}} { \langle }R{ \rangle }_{r,D(s,r)}.$\
$\bullet$ Observe that ${\displaystyle}\frac{\partial^{2}R}{\partial z_{j}\partial z_{l}}=\sum_{k\in{ \mathbb{Z} }^{n}}R_{k\,0\,q_{jl}\,0}\,{ \text{e} }^{ik\cdot\theta}$, then according to Lemma \[lem.1.0\], there exists $C>0$ so that ${\displaystyle}|R_{k\,0\,q_{jl}\,0}|\leq C\frac{{ \langle }R{ \rangle }_{r,D(s,r)}{ \text{e} }^{-|k|s}}{(jl)^{{ \beta }}}$, and thus by and $$\label{Fkq}
|F_{k\,0\,q_{jl}\,0}|\leq C\frac{A_{k}}{\alpha}\frac{{ \langle }R{ \rangle }_{r,D(s,r)}{ \text{e} }^{-|k|s}}{(jl)^{{ \beta }}(1+|j-l|)}.$$ Therefore, as we also have $$\label{D2F}
\frac{\partial^{2}F}{\partial z_{j}\partial z_{l}}=\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,q_{jl}\,0}\,{ \text{e} }^{ik\cdot\theta},$$ we deduce that $$\begin{aligned}
\Big \|\frac{\partial^{2}F}{\partial z_{j}\partial z_{l}}\Big \|_{D(s-{ \sigma },r)} &\leq & \sum_{k\in{ \mathbb{Z} }^{n}}|F_{k\,0\,q_{jl}0}|{ \text{e} }^{|k|(s-{ \sigma })}\nonumber\\
&\leq & \frac{C{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha (jl)^{{ \beta }}(1+|j-l|)}\sum_{k\in{ \mathbb{Z} }^{n}}A_{k}{ \text{e} }^{-|k|{ \sigma }}\nonumber \\
&\leq & \frac{C{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t} (jl)^{{ \beta }}(1+|j-l|)}. \label{F0}
\end{aligned}$$ $\bullet$ We compute $$\label{decomp}
\frac{\partial F}{\partial z_{j}}=\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,q_{j}\,0}{ \text{e} }^{ik\cdot\theta}+\sum_{k\in{ \mathbb{Z} }^{n},\, l\geq 1}F_{k\,0\,q_{j}{ \overline }{q}_{l}}{ \text{e} }^{ik\cdot\theta}{ \overline{z} }_{l}+2\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,2q_{j}\,0}{ \text{e} }^{ik\cdot\theta}z_{j}.$$ Now observe that ${\displaystyle}\big(\frac{\partial R}{\partial z_{j}}\big)_{|z={ \overline{z} }=0}=\sum_{k\in{ \mathbb{Z} }^{n}}R_{k\,0\,q_{j}\,0}\,{ \text{e} }^{ik\cdot\theta},$ then by Lemma \[lem.1.0\], $$\begin{aligned}
|R_{k\,0\,q_{j}\,0}|&\leq& C { \text{e} }^{-|k|s}\, \sup_{|\text{Im}\,\theta|<s}\Big|\big(\frac{\partial R}{\partial z_{j}}\big)_{|z={ \overline{z} }=0}\Big|\\
&\leq &C \,{ \text{e} }^{-|k|s} \Big \|\frac{\partial R}{\partial z_{j}}\Big \|_{D(s,r)} \leq Cr\frac{ \,{ \text{e} }^{-|k|s}}{j^{{ \beta }}} { \langle }R{ \rangle }_{r,D(s,r)}.
$$ From the previous estimate, and we get $$ |F_{k\,0\,q_{j}\,0}|\leq \frac{A_{k}}{\alpha(1+j)}|R_{k\,0\,q_{j}\,0}|\leq \frac{CrA_{k} \,{ \text{e} }^{-|k|s}}{\alpha j^{{ \beta }}(1+j)} { \langle }R{ \rangle }_{r,D(s,r)}$$ and thus $$\begin{aligned}
\Big \|\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,q_{j}0}{ \text{e} }^{ik\cdot\theta}\Big \|_{D(s-{ \sigma },r)} &\leq & \sum_{k\in{ \mathbb{Z} }^{n}}|F_{k\,0\,q_{j}0}|{ \text{e} }^{|k|(s-{ \sigma })}\nonumber \\
&\leq & Cr\frac{{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha j^{{ \beta }}(1+j)}\sum_{k\in{ \mathbb{Z} }^{n}}A_{k}{ \text{e} }^{-|k|{ \sigma }}\nonumber \\
&\leq & \frac{C r { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}j^{{ \beta }}(1+j)}. \label{F1}
\end{aligned}$$ Similarly, we have $ {\displaystyle}|F_{k\,0\,2q_{j}0}| \leq \frac{CrA_{k} \,{ \text{e} }^{-|k|s}}{\alpha j^{{ \beta }}(1+j)} { \langle }R{ \rangle }_{r,D(s,r)},$ which leads to $$\label{F2}
\Big \|\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,2q_{j}0}{ \text{e} }^{ik\cdot\theta}\Big \|_{D(s-{ \sigma },r)} \leq \frac{Cr{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t} j^{{ \beta }}(1+j)}.$$ By Cauchy-Schwarz in the variable $l$ and , $$\begin{aligned}
\Big \|\sum_{k\in{ \mathbb{Z} }^{n},\, l\geq 1}F_{k\,0\,q_{j}{ \overline }{q}_{l}}{ \text{e} }^{ik\cdot\theta}{ \overline{z} }_{l}\Big \|_{D(s-{ \sigma },r)} &\leq & \Big( \sum_{l\geq 1}\Psi^{-2}(l)|\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,q_{j}{ \overline }{q}_{l}}{ \text{e} }^{ik\cdot\theta}|^{2}\Big)^{\frac12}\Big(\sum_{l\geq 1}|z_{l}|^{2}\Psi^{2}(l)\Big)^{\frac12} \nonumber \\
&\leq & \frac{Cr }{\alpha { \sigma }^{t} j^{{ \beta }} } \Big( \sum_{l\geq 1}\frac{1}{ l^{2{ \beta }} \Psi^{2}(l)(1+|j-l|)^{2}}\Big)^{\frac12} { \langle }R{ \rangle }_{r,D(s,r)} \nonumber \\
&\leq & \frac{Cr{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t} j^{{ \beta }}(1+j)}, \label{F3}
\end{aligned}$$ since $\Psi(l)\geq l$.\
Finally, inserting , and in we obtain $$\label{dz}
\Big \|\frac{\partial F}{\partial z_{j}}\Big \|_{D(s-{ \sigma },r)} \leq \frac{Cr{ \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t} j^{{ \beta }}(1+j)}.$$ $\bullet$ We can write ${\displaystyle}\frac{\partial F}{\partial y_{j}}=\sum_{k\in{ \mathbb{Z} }^{n}}F_{km_{j}0\,0}{ \text{e} }^{ik\cdot\theta}$. Hence by and , ${\displaystyle}|F_{km_{j}0\,0}|\leq \frac{A_{k}}{\alpha}|R_{km_{j}0\,0}|$, and thanks to Lemma \[lem.1.0\] applied to the series ${\displaystyle}\frac{\partial R}{\partial y_{j}}=\sum_{k\in{ \mathbb{Z} }^{n}}R_{km_{j}0\,0}{ \text{e} }^{ik\cdot\theta}$, $$\label{Fkm}
|F_{km_{j}0\,0}|\leq C \frac{A_{k}}{\alpha}{ \text{e} }^{-|k|s}{ \langle }R{ \rangle }_{r,D(s,r)},$$ and we obtain $$\label{dy}
\Big \|\frac{\partial F}{\partial y_{j}}\Big \|_{D(s-{ \sigma },r)} \leq \sum_{k\in{ \mathbb{Z} }^{n}}|F_{km_{j}0\,0}|{ \text{e} }^{|k|(s-{ \sigma })} \leq \frac{C }{\alpha { \sigma }^{t}} { \langle }R{ \rangle }_{r,D(s,r)}.$$ $\bullet$ To obtain the bound for $ \|F\|_{D(s-{ \sigma },r)}$ write $$\begin{gathered}
\label{decomp3}
F=\sum_{k\in{ \mathbb{Z} }^{n}}F_{k\,0\,0\,0}{ \text{e} }^{ik\cdot\theta}+\sum_{k\in{ \mathbb{Z} }^{n},1\leq j\leq n}F_{k\,m_{j}\,0\,0}{ \text{e} }^{ik\cdot\theta}y_{j}+\\
\sum_{k\in{ \mathbb{Z} }^{n},j,l\geq 1}F_{k\,0\,q_{jl}\,0}{ \text{e} }^{ik\cdot\theta}z_{j}z_{l}+\sum_{k\in{ \mathbb{Z} }^{n},j,l\geq 1}F_{k\,00\,q_{jl}}{ \text{e} }^{ik\cdot\theta}{ \overline }{z}_{j}{ \overline }{z}_{l}+\sum_{k\in{ \mathbb{Z} }^{n},j,l\geq 1}F_{k\,0q_{j}q_{l}}{ \text{e} }^{ik\cdot\theta}{ \overline }{z}_{j}{z}_{l}.\end{gathered}$$ Since $ {\displaystyle}R_{|y=z={ \overline }{z}=0}=\sum_{k\in{ \mathbb{Z} }^{n}}R_{k\,0\,0\,0}{ \text{e} }^{ik\cdot\theta}$, by Lemmas \[lem.1.0\] and \[lem.Poschel1\] we deduce that $$\label{Fk0}
\big|F_{k0\,0\,0}\big |\leq Cr^{2}\frac{A_{k}}{\alpha}{ \text{e} }^{-|k|s}{ \langle }R{ \rangle }_{r,D(s,r)},$$ hence, thanks to and we can bound the sums of the first line in as in the previous point.\
Now thanks to and to the Cauchy-Schwarz inequality we have $$\begin{aligned}
\Big \|\sum_{k\in{ \mathbb{Z} }^{n},\, j,l\geq 1}F_{k\,0\,q_{jl}0}{ \text{e} }^{ik\cdot\theta}z_{j}z_{l}\Big \|_{D(s-{ \sigma },r)} &\leq &\frac{C { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}} \sum_{j,l\geq 1}\frac{|z_{j}z_{l}|}{(jl)^{{ \beta }}(1+|j-l|)} \nonumber \\
&\leq & \frac{C { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}}\Big( \sum_{j\geq 1}\frac{|z_{j}|}{j^{{ \beta }}} \Big)^{2} \nonumber \\
&\leq & \frac{C { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}}\Big( \sum_{j\geq 1}\Psi^{2}(j)|z_{j}|^{2} \Big)\Big( \sum_{j\geq 1}\frac{1}{j^{2{ \beta }}\Psi^{2}(j) } \Big) \nonumber \\
&\leq &\frac{Cr^{2} { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}}.
\end{aligned}$$ Therefore we proved that $ {\displaystyle}\|F\|_{D(s-{ \sigma },r)} \leq \frac{Cr^{2} { \langle }R{ \rangle }_{r,D(s,r)}}{\alpha { \sigma }^{t}}$.\
This latter estimate together with the estimates , and shows that $${ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}\leq \frac{C }{{ \alpha }{ \sigma }^{t}} { \langle }R{ \rangle }_{r,D(s,r)}.$$ $\spadesuit$ We now show that $$\label{est.N}
{ \langle }{ \widehat }{N}{ \rangle }_{r,D(s-{ \sigma },r)}\leq { \langle }R{ \rangle }_{r,D(s,r)}.$$ Since ${ \widehat }{N}=[R]$ we have $$\label{N1}
{ \widehat }{N}=\sum_{j=1}^{n}R_{0m_{j}00}\,y_{j}+\sum_{j\geq 1}R_{00q_{j}q_{j}}\,z_{j}{ \overline{z} }_{j},$$ and we can observe that $$\label{N2}
R_{0m_{j}00}=\frac{1}{(2\pi)^{n}}\int_{\theta\in { \mathbb{T} }^{n}}\frac{\partial R}{\partial y_{j}}(\theta,0,0,0)\text{d}\theta, \quad R_{00q_{j}q_{j}}=\frac{1}{(2\pi)^{n}}\int_{\theta\in { \mathbb{T} }^{n}}\frac{\partial^{2} R}{\partial z_{j}\partial { \overline{z} }_{j}}(\theta,0,0,0)\text{d}\theta,$$ which imply the bounds ${\displaystyle}|R_{0m_{j}00}|\leq { \langle }R{ \rangle }_{r, D(s,r)}$ and ${\displaystyle}|R_{00q_{j}q_{j}}|\leq { \langle }R{ \rangle }_{r, D(s,r)}/j^{2\beta}$ and thus .\
$\spadesuit$ It remains to check the estimates with the Lipschitz semi norms.\
As in [@Poschel], for $|k|+|q_{j}-{ \overline }{q_{l}}|\neq 0$ define $\delta_{k,jl}=k\cdot { \omega }+\Omega_{j}-\Omega_{l}$. Then by , $$i\Delta_{\xi\eta}F_{kmq_{j}{ \overline }{q}_{l}}=\delta^{-1}_{k,jl}(\eta)\Delta_{\xi\eta}R_{kmq_{j}{ \overline }{q}_{l}}+R_{kmq_{j}{ \overline }{q}_{l}}(\xi) \Delta_{\xi\eta} \delta^{-1}_{k,jl}.$$ By , $| \delta^{-1}_{k,jl}|\leq A_{k}/\alpha$ and thus $$| \Delta_{\xi\eta} \delta^{-1}_{k,jl}|\leq \frac{A^{2}_{k}}{\alpha^{2}}\big(|k||\Delta_{\xi\eta} { \omega }|+|\Delta_{\xi\eta}\Omega_{j}|+|\Delta_{\xi\eta}\Omega_{l}|\big),$$ hence $$\frac{| \Delta_{\xi\eta} \delta^{-1}_{k,jl}|}{|\xi-\eta|}\leq C\frac{kA^{2}_{k}}{\alpha^{2}}\big(|\omega|_{\Pi}^{{ \mathcal{L} }}+|\Omega|^{{ \mathcal{L} }}_{2\beta,\Pi}\big)\leq CM\frac{kA^{2}_{k}}{\alpha^{2}},$$ and we have $$\label{Delta}
\frac{| \Delta_{\xi\eta} F_{kmq_{j}{ \overline }{q}_{l}}|}{|\xi-\eta|}\leq C \frac{kA_{k}}{\alpha}\Big(\frac{| \Delta_{\xi\eta} R_{kmq_{j}{ \overline }{q}_{l}}|}{|\xi-\eta|} + \frac{M}{\alpha}\big|R_{kmq_{j}{ \overline }{q}_{l}}(\xi)\big|\Big).$$ Thanks to the estimate it is easy to obtain .\
Finally, the estimate $ { \langle }{ \widehat }{N}{ \rangle }^{{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}\leq { \langle }R{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}$ is a straightforward consequence of and .
Estimates on the Poisson bracket
--------------------------------
\
\[lem.2\] Let $R\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$ and $F\in { \Gamma }^{{ \beta },+}_{r,D(s,r)}$ be both of degree 2, i.e. of the form . Then there exists $C>0$ so that for any $0<{ \sigma }<s$ $$\label{Est.crochet}
{ \langle }\,\big\{R,F\big\}\,{ \rangle }_{r,D(s-{ \sigma },r)} \leq \frac{C}{{ \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)},$$ and $${ \langle }\,\big\{R,F\big\}\,{ \rangle }^{{ \mathcal{L} }}_{{ \beta },{D}(s-{ \sigma },r)} \leq \frac{C}{{ \sigma }}\Big({ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s,r)}+{ \langle }F{ \rangle }^{+}_{r,D(s,r)}{ \langle }R{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}\Big).$$
The expansion of ${ \big\{\,{ R,F }\, \big\} }$ reads $${ \big\{\,{ R,F }\, \big\} }=\sum_{k=1}^{n}\Big(\frac{\partial R}{\partial \theta_{k}} \frac{\partial F}{\partial y_{k}}-\frac{\partial R}{\partial y_{k}} \frac{\partial F}{\partial \theta_{k}} \Big)+i\sum_{k\geq 1}\Big(\frac{\partial R}{\partial z_{k}} \frac{\partial F}{\partial { \overline{z} }_{k}}-\frac{\partial R}{\partial { \overline{z} }_{k}} \frac{\partial F}{\partial z_{k}} \Big).$$ It remains to estimate each term of this expansion and its derivatives. We will control the derivative with respect to $\theta_{k}$ thanks to the Cauchy formula : $$\label{Cauchy}
\Big \| \frac{\partial P}{\partial \theta_{k}}\Big \|_{D(s-{ \sigma },r)} \leq \frac{C}{{ \sigma }} \big \| P\big \|_{D(s,r)} ,$$ which explains the loss of ${ \sigma }$.\
Notice that if $P$ is of degree 2 (and that is the case for $F$ and $R$) we have $$\label{vanish}
\frac{\partial^{2} P}{\partial z\partial y}=\frac{\partial^{2} P}{\partial y^{2}}=\frac{\partial^{3} P}{\partial z^{3}}=0,$$ fact which will be crucially used in the sequel. Finally observe that $z$ and ${ \overline }{z}$ exactly play the same role, hence we will only take ${\displaystyle}\frac{\partial}{\partial z}$ into consideration.\
$\spadesuit$ We first prove .\
$\bullet$ Since ${\displaystyle}\| P\,Q\|_{D(s,r)}\leq \| P\|_{D(s,r)} \| Q\|_{D(s,r)}$ we have by Cauchy formula $$\begin{aligned}
\big \| { \big\{\,{ R,F }\, \big\} }\big \|_{D(s-{ \sigma },r)}
&\leq& \frac{Cr^{2}}{{ \sigma }} (2n+\sum_{k\geq 1}\frac{1}{k^{2{ \beta }+1}} ){ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}\nonumber \\
&\leq& \frac{Cr^{2}}{{ \sigma }} { \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.\label{dx0}\end{aligned}$$ $\bullet$ With we have $$\begin{aligned}
\Big\| \frac{\partial}{\partial y_{j}} \Big(\frac{\partial R}{\partial \theta_{k}} \frac{\partial F}{\partial y_{k}}\Big)\Big\|_{D(s-{ \sigma },r)}&\leq &\Big\| \frac{\partial}{\partial \theta_{k}} \Big(\frac{\partial R}{\partial y_{j}}\Big) \Big\|_{{D}(s-{ \sigma },r)} \Big\| \frac{\partial F}{\partial y_{k}}\Big\|_{D(s,r)}\\
&\leq& \frac{C}{{ \sigma }} \Big\| \frac{\partial R}{\partial y_{j}} \Big\|_{D(s,r)} \Big\| \frac{\partial F}{\partial y_{k}}\Big\|_{D(s,r)}\\
&\leq &\frac{C}{{ \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)},
\end{aligned}$$ and the same estimate holds interchanging $R$ and $F$. In view of we deduce $$\label{dx1}
\max_{1\leq y\leq n}\Big\|\frac{\partial}{\partial y_{j}} { \big\{\,{ R,F }\, \big\} }\Big\|_{D(s,r)}\leq \frac{C}{{ \sigma }} { \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.$$
$\bullet$ By , $ {\displaystyle}\frac{\partial}{\partial z_{j}} \Big(\frac{\partial R}{\partial y_{k}} \frac{\partial F}{\partial \theta_{k}}\Big)= \frac{\partial R}{\partial y_{k}} \frac{\partial ^{2}F}{ \partial z_{j}\partial\theta_{k}}$, and by $$\begin{aligned}
\Big\| \frac{\partial R}{\partial y_{k}} \frac{\partial ^{2}F}{ \partial z_{j}\partial\theta_{k}} \Big\|_{D(s-{ \sigma },r)}
&\leq & \frac{C}{{ \sigma }} \Big\| \frac{\partial R}{\partial y_{k}}\Big\|_{D(s,r)}\Big\| \frac{\partial F}{\partial z_{j}} \Big\|_{D(s,r)}\\
&\leq &\frac{Cr}{j^{{ \beta }}{ \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.
\end{aligned}$$ Similarly ${\displaystyle}\Big\| \frac{\partial}{\partial z_{j}} \Big(\frac{\partial R}{\partial \theta_{k}} \frac{\partial F}{\partial y_{k}}\Big)\Big\|_{D(s-{ \sigma },r)}\leq \frac{Cr}{j^{{ \beta }}{ \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.$ By the Leibniz rule $$\begin{gathered}
\Big\| \frac{\partial}{\partial z_{j}} \Big(\frac{\partial R}{\partial z_{k}} \frac{\partial F}{\partial { \overline{z} }_{k}}\Big)\Big\|_{D(s,r)}\leq \\
\begin{aligned}
&\leq \Big\| \frac{\partial^{2} R}{\partial z_{k}\partial z_{j}} \Big\|_{{D}(s,r)} \Big\| \frac{\partial F}{\partial { \overline{z} }_{k}}\Big\|_{D(s,r)}+\Big\| \frac{\partial^{2} F}{\partial z_{j} \partial { \overline{z} }_{k}} \Big\|_{{D}(s,r)} \Big\| \frac{\partial R}{\partial z_{k}}\Big\|_{D(s,r)}\\
&\leq \frac{Cr}{j^{{ \beta }}}\Big( \frac{1}{k^{2{ \beta }+1}}+ \frac{1}{k^{2{ \beta }}(1+|j-k|)}\Big){ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)},
\end{aligned}
\end{gathered}$$ and taking the sum in $k$ yields $$\sum_{k\geq 1} \Big\| \frac{\partial}{\partial z_{j}} \Big(\frac{\partial R}{\partial z_{k}} \frac{\partial F}{\partial { \overline{z} }_{k}}\Big)\Big\|_{D(s,r)}\leq \frac{Cr}{j^{{ \beta }} { \sigma }} { \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.$$ The previous estimates imply that $$\label{dx2}
\Big\|\frac{\partial}{\partial z_{j}} { \big\{\,{ R,F }\, \big\} }\Big\|_{D(s-{ \sigma },r)}\leq \frac{Cr}{j^{{ \beta }} { \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.$$ $\bullet$ Thanks to , $ {\displaystyle}\frac{\partial^{2}}{\partial z_{j}\partial z_{l}} \Big(\frac{\partial R}{\partial y_{k}} \frac{\partial F}{\partial \theta_{k}}\Big)=
\frac{\partial R}{\partial y_{k}}\frac{\partial^{3}F}{\partial z_{j}\partial z_{l}\partial \theta_{k}}$, and by we obtain $$\begin{aligned}
\Big\| \frac{\partial^{2}}{\partial z_{j}\partial z_{l}} \Big(\frac{\partial R}{\partial y_{k}} \frac{\partial F}{\partial \theta_{k}}\Big)
\Big\|_{D(s-{ \sigma },r) }&\leq& \Big\| \frac{\partial R}{\partial y_{k}}
\Big\|_{{D}(s,r) }\Big\| \frac{\partial^{3}F}{\partial z_{j}\partial z_{l}\partial \theta_{k}}\Big\|_{D(s-{ \sigma },r) }\nonumber\\
&\leq & \frac{C}{(jl)^{{ \beta }}{ \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}, \label{33}\end{aligned}$$ and the same estimate holds interchanging $R$ and $F$.\
On the other hand, $$\frac{\partial^{2}}{\partial z_{j}\partial z_{l}} \Big(\frac{\partial R}{\partial z_{k}} \frac{\partial F}{\partial { \overline{z} }_{k}}\Big)=
\frac{\partial ^{2}R}{\partial z_{j}\partial z_{k}} \frac{\partial^{2} F}{\partial z_{l}\partial { \overline{z} }_{k}}+\frac{\partial ^{2}R}{\partial z_{l}\partial z_{k}} \frac{\partial^{2} F}{\partial z_{j}\partial { \overline{z} }_{k}},$$ and $$\begin{aligned}
\Big\| \frac{\partial^{2}R}{\partial z_{j}\partial z_{k}} \frac{\partial^{2} F}{\partial z_{l}\partial { \overline{z} }_{k}}\Big\|_{D(s-{ \sigma },r)}&\leq &\Big\| \frac{\partial^{2} R}{\partial z_{j}\partial z_{k}} \Big\|_{D(s,r)} \Big\| \frac{\partial^{2} F}{\partial z_{l}\partial { \overline{z} }_{k}}\Big\|_{D(s,r)}\\
&\leq &\frac{C}{(jlk^{2})^{{ \beta }}(1+|l-k|)}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}.\end{aligned}$$ Hence, with we conclude that $$\label{dx3}
\Big\|\frac{\partial^{2}}{\partial z_{j}\partial z_{l}} { \big\{\,{ R,F }\, \big\} }\Big\|_{D(s-{ \sigma },r)}\leq \frac{C}{(jl)^{{ \beta }} { \sigma }}{ \langle }R{ \rangle }_{r,D(s,r)}{ \langle }F{ \rangle }^{+}_{r,D(s,r)},$$ as the series ${\displaystyle}\sum_{k\geq 1 } \frac1{k^{2{ \beta }}(1+|l-k|)}$ converges.\
Finally, the estimates , , and yield the estimate .\
$\spadesuit$ To prove the estimate with the Lipschitz norms, we can use the previous analysis and the two following facts.\
Firstly, since $\Delta_{\xi\eta}(fg)=f(\xi)\Delta_{\xi\eta}g+g(\eta) \Delta_{\xi\eta}f$, hence $$\|fg\|^{{ \mathcal{L} }}_{D(s,r)}\leq \|f\|_{D(s,r)}\|g\|^{{ \mathcal{L} }}_{D(s,r)}+\|g\|_{D(s,r)}\|f\|^{{ \mathcal{L} }}_{D(s,r)}.$$ Secondly, the operator $\Delta_{\xi\eta}$ commutes with the derivative in any variable.
The canonical transform
-----------------------
\
In this Section we study the Hamiltonian flow generated by a function $F \in { \Gamma }^{{ \beta },+}_{r,D(s-{ \sigma },r)}$ globally of degree 2, i.e. of degree 2 in the variables $z,{ \overline }z$ and of degree 1 in the variable $y$. Namely, we consider the system $$\label{System}
\left\{
\begin{aligned}
&\big(\dot{\theta}(t),\dot{y}(t),\dot{z}(t),\dot{{ \overline{z} }}(t)\big) = X_{F}\big(\big(\theta(t),y(t),z(t),{ \overline{z} }(t)\big)\big) ,\\[3pt]
&\big(\theta(0),y(0),z(0),{ \overline{z} }(0)\big)=\big(\theta^{0},y^{0},z^{0},{ \overline{z} }^{0}\big).
\end{aligned}
\right.$$
\[lem.3\] Let $0<{ \sigma }<s/3$ and $F\in { \Gamma }^{{ \beta },+}_{r,D(s-{ \sigma },r)}$ with $F$ of degree 2. Assume that $ {\displaystyle}{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)} <C{{ \sigma }}$. Then the solution of the equation with initial condition $\big(\theta^{0},y^{0},z^{0},{ \overline{z} }^{0}\big)\in D(s-3{ \sigma },\frac{r}4),$ satisfies $\big(\theta(t),y(t),z(t),{ \overline{z} }(t)\big)\in D(s-2{ \sigma },\frac{r}2)$ for all $0\leq t\leq 1$, and we have the estimates
$$\label{der.y.z}
\sup_{0\leq t\leq 1} \Big| \frac{\partial y_{k}(t)}{\partial w_{j}^{0}}\Big|\leq \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }j^{{ \beta }}}\quad \text{with}\quad w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j},$$
$$\label{der.z.z}
\sup_{0\leq t\leq 1} \Big| \frac{\partial w_{k}(t)}{\partial w_{j}^{0}}\Big| \leq \frac{C{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{(jk)^{{ \beta }}(1+|j-k|)}+\delta_{jk} \quad \text{with}\quad w_{k}=z_{k} \mbox{ or }{ \overline{z} }_{k},\;\;w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j},$$
$$\label{der.y.y}
\sup_{0\leq t\leq 1}\Big| \frac{\partial y_{k}(t)}{\partial y_{j}^{0}}\Big|\leq \frac{C { \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }}+\delta_{jk},$$
$$\label{der.zz}
\sup_{0\leq t\leq 1}\Big| \frac{\partial^{2} y_{k}(t)}{\partial w_{j}^{0}\partial w_{i}^{0}}\Big|\leq \frac{C{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }(ij)^{{ \beta }}(1+|i-j|)} \quad \text{with}\quad w^{0}_{i}=z^{0}_{i} \mbox{ or }{ \overline{z} }^{0}_{i},\;\;w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j}.$$
Before we turn to the proof of Lemma \[lem.3\], we introduce a space of infinite dimensional matrices, with decaying coefficients.\
Let $\|\cdot\|$ be any submultiplicative norm on $\mathcal{M}_{2,2}({ \mathbb{C} })$, the space of the $2\times2$ complex matrices. For $\beta>0$, we say that $B\in \mathcal{M}_{s}^{\beta,+}$ if ${ \langle }{ \langle }\,B\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}<\infty$, where the norm ${ \langle }{ \langle }\,\cdot\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}$ is given by the condition[^3]
$$\sup_{\xi\in \Pi}\sup_{|\text{Im}\, \theta|<s} \|B_{jl}\| \leq \frac{{ \langle }{ \langle }\,B\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}}{(jl)^{{ \beta }}(1+|j-l|)},\quad \forall \,j,l\geq1.$$
Then we have the following result
\[lemm.AB\] Let $A,B \in \mathcal{M}_{s}^{\beta,+}$. Then $AB \in \mathcal{M}_{s}^{\beta,+}$ and $${ \langle }{ \langle }\,AB\,{ \rangle }{ \rangle }^{+}_{{ \beta },s} \leq C{ \langle }{ \langle }\,A\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}{ \langle }{ \langle }\,B\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}.$$
For all $j,l\geq 1$, ${\displaystyle}\big(AB\big)_{jl}=\sum_{k\geq 1}A_{jk}B_{kl}$. Since $\|\cdot\|$ is submultiplicative $$\begin{aligned}
\|\big(AB\big)_{jl}\|&\leq &\sum_{k\geq 1}\|A_{jk}\|\|B_{kl}\| \nonumber \\
& \leq& \frac{ { \langle }{ \langle }\,A\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}{ \langle }{ \langle }\,B\,{ \rangle }{ \rangle }^{+}_{{ \beta },s}}{(jl)^{\beta}}\sum_{k\geq 1}\frac{1}{k^{2{ \beta }}(1+|j-k|)(1+|l-k|)}. \label{A.B}
\end{aligned}$$ Thanks to the triangle inequality, for all $j,l\geq 1$, $$\big\{k\geq 1\big\} \subset \big\{k\geq 1\;:\;|j-k|\geq \frac13|j-l|\big\}\bigcup \big\{k\geq 1\;:\;|l-k|\geq \frac13|j-l|\big\},$$ thus, by splitting the sum in we obtain the desired result.
Here we introduce the notations $Z_{j}=(z_{j},{ \overline }{z}_{j})$ and $Z=(Z_{j})_{j\geq 1}$. Then $F$ reads $$\label{struct}
F(\theta,y,Z)=b_{0}(\theta)+b_{1}(\theta)\cdot y+a(\theta)\cdot Z+\frac12 \big(A(\theta)Z\big)\cdot Z,$$ with $$b_{0}(\theta)=F(\theta,0,0),\qquad b_{1}(\theta)=\nabla_{y}F(\theta,0,0), \qquad a(\theta)=\nabla_{Z}F(\theta,0,0),$$ and $A=(A_{i,j})$ is the infinite matrix so that $$\label{Aij}
A_{i,j}(\theta)=
\begin{pmatrix}
{\displaystyle}\frac{\partial^{2}F}{\partial z_{i}\partial z_{j}}(\theta,0,0) & {\displaystyle}\frac{\partial^{2}F}{\partial z_{i}\partial { \overline }{z}_{j}}(\theta,0,0)\\[12pt]
{\displaystyle}\frac{\partial^{2}F}{\partial { \overline }{z}_{i}\partial z_{j}}(\theta,0,0) & {\displaystyle}\frac{\partial^{2}F}{\partial { \overline }{z}_{i}\partial { \overline }{z}_{j}}(\theta,0,0)
\end{pmatrix}.$$ Observe that $A$ is symmetric.\
By [@Poschel Estimate (9)], the flow $X_{F}^{t}$ exists for $0\leq t\leq 1$ and maps $D(s-3{ \sigma },\frac{r}4)$ into $D(s-2{ \sigma },\frac{r}2)$. Here we have to give a precise description of $X_{F}^{t}$ for $0\leq t\leq 1$. This is possible thanks to the particular structure of F.\
In the sequel we write $(\theta(t),y(t),Z(t))=X_{F}^{t}(\theta^{0},y^{0},Z^{0})$.\
$\spadesuit$ To begin with, the equation for $\theta$ reads $$\label{eq.theta}
\dot{\theta}(t)=\nabla_{y}F(\theta,0,0)= b_{1}(\theta),\quad \theta(0)=\theta^{0}.$$ Since $b_{1}$ is a smooth function (see ), the $n$-dimensional system admits a unique (smooth) local solution $\theta(t)$. By the work of J. Pöschel, this solution exists until time $t=1$, and we have the bound $$\label{sup.theta}
\sup_{0\leq t\leq 1} |\text{Im}\;\theta(t)|<s-2{ \sigma },$$ (this can here be recovered by the usual bootstrap argument, using the smallness assumption on $F$).\
$\spadesuit$ We now turn to the equation in $Z$. We have to solve $$\label{eq.Z}
\dot{Z}(t)=J\nabla_{Z}F(\theta,y,Z)(t),\quad Z(0)=Z^{0},$$ where $$J=\text{diag}\Big\{
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}\Big\}_{j\geq 1}.$$ Notice that by [@Poschel Estimate (9)] we already know that $$\label{est.Z}
\sup_{0\leq t \leq 1} \|Z(t)\|_{\ell^{2}_{\Psi}}<\frac{r}2$$ but we need to precise the behavior of $Z(t)$.\
Since $\theta=\theta(t)$ is known by the previous step, in view of , equation reads $$\label{Zt}
\dot{Z}(t)=b(t)+B(t)\cdot Z(t),\quad Z(0)=Z^{0},$$ where $b(t)=Ja(\theta(t))$ and $B(t)=JA(\theta(t))$.\
We now iterate the integral formulation of the problem $$Z(t)=Z^{0}+\int_{0}^{t}\big(b(t_{1})+B(t_{1})\cdot Z(t_{1}) \big)\text{d}t_{1},$$ and formally obtain $$\label{Zt2}
Z(t)= b^{\infty}(t)+ \big(1+B^{\infty}(t)\big)Z^{0},$$ where $$\label{binf}
b^{\infty}(t)=\sum_{k\geq 1}\int_{0}^{t}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}} \prod_{j=1}^{k-1}B(t_{j})b(t_{k})\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1},$$ and $$\label{def.Binfini}
B^{\infty}(t)=\sum_{k\geq 1}\int_{0}^{t}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}} \prod_{j=1}^{k}B(t_{j})\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1}.$$ By and , there exists $C>0$ so that $$\sup_{0\leq t\leq 1}\|B(t)\|_{\ell^{2}_{\Psi}\to \ell^{2}_{\Psi}}\leq C,$$ and thus, for all $0\leq t\leq 1$ the series converges and $$\begin{aligned}
\|b^{\infty}(t)\|_{\ell^{2}_{\Psi}}&\leq & \sup_{0\leq t\leq 1}\|b(t)\|_{\ell^{2}_{\Psi}}\sum_{k\geq 1}C^{k-1}\int_{0}^{1}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}}\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1}\nonumber\\
&\leq & \sup_{0\leq t\leq 1}\|b(t)\|_{\ell^{2}_{\Psi}}\sum_{k\geq 1}\frac{C^{k-1}}{k\,!}\nonumber\\
&\leq &\sup_{0\leq t\leq 1}\|b(t)\|_{\ell^{2}_{\Psi}} \frac{{ \text{e} }^{C}-1}{C}\nonumber\\
&\leq &C\sup_{0\leq t\leq 1}\|b(t)\|_{\ell^{2}_{\Psi}}.\label{2.46}
\end{aligned}$$ Similarly we have uniformly in $0\leq t\leq 1$ $$\| B^{\infty}(t)\|_{\ell^{2}_{\Psi}\to \ell^{2}_{\Psi}}\leq C.$$ As a conclusion, the formula makes sense.
Indeed, we need more precise estimates on $B^{\infty}$. Recall that $B(t)=A(\theta(t))$, where $A$ is defined by . Then by and , for all $0\leq t\leq 1$, $B(t)\in \mathcal{M}^{{ \beta },+}_{s-{ \sigma }}$ and ${\displaystyle}\sup_{0\leq t\leq 1}{ \langle }{ \langle }\,B(t)\,{ \rangle }{ \rangle }^{+}_{{ \beta },s-{ \sigma }}\leq C { \langle }\,F\,{ \rangle }^{+}_{r,D(s-{ \sigma },r)}$. Hence by Lemma \[lemm.AB\] and $$\label{<B>}
{ \langle }{ \langle }\,B^{\infty}\,{ \rangle }{ \rangle }^{+}_{{ \beta },s-{ \sigma }}\leq { \text{e} }^{C{ \langle }\,F\,{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}-1\leq C { \langle }\,F\,{ \rangle }^{+}_{r,D(s-{ \sigma },r)}.$$ $\spadesuit$ Finally we turn to the equation in $y$ $$\dot{y}(t)=-\nabla_{\theta}F(\theta,y,Z)(t),\quad y(0)=y^{0}.$$ We already know the functions $\theta(t)$ and $Z(t)$. Moreover as the function $F$ is linear in $y$, the previous $n-$dimensional system reads $$\label{eq.y}
\dot{y}(t)=f(t)+g(t)y(t),\quad y(0)=y^{0},$$ with $$f(t)=-\nabla_{\theta}b_{0}(\theta(t))+\nabla_{\theta}a(\theta(t))\cdot Z(t)+\frac12\big(\nabla_{\theta}A(\theta(t))Z(t)\big)\cdot Z(t),$$ and $$g(t)= -\nabla_{\theta}b_{1}(\theta(t)) =-\nabla_{\theta}\nabla_{y}F(\theta,0,0).$$ We can solve the equation with the same techniques as the equation . In fact we have formally $$\label{yt}
y(t)=f^{\infty}(t)+\big(1+g^{\infty}(t)\big)y^{0},$$ where $$\label{def.finf}
f^{\infty}(t)=\sum_{k\geq 1}\int_{0}^{t}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}} \prod_{j=1}^{k-1}g(t_{j})f(t_{k})\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1},$$ and $$g^{\infty}(t)=\sum_{k\geq 1}\int_{0}^{t}\int_{0}^{t_{1}}\cdots \int_{0}^{t_{k-1}} \prod_{j=1}^{k}g(t_{j})\text{d}t_{k} \cdots \text{d}t_{2}\,\text{d}t_{1}.$$ By and the Cauchy formula $$\sup_{0\leq t\leq 1}\|g(t)\|\leq \frac{C}{{ \sigma }}\max_{1\leq j\leq n}\Big\|\frac{\partial F}{\partial y_{j}}\Big\|_{D(s-{ \sigma },r)}\leq \frac{C { \langle }\,F\,{ \rangle }^{+}_{r,D(s-{ \sigma },r)}} {{ \sigma }},$$ and similarly to we have for all $0\leq t\leq 1$ $$|f^{\infty}(t)|\leq C \sup_{0\leq t\leq 1}|f(t)|,$$ and $$\label{norm.inf.g}
\|g^{\infty}(t)\|\leq \frac{C { \langle }\,F\,{ \rangle }^{+}_{r,D(s-{ \sigma },r)}} {{ \sigma }} ,$$ which shows the convergence of the series defining .\
$\spadesuit$ It remains to show the estimates on the solutions of .\
$\bullet$ First we prove . By , $$\nabla_{Z_{j}^{0}}Z_{k}(t)=\left(\begin{array}{cc}1 & 0 \\0 & 1 \end{array}\right)\delta_{kj}+B^{\infty}_{kj}(t),$$ therefore by , for $k\neq j$ we have $$\label{ZZ}
\| \nabla_{Z_{j}^{0}}Z_{k}(t)\| \leq \frac{C{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{(jk)^{{ \beta }}(1+|j-k|)}, \quad \text{and}\quad \| \nabla_{Z_{j}^{0}}Z_{j}(t)\| \leq 1,$$ which was the claim.\
$\bullet$ We prove . By we have $$y_{k}(t)=f_{k}^{\infty}(t)+y_{k}^{0}+\sum_{1\leq j\leq n}g_{jk}^{\infty}(t)y_{j}^{0},$$ hence ${\displaystyle}\frac{\partial y_{k}}{\partial y_{j}^{0}}=\delta_{jk}+g_{jk}^{\infty}(t)$ and the claim follows from ($f^{\infty}$ does not depend on $y^{0}$).\
$\bullet$ We prove . Since $g$ and $g^{\infty}$ do not depend on $Z$, from we deduce that ${\displaystyle}\frac{\partial y}{\partial z_{j}^{0}}=\frac{\partial f^{\infty}}{\partial z_{j}^{0}}$.\
Now by definition of $f^{\infty}$, we get that for all $0\leq t\leq 1$ $$\label{...}
\Big|\frac{\partial y(t)}{\partial z_{j}^{0}}\Big|=\Big|\frac{\partial f^{\infty}(t)}{\partial z_{j}^{0}}\Big|\leq \big|\nabla_{Z^{0}_{j}}f^{\infty}(t)\big|\leq C \sup_{0\leq t\leq 1}| \nabla_{Z^{0}_{j}}f(t)|.$$ For all $1\leq l \leq n$, we compute $$\label{3.54b}
\nabla_{Z_{k}}f_{l}(t)=\partial_{\theta_{l}}a_{k}(\theta(t))+\sum_{i\geq 1}\partial_{\theta_{l}}A_{ki}(\theta(t))Z_{i}(t).$$ As $a_{k}(\theta)=\nabla_{Z_{k}}F(\theta,0,0)$, with the Cauchy formula we deduce $$\sup_{0\leq t\leq 1}\big|\partial_{\theta_{l}}a_{k}(\theta(t))\big|\leq \frac{C}{{ \sigma }}\big\|\nabla_{Z_{k}}F\big\|_{D(s-{ \sigma },r)}\leq \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)} }{{ \sigma }k^{1+\beta}}.$$ Similarly with , $$\sup_{0\leq t\leq 1}\big|\partial_{\theta_{l}}A_{ki}(\theta(t))\big|\leq \frac{C{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)} }{{ \sigma }(ik)^{\beta}(1+|i-k|)}.$$ Inserting the two previous estimates in , we obtain using and the Cauchy-Schwarz inequality $$\begin{aligned}
| \nabla_{Z_{k}} f_{l}(t)|&\leq & \frac{C}{{ \sigma }}\frac{{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{k^{\beta}}\Big(r+\sum_{i\geq 1}\frac{|Z_{i}|}{i^{{ \beta }}(1+|k-i|)}\Big)\nonumber\\
&\leq & \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }k^{{ \beta }}}\label{est.ZZ}.\end{aligned}$$ Since $ {\displaystyle}\nabla_{Z^{0}_{j}}f_{l}(t)=\sum_{k\geq 1} \big( \nabla_{Z^{0}_{j}}Z_{k}(t) \big) \nabla_{Z_{k}}f_{l}(t)$, from and we deduce $$\begin{aligned}
| \nabla_{Z^{0}_{j}}f_{l}(t)|&\leq& \sum_{k\geq 1} \|\nabla_{Z^{0}_{j}}Z_{k}(t)\| \| \nabla_{Z_{k}}f_{l}(t)\|\nonumber \\
&\leq& \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }j^{{ \beta }}}\Big( \sum_{k\geq 1} \frac{1}{k^{2{ \beta }}(1+|j-k|)}+1\Big)\\
&\leq& \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }j^{{ \beta }}},
\end{aligned}$$ and together with , we get that for all $j\geq 1$ $$\sup_{0\leq t\leq 1}\Big|\frac{\partial y(t)}{\partial z_{j}^{0}}\Big|\leq \frac{Cr{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }j^{{ \beta }}}.$$ $\bullet$ It remains to show . first we have $$\Big|\frac{\partial y(t)}{\partial z_{i}^{0}\partial z_{j}^{0}}\Big|\leq \Big|\nabla_{Z^{0}_{i}}\nabla_{Z^{0}_{j}}f^{\infty}(t)\Big|\leq C \sup_{0\leq t\leq 1} \Big|\nabla_{Z^{0}_{i}}\nabla_{Z^{0}_{j}}f(t)\Big| .$$ Then from the very definition of $f$, $\nabla_{Z^{0}_{i}}\nabla_{Z^{0}_{j}}f(t)=\nabla_{\theta}A_{ij}(\theta(t))$, and using the Cauchy estimate in $\theta$ we get, $$\Big|\frac{\partial y(t)}{\partial z_{i}^{0}\partial z_{j}^{0}}\Big|\leq \frac{C{ \langle }F{ \rangle }^{+}_{r,D(s-{ \sigma },r)}}{{ \sigma }(ij)^{{ \beta }}(1+|i-j|)},$$ which was the claim.
In the next result, we denote by $|\cdot|^{{ \mathcal{L} }}$ the Lipschitz norm $$|f|^{{ \mathcal{L} }}=\sup_{\substack{\xi,\eta\in \Pi\\ \xi\neq \eta}}\frac{|f(\xi)-f(\eta)|}{|\xi-\eta|}.$$ We have an analogous result to Lemma \[lem.3\] with Lipschitz norms.
\[lem.3\*\] Under the assumptions of Lemma \[lem.3\] and the condition ${ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}\leq C{ \sigma }$ the solution of satisfies moreover $$\begin{aligned}
\sup_{0\leq t\leq 1} \Big| \frac{\partial y_{k}(t)}{\partial w_{j}^{0}}\Big|^{{ \mathcal{L} }}&\leq &\frac{Cr{ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}}{{ \sigma }j^{{ \beta }}}\quad \text{with}\quad w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j}, \\
\sup_{0\leq t\leq 1} \Big| \frac{\partial w_{k}(t)}{\partial w_{j}^{0}}\Big|^{{ \mathcal{L} }}& \leq& \frac{C{ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}}{(jk)^{{ \beta }}(1+|j-k|)} \quad \text{with}\quad w_{k}=z_{k} \mbox{ or }{ \overline{z} }_{k},\;\;w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j},\\
\sup_{0\leq t\leq 1}\Big| \frac{\partial y_{k}(t)}{\partial y_{j}^{0}}\Big|^{{ \mathcal{L} }}&\leq& \frac{C { \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}}{{ \sigma }},\\
\sup_{0\leq t\leq 1}\Big| \frac{\partial^{2} y_{k}(t)}{\partial w_{j}^{0}\partial w_{i}^{0}}\Big|^{{ \mathcal{L} }}&\leq & \frac{C{ \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s-{ \sigma },r)}}{{ \sigma }(ij)^{{ \beta }}(1+|i-j|)} \quad \text{with}\quad w^{0}_{i}=z^{0}_{i} \mbox{ or }{ \overline{z} }^{0}_{i},\;\;w^{0}_{j}=z^{0}_{j} \mbox{ or }{ \overline{z} }^{0}_{j}.
\end{aligned}$$
We won’t detail the proof, since it is tedious and similar to the proof of Lemma \[lem.3\]. First we define the space $\mathcal{M}^{\beta,+,{ \mathcal{L} }}_{s}$ with norm ${ \langle }{ \langle }\,\cdot\,{ \rangle }{ \rangle }^{+,{ \mathcal{L} }}_{\beta,s}$ similarly to $\mathcal{M}^{\beta,+}_{s}$, but with a Lipschitz norm in $\xi$. Then we have ${\displaystyle}{ \langle }{ \langle }AB{ \rangle }{ \rangle }^{+,{ \mathcal{L} }}\leq C\big({ \langle }{ \langle }A{ \rangle }{ \rangle }^{+,{ \mathcal{L} }}{ \langle }{ \langle }B{ \rangle }{ \rangle }^{+}+{ \langle }{ \langle }B{ \rangle }{ \rangle }^{+,{ \mathcal{L} }}{ \langle }{ \langle }A{ \rangle }{ \rangle }^{+}\big)$. Then one can follow the proof of Lemma \[lem.3\] and use that the different norms (say $\|\,\cdot\,\|$) which appear satisfy $\|fg\|^{{ \mathcal{L} }}\leq C\big(\|f\|^{{ \mathcal{L} }}\|g\|+\|f\|\|g\|^{{ \mathcal{L} }}\big)$.
To conclude this section, we state a result which shows that the Lie transform associated to a quadratic function, is also quadratic. This will be crucial in the proof of Theorem \[thmKAM\] (see Section \[Proof\]).
\[coro\_sol\] The symplectic application $X_{F}^{1}$ reads $$\left(\begin{array}{c}\theta \\y \\Z\end{array}\right)\longmapsto\left(\begin{array}{l} K(\theta) \\L(\theta,Z) + M(\theta)Z+S(\theta)y \\T(\theta)+U(\theta)Z\end{array}\right)$$ where $L(\theta,Z)$ is quadratic in $Z$, $M(\theta)$ and $ U(\theta)$ are bounded linear operators from $\ell^{2}_{\Psi}\times\ell^{2}_{\Psi}$ into itself and $S(\theta)$ is a bounded linear map from ${ \mathbb{R} }^n$ to ${ \mathbb{R} }^n$.
The claim follows from the proof of Lemma \[lem.3\]. The structure of $Z(1)$ follows from , while the structure of $y(1)$ comes from and .
Composition estimates
---------------------
\
In this section we study the new Hamiltonian obtained after composition with the canonical transformation $X_{F}^{1}$.
\[prop.comp\] Let $0<\eta<1/8$ and $0<{ \sigma }<s$, $R\in { \Gamma }^{{ \beta }}_{\eta r,D(s-2{ \sigma },4\eta r)}$ and $F\in { \Gamma }^{{ \beta },+}_{r,D(s-{ \sigma },r)}$ with $F$ of degree 2. Assume that $ {\displaystyle}{ \langle }F{ \rangle }^{+}_{r,D(s,r)}+ { \langle }F{ \rangle }^{+,{ \mathcal{L} }}_{r,D(s,r)}<C{{ \sigma }}$. Then $R \circ X^{1}_{F} \in { \Gamma }^{{ \beta }}_{\eta r,D(s-5{ \sigma }, \eta r)}$ and we have the estimates $$\label{est.comp}
{ \langle }\,R\circ X^{1}_{F}\,{ \rangle }_{\eta r,D(s-5{ \sigma },\eta r)} \leq C{ \langle }\,R\,{ \rangle }_{\eta r,D(s-2{ \sigma },4 \eta r)},$$ $${ \langle }\,R\circ X^{1}_{F}\,{ \rangle }^{{ \mathcal{L} }}_{\eta r,D(s-5{ \sigma },\eta r)} \leq C{ \big( }{ \langle }\,R\,{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}+{ \langle }\,R\,{ \rangle }^{{ \mathcal{L} }}_{\eta r,D(s-2{ \sigma },4\eta r)}{ \big) }.$$
The proof of the first estimate relies on Lemma \[lem.3\]. We omit the proof of the second, which is similar using the estimates of Lemma \[lem.3\*\] instead.\
In the sequel, we use the notation $(\theta,y,z,\bar z) =X^{1}_{F}(\theta^0,y^0,z^0,\bar z^0)$.\
$\spadesuit$ Since $X_{F}^{1}$ maps $D(s-3{ \sigma },\frac{r}4)$ into $D(s-2{ \sigma },\frac{r}2)$, it is clear that $$\label{Est.o.1}
\| R\circ X^{1}_{F}\|_{D(s-5{ \sigma },\eta r)}\leq C{ \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}.$$ $\spadesuit$ By the Leibniz rule, for all $1\leq j\leq n$ $$\frac{\partial( R\circ X^{1}_{F})}{\partial y^{0}_{j}}=\sum_{k=1}^{n} \frac{\partial R(X_{F}^{1})}{\partial y_{k}} \frac{\partial y_{k}}{\partial y^{0}_{j}},
$$ and by we deduce $$\label{Est.o.2}
\Big \| \frac{\partial( R\circ X^{1}_{F})}{\partial y^{0}_{j}}\Big\|_{D(s-5{ \sigma },\eta r)}\leq C{ \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}.$$ $\spadesuit$ For $j\geq 1$, the derivative in $z^{0}_{j}$ reads $$\begin{gathered}
\frac{\partial( R\circ X^{1}_{F})}{\partial z^{0}_{j}}=\\
\begin{aligned}
& \sum_{k=1}^{n} \frac{\partial R(X_{F}^{1})}{\partial y_{k}} \frac{\partial y_{k}}{\partial z^{0}_{j}}+\sum_{k\geq 1}\Big( \frac{\partial R(X_{F}^{1})}{\partial z_{k}}\frac{\partial z_{k}}{\partial z^{0}_{j}}+\frac{\partial R(X_{F}^{1})}{\partial { \overline{z} }_{k}}\frac{\partial { \overline{z} }_{k}}{\partial z^{0}_{j}}\Big).
\end{aligned}
\end{gathered}$$ Therefore, thanks to and we get $$\begin{gathered}
\label{Est.o.3}
\Big\| \frac{\partial( R\circ X^{1}_{F})}{\partial z^{0}_{j}}\Big\|_{D(s-5{ \sigma },\eta r)}\leq \\
\begin{aligned}
& \leq \sum_{k=1}^{n} \Big\|\frac{\partial R(X_{F}^{1})}{\partial y_{k}} \Big\|_{D(s-5{ \sigma },\eta r)} \Big|\frac{\partial y_{k}}{\partial z^{0}_{j}}\Big|+\sum_{k\geq 1}\Big\|\nabla_{Z_{k}}R(X_{F}^{1})\Big\|_{D(s-5{ \sigma },r)} \Big|\frac{\partial Z_{k}}{\partial z^{0}_{j}}\Big|\\
& \leq \frac{C}{j^{\beta}}{ \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}\Big(1+ \sum_{k\geq 1}\frac{1}{k^{2{ \beta }}(1+|j-k|)} \Big) \\
& \leq \frac{C}{j^{\beta}}{ \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}.
\end{aligned}
\end{gathered}$$ $\spadesuit$ We now estimate ${\displaystyle}\Big\| \frac{\partial^{2}( R\circ X^{1}_{F})}{\partial z^{0}_{i}\partial z^{0}_{j}}\Big\|_{D(s-5{ \sigma },\eta r)}$ for $i,j\geq 1$. By the Leibniz rule, the result will follow from the next estimations.\
$\bullet$ Using the Cauchy estimate in $y_{l}$ and $$\Big\| \sum_{1\leq k,l\leq n} \frac{\partial^{2} R(X_{F}^{1})}{\partial y_{k}\partial y_{l}} \frac{\partial y_{k}}{\partial z^{0}_{i}} \frac{\partial y_{l}}{\partial z^{0}_{j}} \Big\|_{D(s-5{ \sigma },\eta r)}\leq
\frac{C { \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}}{(ij)^{{ \beta }}}.
$$ $\bullet$ By $$\Big\| \sum_{1\leq k\leq n} \frac{\partial R(X_{F}^{1})}{\partial y_{k}} \frac{\partial^{2}y_{k}}{\partial z^{0}_{i}\partial z^{0}_{j}} \Big\|_{D(s-5{ \sigma },\eta r)}\leq \frac{C { \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}}{(ij)^{{ \beta }}}.$$ $\bullet$ By $$\Big\| \sum_{k,l\geq 1 } \frac{\partial^{2} R(X_{F}^{1})}{\partial z_{k}(t)\partial z_{l}} \frac{\partial z_{k}}{\partial z^{0}_{i}} \frac{\partial z_{l}}{\partial z^{0}_{j}} \Big\|_{D(s-5{ \sigma },\eta r)}\leq \frac{C { \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}}{(ij)^{{ \beta }}}.$$ $\bullet$ Using the Cauchy estimate in $z_{k}$, and we get $$\Big\| \sum_{ \substack{k\geq 1\\1\leq l\leq n}} \frac{\partial^{2} R(X_{F}^{1})}{\partial z_{k}\partial y_{l}} \frac{\partial y_{l}}{\partial z^{0}_{i}} \frac{\partial z_{k}}{\partial z^{0}_{j}} \Big\|_{D(s-5{ \sigma },\eta r)}\leq \frac{C { \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}}{(ij)^{{ \beta }}}.$$ All these estimates yield $$\label{Est.o.4}
\Big\| \frac{\partial^{2}( R\circ X^{1}_{F})}{\partial z^{0}_{i}\partial z^{0}_{j}}\Big\|_{D(s-5{ \sigma },\eta r)}\leq \frac{C { \langle }R{ \rangle }_{\eta r,D(s-2{ \sigma },4\eta r)}}{(ij)^{{ \beta }}}.$$ \
Finally, follows from , , and .\
Approximation estimates
-----------------------
\
Recall that the notation $\|\cdot\|^{*}$ (respectively ${ \langle }\,\cdot\,{ \rangle }^{*}$) stands either for $\|\cdot\|$ or $\|\cdot\|^{{ \mathcal{L} }}$ (respectively ${ \langle }\,\cdot\,{ \rangle }$ or ${ \langle }\,\cdot\,{ \rangle }^{{ \mathcal{L} }}$).\
First we recall some approximation results [@Poschel Estimate (7)], which show that the second order approximation of $P$ can be controlled by $P$, and that $P-R$ is small when we contract the domain (this contraction is governed by the new parameter $\eta$):
\[lem.taylor.P\] Let $P$ satisfy Assumption \[AS3\] and consider its Taylor approximation $R$ of the form . Then there exists $C>0$ so that for all $\eta>0$ $$\|X_{R} \|^{*}_{r,D(s,r)}\leq C \|X_{P}\|^{*}_{r,D(s,r)},\quad \text{and}\quad \|X_{P}-X_{R} \|^{*}_{\eta r,D(s,4\eta r)}\leq C\eta \|X_{P} \|^{*}_{r,D(s,r)}.$$
We have an analogous result for the norm ${ \langle }\,\cdot\,{ \rangle }_{r,D(s,r)}$.
\[lem.taylor\] Let $P\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$ and consider its Taylor approximation $R$ of the form . Then there exists $C>0$ so that for all $\eta>0$ $${ \langle }R{ \rangle }^{*}_{r,D(s,r)}\leq C { \langle }P{ \rangle }^{*}_{r,D(s,r)},$$ and $${ \langle }P-R{ \rangle }^{*}_{\eta r,D(s,4\eta r)}\leq C\eta { \langle }P{ \rangle }^{*}_{r,D(s,r)}.$$
$\bullet$ We first prove the second estimate. Define the one variable function $f(t)=P(\theta, t^{2}y,tz,t{ \overline{z} })$. Then by the Taylor formula, there exists $0<t_{0}<1$ so that $$f(1)=f(0)+f'(0)+\frac12f''(0)+\frac{1}6f^{(3)}(t_{0}),$$ which reads $$\begin{aligned}
P(\theta,y,z,{ \overline{z} })-R(\theta,y,z,{ \overline{z} })&=&\frac{1}6f^{(3)}(t_{0})\\
&=&\mathcal{O}\Big( \, z^{3} \frac{\partial^{3} P}{\partial z^{3}},\,yz\frac{\partial^{2} P}{\partial y\partial z}, \,y^{2}\frac{\partial^{2} P}{\partial y^{2}}\,\Big).\end{aligned}$$ Using the Cauchy estimates in $z$ or in $y$, we obtain $$\|P-R\|_{D(s,4\eta r)}\leq C \eta\, (\eta r)^{2}{ \langle }P{ \rangle }_{r,D(s, r)}.$$ The estimates of the derivatives are obtained by the same method, with the adequate choice of the function $f$. A derivative in $z$ costs $\eta$ and a derivative in $y$ costs $\eta^{2}$.\
It is then also clear that we have $ { \langle }P-R{ \rangle }^{{ \mathcal{L} }}_{\eta r,D(s,4\eta r)}\leq C\eta { \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}$.\
$\bullet$ The inequality $ { \langle }R{ \rangle }^{*}_{ r,D(s, r)}\leq C { \langle }P{ \rangle }^{*}_{r,D(s,r)}$ is a consequence of the previous point with $\eta=1$.
The KAM step {#Sect.KAM}
============
\
Let $N$ be a Hamiltonian in normal form as in , which reads in the variables $(\theta, y,z,{ \overline{z} })$, $$N=\sum_{1\leq j\leq n}{ \omega }_{j}(\xi)+\sum_{j\geq 1}\Omega(\xi)z_{j}{ \overline{z} }_{j},$$ and suppose that the Assumptions \[AS1\] and \[AS2\] are satisfied.\
Consider a perturbation $P$ which satisfies Assumptions \[AS3\] and \[AS4\] for some $r,s>0$. Then chose $0<\eta <1/8$, $0<{ \sigma }<s,$ and assume that $$\label{Init}
{ \langle }P{ \rangle }_{r,D(s, r)}+\|X_{P}\|_{ r,D(s, r)}+\frac{\alpha}{M}\Big( { \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s, r)}+\|X_{P}\|^{{ \mathcal{L} }}_{ r,D(s, r)} \Big )\leq \frac{\alpha { \sigma }^{t+1} \eta^{2}}{c_{0}},$$ where $t$ is given by Lemmas \[lem.Poschel1\] and \[lem.1\], $c_{0}$ is a large constant depending only on $n$ and $\tau$ (see [@Poschel Estimate (6)].)\
Thus, by Lemmas \[lem.Poschel1\] and \[lem.taylor.P\], the solution $F$ of the homological equation satisfies $$\|X_{F}\|^{*}_{r,D(s-{ \sigma }, r)}\leq \frac{C}{\alpha { \sigma }^{t}} \|X_{P}\|^{{ \mathcal{L} }}_{r,D(s-{ \sigma }, r)}\leq { \sigma }\eta^{2}.$$ Similarly, by Lemmas \[lem.1\] and \[lem.taylor\] $${ \langle }F{ \rangle }^{+,*}_{r,D(s-{ \sigma }, r)}\leq \frac{C}{\alpha { \sigma }^{t}}{ \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}\leq { \sigma }\eta^{2},$$ so that the hypothesis Lemma \[lem.3\] are fulfilled.\
We use the notations of Section \[Strategy\].\
Estimates on the new error term
-------------------------------
\
We estimate the new error term $P_+$ given by .
Assume . Then there exists $C>0$ (independent of $\eta$ and ${ \sigma }$) so that for all ${\displaystyle}0\leq \lambda \leq \frac{\alpha}M$ $$\begin{gathered}
{ \langle }P_{+}{ \rangle }^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}+\|X_{P_{+}}\|^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}\leq \\
\frac{C}{\alpha { \sigma }^{t} \eta^{2}}\Big({ \langle }P{ \rangle }^{\lambda}_{r,D(s, r)}+\|X_{P}\|_{r,D(s,r)}^{\lambda}\Big)^{2}+C\eta \Big({ \langle }P{ \rangle }^{\lambda}_{r,D(s,r)}+\|X_{P}\|_{r,D(s,r)}^{\lambda}\Big)
\end{gathered}$$
By [@Poschel Estimate (13)], we already have $$\label{EstP+}
\|X_{P_{+}}\|^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}\leq \frac{C}{\alpha { \sigma }^{t} \eta^{2}}\big(\|X_{P}\|_{r,D(s,r)}^{\lambda}\big)^{2}+C\eta \|X_{P}\|_{r,D(s,r)}^{\lambda}.$$ It remains to prove a similar estimate for the ${ \langle },{ \rangle }$ norm.\
By Lemmas \[lem.3\] and \[lem.taylor\] $${ \langle }(P-R)\circ X_{F}^{1}{ \rangle }^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}\leq C { \langle }P-R{ \rangle }^{\lambda}_{\eta r,D(s-2{ \sigma },4\eta r)}\leq C\eta { \langle }P{ \rangle }^{\lambda}_{r,D(s, r)}.$$ Then by Lemma \[lem.3\] again $$\begin{aligned}
{ \langle }\int_{0}^{1}{ \big\{\,{ R(t),F }\, \big\} }\circ X^{t}_{F}\,\text{d}t{ \rangle }^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}&\leq &C \int_{0}^{1}{ \langle }{ \big\{\,{ R(t),F }\, \big\} }\circ X^{t}_{F}{ \rangle }^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)}\text{d}t\\
&\leq& C { \langle }{ \big\{\,{ R(t),F }\, \big\} }{ \rangle }^{\lambda}_{\eta r,D(s-2{ \sigma },4\eta r)}.
\end{aligned}$$ Since $R\in { \Gamma }^{{ \beta }}_{r,D(s,r)}$ and $F\in { \Gamma }^{{ \beta },+}_{r,D(s-{ \sigma },r)}$ are both of degree 2 we can apply Lemma \[lem.2\] and write $${ \langle }\int_{0}^{1}{ \big\{\,{ R(t),F }\, \big\} }\circ X^{t}_{F}\,\text{d}t{ \rangle }^{\lambda}_{\eta r,D(s-5{ \sigma },\eta r)} \leq \frac{C}{{ \sigma }} { \langle }R{ \rangle }^{\lambda}_{\eta r,D(s,\eta r)} \,{ \langle }F{ \rangle }^{+,\lambda}_{\eta r,D(s-{ \sigma },\eta r)}.$$ Finally by Lemmas \[lem.1\] and \[lem.taylor\] $${ \langle }R{ \rangle }^{\lambda}_{\eta r,D(s,\eta r)}\, { \langle }F{ \rangle }^{+,\lambda}_{\eta r,D(s-{ \sigma },\eta r)}\leq \frac{C}{\alpha { \sigma }^{t}} \Big({ \langle }R{ \rangle }^{\lambda}_{\eta r,D(s,\eta r)}\Big)^{2} \leq \frac{C}{\alpha{ \sigma }^{t}\eta^{2}} \Big({ \langle }P{ \rangle }^{\lambda}_{r,D(s, r)}\Big)^{2},$$ where we used that ${\displaystyle}{ \langle }\,\cdot \,{ \rangle }_{\eta r,D(s,\eta r)} \leq \eta^{-2}{ \langle }\,\cdot\, { \rangle }_{r,D(s, r)}$. Putting the previous estimates together, we complete the proof.
Estimates on the frequencies {#Sect.4.2.}
----------------------------
\
We turn to the new frequencies given by .
There exists $K>10$ and $\alpha_{+}>0$ so that $$\big |k\cdot \omega^{+}(\xi)+l\cdot \Omega^{+}(\xi)\big|\geq \alpha_{+}\frac{{ \langle }l{ \rangle }}{A_{k}}, \quad |k|\leq K, \quad |l|\leq 2.$$
In fact $K$ can be made explicit, it depends on $n,\tau,c_{0}$ and on all the constants $C$.
On the one hand, since ${\displaystyle}{ \widehat }{{ \omega }}_{j}(\xi)=\frac{\partial { \widehat }{N}}{\partial y_{j}}(0,0,0,0,\xi)$, by Lemma \[lem.taylor.P\] we deduce that $$|{ \widehat }{{ \omega }}|_{\Pi}\leq \sup_{D(s,r)\times \Pi}|\frac{\partial { \widehat }{N}}{\partial y}|\leq \|X_{\widehat{N}}\|_{r,D(s,r)}\leq C \|X_{R}\|_{r,D(s,r)}\leq C \|X_{P}\|_{r,D(s,r)}.$$ On the other hand, ${\displaystyle}{ \widehat }{\Omega}_{j}(\xi)=\frac{\partial^{2} { \widehat }{N}}{\partial z_{j}\partial { \overline{z} }_{j}}(0,0,0,0,\xi)$, thus $$\label{key}
|{ \widehat }{\Omega}|_{2\beta,\Pi}\leq \sup_{D(s,r)\times \Pi}|\frac{\partial^{2} { \widehat }{N}}{\partial z_{j}\partial { \overline{z} }_{j}}|j^{2\beta}\leq { \langle }{ \widehat }{N}{ \rangle }_{r,D(s-{ \sigma },r)}\leq C{ \langle }R{ \rangle }_{r,D(s,r)}\leq C { \langle }P{ \rangle }_{r,D(s,r)},$$ hence by the two previous estimates $$\label{3.2}
|{ \widehat }{{ \omega }}|_{\Pi}+|{ \widehat }{\Omega}|_{2\beta,\Pi}\leq C { \big( }\|X_{P}\|_{r,D(s,r)}+ { \langle }P{ \rangle }_{r,D(s,r)}{ \big) }.$$ Similarly, for the Lipschitz norms we obtain $$|{ \widehat }{{ \omega }}|^{{ \mathcal{L} }}_{\Pi}+|{ \widehat }{\Omega}|^{{ \mathcal{L} }}_{2\beta,\Pi}\leq C { \big( }\|X_{P}\|^{{ \mathcal{L} }}_{r,D(s,r)}+ { \langle }P{ \rangle }^{{ \mathcal{L} }}_{r,D(s,r)}{ \big) }.$$ We follow the analysis done in [@Poschel] to bound the small divisors and thanks to $$\begin{aligned}
|\,k\cdot { \widehat }{{ \omega }}+l\cdot { \widehat }{\Omega}\,|&\leq& |k|{ \langle }l{ \rangle }{ \big( }|{ \widehat }{{ \omega }}|_{\Pi}+|{ \widehat }{\Omega}|_{2\beta,\Pi}{ \big) }\\
&\leq& C |k|{ \langle }l{ \rangle }{ \big( }\|X_{P}\|_{r,D(s,r)}+ { \langle }P{ \rangle }_{r,D(s,r)}{ \big) }.\end{aligned}$$ We now choose ${\displaystyle}{ \widehat }{\alpha}\geq C_{0}K\max_{|k|\leq K}A_{k}( \|X_{P}\|_{r,D(s,r)}+ { \langle }P{ \rangle }_{r,D(s,r)})$ where $C_{0}$ is a large universal constant, and thanks to the estimate given by the frequencies before the iteration we get $$|k\cdot \omega^{+}(\xi)+l\cdot \Omega^{+}(\xi)|\geq \alpha^{+}\frac{{ \langle }l{ \rangle }}{A_{k}}, \quad |k|\leq K,$$ with $\alpha_{+}=\alpha-{ \widehat }{\alpha}$. It remains to show that $\alpha^{+}>0$. This is done in [@Poschel Section 4], and the proof still holds with the new norms.
\[Rq4\] The key point in the previous proof is the estimate , which shows that the perturbations of the external frequencies can be controlled by ${ \langle }P{ \rangle }_{r, D(s,r)}$. In the case of a smoothing perturbation $P$ (case ${ \overline }{p}>p$ in ), the norm ${ \langle }\cdot{ \rangle }_{r, D(s,r)}$ is not needed (more precisely, the decay of the derivatives of $P$ is not needed), because we then have $|{ \widehat }{\Omega}|_{2\beta,\Pi}\leq \|X_{P}\|_{r,D(s,r)}$ with $\beta=({ \overline }{p}-p)/2$.
Iteration and convergence {#Sect.Conv}
==========================
In this section we are exactly in the setting of [@Poschel], and we can make the same choice of the parameters in the iteration. We reproduce here the argument of J. Pöschel.\
The iterative lemma
-------------------
\
Denote $P_{0}=P$ and $N_{0}=N$. Then at the $\nu-$th step of the Newton scheme, we have a Hamiltonian $H_{\nu}=N_{\nu}+P_{\nu}$, so that the new error term $P_{\nu+1}$ is given by the formula and the new normal form $N_{\nu+1}$ is associated with the new frequencies given by .\
Let $c_{1}$ be twice the maximum of all constants obtained during the KAM step.\
Set $r_{0}=r$, $s_{0}=s$, $\alpha_{0}=\alpha$ and $M_{0}=M$. For $\nu\geq 0$ and ${\displaystyle}\kappa=4/3$ set $$\alpha_{\nu}=\frac{\alpha_{0}}{2}(1+2^{-\nu}), \quad M_{\nu}=M_{0}(2-2^{-\nu}),\quad \lambda_{\nu}=\frac{\alpha_{\nu}}{M_{\nu}},$$ $${\varepsilon}_{\nu+1}=\frac{c_{1}{\varepsilon}_{\nu}^{\kappa}}{(\alpha_{\nu}{ \sigma }^{t}_{\nu})^{\kappa-1}},\quad { \sigma }_{\nu+1}=\frac{{ \sigma }_{\nu}}2,\quad \eta_{\nu}^{3}=\frac{{\varepsilon}_{\nu}}{\alpha_{\nu}{ \sigma }^{t}_{\nu}},$$ and $$s_{\nu+1}=s_{\nu}-5{ \sigma }_{\nu},\quad r_{\nu+1}=\eta_{\nu}r_{\nu}.$$ The initial conditions are chosen in the following way : ${ \sigma }_{0}=s_{0}/40\leq 1/4$ so that $s_{0}>s_{1}>\cdots \geq s_{0}/2$, $${\varepsilon}_{0}= \gamma_{0}\alpha_{0}{ \sigma }_{0}^{t}\quad \text{and}\quad \gamma_{0}= { \big( }c_{0}+2^{t+3}c_{1}{ \big) }^{-3},$$ where $c_{0}$ is the constant which appears in . We also define $K_{\nu}=K_{0}2^{\nu}$ with $K^{\tau+1}_{0}=1/(c_{1}\gamma_{0})$.\
With the notation $D_{\nu}=D(s_{\nu},r_{\nu})$ we have
\[Lem.Iterative\] Suppose that $H_{\nu}=N_{\nu}+P_{\nu}$ is given on $D_{\nu}\times \Pi_{\nu}$, where $N_{\nu}={ \omega }_{\nu}(\xi)\cdot y+\Omega_{\nu}(\xi)\cdot z{ \overline{z} }$ is a normal form satisfying ${\displaystyle}|{ \omega }_{\nu}|_{\Pi_{\nu}}^{{ \mathcal{L} }}+|\Omega_{\nu}|^{{ \mathcal{L} }}_{2\beta,\Pi_{\nu}}\leq M_{\nu}$, $$|k\cdot \omega_{\nu}(\xi)+l\cdot \Omega_{\nu}(\xi)|\geq \alpha_{\nu}\frac{{ \langle }l{ \rangle }}{A_{k}}, \quad (k,l)\in \mathcal{Z},$$ on $\Pi_{\nu}$ and $${ \langle }P{ \rangle }_{r_{\nu},D_{\nu}}^{\lambda_{\nu}}+\|X_{P}\|^{\lambda_{\nu}}_{r_{\nu},D_{\nu}}\leq {\varepsilon}_{\nu}.$$ Then there exists a Lipschitz family of real analytic symplectic coordinate transformations $\Phi_{\nu+1}: D_{\nu+1}\times \Pi_{\nu} \longrightarrow D_{\nu}$ and a closed subset $$\Pi_{\nu+1}=\Pi_{\nu}\backslash \bigcup_{|k|>K_{\nu}}\mathcal{R}_{kl}^{\nu+1}(\alpha_{\nu+1}),$$ of $\Pi_{\nu}$, where $$\mathcal{R}_{kl}^{\nu+1}(\alpha_{\nu+1})=\Big\{ \xi\in \Pi_{\nu}\,:\,|k\cdot \omega_{\nu+1}+l\cdot \Omega_{\nu+1}|<\alpha_{\nu+1}\frac{{ \langle }l{ \rangle }}{A_{k}}\Big\},$$ such that for $H_{\nu+1}=H_{\nu}\circ \Phi_{\nu+1}=N_{\nu+1}+P_{\nu+1}$, the same assumptions are satisfied with $\nu+1$ in place of $\nu$.
We don’t give the details of the proof of this result, since it is entirely done in [@Poschel] : it is of course an induction on $\nu\in { \mathbb{N} }$ which essentially relies on the results of the Section \[Sect.KAM\].\
Proof of Theorem \[thmKAM\] {#Proof}
---------------------------
\
The result of Theorem \[thmKAM\] is the convergence of the sequence $H_{\nu}$ to a Hamiltonian in normal form, for parameters $\xi$ in a set $\Pi_{\alpha}$, which is the limit of the sets $\Pi_{\nu}$.\
We again follow the proof of Pöschel and we recall the following Lemma
\[Lem.est\] For $\nu\geq 0$, $$\frac{1}{{ \sigma }_{\nu}}\|\Phi_{\nu+1}-id\|^{\lambda_{\nu}}_{r_{\nu},D_{\nu+1}},\|D\Phi_{\nu+1}-I\|^{\lambda_{\nu}}_{r_{\nu},r_{\nu},D_{\nu+1}}\leq \frac{C{\varepsilon}_{\nu}}{\alpha_{\nu}{ \sigma }^{t}_{\nu}},$$ $$|{ \omega }_{\nu+1}-{ \omega }_{\nu}|^{\lambda_{\nu}}_{\Pi_{\nu}},|\Omega_{\nu+1}-\Omega_{\nu}|^{\lambda_{\nu}}_{2{ \beta },\Pi_{\nu}}\leq C{\varepsilon}_{\nu}.$$
Set $\Pi_{0}=\Pi\backslash \bigcup_{k,l}\mathcal{R}_{kl}^{\alpha_{0}}$ and $\Pi_{\alpha}=\cap_{\nu\geq 1} \Pi_{\nu}$. The proof that $\text{Meas}(\Pi\backslash \Pi_{\alpha})\longrightarrow 0$ when $\alpha\longrightarrow 0$ is done in [@Poschel Section 5] and we do not repeat it here.\
For $\nu\geq 1$ we define the map $$\Phi^{\nu}=\Phi_{1}\circ \cdots \circ \Phi_{\nu}\,:D_{\nu}\times \Pi_{\nu-1}\longrightarrow D_{\nu-1},$$ and thus we have $H_{\nu}=H\circ \Phi^{\nu}$. With the Lemma \[Lem.est\] and since[^4] $\cap_{\nu\geq 1}D_{\nu}\times \Pi_{\nu}=D(s/2)\times \Pi_{\alpha}$, we are then able to show, as in [@Poschel], that $\Phi^{\nu}$ is a Cauchy sequence for the supremum norm on ${D(s/2)}\times \Pi_{\alpha}$. Thus it converges uniformly on ${D(s/2)}\times \Pi_{\alpha}$ and its limit $\Phi$ is real analytic on $D(s/2)$. Further, the estimate holds on $D(s/2)\times \Pi_{\alpha}$.\
It remains to prove that $\Phi$ is indeed defined on $D(s/2, r/2)\times \Pi_{\alpha}$ with the same estimate. By Corollary \[coro\_sol\] all the transforms $\Phi^{\nu}$ are linear in $y$ and quadratic in $z,\bar z$ and thus the same is true for the transform $\Phi$ (this fact was also used in[@Posch] or [@ElKuk1]). This specific form is stable by composition and thus all the $\Phi^{\nu}$ have this form and in particular they are linear in $y$ and quadratic in $z,\bar z$.\
Therefore it suffices to verify that the first derivatives with respect to $y, z,\bar z$ and the second derivatives with respect to $ z,\bar z$ of $\Phi^{\nu}$ are uniformly convergent on $D(s/2)\times \Pi_{\alpha}$ to conclude that $\Phi^{\nu}$ convergences to $\Phi$ (actually an extension of the previously defined $\Phi$) uniformly on $D(s/2, \rho)\times \Pi_{\alpha}$ for any $\rho$. In particular, for $r$ small enough, $$\Phi\ : D(s/2, r/2)\times \Pi_{\alpha}\to D(s, r)$$ and $\Phi$ still satisfies estimate .\
So it remains to analyse the convergence of the derivatives. Using Lemma \[Lem.est\] we obtain successively $\|D\Phi_{\nu}\|_{r_{\nu},r_{\nu},D_{\nu}}\leq 2$ and then uniformly on ${D(s/2)}\times \Pi_{\alpha}$ $$\|D\Phi^{\nu+1}-D\Phi^{\nu}\|_{r_{\nu},r_{\nu},D_{\nu}}\leq \|D\Phi_{\nu}\|_{r_{\nu},r_{\nu},D_{\nu}} \|D\Phi_{\nu}-I|\|_{r_{\nu},r_{\nu},D_{\nu}}$$ and we deduce that uniformly on ${D(s/2)}\times \Pi_{\alpha}$ $$\|D\Phi^{\nu+1}-D\Phi^{\nu}\|_{r_{\nu},r_{\nu},D_{\nu}}\leq c\epsilon^{1/2}_\nu.$$ So again $D\Phi^{\nu}$ converges uniformly on ${D(s/2)}\times \Pi_{\alpha}$. Similarly we obtain the convergence of the second derivatives using the formula $$D^2 \Phi^{\nu+1}= D^2\Phi^{\nu}\cdot (D\Phi^{\nu})^2+ \Phi^{\nu}\cdot D^2\Phi^{\nu}.$$ On the other hand, again using Lemma \[Lem.est\], the frequencies functions ${ \omega }_{\nu}$ and $\Omega_\nu$ converge uniformly on $ \Pi_{\alpha}$ to Lipschitz functions ${ \omega }^\star$ and $\Omega^\star$ satisfying and thus in view of Lemma \[Lem.Iterative\].\
We then deduce that, uniformly on $D(s/2, r/2)\times \Pi_{\alpha}$, $$R_\nu:=H\circ \Phi^\nu -N_\nu \quad \longrightarrow \quad H\circ\Phi -N^\star=:R^\star$$ and since for all $\nu$ the Taylor expansion of $R_\nu$ contains only monomials $y^mz^q\bar z^{\bar q}$ with $2|m|+|q+\bar q|\geq 3$ the same property holds true for $R^\star$.
Application to the nonlinear Schrödinger equation {#Sect.5}
==================================================
Let $n\geq 1$ be an integer and $\nu, {\varepsilon}>0$ be two small parameters so that $\nu\geq C_{0}{\varepsilon}$, where $C_{0}>0$ is a constant which will be defined later. Set $\Pi=[-1,1]^{n}$. We consider a perturbation of the one dimensional Schrödinger equation with harmonic potential $$\label{nls}
i\partial_t u+\partial^{2}_{x} u -x^{2}u -\nu V(\xi,x)u= {\varepsilon}|u|^{2m}u ,\quad
(t,x)\in{ \mathbb{R} }\times {{ \mathbb{R} }},$$ where $m\geq 1$ is an integer and $\big(V(\xi,\cdot)\big)_{\xi\in \Pi}$ is family of a real analytic bounded potentials with $V(0,\cdot)=0$ which will be made explicit below.\
Recall that $T=-\partial^{2}_{x} +x^{2}$ denotes the harmonic oscillator. Its eigenfunctions are the Hermite functions $(h_{j})_{j\geq 1}$, associated to the eigenvalues $(2j-1)_{j\geq 1}$. Now consider the linear operator $A=A(\nu,\xi)=-\partial_{x}^{2} +x^2+\nu V(\xi,x)$. Under the previous assumptions, $A$ is self-adjoint and has pure point spectrum with simple eigenvalues $(\lambda_j(\nu,\xi))_{j\geq 1}$ satisfying $\lambda_j(\nu,\xi)\sim 2j-1$. Its eigenfunctions $\big({ \varphi }_j(\xi,\cdot )\big)_{j\geq 1}$ form an orthonormal basis of $L^2({ \mathbb{R} })$, and ${ \varphi }_{j}(\xi,\cdot) \sim h_{j}$ as $\nu \to 0$ in $L^2$ norm. As a consequence $A$ and $T$ have the same domain and $D(A^{p/2})={ \mathcal{H} }^{p}$. We will prove these facts for the particular class of potentials we will consider (see Lemmas \[lem.61\] and \[lem.asymp\] below).\
The parameter ${\varepsilon}>0$ will be small so that we can apply Theorem \[thmKAM\] and $\nu>0$ will be small too, so that we have a suitable perturbation theory for the operator $A$. We now explain the restriction $\nu \geq C_{0}{\varepsilon}$. The aim of this section is to construct a potential $V$ so that Theorem \[thmKAM\] applies, and in particular, has to be satisfied. Small values of $k$, $l$ in and the asymptotics of Lemma \[lem.asymp\] give $C\nu \geq \alpha$. This together with the condition ${\varepsilon}\leq {\varepsilon}_{0} \alpha$ in Theorem \[thmKAM\] yields the result.\
We fix a finite subset ${ \mathcal{J} }$ of ${ \mathbb{N} }$ of cardinal $n$. Without loss of generality and in order to simplify the presentation, we assume ${ \mathcal{J} }=\{1,\cdots,n\}$. We then expand $u$ and $\bar u$ in the basis of eigenfunctions using the phase space structure of the introduction, namely we write $$\begin{aligned}
u(x)=&\sum_{j=1}^n(y_{j}+I_{j})^{\frac12}{ \text{e} }^{i\theta_{j}}{ \varphi }_{j}(\xi,x)+\sum_{j\geq 1}z_{j}{ \varphi }_{j+n}(\xi,x),\\
\bar u(x) =&\sum_{j=1}^n(y_{j}+I_{j})^{\frac12}{ \text{e} }^{-i\theta_{j}}{ \varphi }_{j}(\xi,x)+\sum_{j\geq 1}{ \overline{z} }_{j}{ \varphi }_{j+n}(\xi,x),\end{aligned}$$ where $(\theta,y,z,\bar z)\in \mathcal{P}^{p}={ \mathbb{T} }^{n}\times { \mathbb{R} }^{n}\times {\ell}^{2}_{p} \times {\ell}^{2}_{p}$ (recall that $\ell^{2}_{p}$ is the space $\ell^{2}_{\Psi}$ with $\Psi(j)=j^{p/2}$) are regarded as variables and $I\in { \mathbb{R} }_+^n$ are regarded as parameters (here ${ \mathbb{R} }_+$ denotes the set of non negative real numbers). In this setting equation reads as the Hamilton equations associated to the Hamiltonian function $H=N+P$ where $$N=\sum_{j=1}^n\lambda_j (\nu,\xi) y_j+ \sum_{j\geq 1}\Lambda_{j}(\nu,\xi) z_j\bar z_j,$$ $\Lambda_{j}(\nu,\xi)= \lambda_{j+n}(\nu,\xi)$, $G(u,\bar u)=(u\,\bar u)^{m+1}$ and $$\begin{aligned}
\begin{split}\label{PNLS}
P(\theta,y,z,{ \overline{z} })={\varepsilon}\int_{{ \mathbb{R} }}G\Big(&\sum_{j=1}^{n}(y_{j}+I_{j})^{\frac12}{ \text{e} }^{i\theta_{j}}{ \varphi }_{j}(\xi,x)+\sum_{j\geq 1}z_{j}{ \varphi }_{j+n}(\xi,x),\\ &\sum_{j=1}^{n}(y_{j}+I_{j})^{\frac12}{ \text{e} }^{-i\theta_{j}}{ \varphi }_{j}(\xi,x)+\sum_{j\geq 1}{ \overline{z} }_{j}{ \varphi }_{j+n}(\xi, x)\Big)\text{d}x.
\end{split} \end{aligned}$$ For the sequel we fix $(I_{j})_{1\leq j\leq n}$. We assume that $(\theta,y,z,\bar z)\in D(s,r)$ for some fixed $s,r>0$ (recall the definition of $D(s,r)$). There is no particular smallness assumption on $s,r$, we only have to take $r>0$ with $r<\min_{1\leq j\leq n}I_{j}$ so that $(y_{j}+I_{j})^{1/2}$ is well-defined.\
We now show that we can construct a class of potentials $V$ so that Theorem \[thmKAM\] applies.\
Definition of the family of potentials $V$
------------------------------------------
\
Let $(f_{j})_{1\leq j\leq n}$ be the dual basis of $(h^{2}_{j})_{1\leq j\leq n}$, i.e. $(f_{j})\in \text{Span}_{{ \mathbb{R} }}(h_{1}^{2},\dots,h_{n}^{2})$ and $\int_{{ \mathbb{R} }}f_{j}h_{k}^{2}=\delta_{jk}$ for all $1\leq j,k\leq n$.\
We say that ${ \alpha }=({ \alpha }_{k})_{k\geq n+1} \in \mathcal{Z}_{n}$ if $-\frac12\leq { \alpha }_{k}\leq \frac12$ for all $k\geq n+1$. We endow the set of such sequences by the probability measure defined as the infinite product $(k\geq n+1)$ of the Lebesgue measure on $[-1/2,1/2]$. Then define $$g(x)=\sum_{k\geq n+1}{ \alpha }_{k}{ \text{e} }^{-k}h_{2k-1}(\sqrt{2}x),$$ and for $\xi=(\xi_{1},\dots , \xi_{n})\in\Pi=[-1,1]^{n}$ and $$\label{def.V}
V(\xi,x)=\sum_{k=1}^{n}\xi_{k}f_{k}(x)+\xi_{1}g(x).$$ The spectral data ${ \varphi }_{j}$ and $\lambda_{j}$ are defined by the spectral equation $$\label{eq.vp}
\big(-\partial^{2}_{x}+x^{2}+\nu V(\xi,x)\big){ \varphi }_{j}(\xi,x)=\lambda_{j}(\nu\xi){ \varphi }_{j}(\xi,x),$$ and we assume that the $({ \varphi }_{j})$ are $L^{2}-$normalised ($\|{ \varphi }_{j}(\xi,\cdot)\|_{L^{2}}=1$ for all $\xi\in \Pi$ and $j\geq 1$). Moreover, in order to define ${ \varphi }_{j}$ uniquely, we impose ${ \langle }{ \varphi }_{j},h_{j}{ \rangle }>0.$\
In the sequel we need a particular case of estimates proved by K. Yajima & G. Zhang [@YajimaZhang1]
For all $2<p<\infty$ there exists $\alpha>0$ and $C>0$ so that for all $\xi \in \Pi$ and $j\geq 1$ $$\label{YZ}
\|{ \varphi }_{j}(\xi,\cdot)\|_{L^{p}({ \mathbb{R} })}\leq Cj^{-{ \alpha }}.$$
The next result is the key estimate in our perturbation theory.
\[lem.61\] There exist ${ \alpha }>0$ and $C>0$ so that for all $\xi \in \Pi$, $\nu>0$ and $j\geq 1$ $$\label{norme.2}
\|{ \varphi }_{j}(\xi,\cdot)-{ \varphi }_{j}(\eta,\cdot)\|_{L^{2}}\leq C\nu |\xi-\eta|j^{-{ \alpha }}.$$
In particular $\|{ \varphi }_{j}(\xi,\cdot)-h_{j}\|_{L^{2}}\leq C\nu |\xi|j^{-{ \alpha }}$, which shows that the ${ \varphi }_{j}$ are close to the Hermite functions in $L^2$ norm.
In the sequel, we write ${ \varphi }_{j}(\xi)$ instead of ${ \varphi }_{j}(\xi,\cdot)$. For $\xi,\eta \in \Pi$, we compute $$A(\nu\,\xi){ \varphi }_{j}(\eta)=\big(-\partial^{2}_{x}+x^{2}+\nu V(\xi,x)\big){ \varphi }_{j}(\eta)=\lambda_{j}(\nu\,\eta){ \varphi }_{j}(\eta)+\nu (V(\xi,x)-V(\eta,x)){ \varphi }_{j}(\eta).$$ Thus by and there exists ${ \alpha }>0$ such that $$\begin{aligned}
\label{CV.0}
\big\|\big(A(\nu\,\xi)-\lambda_{j}(\nu\,\eta)\big){ \varphi }_{j}(\eta)\big\|_{L^{2}}&=&\nu \|(V(\xi)-V(\eta)){ \varphi }_{j}(\eta)\|_{L^{2}}\nonumber\\
&\leq &\nu\|V(\xi)-V(\eta)\|_{L^{4}}\|{ \varphi }_{j}(\eta)\|_{L^{4}}\nonumber \\
&\leq& C\nu |\xi-\eta|{j^{-{ \alpha }}}.\end{aligned}$$ Choosing $\eta=0$ in , and as ${ \varphi }_{j}(0)=h_{j}$ and $\lambda_{j}(0)=2j-1 $, we get $$\begin{aligned}
1=\|h_{j}\|_{L^{2}} &\leq& \big\|\big(A(\nu\,\xi)-(2j-1)\big)^{-1}\big\|_{L^{2}\to L^{2}}\big\|\big(A(\nu\,\xi)-(2j-1)\big)h_{j}\big\|_{L^{2}} \\
&\leq & C\nu{j^{-{ \alpha }}}\big\|\big(A(\nu\,\xi)-(2j-1)\big)^{-1}\big\|_{L^{2}\to L^{2}}.\end{aligned}$$ The previous estimate together with the general formula which holds for any self-adjoint operator $ \|\big(A(\nu\,\xi)-(2j-1)\big)^{-1}\|_{L^{2}\to L^{2}} = \text{dist}\big(2j-1,{ \sigma }(A(\nu\,\xi))\big)^{-1}$ gives $\text{dist}\big(2j-1,{ \sigma }(A(\nu\,\xi))\big) \leq C\nu{j^{-{ \alpha }}}$, where ${ \sigma }(A(\nu\,\xi))$ denotes the spectrum of $A(\nu\,\xi)$. A similar argument, taking $\xi=0$ in , leads to $\text{dist}\big(\lambda_{j}(\nu \eta),{ \sigma }(T)\big) \leq C\nu{j^{-{ \alpha }}}$. Thus for all $j\geq 1$ $$\label{dvp.vp}
\lambda_{j}(\nu \xi)=2j-1+\nu \ O( j^{-{ \alpha }}).$$ Using that $({ \varphi }_{k}(\xi))_{k\geq 1}$ is a Hilbertian basis of $L^{2}({ \mathbb{R} })$, we deduce $$\begin{aligned}
\big\|{ \varphi }_{j}(\eta)-{ \langle }{ \varphi }_{j}(\xi),{ \varphi }_{j}(\eta){ \rangle }{ \varphi }_{j}(\xi)\big\|^{2}_{L^{2}}&=& \sum_{k\geq 1} |{ \langle }{ \varphi }_{j}(\eta)-{ \langle }{ \varphi }_{j}(\xi),{ \varphi }_{j}(\eta){ \rangle }{ \varphi }_{j}(\xi),{ \varphi }_{k}(\xi){ \rangle }|^{2}\nonumber\\
&=& \sum_{k\geq 1, k\neq j} |{ \langle }{ \varphi }_{j}(\eta),{ \varphi }_{k}(\xi){ \rangle }|^{2}.\label{c.1}\end{aligned}$$ With the same decomposition, we can also write $$\begin{aligned}
\|\big(A(\nu\,\xi)-\lambda_{j}(\nu\,\eta)\big){ \varphi }_{j}(\eta)\|^{2}_{L^{2}}&=& \sum_{k\geq 1} |{ \langle }\big(A(\nu\,\xi)-\lambda_{j}(\nu\,\eta)\big){ \varphi }_{j}(\eta),{ \varphi }_{k}(\xi) { \rangle }|^{2}\nonumber \\
&=&\sum_{k\geq 1} |{ \langle }\big(\lambda_{k}(\nu\,\xi)- \lambda_{j}(\nu\,\eta) \big){ \varphi }_{k}(\xi),{ \varphi }_{j}(\eta) { \rangle }|^{2} \nonumber\\
&=&\sum_{k\geq 1}|\lambda_{k}(\nu\,\xi)- \lambda_{j}(\nu\,\eta) |^{2}|{ \langle }{ \varphi }_{k}(\xi),{ \varphi }_{j}(\eta){ \rangle }|^{2}\nonumber \\
&\geq & \sum_{k\geq 1,k\neq j}|{ \langle }{ \varphi }_{k}(\xi),{ \varphi }_{j}(\eta){ \rangle }|^{2} \label{c.2},\end{aligned}$$ because by $| \lambda_{k}(\nu\,\xi)- \lambda_{j}(\nu\,\eta) |\geq 1$ for $k\neq j$ uniformly in $\xi,\eta$ and uniformly in $\nu$ small enough. Now by , and we deduce that $$\big\|{ \varphi }_{j}(\eta)-{ \langle }{ \varphi }_{j}(\xi),{ \varphi }_{j}(\eta){ \rangle }{ \varphi }_{j}(\xi)\big\|^{2}_{L^{2}} \leq C\nu |\xi-\eta| j^{-{ \alpha }} .$$ In particular, taking the scalar product of ${ \varphi }_j(\eta)$ with ${ \varphi }_{j}(\eta)-{ \langle }{ \varphi }_{j}(\xi),{ \varphi }_{j}(\eta){ \rangle }{ \varphi }_{j}(\xi)$, we obtain $$\big| 1-{ \langle }{ \varphi }_{j}(\xi),{ \varphi }_{j}(\eta){ \rangle }^2\big|\leq C\nu |\xi-\eta| j^{-{ \alpha }} .$$ The last two estimates imply $\|{ \varphi }_{j}(\xi)-{ \varphi }_{j}(\eta)\|_{L^{2}}\leq C\nu |\xi-\eta| j^{-{ \alpha }} $ which was the claim.
\[lem.asymp\] We have the following asymptotics when $\nu\longrightarrow 0$ $$\label{dvp1}
\lambda_{j}(\nu\,\xi)=2j-1+\nu \xi_{j}+o(\nu),\quad \forall \, 1\leq j\leq n,$$ $$\label{dvp2}
\Lambda_{j}(\nu\,\xi)=\lambda_{j+n}(\nu \xi)=2(j+n)-1+\nu\sum_{k=1}^{n}\xi_{k} \int_{{ \mathbb{R} }} (f_{k}+\delta_{1k}g)h^{2}_{n+j}+o(\nu),\quad \forall \, j\geq 1.$$
We first prove . We differentiate equation in $\xi_{k}$ $$A(\nu\,\xi)\frac{{ \varphi }_{j}(\xi)}{\partial \xi_{k}}+\nu (f_{k}+\delta_{1k}g){ \varphi }_{j}(\xi)=\lambda_{j}(\nu\,\xi)\frac{{ \varphi }_{j}(\xi)}{\partial \xi_{k}}+\frac{\partial \lambda_{j}(\nu\,\xi)}{\partial \xi_{k}}{ \varphi }_{j}(\xi),$$ take the scalar product with ${ \varphi }_{j}(\xi)$ and the selfadjointness of $A(\nu\,\xi)$ gives $$\label{derivee}
\frac{\partial \lambda_{j}(\nu\,\xi)}{\partial \xi_{k}}=\nu \int_{{ \mathbb{R} }}(f_{k}+\delta_{1k}g){ \varphi }_{j}^{2}(\xi).$$ Now by $$\begin{aligned}
|\int_{{ \mathbb{R} }}(f_{k}+\delta_{1k}g)({ \varphi }_{j}^{2}(\xi)-h^{2}_{j})|&\leq &\|f_{k}+\delta_{1k}g\|_{L^{\infty}}\|{ \varphi }_{j}(\xi)+h_{j}\|_{L^{2}}\|{ \varphi }_{j}(\xi)-h_{j}\|_{L^{2}}\nonumber \\
&\leq & C \|{ \varphi }_{j}(\xi)-h_{j}\|_{L^{2}}\longrightarrow 0\end{aligned}$$ when $\nu\longrightarrow 0$. Thus by definition of the $f_{k}$ and $g$ and by estimate , we obtain that for all $1\leq j\leq n$ $$\begin{aligned}
\lambda_{j}(\nu\,\xi)&=&2j-1+\nu \sum_{k=1}^{n}\xi_{k}\int_{{ \mathbb{R} }}(f_{k}+\delta_{1k}g)h_{j}^{2}+o(\nu)\\
&=& 2j-1 +\nu \xi_{j}+ o(\nu),\end{aligned}$$ which is .\
The asymptotic of is proved in the same way. Observe that we can prove a better estimate on the error term using , but we do not need it here.
Verification of Assumptions \[AS1\] and \[AS2\]
-----------------------------------------------
\
\[Lem.Z\] There exists a null measure set $\mathcal{N}\subset \mathcal{Z}_{n}$ such that for all ${ \alpha }\in \mathcal{Z}_{n} \backslash \mathcal{N}$ we have for all $1\leq p,q$, with $p\neq q$ $$\label{proj1}
\int_{{ \mathbb{R} }}(f_{1}+g)h^{2}_{n+p}\notin { \mathbb{Z} },$$ and $$\label{proj2}
\int_{{ \mathbb{R} }}(f_{1}+g)(h^{2}_{n+p}\pm h^{2}_{n+q})\notin { \mathbb{Z} }.$$
For $j\geq 1$, the Hermite function $h_{j}$ reads $h_{j}(x)=P_{j}(x){ \text{e} }^{-x^{2}/2}$, where $P_{j}$ is a polynomial of degree exactly $(j-1)$, and $P_{j}$ is even (resp. odd) when $(j-1)$ is even (resp. odd). We have $\text{Span}_{{ \mathbb{R} }}(h_{1},\dots,h_{n})={ \text{e} }^{-x^{2}/2}{ \mathbb{R} }_{n-1}[X]$. Thus we deduce that there exist $(\mu_{kj})$ so that $$\label{hj2}
h^{2}_{j}(x)=\sum_{k=1}^{j}\mu_{kj}h_{2k-1}(\sqrt{2}x),$$ with $\mu_{jj}\neq 0$.\
We assume that $q<p$. The application $$({ \alpha }_{n},{ \alpha }_{n+1},\dots)\longmapsto \int_{{ \mathbb{R} }}(f_{1}+g)(h^{2}_{n+p}\pm h^{2}_{n+q})$$ is a linear form. In order to prove , it suffices to check that this linear form is nontrivial. According to and to the definition of $f_{1}$ and $g$, the coefficient of ${ \alpha }_{n+p}$ is $${ \text{e} }^{-(n+p)}\mu_{n+p,n+p}\int_{{ \mathbb{R} }}h^{2}_{2(n+p)-1}(\sqrt{2}x)\text{d}x={ \text{e} }^{-(n+p)}\mu_{n+p,n+p}/\sqrt{2}\neq 0.$$ Therefore for fixed $p,q$ is satisfied on the complementary of a null measure set $\mathcal{N}_{p,q}$. Finally, is satisfied on $\mathcal{Z}_{n} \backslash \mathcal{N}$ where $ \mathcal{N}=\cup_{p,q\geq 1} \mathcal{N}_{p,q}$. The proof of is similar.
In the sequel we fix ${ \alpha }\in \mathcal{Z}_{n} \backslash \mathcal{N}$ so that Lemma \[Lem.Z\] holds true. We are now able to show that Assumption \[AS1\] is satisfied. Recall that in our setting, the internal frequencies are $\lambda(\nu\xi)=(\lambda_{j}(\nu\xi))_{1\leq j\leq n}$ and the external frequencies are $\Lambda(\nu\xi)=(\Lambda_{j}(\nu\xi))_{j\geq 1}$ with $\Lambda_{j}(\nu\xi)=\lambda_{n+j}(\nu\xi)$.
There exists $\nu_{0}>0$ so that for all $0<\nu <\nu_{0}$ we have $$\label{mesnulle}
\text{Meas}\Big(\big\{\,\xi \in \Pi\;:\:k\cdot \lambda(\nu\,\xi)+l\cdot \Lambda(\nu\,\xi)=0\,\big\}\Big) =0,\quad \forall\; (k,l)\in \mathcal{Z},$$ and for all $\xi \in \Pi$ $$\label{non.nul}
l\cdot \Lambda(\nu\,\xi)\neq 0, \quad \forall\, 1\leq |l|\leq 2.$$
We prove by contradiction. Let $(k,l)\in \mathcal{Z}$. In the case $|l|=2$ in we can write $$k\cdot \lambda(\nu\,\xi)+l\cdot \Lambda(\nu\,\xi)=\sum_{j=1}^{n}k_{j}\lambda_{j}(\nu\,\xi)+\lambda_{n+p}(\nu\,\xi)-\lambda_{n+q}(\nu\,\xi):=F(\nu\, \xi),$$ for some $p,q\geq 1$. Now if does not hold, $F: { \mathbb{R} }^{n}\longrightarrow { \mathbb{R} }$ is a $\mathcal{C}^{1}$ function which vanishes on a set of positive measure in any neighbourhood of $0$, thus $F(0)=0$ and for all $1\leq k\leq n$, $\frac{\partial F}{\partial \xi_{k}}(0)=0$. By Lemma \[lem.asymp\] these conditions read $$\begin{aligned}
\sum_{j=1}^{n}(2j-1)k_{j}+2(p-q)&=&0\quad \mbox{ and}\nonumber \\
k_{j}+\int_{{ \mathbb{R} }}(f_{j}+\delta_{ij}g)(h^{2}_{n+p}-h^{2}_{n+q})&=&0, \quad \forall \;1\leq j\leq n. \label{dl}\end{aligned}$$ In particular for $j=1$, is in contradiction with .\
The case $|l|=1$ is similar, using .\
It remains to prove . For all $j\geq 1$, $\Lambda_{j}(\nu\,\xi)\longrightarrow 2j-1$ when $\nu \longrightarrow0$. Hence holds true if $\nu$ is small enough.
We now check Assumption \[AS2\]. Firstly, thanks to we have that for $j,k\geq 1$, $|\Lambda_{j}(\nu \xi)-\Lambda_{k}(\nu \xi)|\geq |j-k|$ and $|\Lambda_{j}(\nu \xi)|\geq j$. Then by and $$\begin{aligned}
|\Lambda_{j}(\nu \xi)-\Lambda_{j}(\nu \eta)| &\leq & \nu |\xi-\eta|\,\sup_{\xi \in\Pi}\int_{{ \mathbb{R} }}\big| (f_{k}+\delta_{1k}g){ \varphi }^{2}_{j+n}(\xi,\cdot) \big|\\
&\leq & \nu |\xi-\eta| \big\|f_{k}+\delta_{1k}g\big\|_{L^{2}} \sup_{\xi \in\Pi}\|{ \varphi }_{j+n}(\xi,\cdot)\|^{2}_{L^{4}}\\
&\leq &C \nu |\xi-\eta|j^{-{ \alpha }},\end{aligned}$$ and Assumption \[AS2\] is fulfilled.\
Verification of Assumptions \[AS3\] and \[AS4\]
-----------------------------------------------
\
Recall that for $p\geq 0$, ${ \mathcal{H} }^{p}=D(T^{p/2})$ is the Sobolev space based on the harmonic oscillator. Thanks to and , we also have ${ \mathcal{H} }^{p}=D(A^{p/2}(\nu\,\xi))$ for all $\nu>0$ small enough and $\xi\in \Pi$. Observe that ${ \mathcal{H} }^{p}$ is an algebra and the Sobolev embeddings which hold for the usual Sobolev space $H^{p}$ are also true here, since $H^{p}\subset { \mathcal{H} }^{p}$.\
Let $u=\sum_{j\geq 1} { \alpha }_{j}{ \varphi }_{j}$. Then $u\in { \mathcal{H} }^{p}$ if and only if ${ \alpha }_{j}\in \ell^{2}_{p}$.\
We now check the smoothness of $P$ and the decay of the vector field $X_{P}$.\
Let $p\geq 2$ so that we are in the framework of Theorem \[thmKAM\]. Since $G(u,{ \overline }{u})=(u{ \overline }{u})^{m+1}$ in , we have $$\label{def.PI}
P={\varepsilon}\int_{{ \mathbb{R} }}|u|^{2(m+1)}.$$ We first show that $\frac{\partial P}{\partial z_{j}} \in \ell^{2}_{p}$. We have $$\label{na.z}
\frac{\partial P}{\partial z_{j}}={\varepsilon}(m+1)\int_{{ \mathbb{R} }}{ \varphi }_{j+n}u^{m}\,{ \overline }{u}^{m+1},$$ thus $\frac{\partial P}{\partial z_{j}}$ is (up to a constant factor) the $(j+n)$th coefficient of the decomposition of $u^{m}\,{ \overline }{u}^{m+1}$, and this latter term is in ${ \mathcal{H} }^{p}$ (because ${ \mathcal{H} }^{p}$ is an algebra), hence the result. The other components of $X_{P}$ can be handled in the same way, and we get $X_{P}\in \mathcal{P}^{p}$.\
By and Sobolev embeddings $$\sup_{D(s,r)\times \Pi}|P|\leq {\varepsilon}\|u\|_{L^{2(m+1)}}^{2(m+1)}\leq {\varepsilon}\|u\|_{{ \mathcal{H} }^{p}}^{2(m+1)}.$$ Similarly, using and $$\begin{aligned}
\frac{\partial P}{\partial \theta_{j}}&=&{\varepsilon}i(m+1)(y_{j}+I_{j})^{\frac12}\Big[ { \text{e} }^{i\theta_{j}} \int_{{ \mathbb{R} }}{ \varphi }_{j}u^{m}\,{ \overline }{u}^{m+1}+ { \text{e} }^{-i\theta_{j}} \int_{{ \mathbb{R} }}{ \varphi }_{j}u^{m+1}\,{ \overline }{u}^{m} \Big]\\
\frac{\partial P}{\partial y_{j}}&=&{\varepsilon}\frac {m+1}2 (y_{j}+I_{j})^{-\frac12} \Big[{ \text{e} }^{i\theta_{j}}\int_{{ \mathbb{R} }}{ \varphi }_{j}u^{m}\,{ \overline }{u}^{m+1}+{ \text{e} }^{-i\theta_{j}}\int_{{ \mathbb{R} }}{ \varphi }_{j}u^{m+1}\,{ \overline }{u}^{m} \Big]\end{aligned}$$ it is easy to see that $\sup_{D(s,r)\times \Pi}| X_{P}|_{r}\leq C{\varepsilon}$. We now turn to the Lipschitz norms. Let $\xi,\eta \in \Pi$ $$\begin{aligned}
|P(\xi)-P(\eta)|&\leq &C {\varepsilon}\|u(\xi)-u(\eta)\|_{L^{2}}(\|u(\xi)\|^{2m+1}_{L^{4(2m+1)}}+\|u(\eta)\|^{2m+1}_{L^{4(2m+1)}})\nonumber \\
&\leq &C{\varepsilon}\|u(\xi)-u(\eta)\|_{L^{2}} \|u\|_{{ \mathcal{H} }^{p}}^{2m+1}.\label{a1}\end{aligned}$$ Now by $$\begin{aligned}
\|u(\xi)-u(\eta)\|_{L^{2}} &\leq & C\sum_{j=1}^{n}\| { \varphi }_{j}(\xi)- { \varphi }_{j}(\eta)\|_{L^{2}}+\sum_{j\geq 1} j^{p}|z_{j}|\| { \varphi }_{j+n}(\xi)- { \varphi }_{j+n}(\eta)\|_{L^{2}}\nonumber \\
&\leq & C|\xi-\eta|,\label{a2}\end{aligned}$$ where in the last line we used Cauchy-Schwarz and the fact that $(z_{j})_{j\geq 1}\in l^{2}_{p}$ with $p\geq 2$. Then and show the Lipschitz regularity of $P$. We can proceed similarly for $X_{P}$.\
It remains to prove the decay estimates of Assumption \[AS4\]. Using , and the Sobolev embeddings, we obtain $$\Big|\frac{\partial P}{\partial z_{j}} \Big|\leq{\varepsilon}(m+1)\|{ \varphi }_{j+n}\|_{L^{\infty}({ \mathbb{R} })}\|u\|^{2m+1}_{L^{2m+1}}\leq C{\varepsilon}j^{-{ \alpha }} \|u\|^{2m+1}_{{ \mathcal{H} }^{p}},$$ and similarly, from $$\frac{\partial^{2} P}{\partial z_{j}\partial z_{l}}={\varepsilon}m(m+1)\int_{{ \mathbb{R} }}{ \varphi }_{j+n}{ \varphi }_{l+n}u^{m-1}\,{ \overline }{u}^{m+1},$$ we deduce $$\Big|\frac{\partial^{2} P}{\partial z_{j}\partial z_{l}} \Big|\leq C{\varepsilon}\|{ \varphi }_{j+n}\|_{L^{\infty}({ \mathbb{R} })}\|{ \varphi }_{l+n}\|_{L^{\infty}({ \mathbb{R} })}\|u\|^{2m}_{L^{2m}}\leq {\varepsilon}C(jl)^{-{ \alpha }} \|u\|^{2m}_{{ \mathcal{H} }^{p}}.$$ The estimates of the Lipschitz norms are obtained as in , and using .
As a conclusion Assumptions \[AS1\] - \[AS4\] are satisfied and we can apply Theorem \[thmKAM\] with some ${ \beta }>0$ if ${\varepsilon}>0$ is small enough. Recall that $\Pi=[-1,1]^n$.
\[main1\] Let $m\geq 1$ and $n\geq 1$ be two integers. Let $V(\xi, \cdot)$ be the $n$ parameters family of potentials defined by . There exist ${\varepsilon}_0>0$, $\nu_0>0$, $C_{0}>0$ and, for each ${\varepsilon}<{\varepsilon}_0$, a Cantor set $\Pi_{{\varepsilon}}\subset \Pi$ of asymptotic full measure when ${\varepsilon}\to 0$, such that for each $\xi\in \Pi_{{\varepsilon}}$ and for each $C_{0}{\varepsilon}\leq \nu<\nu_0$, the solution of $$\label{nls2}
i\partial_t u+\partial^{2}_{x} u -x^{2}u -\nu V(\xi,x)u= {\varepsilon}|u|^{2m}u ,\quad
(t,x)\in{ \mathbb{R} }\times {{ \mathbb{R} }}$$ with initial datum $$\label{CI}
u_0(x)=\sum_{j=1}^n I_j^{1/2}e^{i\theta_j}{ \varphi }_j(\xi,x),$$ with $(I_1,\cdots,I_n)\subset (0,1]^n$ and $\theta\in { \mathbb{T} }^n$, is quasi periodic with a quasi period $\omega^*$ close to $\omega_0
=(2j-1)_{j=1}^n$: $|\omega^*-\omega_0|<C\nu$.\
More precisely, when $\theta$ covers ${ \mathbb{T} }^n$, the set of solutions of with initial datum covers a $n$ dimensional torus which is invariant by . Furthermore this torus is linearly stable.
From the proof it is clear that our result also applies to any non linearity which is a linear combination of $|u|^{2m}u$. Moreover, under ad hoc conditions on the derivatives of $G$, we can admit some non linearities of the form $\frac{\partial G}{\partial {{ \overline }{u}}}(x,u,{ \overline }{u})$ (i.e. depending on $x$) in . Also we can replace the set $\{1,\cdots,n\}$ by any finite set of ${ \mathbb{N} }$ of cardinality $n$.
Application to the linear Schrödinger equation {#Sect.6}
==============================================
In this section we prove Theorem \[theo:LS\] following the scheme developed by H. Eliasson and S. Kuksin in [@ElKuk2] for the linear Schrödinger equation on the torus with quasi-periodic in time potentials.\
The setting differs slightly from Section \[Sect.5\] since now we are not considering a perturbation around a finite dimensional torus but we want to construct a linear change of variable defined on all the phase space. Consider the equation $$\label{LS*}
i\partial_t u=-\partial^{2}_{x} u +x^2u +\epsilon V(t\omega,x)u$$ where $V$ satisfies the condition . Recall the definition of the phase space $ \mathcal{P}^{p}={ \mathbb{T} }^{n}\times { \mathbb{R} }^{n}\times {\ell}^{2}_{p} \times {\ell}^{2}_{p}$. Recall also that $h_j$, $j\geq 1$ denote the eigenfunctions of the quantum harmonic oscillator $T=-\partial^{2}_{x} +x^2$ and that we have $Th_j= (2j-1) h_j$, $j\geq 1$. Expanding $u$ and $\bar u$ on the Hermite basis, $u=\sum_{j\geq 1} z_j h_j$, $\bar u=\sum_{j\geq 1} \bar z_j h_j$, equation reads as a non autonomous Hamiltonian system $$\label{ls1}
\left\{ \begin{array}{ll}
\dot z_j= -i(2j-1) z_j- i{\varepsilon}\frac{\partial}{\partial \bar z_j}\widetilde{Q}(t,z,\bar z), & j\geq 1\\[4pt]
\dot{\bar z}_j=i(2j-1)\bar z_j+ i{\varepsilon}\frac{\partial}{\partial z_j}\widetilde Q(t,z,\bar z), & j\geq 1
\end{array}\right.$$ where $$\widetilde Q(t,z,\bar z)=\int_{ \mathbb{R} }V(\omega t,x)\big(\sum_{j\geq 1} z_j h_j(x)\big)\big(\sum_{j\geq 1} \bar z_j h_j(x)\big)\text{d}x$$ and[^5] $(z,\bar z)\in \ell^2_{2}\times \ell^2_{2}$. We then re-interpret as an autonomous Hamiltonian system in an extended phase space $\mathcal{P}^{2}$ $$\label{ls2}
\left\{ \begin{array}{ll}
\dot z_j=-i(2j-1) z_j- i{\varepsilon}\frac{\partial}{\partial \bar z_j}Q(\theta,z,\bar z) & j\geq 1 \\
\dot{\bar z}_j= i(2j-1)\bar z_j+ i{\varepsilon}\frac{\partial}{\partial z_j}Q(\theta,z,\bar z) & j\geq 1\\
\dot \theta_j= \omega_j & j=1,\cdots, n\\
\dot y_j= -{\varepsilon}\frac{\partial}{\partial \theta_j}Q(\theta,z,\bar z) & j=1,\cdots, n
\end{array}\right.$$ where $$Q(\theta,z,\bar z)=\int_{ \mathbb{R} }V(\theta,x)\big(\sum_{j\geq 1} z_j h_j(x)\big)\big(\sum_{j\geq 1} \bar z_j h_j(x)\big)\text{d}x$$ is quadratic in $(z,\bar z)$. We notice that the first three equations of are independent of $y$ and are equivalent to . Furthermore reads as the Hamiltonian equations associated with the Hamiltonian function $H=N+Q$ where $$N(\omega)= \sum_{j=1}^n \omega_j y_j +\sum_{j\geq 1} (2j-1) z_j\bar z_j.$$ Here the external parameters are directly the frequencies ${\displaystyle}{ \omega }=(\omega_j)_{1\leq j\leq n}\in [0,2\pi)^n=:\Pi$ and the normal frequencies $\Omega_j=2j-1$ are constant.\
Statement of the results and proof
----------------------------------
\
\[thmKAM2\] There exists ${\varepsilon}_0>0$ such that if $0<{\varepsilon}<{\varepsilon}_{0}$, there exist
(i) a Cantor set $\Pi_{\varepsilon}\subset \Pi$ with $\text{Meas}(\Pi\backslash \Pi_{\varepsilon})\rightarrow 0$ as ${\varepsilon}\rightarrow 0$ ;
(ii) a Lipschitz family of real analytic, symplectic and linear coordinate transformation $\Phi: \Pi_{\varepsilon}\times \mathcal P^{0} \rightarrow \mathcal P^{0}$ of the form $$\label{specific}
\Phi_{ \omega }(y,\theta,Z)= (y+\frac 1 2 Z\cdot M_{ \omega }(\theta)Z,\theta,L_{ \omega }(\theta)Z)$$ where $Z=(z,\bar z)$, $L_{ \omega }(\theta)$ and $M_{ \omega }(\theta)$ are linear bounded operators from $\ell^2_{p}\times\ell^2_{p} $ into itself for all $p\geq 0$ and $L_{ \omega }(\theta)$ is invertible ;
(iii) a Lipschitz family of new normal forms $$N^\star({ \omega })=\sum_{j=1}^n \omega_j y_j+ \sum_{j\geq 1}\Omega_j^\star({ \omega }) z_j\bar z_j \;;$$
such that on $\Pi_{{\varepsilon}}\times \mathcal{P}^{0}$ $$H\circ \Phi = N^\star.$$ Moreover the new external frequencies are close to the original ones $$|\Omega^\star-\Omega|_{2\beta,\Pi_{\varepsilon}}\leq c{\varepsilon},$$ and the new frequencies satisfy a non resonant condition, there exists $\alpha >0$ such that for all ${ \omega }\in \Pi_{{\varepsilon}}$ $$\big| k\cdot { \omega }+l \cdot \Omega^\star({ \omega }) \big|\geq {\alpha} \ \frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}},\quad (k,l)\in \mathcal{Z} .$$
Notice that in the new coordinates, $(y',\theta',z',\bar z')=\Phi_{{ \omega }}^{-1}(y,\theta,z,\bar z)$, the dynamic is linear with $y'$ invariant : $$\label{ls3}
\left\{ \begin{array}{ll}
\dot z'_j=i\Omega^\star_j z'_j & j\geq 1 \\
\dot{\bar z}'_j= -i\Omega^\star_j\bar z'_j & j\geq 1\\
\dot \theta'_j= \omega_j & j=1,\cdots, n\\
\dot y'_j= 0 & j=1,\cdots, n.
\end{array}\right.$$ As is equivalent to , this theorem implies Theorem \[theo:LS\]. In particular the solutions $u(t,x)$ of with initial datum $u_0(x)=\sum_{j\geq 1} z_{j}(0)h_j(x)$ read $u(t,x)=\sum_{j\geq 1} z_j(t)h_j(x)$ with $$\label{fin}
(z,\bar z)(t)=L_{ \omega }({ \omega }t)(z'(0)e^{i\Omega^\star t},\bar z'(0)e^{-i\Omega^\star t})$$ and $(z'(0),\bar z'(0))= L_{ \omega }^{-1}(0)(z(0),\bar z(0))$.\
Thus $$u(t,x)=\sum_{j\geq 1} \psi_j({ \omega }t,x)e^{i\Omega^\star_j t}$$ where $\psi_j (\theta,x)=\sum_{\ell\geq 1} [L_{ \omega }(\theta)L_{ \omega }^{-1}(0)(z(0),\bar z(0))]_\ell h_\ell(x)$.\
In particular the solutions are all almost periodic in time with a non resonant frequencies vector $({ \omega },\Omega^\star)$. Furthermore we observe that $ \psi_j({ \omega }t,x)e^{i\Omega^\star_j t}$ solves if and only if $\Omega^\star_j +k\cdot { \omega }$ is an eigenvalue of (with eigenfunction $ \psi_j(\theta,x)e^{i\theta\cdot k}$). This shows that the spectrum of the Floquet operator equals $\{\Omega^\star_j +k\cdot { \omega }\mid k\in{ \mathbb{Z} }^n, \ j\geq 1 \}$ and thus Corollary 1.4 is proved.
Although $\Phi$ is defined on $ \mathcal P^{0} $, the normal forms $N$ and $N^\star$ are well defined on $ \mathcal P^{p} $ only when $p\geq 1/2$. Nevertheless their flows are well defined and continuous from $\mathcal P^{0}$ into itself (cf. ).
Let $\tilde \Pi\subset \Pi$ be the subset of Diophantine vector of frequencies ${ \omega }$, i.e. having the property that there exists $0< \alpha\leq 1$ such that $$\label{diophantine}
\big|k\cdot { \omega }-b\big|\geq \frac{{2\pi\alpha}}{|k|^{\tau-1}},\quad k\in { \mathbb{Z} }^n\setminus\{0\}, \ b\in { \mathbb{Z} }$$ for some $\tau>n+2$. It is well known that $\text{Meas}(\Pi\backslash \tilde \Pi)=0$. Further this Diophantine condition implies that $$\big|k\cdot { \omega }+l\cdot \Omega \big|\geq {\alpha} \ \frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}},\quad (k,l)\in \mathcal{Z},$$ since $l\cdot\Omega \in { \mathbb{Z} }$ and if $ { \langle }l{ \rangle }\leq 2\pi |k|$ then $\frac{2{\pi\alpha}}{|k|^{\tau-1}}\geq \alpha \frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}}$ while if $ { \langle }l{ \rangle }\geq 2\pi|k|$ then $\big|k\cdot { \omega }+l\cdot\Omega \big|\geq 2{ \langle }l{ \rangle }-2\pi|k|\geq { \langle }l{ \rangle }\geq {\alpha} \ \frac{ { \langle }l{ \rangle }}{1+|k|^{\tau}}$. Thus Assumptions \[AS1\] holds true. Further as the normal frequencies $\Omega_j=2j-1$ are constant, \[AS2\] is satisfied.\
We now show that Assumption \[AS3\] holds. Because of the assumptions on the smoothness of $V$, the only condition which needs some care is that $(\frac{\partial Q}{\partial z_{k}})_{k\geq 1} \in \ell^{2}_{p}$. We have $$\frac{\partial Q}{\partial z_{k}}=\int_{{ \mathbb{R} }}V(\theta,x)h_{k}{ \overline }{u} \,\text{d}x,$$ which is the $k$th coefficient of the decomposition of $V(\theta,x){ \overline }{u}$ in the Hermite basis. Thus $(\frac{\partial Q}{\partial z_{k}})_{k\geq 1} \in \ell^{2}_{2}$ if and only if $V(\theta,x){ \overline }{u} \in { \mathcal{H} }^{2}$ which is true since $\bar u\in { \mathcal{H} }^{2}$ and $V$ and $\partial_x V$ are bounded.\
We turn to Assumption \[AS4\]. Recall that by , for all $2<r\leq +\infty$, there exists $\beta>0$ so that $\|h_{j}\|_{L^{r}({ \mathbb{R} })}\leq C j^{-\beta}$. On the other hand, by assumption $V$ is real analytic in $\theta$ and $L^q$ in $x$ for some $1\leq q<+\infty$. Consider $1<{ \overline }{q}\leq +\infty$ so that $\frac1q+\frac1{{ \overline }{q}}=1$, then with Hölder, we compute $$\begin{aligned}
\Big|\frac{\partial Q}{\partial z_{k}}\Big|=\big|\int_{{ \mathbb{R} }}V(\theta,x)h_{k}{ \overline }{u} \,\text{d}x\big|&\leq &\sup_{\theta\in [0,2\pi]^{n}}\|V(\theta,\cdot)\|_{L^{q}({ \mathbb{R} })}\|h_{k}\,u\|_{L^{{ \overline }{q}}({ \mathbb{R} })}\\
&\leq &\sup_{\theta\in [0,2\pi]^{n}}\|V(\theta,\cdot)\|_{L^{q}({ \mathbb{R} })}\|h_{k}\|_{L^{2{ \overline }{q}}({ \mathbb{R} })}\|u\|_{L^{2{ \overline }{q}}({ \mathbb{R} })}\\
&\leq &C k^{-\beta}.\end{aligned}$$ Similarly, $$\begin{aligned}
\Big|\frac{\partial^{2} Q}{\partial z_{k}\partial { \overline }{z}_{l}}\Big|=\big|\int_{{ \mathbb{R} }}V(\theta,x)h_{k}h_{l}\,\text{d}x\big|&\leq& \sup_{\theta\in [0,2\pi]^{n}}\|V(\theta,\cdot)\|_{L^{q}({ \mathbb{R} })}\|h_{k}\|_{L^{2{ \overline }{q}}({ \mathbb{R} })}\|h_{l}\|_{L^{2{ \overline }{q}}({ \mathbb{R} })}\\
&\leq &C (jl)^{-\beta}. \end{aligned}$$ Therefore, Theorem \[thmKAM\] applies (with $p=2$) and we almost obtain the conclusions of Theorem \[thmKAM2\]. Indeed, comparing with Theorem \[thmKAM\], we have to prove:
- the symplectic coordinate transformation $\Phi$ is quadratic (and thus it is defined on the whole phase space) and have the specific form ;
- the new normal form still have the same frequencies vector ${ \omega }$ ;
- the new Hamiltonian reduces to the new normal form, i.e. $R^\star=0$ ;
- the symplectic coordinate transformation $\Phi$, which is defined by Theorem \[thmKAM2\] on each $\mathcal{P}^{2}$, extends to $\mathcal{P}^{0}={ \mathbb{T} }^{n}\times { \mathbb{R} }^{n}\times \ell_{2}^2\times\ell_{2}^2$.
Actually, at the principle $Q$ is homogeneous of degree 2 in $Z$ and independent of $y$ and the same is true for $F$ the solution of the first homological equation $$\{F,N\}+\widehat N={\varepsilon}Q.$$ As a first consequence, $\widehat N$ does not contain linear terms in $y$ and thus $\omega$ remains unchanged by the first iterative step (cf. ). Now going to Lemma \[lem.3\] we notice that following notations , $b_0=b_1=a=0$. Therefore $\theta$ remains unchanged ($\dot \theta=0$) and the equation for $Z$ reads $\dot Z= JA(\theta)Z$ which leads to $Z(\tau)=e^{\tau JA(\theta)}Z(0)$ (see ). Thus $Z(1)=L^{(1)}_\omega(\theta) Z(0)$ where $L^{(1)}_\omega(\theta)=e^{JA(\theta)}$ is invertible from $\mathcal{P}^{2}$ onto itself.\
In the same way, $\dot y(\tau)=-\frac 1 2 \nabla_\theta A(\theta)Z(\tau)\cdot Z(\tau)$ (see ) which leads to $y(1)=y(0)+ \frac 1 2 Z(0)\cdot M_{ \omega }(\theta)Z(0)$ for some linear operator $M_{ \omega }(\theta)$. Finally the new error term (cf. ) ${\displaystyle}Q_+=\int_0^1\{Q(t),F\}\circ X^t_F \,\text{d} t$ is still homogeneous of degree 2 in $Z$ and independent of $y$. Thus properties (i), (ii) are satisfied after the first step and the new error term conserves the same form. Therefore we can iterate the process and the limiting transformation $\Phi=\Phi^1\circ \Phi^2\circ \cdots$ also satisfies (i) and (ii). Furthermore the transformed Hamiltonian as well as the original one is linear in $y$ and quadratic in $Z$ and thus (iii) holds true.\
It remains to check (iv). This follows from the fact that $\Phi$ is a [*linear*]{} symplectomorphism and thus, as remarked in [@Kuk3 Proposition 1.3’], extends by duality on $\ell^2_p\times\ell^2_p$ for all $p\in [-2,2]$ and in particular for $p=0$.
[**]{} The point is that, when $V$ is smooth with bounded derivatives, the perturbation $Q$ satisfies Assumption 3 for all $p\geq 0$. That is $X_Q$ maps smoothly $\mathcal P^{p}$ into itself. Therefore Theorem \[thmKAM\] applies for all $p\geq 2$ and by , the canonical transformation $\Phi$ is close to the identity in the $\mathcal P^{p}$-norm. Since in the new variables, $(y',\theta',z',\bar z')=\Phi^{-1}(y,\theta,z,\bar z)$, the modulus of $z'_j$ is invariant, we deduce that there exist a constant $C$ such that $$(1-C{\varepsilon}) \|z(0)\|_{p}\leq \|z(t)\|_{p}\leq (1+C{\varepsilon}) \|z(0)\|_{p}$$ which in turn implies $$(1-{\varepsilon}C)\|u_{0}\|_{{ \mathcal{H} }^{p}}\leq \|u(t)\|_{{ \mathcal{H} }^{p}}\leq (1+{\varepsilon}C)\|u_{0}\|_{{ \mathcal{H} }^{p}}, \quad \forall \,t\in { \mathbb{R} }.$$
An explicit example
-------------------
\
Consider the linear equation $$\label{lin.bis}
i\partial_t u=-\partial^{2}_{x} u +x^2u +\epsilon V(t\omega)u$$ where $V:{ \mathbb{T} }^{n}\longrightarrow { \mathbb{R} }$ is real analytic and independent of $x\in { \mathbb{R} }$. Up to a translation of the spectrum, we can assume that $V(0)=0$. Notice that this case is not in the scope of Theorem \[thmKAM2\], since $V$ does not satisfy .\
We suppose moreover that $\int_{{ \mathbb{T} }^n}V=0$ and that ${ \omega }\in [0,2\pi)^{n}$ is Diophantine (see ).
Define $v(t,x)={ \text{e} }^{-i{\varepsilon}\int_{0}^{t}V({ \omega }s)\text{d}s}u(t,x)$. The function $u$ satisfies iff $v$ satisfies $ i\partial_t v=-\partial^{2}_{x} v +x^2v$. This latter equation is explicitly solvable using the Hermite basis, and the solution of with initial condition $u_{0}(x)=\sum_{j=1}^{\infty}{ \alpha }_{j}h_{j}(x)$ then reads $$u(t,x)={ \text{e} }^{i{\varepsilon}\int_{0}^{t}V({ \omega }s)\text{d}s}\sum_{j=1}^{\infty}{ \alpha }_{j}h_{j}(x){ \text{e} }^{i(2j-1)t}.$$ Write ${\displaystyle}V(\theta)=\sum_{k\in { \mathbb{Z} }^{n},k\neq 0} a_{k}{ \text{e} }^{ik\cdot \theta}$. Then, as ${ \omega }$ is Diophantine, we can compute $ {\displaystyle}\int_{0}^{t}V({ \omega }s)\text{d}s=-i\sum_{k\in { \mathbb{Z} }^{n},k\neq 0} \frac{a_{k}}{k\cdot { \omega }}({ \text{e} }^{ik\cdot { \omega }t }-1)$, and $W$ defined by ${\displaystyle}W(\theta)=\exp\big( {\varepsilon}\sum_{k\in { \mathbb{Z} }^{n},k\neq 0} \frac{a_{k}}{k\cdot { \omega }}({ \text{e} }^{ik\cdot \theta }-1) \big)$ is a periodic and analytic function in $\theta$. Finally, ${\displaystyle}u(t,x)=\sum_{j=1}^{\infty}{ \alpha }_{j}W({ \omega }t)h_{j}(x){ \text{e} }^{i(2j-1)t} $ is an almost periodic function in time (as an infinite sum of quasi-periodic functions).\
We can explicitly compute the transformation $\Phi$ in . Here the Hamiltonian reads $H=N+Q$ with $Q=V(\theta)\sum_{k\geq 1}|z_{k}|^{2}$. Set $\Phi(y',\theta',z',{ \overline }{z}')=(y,\theta,z,{ \overline }{z})$ where $$\left\{ \begin{array}{ll}
{\displaystyle}z_j={\displaystyle}W(\theta)\,z'_{j},\quad {\bar z}_j={ \overline }{W(\theta)}\,{\bar z}'_{j}, & j\geq 1 \\[3pt]
{\displaystyle}\theta_j= \theta'_j,\quad y_j=y'_{j}-{\varepsilon}k_{j}\sum_{k\in { \mathbb{Z} }^{n},k\neq 0} \frac{a_{k}}{k\cdot { \omega }}{ \text{e} }^{ik\cdot \theta }\sum_{l\geq 1}|z_{l}|^{2}, & 1\leq j\leq n.
\end{array}\right.$$ Then a straightforward computation gives $$H\circ \Phi(y',\theta',z',{ \overline }{z}')=\sum_{j=1}^n \omega_j y'_j +\sum_{j\geq 1} (2j-1) z'_j\bar z'_j.$$ Therefore in this case $\Omega^{\star}_{j}({ \omega })=2j-1$.\
Finally we study the spectrum of the Floquet operator associated to the equation . Observe that $W({ \omega }t)h_{j}(x){ \text{e} }^{i(2j-1)t}$ solves if and only if any $2j-1+k\cdot { \omega }$ (with $j\geq 1$ and $k\in { \mathbb{Z} }^{n}$) is an eigenvalue of (with eigenfunction $W(\theta)h_{j}(x){ \text{e} }^{i\theta \cdot k}$). This shows that the Floquet spectrum is pure point, since linear combinations of $W(\theta)h_{j}(x){ \text{e} }^{i\theta \cdot k}$ are dense in $L^2({ \mathbb{R} })\otimes L^2({ \mathbb{T} }^n)$.
Appendix
========
We show here how we can construct periodic solutions to the equation $$\label{nls1}
\left\{
\begin{aligned}
&i\partial_t u+\partial^{2}_{x} u -x^{2}u = |u|^{p-1}u ,\quad p\geq 1\quad
(t,x)\in{ \mathbb{R} }\times {{ \mathbb{R} }},\\
&u(0,x)= f(x),
\end{aligned}
\right.$$ thanks to variational methods. This is classical, see e.g. [@Struwe] and [@BereLions] for more details. See also [@Carles]. Recall that for $s\geq 0$ we have defined the Sobolev space ${ \mathcal{H} }^{s}({ \mathbb{R} })=D(T^{s/2})$, where $T=-\partial^{2}_{x}+x^{2}$ is the harmonic oscillator. We also define ${ \mathcal{H} }^{\infty}({ \mathbb{R} })=\cap_{s>0}{ \mathcal{H} }^{s}({ \mathbb{R} })$. We then have the following result.
Let $\mu>0$. Then there exists an $L^{2}({ \mathbb{R} })$-orthogonal family $({ \varphi }^{j})_{j\geq 1}\in \mathcal{H}^{\infty}({ \mathbb{R} })$ with $\|{ \varphi }^{j}\|_{L^{2}({ \mathbb{R} })}=\mu$ and a sequence of positives numbers $(\lambda_{j})_{j\geq 1}$ so that for all $j\geq 1$, $u(t,x)={ \text{e} }^{-i\lambda_{j}t}{ \varphi }^{j}(x)$ is a solution of .
We look for a solution of of the form ${\displaystyle}u(t,x)={ \text{e} }^{-i\lambda t}{ \varphi }(x) $, hence ${ \varphi }$ has to satisfy $$\label{Elliptique}
\big(-\partial_{x}^{2}+x^{2}\big){ \varphi }=\lambda { \varphi }-|{ \varphi }|^{p-1}{ \varphi }.$$ Let $\mu>0$, denote by $E_{\mu}$ the set $$E_{\mu}=\Big\{ { \varphi }\in { \mathcal{H} }^{1}({ \mathbb{R} }),\;s.t.\;\|{ \varphi }\|_{L^{2}({ \mathbb{R} })}=\mu \Big\},$$ and define the functional $$J({ \varphi })=\int\frac12\big((\partial_{x} { \varphi })^{2}+x^{2}{ \varphi }^{2}\big)+\frac{1}{p+1}|{ \varphi }|^{p+1}\big)\text{d}x,$$ which is $\mathcal{C}^{1}$ on $E_{\mu}$.\
Then the problem ${\displaystyle}\min_{{ \varphi }\in E_{\mu}} J({ \varphi })$ admits a solution ${ \varphi }^{1}$, and ${ \varphi }^{1}$ solves for some $\lambda=\lambda_{1}>0$.\
Indeed, by Rellich’s theorem (see e.g. [@ReedSimon4 page 247]), for all $C>0$, the set $$\begin{gathered}
\Big\{ { \varphi }\in { \mathcal{H} }^{1}({ \mathbb{R} }),\;s.t.\;\|{ \varphi }\|_{L^{2}({ \mathbb{R} })}=\mu,\\
\int\frac12\big((\partial_{x} { \varphi })^{2}+x^{2}{ \varphi }^{2}\big)+\frac{1}{p+1}|{ \varphi }|^{p+1}\leq C \Big\},
\end{gathered}$$ is compact in $L^{2}({ \mathbb{R} })$ (observe that we have used the Sobolev embedding ${ \mathcal{H} }^{1}\subset L^{p+1}$ which holds for any $p\geq 1$). Then, if ${ \varphi }_{n}$ is a minimising sequence of $J$, up to a sub-sequence, we can assume that ${ \varphi }_{n}\longrightarrow { \varphi }^{1} \in E_{\mu}$ in $L^{2}({ \mathbb{R} })$. Finally, the lower semicontinuity of $J$ ensures that ${ \varphi }^{1}$ is a minimum of $J$ in $E_{\mu}$, and the claim follows. Moreover, $\lambda_{1}$ is given by $$\lambda_{1}=\frac{1}{\mu}\int(\partial_{x} { \varphi }^{1})^{2}+x^{2}({ \varphi }^{1})^{2}+|{ \varphi }^{1}|^{p+1}.$$ Now we define the set ${\displaystyle}E_{\mu}^{1}=E_{\mu}\cap \big\{ { \langle }{ \varphi },{ \varphi }^{1}{ \rangle }_{L^{2}({ \mathbb{R} })}=0\big\}$. Similarly, we may construct ${ \varphi }^{2}\in E^{1}_{\mu}$ so that ${\displaystyle}J ({ \varphi }^{2})=\min_{{ \varphi }\in E^{1}_{\mu}}J({ \varphi })$. The orthogonality condition implies in particular that ${ \varphi }^{2}\neq { \varphi }^{1}$. Let $k\geq 1$, and assume that we have constructed ${\displaystyle}({ \varphi }^{j})_{1\leq j \leq k}$ so that ${ \langle }{ \varphi }^{i},{ \varphi }^{j}{ \rangle }_{L^{2}}=\mu^{2} \delta_{ij}$ for all $1\leq i,j\leq k$. Define the set $$E^{k}_{\mu}=E_{\mu}\cap \big\{ { \langle }{ \varphi }, { \varphi }^{j}{ \rangle }_{L^{2}}=0,\;1\leq j\leq k\big\}.$$ By Rellich’s theorem, the set $$\begin{gathered}
\Big\{ { \varphi }\in { \mathcal{H} }^{1}({ \mathbb{R} }),\;s.t.\;\|{ \varphi }\|_{L^{2}({ \mathbb{R} })}=\mu,\\
\int\frac12\big((\partial_{x} { \varphi })^{2}+x^{2}{ \varphi }^{2}\big)+\frac{1}{p+1}|{ \varphi }|^{p+1}\leq C, \;\; { \langle }{ \varphi }, { \varphi }^{j}{ \rangle }_{L^{2}}=0,\;1\leq j\leq k\Big\},
\end{gathered}$$ is compact in $L^{2}({ \mathbb{R} })$ and we can construct ${ \varphi }^{k+1}\in E^{k}_{\mu}$ so that ${\displaystyle}J ({ \varphi }^{k+1})=\min_{{ \varphi }\in E^{k}_{\mu}}J({ \varphi })$. Then ${ \varphi }^{k+1}$ is a nontrivial solution of with $$\lambda_{k+1}=\frac{1}{\mu}\int(\partial_{x} { \varphi }^{k+1})^{2}+x^{2}({ \varphi }^{k+1})^{2}+|{ \varphi }^{k+1}|^{p+1}.$$ The regularity ${ \varphi }^{j}\in \mathcal{H}^{\infty}$ is a direct consequence of the ellipticity of the operator $-\partial_{x}^{2}+x^{2}$.
Of course, the proof can be generalised to a larger class of nonlinearities in . In particular, we can deal with the nonlinearity $-{\varepsilon}|u|^{p-1}u$ with ${\varepsilon}>0$ provided that $p<5$ and that ${\varepsilon}\mu^{\frac{p+3}2}>0$ is small enough. Indeed in that case, thanks to the Gagliardo-Nirenberg inequality we have $$\frac{{\varepsilon}}{p+1}\int |{ \varphi }|^{p+1} \leq C {\varepsilon}\mu^{\frac{p+3}2} \Big(\int(\partial_{x} { \varphi })^{2}+x^{2}{ \varphi }^{2}\Big)^{(p-1)/4},$$ and the nonlinear part of the energy can be controlled by the linear part, which enables us the perform the same arguments as previously.
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[^1]: The first author was supported in part by the grant ANR-06-BLAN-0063.\
The second author was supported in part by the grant ANR-07-BLAN-0250.
[^2]: This means that ${ \langle }P{ \rangle }_{r,D(s,r)}$ is the smallest real number which satisfies the mentioned conditions : this defines a norm.
[^3]: This means that ${ \langle }{ \langle }\,\cdot\,{ \rangle }{ \rangle }^{+}_{\beta,s}$ is the smallest real number which satisfies the mentioned conditions : this defines a norm.
[^4]: here we use the notation $D(s/2)=D(s/2, 0)$.
[^5]: For the moment we work in $\ell^2_{2}\times \ell^2_{2}$, the largest phase space in which our abstract result applies.
|
---
abstract: 'We compare asynchronous vs. synchronous update of discrete dynamical networks and find that a simple time delay in the nodes may induce a reproducible deterministic dynamics even in the case of asynchronous update in random order. In particular we observe that the dynamics under synchronous parallel update can be reproduced accurately under random asynchronous serial update for a large class of networks. This mechanism points at a possible general principle of how computation in gene regulation networks can be kept in a quasi-deterministic “clockwork mode” in spite of the absence of a central clock. A delay similar to the one occurring in gene regulation causes synchronization in the model. Stability under asynchronous dynamics disfavors topologies containing loops, comparing well with the observed strong suppression of loops in biological regulatory networks.'
author:
- Konstantin Klemm
- Stefan Bornholdt
title: |
Robust gene regulation:\
Deterministic dynamics from asynchronous networks with delay
---
Erwin Schrödinger in his lecture “What is life?” held in 1943 [@Schroedinger] was one of the first to notice that the information processing performed in the living cell has to be extremely robust and therefore requires a quasi-deterministic dynamics (which he called “clockwork mode”). The discovery of a “digital” storage medium for the genetic information, the double-stranded DNA, confirmed one important part of this picture. Today, new experimental techniques allow to observe the dynamics of regulatory genes in great detail, which motivates us to reconsider the other, dynamical part of Schrödinger’s picture of a “clockwork mode”. While the dynamical elements of gene regulation often are known in great detail, the complex dynamical patterns of the vast network of interacting regulatory genes, while highly reproducible between identical cells and organisms under similar conditions, are largely not understood. Most remarkably, these virtually deterministic activation patterns are often generated by asynchronous genetic switches without any central clock. In this Letter we address this astonishing fact with a toy model of gene regulation and study the conditions of when deterministic dynamics could occur in asynchronous circuits. Let us start from the observed dynamics of small circuits of regulatory genes, then derive a discrete dynamical model gene, followed by a study of networks of such genetic switches, with a focus on comparing their asynchronous and synchronous dynamics.
Recently, several small gene regulation circuits have been described in terms of a detailed picture of their dynamics [@Elowitz; @Hes1; @Baltimore; @p53; @Smolen]. A particularly simple motif is the single, self-regulating gene [@Rosenfeld; @Hes1] that allows for a detailed modeling of its dynamics. A set of two differential equations, for the temporal evolution of the concentrations of messenger RNA and protein, respectively, and an explicit time delay for transmission delay provide a quantitative model for the observed dynamics in this minimal circuit [@Jensen03]. The equations of this model take the basic form $$\begin{aligned}
\label{eq:originaldiff1}
\frac {{{\rm d}}c}{{{\rm d}}t} &=& \alpha [f(s(t-\vartheta)) - c(t)] \\
\frac {{{\rm d}}b}{{{\rm d}}t} &=& \beta [c(t)-b(t)]\end{aligned}$$ for the the dynamics of the concentrations $c$ of mRNA and $b$ of protein, with some non-linear transmission function $f(s)$ of an input signal $s$, a time delay $\vartheta$, and the time constants $\alpha$ and $\beta$. In order to define a minimal discrete gene model let us keep the basic features (delay, low pass filter characteristics), omit the second filter, and write the difference equation for one gene $i$ as $$\Delta c_i = \alpha [f(s_i(t-\vartheta)) - c_i(t)] \Delta t~.$$ The non-linear function $f$ is typically a steep sigmoid. We approximate it as a step function $\Theta$ with $\Theta(s)=0$ for $s<0$ and $\Theta(s)=1$ otherwise. Rescaling time with $\epsilon = \alpha \Delta t$ and $\tau = \vartheta / \Delta t$ this reads $$\Delta c_i = \epsilon [\Theta(s_i(t-\tau)) - c_i(t)]~.$$ For simplicity let us update $c_i$ by equidistant steps according to $$\label{eq:cupdatefinal}
\Delta c_i = \left\{
\begin{array}{rl}
+\epsilon, & {\rm if }\;s_i(t-\tau) \ge 0\; {\rm and}\;c_i \le 1-\epsilon \\
-\epsilon, & {\rm if }\;s_i(t-\tau) < 0 \; {\rm and}\;c_i \ge \epsilon \\
0, & {\rm otherwise}
\end{array}
\right.$$ The coupling between nodes is defined by $$\label{eq:xsum}
s_i (t) = \sum_j w_{ij} x_j (t) - a_i~,$$ with discrete output states $x_j (t)$ of the nodes defined as $$\label{eq:xdefinition}
x_j (t) = \Theta(c_j (t) - 1/2)~.$$ The influence of node $j$ on node $i$ can be activating ($w_{ij}=1$), inhibitory ($w_{ij}=-1$), or absent ($w_{ij}=0$). A constant bias $a_i$ is assigned to each node.
In the following let us consider a network model of such nodes. Consider $N$ nodes with concentration variables $c_i$, state variables $x_i$, biases $a_i$ and a coupling matrix $(w_{ij})$. Given initial values $x_i(0)=c_i(0)\in\{0,1\}$ the time-discrete dynamics is obtained by iterating the following update steps:
\(1) Choose a node $i$ at random. (2) Calculate $s_i$ according to Eq. (\[eq:xsum\]). (3) Update $c_i$ according to Eq. (\[eq:cupdatefinal\]).
For $\tau=0$ and $\epsilon=1$ random asynchronous update is recovered. For $\tau>0$ there is an explicit transmission delay from the output of node $j$ to the input of node $i$. To be definite, at $t=0$ we assume that nodes have not flipped during the previous $\tau$ time steps.
Let us first explore the dynamics of a simple but non-trivial interaction network with $N=3$ sites and
non-vanishing couplings $w_{01} = w_{21} = -1$ and $w_{10}= w_{12} = +1$, see Fig. \[fig:combined\]. Note that under [*asynchronous*]{} update the sequence of states reached by the dynamics is not unique. The system may branch off to different configurations depending on node update ordering. This is illustrated in Fig. \[fig:singletsz\_0\](a): Without delay ($\tau=0$) and filter ($\epsilon=1$) the dynamics is irregular, [[*i.e. *]{}]{}non-periodic.
With filter only ($\tau=0$, $\epsilon=0.01$, Fig. \[fig:singletsz\_0\](b)), the dynamics is periodic at times, but also intervals of fast irregular flipping occur. Finally, in the presence of delay ($\tau=100$, $\epsilon=1$, Fig. \[fig:singletsz\_0\](c)) we obtain perfectly ordered dynamics with synchronization of flips. Nodes 0 and 2 change states practically at the same (macro) time, followed by a longer pause until node 1 changes state, etc. With increasing delay time $\tau$ the dynamics under asynchronous update approaches the dynamics under synchronous update (cf. Fig. \[fig:combined\]) when viewed on a coarse-grained (macro) time scale.
Let us further quantify the difference between synchronous and asynchronous dynamics. First, a definition of equivalence between the two dynamical modes has to be given. Let us start from the time series ${{\bf x}}(t)$ of configurations ${{\bf x}}= (x_0,\dots, x_{N-1})$ produced by the asynchronous (random serial) update of the model and the respective time series ${{\bf y}}(u)$ produced by synchronous (parallel) update, using identical initial condition ${{\bf y}}(0) = {{\bf x}}(0)$. These time series live on different time scales, which we call the micro time scale of single site updates in the asynchronous case, and the macro time scale where each time step is an entire sweep of the system. Assume that at time $t_u$ the asynchronous system is in state ${{\bf x}}(t_u) = {{\bf y}}(u)$. In order to follow the synchronous update it has to subsequently reach the state ${{\bf y}}(u+1)$ on a shortest path in phase space. Formally, let us require that there is a micro time $t_{u+1}>t_u$ such that ${{\bf x}}(t_{u+1})= {{\bf y}}(u+1)$ and each node flips at most once in the time interval $[t_u,t_{u+1}]$. Once this is violated we say that an error has occured at the particular macro time step $u$. This error allows to define a numerical measure of discrepancy between asynchronous and synchronous dynamics. Starting from identical initial conditions, the system is iterated in synchronous and asynchronous modes (here for $u_{\rm total} = 10^7$ macro time steps). Whenever the resulting time series are no longer equivalent, an error counter is incremented and the system reset to initial condition. The total error $E$ of the run is the number of errors divided by $u_{\rm total}$.
For the network in Fig. \[fig:combined\] and the initial condition $x_i=c_i=0$ for $i=1,2,3$ the error $E$ is exponentially suppressed with delay time $\tau$ (Fig. \[fig:singlez\_0\]).
The asynchronous dynamics with delay follows the attractor during a time span that increases exponentially with the given delay time. Note that there is only one possibility for the asynchronous dynamics to leave the attractor: When the system is in configuration $(1,1,0)$ or $(0,1,1)$, node $2$ may change state such that the system goes to configuration $(1,0,0)$ or $(0,0,1)$ respectively, whereas the correct next configuration on the attractor is $(0,1,0)$. Consider the case $\epsilon=1$ where $c_i=x_i$ for all $i$. Let us assume that the system is in configuration $(1,1,1)$ and at time $t_0$ node 0 changes state, thereby generating configuration $(0,1,1)$. This decreases the input sum $s_1$ below zero such that for $\tau=0$ node $0$ would change state immediately in its next update. With explicit transmission delay $\tau>0$, however, node 1 still “sees” the input sum $s_i=0$ generated by the configuration $(1,1,1)$ until time step $t_0+\tau$. If node $2$ is chosen for update in this time window $t_0+1,\dots,t_0+\tau$ it changes state immediately and updates are performed in correct order. The opposite case, that node 2 does not receive an update in any of the $\tau$ time steps, happens with probability $(2/3)^\tau$, yielding the correct error decay of the simulation (Fig. \[fig:singlez\_0\]).
Next we demonstrate that there are cases where also low-pass filtering, $\epsilon \ll 1$, is needed for the asynchronous dynamics to follow the deterministic attractor. Consider a network of $N=5$ nodes with bias values $a_0 = a_4 = 0$ and $a_1 = a_2 = a_3 = 1$. The only non-zero couplings are $w_{10} = w_{21} = w_{31} = w_{42} = +1 $ and $w_{01} = w_{43} = -1$. Nodes 0 and 1 form an oscillator, [[*i.e. *]{}]{}$(x_0,x_1)$ iterate the sequence $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. Nodes $2$ and $3$ simply “copy” the state of node $1$ such that under synchronous update always $x_3(t)=x_2(t)=x_1(t-1)$. Consequently, under synchronous update the input sum of node $4$ never changes because the positive contribution from node 2 and the negative contribution from node 3 cancel out. Under asynchronous update, however, the input sum of node 4 may fluctuate because nodes 2 and 3 do not flip precisely at the same time. The effect of the low-pass filter $\epsilon \ll 1$ is to suppress the spreading of such fluctuations on the micro time scale. The influence of the filter is seen in Fig. \[fig:five\_0\].
When $\tau$ is kept constant, the error drops algebraically with decreasing $\epsilon$. An exponential decay $E \sim \exp(-\alpha /\epsilon)$ is obtained when $\tau \propto 1/\epsilon$ (the filter can take full effect only in the presence of sufficient delay).
Let us finally consider an example of a larger network with $N=16$ nodes and $L=48$ non-vanishing couplings (chosen randomly from the off-diagonal elements in the matrix $(w_{ij})$ and assigned values $+1$ or $-1$ with probability $1/2$ each; biases are chosen as $a_i = \sum_j w_{ij}/2$). Simulation runs under pure asynchronous update ($\tau=0$, $\epsilon=1$) typically yield dynamics as in Fig. \[fig:largets\_1\](a).
The time series ${{{\bf x}}(t)}$ is non-periodic and non-reproducible, [[*i.e. *]{}]{}under different order of updates a different series is obtained. For the same initial condition, periodic dynamics is observed in the presence of sufficent transmission delay and filtering, Fig. \[fig:largets\_1\](b). In this case, the system follows precisely the attractor of period 28 found under synchronous update. As seen in Fig. \[fig:largets\_1\](c), the error decays exponentially as a function of the delay time $\tau$.
Let us now turn to the dangers of asynchronous update: There is a fraction of attractors observed under synchronous update that cannot be realized under asynchronous update. Synchronization cannot be sustained if the dynamics is separable. In the trivial case, separability means that the set of nodes can be divided into two subsets that do not interact with each other. Then there is no signal to synchronize one set of nodes with the other and they will go out of phase. In general, synchronization is impossible if the set of flips itself is separable. Consider, as the simplest example, a network of $N=2$ nodes with the couplings $w_{01} = w_{10} = +1$, biases $a_0 = a_1 =
1$ and the initial condition $(y_0(0),y_1(0)) = (0,1)$. Under synchronous update, the state alternates between vector $(0,1)$ and $(1,0)$. Under asynchronous update with delay time $\tau$, the transition of one node $i$ from $x_i=0$ to $x_i=1$ causes the other node $j$ to switch from $x_j=0$ to $x_j=1$ approximately $\tau$ time steps later. The “on”-transitions only trigger subsequent “on”-transitions and, analogously, the “off”-transitions only trigger subsequent “off”-transitions. The dynamics can be divided into two distinct sets of events that do not influence each other. Consequently, synchronization between flips cannot be sustained, as illustrated in Fig. \[fig:cycle\_illu\].
When the phase difference reaches the value $\tau$, on- and off-transitions annihilate. Then the system leaves the attractor and reaches one of the fixed points with $x_0 = x_1$.
These observations have important implications for robust topological motifs in asynchronous networks. First of all, the above example of a small excitatory loop can be quickly generalized to any larger loop with excitatory interactions, as well as to loops with an even number of inhibitory couplings, where in principle similar dynamics could occur. Higher order structures that fail to synchronize include competing modules, e.g. two oscillators (loops with odd number of inhibitory links) that interact with a common target.
In conclusion we find that asynchronously updated networks of autonomous dynamical nodes are able to exhibit a reproducible and quasi-deterministic dynamics under broad conditions if the nodes have transmission delay and low pass filtering as, e.g., observed in regulatory genes. Timing requirements put constraints on the topology of the networks (e.g. suppression of certain loop motifs). With respect to biological gene regulation networks where indeed strong suppression of loop structures is observed [@Shen-Orr02; @Milo02], one may thus speculate about a new constraint on topological motifs of gene regulation: The requirement for deterministic dynamics from asynchronous dynamical networks.
[**Acknowledgements**]{} S.B. thanks D. Chklovskii, M.H. Jensen, S. Maslov, and K. Sneppen for discussions and comments, and the Aspen Center for Physics for hospitality where part of this work has been done.
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|
---
author:
- |
Yuchen Liu, David Wentzlaff, S.Y. Kung\
Department of Electrical Engineering\
Princeton University\
`{yl16, wentzlaf, kung}@princeton.edu`\
bibliography:
- 'Style\_Ref/refs.bib'
title: 'Rethinking Class-Discrimination Based CNN Channel Pruning'
---
|
---
abstract: 'We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in $\R^3$. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to $\frac1N$ where $N$ is the expected particle number. Assuming that the mass of the tracer particle is proportional to $N$, we derive generalized Hartree equations in the limit $N\rightarrow\infty$. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials.'
address:
- 'Department of Mathematics, University of Texas at Austin, Austin TX 78712, USA'
- 'Department of Mathematics, Rutgers University, Piscataway, NJ 08854'
author:
- Thomas Chen
- Avy Soffer
title: Mean field dynamics of a quantum tracer particle interacting with a boson gas
---
Introduction of the model
=========================
We consider a heavy quantum mechanical tracer particle coupled to a field of identical scalar bosons with two-particle interactions. The Hilbert space for the quantum tracer particle (with position variable $X\in\R^3$) is given by $L^2(\R^3)$. The boson Fock space is given by =\_[n1]{}\_n where \_n := (L\^2(\^3))\^[\_[sym]{}n]{} is the $n$-particle Hilbert space. We denote the Fock vacuum by $\vac\in\cF$, and introduce creation- and annihilation operators satisfying the canonical commutation relations \[a\_x,a\_y\^+\]=(x-y) , \[a\_x,a\_y\]= 0 , \[a\_x\^\*,a\_y\^\*\]= 0 , where $a_x\vac=0$ for all $x\in\R^3$. The Hilbert space of the coupled system is given by = L\^2(\^3). We will study the time evolution of this system for initial data $\Phi_0\in\cH$ with a large but finite expected particle number, $\Bra\Phi_0,\1\otimes\Nb\Phi_0\Ket=N$, where := dx a\_x\^+ a\_x is the boson number operator. We assume that the bosons interact with one another via a mean field interaction potential $\frac\lambda {2N} v$, where $\lambda>0$ is a coupling constant. Moreover, we assume that the mass of the heavy tracer particle is $N$. Accordingly, the Hamiltonian of the system is given by \[eq-cHN-def-1\] \_N &:=& -1[2N]{}\_X+T + dx w(X-x)a\_x\^+ a\_x\
&&+dx dy a\_x\^+a\_x v(x-y) a\_y\^+ a\_y where T := 12 dx a\_x\^+(-\_x a\_x) is the kinetic energy operator for the boson field, $w$ is the potential energy accounting for the coupling between the tracer particle and the bosons, and $\frac\lambda {2N} v$ is the potential accounting for pair interactions between bosons.
This system exhibits a close formal similarity to the translation-invariant model in non-relativistic Quantum Electrodynamics (QED) describing a quantum mechanical electron coupled to the quantized electromagnetic radiation field. The framework that we will use in our analysis is strongly inspired by [@fr-1; @fr-2] and [@bcffs2; @bcfs-2; @ch-4; @chfro; @cfp-1; @cfp-2].
We define the conserved total momentum operator P\_[tot]{} = i\_X + where := dx a\_x\^+ (i\_x a\_x) is the momentum operator for the boson field. The Hamiltonian is translation invariant, $[\cH_N,P_{tot}]=0$. Accordingly, we consider the decomposition of $\frH$ as a fiber integral = \_[\^3]{}\^dk \_k with respect to $P_{tot}$ where the fiber Hilbert spaces $\frH_k$ are isomorphic to $\cF$, and invariant under the dynamics generated by the Hamiltonian $\cH_N$.
For each fixed $k\in\R^3$, we consider the value $Nk$ of the conserved total momentum $P_{tot}$. The restriction of $\cH_N$ to $\frH_k$ is given by the fiber Hamiltonian \[eq-cHN-def-1\] \_N(k) &:=& 1[2N]{}(Nk-)\^2 + T + dx w(x) a\_x\^+ a\_x\
&& + dx dy a\_x\^+a\_x v(x-y) a\_y\^+ a\_y\
&=:&1[2N]{}(Nk-)\^2 + T + W\_1(0)+W\_2 , where W\_1(y) := dx w(x-y) a\_x\^+ a\_x . We also introduce W := W\_1(0)+ W\_2 for notational convenience. We note that here, $x$ (under a slight abuse of notation) stands for the relative coordinate $x-X$, with origin located at $X=0$. For a more detailed introduction to the fiber decomposition with respect to the conserved total momentum, we refer to [@bcffs2].
We will in the sequel identify $\frH$ with $L^2(\R^3,\cF)$, and omit the tensor products from the notation in . The solution of the Schrödinger equation on $\frH$ has the following form.
Given $u\in L^2(\R^3)$ and $\Psi_{k,0}^\cF\in\cF$, let \[eq-Phi-u0-init-1\] \_[u,0]{}(X) := dk u(k) e\^[iX(Nk-)]{}\_[k,0]{}\^ . Then, \_u(t,X) := dk u(k) e\^[i X(Nk-)]{}\_k\^(t) solves \[eq-Phiu-Schrod-1\] i\_t\_u=\_N\_u on $\frH$ with initial data $\Phi_u(0,X)= \Phi_{u,0}(X)\in\frH$, iff $\Psi^\cF_k(t)\in\cF$ solves \[eq-fiber-dyn-1\] i\_t\_k\^(t) = \_N(k)\_k\^(t) on $\cF$ with initial data $\Psi_k^\cF(0)=\Psi_{k,0}^\cF\in\cF$.
We have\
&=&dk u(k) e\^[i X(Nk-)]{}(-1[2N]{}(Nk-)\^2)\
&& +1[2N]{}(Nk-)\^2+T+W)\_k\^(t)\
&=&dk u(k) (e\^[-i X]{}(T+W)e\^[i X]{})e\^[i X(Nk-)]{}\_k\^(t) . Clearly, e\^[-i X]{} T e\^[i X]{} = T and e\^[-i X]{} a\_x e\^[i X]{} = a\_[x+X]{} , e\^[-i X]{} a\_x\^+ e\^[i X]{} = a\_[x+X]{}\^+ , as one sees from e\^[-i X]{} a\_x\^+ e\^[i X]{} &=& dk e\^[-ikx]{}e\^[-i X]{} a\_k\^+ e\^[i X]{}\
&=&dk e\^[-ikx]{}a\_k\^+ e\^[-i X(+k)]{} e\^[i X]{}\
&=&dk e\^[-ik(x+X)]{}a\_k\^+ . Therefore, e\^[-i X]{} W\_1 e\^[i X]{} &=& w(x) a\_[x+X]{}\^+a\_[x+X]{} dx\
&=& w(x-X) a\_[x]{}\^+a\_[x]{} dx = W\_1(X) , and e\^[-i X]{} W\_2 e\^[i X]{} &=& a\_[x+X]{}\^+a\_[x+X]{} v(x-y) a\_[y+X]{}\^+a\_[y+X]{} dx dy\
&=& a\_[x]{}\^+a\_[x]{} v(x-X-(y-X)) a\_[y]{}\^+a\_[y]{} dx dy\
&=& W\_2 . We thus conclude that (i\_t+1[2N]{}\_X )\_u(t,X) = ( T+W\_1(X)+W\_2 ) \_u(t,X) , as claimed in .
The main results proven in this paper can be summarized as follows.
The fiber ground state on $\cF_N$ for large $N$
-----------------------------------------------
The fiber Hamiltonian $\cH_N(k)$ commutes with the number operator $\Nb$. In Section \[sec-cFN-gs-1\], we study its restriction to the $N$-particle Fock space $\cF_N$, and prove that the fiber ground state energy, E\_N(k) &:=& [infspec]{}( H\_N(k)|\_[\_N]{} )\
&=& \_[\_N\_N;\_[\_N]{}=1]{} \_N , H\_N(k) \_N . satisfies the asymptotics \_[N]{} = 4 + \_[\_[L\^2]{}=1]{}\_0\[\] where \[eq-HEn-def-0\] \_0\[\]&:=& 12dx |(x)|\^2 + dx w(x) |(x)|\^2\
&& + 2 dx dy |(x)|\^2 v(x-y) |(y)|\^2 is the Hartree energy functional; see Proposition \[prop-EN-asympt-1\].
The problem considered here corresponds to a mean field limit; see [@lenaro] and the references quoted therein. For the much more difficult case of a dilute gas and the Gross-Pitaevskii limit, we refer to [@lise; @liseyn; @lisesoyn].
Coherent states and mean field limit as $N\rightarrow\infty$
------------------------------------------------------------
In Section \[sec-coh-mean-lim-1\], we derive the mean field limit of in the following sense. We define the Weyl operator &:=& ( N dx ( (x)a\_x\^+ - a\_x ) ) , and consider the solution of the Schrödinger equation with initial data given by a coherent state of the form = (-N2\_0\_2\^2) ( N dx \_0(x)a\_x\^+ ) , for $\phi_0\in H^1(\R^3)$.
For an arbitrary but fixed value $k\in\R^3$ of the conserved momentum, we assume that for some $T>0$, $\phi_t\in L^\infty_{t}H_x^3([0,T)\times\R^3)$ is the solution of \[eq-phi-pde-0\] i\_t\_t = - (k-(\_t,i\_t)) i\_t -12 \_t +w \_t + (v\*|\_t|\^2)\_t , with initial data $\phi_0\in H_x^3(\R^3)$. We introduce the scalar \[eq-Sfact-def-0\] S(t,t’) &:=& N\_[t’]{}\^t ds ( - k\^2 + 12 (\_s,i\_s)\^2\
&& + 2|\_s(x)|\^2 v(x-y) |\_s(y)|\^2 dx dy ) , and a self-adjoint Hamiltonian quadratic in creation- and annihilation operators, of the form \[eq-cHmf-def-0\] (k)&:=&(k)+where (k)&:=&-(k-(\_t,i\_t)) + T + W\_1(0)\
&& + | \_t(x)|\^2 v(x-y) a\_y\^+ a\_y dx dy is a generalized Hartree Hamiltonian which commutes with the particle number operator $\Nb$, and where &:=& 12(a\^+(i\_t)+a(i\_t))\^2\
&& + v(x-y) \_t(x) a\^+\_x a\_y dx dy\
&& + v(x-y) ( \_t(x) \_t(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy , includes correlations which do not preserve the particle number. Then, \[eq-cV-def-0\] i(t,s) = (k) (t,s) , (s,s)= , determines the unitary flow $\cV(t,s)$ generated by $\cHmf(k)$.
Our main result in section \[sec-coh-mean-lim-1\] states that the limit \[eq-meanfield-N-lim-0\] \_[N]{} e\^[-it\_N(k)]{} - e\^[-iS(t,0)]{}(t,0)\_ = 0 holds, under the assumption $v\in C^2(\R^3)$; see Theorem \[thm-cohconv-1\]. The more technical parts of the proof are presented in Section \[sec-meanfield-proof-1\]. Therefore, the solution of , with initial data characterized by a coherent state $\Psi_{k,0}^\cF= \cW[\sqrt N\phi_0]\,\vac$, is given by \_u(t,X) = dk u(k) e\^[i X(Nk-)]{} e\^[-iS(t,0)]{} (t,0)+ o\_N(1) asymptotically, as $N\rightarrow\infty$.
The convergence in Fock space proven in our work is closely related to [@grmama; @grma-1; @grma-2]. In our construction, the generator $\cHcor$ of $\cV(t,s)$ includes both a diagonal and an off-diagonal part, while in [@grmama; @grma-1; @grma-2] the term analogous to $\cV(t,s)$ has a purely off-diagonal generator. In the case considered here, the choice with allows us to efficiently control the operator $\frac1N(Nk- \Pb)^2$ in $\cH_N$ which is not present in the above mentioned works. The Hamiltonian $\cHcor$ comprises the $O_N(1)$ terms of the quasifree reduction of $\cH_N$, in the sense of [@BBCFS]. After completing this work, we noticed that a construction was used in [@lenasc] that is in part similar.
There is a vast literature on the derivation of time-dependent Hartree or nonlinear Schrödinger equations from an interacting boson field with mean field or Gross-Pitaevskii scaling. The first rigorous results were obtained in the pioneering papers [@he] and (a few years later) [@sp; @Spohn81]; more recently, the works [@esy1; @esy2; @esy3; @esy4], and subsequently [@klma] and [@rosc], motivated much of the current activity in the research area; we refer to [@CHPS-1; @chpa2; @CPBBGKY; @xchho1; @xchho2; @frknpi; @frknsc; @frtsya; @GreSohSta-2012; @grmama; @grma-1; @grma-2; @kiscst; @lenaro; @lenasc; @pick2].
The mean field limit of a [*classical*]{} tracer particle coupled to a nonlinear Hartree equation was derived in [@defrpipi-1].
Analysis of the mean field equations
------------------------------------
In sections \[ssec-NLH-gs-1\] and \[ssec-NLH-timedep-1\], we analyze the mean field equation . In Section \[ssec-NLH-gs-1\], we determine the ground state for a conserved total momentum $k\in\R^3$, under the assumption that the Hartree energy functional for $k=0$ admits a minimizer; see Proposition \[prop-Hartree-gs-k-1\].
In Section \[ssec-NLH-timedep-1\], we study dispersive solutions to . In particular, we show that is unitarily equivalent to the nonlinear Hartree equation \[eq-psi-main-eq-0\] i\_t = -12+ w\_+ (v\*||\^2) ,(t=0)=\_0\_0 H\_x\^1(\^3) where w\_(t,x) := w(x-X\_(t)) and X\_(t) := \_0\^t ds ( k - (, i)(s)) is the expected trajectory of the tracer particle.
In Theorem \[thm-psi-gwp-1\], we prove the global well-posedness for in the space L\_t\^H\_x\^1(\^3)L\_t\^ W\_x\^[1,]{}(\^3) , under the assumption that $\|w\|_{W_x^{2,\frac32}}<\infty$, and that $\|w\|_{W_x^{1,\frac32}}$, $\|\lambda v\|_{W_x^{1,\frac32}}$ are sufficiently small. As a corollary, we obtain global well-posedness for , in the sense stated in Theorem \[thm-phi-gwp-1\]; in particular, $\phi(t,x)=\psi(t,x+X_\psi(t))$. We note that under less restrictive assumptions on $w$ and $\lambda v$, the problem can be controlled with methods developed in [@beso]. We also note that $X_\psi(t)$ can be written as $X_\psi(t) = k t - (\psi, x \psi)$, and that it satisfies the Ehrenfest dynamics \[eq-Xpsi-Ehrenfest-1\] \_t\^2 X\_(t) &=& ( , (w\_+v\*||\^2) )\
&=& dx(w)(x-X\_(t)) |(x)|\^2 . The term involving $v$ is zero because $v$ is even, see Remark \[rem-Ehrenfest-1\], below.
In particular, we find $\partial_t^2 X_\psi(t)=0$ in the special case where $\phi_0=Q_k$ is the nonlinear ground state of , with $\|Q_k\|_{L_x^2}=1$. This is because we have $Q_k=e^{-i\frac k2 x}Q_0$ where $Q_0$ is the rotationally symmetric minimizer of the Hartree functional , with $\|Q_0\|_{L_x^2}=1$; see Proposition \[prop-Hartree-gs-k-1\]. Due to rotational symmetry of $Q_0$, we find that $X_\psi(t)=\frac k2 t$, and that with $\psi(t,x)=Q_k(x-\frac k2t)$, the r.h.s of is zero, so that $\partial_t^2 X_\psi(t)=0$.
Clearly, describes a [*classical*]{} tracer particle moving along the trajectory $X_\psi(t)\in\R^3$, coupled to a boson field described by the Hartree equation . A model of a similar type has been analyzed in [@frzh-1], for which the emergence of Hamiltonian friction was established in certain cases in [@frzh-2; @frzhso-1].
The fiber ground state on $\cF_N$ for large $N$ {#sec-cFN-gs-1}
===============================================
The fiber Hamiltonian $\cH_N(k)$ commutes with the number operator $\Nb$, and its restriction to the $N$-particle Fock space $\cF_N$ is given by H\_N(k) := \_N(k)|\_[\_N]{} . In this section, we will determine the asymptotics of its ground state energy in the limit of large $N$. We define E\_N(k) &:=& [infspec]{}( H\_N(k)|\_[\_N]{} )\
&=& \_[\_N\_N;\_[\_N]{}=1]{} \_N , H\_N(k) \_N . For large $N$, the following asymptotics hold.
\[prop-EN-asympt-1\] The ground state energy of the fiber Hamiltonian $H_N(k)$ satisfies \_[N]{} = 4 + \_[\_[L\^2]{}=1]{}\_0\[\] where \[eq-HEn-def-1\] \_0\[\]&:=& 12dx |(x)|\^2 + dx w(x) |(x)|\^2\
&& + 2 dx dy |(x)|\^2 v(x-y) |(y)|\^2 is the Hartree energy functional.
Let (\_[k2]{}\_N)(x\_1,…,x\_N) := (-\_[j=1]{}\^N x\_j) \_N(x\_1,…,x\_N) . Then, the kinetic energy part in yields\
&=& ((Nk-2-\_[j=1]{}\^N k\_j)\^2 +12\_[j=1]{}\^N (k\_j+k2)\^2) |\_N(\_N)|\^2 d\_N\
&=& (4+(\_[j=1]{}\^N k\_j)\^2+12\_[j=1]{}\^N k\_j\^2) |\_N(\_N)|\^2 d\_N\
&=&4 + \_N , ( 1[2N]{}\^2 + T ) \_N , where $\uk_N:=(k_1,\dots,k_N)$ and $d\uk_N:=dk_1\cdots dk_N$, while the interaction part yields \_[k2]{}\_N , W \_[k2]{}\_N = \_N , W \_N . Thus, we obtain the lower bound E\_N(k) &=& \_[\_N\_N;\_N\_[\_N]{}=1]{} \_N , H\_N(k) \_N\
&=& \_[\_N\_N;\_N\_[\_N]{}=1]{} \_[k2]{}\_N , H\_N(k) \_[k2]{}\_N\
&=&4 + \_[\_N\_N;\_N\_[\_N]{}=1]{} \_N , ( 1[2N]{}\^2 + H\_N(0) ) \_N\
&&4 + E\_N(0) . Next, we determine an upper bound.
Let \_[N,]{} := (a\^+())\^Nwhere $\phi\in H^1(\R^3)$ and $\|\phi\|_{L^2}=1$. We choose \[eq-Qk-def-1\] = e\^[-ik2 x]{} Q\_0 where $Q_0 $ is the minimizer of the Hartree functional with mass $\|Q_0 \|_{L^2}=1$. Then, we find that \[eq-EN0-upperbd-1\]\
&=& 4 + \_[N,Q\_0 ]{} , 1[2N]{}\^2 \_[N,Q\_0 ]{} + \_[N,Q\_0 ]{} , H\_N(0) \_[N,Q\_0 ]{}\
&=& 4 + \_[N,Q\_0 ]{} , 1[2N]{}\^2 \_[N,Q\_0 ]{} + N\_0\[Q\_0 \] . The minimizer of $\cE_0[\phi]$ is rotation symmetric, therefore \[eq-phi-rot-zero-1\] k |Q\_0 (k) |\^2 dk = 0 , and we have \[eq-Pbexp-id-1\]\
&=& 1[2N]{}(\_[j=1]{}\^N k\_j)\^2 |Q\_0 (k\_1) |\^2|Q\_0 (k\_N) |\^2 d\_N\
&=& 1[2N]{}(\_[j=1]{}\^N k\_j\^2 ) |Q\_0 (k\_1) |\^2|Q\_0 (k\_N) |\^2 d\_N\
&=& 1[2]{} k\^2 |Q\_0 (k) |\^2 dk < . Passing to the third line, we used that all terms whose integrands are proportional to $k_j \cdot k_\ell$, with $j\neq\ell$, vanish, due to . Notably, the term is $O(\frac1N)$ smaller than the other two terms on the last line of , and we conclude that \[eq-EN-mainbd-1\] 4 + N 4 + 1[2N]{} k\^2 |(k) |\^2 dk + \_0\[Q\_0 \] . But as was proven in [@lenaro], \_[N]{}N = \_0\[Q\_0 \] . Hence, implies that \_[N]{} = 4 + \_0\[Q\_0 \] , as claimed.
Coherent states and mean field limit as $N\rightarrow\infty$ {#sec-coh-mean-lim-1}
============================================================
In this section, we derive the dynamical mean field limit on Fock space. In order to render the exposition more readable, we are presenting the core of the proof here, but provide the more technical parts of the proof later, in Section \[sec-meanfield-proof-1\]. In this paper, we are neither attempting to optimize the bounds on the convergence rates, nor the requirements on the potentials $w$ and $v$.
Let $\phi\in L^2(\R^3)$. In this section, it will be convenient to use the notation $\phi_t(x)\equiv \phi(t,x)$. We define the Weyl operator &:=& ( N dx ( (x)a\_x\^+ - a\_x ) ) We consider the solution of the Schrödinger equation with initial data given by a coherent state of the form = (-N2\_0\_2\^2) ( N dx \_0(x)a\_x\^+ ) , for $\phi_0\in H^1(\R^3)$.
Moreover, we define a time-dependent mean-field Hamiltonian which is self-adjoint and quadratic in creation- and annihilation operators, of the form (k)&:=&(k)+where the “diagonal term” \[eq-cHHar-def-1\] (k)&:=&-(k-(\_t,i\_t)) + T + W\_1(0)\
&& + | \_t(x)|\^2 v(x-y) a\_y\^+ a\_y dx dy is the Hartree Hamiltonian which commutes with the particle number operator $\Nb$, and where the “off-diagonal term” \[eq-cHcor-def-1\] &:=& 12(a\^+(i\_t)+a(i\_t))\^2\
&& + v(x-y) \_t(x) a\^+\_x a\_y dx dy\
&& + v(x-y) ( \_t(x) \_t(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy , includes correlations which do not preserve the particle number. We obtain the unitary flow $\cV(t,s)$, \[eq-cV-def-1\] i\_t (t,s) = (k) (t,s) , (s,s)= , generated by $\cHmf(k)$.
\[thm-cohconv-1\] Let $k\in\R^3$. We assume that $v\in C^2(\R^3)$, and that for some $T>0$, $\phi_t\in L^\infty_{t}H_x^3([0,T)\times\R^3)$ is the solution of \[eq-phi-pde-1\] i\_t\_t = - (k-(\_t,i\_t)) i\_t -12 \_t +w \_t + (v\*|\_t|\^2)\_t , with initial data $\phi_0\in H_x^3(\R^3)$. Let \[eq-Sfact-def-1\] S(t,t’) &:=& N\_[t’]{}\^t ds ( - k\^2 + 12 (\_s,i\_s)\^2\
&& + 2|\_s(x)|\^2 v(x-y) |\_s(y)|\^2 dx dy ) . Then, the limit \[eq-meanfield-N-lim-1\] \_[N]{} e\^[-it\_N(k)]{} - e\^[-iS(t,0)]{}(t,0)\_ = 0 holds.
We have \[eq-cFdiff-bd-1\] e\^[-it\_N(k)]{} - e\^[-iS(t,0)]{}(t,0) \_\^2 = 2( 1- M(t) ) where M(t) &:=& Re e\^[-it\_N(k)]{} , e\^[-iS(t,0)]{}(t,0)\
&=& Re , \^\*\[N\_0\] e\^[it\_N(k)]{} e\^[-iS(t,0)]{}(t,0) . One can easily verify that given , we have i\_t = \[ (k) , \] . We consider the unitary flow (t,s) := \^\*\[N\_s\] e\^[i(t-s)\_N(k)-iS(t,s)]{} and introduce the selfadjoint operator \[eq-cL-def-1\] \_N\^[\_t]{}(k) &:=&\^\*\[N\_t\](\_N(k)-\_t S(t,0) )\
&& -\^\*\[N\_t\]\[(k),\]- (k)\
&=&\^\*\[N\_t\] \_N(k) - \_t S(t,0)\
&&-\^\*\[N\_t\](k)-(k) . Then, it is clear that \[eq-cL-def-1\] i\_t ( \_N(t,0) (t,0) ) = - \_N(t,0) \_N\^[\_t]{}(k) (t,0) . A straightforward but somewhat lengthy calculation shows that \[eq-cL-expl-1\] \_N\^[\_t]{}(k)&=& 1[2N]{}(( a\^+(i\_t) + a(i\_t) ) +( a\^+(i\_t) + a(i\_t) ) )\
&&+ 1[2N]{}\^2\
&& + v(x-y) a\_x\^+( a\_y + \_t(y)a\^+\_y ) a\_x dx dy\
&& + v(x-y) a\^+\_x a\^+\_y a\_y a\_x dx dy . This follows from Lemma \[lm-cL-1\].
Evidently, M(t) &=& Re , \_N(t,0) (t,0)\
&=&M(0)+Re\_0\^t ds \_s M(s)\
&=&1-Re { i\_0\^t ds , \_N(w,0) \_N\^[\_s]{}(k) (s,0) } . It follows from the unitarity of $\cU_N(t,0)$ that | , \_N(t,0) \_N\^[\_t]{}(k) (t,0) | \_N\^[\_t]{}(k) (t,0) \_ . We prove in Lemma \[lm-cL-bd-1\] that \[eq-cL-bd-1\] \_N\^[\_t]{}(k) (t,0) \_ C\_0 , for some constants $C_0$, $C_1$ depending on $\|v\|_{ C^2(\R^3)}$ and $\|\phi_t\|_{L^\infty_{t}H_x^3([0,T)\times\R^3)}$. Hence, we find that |M(t)-1| C\_0 \_0\^t ds < . We therefore conclude that for any $t>0$, the lhs of converges to zero in the limit $N\rightarrow\infty$.
For the convergence to the mean field dynamics in Theorem , we required that the solution to the mean field equations obtained in , \[eq-phi-pde-1-2\] i\_t\_t = - (k-(\_t,i\_t)) i\_t -12 \_t +w \_t + (v\*|\_t|\^2)\_t , has regularity $\phi\in L_t^\infty H_x^3([0,T)\times\R^3)$, for $T>0$ (possibly $T=\infty$).
In Section \[ssec-NLH-gs-1\] and Section \[ssec-NLH-timedep-1\], we study solutions to under less strict assumptions on the regularity, $\phi\in L_t^\infty H_x^1$, in the following two cases. In Section \[ssec-NLH-gs-1\], we assume that the interaction potentials $w$ and $v$ are such that the standard Hartree equation with external potential $w$ possesses a ground state, and we construct the ground state solution to . In Section \[ssec-NLH-timedep-1\], we assume that $w$ and $v$ do not allow for bound states. In this situation, we are considering dispersive solutions of , and prove local and global well-posedness under the assumption that $\|w\|_{W^{2,3/2}}<\infty$, and that $\|w\|_{W^{1,3/2}}$, $\|\lambda v\|_{ W^{1,3/2}}$ are sufficiently small.
Global dispersive solutions to with regularity $\phi\in L_t^\infty H_x^3(\R\times\R^3)$ can be constructed along the same lines as in our analysis in section \[ssec-NLH-timedep-1\], under the assumption that $\|w\|_{W^{4,3/2}}<\infty$, and that $\|w\|_{W^{3,3/2}}$, $\|\lambda v\|_{ W^{3,3/2}}$ are sufficiently small. Because the arguments are straightforward, we leave this part as an exercise.
For comparison, we note that the model usually encountered in the literature, describing the boson gas without tracer particle, is given by \[eq-cHN-def-2-0\] \_N:=T + dx w(x) a\_x\^+ a\_x + dx dy a\_x\^+a\_x v(x-y) a\_y\^+ a\_y . That is, no term $\frac1{2N}(Nk-\Pb)^2$ appears here, [@he; @rosc; @grmama].
Our constructions in this section apply to this case, too, but simplify. The mean field equation for $\phi$ is the standard Hartree equation, $$i\partial_t\phi=-\Delta\phi+w\phi+\lambda (v*|\phi|^2)\phi$$ with $\phi(0)=\phi_0\in H_x^1$.
The first term on the right hand side of and the first term on the right hand side of are then absent. Moreover, the terms on the first two lines on the r.h.s. of are absent, yielding \[eq-cL-expl-2-0\] \_N\^[\_t]{}&:=& v(x-y) a\_x\^+( a\_y + \_t(y)a\^+\_y ) a\_x dx dy\
&& + v(x-y) a\^+\_x a\^+\_y a\_y a\_x dx dy . As a consequence, a bound of the form , \[eq-cL-bd-2-0\] \_N\^[\_t]{} (t,0) \_ C\_0 , follows from Lemma \[lm-Nb-bd-1\], but Lemma \[lm-cL-bd-1\] does not need to be invoked. In particular, the constants $C_0,C_1$ depend only on $\|\phi\|_{L_t^\infty H_x^1(I\times\R^3)}$; that is, only $H_x^1$-regularity of $\phi$ is required, not $H_x^3$-regularity. This implies a convergence result analogous to , but only requiring that $\|\phi\|_{L_t^\infty H_x^1(I\times\R^3)}<\infty$. Clearly, $I=\R$, if the Hartree equation for $\phi$ is globally well-posed.
Ground state for the generalized Hartree functional {#ssec-NLH-gs-1}
===================================================
We introduce the generalized Hartree energy functional corresponding to an arbitrary but fixed conserved momentum $k\in\R^3$, \[eq-cEk-def-1\] \_[k]{}\[\] &:=& 1N \_[N,]{} , \_N(k) \_[N,]{}\
&=& 12(k- dx i\_x (x))\^2 + 12dx |(x)|\^2\
&& + dx w(x) |(x)|\^2 + 2 dx dy |(x)|\^2 v(x-y) |(y)|\^2\
&=&12(k- (, i))\^2 + \_0\[\] . Here, $\cE_0[\phi]$ is the standard Hartree energy functional with external potential $w$. We note that the scaling in the model is chosen in such a manner that $\cE_{k}[\phi]$ is independent of $N$.
Let $Q_0^{(\mu)}$ denote the minimizer of \[eq-cSk-def-0\] \_0\[\]=\_0\[\]-\_[L\^2]{}\^2 where $\mu$ is a Lagrange multiplier (the chemical potential) implementing the constraint that the $L^2$-mass $\|\phi\|_{L^2}^2=M$ is constant. Accordingly, $\mu$ depends on $M$, and we denote by $\mu_0$ the value of $\mu$ for which $M=\|Q_0^{(\mu_0)}\|_{L^2}^2=1$. For brevity, we write $Q_0 :=Q_0^{(\mu_0)}$, as we will only consider the case $M=1$. By variation of in $\phi$, if follows that $Q_0 $ satisfies the stationary Hartree equation \[eq-HartreeEq-1\] \_0 Q\_0 &=& -12Q\_0 + wQ\_0 +(v\*|Q\_0 |\^2)Q\_0 where the value of $\mu_0$ is obtained from taking the inner product of with $Q_0 $.
Likewise, for a nonzero total conserved momentum $k\in\R^3$, we consider the minimizer $Q_k $ of \[eq-cSk-def-1\] \_k\[\]:=\_k\[\] - \_[L\^2]{}\^2, under the constraint condition $\|Q_k \|_{L^2}^2=1$, and we denote the corresponding value of the chemical potential by $\mu_k$. By variation in $\phi$, it follows that the minimizer $Q_k $ satisfies \[eq-Hartreek-def-1\] \_k Q\_k =-(k- (Q\_k , iQ\_k ))iQ\_k -12Q\_k + wQ\_k +(v\*|Q\_k |\^2)Q\_k with $\|Q_k \|_{L^2}=1$. The value of $\mu_k$ is obtained from taking the inner product of with $Q_k $.
\[prop-Hartree-gs-k-1\] The vector \[eq-phik-def-1\] Q\_k :=e\^[-i2]{}Q\_0 minimizes $\cS_0[\phi]=\cE_k[\phi]-\mu\|\phi\|_{L^2}^2$ with constraint $\|Q_k \|_{L^2}=1$, and satisfies with chemical potential \_k = \_0+.
We remark that $\frac{k^2}4$ is the kinetic energy of a dressed particle, consisting of the tracer particle together with a cloud of bosons, of total mass 2.
First of all, we verify from straightforward calculation that \_k\[e\^[-i2]{}\] &=& + 12 (,i)\^2 + \_0\[\] - \_[L\^2]{}\^2\
&& + \_0\[\] - \_[L\^2]{}\^2 , noting that for the choice of frequency $\frac k2$ in the exponent of , terms linear in $i\nabla Q_0 $ on the right hand side cancel. Therefore, \_[\_[L\^2]{}=1]{} { \_k\[e\^[-i2]{}\] } && + \_[\_[L\^2]{}=1]{} {\_0\[\] - \_[L\^2]{}\^2}\
&=& + \_0\[Q\_0 \] - \_0Q\_0 \_[L\^2]{}\^2 . On the other hand, because $Q_0 $ is spherically symmetric, $(Q_0 ,i\nabla Q_0 )=0$. Therefore, \_k\[e\^[-i2]{}Q\_0 \] &=& + 12 (Q\_0 ,iQ\_0 )\^2 + \_0\[Q\_0 \] - Q\_0 \_[L\^2]{}\^2\
&=& + \_[\_[L\^2]{}=1]{} {\_0\[\] - \_[L\^2]{}\^2} saturates the lower bound.
Furthermore, substituting $e^{-i\frac{k\cdot x}2}Q_0 $ for $Q_k $ in yields \[eq-Hartreek-def-2\] \_k Q\_0 &=& -12 Q\_0 + wQ\_0 +(v\*|Q\_0 |\^2)Q\_0 where we note that all terms proportional to $\nabla Q_0 $ cancel. Comparing with , we conclude that $\mu_k=\mu_0+\frac{k^2}{4}$, as claimed.
Mean field limit for the ground state
-------------------------------------
Given $\phi_t=Q_k$ for some $k\in\R^3$, $\forall t\in\R$, is evidently solved by , and the expressions appearing in Theorem \[thm-cohconv-1\] simplify as follows. The Hamiltonian $\cHmf(k)=\cHHar(k)+\cHcor$ becomes time-independent, with (k)&=& T + W\_1(0) + | Q\_0(x)|\^2 v(x-y) a\_y\^+ a\_y dx dy , and \_[cor]{}\^[ Q\_k]{} &=& 12(a\^+(iQ\_k)+a(iQ\_k))\^2\
&& + v(x-y) Q\_k(x) a\^+\_x a\_y dx dy\
&& + v(x-y) ( Q\_k(x) Q\_k(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy . Consequently, simplifies to (t,s) = ( - i(t-s)\_[mf]{}\^[ Q\_k]{}(k) ), and simplifies to S(t,t’) = N (t-t’) ( 2| Q\_0(x)|\^2 v(x-y) | Q\_0(y)|\^2 dx dy ) , inside the expression in . The nonlinear ground state $Q_0$ of the Hartree functional, normalized by $\| Q_0\|_{L_x^2}=1$ is, for $w$, $v\in C^1(\R^3)$, an element of $H_x^3(\R^3)$, see Lemma \[lm-Q0-H3-1\]. Accordingly, $ Q_k\in H_x^3(\R^3)$, as required in Theorem \[thm-cohconv-1\].
\[lm-Q0-H3-1\] Assume that $w$, $v\in C^1(\R^3)$. Let $Q_0$ denote the minimizer of the Hartree functional $\cE_0[\,\cdot\,]$, satisfying with $\mu_0<0$, and $\|Q_0\|_{L_x^2}=1$. Then, $ Q_0\in H_x^3(\R^3)$.
Given $\mu_0<0$ in , we have the identity Q\_0 = - (|\_0|-)\^[-1]{} (w Q\_0 + (v\*|Q\_0|\^2)Q\_0) where $|\mu_0|-\Delta\geq|\mu_0|$ is strictly positive. Therefore, Q\_0\_[H\_x\^3]{}&=&(1-)\^[32]{}(|\_0|-)\^[-1]{} (w Q\_0 + (v\*|Q\_0|\^2)Q\_0)\_[L\_x\^2]{}\
&& |\_0|\^[-1]{} w Q\_0 + (v\*|Q\_0|\^2)Q\_0 \_[H\_x\^1]{}\
&&|\_0|\^[-1]{} (w\_[C\^1]{}+v\_[C\^1]{})Q\_0\_[H\_x\^1]{} < , using that $\|\lambda (v*|Q_0|^2)\|_{C^1}\leq \lambda \|v\|_{C^1}\|\,\|Q_0\|_{L_x^2}^2$, and $\|Q_0\|_{L_x^2}^2=1$.
Time-dependent mean field equations {#ssec-NLH-timedep-1}
===================================
The nonlinear dispersive PDE with conserved energy functional $\cE_k[\phi]$, for given $k\in\R^3$, is given by \[eq-phi-pde-2\] i\_t= -(k-j\_(t)) i-12 +w + (v\*||\^2). where j\_(t):= ((t),i(t)) is the expected momentum (respectively, the current) determined by $\phi$. The quantity $k-j_\phi$ is the momentum of the tracer particle, given that the bosons are in the coherent state parametrized by $\phi$, and the total conserved momentum is $k$. We will prove the following global well-posedness result for the corresponding Cauchy problem.
\[thm-phi-gwp-1\] Let $\phi_0\in H_x^1$. Then, there exists a unique global in time mild solution to , i\_t= - (k-j\_(t)) i-12 +w + (v\*||\^2), with initial data $\phi(t=0)=\phi_0$, satisfying \_[L\^\_t H\_x\^1(\^3)]{} + \_[X\_]{}\_[L\_t\^W\_x\^[1, ]{}(\^3)]{} < where $(\tau_{X_\phi}\phi)(t,x):=\phi(x+X_\phi(t))$, and $X_\phi(t)=\int_0^t ds (k-(\phi,i\nabla\phi)(s))$. In particular, $|\partial_t X_\phi(t)|<C\|\phi_0\|_{H_x^1}$, uniformly in $t\in\R$.
For any $\phi\in L^\infty_t H_x^1(\R\times\R^3)$, |j\_(t)| \_[L\^\_t H\_x\^1(\^3)]{} < C is bounded, and therefore, X\_(t) := \_0\^t ds(k-j\_(s)) is finite for every $t\in\R$. Consequently, $e^{iX_\phi(t) \cdot i\nabla}:H_x^1\rightarrow H_x^1$ is unitary for every $t\in\R$, and we may define \[eq-psi-def-1\] (t,x) := e\^[iX\_(t) i]{}(t,x) . The exponential generates translations in $x$-space by $X_\phi(t)$, yielding \[eq-psi-def-1-2\] (t,x) = (t,x-X\_(t) ) as an equivalent expression. Clearly, j\_(t) = j\_(t) , by unitarity of $e^{i\int_0^t ds(k-j_\phi(s))\cdot x}$ on $L^2(\R^3)$. Therefore, \[eq-Xphi-Xpsi-id-1\] X\_(t) = X\_(t) , and \[eq-psi-Hartree-def-1\] i\_t = e\^[iX\_(t) i]{} ( -12 +w + (v\*||\^2) ) e\^[-iX\_(t) i]{} , where we note that the first term on the r.h.s. of has been canceled by $(i\partial_tX_\phi(t))\phi$ obtained from the time derivative. Noting that the operator $-\Delta$ is translation invariant, and\
&=& v(x-X\_(t)-y) |(t,y)|\^2 dy\
&=& v(x-y) |(t,y-X\_(t))|\^2 dy\
&=& ( v \* ||\^2 )(t,x) , we find that $\psi$ satisfies the nonlinear Hartree equation \[eq-psi-main-eq-1\] i\_t = -12+ w\_+ (v\*||\^2) ,(t=0)=\_0\_0 where w\_(t,x) := w(x-X\_(t)) is the potential $w$, translated by $X_\psi(t)$.
The claim of the theorem therefore follows from the global well-posedness of established in Theorem \[thm-psi-gwp-1\], below.
The physical interpretation of is as follows. The field $\psi$ describes a self-interacting boson gas in mean field description which interacts with a point-like tracer particle traveling along a trajectory $X_\psi(t)$. The tracer particle creates an interaction potential $w$ which moves along $X_\psi(t)$, here denoted by $w_\psi$. The momentum of the tracer particle, $\partial_t X_\psi(t)=k-(\psi,i\nabla\psi)$, together with the expected momentum of the boson field, $(\psi,i\nabla\psi)$, adds up to the conserved total momentum $k$. We note the close similarity to the equations studied in [@frzh-1] for a classical particle coupled to a boson gas.
\[rem-Ehrenfest-1\] The Ehrenfest dynamics of $X_\psi(t)$ is given by \[eq-Xpsi-Ehrenfest-2\] \_t\^2 X\_(t) = ( , (w\_+v\*||\^2) ) , as stated in . The term depending on $v$ is zero because, first of all, \[eq-vterm-1\]\
&=& dx ||\^2 (v\* ||\^2 )\
&=& dx ||\^2 (v\* ||\^2 ) . On the other hand, using integration by parts, \[eq-vterm-2\]\
&=& - dx (||\^2) (v\* ||\^2)\
&=& - dx (v\*||\^2) ||\^2 where the last line holds because $v$ is a radial function, and thus even. Comparing and , we find that $\int dx \, |\psi|^2 \nabla(v* |\psi|^2 )=-\int dx \, |\psi|^2 \nabla(v* |\psi|^2 )=0$. Therefore, \[eq-Xpsi-Ehrenfest-3\] \_t\^2 X\_(t) &=& ( , w\_)\
&=& dx (w)(x-X\_(t)) |(x)|\^2 , as claimed.
A key advantage of over is that the Cauchy problem can largely be controlled with known dispersive tools for the analysis of the Hartree equation. A main difficulty is introduced by the dependence of the potential $w_\psi$ on $\psi$. Another complication arises from the fact that the energy is not conserved.
We construct local and global in time dispersive solutions under the assumption that $\|w\|_{W^{2,3/2}}<\infty$, and that $\|w\|_{W^{1,3/2}}$, $\|\lambda v\|_{ W^{1,3/2}}$ are sufficiently small. These conditions ensure, in accordance with the Birman-Schwinger principle, that neither $w$ nor $\lambda v$ create bound states. We require that $w$ has one more derivative than $v$, to control the dependence of $w_\psi$ on $\psi$. First, we prove local well-posedness.
\[thm-lwp-main-1\] (Local well-posedness) Let $\psi_0\in H^1(\R^3)$ with $\|\psi_0\|_{L_x^2}=1$. Assume that $\|w\|_{W_x^{2,\frac32}}<\infty$, and that w\_[W\_x\^[1,32]{}]{} + 3 v\_[W\_x\^[1,32]{}]{} <1 . Then, there exists a unique mild solution L\_t\^H\_x\^1(\[0,T\]\^3)L\_t\^ W\_x\^[1,]{}(\[0,T\]\^3) to with initial condition $\psi(t=0)=\psi_0$, provided that $T>0$ is sufficiently small.
We consider the map : e\^[it]{}\_0 + i\_0\^t ds e\^[i(t-s)]{} ((w\_)(s)+((v\*||\^2)) (s)) , where we may assume that $\|\psi\|_{L_x^2}=1$. Clearly, using the Strichartz and Hölder inequalities as in \_0\^t ds e\^[i(t-s)]{} ( w\_)(s) \_[L\^q\_t L\^r\_x]{} && w\_\_[L\_t\^[q’]{}L\_x\^[r’]{}]{}\
&&w\_\_[L\_t\^L\_x\^]{}\
&& w\_\_[L\^\_t L\_x\^[32]{}]{} \_[L\_[t,x]{}\^]{} , with $(q,r)$ and $(\tilde q, \tilde r)$ denoting arbitrary Strichartz admissible pairs, we find that, under inclusion of a derivative,\
&&\_0\_[H\^1]{} + w\_\_[L\^\_t W\^[1,32]{}]{}\_[L\_t\^W\_x\^[1, ]{}]{} + v\*||\^2\_[L\^\_t W\^[1,32]{}]{}\_[L\_t\^W\_x\^[1, ]{}]{} . We use Young’s inequality for convolutions in v\*||\^2 \_[L\^\_t W\^[1,32]{}]{} v\_[W\^[1,32]{}]{} \_[L\^2]{}\^2 , and observe that w\_\_[L\^\_t W\^[1,32]{}]{} \_[X]{} w(-X)\_[W\^[1,32]{}]{} = w\_[W\^[1,32]{}]{} . Therefore, \[eq-cM-Strich-bd-1\] \_[L\_t\^q W\_x\^[1,r]{}]{} \_0\_[H\^1]{} + (w\_[W\^[1,32]{}]{}+v\_[W\^[1,32]{}]{}) \_[L\_t\^W\_x\^[1, ]{}]{} , for any Strichartz admissible pair $(q,r)$. Consequently, writing $I:=[0,T]$, and defining the Banach space \[eq-Yspace-def-1\] Y(I) := L\_t\^H\_x\^1(I\^3)L\_t\^ W\_x\^[1,]{}(I\^3) endowed with the norm f\_[Y(I)]{} := f\_[L\_t\^H\_x\^1(I\^3)]{} + f\_[L\_t\^ W\_x\^[1, ]{}(I\^3)]{} , we find \[eq-cM-apriori-bd-1\] \_[Y(I)]{} 2\_0\_[H\_x\^1]{} + (w\_[W\_x\^[1,32]{}]{}+v\_[W\_x\^[1,32]{}]{}) \_[Y(I)]{} . Assuming that $\|w\|_{W_x^{1,\frac32}}+\|\lambda v\|_{W_x^{1,\frac32}}<1-\delta$, for some $\delta\in(0,1)$, and defining $R:=2\delta^{-1}\|\psi_0\|_{H^1}$, we find that \_[Y(I)]{} R + (1-) \_[Y(I)]{} . Hence, the image of the ball $B_R(0)\subset Y$ under the map $\cM$ is contained in itself.
Next, we prove the contractivity of $\cM$. Given $\psi_1,\psi_2\in B_R(0)\subset Y$, we have \[eq-cM-contr-1\] -\_[Y(I)]{} && w\_[\_1]{}-w\_[\_2]{}\_[L\^\_t W\_x\^[1,32]{}]{}\_1\_[Y(I)]{}\
&& + w\_[\_2]{}\_[L\^\_t W\_x\^[1,32]{}]{}\_1-\_2\_[Y(I)]{}\
&& + v\*(|\_1|\^2-|\_2|\^2)\_[L\^\_t W\_x\^[1,32]{}]{}\_1\_[Y(I)]{}\
&& + v\*|\_2|\^2\_[L\^\_t W\_x\^[1,32]{}]{}\_1-\_2\_[Y(I)]{} . To control the first term on the r.h.s. of , we have w\_[\_1]{}-w\_[\_2]{}\_[L\^\_t W\_x\^[1,32]{}]{} && w\_[W\_x\^[2,32]{}]{}\_t|X\_[\_1]{}(t)-X\_[\_2]{}(t)|, where |X\_[\_1]{}(t)-X\_[\_2]{}(t)| && \_0\^t ds |(\_1,i\_1)-(\_2,i\_2)|\
&&\_0\^t ds |(i(\_1-\_2),\_1)+(\_2,i(\_1-\_2))|\
&&t \_1-\_2\_[L\^\_t H\_x\^1]{} (\_1\_[L\^\_t L\_x\^2]{}+\_2\_[L\^\_t L\_x\^2]{})\
&& 2 T \_1-\_2\_[Y(I)]{} for $t\in I=[0,T]$.
To control the term on the third line on the r.h.s. of , we use v\*(|\_1|\^2-|\_2|\^2)\_[L\^\_t W\_x\^[1,32]{}]{} && v\_[W\_x\^[1,32]{}]{} |\_1|\^2-|\_2|\^2\_[L\^\_t L\_x\^1]{}\
&& v\_[W\_x\^[1,32]{}]{} |\_1|+|\_2|\_[L\^\_t L\_x\^2]{}\_1-\_2\_[L\^\_t L\_x\^2]{}\
&&2 v\_[W\_x\^[1,32]{}]{} \_1-\_2\_[Y(I)]{} where $\||\psi_1|+|\psi_2|\|_{L_x^2}\leq\|\psi_1\|_{L_x^2}+\|\psi_2\|_{L_x^2}=2$.
Summarizing, we have \[eq-cM-contr-2\]\
&& 2 T w\_[W\_x\^[2,32]{}]{} \_1\_[Y(I)]{} \_1-\_2\_[Y(I)]{}\
&& + w\_[W\_x\^[1,32]{}]{}\_1-\_2\_[Y(I)]{}\
&& + 2 v\_[W\_x\^[1,32]{}]{} \_1-\_2\_[Y(I)]{}\
&& + v \_[W\_x\^[1,32]{}]{}\_1-\_2\_[Y(I)]{}\
&& (2T R w\_[W\_x\^[2,32]{}]{} + w\_[W\_x\^[1,32]{}]{} + 3 v\_[W\_x\^[1,32]{}]{} ) \_1-\_2\_[Y(I)]{} . Therefore, $\cM$ is contractive on the ball $B_R(0)\subset Y$ if 2T R w\_[W\_x\^[2,32]{}]{} + w\_[W\_x\^[1,32]{}]{} + 3 v\_[W\_x\^[1,32]{}]{} <1 . To this end, we require that w\_[W\_x\^[1,32]{}]{} + 3 v\_[W\_x\^[1,32]{}]{} <1 , and that $T>0$ is sufficiently small (depending on $R$).
We remark that the only place in the proof that requires a finite time $T>0$ is the control of $w_\psi$. The Strichartz estimates employed here remain valid with $\R$ instead of $I$. We may therefore patch together local in time solutions using a global Strichartz inequality.
\[thm-psi-gwp-1\] (Global well-posedness) Let $\psi_0\in H^1(\R^3)$ with $\|\psi_0\|_{L_x^2}=1$. Assume that $\|w\|_{W_x^{2,\frac32}}<\infty$, and that w\_[W\_x\^[1,32]{}]{} + 3 v\_[W\_x\^[1,32]{}]{} <1 . Then, there exists a unique global mild solution $\psi\in Y(\R)$ to with initial condition $\psi(t=0)=\psi_0$. In particular, it satisfies \[eq-psisol-unif-bd-1\] \_[Y()]{} 2(1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} . Moreover, \[eq-parttXpsi-bd-1\] |\_t X\_(t)| && |k| + (1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} ,t; that is, the momentum of the tracer particle is uniformly bounded in time.
The Strichartz estimate obtained in \[eq-cM-Strich-bd-1\] holds globally in time. With $(q,r)=(\frac{10}3,\frac{10}3)$, it implies \[eq-cM-Strich-bd-2\] \_[L\_t\^W\_x\^[1, ]{}(\^3)]{} \_0\_[H\^1]{} + (w\_[W\^[1,32]{}]{}+v\_[W\^[1,32]{}]{}) \_[L\_t\^W\_x\^[1, ]{}(\^3)]{} , respectively, \[eq-psi-Strich-bd-1\] \_[L\_t\^W\_x\^[1, ]{}(\^3)]{} (1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} . We use this a priori bound to control the $L^\infty H_x^1$ norm of $\psi$.
Let $I_j:=[(j-1)T, jT]$. The estimate with $(q,r)=(\infty,2)$, combined with , implies that \[eq-psisol-unif-bd-1\] \_[L\_t\^H\_x\^1(I\_1\^3)]{} 2(1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{}, and hence, \[eq-psisol-unif-bd-2\] (t=2T)\_[H\_x\^1(I\_1\^3)]{} (1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} . Applying Theorem \[thm-lwp-main-1\] for $I_2$ with initial data $\psi(t=2T)$ yields local well-posedness on $Y(I_2)$ with the same upper bound on $\|\psi(t=3T)\|_{H_x^1(I_1\times\R^3)}$ as in .
Iterating this argument for $I_j$, $j\in \Z$, we find that \[eq-Linfty-H1-psi-1\] \_[L\_t\^H\_x\^1(\^3)]{} (1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} , globally in time.
Therefore, we obtain \[eq-cM-apriori-bd-2\] \_[Y()]{} &=& \_[L\_t\^H\_x\^1(\^3)]{}+\_[L\_t\^W\_x\^[1, ]{}(\^3)]{}\
&& 2(1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} , as claimed in .
Finally, we note that the bound implies that |\_t X\_(t)| &=& |k-(,i)(t)|\
&& |k|+\_[L\_t\^H\_x\^1(\^3)]{}\
&& |k| + (1- w\_[W\_x\^[1,32]{}]{} - v\_[W\_x\^[1,32]{}]{})\^[-1]{}\_0\_[H\_x\^1]{} ,t. Thus, the momentum of the tracer particle is uniformly bounded in time.
Proof of convergence to the mean field dynamics {#sec-meanfield-proof-1}
===============================================
Determination of $\cL_N^{\phi_t}(k)$ {#ssec-cL-1}
------------------------------------
In the following Lemma, we determine the explicit form of the operator . We will use the notation $\phi_t(x)\equiv \phi(t,x)$, similarly as in Section \[sec-coh-mean-lim-1\].
\[lm-cL-1\] The selfadjoint operator $\cL_N^{\phi_t}(k)$ in is given by \[eq-cL-expl-1\] \_N\^[\_t]{}(k)&=& 1[2N]{}(( a\^+(i\_t) + a(i\_t) ) +( a\^+(i\_t) + a(i\_t) ) )\
&&+ 1[2N]{}\^2\
&& + v(x-y) a\_x\^+( a\_y + \_t(y)a\^+\_y ) a\_x dx dy\
&& + v(x-y) a\^+\_x a\^+\_y a\_y a\_x dx dy .
We recall from that \_N\^[\_t]{}(k) &=&\^\*\[N\_t\] \_N(k) - \_t S(t,0)\
&&-\^\*\[N\_t\](k)-(k) . The explicit expressions for the terms on the right hand side are given by \[eq-WcHNW-1\]\
&=& (k\^2 - (\_t,i\_t)\^2) + 2 |\_t(x)|\^2 v(x-y) |\_t(y)|\^2 dx dy\
&&+ \^\*\[N\_t\] ( -(k-(\_t,i\_t)) + T + W\_1(0) )\
&& +12(a\^+(i\_t)+a(i\_t))\^2\
&&+ 1[2N]{}(( a\^+(i\_t) + a(i\_t) ) +( a\^+(i\_t) + a(i\_t) ) )\
&&+ 1[2N]{}\^2\
&& + N v(x-y) |\_t(x)|\^2 ( a\^+\_y \_t(y) + a\_y )dx dy\
&& + v(x-y) |\_t(x)|\^2 a\^+\_y a\_y dx dy\
&& + v(x-y) \_t(x) a\^+\_x a\_y dx dy\
&& + v(x-y) ( \_t(x) \_t(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy\
&& + v(x-y) a\_x\^+( a\_y + \_t(y)a\^+\_y ) a\_x dx dy\
&& + v(x-y) ( a\^+\_x a\^+\_y a\_y a\_x )dx dy , and \[eq-WcHmfW-1\]\
&=& \^\*\[N\_t\] ( -(k-(\_t,i\_t)) + T + W\_1(0) )\
&& + N|\_t(x)|\^2 v(x-y) |\_t(y)|\^2 dx dy\
&& + N v(x-y) |\_t(x)|\^2 ( a\^+\_y \_t(y) + a\_y )dx dy\
&& + v(x-y) |\_t(x)|\^2 a\^+\_y a\_y dx dy , and \[eq-cHcor-def-2\] &=& 12(a\^+(i\_t)+a(i\_t))\^2\
&& + v(x-y) \_t(x) a\^+\_x a\_y dx dy\
&& + v(x-y) ( \_t(x) \_t(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy , and \_t S(t,0) &=& N ( - k\^2 + (\_t,i\_t)\^2\
&& + 2|\_t(x)|\^2 v(x-y) |\_t(y)|\^2 dx dy ) . We thus obtain \[eq-cL-def-1\] \_N\^[\_t]{}(k)&=& 1[2N]{}(( a\^+(i\_t) + a(i\_t) ) +( a\^+(i\_t) + a(i\_t) ) )\
&&+ 1[2N]{}\^2\
&& + v(x-y) a\_x\^+( a\_y + \_t(y)a\^+\_y ) a\_x dx dy\
&& + v(x-y) a\^+\_x a\^+\_y a\_y a\_x dx dy , as claimed in .
We note that in order to obtain , we used the following. Introducing the abbreviated notations := , V := a\^+(i\_t)+a(i\_t) , D := (,i) , it is clear that \^\* = +N V + N D . Therefore,\
&=& (N(k-D)-\^\*(-ND))\^2\
&=& (k-D)\^2 - (k-D)\^\*(-ND)+1[2N]{}(+N V)\^2\
&=& (k\^2-D\^2) - (k-D)\^\*+1[2N]{}\^2 +1[2N]{}(V+V)+1[2]{}V\^2 . The terms on the last line are contained in the first five lines on the rhs of .
Estimates on $M(t)$ {#ssec-cL-2}
-------------------
In this subsection, we prove the estimate .
\[lm-cL-bd-1\] The following estimate holds, \_N\^[\_t]{}(k) (t,0) \_ e\^[C\_1 t]{} , for constants $C_0$, $C_1$ depending on $\|v\|_{C^2}$ and $\|\phi_t\|_{L_t^\infty H_x^3([0,T)\times\R^3)}$.
Let := dk k a\_k\^+ a\_k where $\langle k \rangle^2 := 1+k^2$. Then, for any $\Psi\in\cF$, and $\alpha\geq1$, ||\^ , \^ \^ . Thus, we obtain the following bounds on the individual terms in ,\
&& \^[3/2]{} \_t 1[2 N]{} \^2 \_t && 1[2N]{}\^2 \_t\
&& \^[3/2]{} \_t ( v(x-y) a\^+\_x a\^+\_y a\_y a\_x dx dy ) \_t \^[2]{} \_t Therefore, we obtain that \_N\^[\_t]{}(k) (t,0) \_ & & ( + 1[2N]{} ) \^2\_t + \^2\_t\
&& e\^[C\_1 t]{} , using Lemma \[lm-Nb-bd-1\] and Lemma \[lm-Pb-bd-1\].
\[lm-Nb-bd-1\] The following estimate holds, \^ (t,0) \_ e\^[C\_1 t]{} , =1,2 , for a constants $C_1$ depending only on $\|\phi_0\|_{H^1}$.
We split $\cHcor$ (see for the definition) into =+where the diagonal part $\cHcd$ commutes with the number operator, \[eq-cHcd-Nb-comm-1\] \[,\] = 0 , and where &=& 12(a\^+(i\_t)a\^+(i\_t)+a(i\_t)a(i\_t))\
&& + v(x-y) ( \_t(x) \_t(y) a\^+\_x a\^+\_y + a\_y a\_x ) dx dy\
&=:&+is the off-diagonal part.
Defining (t) := \^\*\_t \_t , where for brevity, $\cV_t:=\cV(t,0)$, we have, for $\alpha=1,2$, i\_t \^(t) &=& \^\*\_t \[\^,\] \_t\
&=& \^\*\_t (+ (-1)\^) \_t, due to .
This implies that \[eq-Nbalph-bd-1\] \^\_t &=& \^(t)\
&& \^(0) + \_0\^t ds \_s(s)\
&& \_0\^t ds \^\*\_s (+ (-1)\^) \_s . Writing \[eq-K-def-1\] K\_t(x,x’):= 12(i\_t)(x)(i\_t)(x’) + 2 v(x-y) \_t(x) \_t(y) , we have \[eq-cHcodpm-def-1\] = dx dx’ K\_t(x,x’) a\_x\^+a\_[x’]{}\^+ , = dx dx’ K\_t(x,x’) a\_xa\_[x’]{} , and we will next prove that \[eq-cHcodms-bd-1\] \^\*\_s \_s && K\_s \_[L\^2\_[x,x’]{}]{} (s) and \[eq-cHcodps-bd-1\] \^\*\_s \_s && 2 K\_s \_[L\^2\_[x,x’]{}]{} (1+(s))\
&& 2 K\_s \_[L\^2\_[x,x’]{}]{} ( 1 + (s) ) . To prove , we note that, using $K_s(x,x')=K_s(x',x)$,\
&=& ,\^\*\_s \_s\
&=&dxdx’dydy’ K\_s(y,y’) , \_s\^\*(2(x-y)(x’-y’)\
&& + 4(x’-y’)a\_y\^+a\_x + a\_x\^+a\_y\^+a\_[y’]{}a\_[x’]{}) \_s\
&=:&(I)+(II)+(III) . We have (I) = 2 K\_s \_[L\^2\_[x,x’]{}]{}\^2 (II) &=& 4dx dx’ K\_s(x,x’)a\_[x’]{} \_s \^2\
&& 4(dx K(x,x’)\_[L\^2\_[x’]{}]{}\^2) a\_[x’]{} \_s \_[L\^2\_[x’]{}]{}\^2\
&=& 4 K\_s \_[L\^2\_[x,x’]{}]{}\^2 \^[12]{}(s) \^2 (III) &=& |dxdx’dydy’ K\_s(y,y’) a\_y a\_x \_s , a\_[y’]{}a\_[x’]{} \_s |\
&& (dxdx’dydy’ |K\_s(x,x’)|\^2 |K\_s(y,y’)|\^2 )\^[12]{}\
&& (dxdx’dydy’ a\_y a\_x \_s \^2 a\_[y’]{}a\_[x’]{} \_s \^2 )\^[12]{}\
&=& ( dxdx’ |K\_s(x,x’)|\^2 )\
&& dxdx’ \_s , (a\_x\^+a\_xa\_[y]{}\^+a\_[y]{}-(x-y)a\_x\^+a\_y) \_s\
&=& K\_s \_[L\^2\_[x,x’]{}]{}\^2 ( (s) \^2 - \^[12]{}(s) \^2) so that, using $\| \Nb^{\frac12}(s) \vac \|^2\leq \|\Nb(s)\vac\|$ from Cauchy-Schwarz, \^\*\_s \_s \^2 2 K\_s \_[L\^2\_[x,x’]{}]{}\^2 ( (s) + 1 )\^2 . This implies . Similarly, one arrives at .
Therefore, implies that \[eq-Nbalph-bd-2\] 1+\^\_t &=& 1+\^(t)\
&& 1+\^(0) + \_0\^t ds \_s(s)\
&& 1+ 2 \_0\^t ds K\_s \_[L\^2\_[x,x’]{}]{}\^2 ( 1 + (s) ) , and by the Gronwall inequality, \[eq-Nbalph-bd-3\] 1+\^\_t && (4 \_0\^t ds K\_s \_[L\^2\_[x,x’]{}]{}\^2 ) , for $\alpha=1,2$.
Finally,\
&& 12\_s\_[L\_x\^2]{}\^2+ 2(|\_s(x)|\^2 |v(x-y)|\^2 |\_s(y)|\^2 dx dy)\^[12]{}\
&&(1+v\_[L\_x\^]{}) \_t\_[L\_x\^H\^1\_[x]{}(\[0,T)\^3)]{}\^2 . This implies the claim of the lemma.
\[lm-Pb-bd-1\] Let = dk k a\_k\^+ a\_k where $\langle k \rangle^2 = 1+k^2$. Then, the following estimate holds, \^ (t,0) \_ e\^[C\_1 t]{} , =1,2 , for a constant $C_1$ depending only on $\|v\|_{C^2}$ and $\|\phi\|_{L^\infty_t H^3([0,T)\times\R^3)}$.
We define (t) := \^\*\_t \_t , where $\cV_t:=\cV(t,0)$ as before. For $\alpha=1,2$, we find i\_t \^(t) &=& \^\*\_t \[\^,+\] \_t . Similarly to and , we define J\_t(x,x’):= (i\_t)(x’) + v(x-y) \_t(y) , so that = 12\_[L\_x\^2]{}\^2 + dx dx’ J\_t(x,x’) a\_x\^+a\_[x’]{} . Since \[eq-QbcH-comm-1\]\
&&+ dx dx’ { a\_x\^\* a\^\*\_[x’]{} (\_x+\_[x’]{}) K\_t(x,x’) - a\_x a\_[x’]{} (\_x+\_[x’]{}) } Then, the estimate \[eq-partt0-Qb-bd-1\]\
&& C ( (\_x-\_[x’]{}) J\_t\_[L\^2\_[x,x’]{}]{}\
&& + (\_x+\_[x’]{}) K\_t\_[L\^2\_[x,x’]{}]{} ) ((t)+1)\
&& C ( (\_x-\_[x’]{}) J\_t\_[L\^2\_[x,x’]{}]{} + (\_x+\_[x’]{}) K\_t\_[L\^2\_[x,x’]{}]{} ) ((t)+1) follows from the same arguments as the proof of , .
For $\alpha=2$, we have \_t\^2\_t && \_t . Taking the commutator with an operator acts as a derivation, thus\
&=& \[ , +\] + \[ , +\]\
&=& 2 \[ , +\] + \[ , \[ , +\]\] where, similarly to ,\
&=& dx dx’ a\_x\^\* a\_[x’]{} (\_x-\_[x’]{})\^2 J\_t(x,x’)\
&&+ dx dx’ { a\_x\^\* a\^\*\_[x’]{} (\_x+\_[x’]{})\^2 K\_t(x,x’)\
&& + a\_x a\_[x’]{} (\_x+\_[x’]{})\^2 } . In analogy to , we therefore find that \_t\^2\_t &&A\_1(t)( \^2 \_t +1)+A\_2(t)( \_t +1) where A\_1(t) &:=& C ( (\_x-\_[x’]{}) J\_t\_[L\^2\_[x,x’]{}]{} + (\_x+\_[x’]{}) K\_t\_[L\^2\_[x,x’]{}]{} ) and A\_2(t) &:=& C ( (\_x-\_[x’]{})\^2 J\_t\_[L\^2\_[x,x’]{}]{} + (\_x+\_[x’]{})\^2 K\_t\_[L\^2\_[x,x’]{}]{} ) . Since $\|\Qb\cV_t\vac\|\leq\|\Qb^2\cV_t\vac\|$, we conclude that \_t\^2\_t &&(A\_1(t)+A\_2(t))( \^2 \_t +1) and hence, \^2\_t && ( \_0\^t ds (A\_1(s)+A\_2(s)) )\
&& ( t A\_1+A\_2\_[L\_t\^(\[0,T))]{} ) , using that $\|\Qb^2\cV_0\vac\|=0$, due to $\cV_0=\1$.
Finally, we have, for $\alpha=1,2$,\
&& + (\_x+\_[x’]{})\^K\_t\_[L\_t\^L\^2\_[x,x’]{} (\[0,T)\^3\^3)]{}\
&& (1+v\_[C\^(\^3)]{})\_t\_[L\_t\^H\_x\^[1+]{}(\[0,T)\^3)]{}\^2 , as can be easily checked.
[**Acknowledgements:**]{} Part of this work was done while A. Soffer was a visiting professor at CCNU (Central China Normal University). A. Soffer is partially supported by NSFC grant No.11671163 and NSF grant DMS01600749. Part of this work was done when T. Chen was visiting CCNU. T. Chen thanks A. Soffer for his hospitality at CCNU in Wuhan. T. Chen gratefully acknlowledges support by the NSF through grants DMS-1151414 (CAREER) and DMS-1716198.
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author:
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M. M. Stetsko[^1]\
[Department for Theoretical Physics, Ivan Franko National University of Lviv,]{}\
[12 Drahomanov Str., Lviv, UA-79005, Ukraine ]{}
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Introduction
============
General Relativity is extremely successful theory which explains vast range of gravitational phenomena starting from the planetary motion and up to the evolution of the Universe [@Will_LRR2014; @Berti_CQG2015]. Numerous observations in astrophysics go hand in hand with theoretically predicted values [@Abbott_PRL16-17], but nonetheless there are still some open issues that give a chance for new theories which can be treated as generalization of Einsteinian General Relativity [@Clifton_PhysRept2012; @Heisenberg]. The most puzzling questions which still remain unsolved are the problems related to the origin (or existence) of cosmological singularities, Dark Energy and Dark Matter issues, the problems of evolution of the early Universe, for instance the problem of Inflation which is explained by approaches that take into account terms of the higher order of curvature [@Starobinsky_model].
To find solutions of the mentioned above problems different sorts of modification of General Relativity were used. Among them we distinguish for example $F(R)$ theory, Lovelock Gravity, nonlocal modifications of the gravitational action, approaches which incorporate torsion tensor, namely Teleparallel Gravity, Scalar-Tensor Gravity. There are some interrelations between different of these approaches, but certainly any of them has its own peculiarities, more detailed description of the mentioned modifications of General Relativity is given in recent review [@Heisenberg].
Scalar-Tensor theories are probably the most conservative modification of General Relativity, because the departing point of all these theories is still General Relativity, but additional scalar fields are also included. The scalar fields can be coupled with gravitational degrees of freedom in different ways and due to the character of coupling we can classify all these theories. Important feature of the scalar-tensor theories is also coupling between the scalar and additional material fields which might be taken into consideration, but it should be pointed out here that usually it is supposed to be just minimal coupling between the material fields and gravity in order to obey the equivalence principle.
Among the various types of the scalar-tensor theories we would like to focus on the so-called Horndeski Gravity [@Horndeski_IJTP74]. One of the most remarkable features of this theory is related to the fact that equations of motion in Horndeski Gravity are of the second order, so it is free from ghost instabilities. Being beyond the scope in the gravity community for several decades it has been studied intensively since the time when relation between Horndeski Gravity and some scenarios in String Theory was established. Namely, this revival of interest in Horndeski Gravity is caused by the investigation of the so called Galileon Theories [@Nicolis_PRD09; @Deffayet_PRD09] which are the scalar field theories that posses shift or Galilean symmetry, they are ghost free and those studies also renewed interest to DGP-model [@Dvali_PLB00], which originally suffered from the ghost instabilities. The equivalence between Horndeski and Galileon Theories was established [@Kobayashi_PTEP11]. Another approach which also gives rise to Galileon Theory takes its origin in Kaluza-Klein dimensional reduction procedure [@Acoleyen_PRD11; @Charmousis_LNP15]. Some other approaches related to String Theory also give rise to Galileon-like models [@Cartier_PRD01; @deRham_PRD11]. We note that multiscalar versions of Horndeski Gravity were also studied [@Deffayet_PRD10; @Padilla_JHEP13; @Charmousis_JHEP14; @Ohashi_JHEP15] and approaches that go beyond Horndeski Theory, but keeping their main attractive features were considered [@Zumala_PRD14; @Gleyzes_PRL15; @Langlois_JCAP16; @Crisostomi_JCAP16; @BenAchour_PRD16].
Since its second revival Hondeski Gravity has been applied to vast range of problems in Cosmology and Physics of Black Holes. In particular, cosmological solutions and various cosmological scenarios were studied [@Sushkov_PRD09; @Skugoreva_PRD13; @Starobinsky_JCAP16]. Dynamics of Dark Energy/ Dark Matter models was investigated [@Granda_JCAP10; @Gao_JCAP10; @Sadjadi_PRD11]. Slow-roll inflation mechanism without violation of unitarity bounds can be achieved in case of nonminimally coupled theories [@Germani_PRL10]. Various aspects of inflation were studied in theories with nonminimal derivative coupling, in particular reheating process during rapid oscillations and curvaton scenario were examined [@Sadjadi_JCAP13; @Dalianis_JCAP17; @Feng_PLB14; @Feng_PRD14; @Qiu_EPJC17]. Effective field theory of Dark Energy was considered and its relation to Horndeski Gravity was established [@Kennedy_PRD17]. Predictions of Horndeski Gravity and constraints on various modifications of General Relativity due to modern experimentally obtained results were discussed [@Heisenberg; @Hou_EPJC17; @Crisostomi_PRD18; @Langlois_PRD18].
A lot of attention has been paid to the investigation of compact objects such as neutron stars and black holes in Horndeski Gravity. Horndeski Gravity in its most general setting is known to have very complicated structure, thus the studies of the compact objects in this case are very difficult to perform. As a consequence, the investigation of the black holes and the neutron stars even in particular cases of Horndeski Gravity is of paramount importance. Rinaldi was the first who derived a static black hole solution in four dimensional case [@Rinaldi_PRD12]. Then static black holes’ solutions were obtained and investigated for various dimensions and in more general setup of Horndeski Gravity [@Minamitsuji_PRD14; @Anabalon_PRD14; @Cisterna_PRD14; @Kobayashi_PTEP14; @Babichev_JHEP14; @Bravo-Gaete_PRD14; @Giribet_PRD15; @Clement_CQG18]. Some attention has been paid to the examination of slowly rotating neutron stars and black holes [@Cisterna_PRD16; @Maselli_PRD15; @Stetsko_slr]. Some aspects of black hole thermodynamics were studied [@Feng_JHEP15; @Feng_PRD16; @Stetsko_PRD19]. Stability problem for different types of black holes, for instance with respect to odd-parity perturbations, was investigated [@Cisterna_PRD15; @Takahashi_PRD17; @Tretyakova_CQG17; @Ganguly_CQG18; @Babichev_PRL18]. Important problem also related to stability of black holes, is the causal structure which was studied in [@Benkel_PRD18]. Boson and neutron stars in Horndeski Gravity were examined [@Cisterna_PRD16; @Cisterna_PRD15; @Cisterna_PRD15_2; @Brihaye_PRD16; @Verbin_PRD18]. Several other aspects of black hole’s physics in Horndeski Theory such as the existence of hair [@Sotiriou_PRL14], or influence of higher order terms over curvature [@Antoniou_PRL18] were investigated.
In this work we consider a particular case of Horndeski gravity, namely the theory with nonminimal derivative coupling and we also take into account nonlinear electromagnetic field of Born-Infeld type which is minimally coupled to gravity. Within this framework we find static solutions of field equations which represent black holes. We point out here that charged static black hole solution in case of standard linear Maxwell field was studied in [@Cisterna_PRD14; @Feng_PRD16]. In our previous work [@Stetsko_PRD19] we studied charged black hole, but with nonlinearity of other type, namely the so-called power-law nonlinearity. Some aspects of the present work were also examined in our recent work [@Stetsko_prep] where comparison between Born-Infeld and power-law cases was shown. Here we give deeper analysis of Born-Infeld case, with thorough investigation of the obtained solutions. We also study thermodynamics of the black holes, derive and examine their temperature and obtain the first law of black hole’s thermodynamics.
The organization of the present paper is the following: in the second section obtain and investigate the black holes solutions, it should be pointed out here that since we take into account cosmological constant apart of spherically symmetric black hole solutions with non-spherical topology of horizon are considered in the present work. Small subsection of the second section is devoted to the investigation of the gauge potential for the corresponding black holes. In the third section we investigate some aspects of thermodynamics of the black hole, namely the behaviour of its temperature is studied and it is compared with the results derived for the linear Maxwell field [@Feng_PRD16; @Stetsko_PRD19], in the second part of this section we obtain the first law of black hole’s thermodynamics using Wald’s approach. In the forth section we give some conclusions.
Field equations and black hole’s solution
=========================================
The action for the system we consider comprises of several terms, namely the standard Einstein-Hilbert part with cosmological constant, the terms describing the scalar field minimally and nonminimally coupled to gravity and finally the electromagnetic part, minimally coupled to gravity and which is represented by the action of Born-Infeld type. So this action can be written the form: $$\label{action}
S=\frac{1}{16\pi}\int d^{n+1}x\sqrt{-g}\left( R-2\Lambda-\frac{1}{2}\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\partial_{\mu}\vp\partial_{\nu}\vp +4\b^2\left(1-\sqrt{1+\frac{F_{\mu\nu}F^{\mu\nu}}{2\b^2}}\right)\right)+S_{GHY}$$ and here $g_{\mu\nu}$ is the metric tensor and $g=det{g_{\mu\nu}}$ denotes its determinant, $G_{\mu\nu}$ and $R$ are the Einstein tensor and Ricci scalar respectively, $\L$ is the cosmological constant and $\vp$ is the scalar field, nonminimally coupled to gravity, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the Maxwell field tensor and $\b$ denotes the Born-Infeld coupling constant. It should be noted that when $\b\rightarrow\infty$ the Born-Infeld term gets transformed into standard linear Maxwell term, we also remark that the parameter $\beta$ is supposed to be positive to obtain reasonable physical results which are in agreement with linear field case. Finally, $S_{GHY}$ denotes boundary Gibbons-Hawking-York term, which is taken into account to have the variational problem well defined, and in case of the theory with nonminimal derivative coupling it can be represented in the form: $$\label{GHY_nm}
S_{GHY}=\frac{1}{8\pi}\int d^nx\sqrt{|h|}\left(K+\frac{\e}{4}\left[\nb^{\mu}\vp\nb^{\nu}\vp K_{\mu\nu}+(n^{\mu}n^{\nu}\nb_{\mu}\vp \nb_{\nu}\vp+(\nb\vp)^2)K\right]\right),$$ and here $h$ denotes the determinant of the boundary metric $h_{\mu\nu}$, $K_{\mu\nu}$ and $K$ are the extrinsic curvature tensor and its trace respectively, $n_{\mu}$ is the normal to the boundary hypersurface. As it is known Gibbons-Hawking-York term does not affect on the equations of motion in the bulk. We also point out here that gauge field part of the Lagrangian in the action (\[action\]) can be considered as a particular case of a more general Born-Infeld-type Lagrangian, but the given form of the Lagrangian allows to catch the most important features of the Born-Infeld Theory and it was used in numerous papers where different aspects of Black Holes’ Physics were studied [@BI_diff].
The principle of the least action, applied to the action (\[action\]) gives rise to equations of motion for the system. Namely, for the gravitational field the equations of motion can be written in the form: $$\label{eom}
G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{1}{2}(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu})+T^{(3)}_{\mu\nu}$$ where the following notations are used: $$\label{scal_min}
T^{(1)}_{\mu\nu}=\nb_{\mu}\vp\nb_{\nu}\vp-\frac{1}{2}g_{\mu\nu}\nb^{\lambda}\vp\nb_{\lambda}\vp,$$ $$\begin{aligned}
\label{scal_nm}
\nonumber T^{(2)}_{\mu\nu}=\frac{1}{2}\nb_{\mu}\vp\nb_{\nu}\vp R-2\nb^{\lambda}\vp\nb_{\nu}\vp R_{\lambda\mu}+\frac{1}{2}\nb^{\lambda}\vp\nb_{\lambda}\vp G_{\mu\nu}-g_{\mu\nu}\left(-\frac{1}{2}\nb_{\lambda}\nb_{\kappa}\vp\nb^{\lambda}\nb^{\kappa}\vp\right.\\\left.+\frac{1}{2}(\nb^2\vp)^2-R_{\lambda\kappa}\nb^{\lambda}\vp\nb^{\kappa}\vp\right)
-\nb_{\mu}\nb^{\lambda}\vp\nb_{\nu}\nb_{\lambda}\vp+
\nb_{\mu}\nb_{\nu}\vp\nb^2\vp-R_{\lambda\mu\kappa\nu}\nb^{\lambda}\vp\nb^{\kappa}\vp\end{aligned}$$ $$\label{max_tr_nlin}
T^{(3)}_{\mu\nu}=2\b^2g_{\mu\nu}\left(1-\sqrt{1+\frac{F_{\k\l}F^{\k\l}}{2\b^2}}\right)+\frac{2F_{\mu\rho}{F_{\nu}}^{\rho}}{\sqrt{1+\frac{F_{\k\l}F^{\k\l}}{2\b^2}}}$$ It is worth being remarked that the term $T^{(1)}_{\mu\nu}$ is the standard form of stress-energy tensor for a minimally coupled scalar field and $T^{(2)}_{\mu\nu}$ corresponds to the stress-energy tensor of the nonminimally coupled part. Finally, the term $T^{(3)}_{\mu\nu}$ denotes the stress-energy tensor for the electromagnetic field given by the Born-Infeld action. Varying the action (\[action\]) with respect to the scalar field $\vp$ one arrives at the following equation: $$\label{scal_f_eq}
(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nb^{\mu}\nb^{\nu}\vp=0.$$ Taking the variation of the action (\[action\]) with respect to the gauge potential $A_{\mu}$ we obtain the equations for the gauge field: $$\label{Maxwell_eq}
\nb_{\mu}\left(\frac{F^{\mu\nu}}{\sqrt{1+\frac{F_{\k\l}F^{\k\l}}{2\b^2}}}\right)=0.$$ It should be noted that in the limit $\b\rightarrow\infty$ standard Maxwell equations are recovered.
Since the action (\[action\]) includes the cosmological constant $\L$ and it means that topological solutions might exist. Here we will consider static topological solutions and consequently suppose that the metric takes the form: $$\label{metric}
ds^2=-U(r)dt^2+W(r)dr^2+r^2d\Omega^{2(\ve)}_{(n-1)},$$ where $d\Omega^{2(\ve)}_{n-1}$ is the line element of a $n-1$–dimensional hypersurface of a constant curvature which can be represented in the form: $$\begin{aligned}
d\Omega^{2(\ve)}_{(n-1)}=
\begin{cases}
d\th^2+\sin^2{\th}d\Omega^2_{(n-2)}, \quad \ve=1,\\
d\th^2+{\th}^2 d\Omega^2_{(n-2)},\quad \ve=0,\\
d\th^2+\sinh^2{\th}d\Omega^2_{(n-2)},\quad \ve=-1,
\end{cases}\end{aligned}$$ and here $d\Omega^2_{(n-2)}$ is the line element of a $n-2$–dimensional hypersphere. Thus, the expression $d\Omega^{2(\ve)}_{(n-1)}$ represents the line element on a hypersurface of positive, null or negative curvature for given values of parameter $\ve$. We point out here that in the present work we consider dimensions $n\geqslant 3$.
As it has been mentioned above we are going to obtain the static solutions and it means that we can choose the gauge field form as follows: $A=A_{0}(r)dt$ (only a scalar component of electromagnetic potential is nonzero). Taking the chosen form of gauge field and the metric (\[metric\]) into account we solve the equations (\[Maxwell\_eq\]) and obtain the electromagnetic field tensor of the form: $$\label{EM_field}
F_{rt}=-\frac{q\b}{\sqrt{q^2+\b^2r^{2(n-1)}}}\sqrt{UW},$$ where $q$ is an integration constant related to the black hole’s charge. We note here that the behaviour of the electromagnetic field depends on the product of metric functions $UW$ which is a function of the radial coordinate $r$.
Taking into consideration the explicit form of the metric (\[metric\]) and integrating the equation (\[scal\_f\_eq\]) for a once we arrive at the relation: $$\sqrt{\frac{U}{W}}r^{n-1}\left[\al-\e\frac{(n-1)}{2rW}\left(\frac{U\rq{}}{U}-\frac{(n-2)}{r}(\ve W-1)\right)\right]\vp'=C$$ In the following we assume that obtained in the latter relation constant $C$ is equal to zero and it simplifies the procedure of solving of the field equations (\[eom\]). The imposed condition on the constant $C$ is equivalent to the assumption about the relation between a component of the metric tensor $g_{\mu\nu}$ and Einstein tensor $G_{\mu\nu}$, namely: $$\label{cond_nm}
\al g_{rr}-\e G_{rr}=0$$ We note that the relation (\[cond\_nm\]) was used in most papers where black holes in the theory with nonminimal derivative coupling were examined, in particular it appears in [@Rinaldi_PRD12; @Minamitsuji_PRD14; @Feng_JHEP15; @Feng_PRD16; @Stetsko_PRD19].
Taking into account the evident form for the metric (\[metric\]) and the obtained above expression for electromagnetic field (\[EM\_field\]) we can represent the equations (\[eom\]) in the form: $$\begin{aligned}
\label{G_0}
\nonumber \frac{(n-1)}{2rW}\left(\frac{W\rq{}}{W}+\frac{(n-2)}{r}(\ve W-1)\right)\left(1+\frac{3}{4}\e\frac{(\vp\rq{})^2}{W}\right)-\L=\frac{\al}{4W}(\vp\rq{})^2+\\\frac{\e}{2}\left(\frac{(n-1)}{rW^2}{\vp''}\vp\rq{}+\frac{(n-1)(n-2)}{r^2W^2}(\vp\rq{})^2\left(\ve W-\frac{1}{2}\right)\right)-2\b^2+\frac{2\b}{r^{n-1}}\sqrt{q^2+\b^2 r^{2(n-1)}};\end{aligned}$$ $$\begin{aligned}
\label{G_1}
\nonumber \frac{(n-1)}{2rW}\left(\frac{U\rq{}}{U}-\frac{(n-2)}{r}(\ve W-1)\right)\left(1+\frac{3}{4}\e\frac{(\vp\rq{})^2}{W}\right)+\L=\\\frac{\al}{4W}(\vp\rq{})^2-\frac{\e}{2}\ve\frac{(n-1)(n-2)}{2r^2W}(\vp\rq{})^2+2\b^2-\frac{2\b}{r^{n-1}}\sqrt{q^2+\b^2 r^{2(n-1)}};\end{aligned}$$
$$\begin{aligned}
\label{G_2}
\nonumber\left[\frac{1}{2UW}\left(U\rq{}\rq{}-\frac{(U\rq{})^2}{2U}-\frac{U\rq{}W\rq{}}{2W}\right)+\frac{n-2}{2rW}\left(\frac{U\rq{}}{U}-\frac{W\rq{}}{W}\right)-\frac{(n-2)(n-3)}{2r^2W}(\ve W-1)\right]\times \\\nonumber\left(1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\right)+\L=-\frac{\al}{4W}(\vp\rq{})^2-\frac{\e}{2W^2}\vp\rq{}\rq{}\vp\rq{}\left(\frac{U\rq{}}{2U}+\frac{n-1}{r}\right)+\\\frac{\e}{2}\frac{(\vp\rq{})^2}{W}\left(\frac{U\rq{}W\rq{}}{4UW^2}+\frac{(n-2)W\rq{}}{2rW^2}-\ve\frac{(n-2)(n-3)}{2r^2}\right)+2\b^2\left(1-\frac{\b r^{n-1}}{\sqrt{q^2+\b^2r^{2(n-1)}}}\right).\end{aligned}$$
It is worth noting that we have four equations, namely (\[cond\_nm\]), (\[G\_0\]), (\[G\_1\]) and (\[G\_2\]) for three unknown functions $U(r)$, $W(r)$ and $\vp'$, it means the we might choose three of them to find the unknown functions and the fourth equation should be satisfied as an identity for the obtained solution. From the reason of simplicity we take the equations (\[cond\_nm\]), (\[G\_0\]) and (\[G\_1\]). Having used the equations (\[G\_0\]) and (\[G\_1\]) we can write: $$\label{fi_der_2}
(\vp')^2=-\frac{4r^2W}{\e(2\al r^2+\ve\e(n-1)(n-2))}\left(\al+\L\e-2\b^2\e+2\b\e r^{1-n}\sqrt{q^2+\b^2r^{2(n-1)}}\right);$$ $$\label{UW_prod}
UW=\frac{\left((\al-\L\e+2\b^2\e)r^2+\ve\e(n-1)(n-2)-2\b\e r^{3-n}\sqrt{q^2+\b^2r^{2(n-1)}}\right)^2}{(2\al r^2+\ve\e(n-1)(n-2))^2}.$$ We point out here that the right hand side of the relation (\[fi\_der\_2\]) has to take positive values outside black hole’s horizon. This condition leads to some restrictions on the parameters of coupling $\al$, $\e$, cosmological constant $\L$, charge parameter $q$ and Born-Infeld parameter $\b$. Having supposed that the coupling parameters $\al$ and $\e$ are positive (and assuming that the expression $2\al r^2+\ve\e(n-1)(n-2)$ is positive in the outer domain) one arrives at the conclusion that the expression $\al+\L\e-2\b^2\e+2\b\e r^{1-n}\sqrt{q^2+\b^2r^{2(n-1)}}$ should be negative outside the horizon, what can be achieved if the cosmological constant $\L$ is negative. We point out here that similar condition on the cosmological constant was imposed in case of neutral [@Stetsko_slr] and charged [@Stetsko_PRD19] black holes in Horndeski gravity. Since the metric function $W(r)$ diverges on the horizon changing its sign while crossing the horizon, the same is true for the function $(\vp')^2$, unless the expression $\al+\L\e-2\b^2\e+2\b\e r^{1-n}\sqrt{q^2+\b^2r^{2(n-1)}}$ also changes it sign on the horizon. It leads to the consequence that in the inner domain $\vp'$ becomes purely imaginary (phantom-like behaviour), similar situation takes place in case of neutral and charged black holes [@Stetsko_slr; @Stetsko_PRD19]. We point out there that the kinetic energy of the scalar field $K=\nb_{\mu}\vp\nb^{\mu}\vp$ is finite at the horizon and is positive in the inner domain up to the moment when the expression $\al+\L\e-2\b^2\e+2\b\e r^{1-n}\sqrt{q^2+\b^2r^{2(n-1)}}$ changes it sign. Similar analysis performed under assumption that $\e<0$ while $\al>0$ shows that in this case the cosmological constant $\L$ should be taken negative as well. Here we also note that if $\al+\L\e-2\b^2\e=0$, the right hand side of the relation (\[fi\_der\_2\]) becomes negative in the outer domain and it is not acceptable from the physical point of view. If one imposes the condition $\al-\L\e+2\b^2\e=0$ to provide positivity of the right hand side of the relation (\[fi\_der\_2\]) the parameter $\e$ should take negative values, this condition allows to obtain the following relations for the metric function in a bit simpler form, but it does not change the character of asymptotic behaviour of the metric function $U(r)$ for small and large distances in comparison with the general case as it might be shown from the relations to be obtained. The right hand side of the relation (\[UW\_prod\]) is always nonnegative and it demonstrates that the metric functions $U(r)$ and $W(r)$ always have the same sign as it should be for a black hole’s solution. It is worth emphasizing that in the limit $\beta\rightarrow\infty$ the relations (\[fi\_der\_2\]), (\[UW\_prod\]) as well as the relations for the metric function $U(r)$ that will be obtained below, are reduced to corresponding relations derived for linear field [@Cisterna_PRD14; @Feng_PRD16; @Stetsko_PRD19].
The metric function $U(r)$ can be written as follows: $$\begin{aligned}
\label{U_int}
\nonumber U(r)=\ve-\frac{\mu}{r^{n-2}}-\frac{2(\L-2\b^2)}{n(n-1)}r^2-\frac{2\b(\al-\L\e+2\b^2\e)}{\al(n-1)r^{n-2}}\int\sqrt{q^2+\b^2r^{2(n-1)}}dr+\frac{(\al+\L\e-2\b^2\e)^2}{2(n-1)\al\e r^{n-2}}\times\\\int\frac{r^{n+1}}{r^2+d^2}dr-\frac{2\b(\al+\L\e-2\b^2\e)d^2}{(n-1)\al r^{n-2}}\int\frac{\sqrt{q^2+\b^2r^{2(n-1)}}}{r^2+d^2}dr+\frac{2\b^2\e}{\al(n-1)r^{n-2}}\int\frac{r^{3-n}(q^2+\b^2r^{2(n-1)})}{r^2+d^2}dr,\end{aligned}$$ where $d^2=\ve\e(n-1)(n-2)/2\al$ and we point out that in the following relations we assume that $d^2>0$. It should be noted that the second and the fourth integrals in the relation (\[U\_int\]) have some differences for odd and even dimensions of space $n$ and the first and third ones cannot be expressed in terms of elementary functions. Namely the first integral when $q^2r^{2(1-n)}/\b^2<1$ can be represented as follows [@Bateman]: $$\label{hyp_large}
\int\sqrt{q^2+\b^2r^{2(n-1)}}dr=\frac{\b}{n}r^n{_{2}F_{1}}\left(-\frac{1}{2},\frac{n}{2(1-n)};\frac{2-n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)$$ and we additionally stress here that the right hand side of the written above relation is valid for large distances ($r^{2(n-1)}>q^2/\b^2$), whereas for small distances $r^{2(n-1)}<q^2/\b^2$ we can write: $$\label{hyp_small}
\int\sqrt{q^2+\b^2r^{2(n-1)}}dr=qr{_{2}F_{1}}\left(-\frac{1}{2},\frac{1}{2(n-1)};\frac{2n-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right).$$ The third integral in (\[U\_int\]) needs also special care and due to its importance we pay attention to it. We point out here that the result of calculation of the mentioned integral depends on the parity of $n$ and similarly to the written above relations (\[hyp\_large\]) and (\[hyp\_small\]) its evident form is defined by the condition whether $r^{2(n-1)}>q^2/\b^2$ or $r^{2(n-1)}<q^2/\b^2$ , but it also depends on the relation between $r$ and $d$ (takes a bit different form for $r>d$ and $r<d$). Namely, for odd $n$ when $r^{2(n-1)}>q^2/\b^2$ and $r>d$ (large distances) this integral can be represented in the form: $$\label{int_1}
\int\frac{\sqrt{q^2+\b^2r^{2(n-1)}}}{r^2+d^2}dr=\b r^{n-2}\sum^{+\infty}_{j=0}\frac{(-1)^j}{n-2(j+1)}\left(\frac{d}{r}\right)^{2j}{_{2}}F_{1}\left(-\frac{1}{2},\frac{n-2(j+1)}{2(1-n)};\frac{n+2j}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right),$$ if $r<d$, but still $r^{2(n-1)}>q^2/\b^2$ we can represent the latter integral in the following form: $$\label{int_2}
\int\frac{\sqrt{q^2+\b^2r^{2(n-1)}}}{r^2+d^2}dr=\frac{\b r^{n}}{d^2}\sum^{+\infty}_{j=0}\frac{(-1)^j}{n+2j}\left(\frac{r}{d}\right)^{2j}{_{2}}F_{1}\left(-\frac{1}{2},\frac{n+2j}{2(1-n)};\frac{n-2(j+1)}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right).$$ There is no difficulties with the latter integral when two other options for distance are given, namely when $r^{2(n-1)}<q^2/\b^2$ and $r<d$ or $r>d$. In contrast to odd $n$, the latter integral has some subtleties when $n$ is even. If $r^{2(n-1)}>q^2/\b^2$ and $r>d$ it can be written in the form: $$\begin{aligned}
\label{int_1e}
\nonumber\int\frac{\sqrt{q^2+\b^2r^{2(n-1)}}}{r^2+d^2}dr=\b r^{n-2}\mathop{\sum^{+\infty}_{j=0}}_{ j\neq \frac{n}{2}-1}\frac{(-1)^j}{n-2(j+1)}\left(\frac{d}{r}\right)^{2j}{_{2}}F_{1}\left(-\frac{1}{2},\frac{n-2(j+1)}{2(1-n)};\frac{n+2j}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)\\+(-1)^{\frac{n}{2}-1}\b d^{n-2}\left(\sum^{+\infty}_{l=1}\frac{(-1)^l}{l!}\left(-\frac{1}{2}\right)_{l}\left(\frac{q}{\b}\right)^{2l}\frac{r^{2(1-n)l}}{2(1-n)l}+\ln{\left(\frac{r}{d}\right)}\right),\end{aligned}$$ where $(a)_l$ denotes the Pochhammer symbol of $a$. When $r<d$ but still $r^{2(n-1)}>q^2/\b^2$ we can write: $$\begin{aligned}
\label{int_2e}
\nonumber\int\frac{\sqrt{q^2+\b^2r^{2(n-1)}}}{r^2+d^2}dr=\b\frac{r^n}{d^2}\mathop{\sum^{+\infty}_{j=0,}\sum^{+\infty}_{l=0}}_{j\neq (n-1)l-\frac{n}{2}}\frac{(-1)^{j+l}}{l!(n+2j+2(1-n)l)}\left(-\frac{1}{2}\right)_l\left(\frac{r}{d}\right)^{2l}\left(\frac{q}{\b}r^{1-n}\right)^{2l}\\+\b d^{n-2}\sum^{+\infty}_{l=1}\frac{(-1)^{nl-\frac{n}{2}}}{l!}\left(-\frac{1}{2}\right)_l\left(\frac{q}{\b}d^{1-n}\right)^{2l}\ln{\left(\frac{r}{d}\right)}.\end{aligned}$$ If $r^{2(n-1)}<q^2/\b^2$ and $r<d$ or $r>d$ one can also write the evident form for the above integral. It should be pointed out that the main difference between odd and even $n$ cases for given above integral is the presence of logarithmic terms $\sim\ln(r/d)$ for even $n$, whereas for odd $n$ such a term does not appear.
Considering large distance case when $r^{2(n-1)}>q^2/\b^2$ and $r>d$ and taking into account the written above relations (\[hyp\_large\]) and (\[int\_1\]) or (\[int\_1e\]) we can write the explicit form for the metric function $U$, namely for odd $n$ it takes the form: $$\begin{aligned}
\label{U_odd}
\nonumber U(r)=\ve-\frac{\mu}{r^{n-2}}-\frac{2(\L-2\b^2)}{n(n-1)}r^2-\frac{2\b^2(\al-\L\e+2\b^2\e)}{\al n(n-1)}r^2{_{2}F_{1}}\left(-\frac{1}{2},\frac{n}{2(1-n)};\frac{n-2}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\nonumber\frac{(\al+\L\e-2\b^2\e)^2}{2\al\e(n-1)}\left[\sum^{(n-1)/2}_{j=0}(-1)^j\frac{d^{2j}r^{2(1-j)}}{n-2j}+(-1)^{\frac{n+1}{2}}\frac{d^n}{r^{n-2}}\arctan{\left(\frac{r}{d}\right)}\right]-\frac{2\b^2(\al+\L\e-2\b^2\e)d^2}{\al(n-1)}\times\\\nonumber\sum^{+\infty}_{j=0}\frac{(-1)^j}{n-2(j+1)}\left(\frac{d}{r}\right)^{2j}{_{2}F_{1}}\left(-\frac{1}{2},\frac{n-2(j+1)}{2(1-n)};\frac{n+2j}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\frac{2\b^2\e}{\al(n-1)}\times\\\left[q^2\sum^{(n-5)/2}_{j=0}\frac{(-1)^jr^{6-2n+2j}}{(4-n+2j)d^{2(j+1)}}+\frac{(-1)^{\frac{n-3}{2}}}{r^{n-2}}\left(\frac{q^2}{d^{n-2}}+\b^2d^n\right)\arctan\left(\frac{r}{d}\right)+\b^2\sum^{(n-1)/2}_{j=0}(-1)^j\frac{d^{2j}r^{2(1-j)}}{n-2j}\right];\end{aligned}$$ and for even $n$ we obtain: $$\begin{aligned}
\label{U_even}
\nonumber U(r)=\ve-\frac{\mu}{r^{n-2}}-\frac{2(\L-2\b^2)}{n(n-1)}r^2-\frac{2\b^2(\al-\L\e+2\b^2\e)}{\al n(n-1)}r^2{_{2}F_{1}}\left(-\frac{1}{2},\frac{n}{2(1-n)};\frac{n-2}{2(n-1)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\nonumber\frac{(\al+\L\e-2\b^2\e)^2}{2\al\e(n-1)}\left[\sum^{n/2-1}_{j=0}(-1)^j\frac{d^{2j}r^{2(1-j)}}{n-2j}+(-1)^{\frac{n}{2}}\frac{d^n}{2r^{n-2}}\ln{\left(\frac{r^2}{d^2}+1\right)}\right]-\frac{2\b^2(\al+\L\e-2\b^2\e)d^2}{\al(n-1)}\times\\\nonumber\left[
\mathop{\sum^{+\infty}_{j=0}}_{j\neq\frac{n}{2}-1}\frac{(-1)^j}{n-2(j+1)}\left(\frac{d}{r}\right)^{2j}{_{2}F_{1}}\left(-\frac{1}{2},\frac{n-2(j+1)}{2(1-n)};-\frac{n+2j}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+(-1)^{\frac{n}{2}}\frac{d^{n-2}}{r^{n-2}}\times\right.\\\nonumber\left.\left(\sum^{+\infty}_{j=1}\frac{(-1)^j}{j!}\left(-\frac{1}{2}\right)_{j}\left(\frac{q}{\b}\right)^{2j}\frac{r^{2(1-n)j}}{2(n-1)j}-\ln{\left(\frac{r}{d}\right)}\right)\right]+\frac{2\b^2\e}{\al(n-1)}\left[q^2\sum^{(n-6)/2}_{j=0}\frac{(-1)^jr^{6-2n+2j}}{(4-n+2j)d^{2(j+1)}}+\right.\\\left.\frac{(-1)^{\frac{n-2}{2}}}{2r^{n-2}}\left(\frac{q^2}{d^{n-2}}\ln\left(1+\frac{d^2}{r^2}\right)-\b^2d^n\ln\left(1+\frac{r^2}{d^2}\right)\right)+\b^2\sum^{n/2-1}_{j=0}(-1)^j\frac{d^{2j}r^{2(1-j)}}{n-2j}\right].\end{aligned}$$ It should be stressed that the infinite sums in the written above relations (\[U\_odd\]) and (\[U\_even\]) are convergent when $r>d$ and $r^{2(n-1)}>q^2/\b^2$ (which also takes place for large distances), but there is no difficulty in writing the evident form for the metric function $U(r)$ when $r^{2(n-1)}>q^2/\b^2$ and $d<r$ or for other two possible options for the distance (small distances). As we have noted above, the condition $d^2>0$ is imposed after the integral form for the metric function $U(r)$ is written, we point out here that solutions with $d^2<0$ might be studied, but similarly as it was shown for neutral black hole [@Stetsko_slr] or power-law field [@Stetsko_PRD19] the corresponding solutions do not represent a black hole.
For the flat horizon case ($\ve=0$) the metric function $U(r)$ can be written in a simpler form: $$\begin{aligned}
\label{U_flat}
\nonumber U(r)=-\frac{\mu}{r^{n-2}}+\frac{(\al-\L\e+2\b^2\e)^2+4\b^4\e^2}{2\al\e n(n-1)}r^2-\frac{2\b^2(\al-\L\e+2\b^2\e)}{\al n(n-1)}r^2\times\\{_{2}F_{1}}\left(-\frac{1}{2},\frac{n}{2(1-n)};\frac{2-n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)-\frac{2\b^2\e q^2}{\al(n-1)(n-2)}r^{2(2-n)}. \end{aligned}$$ We note that the written above relation takes place for large $r$, for small $r$ instead of the relation (\[hyp\_large\]) the relation (\[hyp\_small\]) should be used, completely in the same way as it was done for the previously considered cases when $\ve\neq 0$.
Taking into account all the obtained expressions for the metric function $U$ and the product $UW$ we might investigate their behaviour for some specific values of $r$. Firstly, we examine the function $U(r)$ for small values of $r$. Having used the relations (\[U\_odd\]) and (\[U\_even\]) rewritten for small distances and considering only the leading terms we write: $$\label{u_r0_c}
U(r)\simeq\frac{4\b^2q^2}{\ve(n-1)^2(n-2)(4-n)}r^{2(3-n)}$$ and here we omit all the subleading terms for small $r$. It should be pointed out that the relation (\[u\_r0\_c\]) is valid when $n\geqslant 5$. If $n=3$ the leading term is of the form: $$\label{u_r0_3}
U(r)\simeq-\frac{\mu}{r},$$ in this case the dominant term for small distances is the same as for the nonminimally coupled theory without any electromagnetic field [@Stetsko_slr]. This fact can be explained by nonsingular behaviour of the electromagnetic potential at the origin of coordinates for the case $n=3$ and what is not true for higher dimensions. When $n=4$ the leading term can be written in the following form: $$\label{u_r0_c4}
U(r)\simeq -\frac{\b^2q^2}{9\ve r^2}\ln{\left(1+\frac{d^2}{r^2}\right)},$$ as it is easy to conclude for $n=4$ the leading term (\[u\_r0\_c4\]) is caused by the term of the same origin as in the relation (\[u\_r0\_c\]), but due to different powers of $r$ under integral from which they are derived they can have either power-law or logarithmic dependences. The product of the metric functions $UW$ for small $r$ and $n\geqslant 4$ takes the form: $$\label{UW_r0_c}
UW\simeq\frac{4\b^2q^2}{(n-1)^2(n-2)^2}r^{2(3-n)},$$ whereas for $n=3$ the product $UW\simeq\left(1-\frac{\b q}{\ve}\right)^2$ and when $q=0$ it goes to the limit that is typical for static black holes in standard General Relativity. We note that singular behaviour of the product $UW$ for small $r$ takes place for the black holes with linear and power-law nonlinear electromagnetic filed in the theory with nonminimal derivative coupling [@Feng_PRD16; @Stetsko_PRD19].
In the above analysis we have not considered the solution with flat horizon surface (\[U\_flat\]). As it is easy to see, the leading term for small distances takes the form as follows: $$\label{u_r0_f}
U(r)\simeq-\frac{2\b^2\e q^2}{\al(n-1)(n-2)}r^{2(2-n)},$$ here we note that the latter equation takes place for all $n\geqslant 3$. Having compared the relations (\[u\_r0\_c\]) and (\[u\_r0\_f\]) one can conclude that for the case of flat horizon ($\ve=0$) the singularity of the metric function $U(r)$ when $r\rightarrow 0$ is stronger than for nonflat horizon surface ($\ve=\pm 1$). We also remark that for small distances for $\ve=0$ and $\ve=1$ the leading term is negative and for $\ve=-1$ it might be positive.
For large $r$ the metric functions (\[U\_odd\]), (\[U\_even\]) and (\[U\_flat\]) have similar dependence given by the leading term of the form: $$U\simeq \frac{(\al-\L\e)^2}{2n(n-1)\al\e}r^2,$$ and here similarly to small distances we do not write subleading terms. It is easy to see that for large $r$ the metric function is always of anti-de Sitter type. The product of the metric functions for large $r$ is as follows $UW\simeq (\al-\L\e)^2/4\al^2$.
We also consider the regime of large $\e$, namely when the terms which correspond to the minimal coupling (terms related to the parameter $\al$) are supposed to be considerably smaller than the terms which appear due to the presence of nonminimal coupling (terms related to $\eta$). The metric function $U(r)$ takes the form: $$\begin{aligned}
\label{U_large_e}
\nonumber U(r)\simeq\ve-\frac{\mu}{r^{n-2}}+\frac{4\b^2 q^2}{\ve(n-1)^2(n-2)(4-n)}r^{2(3-n)}-\frac{2(\L-2\b^2)}{n(n-1)}r^2+\\\nonumber\frac{(\L-2\b^2)^2+4\b^4}{\ve(n-1)^2(n^2-4)}r^4-\frac{4\b^2}{n(n-1)}r^2{_{2}F_{1}}\left(-\frac{1}{2},\frac{n}{2(1-n)};\frac{2-n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\frac{4\b^2(\L-2\b^2)r^4}{\ve(n-1)^2(n^2-4)}{_{2}F_{1}}\left(-\frac{1}{2},\frac{n+2}{2(1-n)};\frac{4-n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+{\cal O}\left(\frac{1}{\e}\right).\end{aligned}$$ Having compared the expression (\[U\_large\_e\]) with (\[U\_flat\]) one can conclude that for the cases of nonflat horizon ($\ve=\pm 1$) the written above relation does not contain the terms proportional to $\e$ whereas the relation (\[U\_flat\]) does, so they have completely different behavior in this regime. We also remark that similar situation took place for nonlinear electromagnetic field with power-law dependence [@Stetsko_PRD19]. The product of the metric functions in regime of large $\e$ for nonflat topology of horizon takes the form: $$UW\simeq\left(1-\frac{\L-2\b^2}{\ve(n-1)(n-2)}r^2-\frac{2\b r^{3-n}}{\ve(n-1)(n-2)}\sqrt{q^2+\b^2r^{2(n-1)}}\right)^2+{\cal O}\left(\frac{1}{\e}\right).$$ For flat horizon solution ($\ve=0$) we obtain: $$UW\simeq\e^2\left(\frac{\L-2\b^2}{2\al}+\frac{\b\e}{\al}r^{1-n}\sqrt{q^2+\b^2r^{2(n-1)}}\right)^2+{\cal O}(\e).$$
We have analyzed the behaviour of the metric function $U(r)$ for different distances $r$ and various values of some parameters. Here we also give graphical representation which might make the given above analysis more transparent. The figure \[\[metr\_f\_graph\]\] shows that the change of the parameter $\beta$ (parameter of nonlinearity of the gauge field) affects weakly on the behaviour of the metric function (at least this effect is small in the domain of change of the parameters we have used here). The influence of the cosmological constant $\L$ as it follows even from the above analysis is more important for large distances, where the AdS-term becomes dominant, but for small distances the influence of the AdS-term is negligibly small.
Some particular interest might be also in considering of the regime of small $\beta$, namely for the metric function $U(r)$ we have: $$\label{metr_small_b}
U(r)\simeq\ve-\frac{\mu}{r^{n-2}}-\frac{2\L}{n(n-1)}r^2-\frac{2\b(\al-\L\e)}{\al(n-1)}\frac{q}{r^{n-3}}-\frac{2\b(\al+\L\e)}{\al(n-1)}\frac{qd}{r^{n-2}}\arctan{\left(\frac{r}{d}\right)}+{\cal{O}}(\b^2),$$ and for the product of the metric functions we write: $$UW\simeq\frac{\left((\al-\L\e)r^2+\ve\e(n-1)(n-2)\right)}{(2\al r^2+\ve\e(n-1)(n-2))^2}\left((\al-\L\e)r^2+\ve\e(n-1)(n-2)-4\e q r^{3-n}\right)+{\cal{O}}(\b^2).$$ We point out that for the flat horizon case ($\ve=0$) the last term in the asymptotic relation (\[metr\_small\_b\]) does not appear. It should be also noted that given above two relations are valid for small and intermediate distances, since the product $\b r^{2(n-1)}$ which is present in general relation for the metric functions might become large for corresponding large values of $r$.
![Metric functions $U(r)$ for different values of parameter $\beta$ (the left graph) and different values of the cosmological constant $\L$ (the right one). For all the graphs we have $n=3$, $\ve=1$, $\al=0.2$, $\eta=0.4$, $\mu=1$, $q=0.2$. For the left graph the other parameters are equal to $\L=-2$, $\beta=1$ (solid curve) $\beta=10$ (dashed curve). For the right graph $\beta=1$, and $\L=-1$ (dotted curve), $\L=-2$ (solid curve) and $\L=-3$ (dashed curve).[]{data-label="metr_f_graph"}](BI_metr_funct_diff_b.eps "fig:")![Metric functions $U(r)$ for different values of parameter $\beta$ (the left graph) and different values of the cosmological constant $\L$ (the right one). For all the graphs we have $n=3$, $\ve=1$, $\al=0.2$, $\eta=0.4$, $\mu=1$, $q=0.2$. For the left graph the other parameters are equal to $\L=-2$, $\beta=1$ (solid curve) $\beta=10$ (dashed curve). For the right graph $\beta=1$, and $\L=-1$ (dotted curve), $\L=-2$ (solid curve) and $\L=-3$ (dashed curve).[]{data-label="metr_f_graph"}](BI_metr_funct_diff_L.eps "fig:")
To obtain information about coordinate and physical singularities of the metric Kretschmann scalar at different points should be examined. In general it takes the form: $$\label{Kr_scalar}
R_{\mu\nu\k\l}R^{\mu\nu\k\l}=\frac{1}{UW}\left(\frac{d}{dr}\left[\frac{U'}{\sqrt{UW}}\right]\right)^2+\frac{(n-1)}{r^2W^2}\left(\frac{(U')^2}{U^2}+\frac{(W')^2}{W^2}\right)+\frac{2(n-1)(n-2)}{r^4W^2}(\ve W-1)^2.$$ One can verify easily that at the horizon point $r_+$, namely when $U(r_+)=0$ the Kretschmann scalar (\[Kr\_scalar\]) is nonsingular, it means that the horizon points are the points with a coordinate singularity as it should be for a black hole. To investigate the behavior of the metric at the origin and at the infinity one should use corresponding asymptotic relations that has been obtained previously. At large distances when $r\rightarrow\infty$ the Kretschmann scalar is as follows: $$R_{\mu\nu\k\l}R^{\mu\nu\k\l}\simeq \frac{8(n+1)\al^2}{n(n-1)^2\e^2}$$ It should be noted that the relation for Kretschmann scalar at the infinity is completely the same as it is for a chargeless solution [@Stetsko_slr] or with nonlinear electromagnetic field [@Stetsko_PRD19]. This similarity is caused by the same asymptotic behaviour of all the metrics at the infinity. Since the metric functions have different behaviour in the domain close to the origin of coordinates for diverse dimensions of space $n$ and values of $\ve$ it means that the Kretschmann scalar (\[Kr\_scalar\]) should be examined separately for all these cases. Firstly, we consider the situation when $n\geqslant 5$ and $\ve=\pm 1$. Having substituted the leading terms of the metric functions given by the relations (\[u\_r0\_c\]) and (\[UW\_r0\_c\]) into the relation (\[Kr\_scalar\]) and after little algebra we obtain: $$R_{\mu\nu\k\l}R^{\mu\nu\k\l}\simeq \frac{4(n-1)^2(n-2)(n-3)^2}{(n-4)r^4}.$$ It should be pointed out that the obtained relation does not contain the integration constant $q$ nor the parameter $\b$ which are present in the leading terms of the metric functions. We also remark here that for power-law field as well as for the linear one [@Stetsko_PRD19] the Kretschmann scalar at the origin has similar dependence $\sim 1/r^4$ and does not depend on the charge parameter $q$. Now we derive the Kretschmann scalar when $n=4$ and $\ve\neq 0$. Here it is necessary to utilize the relations (\[u\_r0\_c4\]) and (\[UW\_r0\_c\]). As a result we arrive at the expression: $$\label{Kr_sc_4_c}
R_{\mu\nu\k\l}R^{\mu\nu\k\l}\simeq\frac{40}{r^4}\ln^2\left(1+\frac{d^2}{r^2}\right).$$ We note here that the written above expression does not depend on the parameters $q$ and $\b$ analogously as it was in the previous case ($n\geqslant 5$), but it has a bit stronger singular behaviour at the origin due to the presence of a divergent logarithmic factor. In three dimensional space ($n=3$) we have to use the the relation (\[u\_r0\_3\]) and the corresponding relation for the function $W$ and as a result the Kretschmann scalar in the vicinity of the origin takes the form: $$\label{Kr_sc_n3}
R_{\mu\nu\k\l}R^{\mu\nu\k\l}\simeq\frac{\mu^2}{\left(1-\frac{\b q}{\ve}\right)^4 r^6}$$ We conclude that in three dimensional space the singularity of the Kretschmann scalar is the strongest and it is caused by the other term than in the previously analyzed cases when $n\geqslant 4$. When $q=0$ the relation (\[Kr\_sc\_n3\]) is completely the same as it was for the neutral case [@Stetsko_slr].
Now we consider flat horizon geometry ($\ve=0$) and we should use the asymptotic relation (\[u\_r0\_f\]) and corresponding relation for the metric function $W$. We remark that for the flat horizon geometry we do not consider the cases of various dimensions separately because the behaviour of the metric functions is defined by similar relation for all the dimensions. As a result we obtain: $$\label{Kr_sc_r_f}
R_{\mu\nu\k\l}R^{\mu\nu\k\l}\simeq\frac{2(n-1)(n-2)}{r^4}$$ and here similarly to the investigated above cases of nonflat geometry the behaviour of the Kretschmann scalar does not depend on the parameters $q$ and $\b$, but in contrast with the nonflat cases the relation (\[Kr\_sc\_r\_f\]) is valid for all the dimensions $n\geqslant 3$.
Electromagnetic field potential
-------------------------------
The important feature of a charged black hole is its gauge potential, namely it has key role when one tries to derive the first law of black holes thermodynamics. The evident form of the gauge potential can be found easily when one uses the relation for the gauge field (\[EM\_field\]). So, we can write: $$\label{gauge_pot}
A_0(r)\equiv\psi=\psi_0-\int\frac{q\b}{\sqrt{q^2+\b^2r^{2(n-1)}}}\sqrt{UW}dr.$$ Taking into account the relation for the product of the metric functions (\[UW\_prod\]) one can perform the integration and write the gauge potential in the form: $$\begin{aligned}
\label{gauge_pot_odd}
\nonumber A_0(r)\equiv\psi(r)=\psi_0+\frac{q}{2(n-2)\al}\frac{(\al-\L\e+2\b^2\e)}{r^{n-2}}{_{2}F_{1}}\left(\frac{1}{2},\frac{2-n}{2(1-n)};\frac{4-3n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\nonumber\frac{qd^2}{2\al}\frac{(\al+\L\e-2\b^2\e)}{r^n}\sum^{+\infty}_{j=0}\frac{(-1)^j}{n+2j}\left(\frac{d}{r}\right)^{2j}{_{2}F_{1}}\left(\frac{1}{2},-\frac{2j+n}{2(1-n)};\frac{2-3n-2j}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\frac{\b^2\e q}{\al}\left(\sum^{(n-5)/2}_{j=0}\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\frac{(-1)^{\frac{n-3}{2}}}{d^{n-2}}\arctan\left(\frac{r}{d}\right)\right)\end{aligned}$$ for odd $n$ and large $r$ and $$\begin{aligned}
\label{gauge_pot_even}
\nonumber A_0(r)\equiv\psi(r)=\psi_0+\frac{q}{2(n-2)\al}\frac{(\al-\L\e+2\b^2\e)}{r^{n-2}}{_{2}F_{1}}\left(\frac{1}{2},\frac{2-n}{2(1-n)};\frac{4-3n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\nonumber\frac{qd^2}{2\al}\frac{(\al+\L\e-2\b^2\e)}{r^n}\sum^{+\infty}_{j=0}\frac{(-1)^j}{n+2j}\left(\frac{d}{r}\right)^{2j}{_{2}F_{1}}\left(\frac{1}{2},-\frac{2j+n}{2(1-n)};\frac{2-3n-2j}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)+\\\frac{\b^2\e q}{\al}\left(\sum^{(n-6)/2}_{j=0}\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\frac{(-1)^{\frac{n-2}{2}}}{2d^{n-2}}\ln\left(1+\frac{d^2}{r^2}\right)\right)\end{aligned}$$ for even $n$ and large $r$ respectively. We note that almost all the terms which depend on the radial coordinate $r$ go to zero at infinity and the only exclusion is the $\arctan(r/d)$ term in the case of odd $n$ but this term is finite at the infinity, so the value of the gauge potential at the infinity is mainly defined by the constant of integration $\psi_0$ that we have introduces in our relation. This constant can be taken arbitrary, but here we take it nonzero to provide the condition that the gauge potential (\[gauge\_pot\]) is equal to zero at the horizon. This requirement is not necessary, but it is quite convenient when one uses the Wald procedure to obtain the first law of black hole thermodynamics [@Feng_PRD16; @Stetsko_PRD19]. We note that for small $r$ another representation for hypergeometric function should be used. Namely, for odd $n$ we can write: $$\begin{aligned}
\label{g_pot_odd_sm}
\nonumber A_0(r)\equiv\psi(r)=\psi_0-\frac{\b(\al-\L\e+2\b^2\e)}{2\al}r{_{2}F_{1}}\left(\frac{1}{2},\frac{1}{2(n-1)};\frac{2n-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right)+\\\nonumber\frac{\b(\al+\L\e-2\b^2\e)}{2\al}r\sum^{+\infty}_{j=0}\frac{(-1)^j}{2j+1}\left(\frac{r}{d}\right)^{2j}{_{2}F_{1}}\left(\frac{1}{2},-\frac{2j+1}{2(n-1)};\frac{2(j+n)-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right)+\\\frac{\b^2\e q}{\al}\left(\sum^{(n-5)/2}_{j=0}\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\frac{(-1)^{\frac{n-3}{2}}}{d^{n-2}}\arctan\left(\frac{r}{d}\right)\right)\end{aligned}$$ and for even $n$ we obtain: $$\begin{aligned}
\label{g_pot_even_sm}
\nonumber A_0(r)\equiv\psi(r)=\psi_0-\frac{\b(\al-\L\e+2\b^2\e)}{2\al}r{_{2}F_{1}}\left(\frac{1}{2},\frac{1}{2(n-1)};\frac{2n-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right)+\\\nonumber\frac{\b(\al+\L\e-2\b^2\e)}{2\al}r\sum^{+\infty}_{j=0}\frac{(-1)^j}{2j+1}\left(\frac{r}{d}\right)^{2j}{_{2}F_{1}}\left(\frac{1}{2},-\frac{2j+1}{2(n-1)};\frac{2(j+n)-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right)+\\\frac{\b^2\e q}{\al}\left(\sum^{(n-6)/2}_{j=0}\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\frac{(-1)^{\frac{n-2}{2}}}{2d^{n-2}}\ln\left(1+\frac{d^2}{r^2}\right)\right)\end{aligned}$$ The written above relations show that near the origin of coordinates the gauge potential $A_0(r)$ is singular when $n\geqslant 4$, namely when $n=4$ the potential has a logarithmic divergence at the origin whereas for $n>4$ we have power-law singularity. It should be pointed out that even the gauge field (\[EM\_field\]) is singular at the origin if $n\geqslant 4$, the only case when the gauge field and the potential are not singular is $n=3$. We also emphasize that the behaviour of the gauge field and potential in our case is completely different from the situation which takes place in standard General Relativity with a gauge field described by a Born-Infeld type of action, where the gauge field and potential have nonsingular behaviour for all values of $n$.
If $\ve=0$ the gauge potential has a simpler form, namely for large $n$ we can write: $$\label{g_pt_e0_lar}
A_0(r)\equiv\psi=\psi_0-\frac{\b^2\e q}{(n-2)\al}r^{2-n}+\frac{(\al-\L\e+2\b^2\e)q}{2(n-2)\al r^{n-2}}{_{2}F_{1}}\left(\frac{1}{2},\frac{2-n}{2(1-n)};\frac{4-3n}{2(1-n)};-\frac{q^2}{\b^2}r^{2(1-n)}\right)$$ and for small $r$ we obtain: $$\label{g_pt_e0_sm}
A_0(r)\equiv\psi=\psi_0-\frac{\b^2\e q}{(n-2)\al}r^{2-n}-\frac{\b(\al-\L\e+2\b^2\e)}{2\al}r{_{2}F_{1}}\left(\frac{1}{2},\frac{1}{2(n-1)};\frac{2n-1}{2(n-1)};-\frac{\b^2}{q^2}r^{2(n-1)}\right).$$ It follows from the latter two relations that the gauge potential has power-law singularity at the origin for all $n\geqslant3$ and at the infinity it equals to the constant $\psi_0$.
Having the explicit form of the gauge field (\[EM\_field\]) and using the Gauss law we calculate total charge of the black hole which is of the crucial importance for black hole thermodynamics. The Gauss law for the Born-Infeld electrodynamics takes the following form: $$Q=\frac{1}{4\pi}\int_{\Sigma}\left(1+\frac{F_{\k\l}F^{\k\l}}{2\b}\right)^{-\frac{1}{2}}*F$$ and here $*F$ denotes the Hodge dual of electromagnetic field form $F$ and the integral is taken over a closed $n-1$–dimensional hypersurface $\Sigma$. After calculation of the latter integral one arrives at: $$Q=\frac{\omega_{n-1}}{4\pi}q,$$ where $\omega_{n-1}$ is the hypersurface area of a “unit” hypersurface of constant curvature (it would be surface area of a unit hypersphere in case of spherical symmetry). For nonspherical geometry of horizon it is convenient to define the total electric charge per unit area which can be written in the form: $$\bar{Q}=\frac{1}{\omega_{n-1}}Q.$$ The electric potential measured at the infinity with respect to the horizon can be represented in the form: $$\Phi_q=A_{\mu}\chi^{\mu}\Big|_{+\infty}-A_{\mu}\chi^{\mu}\Big|_{r_+},$$ where $\chi^{\mu}$ is a timelike Killing vector null on the event horizon and we have taken it to be the time translation vector $\chi^{\mu}=\partial/\partial t$. After calculation we obtain: $$\Phi_q=\psi_0,$$ so the potential $\Phi_q$ is completely defined by the integration constant $\psi_0$. It should be pointed out that one can impose that $A_{\mu}$ is zero at the horizon, but it would give rise to the same expression for the measured electric potential $\Phi_q$.
Black hole thermodynamics
=========================
In this section we derive and investigate main thermodynamic relations of the black hole solutions obtained in the previous section. One of the most important thermodynamic quantities of the black hole is its temperature which can be obtained in the same way as it is done in General Relativity, namely its definition is based on the notion of surface gravity which can represented in the following form: $$\label{surf_grav}
\kappa^2=-\frac{1}{2}\nabla_{a}{\chi}_b\nabla^{a}{\chi}^{b},$$ and here $\bar{\chi}_a$ is a Killing vector, which should be null on the event horizon. Similarly as in the previous section we take the vector of time translation ${\chi}^a=\partial/\partial t$. One can calculate the surface gravity (\[surf\_grav\]) and substituting it in the definition of the temperature one arrives at the relation: $$\label{BH_temp}
T=\frac{\k}{2\pi}=\frac{1}{4\pi}\frac{U\rq{}(r_+)}{\sqrt{U(r_+)W(r_+)}}$$ where $r_+$ is the radius of the event horizon of the black hole. It should be pointed out that nonetheless on the complicated structure of the metric function $U(r)$ the evident form of which contains hypergeometric functions, the temprature can be represented in a relatively compact form which comprises just of rational and irrational functions of the horizon radius $r_+$. After all the calculations the temperature can be represented in the following form: $$\begin{aligned}
\label{temp_BI_gen}
\nonumber T=\frac{1}{4\pi}\frac{2\al r^2_{+}+\ve\e(n-1)(n-2)}{(\al-\L\e+2\b^2\e)r^2_{+}+\ve\e(n-1)(n-2)-2\b\e r^{3-n}_{+}\sqrt{q^2+\b^2r^{2(n-1)}_{+}}}\times\\\nonumber\left[\frac{(n-2)\ve}{r_+}-\frac{2\b(\al-\L\e+2\b^2\e)}{\al(n-1)r^{n-2}_+}\sqrt{q^2+\b^2r^{2(n-1)}_+}+\frac{(\al+\L\e-2\b^2\e)^2}{2(n-1)\al\e}\frac{r^3_+}{r^2_{+}+d^2}-\right.\\\left.\frac{2(\L-2\b^2)}{n-1}r_{+}-\frac{2\b(\al+\L\e-2\b^2\e)d^2}{\al(n-1)r^{n-2}_+}\frac{\sqrt{q^2+\b^2r^{2(n-1)}_+}}{r^2_{+}+d^2}+\frac{2\b^2\e\left(q^2+\b^2r^{2(n-1)}_+\right)}{\al(n-1)r^{2n-5}_{+}(r^2_{+}+d^2)}\right].\end{aligned}$$ In the limit when $\b\rightarrow\infty$ we recover the relation for the temperature for linear Maxwell field, namely we arrive at [@Stetsko_PRD19]: $$\begin{aligned}
\label{temp_lin_f}
\nonumber T=\frac{1}{4\pi}\frac{2\al r^2_{+}+\ve\e(n-1)(n-2)}{(\al-\L\e)r^2_{+}+\ve\e(n-1)(n-2)-\e q^2r^{2(2-n)}_{+}}\left[\frac{(n-2)\ve}{r_+}+\frac{(\al-\L\e)^2}{2\al\e}r_{+}-\frac{2q^2}{(n-1)r^{2n-3}_{+}}\right.\\\left.+\frac{r_+}{\al(n-1)(r^2_{+}+d^2)}\left((\al+\L\e)q^2r^{2(2-n)}_{+}-\frac{(\al+\L\e)^2}{2\e}d^2+\frac{\e q^4}{2}r^{2(3-2n)}_{+}\right)\right].\end{aligned}$$ Here we remark, that the relation for the temperature for linear Maxwell field given in our previous work [@Stetsko_PRD19] was written in a bit different form, but some simple transformations allows us to obtain the relation (\[temp\_lin\_f\]).
The obtained relations for the temperature (\[temp\_BI\_gen\]) for arbitrary $\b$, as well as its particular case (\[temp\_lin\_f\]) for linear field are not so simple to comprehend their behaviour in full details, but nevertheless some general conclusions can be made just looking at the given above relations. First of all for large horizon radii $r_+$ the dominant term in both cases is linear over $r_+$, namely we arrive at the relation $T\simeq (\al-\L\e)r_+/4\pi\e$ for arbitrary $\b$. The fact that the leading term of the asymptotic relation does not depend on the parameter $\b$ can be explained by the domination of the AdS-term in this case and also thanks to the circumstance that Born-Infeld modification of gauge action was introduced to eliminate divergent behaviour of the gauge field for small distances, whereas for the large ones the electromagnetic field behaves almost in the same way as for the linear field case. For small horizon radius the behaviour of the temperature is different for arbitrary finite $\b$ and for the limit case of linear field ($\b\rightarrow\infty$), namely for the first case we have $T\sim\b r^{2-n}_+$ whereas for the latter one it behaves as $T\sim r^{3-2n}_+$ so we conclude that for linear field the dependence of the temperature for small radius of horizon is stronger for linear field, but this conclusion is expectable since as we have already mentioned Born-Infeld theory was introduced to modify the electromagnetic field in a way to make it finite at the origin, thus it does not takes place here in the theory with nonminimal derivative coupling apart of the case when $n=3$. We also pay attention to the case $n=3$ here, namely when $\b\neq 0$ and $\ve\neq 0$ for small horizon radius the temperature might change its sign depending on the relation between $\b$ and $q$.
We have analyzed the dependences for the temperatures (\[temp\_BI\_gen\]) and (\[temp\_lin\_f\]) for some particular cases of $r_+$, namely for its extremely large and small values, to understand behaviour of the temperatures better for some intermediate values of $r_+$ we demonstrate it graphically. Figure \[\[temp\_graph\]\] shows this dependence for various values of $\b$, when the other parameters are held fixed (the left graph) and for various values of the cosmological constant $\L$ (the right graph). We can conclude that the variation of the parameter $\b$ affects considerably on the temperature for small horizon radius, whereas for large values of $r_+$ this influence is negligibly small and the behaviour is mainly defined by the AdS-terms in all the cases. Variation of the parameter $\L$ has substantial influence on the temperature for large $r_+$, whereas for small $r_+$ its contribution becomes negligibly small. The function $T(r_+)$ might be nonmonotonous, what is better seen on the right graph, this fact might give us some critical behaviour in extended thermodynamic phase space similarly as it is done for charged black holes in the framework of standard General Relativity [@Kubiznak_JHEP12; @Gunasekaran_JHEP12], but this issue will be investigated elsewhere.
![Temperature $T$ as a function of the horizon radius $r_+$ for different values of parameter $\beta$ (the left graph) and different values of the cosmological constant $\L$ (the right one). For all the graphs we have $n=3$, $\ve=1$, $\al=0.2$, $\eta=0.4$, $q=0.2$. For the left graph the other parameters are equal to $\L=-8$, $\beta=8$ (dashed line), $\beta=50$ (dotted line) and solid line represents linear field case ($\b\rightarrow\infty$). For the right graph $\beta=8$, and $\L=-2$ (solid line), $\L=-4$ (dashed line) and $\L=-8$ (dotted line).[]{data-label="temp_graph"}](Temp_diff_beta.eps "fig:")![Temperature $T$ as a function of the horizon radius $r_+$ for different values of parameter $\beta$ (the left graph) and different values of the cosmological constant $\L$ (the right one). For all the graphs we have $n=3$, $\ve=1$, $\al=0.2$, $\eta=0.4$, $q=0.2$. For the left graph the other parameters are equal to $\L=-8$, $\beta=8$ (dashed line), $\beta=50$ (dotted line) and solid line represents linear field case ($\b\rightarrow\infty$). For the right graph $\beta=8$, and $\L=-2$ (solid line), $\L=-4$ (dashed line) and $\L=-8$ (dotted line).[]{data-label="temp_graph"}](Temp_diff_lambda.eps "fig:")
Wald’s approach and entropy of black hole
-----------------------------------------
There are several approaches to define entropy of a black hole, some of them take their roots in earlier work of Gibbons and Hawking [@Gibbons_PRD77], other approaches are based on Wald’s procedure (or method) which can be treated as a generalization of Noether method to derive conserved quantities in gravity [@Wald_PRD93; @Iyer_PRD94]. It should be pointed out that Wald’s procedure applicable to quite general diffeomorphism-invariant theories. It has been applied to numerous black hole solutions in the framework of standard General Relativity as well as its generalizations [@Feng_JHEP15; @Feng_PRD16; @Stetsko_PRD19; @Liu_PLB14; @Lu_JHEP15; @Fan_JHEP15]. In Horndeski Gravity the Wald’s procedure appears to be a consistent approach to obtain the first law of black hole thermodynamics. We point out here that the scalar potential in Horndeski Gravity has singular behaviour, but the Wald’s procedure takes this fact into account [@Feng_JHEP15; @Feng_PRD16]. We also note that Wald’s procedure allowed to derive reasonable and in some sense universal relations for back hole’s entropy [@Feng_JHEP15; @Feng_PRD16; @Stetsko_PRD19].
Here we also utilize Wald’s procedure to derive a relation for black hole’s entropy and firstly we describe the keypoints of this approach. Assuming that we have a Lagrangian ${\cal L}$ of a system, we can perform its variation and as a result we write: $$\delta {\cal L}=e.o.m.+\sqrt{-g}\nabla_{\mu}J^{\mu},$$ where $e.o.m.$ represents the terms which give equations of motion for the system and the term $\sqrt{-g}\nabla_{\mu}J^{\mu}$ gives rise to the so called surface term in the action integral because of full divergence of the last term, the $J^{\mu}$ denotes the surface “current”. Using the obtained relation for the current $J^{\mu}$ one can construct “current” one-form: $J_{(1)}=J_{\nu}dx^{\nu}$ and its Hodge dual: $\Theta_{(n)}=*J_{(1)}$. Having utilized infinitesimal diffeomorphism given by the vector $\delta x^{\mu}=\xi^{\mu}$ one can define the form: $$\label{diff_forms}
J_{(n)}=\Theta_{(n)}-i_{\xi}*{\cal L}=e.o.m.-d*J_{(2)},$$ and here $i_{\xi}*{\cal L}$ denotes contraction of the vector field $\xi^{\mu}$ with the dual of the form $*{\cal L}$. In case the equations of motion are fulfilled (on-shell condition) we can conclude that the form $J_{(n)}$ is exact, namely $J_{(n)}=dQ_{(n-1)}$, where $Q_{(n-1)}=-*J_{(2)}$. The written above relation (\[diff\_forms\]) allows to obtain thermodynamic relations, namely the first law of black hole thermodynamics if the infinitesimal diffeomorphism vector $\xi^{\mu}$ is taken to be a Killing vector null at the horizon. It was shown by Wald that the variation of gravitational Hamiltonian can be represented in the form: $$\label{var_hamilt}
\delta{\cal H}=\frac{1}{16\pi}\left(\delta \int_{c}J_{(n)}-\int_{c}d(i_{\xi}\Theta_{(n)})\right)=\frac{1}{16\pi}\int_{\Sigma_{(n-1)}}(\delta Q-i_{\xi}\Theta_{(n)}),$$ where $c$ denotes $n$–dimensional Cauchy surface and $\Sigma_{(n-1)}$ is its $n-1$–dimensional boundary which consist of two parts: one on the event horizon and the other at the infinity. The first law of black hole thermodynamics can be obtained from the relation: $$\label{variat_inf_hor}
\delta{\cal H}_{\infty}=\delta{\cal H}_{+},$$ we note that the in the latter relation the variation in the left hand side is taken at the infinity and in the right hand side is taken at the events horizon.
Using written above relations we can calculate the variation of Hamiltonian and consequently derive the first law. For minimally coupled part of the action with Born-Infeld term we can write: $$\label{var_min}
(\delta Q-i_{\xi}\Theta)_{min}=r^{n-1}\sqrt{UW}\left(\frac{(n-1)}{rW^2}\delta W+\frac{2}{UW}\frac{1}{\left(1-\frac{(\psi')^2}{\beta^2UW}\right)^{\frac{3}{2}}}\left[\psi'\psi\left(\frac{\delta U}{U}+\frac{\delta W}{W}\right)-2\psi\delta\psi'\right]-\frac{\al\vp'}{W}\delta\vp\right)\Omega_{(n-1)},$$ here $\Omega_{(n-1)}$ denotes surface $n-1$–form. For nonminimally coupled part of the action we obtain: $$\label{var_nm}
(\delta Q-i_{\xi}\Theta)_{nm}=\frac{\eta(n-1)}{2}r^{n-2}\sqrt{\frac{U}{W}}\left(\frac{(\vp')^2}{2W^2}\delta W-\delta\left(\frac{(\vp')^2}{W}\right)+\frac{2\al r}{(n-1)\e}\vp'\delta\vp\right)\Omega_{(n-1)}.$$ Having combined latter two relations we can obtain total variation: $$\begin{aligned}
\label{var_tot}
\nonumber(\delta Q-i_{\xi}\Theta)_{tot}=r^{n-1}\sqrt{UW}\left((n-1)\left(1+\frac{\e(\vp')^2}{4W}\right)\frac{\delta W}{rW^2}+\frac{2}{UW}\frac{1}{\left(1-\frac{(\psi')^2}{\beta^2UW}\right)^{\frac{3}{2}}}\times\right.\\\left.\left[\psi'\psi\left(\frac{\delta U}{U}+\frac{\delta W}{W}\right)-2\psi\delta\psi'\right]-\frac{\eta(n-1)}{2rW}\delta\left(\frac{(\vp')^2}{W}\right)\right)\Omega_{(n-1)}.\end{aligned}$$ We note that in the limit $\beta\rightarrow\infty$ the given above relation is reduced to the corresponding relation for standard linear Maxwell field [@Feng_PRD16]. Using the written above relation (\[var\_tot\]) we can calculate total variation at the infinity as well as on the horizon. As a result at the infinity we write: $$\label{var_inf}
(\delta Q-i_{\xi}\Theta)_{tot}=((n-1)\delta\mu+4\psi_0\delta q)\Omega_{(n-1)}.$$ The obtained relation is very simple and completely coincides with corresponding relation obtained for linear Maxwell field [@Feng_PRD16]. Taking into account relations for the total charge and gauge potential and performing integration over angular variables we can obtain relation for the variation of the gravitational Hamiltonian at the infinity $\delta{\cal H}_{\infty}$: $$\label{H_inf_calc}
\delta{\cal H}_{\infty}=\delta M-\Phi_q\delta Q$$ where $\delta M$ is the variation of the black hole’s mass which can be written in the form: $$\label{BH_mass}
M=\frac{(n-1)\omega_{n-1}}{16\pi}\mu$$ It should be pointed out that for non-spherical topology of horizon the relation (\[H\_inf\_calc\]) should be treated as the variation per unit volume. We would also like to emphasize that obtained relation for the mass (\[BH\_mass\]) takes completely the same form as in case of standard General Relativity.
Taking the variation of the Hamiltonian at the horizon we arrive at the relation: $$\label{TD_diff}
\delta {\cal H}_{+}=\frac{(n-1)\omega_{n-1}}{16\pi}U\rq{}(r_+)r^{n-2}_{+}\delta r_{+}=\sqrt{U(r_+)W(r_+)}T\delta\left(\frac{{\cal A}}{4}\right)=\left(1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\Big|_{r_+}\right)T\delta\left(\frac{{\cal A}}{4}\right).$$ where ${\cal A}=\omega_{n-1}r^{n-1}_+$ is the horizon area of the black hole. Here we note that variation of the Hamiltonian at the horizon does not include a contribution from the gauge field due to the fact that gauge potential equals to zero at the horizon. Latter relation for the variation of the Hamiltonian at the horizon can’t be represented in the form $T\delta S$ which takes place in the standard General Relativity, in other words the form in the right hand side of the relation (\[TD\_diff\]) is not exact, this fact was noted in [@Feng_PRD16] and a specific procedure was proposed to derive a relation for black hole’s entropy. To write the first law of black hole mechanics it was proposed to introduce specific “scalar charge”, related to the scalar field [@Feng_PRD16]. But as it was shown in our earlier works [@Stetsko_slr; @Stetsko_PRD19] the “scalar charges” can be chosen in different way, and here we take the same form for them as in our previous works and rewrite the latter relation in the form: $$\label{var_horizon}
\delta{\cal H}_{+}=T\delta S+\Phi^{+}_{\vp}\delta Q^{+}_{\vp},$$ where $S$ is the entropy of the black hole, $Q^{+}_{\vp}$ and $\Phi^{+}_{\vp}$ denote introduced “scalar charge” and related to it potential. These introduced values can be chosen in the form: $$\label{entropy}
S=\left(1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\Big|_{r_+}\right)\frac{{\cal A}}{4},$$ $$\label{sc_pot}
Q^{+}_{\vp}=\omega_{n-1}\sqrt{1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\Big|_{r_+}}, \quad \Phi^{+}_{\vp}=-\frac{{\cal A}T}{2\omega_{n-1}}\sqrt{1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\Big|_{r_+}}.$$ As we mentioned above the scalar “charge” $Q^{+}_{\vp}$ and conjugate potential $\Psi^{+}_{\vp}$ might be defined in other way, but in our case it allows to derive relation between the temperature and entropy from one side and introduced scalar “charge” and potential from the other one: $$\label{rel_entr_pot}
\Phi^{+}_{\vp}Q^{+}_{\vp}=-\frac{{\cal A}T}{2}\left(1+\frac{\e}{4}\frac{(\vp\rq{})^2}{W}\Big|_{r_+}\right)=-2TS$$
Using relations (\[variat\_inf\_hor\]) and corresponding results for the variations at the horizon and at the infinity we can write the first law in the following form: $$\label{first_law}
\delta M=T\delta S+\Phi^{+}_{\vp}\delta Q^{+}_{\vp}+\Phi_{q}\delta Q$$ The obtained relation (\[first\_law\]) has very simple form, similar to corresponding relation in standard General Relativity, but nevertheless the introduced definition of the entropy is not supported by some independent way of calculation. It should be pointed out that several attempts to calculate entropy with help of Euclidean methods were made, but there they were related mainly to planar geometry [@Caceres_JHEP17] or chargeless case. It should be pointed out here that accurate application of Euclidean action requires properly regularized and renormalized action, this regularization means that we take into account Gibbons-Hawking-York boundary term (\[GHY\_nm\]) and renormalization or the regularized action should be performed to make the action finite, since we deal with asymptotically AdS solutions, but this issue is a subject of independent investigation and will be performed elsewhere.
Conclusions
===========
In this work we consider particular case of general Horndeski gravity, namely the theory with nonminimal derivative coupling and we also take into account gauge filed minimally coupled just to gravity sector and given by a Lagrangian of Born-Infeld type. We obtain static solutions which represent black holes. Since the cosmological constant is taken into account it allowed us to consider not only spherically symmetric solution, but also to obtain topological solutions with nonspherical horizon, namely with flat ($\ve=0$) and hyperbolic ($\ve=-1$) ones.
In general the structure of the obtained solutions is complicated, but nevertheless they share some common features with black holes’ solutions derived in the framework of General Relativity as well as in Horndeski Gravity. Firstly, all the obtained solutions have AdS-like behaviour at large distances, because of the presence of the cosmological constant $\L$ and due to the influence of the scalar field, but in contrast with the ordinary General Relativity, where the metric at large distances is completely defined by a term with bare cosmological constant $\Lambda$ in our case we have some effective constant which is defined by the bare one and coupling constants for the scalar field $\al$ and $\eta$. The behaviour of the metric for very small distances ($r\rightarrow 0$) depends on the type of topology as well as on the dimension of space. Namely, when $\ve\neq 0$ there are three different types of the behaviour of the metric function $U(r)$, for $n=3$, $n=4$ and $n\geqslant 5$, what is demonstrated by the relations (\[u\_r0\_3\]), (\[u\_r0\_c4\]) and (\[u\_r0\_c\]) respectively, whereas for the flat horizon solution ($\ve=0$) the character of the function $U(r)$ for all $n\geqslant 3$ is the same (\[u\_r0\_f\]). We also stress here that the leading terms for flat geometry ($\ve=0$) as well as nonflat geometry $\ve\neq 0$ if $n\geqslant 4$ is completely defined by the gauge field term and for the particular case $\ve\neq 0$ and $n=3$ the dominant term is of Schwarzschild type. In addition we would like to emphasize that the Kretschmann scalar (\[Kr\_scalar\]), which defines the character of singularity at the origin for the flat case $\ve=0$ and for the nonflat one $\ve\neq 0$ if $n\geqslant 5$ for small distances show the dependence $\sim 1/r^4$ and does not depend on the charge $q$, the same situation takes place for linear and power-law fields [@Stetsko_PRD19]. For $n=4$ and $\ve\neq 0$ the Kretshmann scalar has additional peculiarity of a logarithmic character (\[Kr\_sc\_4\_c\]) and for the case $n=3$ and $\ve\neq 0$ due to domination of the Schwarzschild term the Kretshmann scalar has completely different behaviour (\[Kr\_sc\_n3\]). We also obtained and examined the relations for the gauge field and gauge potential and it was shown that for $n=3$ and $\ve\neq 0$ the field and potential are nonsingular at the origin, whereas for other considered dimensions and types of geometry they are singular at this point. We point out here that in standard General Relativity with Born-Infeld field where the field and potential are nonsingular at all distances for all dimensions.
We have also examined some aspects of black hole’s thermodynamics. First of all we have obtained relation for the temperature (\[temp\_BI\_gen\]) and we show that in the limit $\b\rightarrow\infty$ we recover the relation (\[temp\_lin\_f\]) which was derived in our previous work [@Stetsko_PRD19]. Careful analysis of the obtained relation shows that for large radius of the horizon $r_+$ the temperature increases almost linearly due to domination of AdS-term in this case. For small radius of horizon the situation is completely different, namely the leading term in this case is related to the gauge field, but it should be pointed out here that in case of finite $\b$ the character of dependence $T(r_+)$ is of the type $\sim \b r^{2-n}_+$, whereas for linear field one arrives at the asymptotic $\sim r^{3-2n}_+$ this difference can be explained by the fact that Born-Infeld gauge field has more moderate dependence of the $r_+$ than the linear one. To obtain the first law of black hole thermodynamics we have utilized Wald’s approach which is applicable to general diffeomorphism-invariant theories. Wald’s method is well-posed, but nevertheless the definition of entropy is not an easy task, to introduce the entropy we followed the approach suggested in [@Feng_PRD16] and used in our work [@Stetsko_PRD19] where additional “scalar charges” were introduced. Here we point out that the choice we made to introduce “scalar charge” is not unique, in the paper [@Feng_PRD16] it was taken in a bit different form, this ambiguity and the fact that we do not have a corresponding “charge” related to the scalar field which appears due to integration of equations of motion makes this final step in Wald’s procedure a bit unsatisfactory. To make this step well grounded we should have independent approach to define entropy and as a consequence to write the first law. As it is known the relation for entropy can be derived with help of Euclidean approach. In order to use Euclidean approach one should renormalize the total gravity action (\[action\]) to make it finite at infinity, we also point out here that several attempts to use Euclidean techniques were made, but they were mainly related to uncharged black hole [@Minamitsuji_PRD14] or black holes with flat topology of horizon [@Caceres_JHEP17].
Acknowledgments
===============
This work was partly supported by Project FF-83F (No. 0119U002203) from the Ministry of Education and Science of Ukraine.
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[^1]: E-mail: mstetsko@gmail.com
|
---
abstract: 'The instanton Floer homology of a knot in $S^{3}$ is a vector space with a canonical mod $2$ grading. It carries a distinguished endomorphism of even degree, arising from the $2$-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces of this endomorphism. We show that the Euler characteristics of these generalized eigenspaces are the coefficients of the Alexander polynomial of the knot. Among other applications, we deduce that instanton homology detects fibered knots.'
address: |
Harvard University, Cambridge MA 02138\
Massachusetts Institute of Technology, Cambridge MA 02139
author:
- 'P. B. Kronheimer and T. S. Mrowka [^1]'
bibliography:
- 'alexander.bib'
title: Instanton Floer homology and the Alexander polynomial
---
=1
Introduction
============
For a knot $K\subset S^{3}$, the authors defined in [@KM-sutures] a Floer homology group $\KHI(K)$, by a slight variant of a construction that appeared first in [@Floer-Durham-paper]. In brief, one takes the knot complement $S^{3} \setminus N^{\circ}(K)$ and forms from it a closed $3$-manifold $Z(K)$ by attaching to $\partial N(K)$ the manifold $F\times S^{1}$, where $F$ is a genus-1 surface with one boundary component. The attaching is done in such a way that $\{\text{point}\}\times S^{1}$ is glued to the meridian of $K$ and $\partial F \times \{\text{point}\}$ is glued to the longitude. The vector space $\KHI(K)$ is then defined by applying Floer’s instanton homology to the closed 3-manifold $Z(K)$. We will recall the details in section \[sec:background\]. If $\Sigma$ is a Seifert surface for $K$, then there is a corresponding closed surface $\bar\Sigma$ in $Z(K)$, formed as the union of $\Sigma$ and one copy of $F$. The homology class $\bar\sigma=[\bar\Sigma]$ in $H_{2}(Z(K))$ determines an endomorphism $\mu(\bar\sigma)$ on the instanton homology of $Z(K)$, and hence also an endomorphism of $\KHI(K)$. As was shown in [@KM-sutures], and as we recall below, the generalized eigenspaces of $\mu(\bar\sigma)$ give a direct sum decomposition, $$\label{eq:eigenspace-decomposition}
\KHI(K) = \bigoplus_{j=-g}^{g} \KHI(K,j).$$ Here $g$ is the genus of the Seifert surface. In this paper, we will define a canonical $\Z/2$ grading on $\KHI(K)$, and hence on each $\KHI(K,j)$, so that we may write $$\KHI(K,j) = \KHI_{0}(K,j) \oplus \KHI_{1}(K,j).$$ This allows us to define the Euler characteristic $\chi(\KHI(K,j))$ as the difference of the ranks of the even and odd parts. The main result of this paper is the following theorem.
\[thm:main\] For any knot in $S^{3}$, the Euler characteristics $\chi(\KHI(K,j))$ of the summands $\KHI(K,j)$ are minus the coefficients of the symmetrized Alexander polynomial $\Delta_{K}(t)$, with Conway’s normalization. That is, $$\Delta_{K}(t) = - \sum_{j} \chi(\KHI(K,j)) t^{j}.$$
The Floer homology group $\KHI(K)$ is supposed to be an “instanton” counterpart to the Heegaard knot homology of Ozsváth-Szabó and Rasmussen [@Ozsvath-Szabo-knotfloer; @Rasmussen-thesis]. It is known that the Euler characteristic of Heegaard knot homology gives the Alexander polynomial; so the above theorem can be taken as further evidence that the two theories are indeed closely related.
![\[fig:Oriented-Skein\] Knots $K_{+}$, $K_{-}$ and $K_{0}$ differing at a single crossing.](Oriented-Skein)
The proof of the theorem rests on Conway’s skein relation for the Alexander polynomial. To exploit the skein relation in this way, we first extend the definition of $\KHI(K)$ to links. Then, given three oriented knots or links $K_{+}$, $K_{-}$ and $K_{0}$ related by the skein moves (see Figure \[fig:Oriented-Skein\]), we establish a long exact sequence relating the instanton knot (or link) homologies of $K_{+}$, $K_{-}$ and $K_{0}$. More precisely, if for example $K_{+}$ and $K_{-}$ are knots and $K_{0}$ is a $2$-component link, then we will show that there is along exact sequence $$\cdots \to \KHI(K_{+}) \to \KHI(K_{-}) \to \KHI(K_{0}) \to \cdots .$$ (This situation is a little different when $K_{+}$ and $K_{-}$ are $2$-component links and $K_{0}$ is a knot: see Theorem \[thm:skein\].)
Skein exact sequences of this sort for $\KHI(K)$ are not new. The definition of $\KHI(K)$ appears almost verbatim in Floer’s paper [@Floer-Durham-paper], along with outline proofs of just such a skein sequence. See in particular part ($2'$) of Theorem 5 in [@Floer-Durham-paper], which corresponds to Theorem \[thm:skein\] in this paper. The material of Floer’s paper [@Floer-Durham-paper] is also presented in [@Braam-Donaldson]. The proof of the skein exact sequence which we shall describe is essentially Floer’s argument, as amplified in [@Braam-Donaldson], though we shall present it in the context of sutured manifolds. The new ingredient however is the decomposition of the instanton Floer homology, without which one cannot arrive at the Alexander polynomial.
The structure of the remainder of this paper is as follows. In section \[sec:background\], we recall the construction of instanton knot homology, as well as instanton homology for sutured manifolds, following [@KM-sutures]. We take the opportunity here to extend and slightly generalize our earlier results concerning these constructions. Section \[sec:skein\] presents the proof of the main theorem. Some applications are discussed in section \[sec:applications\]. The relationship between $\Delta_{K}(t)$ and the instanton homology of $K$ was conjectured in [@KM-sutures], and the result provides the missing ingredient to show that the $\KHI$ detects fibered knots. Theorem \[thm:main\] also provides a lower bound for the rank of the instanton homology group:
\[cor:alexander-vs-rank\] If the Alexander polynomial of $K$ is $\sum_{-d}^{d}a_{j} t^{j}$, then the rank of $\KHI(K)$ is not less than $\sum_{-d}^{d}|a_{j}|$.
The corollary can be used to draw conclusions about the existence of certain representations of the knot group in $\SU(2)$.
##### Acknowledgment.
As this paper was being completed, the authors learned that essentially the same result has been obtained simultaneously by Yuhan Lim [@Lim]. The authors are grateful to the referee for pointing out the errors in an earlier version of this paper, particularly concerning the mod $2$ gradings.
Background {#sec:background}
==========
Instanton Floer homology
------------------------
Let $Y$ be a closed, connected, oriented $3$-manifold, and let $w\to Y$ be a hermitian line bundle with the property that the pairing of $c_{1}(w)$ with some class in $H_{2}(Y)$ is odd. If $E\to Y$ is a $U(2)$ bundle with $\Lambda^{2}E \cong w$, we write $\bonf(Y)_{w}$ for the space of $\PU(2)$ connections in the adjoint bundle $\ad(E)$, modulo the action of the gauge group consisting of automorphisms of $E$ with determinant $1$. The instanton Floer homology group $I_{*}(Y)_{w}$ is the Floer homology arising from the Chern-Simons functional on $\bonf(Y)_{w}$. It has a relative grading by $\Z/8$. Our notation for this Floer group follows [@KM-sutures]; an exposition of its construction is in [@Donaldson-book]. We will always use complex coefficients, so $I_{*}(Y)_{w}$ is a complex vector space.
If $\sigma$ is a $2$-dimensional integral homology class in $Y$, then there is a corresponding operator $\mu(\sigma)$ on $I_{*}(Y)_{w}$ of degree $-2$. If $y\in Y$ is a point representing the generator of $H_{0}(Y)$, then there is also a degree-$4$ operator $\mu(y)$. The operators $\mu(\sigma)$, for $\sigma\in H_{2}(Y)$, commute with each other and with $\mu(y)$. As shown in [@KM-sutures] based on the calculations of [@Munoz], the simultaneous eigenvalues of the commuting pair of operators $(\mu(y),\mu(\sigma))$ all have the form $$\label{eq:eigenvalue-pairs}
(2, 2k) \qquad\text{or}\qquad (-2, 2k\sqrt{-1}),$$ for even integers $2k$ in the range $$| 2k | \le |\sigma|.$$ Here $|\sigma|$ denotes the Thurston norm of $\sigma$, the minimum value of $-\chi(\Sigma)$ over all aspherical embedded surfaces $\Sigma$ with $[\Sigma]=\sigma$.
Instanton homology for sutured manifolds
----------------------------------------
We recall the definition of the instanton Floer homology for a balanced sutured manifold, as introduced in [@KM-sutures] with motivation from the Heegaard counterpart defined in [@Juhasz-1]. The reader is referred to [@KM-sutures] and [@Juhasz-1] for background and details.
Let $(M,\gamma)$ be a balanced sutured manifold. Its oriented boundary is a union, $$\partial M = R_{+}(\gamma) \cup A(\gamma) \cup
(- R_{-}(\gamma))$$ where $A(\gamma)$ is a union of annuli, neighborhoods of the sutures $s(\gamma)$. To define the instanton homology group $\SHI(M,\gamma)$ we proceed as follows. Let $([-1,1]\times T,\delta)$ be a product sutured manifold, with $T$ a connected, oriented surface with boundary. The annuli $A(\delta)$ are the annuli $[-1,1]\times \partial T$, and we suppose these are in one-to-one correspondence with the annuli $A(\gamma)$. We attach this product piece to $(M,\gamma)$ along the annuli to obtain a manifold $$\label{eq:barM}
\bar{M} = M \cup \bigl( [-1,1]\times T \bigr).$$ We write $$\label{eq:boundary-barM}
\partial \bar{M} = \bar{R}_{+} \cup (-\bar{R}_{-}).$$ We can regard $\bar{M}$ as a sutured manifold (not balanced, because it has no sutures). The surface $\bar{R}_{+}$ and $\bar{R}_{-}$ are both connected and are diffeomorphic. We choose an orientation-preserving diffeomorphism $$h : \bar{R}_{+} \to \bar{R}_{-}$$ and then define $Z=Z(M,\gamma)$ as the quotient space $$Z = \bar{M}/\sim,$$ where $\sim$ is the identification defined by $h$. The two surfaces $\bar{R}_{\pm}$ give a single closed surface $$\bar{R}\subset Z.$$
We need to impose a side condition on the choice of $T$ and $h$ in order to proceed. We require that there is a closed curve $c$ in $T$ such that $\{1\}\times c$ and $\{-1\}\times c$ become non-separating curves in $\bar{R}_{+}$ and $\bar{R}_{-}$ respectively; and we require further that $h$ is chosen so as to carry $\{1\}\times c$ to $\{-1\}\times c$ by the identity map on $c$.
We say that $(Z,\bar{R})$ is an admissible closure of $(M,\gamma)$ if it arises in this way, from some choice of $T$ and $h$, satisfying the above conditions.
In [@KM-sutures Definition 4.2], there was an additional requirement that $\bar{R}_{\pm}$ should have genus $2$ or more. This was needed only in the context there of Seiberg-Witten Floer homology, as explained in section 7.6 of [@KM-sutures]. Furthermore, the notion of closure in [@KM-sutures] did not require that $h$ carry $\{1\}\times c$ to $\{-1\}\times c$, hence the qualification “admissible” in the present paper.
In an admissible closure, the curve $c$ gives rise to a torus $S^{1}\times c$ in $Z$ which meets $\bar{R}$ transversely in a circle. Pick a point $x$ on $c$. The closed curve $S^{1}\times \{x\}$ lies on the torus $S^{1}\times c$ and meets $\bar{R}$ in a single point. We write $$w \to Z$$ for a hermitian line bundle on $Z$ whose first Chern class is dual to $S^{1}\times\{x\}$. Since $c_{1}(w)$ has odd evaluation on the closed surface $\bar{R}$, the instanton homology group $I_{*}(Z)_{w}$ is well-defined. As in [@KM-sutures], we write $$I_{*}(Z|\bar{R})_{w} \subset I_{*}(Z)_{w}$$ for the simultaneous generalized eigenspace of the pair of operators $$(\mu(y),\mu(\bar{R}))$$ belonging to the eigenvalues $(2,2g-2)$, where $g$ is the genus of $\bar{R}$. (See .)
For a balanced sutured manifold $(M,\gamma)$, the instanton Floer homology group $\SHI(M,\gamma)$ is defined to be $I_{*}(Z|\bar{R})_{w}$, where $(Z,\bar{R})$ is any admissible closure of $(M,\gamma)$. .
It was shown in [@KM-sutures] that $\SHI(M,\gamma)$ is well-defined, in the sense that any two choices of $T$ or $h$ will lead to isomorphic versions of $\SHI(M,\gamma)$.
Relaxing the rules on $T$ {#subsec:disconnected-T}
-------------------------
As stated, the definition of $\SHI(M,\gamma)$ requires that we form a closure $(Z,\bar{R})$ using a *connected* auxiliary surface $T$. We can relax this condition on $T$, with a little care, and the extra freedom gained will be convenient in later arguments.
So let $T$ be a possibly disconnected, oriented surface with boundary. The number of boundary components of $T$ needs to be equal to the number of sutures in $(M,\gamma)$. We then need to choose an orientation-reversing diffeomorphism between $\partial T$ and $\partial R_{+}(\gamma)$, so as to be able to form a manifold $\bar{M}$ as in , gluing $[-1,1]\times \partial T$ to the annuli $A(\gamma)$. We continue to write $\bar{R}_{+}$, $\bar{R}_{-}$ for the “top” and “bottom” parts of the boundary of $\partial \bar{M}$, as at . Neither of these need be connected, although they have the same Euler number. We shall impose the following conditions.
1. On each connected component $T_{i}$ of $T$, there is an oriented simple closed curve $c_{i}$ such that the corresponding curves $\{1\}\times c_{i}$ and $\{-1\}\times c_{i}$ are both non-separating on $\bar{R}_{+}$ and $\bar{R}_{-}$ respectively.
2. \[item:T-condition-2\] There exists a diffeomorphism $h :
\bar{R}_{+}\to\bar{R}_{-}$ which carries $\{1\}\times c_{i}$ to $\{-1\}\times c_{i}$ for all $i$, as oriented curves.
3. There is a $1$-cycle $c'$ on $\bar{R}_{+}$ which intersects each curve $\{1\}\times c_{i}$ once.
We then choose any $h$ satisfying \[item:T-condition-2\] and use $h$ to identify the top and bottom, so forming a closed pair $(Z,\bar{R})$ as before. The surface $\bar{R}$ may have more than one component (but no more than the number of components of $T$). No component of $\bar{R}$ is a sphere, because each component contains a non-separating curve. We may regard $T$ as a codimension-zero submanifold of $\bar{R}$ via the inclusion of $\{1\}\times T$ in $\bar{R}_{+}$.
For each component $\bar{R}_{k}$ of $\bar{R}$, we now choose one corresponding component $T_{i_{k}}$ of $T\cap\bar{R}_{k}$. We take $w\to Z$ to be the complex line bundle with $c_{1}(w)$ dual to the sum of the circles $S^{1}\times \{x_{k}\}\subset S^{1}\times c_{i_{k}}$. Thus $c_{1}(w)$ evaluates to $1$ on each component $\bar{R}_{k}\subset\bar{R}$. We may then consider the instanton Floer homology group $I_{*}(Z|\bar{R})_{w}$.
\[lem:relaxed-independence\] Subject to the conditions we have imposed, the Floer homology group $I_{*}(Z|\bar{R})_{w}$ is independent of the choices made. In particular, $I_{*}(Z|\bar{R})_{w}$ is isomorphic to $\SHI(M,\gamma)$.
By a sequence of applications of the excision property of Floer homology [@Floer-Durham-paper; @KM-sutures], we shall establish that $I_{*}(Z|\bar{R})_{w}$ is isomorphic to $I_{*}(Z'|\bar{R}')_{w'}$, where the latter arises from the same construction but with a *connected* surface $T'$. Thus $I_{*}(Z'|\bar{R}')_{w'}$ is isomorphic to $\SHI(M,\gamma)$ by definition: its independence of the choices made is proved in [@KM-sutures].
We will show how to reduce the number of components of $T$ by one. Following the argument of [@KM-sutures section 7.4], we have an isomorphism $$\label{eq:u-to-w}
I_{*}(Z|\bar{R})_{w} \cong I_{*}(Z|\bar{R})_{u},$$ where $u\to Z$ is the complex line bundle whose first Chern class is dual to the cycle $c'\subset Z$. We shall suppose in the fist instance that at least one of $c_{i}$ or $c_{j}$ is non-separating in the corresponding component $T_{i}$ or $T_{j}$. Since $c_{1}(u)$ is odd on the $2$-tori $S^{1}\times c_{i}$ and $S^{1}\times c_{j}$, we can apply Floer’s excision theorem (see also [@KM-sutures Theorem 7.7]): we cut $Z$ open along these two $2$-tori and glue back to obtain a new pair $(Z' | \bar{R}')$, carrying a line bundle $u'$, and we have $$I_{*}(Z|\bar{R})_{u} \cong I_{*}(Z'|\bar{R}')_{u'}.$$ Reversing the construction that led to the isomorphism , we next have $$I_{*}(Z'|\bar{R}')_{u'} \cong I_{*}(Z'|\bar{R}')_{w'},$$ where the line bundle $w'$ is dual to a collection of circles $S^{1}\times\{x'_{k'}\}$, one for each component of $\bar{R}'$. The pair $(Z',\bar{R}')$ is obtained from the sutured manifold $(M,\gamma)$ by the same construction that led to $(Z,R)$, but with a surface $T'$ having one fewer components: the components $T_{i}$ and $T_{j}$ have been joined into one component by cutting open along the circles $c_{i}$ and $c_{j}$ and reglueing.
If both $c_{i}$ and $c_{j}$ are separating in $T_{i}$ and $T_{j}$ respectively, then the above argument fails, because $T'$ will have the same number of components as $T$. In this case, we can alter $T_{i}$ and $c_{i}$ to make a new $T'_{i}$ and $c'_{i}$, with $c'_{i}$ non-separating in $T'_{i}$. For example, we may replace $Z$ by the disjoint union $Z \amalg Z_{*}$, where $Z_{*}$ is a product $S^{1}\times T_{*}$, with $T_{*}$ of genus $2$. In the same manner as above, we can cut $Z$ along $S^{1}\times c_{i}$ and cut $Z_{*}$ along $S^{1}\times c_{*}$, and then reglue, interchanging the boundary components. The effect of this is to replace $T_{i}$ be a surface $T'_{i}$ of genus one larger. We can take $c'_{i}$ to be a non-separating curve on $T_{*}
\setminus c_{*}$.
Instanton homology for knots and links {#subsec:inst-homology-link}
--------------------------------------
Consider a link $K$ in a closed oriented $3$-manifold $Y$. Following Juhász [@Juhasz-1], we can associate to $(Y,K)$ a sutured manifold $(M,\gamma)$ by taking $M$ to be the link complement and taking the sutures $s(\gamma)$ to consist of two oppositely-oriented meridional curves on each of the tori in $\partial M$. As in [@KM-sutures], where the case of knots was discussed, we take Juhász’ prescription as a definition for the instanton knot (or link) homology of the pair $(Y,K)$:
We define the instanton homology of the link $K\subset Y$ to be the instanton Floer homology of the sutured manifold $(M,\gamma)$ obtained from the link complement as above. Thus, $$\KHI(Y,K) = \SHI(M,\gamma).$$
Although we are free to choose any admissible closure $Z$ in constructing $\SHI(M,\gamma)$, we can exploit the fact that we are dealing with a link complement to narrow our choices. Let $r$ be the number of components of the link $K$. Orient $K$ and choose a longitudinal oriented curve $l_{i}\subset \partial M$ on the peripheral torus of each component $K_{i}\subset K$. Let $F_{r}$ be a genus-1 surface with $r$ boundary components, $\delta_{1},\dots,\delta_{r}$. Form a closed manifold $Z$ by attaching $F_{r}\times S^{1}$ to $M$ along their boundaries: $$\label{eq:special-closure}
Z = (Y\setminus N^{o}(K))
\cup (F_{r}\times S^{1}).$$ The attaching is done so that the curve $p_{i}\times
S^{1}$ for $p_{i}\in \delta_{i}$ is attached to the meridian of $K_{i}$ and $\delta_{i}\times \{q\}$ is attached to the chosen longitude $l_{i}$. We can view $Z$ as a closure of $(M,\gamma)$ in which the auxiliary surface $T$ consists of $r$ annuli, $$T = T_{1}\cup \dots \cup T_{r}.$$ The two sutures of the product sutured manifold $[-1,1]\times T_{i}$ are attached to meridional sutures on the components of $\partial M$ corresponding to $K_{i}$ and $K_{i-1}$ in some cyclic ordering of the components. Viewed this way, the corresponding surface $\bar{R}\subset Z$ is the torus $$\bar{R} = \nu \times S^{1}$$ where $\nu\subset F_{r}$ is a closed curve representing a generator of the homology of the closed genus-1 surface obtained by adding disks to $F_{r}$. Because $\bar{R}$ is a torus, the group $I_{*}(Z|\bar{R})_{w}$ can be more simply described as the generalized eigenspace of $\mu(y)$ belonging to the eigenvalue $2$, for which we temporarily introduce the notation $I_{*}(Z)_{w,+2}$. Thus we can write $$\KHI(Y,K) = I_{*}(Z)_{w,+2}.$$
An important special case for us is when $K \subset Y$ is null-homologous in $Y$ with its given orientation. In this case, we may choose a Seifert surface $\Sigma$, which we regard as a properly embedded oriented surface in $M$ with oriented boundary a union of longitudinal curves, one for each component of $K$. When a Seifert surface is given, we have a *uniquely preferred* closure $Z$, obtained as above but using the longitudes provided by $\partial
\Sigma$. Let us fix a Seifert surface $\Sigma$ and write $\sigma$ for its homology class in $H_{2}(M,\partial M)$. The preferred closure of the sutured link complement is entirely determined by $\sigma$.
The decomposition into generalized eigenspaces
----------------------------------------------
We continue to suppose that $\Sigma$ is a Seifert surface for the null-homologous oriented knot $K\subset Y$. We write $(M,\gamma)$ for the sutured link complement and $Z$ for the preferred closure.
The homology class $\sigma = [\Sigma]$ in $H_{2}(M,\partial M)$ extends to a class $\bar\sigma = [\bar\Sigma]$ in $H_{2}(Z)$: the surface $\bar\Sigma$ is formed from the Seifert surface $\Sigma$ and $F_{r}$, $$\bar\Sigma = \Sigma\cup F_{r}.$$ The homology class $\bar\sigma$ determines an endomorphism $$\mu(\bar\sigma) : I_{*}(Z)_{w,+2} \to I_{*}(Z)_{w,+2}.$$ This endomorphism is traceless, a consequence of the relative $\Z/8$ grading: there is an endomorphism $\epsilon$ of $I_{*}(Z)_{w}$ given by multiplication by $(\sqrt{-1})^{s}$ on the part of relative grading $s$, and this $\epsilon$ commutes with $\mu(y)$ and anti-commutes with $\mu(\bar\sigma)$. We write this traceless endomorphism as $$\label{eq:mu-o}
\mu^{o}(\sigma) \in \sl( \KHI(Y,K)).$$ Our notation hides the fact that the construction depends (a priori) on the existence of the preferred closure $Z$, so that $\KHI(Y,K)$ can be canonically identified with $I_{*}(Z)_{w,+2}$.
It now follows from [@KM-sutures Proposition 7.5] that the eigenvalues of $\mu^{o}(\sigma)$ are even integers $2j$ in the range $-2\bar{g}+2 \le 2j \le 2\bar{g}-2$, where $\bar{g}=g(\Sigma)+r $ is the genus of $\bar{\Sigma}$. Thus:
For a null-homologous oriented link $K\subset Y$ with a chosen Seifert surface $\Sigma$, we write $$\KHI(Y,K,[\Sigma], j)
\subset \KHI(Y,K)$$ for the generalized eigenspace of $\mu^{o}([\Sigma])$ belonging to the eigenvalue $2j$, so that $$\KHI(Y,K) =
\bigoplus_{j=-g(\Sigma)+1-r}^{g(\Sigma)-1+r}
\KHI(Y,K,[\Sigma], j),$$ where $r$ is the number of components of $K$. If $Y$ is a homology sphere, we may omit $[\Sigma]$ from the notation; and if $Y$ is $S^{3}$ then we simply write $\KHI(K,j)$.
The authors believe that, for a general sutured manifold $(M,\gamma)$, one can define a unique linear map $$\mu^{o} : H_{2}(M,\partial M) \to \sl ( \SHI(M,\gamma))$$ characterized by the property that for any admissible closure $(Z,\bar{R})$ and any $\bar{\sigma}$ in $H_{2}(Z)$ extending $\sigma \in H_{2}(M,\partial M)$ we have $$\mu^{o}(\sigma) = \text{traceless part of $\mu(\bar\sigma)$},$$ under a suitable identification of $I_{*}(Z| \bar{R})_{w}$ with $\SHI(M,\gamma)$. The authors will return to this question in a future paper. For now, we are exploiting the existence of a preferred closure $Z$ so as to side-step the issue of whether $\mu^{o}$ would be well-defined, independent of the choices made.
The mod 2 grading {#subsec:mod-2-grading}
-----------------
If $Y$ is a closed $3$-manifold, then the instanton homology group $I_{*}(Y)_{w}$ has a canonical decomposition into parts of even and odd grading mod $2$. For the purposes of this paper, we normalize our conventions so that the two generators of $I_{*}(T^{3})_{w}=\C^{2}$ are in *odd* degree. As in [@KM-book section 25.4 ], the canonical mod $2$ grading is then essentially determined by the property that, for a cobordism $W$ from a manifold $Y_{-}$ to $Y_{+}$, the induced map on Floer homology has even or odd grading according to the parity of the integer $$\label{eq:iota-W} \iota(W) = \frac{1}{2}
\Bigl( \chi(W) + \sigma(W) +
b_1(Y_+) - b_0(Y_+) - b_1(Y_-) + b_0(Y_-)\Bigr).$$ (In the case of connected manifolds $Y_{+}$ and $Y_{-}$, this formula reduces to the one that appears in [@KM-book] for the monopole case. There is more than one way to extend the formula to the case of disconnected manifolds, and we have simply chosen one.) By declaring that the generators for $T^{3}$ are in odd degree, we ensure that the canonical mod $2$ gradings behave as expected for disjoint unions of the $3$-manifolds. Thus, if $Y_{1}$ and $Y_{2}$ are the connected components of a $3$-manifold $Y$ and $\alpha_{1}\otimes \alpha_{2}$ is a class on $Y$ obtained from $\alpha_{i}$ on $Y_{i}$, then $\gr(\alpha_{1}\otimes
\alpha_{2})$ is $\gr(\alpha_{1}) + \gr(\alpha_{2})$ in $\Z/2$ as expected.
Since the Floer homology $\SHI(M,\gamma)$ of a sutured manifold $(M,\gamma)$ is defined in terms of $I_{*}(Z)_{w}$ for an admissible closure $Z$, it is tempting to try to define a canonical mod $2$ grading on $\SHI(M,\gamma)$ by carrying over the canonical mod $2$ grading from $Z$. This does not work, however, because the result will depend on the choice of closure. This is illustrated by the fact that the mapping torus of a Dehn twist on $T^{2}$ may have Floer homology in *even* degree in the canonical mod $2$ grading (depending on the sign of the Dehn twist), despite the fact that both $T^{3}$ and this mapping torus can be viewed as closures of the same sutured manifold.
We conclude from this that, without auxiliary choices, there is no *canonical* mod $2$ grading on $\SHI(M,\gamma)$ in general: only a relative grading. Nevertheless, in the special case of an oriented null-homologous knot or link $K$ in a closed $3$-manifold $Y$, we *can* fix a convention that gives an absolute mod $2$ grading, once a Seifert surface $\Sigma$ for $K$ is given. We simply take the preferred closure $Z$ described above in section \[subsec:inst-homology-link\], using $\partial\Sigma$ again to define the longitudes, so that $\KHI(Y,K)$ is identified with $I_{*}(Z)_{w,+2}$, and we use the canonical mod $2$ grading from the latter.
With this convention, the unknot $U$ has $\KHI(U)$ of rank $1$, with a single generator in odd grading mod $2$.
The skein sequence {#sec:skein}
==================
The long exact sequence
-----------------------
Let $Y$ be any closed, oriented $3$-manifold, and let $K_{+}$, $K_{-}$ and $K_{0}$ be any three oriented knots or links in $Y$ which are related by the standard skein moves: that is, all three links coincide outside a ball $B$ in $Y$, while inside the ball they are as shown in Figure \[fig:Oriented-Skein\]. There are two cases which occur here: the two strands of $K_{+}$ in $B$ may belong to the same component of the link, or to different components. In the first case $K_{0}$ has one more component than $K_{+}$ or $K_{-}$, while in the second case it has one fewer.
\[thm:skein\] Let $K_{+}$, $K_{-}$ and $K_{0}$ be oriented links in $Y$ as above. Then, in the case that $K_{0}$ has one more component than $K_{+}$ and $K_{-}$, there is a long exact sequence relating the instanton homology groups of the three links, $$\label{eq:skein-first}
\cdots\to \KHI(Y,K_{+}) \to \KHI(Y,K_{-}) \to \KHI(Y,K_{0}) \to
\cdots.$$ In the case that $K_{0}$ has fewer components that $K_{+}$ and $K_{-}$, there is a long exact sequence $$\label{eq:skein-second}
\cdots\to \KHI(Y,K_{+}) \to \KHI(Y,K_{-}) \to
\KHI(Y,K_{0})\otimes V^{\otimes 2} \to
\cdots$$ where $V$ a 2-dimensional vector space arising as the instanton Floer homology of the sutured manifold $(M,\gamma)$, with $M$ the solid torus $S^{1}\times D^{2}$ carrying four parallel sutures $S^{1}\times \{p_{i}\}$ for four points $p_{i}$ on $\partial D^{2}$ carrying alternating orientations.
Let $\lambda$ be a standard circle in the complement of $K_{+}$ which encircles the two strands of $K_{+}$ with total linking number zero, as shown in Figure \[fig:K-plus-with-lambda\].
![\[fig:K-plus-with-lambda\] The knot $K_{+}$, with a standard circle $\lambda$ around a crossing, with linking number zero.](K-plus-with-lambda)
Let $Y_{-}$ and $Y_{0}$ be the $3$-manifolds obtained from $Y$ by $-1$-surgery and $0$-surgery on $\lambda$ respectively. Since $\lambda$ is disjoint from $K_{+}$, a copy of $K_{+}$ lies in each, and we have new pairs $(Y_{-1},K_{+})$ and $(Y_{0},K_{+})$. The pair $(Y_{-1},K_{+})$ can be identified with $(Y,K_{-})$.
![\[fig:Skein-Tubes\] Sutured manifolds obtained from the knot complement, related by a surgery exact triangle.](Skein-Tubes)
Let $(M_{+},\gamma_{+})$, $(M_{-},\gamma_{-})$ and $(M_{0},\gamma_{0})$ be the sutured manifolds associated to the links $(Y,K_{+})$, $(Y,K_{-})$ and $(Y_{0},K_{0})$ respectively: that is, $M_{+}$, $M_{-}$ and $M_{0}$ are the link complements of $K_{+}\subset Y$, $K_{-}\subset Y$ and $K_{0}\subset Y_{0}$ respectively, and there are two sutures on each boundary component. (See Figure \[fig:Skein-Tubes\].) The sutured manifolds $(M_{-},\gamma_{-})$ and $(M_{0}, \gamma_{0})$ are obtained from $(M_{+},\gamma_{+})$ by $-1$-surgery and $0$-surgery respectively on the circle $\lambda\subset M_{+}$. If $(Z,\bar{R})$ is any admissible closure of $(M_{+},\gamma_{+})$ then surgery on $\lambda\subset Z$ yields admissible closures for the other two sutured manifolds. From Floer’s surgery exact triangle [@Braam-Donaldson], it follows that there is a long exact sequence $$\label{eq:SHI-long-exact}
\cdots\to \SHI(M_{+},\gamma_{+}) \to
\SHI(M_{-},\gamma_{-}) \to
\SHI(M_{0},\gamma_{0}) \to
\cdots$$ in which the maps are induced by surgery cobordisms between admissible closures of the sutured manifolds.
By definition, we have $$\begin{aligned}
\SHI(M_{+},\gamma_{+}) &= \KHI(Y,K_{+}) \\
\SHI(M_{-},\gamma_{-}) &= \KHI(Y,K_{-}) .
\end{aligned}$$
![\[fig:Decompose-M0\] Decomposing $M_{0}$ along a product annulus to obtain a link complement in $S^{3}$.](Decompose-M0)
However, the situation for $(M_{0}, \gamma_{0})$ is a little different. The manifold $M_{0}$ is obtained by zero-surgery on the circle $\lambda$ in $M_{+}$, as indicated in Figure \[fig:Skein-Tubes\]. This sutured manifold contains a product annulus $S$, consisting of the union of the twice-punctured disk shown in Figure \[fig:Decompose-M0\] and a disk $D^{2}$ in the surgery solid-torus $S^{1}\times D^{2}$. As shown in the figure, sutured-manifold decomposition along the annulus $S$ gives a sutured manifold $(M'_{0},\gamma'_{0})$ in which $M'_{0}$ is the link complement of $K_{0}\subset Y$: $$(M_{0},\gamma_{0}) \decomp{S} (M'_{0}, \gamma'_{0}).$$ By Proposition 6.7 of [@KM-sutures] (as adapted to the instanton homology setting in section 7.5 of that paper), we therefore have an isomorphism $$\SHI (M_{0},\gamma_{0})\cong \SHI (M'_{0},
\gamma'_{0}).$$
We now have to separate cases according to the number of components of $K_{+}$ and $K_{0}$. If the two strands of $K_{+}$ at the crossing belong to the same component, then every component of $\partial M'_{0}$ contains exactly two, oppositely-oriented sutures, and we therefore have $$\SHI (M'_{0},
\gamma'_{0}) = \KHI(Y, K_{0}).$$ In this case, the sequence becomes the sequence in the first case of the theorem.
![\[fig:Remove-Sutures\] Removing some extra sutures using a decomposition along a product annulus. The solid torus in the last step has four sutures.](Remove-Sutures)
If the two strands of $K_{+}$ belong to different components, then the corresponding boundary components of $M_{+}$ each carry two sutures. These two boundary components become one boundary component in $M'_{0}$, and the decomposition along $S$ introduces two new sutures; so the resulting boundary component in $M'_{0}$ carries six meridional sutures, with alternating orientations. Thus $(M'_{0}, \gamma'_{0})$ fails to be the sutured manifold associated to the link $K_{0}\subset Y$, on account of having four additional sutures. As shown in Figure \[fig:Remove-Sutures\] however, the number of sutures on a torus boundary component can always be reduced by $2$ (as long as there are at least four to start with) by using a decomposition along a separating annulus. This decomposition results in a manifold with one additional connected component, which is a solid torus with four longitudinal sutures. This operation needs to be performed twice to reduce the number of sutures in $M'_{0}$ by four, so we obtain two copies of this solid torus. Denoting by $V$ the Floer homology of this four-sutured solid-torus, we therefore have $$\SHI (M'_{0},
\gamma'_{0}) = \KHI(Y, K_{0})\otimes V\otimes V$$ in this case. Thus the sequence becomes the second long exact sequence in the theorem.
At this point, all that remains is to show that $V$ is $2$-dimensional, as asserted in the theorem. We will do this indirectly, by identifying $V\otimes V$ as a $4$-dimensional vector space. Let $(M_{4},\gamma_{4})$ be the sutured solid-torus with $4$ longitudinal sutures, as described above, so that $\SHI(M_{4},\gamma_{4})=V$. Let $(M,\gamma)$ be two disjoint copies of $(M_{4},\gamma_{4})$, so that $$\SHI(M,\gamma) = V\otimes V.$$
We can describe an admissible closure of $(M,\gamma)$ (with a disconnected $T$ as in section \[subsec:disconnected-T\]) by taking $T$ to be four annuli: we attach $[-1,1]\times T$ to $(M,\gamma)$ to form $\bar{M}$ so that $\bar{M}$ is $\Sigma\times
S^{1}$ with $\Sigma$ a four-punctured sphere. Thus $\partial\bar{M}$ consists of four tori, two of which belong to $\bar{R}_{+}$ and two to $\bar{R}_{-}$. The closure $(Y,\bar{R})$ is obtained by gluing the tori in pairs; and this can be done so that $Y$ has the form $\Sigma_{2}\times S^{1}$, where $\Sigma_{2}$ is now a closed surface of genus $2$. The surface $\bar{R}$ in $\Sigma_{2}\times S^{1}$ has the form $\gamma\times S^{1}$, where $\gamma$ is a union of two disjoint closed curves in independent homology classes. The line bundle $w$ has $c_{1}(w)$ dual to $\gamma'$, where $\gamma'$ is a curve on $\Sigma_{2}$ dual to one component of $\gamma$.
Thus we can identify $V\otimes V$ with the generalized eigenspace of $\mu(y)$ belonging to the eigenvalue $+2$ in the Floer homology $I_{*}(\Sigma_{2}\times S^{1})_{w}$, $$\label{eq:VVisSigma2}
V\otimes V = I_{*}(\Sigma_{2} \times S^{1})_{w,+2},$$ where $w$ is dual to a curve lying on $\Sigma_{2}$. Our next task is therefore to identify this Floer homology group. This was done (in slightly different language) by Braam and Donaldson [@Braam-Donaldson Proposition 1.15]. The main point is to identify the relevant representation variety in $\bonf(Y)_{w}$, for which we quote:
\[lem:Sigma2-calc\] For $Y=\Sigma_{2}\times S^{1}$ and $w$ as above, the critical-point set of the Chern-Simons functional in $\bonf(Y)_{w}$ consists of two disjoint $2$-tori. Furthermore, the Chern-Simons functional is of Morse-Bott type along its critical locus.
To continue the calculation, following [@Braam-Donaldson], it now follows from the lemma that $I_{*}(\Sigma_{2}\times S^{1})_{w}$ has dimension at most $8$ and that the even and odd parts of this Floer group, with respect to the relative mod 2 grading, have equal dimension: each at most $4$. On the other hand, the group $I_{*}(\Sigma_{2}\times S^{1}| \Sigma_{2})_{w}$ is non-zero. So the generalized eigenspaces belonging to the eigenvalue-pairs $((-1)^{r}2, i^{r}2)$, for $r=0,1,2,3$, are all non-zero. Indeed, each of these generalized eigenspaces is $1$-dimensional, by Proposition 7.9 of [@KM-sutures]. These four 1-dimensional generalized eigenspaces all belong to the same relative mod-2 grading. It follows that $I_{*}(\Sigma_{2}\times S^{1})_{w}$ is 8-dimensional, and can be identified as a vector space with the homology of the critical-point set. The generalized eigenspace belonging to $+2$ for the operator $\mu(y)$ is therefore $4$-dimensional; and this is $V\otimes V$. This completes the argument.
Tracking the mod 2 grading
--------------------------
Because we wish to examine the Euler characteristics, we need to know how the canonical mod 2 grading behaves under the maps in Theorem \[thm:skein\]. This is the content of the next lemma.
\[lem:mod-2-sequence\] In the situation of Theorem \[thm:skein\], suppose that the link $K_{+}$ is null-homologous (so that $K_{-}$ and $K_{0}$ are null-homologous also). Let $\Sigma_{+}$ be a Seifert surface for $K_{+}$, and let $\Sigma_{-}$ and $\Sigma_{0}$ be Seifert surfaces for the other two links, obtained from $\Sigma_{+}$ by a modification in the neighborhood of the crossing. Equip the instanton knot homology groups of these links with their canonical mod $2$ gradings, as determined by the preferred closures arising from these Seifert surfaces. Then in the first case of the two cases of the theorem, the map from $\KHI(Y,K_{-})$ to $\KHI(Y,K_{0})$ in the sequence has odd degree, while the other two maps have even degree, with respect to the canonical mod 2 grading.
In the second case, if we grade the 4-dimensional vector space $V\otimes V$ by identifying it with $I_{*}(\Sigma_{2}\times
S^{1})_{w,+2}$ as in , then the map from $\KHI(Y,K_{0})\otimes V^{\otimes 2}$ to $\KHI(Y,K_{+})$ in has odd degree, while the other two maps have even degree.
We begin with the first case. Let $Z_{+}$ be the preferred closure of the sutured knot complement $(M_{+},\gamma_{+})$ obtained from the knot $K_{+}$, as defined by . In the notation of the proof of Theorem \[thm:skein\], the curve $\lambda$ lies in $Z_{+}$. Let us write $Z_{-}$ and $Z_{0}$ for the manifolds obtained from $Z_{+}$ by $-1$-surgery and $0$-surgery on $\lambda$ respectively. It is a straightforward observation that $Z_{-}$ and $Z_{0}$ are respectively the preferred closures of the sutured complements of the links $K_{-}$ and $K_{0}$. The surgery cobordism $W$ from $Z_{+}$ to $Z_{-}$ gives rise to the map from $\KHI(Y,K_{+})$ to $\KHI(Y,K_{-})$. This $W$ has the same homology as the cylinder $[-1,1]\times Z_{+}$ blown up at a single point. The quantity $\iota(W)$ in is therefore even, and it follows that the map $$\KHI(Y,K_{+}) \to \KHI(Y,K_{-})$$ has even degree. The surgery cobordism $W_{0}$ induces a map $$\label{eq:second-cobordism}
I_{*}(Z_{-})_{w} \to I_{*}(Z_{0})_{w}$$ which has odd degree, by another application of . This concludes the proof of the first case.
In the second case of the theorem, we still have a long exact sequence $$\to I_{*}(Z_{+})_{w} \to I_{*}(Z_{-})_{w} \to
I_{*}(Z_{0})_{w} \to$$ in which the map $I_{*}(Z_{-})_{w} \to I_{*}(Z_{0})_{w}$ is odd and the other two are even. However, it is no longer true that the manifold $Z_{0}$ is the preferred closure of the sutured manifold obtained from $K_{0}$. The manifold $Z_{0}$ can be described as being obtained from the complement of $K_{0}$ by attaching $G_{r}\times S^{1}$, where $G_{r}$ is a surface of genus $2$ with $r$ boundary components. Here $r$ is the number of components of $K_{0}$, and the attaching is done as before, so that the curves $\partial G_{r}\times
\{q\}$ is attached to the longitudes and the curves $\{p_{i}\}\times S^{1}$ are attached to the meridians. The *preferred* closure, on the other hand, is defined using a surface $F_{r}$ of genus $1$, not genus $2$. We write $Z'_{0}$ for the preferred closure, and our remaining task is to compare the instanton Floer homologies of $Z_{0}$ and $Z'_{0}$, with their canonical $\Z/2$ gradings.
An application of Floer’s excision theorem provides an isomorphism $$I_{*}(Z_{0})_{w,+2} \to I_{*}(Z'_{0})_{w,+2} \otimes
I_{*}(\Sigma_{2}\times S^{1})_{w,+2}$$ where (as before) the class $w$ in the last term is dual to a non-separating curve in the genus-2 surface $\Sigma_{2}$.
![\[fig:F-and-G\] The surfaces $G_{r}$ and $F_{r} \amalg \Sigma_{2}$, used in constructing $Z_{0}$ and $Z'_{0}$ respectively.](F-and-G-new)
(See Figure \[fig:F-and-G\] which depicts the excision cobordism from $G_{r}\times S^{1}$ to $(F_{r}\amalg \Sigma_{2})\times
S^{1}$, with the $S^{1}$ factor suppressed.) The isomorphism is realized by an explicit cobordism $W$, with $\iota(W)$ odd, which accounts for the difference between the first and second cases and concludes the proof.
Tracking the eigenspace decomposition
-------------------------------------
The next lemma is similar in spirit to Lemma \[lem:mod-2-sequence\], but deals with eigenspace decomposition rather than the mod $2$ grading.
\[lem:eigenspace-sequence\] In the situation of Theorem \[thm:skein\], suppose again that the links $K_{+}$, $K_{-}$ and $K_{0}$ are null-homologous. Let $\Sigma_{+}$ be a Seifert surface for $K_{+}$, and let $\Sigma_{-}$ and $\Sigma_{0}$ be Seifert surfaces for the other two links, obtained from $\Sigma_{+}$ by a modification in the neighborhood of the crossing. Then in the first case of the two cases of the theorem, the maps in the long exact sequence intertwine the three operators $\mu^{o}([\Sigma_{+}])$, $\mu^{o}([\Sigma_{-}])$ and $\mu^{o}([\Sigma_{0}])$. In particular then, we have a long exact sequence $$\to \KHI(Y,K_{+},[\Sigma_{+}],j) \to
\KHI(Y,K_{-},[\Sigma_{-}],j) \to
\KHI(Y,K_{0},[\Sigma_{0}],j) \to$$ for every $j$.
In the second case of Theorem \[thm:skein\], the maps in the long exact sequence intertwine the operators $\mu^{o}([\Sigma_{+}])$ and $\mu^{o}([\Sigma_{-}])$ on the first two terms with the operator $$\mu^{o}([\Sigma_{0}]) \otimes 1 +
1 \otimes \mu([\Sigma_{2}])$$ acting on $$\KHI(Y,K_{0})\otimes I_{*}(\Sigma_{2}\times
S^{1})_{w,+2}\cong \KHI(Y,K_{0})\otimes V^{\otimes 2}.$$
The operator $\mu^{o}([\Sigma])$ on the knot homology groups is defined in terms of the action of $\mu([\bar\Sigma])$ for a corresponding closed surface $\bar\Sigma$ in the preferred closure of the link complement. The maps in the long exact sequences arise from cobordisms between the preferred closures. The lemma follows from the fact that the corresponding closed surfaces are homologous in these cobordisms.
Proof of the main theorem
-------------------------
For a null-homologous link $K \subset Y$ with a chosen Seifert surface $\Sigma$, let us write $$\begin{aligned}
\chi (Y,K,[\Sigma])&= \sum_{j}
\chi(\KHI(Y,K,[\Sigma],j))t^{j} \\
&= \sum_{j} \bigl ( \dim\KHI_{0}(Y,K,[\Sigma],j) -
\dim\KHI_{1}(Y,K,[\Sigma],j)\bigr) t^{j} \\
&= \str( t^{\mu^{o}(\Sigma)/2}),
\end{aligned}$$ where $\str$ denotes the alternating trace. If $K_{+}$, $K_{-}$ and $K_{0}$ are three skein-related links with corresponding Seifert surfaces $\Sigma_{+}$, $\Sigma_{-}$ and $\Sigma_{0}$, then Theorem \[thm:skein\], Lemma \[lem:mod-2-sequence\] and Lemma \[lem:eigenspace-sequence\] together tell us that we have the relation $$\chi (Y,K_{+},[\Sigma_{+}]) - \chi (Y,K_{-},[\Sigma_{-}]) + \chi
(Y,K_{0},[\Sigma_{0}]) = 0$$ in the first case of Theorem \[thm:skein\], and $$\chi (Y,K_{+},[\Sigma_{+}]) - \chi (Y,K_{-},[\Sigma_{-}]) - \chi
(Y,K_{0},[\Sigma_{0}]) r(t) = 0$$ in the second case. Here $r(t)$ is the contribution from the term $I_{*}(\Sigma_{2}\times S^{1})_{w,+2}$, so that $$r(t) = \str (t^{\mu([\Sigma_{2}])/2}).$$ From the proof of Lemma \[lem:Sigma2-calc\] we can read off the eigenvalues of $[\Sigma_{2}]/2$: they are $1$, $0$ and $-1$, and the $\pm 1$ eigenspaces are each $1$-dimensional. Thus $$r(t) = \pm (t - 2 + t^{-1}).$$
To determine the sign of $r(t)$, we need to know the canonical $\Z/2$ grading of (say) the $0$-eigenspace of $\mu([\Sigma_{2}])$ in $I_{*}(\Sigma_{2}\times S^{1})_{w,+2}$. The trivial $3$-dimensional cobordism from $T^{2}$ to $T^{2}$ can be decomposed as $N^{+}\cup
N^{-}$, where $N^{+}$ is a cobordism from $T^{2}$ to $\Sigma_{2}$ and $N_{-}$ is a cobordism the other way. The $4$-dimensional cobordisms $W^{\pm}= N^{\pm}\times S^{1}$ induce isomorphisms on the $0$-eigenspace of $\mu([T^{2}])=\mu([\Sigma_{2}])$; and $\iota(W^{\pm})$ is odd. Since the generator for $T^{3}$ is in odd degree, we conclude that the $0$-eigenspace of $\mu([\Sigma_{2}])$ is in even degree, and that $$\begin{aligned}
r(t) &= - (t - 2 + t^{-1}) \\
&= - q(t)^{2}
\end{aligned}$$ where $$q(t) = (t^{1/2}-t^{-1/2}).$$
We can roll the two case of Theorem \[thm:skein\] into one by defining the “normalized” Euler characteristic as $$\label{eq:renormalized}
\tilde\chi(Y,K,[\Sigma]) =
q(t)^{1-r}\chi(Y,K,[\Sigma])$$ where $r$ is the number of components of the link $K$. With this notation we have:
For null-homologous skein-related links $K_{+}$, $K_{-}$ and $K_{0}$ with corresponding Seifert surface $\Sigma_{+}$, $\Sigma_{-}$ and $\Sigma_{0}$, the normalized Euler characteristics satisfy $$\tilde \chi (Y,K_{+},[\Sigma_{+}]) - \tilde \chi
(Y,K_{-},[\Sigma_{-}])= (t^{1/2}-t^{-1/2})\,\tilde \chi
(Y,K_{0},[\Sigma_{0}]).$$
In the case of classical knots and links, we may write this simply as $$\tilde \chi (K_{+}) - \tilde\chi
(K_{-})= (t^{1/2}-t^{-1/2})\,\tilde\chi
(K_{0}).$$ This is the exactly the skein relation of the (single-variable) normalized Alexander polynomial $\Delta$. The latter is normalized so that $\Delta=1$ for the unknot, whereas our $\tilde\chi$ is $-1$ for the unknot because the generator of its knot homology is in odd degree. We therefore have:
For any link $K$ in $S^{3}$, we have $$\tilde\chi(K) = - \Delta_{K}(t),$$ where $\tilde\chi(K)$ is the normalized Euler characteristic and $\Delta_{K}$ is the Alexander polynomial of the link with Conway’s normalization.
In the case that $K$ is a knot, we have $\tilde\chi(K)=\chi(K)$, which is the case given in Theorem \[thm:main\] in the introduction.
The equality $r(t)= - q(t)^{2}$ can be interpreted as arising from the isomorphism $$I_{*}(\Sigma_{2} \times S^{1})_{w,+2} \cong V\otimes V,$$ with the additional observation that the isomorphism between these two is odd with respect to the preferred $\Z/2$ gradings.
Applications {#sec:applications}
============
Fibered knots
-------------
In [@KM-sutures], the authors adapted the argument of Ni [@Ni-A] to establish a criterion for a knot $K$ in $S^3$ to be a fibered knot: in particular, Corollary 7.19 of [@KM-sutures] states that $K$ is fibered if the following three conditions hold:
1. the Alexander polynomial $\Delta_{K}(T)$ is monic, in the sense that its leading coefficient is $\pm 1$;
2. the leading coefficient occurs in degree $g$, where $g$ is the genus of the knot; and
3. the dimension of $\KHI(K,g)$ is $1$.
It follows from our Theorem \[thm:main\] that the last of these three conditions implies the other two. So we have:
\[prop:fibered-knot\] If $K$ is a knot in $S^{3}$ of genus $g$, then $K$ is fibered if and only if the dimension of $\KHI(K,g)$ is $1$.
Counting representations
------------------------
We describe some applications to representation varieties associated to classical knots $K\subset S^{3}$. The instanton knot homology $\KHI(K)$ is defined in terms of the preferred closure $Z=Z(K)$ described at , and therefore involves the flat connections $$\Rep(Z)_{w} \subset \bonf(Z)_{w}$$ in the space of connections $\bonf(Z)_{w}$: the quotient by the determinant-1 gauge group of the space of all $\PU(2)$ connections in $\PP(E_{w})$, where $E_{w}\to Z$ is a $U(2)$ bundle with $\det(E)=w$. If the space of these flat connections in $\bonf(Z)_{w}$ is non-degenerate in the Morse-Bott sense when regarded as the set of critical points of the Chern-Simons functional, then we have $$\dim I_{*}(Z)_{w} \le \dim H_{*}(\Rep(Z)_{w}).$$ The generalized eigenspace $I_{*}(Z)_{w,+2}\subset I_{*}(Z)_{w}$ has half the dimension of the total, so $$\dim \KHI(K) \le \frac{1}{2} \dim H_{*}(\Rep(Z)_{w}).$$
As explained in [@KM-sutures], the representation variety $\Rep(Z)_{w}$ is closely related to the space $$\Rep(K,\bi) = \{ \, \rho: \pi_{1}(S^{3}
\setminus K) \to \SU(2) \mid \rho(m) =
\bi \,\},$$ where $m$ is a chosen meridian and $$\bi =
\begin{pmatrix}
i & 0 \\ 0 & -i
\end{pmatrix}.$$ More particularly, there is a two-to-one covering map $$\label{eq:covering}
\Rep(Z)_{w} \to \Rep(K,\bi).$$ The circle subgroup $\SU(2)^{\bi}\subset \SU(2)$ which stabilizes $\bi$ acts on $\Rep(K,\bi)$ by conjugation. There is a unique reducible element in $\Rep(K,\bi)$ which is fixed by the circle action; the remaining elements are irreducible and have stabilizer $\pm 1$. The most non-degenerate situation that can arise, therefore, is that $\Rep(K,\bi)$ consists of a point (the reducible) together with finitely many circles, each of which is Morse-Bott. In such a case, the covering is trivial. As in [@KM-knot-singular], the corresponding non-degeneracy condition at a flat connection $\rho$ can be interpreted as the condition that the map $$H^{1}(S^{3}\setminus K; \g_{\rho}) \to H^{1}(m ;
\g_{\rho}) = \R$$ is an isomorphism. Here $\g_{\rho}$ is the local system on the knot complement with fiber $\su(2)$, associated to the representation $\rho$. We therefore have:
Suppose that the representation variety $\Rep(K,\bi)$ associated to the complement of a classical knot $K\subset S^{3}$ consists of the reducible representation and $n(K)$ conjugacy classes of irreducibles, each of which is non-degenerate in the above sense. Then $$\dim \KHI(K) \le 1 + 2n(K).$$
Under the given hypotheses, the representation variety $\Rep(K,\bi)$ is a union of a single point and $n(K)$ circles. Its total Betti number is therefore $1 + 2 n(K)$. The representation variety $\Rep(Z)_{w}$ is a trivial double cover , so the total Betti number of $\Rep(Z)_{w}$ is twice as large, $2 +
4n(K)$.
Combining this with Corollary \[cor:alexander-vs-rank\], we obtain:
Under the hypotheses of the previous corollary, we have $$\sum_{j=-d}^{d} |a_{j}| \le 1 + 2n(K)$$ where the $a_{j}$ are the coefficients of the Alexander polynomial.
Among all the irreducible elements of $\Rep(K,\bi)$, we can distinguish the subset consisting of those $\rho$ whose image is binary dihedral: contained, that is, in the normalizer of a circle subgroup whose infinitesimal generator $J$ satisfies $\mathrm{Ad}(\bi)(J)=-J$. If $n'(K)$ denotes the number of such irreducible binary dihedral representations, then one has $$| \det(K) | = 1 + 2n'(K).$$ (see [@Klassen]). On the other hand, the determinant $\det(K)$ can also be computed as the value of the Alexander polynomial at $-1$: the alternating sum of the coefficients. Thus we have:
Suppose that the Alexander polynomial of $K$ fails to be alternating, in the sense that $$\left| \sum_{j=-d}^{d} (-1)^{j} a_{j} \right|
< \sum_{j=-d}^{d} | a_{j}|.$$ Then either $\Rep(K,\bi)$ contains some representations that are not binary dihedral, or some of the binary-dihedral representations are degenerate as points of this representation variety.
This last corollary is nicely illustrated by the torus knot $T(4,3)$. This knot is the first non-alternating knot in Rolfsen’s tables [@Rolfsen], where it appears as $8_{19}$. The Alexander polynomial of $8_{19}$ is not alternating in the sense of the corollary; and as the corollary suggests, the representation variety $\Rep(8_{19}; \bi)$ contains representations that are not binary dihedral. Indeed, there are representations whose image is the binary octahedral group in $\SU(2)$.
[^1]: The work of the first author was supported by the National Science Foundation through NSF grant number DMS-0405271. The work of the second author was supported by NSF grants DMS-0206485, DMS-0244663 and DMS-0805841.
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[CERN-TH/2003-235]{}
[**BOUND STATES IN TWO SPATIAL DIMENSIONS IN THE NON-CENTRAL CASE**]{}\
\
Theoretical Physics Division, CERN\
CH - 1211 Geneva 23, Switzerland\
and\
Laboratoire de Physique Théorique ENSLAPP\
F - 74941 Annecy-le-Vieux, France\
[^1]\
Gordon McKay Laboratory, Harvard University\
Cambridge, MA 02138-2901, U.S.A.\
and\
Theoretical Physics Division, CERN\
CH - 1211 Geneva 23, Switzerland\
[**ABSTRACT**]{}\
We derive a bound on the total number of negative energy bound states in a potential in two spatial dimensions by using an adaptation of the Schwinger method to derive the Birman-Schwinger bound in three dimensions. Specifically, counting the number of bound states in a potential $gV$ for $g = 1$ is replaced by counting the number of $g_i$’s for which zero energy bound states exist, and then the kernel of the integral equation for the zero-energy wave functon is symmetrized. One of the keys of the solution is the replacement of an inhomogeneous integral equation by a homogeneous integral equation.
CERN-TH/2003-235\
August 2003
Introduction
============
In a previous paper [@aaa], K. Chadan, N.N. Khuri and ourselves (A.M. and T.T.W.) obtained a bound on the number of bound states in a two-dimensional central potential. This bound has the merit that, for a potential $gV$, the coupling constant dependence for large $g$ is optimal, i.e., the same as the one of the semiclassical estimate [@bb]. Previous work on the subject was done by Newton [@cc] and Seto [@dd]. We also obtained a bound for the non-central case, but only by using a rather brutal method which consists of replacing the potential by a central potential which is defined, after choosing a certain origin, by V\_c(r) & = & In f V()\
& & || = r \[1\] Because of the monotonicity of the energy levels with respect to the potential, putting $V_c$ in our formulae will give a bound for the potential $V$. However, for potentials with singularities outside the given origin, this may lead to no bound at all. Our attention has been attracted by the fact that in condensed matter physics problems exist, where counting the bound states on a surface may be useful [@ee], but where it is very unlikely that the potential will be central, even approximately.
In the present paper, we obtain a bound on the number of bound states in a non-central two-dimensional potential, using an adaptation of the Schwinger method to derive the Birman-Schwinger bound [@ff] in the three-dimensional case. The condition under which we obtain a bound is d\^2x d\^2y |V(x)| (n |x-y|)\^2 |V(y)| < . \[2\] This condition is , just like that of Birman and Schwinger, but we show in the Appendix that it follows from the linear conditions d\^2 x & ( n (2+|x|) )\^2 |V(x)| <\
[and]{}\
d\^2 x & V\_R (||) n\^-|x| < , \[3\] where $V_R$ is the circular decreasing rearrangement of $|V|$ (for the definition of $V_R$, see the Appendix).
Condition (2) has already been proposed by Sabatier [@ggg]. Condition (3) appears in a forthcoming work by N.N. Khuri, A. Martin, P. Sabatier and T.T. Wu, dealing with the scattering problem. It has the advantage of showing more clearly what kind of behaviour the potential is allowed to have at short and large distances.
The strategy of Schwinger consists of counting the number of zero-energy bound states for a potential $gV$ for $0 < g < 1$ instead of the actual number of negative energy bound states for $g = 1$. In three dimensions these two numbers are equal. Indeed, let $0 < g_1, g_2, \dots g_n < 1$ be the coupling constants for which we have zero-energy bound states. Each $g_i$ is the origin of a bound state trajectory in the $E-g$ plane, $E_i(g)$, with $E_i(g_i) = 0$. These trajectories are monotonous decreasing: = V \^2 d\^nx, \[4\] by the Feynman-Hellmann theorem, but g\_i V \^2 d\^n x = E - ||\^2 d\^n x < 0 E < 0. \[5\]
This shows that the number of negative-energy bound states is exactly the same as the number of $g_i$’s $< 1$. At the crossing of any pair of trajectories there is no problem because of their monotonicity.
The same result holds in two dimensions with one modification: any attractive potential (i.e., $\int \, d^2x \, V < 0$), has a bound state for arbitrarily small $g$, with a binding energy going to zero for $g \to 0$ like $- \exp - (C/g)$ [@aaa]. At $E = 0$ it disappears and is not included in Schwinger’s accounting, so we have to add one unit.
Since we only want a on the number of bound states, we can always replace $V(y)$ by $-V^-(y)$: V\^-(y) & = & 0 [for]{} V > 0\
V\^-(y) & = & -V(y) [for]{} V 0 \[6\] Using $-|V(y)|$ instead of $-V^-(y)$ gives a more crude bound.
It can be shown that the general solution of the zero-energy Schrödinger equation - - V\^- = 0 \[7\] in the equivalent integral form (x) = C - n k\_0 |x-y| V\^-(y) (y) d\^2y \[8\] has a general asymptotic behaviour for $|x| \to \infty$ (x) - C \~- n k\_0 |x| V\^-(y) (y) d\^2 y + o (1) \[9\] under the condition V\^-(y) ( n (2+|y|) )\^2 d\^2y < \[10\] which follows from condition (1) as shown in the Appendix.
Zero-energy bound states are characterized by the fact that $\psi$ is . Hence we get the necessary condition: V\^- (y) (y) d\^2y = 0 \[11\] Now we have two possibilities:\
- I. At infinity $\psi (\vec{x}) \to 0$ and hence $C = 0$ and those bound states wave functions satisfy a integral equation \_i (x) = - d\^2 y n (k\_0 |x-y|) V\^-(y) \_i (y) \[12\] (notice that the scale factor $k_0$ disappears because of condition (11)).
This is what happens in the case of a central potential for a non-zero azimuthal angular momentum $m$.
- II\. At infinity, $\psi (\vec{x}) \to C$, with $C \not= 0$. In this case, the bound state wave functions satisfy an inhomogeneous integral equation. This case has been described in Ref. [@hh], where it is shown that for a central potential in two dimensions, the $m = 0$ phase shift has the universal behaviour (k) \~ , k 0 \[13\] if there is a zero-energy bound state of type II. Then (k) n k 0 . \[14\] In Ref. [@hh], a much stronger result is stated. This much stronger result, however, holds only for a very rapidly decreasing potential.
Counting Bound States in Case I
===============================
Following Schwinger, we symmetrize the kernel of the integral equation: \_i (x) = g\_i K (x, y) \_i (y) d\^2 y \[15\] with \_i (x) & = & \_i(x)\
K (x , y) & = &- n k\_0 |x-y| . \[16\] If $V^-(x)$ vanishes in some regions, it seems impossible to go back from $\phi_i$ to $\psi_i$. However, this can be remedied by defining V\^-\_(x) = V\^- (x) + - |x| . \[17\] Since the bounds we shall get are continuous in $V$, we can take the limit $\epsilon \to 0$ at the end.
$K$ can be written as K = |\_i> <\_i | + R \[18\] $R$ is a sum over states which do not satisfy (11). The $\phi_i$’s in themselves do not form a complete set. If we define $a$ by a(x) = \[19\] we have = 0 \[20\] from property (11), and naturally $<a | a> = 1$.
If we define $\hat{Tr}$, a trace restricted to the $\phi_i$’s, we have $$\hat{Tr} K = \Sigma \, \frac{1}{g_i} > \sum_{g_i \leq 1} \; \frac{1}{g_i}
> N_I,$$ $N_I$ being the number of bound states of type I. However, this trace turns out to be divergent because of the logarithmic singularity of the kernel (the same happened in Schwinger’s original work!), and we follow Schwinger to iterate the integral equation (15): $$\phi_i(x) = g^2_i \int \, K (x , y) K (y, z) \, \phi_i (z) d^2y d^2z$$ and then K\^2 = > N\_I \[21\] Forgetting the “hat" on the trace still gives a bound because $K^2$ is a positive operator (contrary to K!), but this bound depends on the scale parameter $k_0$ entering in the logarithm. Among the missing states in $\hat{Tr}$ is the state $|a>$, orthogonal to the $\phi_i$’s, and this one should be removed from the complete trace. In this way, we get $$N_I <
TrK^2 - <a|K^2|a> ,$$ or more explicitly & N\_I & < V\^-(x) (n k\_0 |x-y|)\^2 V\^-(y) d\^2x d\^2y\
& - & V\^- (x) (n k\_0 |x-z|) V\^-(z) (n k\_0 |z-y| ) V\^-(y) d\^2x d\^2y d\^2z \[22\] It is visible that the second term is negative as we announced.
Rewriting $N_I$ as $$\frac{1}{\int V^- (z) d^2 z} \;
\int d^2x d^2y d^2z\, V^-(x) V^-(y) V^-(z) \;
\left[ (\ell n \, k_0 |x-y|)^2 - \ell n \, k_0 |x-z| \ell n \, k_0 |y-z|
\right ],$$ we see that (22) is manifestly of the scale factor $k_0$.
Counting Bound States in Case II
================================
At first it would seem that Schwinger’s technique will not work because, in Eq. (8), the constant is not zero and therefore we deal with an integral equation which can be written, after the same changes of variables as in Section 2, given by (16) and (19): \_i = C\_i a + g\_i K \_i , \[23\] with, again, = 0 < a | a> = 1 \[24\] Equation (24) is precisely the key property which will make it possible to replace (23) by a .
Again, the $\phi_i$’s corresponding to different $g_i$’s are orthogonal because = C\_j < \_i | a> + g\_j < \_i | C \_j>\
= C\_j < \_j | a> + g\_i < \_i | C \_j> . Hence, from (24): ( - ) < \_i | \_j > = 0. \[25\]
Let us call ${\cal S}$ the Hilbert space associated to the integral equation (23), and construct a new Hilbert space by removing the element $a$: =[S ’]{} { a } \[26\] We want to define a new operator $K'$ acting in ${\cal S '}$. Let b = K a. \[27\] Notice that = - <a|a> = - . \[28\] We try $$K' = K - |b><a| - |a><b| + C |a><a|$$ where $C$ will be chosen so that K’ a = 0. \[29\] We have $$K'|a> = |b> - |b> - <b|a> |a> + C | a>,$$ and hence we take C = <b|a> = <a |K| a> \[30\] $K'$ is Hermitian like $K$, and we get g\_i K’ |\_i > & = & g\_i |K|\_i > - g\_i <b|\_i > |a>\
& = & g\_i |K|\_i > + C\_i |a> .Hence g\_i K’ |\_i > = |\_i > \[31\] which is .
To get a bound on the number of bound states of type II, we have to get a bound on trace $K'^2$ (not surprisingly, trace $K'$ is divergent). It is a lengthy but straightforward exercise to calculate that trace, which gives N\_[II]{} < [tr]{} K\^2 - 2 <a|K\^2 |a> + (<a|K|a>)\^2 \[32\] The last two terms give an overall negative contribution. The first term is the same as the one appearing in $N_I$. It is easy to see that the right-hand side of (32) is independent of the scale parameter $k_0$ entering into the kernel $K$. Finally, let us notice that the treatment of case II case I because, in the argument, it has never been said that $C_i \not= 0$. Equation (31) holds irrespective of whether $C_i = 0$ or $C_i \not= 0$. Notice that the bound on $N_{I}$ is larger than the bound on $N_{II}$. Therefore the bound on $N_I$ becomes completely obsolete.
Concluding Remarks
==================
If we include the bound state with evanescent energy for zero coupling constant, we get the bound $$N < 1 + \, {\rm tr} \, K^2 - 2 <a |K^2|a> + (<a|K|a>)^2.$$ Dropping the last two terms still gives a scale-dependent bound - which can be minimized with respect to the scale - which precisely appears in condition (2), itself following from the linear condition (3) as shown in the Appendix.
Conditions (2) and (3) both allow a potential behaving like $$\frac{1}{r^2 (\ell n r)^{3+\epsilon}}$$ at infinity, with local singularities not worse than $$- \frac{1}{|r - r_0|^2 (| \ell n |r - r_0||)^{2+\epsilon}},$$ $\epsilon$ positive, arbitrarily small. Both conditions are violated for $\epsilon < 0$. However, we shall see in the Appendix that (2) is definitely weaker than (3).
Our bound has the merit of being valid for the non-central case, which, as we said in the Introduction, is important for solid-state physics. However, for a potential $gV$, it behaves like $g^2$ for large $g$, while in Ref. [@aaa], in the central case, we get a bound behaving like $g$. In Ref. [@aaa] we make a conjecture which is very far from being proved, but clever mathematical physicists might prove it or something similar. The present work should be considered only as a first step which could possibly give reasonable results for not too large $g$.
[**Acknowledgements**]{}
We are grateful to P. Sabatier for suggesting the use of the non-linear expression defined by (2) for potentials in two dimensions. Our work was stimulated by discussions with our colleagues K. Chadan and N.N. Khuri. One of us (T.T.W.) would also like to thank the CERN Theoretical Physics Division for hospitality. This paper was put in final form while one of us (A.M.) was visiting the “Institut des Hautes Etudes Scientifiques", whose hospitality is acknowledged.
=
Condition (2) is I = d\^2 x d\^2 y V\^-(x) (n |x-y|)\^2 V\^- (y) < \[A1\] Condition (3) is a set of two conditions: d\^2 x ( n (2+|x|) )\^2 V\^- (x) < \[A2\] d\^2 x V\_R (|x|) n\^-(|x|) < . \[A3\] In (A.3) we use: - n\^- (|x|) & = & 0 |x| > 1\
& = & - n |x| |x| < 1 .
$- V_R (|x|)$, the circular decreasing rearrangement of $V^- (x)$. Since this notion is not very well known among physicists, let us remind the reader that $V_R (|x|)$ is a decreasing function of $|x|$, such that $$\mu (V_R (|x|) > A) = \mu (V^- (x) > A), \, \forall A,$$ where $\mu$ is the Lebesque measure. In more familiar terms, the rearranged Mont Blanc would be a mountain with axial symmetry, with a single peak, such that the surface between the level lines would be the same as the surface between the level lines of the original Mont Blanc (rather awfully dull!).
We shall prove first that the convergence of $I$ in (A.1) follows from the convergence of (A.2) and (A.3). More exactly, we shall get an explicit bound on (A.1) in terms of (A.2) and (A.3). We write I = I\_+ + I\_- \[A4\] with I\_+ = d\^2x d\^2y V\^- (x) (n\^+ |x-y|)\^2 V\^- (y) \[A5\] I\_- = d\^2x d\^2y V\^- (x) (n\^- |x-y|)\^2 V\^- (y) \[A6\] $\ell n^-$ has already been defined. $\ell n^+ (a) = \ell n a$ for $a
\geq 1$, $= 0$ for $a < 1$. It is elementary to get a bound on $I_+$ from (A.2) only. Indeed, $$0 < \ell n^+ |x-y| < \ell n^+ (|x|+|y|) < \ell n (2 +|x|) + \ell n (2 + |y|)$$ and thus ( n\^+ |x-y| )\^2 < 2 \[A7\] Hence I\_+ < 4 d\^2 x V\^- (x) d\^2 y V\^- (y) (n (2+|y|))\^2 . \[A8\] The convergence of the right-hand side of (A.8) follows directly from (A.2).
Concerning $I_-$, we use a rearrangement inequality due to Luttinger and Friedberg [@jj], which says A (x) B (|x-y|) C (y) d\^2x d\^2y A\_R (|x|) B\_R (|x-y|) C\_R (|y|) d\^2x d\^2y \[A9\] where $A, B, C$ are non-negative functions and $A_R, B_R, C_R$ are their decreasing rearrangements. Since $\ell n^-$ and $(\ell n^-)^2$ are decreasing functions of their argument they are their own rearrangement. Hence I\_- < d\^2x d\^2 y V\_R (|x|) ( n\^- (|x-y|) )\^2 V\_R (|y|) \[A10\]
In (A.10), we can carry out first the angular integration, the angle ($\vec{x}, \vec{y})$ appearing only in $\ell n^-$. However, to be able to do that easily we have to sacrifice some information, i.e., use $(\ell n^- |x-y|)^2 \leq (\ell n (|x-y|))^2$. We have to calculate $$\int \, \frac{d \theta}{2\pi} \, \left ( \ell n (|x|^2 + |y|^2 -
2|x||y| \, \cos \theta) \right )^2 .$$ We have $$\ell n (|x|^2 + |y|^2 - 2 |x||y| \, \cos \theta ) =
\ell n (|x| - |y| e^{i \theta}) + \ell n (|x| - |y| e^{-i \theta}).$$ Assume $|x| > |y|$. Then we get n (|x|\^2 + |y|\^2 - 2 |x||y| ) = 2 \[A11\] Hence, if $|x| > |y|$, using the orthogonality of the $\cos \, n \, \theta$: $$\int \, \frac{d \theta}{2\pi} \, (\ell n (|x|^2 + |y|^2 - 2 |x||y| \, \cos
\theta ))^2 =
4 ( \ell n |x|)^2 + 2 \sum^{\infty}_{n=1} \left( \frac{|y|}{|x|} \right
)^{2n} \,
\frac{1}{n^2}$$ We see a dilogarithm, or Spence function, appearing on the right-hand side. However, we only need to notice that \_[|y| |x|]{} ( n |x|\^2 + |y|\^2 - 2 |y||x| )\^2 = 4 (n |x|)\^2 + 2 \^\_[n=1]{} = 4 (n |x| )\^2 + . \[A12\] In this way we get $$I_{-} < (2\pi)^2 \times 2 \int_{|x|>|y|} |x|d|x| \, |y|d|y| \, V_R (|x|)
V_R (|y|) \;
\left [ 4 (\ell n |x|)^2 + \frac{\pi^2}{3} \right ]$$ Again, we split the integral into & 32 \^2 & \_[|x|>|y|]{} |x|d|x| |y|d|y| V\_R(|x|) V\_R(|y|) (n\^- |x|)\^2\
+ & 32 \^2 & \_[|x|>|y|]{} |x|d|x| |y|d|y| V\_R(|x|) V\_R(|y|) (n\^+ |x|)\^2\
+ & &\^2 \[A13\] In the first term of (A.13) we can replace $(\ell n^- (|x|))^2$ by $\ell
n^-|x| \, \ell n^- |y|$, since $|x| > |y|$ and since $\ell n^-$ is decreasing.
In the second term, we can drop the restriction $|x| > |y|$ and notice that $$\int d^2x V_R (|x|) \, \left ( \ell n^+ (|x|) \right )^2 < \int d^2x V_-(x)
\left ( \ell n^+ (|x|) \right
)^2 .$$ Indeed, $\int d^2x A_R (x) \phi (|x|)$, where $\phi (|x|)$ is , is less than $\int d^2x \,
A(x) \, \phi(|x|)$. Suppose that $\phi (|x|) \to L$. Then $$\int d^2x A_R (x) \phi (|x|) = \int d^2x A_R (x)L - \int d^2x A_R (x)
(L-\phi(|x|)).$$ $L - \phi(|x|)$ is its own decreasing rearrangement and following the well-known properties $$\int A_R B_R d^2x \geq \int A(x) B(x) d^2x$$ and $$\int A_R (|x|) d^2x = \int A(x) d^2x,$$ we get the desired property. If $L$ is infinite, we can use a limiting procedure. Finally, we get I\_- & < & 16 \^2 \^2\
& + & 32 \^2 d\^2 x V\_-(x) d\^2 y V\_-(y) (n (2+|y|))\^2\
& + & \^2 \[A14\] From (A.2) and (A.3) we see that $I_-$ is bounded. This concludes the proof.
One question is: can we go in the opposite direction? Assume that we know that (A.1) holds. There exists certainly a region $|x -x_o |< R$ where ${\rm Inf} \, V^- = m > 0$. If such a region did not exist, $V^-$ would be zero almost everywhere! So I > R\^2 m \_[|x|>|x\_o|+R+2]{} V\^- (x) n \[2+|x|\]\^2 d\^2 x \[A15\] Now we choose $y_o > 4 + |x_o| + R + 2$ such that $${\rm Inf}_{|y-y_o| < R'} \, V_-(y) = m' > 0 ,$$ then I > R’\^2 m’ \_[|x|<|x\_o|+R+2]{} V\^- (x) (n (4-R’))\^2 d\^2 x. \[A16\] This proves that the convergence of (A.1) implies the convergence of (A.2).
It is not possible to deduce (A.3) from (A.1) because (A.3) involves $V_R$ and (A.1) does not. However, in practice the conditions are very similar. Nevertheless, the following example shows that (A.3) is stronger than (A.1), even for a potential which does not need rearrangement: take the central potential V(|x|) & = & - |x| < ,\
& = & 0 |x| . \[A17\] For $\gamma \leq \frac{1}{2}$ (A.1) and (A.3) are divergent, for $\frac{1}{2} < \gamma \leq 1$ (A.1) is convergent and (A.3) is divergent, for $\gamma > 1$ (A.1) and (A.3) are convergent.
[99]{} K. Chadan, N.N. Khuri, A. Martin and T.T. Wu, [*J. Math. Phys.*]{} [**44**]{} (2003) 406. A. Martin, (1972) 140,\
H. Tamura, [*Proc. Jpn. Acad.*]{} [**50**]{} (1974) 19. R.G. Newton, [*J. Math. Phys.*]{} [**3**]{} (1962) 867, [*J. Operator Theory*]{} [**10**]{} (1983) 119. N. Seto, Publ. RIMS, Kyoto University 9179) 429. F. Bassani, T. Martin, private communications. J. Schwinger, [*Proc. Nat. Acad. Sci. USA*]{} [**47**]{} (1961) 122;\
M. Birman, [*Math. Sb.*]{} [**55**]{} (1961) 124;\
English translation [*Amer. Math. Soc. trans.*]{} [**53**]{} (1966) 23. See also:\
M.S. Birman, [*Dokl. Acad. Nauk. SSSR*]{}, [**129**]{} (1959) 239 (in Russian. No English translation available.) P. Sabatier, private communication. K. Chadan, N.N. Khuri, A. Martin and T.T. Wu, [*Phys. Rev.*]{} [**D58**]{} (1998) 025014. J.M. Luttinger and R. Friedberg, quoted in:\
J.M. Luttinger, [*J. Math. Phys.*]{} [**14**]{} (1973) 1450.
[^1]: Work supported in part by the U.S. Department of Energy under Grant No. DE-FG02-84-ER40158
|
---
abstract: 'An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudo-cycle and a well-defined integral homology class in singular homology.'
address: 'Department of Mathematics, University of Chicago, Chicago, IL 60637'
author:
- Matthias Schwarz
date: 'March 9. 1999'
title: Equivalences for Morse homology
---
[^1]
Introduction
============
The aim of this paper is to give an explicit construction of an isomorphism between Morse homology and singular homology. Morse homology is a Morse-theoretical approach to the homology of a smooth manifold which goes back already to Thom and plays a crucial role in Smale’s proof of the $h$-cobordism theorem, cf. also [@Mil-hcobordism]. It was studied by J. Franks [@Franks-MS], rediscovered by Witten [@Witten-82] in terms of a deformation of the de Rham complex and generalized by Floer [@F-Witten] as an approach to solve a conjecture by Arnold. In [@Sch-Morse] the author developed a comprehensive approach to Morse homology as an axiomatic homology theory for the category smooth manifolds (not necessarily compact) satisfying all Eilenberg-Steenrod axioms. Moreover, this approach used the purely relative “Floer-theoretical” definition of Morse homology in terms of moduli spaces of trajectories for the gradient flow equation connecting critical points. However, [@Sch-Morse] did not present an explicit isomorphism to other axiomatic homology theories like for instance the de Rham theorem between de Rham cohomology and singular cohomology. That Morse homology is isomorphic to other homology theories is proved in [@Sch-Morse] by extending it to a slightly larger category of certain CW-spaces compatible with the manifold structure in which the isomorphism is deduced inductively, based on the Eilenberg-Steenrod axioms. That is, in such a category existence and uniqueness of the isomorphism follows by abstract application of the axioms.
In the approach of Smale and Milnor, used similarly also in Floer’s description, a direct isomorphism between Morse homology and singular homology is obtained by choosing a special, namely self-indexing Morse function, such that the boundary map in the Morse chain complex can be related to the connecting homomorphism $\partial_\ast$ in the long exact sequence of the cell decomposition induced by the Morse function (see also Section \[ssc other\] below).
The approach of this paper is to show that, given [*any*]{} Morse function $f$ with a generic Riemannian metric $g$, one can construct singular cycles explicitly from the given Morse cycle. The main objects which have to be considered as intermediate tools are so-called [ *pseudo-cycles*]{}. This is a geometric differential-topological way to represent homological (integral!) cycles which plays also a role in the definition of quantum cohomology (see e.g. [@McD-S]). In this paper, we give a short proof that integral homology classes can be represented by pseudo-cycles and that every pseudo-cycle in fact leads to an integral homology class.
The purpose of this paper is also to provide a detailed construction of this equivalence between Morse homology and singular homology via pseudo-cycles, which to the author’s knowledge has not yet been carried out elsewhere, but which already has been used several times, in particular in the theory of quantum cohomology and Floer homology, e.g. [@PSS], [@Sch-qcl], [@Sei], [@Sch-QH].
After a short account on the definition of Morse homology pseudo-cycles are defined in Section \[sc pc\], where it is proven that pseudo-cycles represent integral homology classes and every class can be represented as such. Section \[sc expl\] contains the construction of the explicit isomorphism between Morse homology and singular homology. In the first part we show how to obtain a well-defined pseudo-cycle from a given Morse cycle and that the induced singular class is uniquely associated to the Morse homology class. The idea is to glue all unstable manifolds of critical points, which occur in the given Morse cycle, along the $1$-codimensional strata of their suitable compactifications. In the second part we construct the inverse homomorphism in terms of intersections of pseudo-cycles representing singular classes and stable manifolds of critical points.
Morse homology
==============
Definition
----------
Let $M$ be an oriented[^2] smooth manifold, $f\in
C^\infty(M,\bbr)$ an exhausting[^3] Morse function and $g$ be a complete Riemannian metric. Consider the critical set $\operatorname{Crit}_\ast f$ of $f$ as graded by the Morse index $\mu\colon \operatorname{Crit}f\to\bbz$ and define the stable and unstable manifolds of the negative gradient flow in terms of spaces of curves, $$\label{eq unstable}\begin{split}
W^u(x) &= \{\, \gamma\colon (-\infty,0]\to M
\,|\,\dot\gamma + \nabla_g f\circ\gamma=0,\,
\gamma(-\infty)=x \,\},\\
W^s(y) &= \{\, \gamma\colon [0,\infty) \to M
\,|\,\dot\gamma + \nabla_g f\circ\gamma=0,\,
\gamma(+\infty)=y \,\}
\end{split}$$ for $x,y\in\operatorname{Crit}f$. The curves $\gamma$ are smooth and $\gamma(\pm\infty)$ denotes the limit for $t\to\pm\infty$. The spaces $W^u(x)$ and $W^s(y)$ are finite-dimensional manifolds with $$\dim W^u(x)=\mu(x) \quad\text{and}\quad
\dim W^s(y)=\dim M -\mu(y)$$ and the evaluation mapping $\gamma\to\gamma(0)$ induces smooth embeddings into $M$, i.e. diffeomorphisms onto the image, $$E_x\colon W^u(x)\hookrightarrow M,\quad
E_y\colon W^s(y)\hookrightarrow M\,.$$ However, in general, these maps are not proper. Choosing a generic Riemannian metric we obtain Morse-Smale transversality, namely $W^u(x)$ and $W^s(y)$ intersect transversely in $M$ with respect to $E_x$ and $E_y$. If this transversality holds for all $x,y\in\operatorname{Crit}f$, $(f,g)$ is called a [*Morse-Smale*]{} pair. We obtain the manifold of connecting orbits $$\begin{aligned}
M_{x,y}(f,g) &= W^u(x)\pitchfork W^s(y)\\
&= \{\, \gamma\colon\bbr\to M\,|\,
\dot\gamma+\nabla_g f\circ\gamma=0,\;
\gamma(-\infty)=x,\,\gamma(+\infty)=y\,
\},\\
\dim M_{x,y}(f,g) &=\mu(x)-\mu(y),\end{aligned}$$ on which, if $x\not=y$, $\bbr$ acts freely and properly by shifting $$(\tau\ast\gamma)(t)=\gamma(t+\tau)\,.$$
Let us fix orientations for all unstable manifolds $W^u(x)$, then the orientation of $M$ induces orientations for $W^s(y)$ and $M_{x,y}$. We call an unparameterized trajectory $\hat\gamma\in M_{x,y}/\bbr$ for relative index $1$ positively oriented if the orbit $\bbr\cdot\hat\gamma\subset M_{x,y}$ is positively oriented by the action of $\bbr$ which corresponds to the action by the negative gradient flow. Thus, for relative index $1$, the moduli spaces of unparameterized trajectories $$\widehat M_{x,y}=M_{x,y}/\bbr,\quad \mu(x)-\mu(y)=1,$$ are compact, that is finite, and every element $\hat\gamma$ carries a sign $\tau(\hat\gamma)\in\{\pm1\}$. We define the intersection numbers $$n(x,y) = \sum_{\hat\gamma\in\widehat M_{x,y}}
\tau(\hat\gamma)$$ and an operator on the module over $\bbz$ generated by the critical points of index $k$, $$\begin{gathered}
C_k(f) = \bbz\otimes \operatorname{Crit}_k f,\quad
\partial=\partial(f,g),\\
\partial\colon C_k(f) \to C_{k-1}(f),\quad
\partial x = \sum_{y} n(x,y)\,y\,.\end{gathered}$$ The fundamental theorem of Morse homology is
\[thm boundary\] $\partial$ is a chain boundary operator, i.e. $\partial\circ\partial=0$.
Hence, the homology $H_k(f,g;\bbz)=H_k(C_\ast(f),\partial(f,g))$ is well-defined as the quotient of the module of [*Morse-cycles*]{} $$Z_k(f,g) = \{\, a=\hspace{-1em}\sum_{x\in\operatorname{Crit}_k f}\hspace{-1em} a_x x \,|\,
\partial a=0\,\}\,.$$ modulo the boundaries $B_k(f,g)=\operatorname{im}\partial$.
Let us now recall the homotopy invariance result in Morse homology. It is based on Conley’s continuation principle (see [@Con-isolated]).
\[thm continuation\] Given two Morse-Smale pairs $(f^0,g^0)$ and $(f^1,g^1)$ there exists a canonical homomorphism $$\Phi_{10}\colon H_\ast(f^0,g^0) \to H_\ast(f^1,g^1)$$ such that $$\Phi_{21}\circ\Phi_{10} = \Phi_{20}\quad
\text{and}\quad \Phi_{00}=\operatorname{id}\,.$$ In particular, every $\Phi_{ji}$ is an isomorphism.
This continuation theorem implies that we have well-defined Morse homology groups $$\label{eq Morse homology}\begin{split}
&H^{\text{Morse}}_\ast(M;\bbz) \stackrel{\mathsf{def}}{=} \big\{\,
(a_i)\in\prod H_\ast(f^i,g^i)\,|\,
a_j=\Phi_{ji}a_i\,\big\}\\
&\rho_i\colon
H_\ast(f^i,g^i)\stackrel{\cong}{\longrightarrow}H^{\text{Morse}}_\ast(M;\bbz),\quad
\rho_i\circ\Phi_{ij}=\rho_j\,.
\end{split}$$ Let us recall the construction of $\Phi_{10}$ from [@Sch-Morse]. Given the Morse-Smale pairs $(f^i,g^i)$, $i=0,1$, we choose an asymptotically constant homotopy over $\bbr$, $(f_s,g_s)$, $s\in\bbr$ with $$(f_s,g_s) = \begin{cases}
(f^0,g^0) ,& s\leqslant -R,\\
(f^1,g^1) ,& s\geqslant R,
\end{cases}$$ for $R$ large enough. This gives rise to the trajectory spaces $$\begin{aligned}
M_{x_o,x_1}(f_s,g_s) = \{\,\gamma\,|\, &\dot\gamma(s)
+ \nabla_{g_s}f_s(\gamma(s))=0,\\
&\gamma(-\infty)=x_o,\,\gamma(\infty)=x_1\,\}\,.\end{aligned}$$ For a generic choice of the homotopy $(f_s,g_s)$, these spaces are finite dimensional manifolds with $$\dim M_{x_o,x_1}=\mu(x_o)-\mu(x_1)$$ and compact in dimension $0$. As in the definition of the boundary operator $\partial$ we define $$\begin{gathered}
\Phi_{10}\colon C_\ast(f^0,g^0) \to C_\ast(f^1,g^1),\\
\Phi_{10} x_o = \sum_{x_1} n(x_o,x_1) x_1,\\
n(x_o,x_1)=\#_{\text{alg}}M_{x_o,x_1}(f_s,g_s)\,,\end{gathered}$$ where $\#_{\text{alg}}$ means counting with signs $\tau(u)=\pm1$, analogously to above. That $\Phi_{10}$ is well-defined on the level of homology follows from a theorem stating that $$\Phi_{10}\circ\partial_0 = \partial_1\circ\Phi_{10}\,.$$ Moreover, it is shown in [@Sch-Morse] that the homomorphism $\Phi_{10}$ on homology level does not depend on the choice of the homotopy $(f_s,g_s)$.
Isomorphism via axiomatic approach {#ssc axiom}
----------------------------------
In [@Sch-Morse], Morse homology is extended towards an axiomatic homology theory for the category of smooth manifolds. It is functorial with respect to smooth maps, there exists a relative version so that we have an associated long exact sequence, and all axioms of Eilenberg and Steenrod are satisfied. However, in order to derive a natural isomorphism with any other axiomatic homology theory, an extension to a larger category of spaces is required, e.g. towards the subcategory of CW-pairs which are embedded smoothly into finite-dimensional manifolds as strong deformation retracts of open subsets. This approach is adopted in [@Sch-Morse] in order to prove the equivalence with other homology theories.
Pseudo-cycle homology {#sc pc}
=====================
In [@McD-S], pseudo-cycles were defined in order to find a suitable differential-topological representation of homology cycles. However, this was only used with rational coefficients so that every cycle can be represented as a closed submanifold. Here, we consider integral homology classes.
Let $M$ be a compact[^4] $n$-dimensional manifold. We consider an oriented smooth $k$-dimensional manifold without boundary $V$ together with a smooth map $f\colon V\to M$. Let the set $f(V^\infty)$ be defined as in [@McD-S], $$\label{eq pseudocycle}
f(V^\infty)\stackrel{\mathsf{def}}{=} \bigcap_{K\subset V\text{ cpt.}}
\overline{f(V\setminus K)}\,.$$ According to [@McD-S], $f\colon V \to M$ is a [**pseudo-cycle**]{} if $f(V^\infty)$ can be covered by the image of a smooth map $g\colon P \to M$ which is defined on a manifold $P$ of dimension not larger than $\dim
V-2$.
Moreover, let $W$ be an oriented smooth $(k+1)$-dimensional manifold with boundary $\partial W$, such that the inclusion $i\colon \partial
W\hookrightarrow W$ is proper, and let $F\colon W\to M$ be a smooth map.
\[prop cycle1\] Let $(f,V)$ be a pseudo-cycle and $(F,W)$ as above.
- If $H_k(f(V^\infty);\bbz)=H_{k-1}(f(V^\infty);\bbz)=0$ and $f(V)\not\subseteq f(V^\infty)$, then $(f,V)$ induces a unique integral homology class $\alpha_f\in H_k(M;\bbz)$.
- Let $\partial W=U$ be an open subset of $V$ such that $f(V^\infty)\subseteq f(U^\infty)$ and $F(W^\infty)\cap
f(U)=\emptyset$. If $H_k(F(W^\infty);\bbz)=0$ the homology class $\alpha_f$ vanishes.
In view of part (b) let us consider two pseudo-cycles $f_1\colon V_1\to
M$ and $f_2\colon V_2\to M$ to be [**cobordant**]{} if their disjoint union $V=V_1\amalg V_2^\ast$ with orientation on $V_2$ reversed forms a pseudo-cycle $f\colon V\to M$ such that there exists $F\colon W\to M$ satisfying the condition in (b).
In this section we are using Alexander-Spanier homology theory for locally compact Hausdorff spaces, cf. [@Massey]. The homology theory with arbitrary supports[^5], i.e. not necessarily compact supports, is denoted by $H^\infty_\ast(X)$. Note that, however, homology theory with arbitrary supports, which is functorial with respect to proper maps, agrees with any homology theory with compact supports, as for instance singular homology, when restricted to compact sets as $M$, $f(V^\infty)$ and $F(W^\infty)$.
Every oriented $k$-dimensional manifold $X$ without boundary carries a uniquely defined fundamental class $[X]\in H^\infty_k(X;\bbz)$. If $X$ is a manifold with boundary $\partial X$ then $[X]$ is well-defined in $H^\infty_k(X\setminus\partial X)=H_k^\infty(X,\partial X)$. Every open subset $U\subset X$ inherits an orientation from $X$ so that the natural restriction map $\rho\colon H^\infty_k(M;\bbz) \to H^\infty_k(U;\bbz)$ gives $\rho([U])=[X]$. Without loss of generality we may assume that $$f(V)\cap f(V^\infty) =\emptyset\,.$$ Otherwise, we replace $V$ by the open, nonempty subset $V\setminus
f^{-1}(f(V^\infty))$. Since Alexander-Spanier homology with arbitrary supports is functorial with respect to proper maps of locally compact Hausdorff spaces we redefine the map $$f\colon V \to M\setminus f(V^\infty)\,.$$ By definition of $f(V^\infty)$, $f$ is proper. The integral class $$(f)_\ast([V])\in H^\infty_k(M\setminus f(V^\infty);\bbz)$$ is well-defined. From the exact homology sequence for $H^\infty_\ast$ and the pair $(M,f(V^\infty))$, $$H_k(f(V^\infty)) \to
H_k(M) \stackrel{j_\ast}{\longrightarrow}
H^\infty_k(M\setminus f(V^\infty)) \to
H_{k-1}(f(V^\infty))\,,$$ we obtain by assumption the isomorphism $j_\ast$. Hence, $$\alpha_f\equiv j_\ast^{-1}(f)_\ast([V])
\in H_k(M;\bbz)$$ is well-defined.
We consider now an open subset $U\subset V$ such that $f(V^\infty)\subset f(U^\infty)$ and $f(U)\cap
f(U^\infty)=\emptyset$. We carry out the same procedure as before for the proper map $$f_U\colon U \to M\setminus f(U^\infty)$$ and relate it to $\alpha_f$ by the following commutative diagram with respect to the natural restriction homomorphism $\rho$, $$\begin{CD}
H^\infty_k(V) @>{f_\ast}>> H^\infty_k(M\setminus f(V^\infty))\\
@VV{\rho}V @VV{\rho}V\\
H^\infty_k(U) @>{(f_U)_\ast}>>
H^\infty_k(M\setminus f(U^\infty))\,.
\end{CD}$$ Since $\rho\circ j_\ast=j^U_\ast$ it follows that $$\label{eq commut}
j^U_\ast(\alpha_f) = \rho\circ f_\ast([V])
= (f_U)_\ast([U])\,.$$
Let us consider now the bordism $\partial W=U$. Without loss of generality we can assume that $F(W^\infty)\cap F(W)=\emptyset$ so that we have the proper map $$F\colon W\to M\setminus F(W^\infty)\,.$$ In Alexander-Spanier homology theory, we know that the fundamental class $[U]$ is the image of the fundamental class of the manifold with boundary $W$ under the boundary homomorphism $\partial_\ast$ in the exact homology sequence of the pair $(W,U)$. We obtain the commutative diagram $$\begin{CD}
H_{k+1}^\infty(W,U) @>{\partial_\ast}>>
H_k^\infty(U) @>{i_\ast}>> H_k^\infty(W)\\
&& @VV{(f_U)_\ast}V @VV{F_\ast}V\\
&& H^\infty_k(M\setminus f(U^\infty)) @>{\rho}>>
H^\infty_k(M\setminus F(W^\infty))\\
&& @AA{j^U_\ast}A @AA{j^W_\ast}A\\
&& H_k(M) @>{\operatorname{id}}>> H_k(M)\,,
\end{CD}$$ therefore by (\[eq commut\]) $$j^W_\ast(\alpha_f)=\rho\circ j^U_\ast(\alpha_f)
=F_\ast\circ i_\ast([U])=0,$$ because $[U]\in\operatorname{im}\partial_\ast$. Since $j^W_\ast$ is injective due to $H^\infty_k(F(W^\infty))=0$ it follows that $\alpha_f$ vanishes.
A topological space $S$ is said to have [**covering dimension**]{} at most $n$ if every open cover $\mathfrak{U}=\{U_\alpha\}$ has a refinement $\mathfrak{U}'=\{U_\alpha'\}$ for which all the $(n+2)$-fold intersections are empty[^6], $$U'_B = \bigcap _{\beta\in B} U'_\beta = \emptyset
\quad\text{if } |B| \geqslant n+2\,.$$ We say then ${\dim_{\text{\rm cov}}} S \leqslant n$. Clearly, $S$ having covering dimension at most $n$ implies that $\check H^m(S)=0$ for $m>n$. For a compact space $S$ this implies $H_m(S)=0$ for $m>n$.
We have the following simple
\[lm covering\] Let $f\colon P\to M$ be a smooth map between manifolds and $S$ be a compact subset of $M$ such that $S\subset f(P)$. Then $$\operatorname{\dim_{\text{\rm cov}}}S \leqslant \dim P\,.$$
The final result is
\[cor pseudo-cycle\] Every pseudo-cycle $f\colon V\to M$ of dimension $k$ induces a well-defined integral homology class $\alpha_f\in
H_k(M;\bbz)$. Moreover, any singular cycle $\alpha\in
Z^{\mathsf{sing}}_k(M;\bbz)$ gives rise to a $k$-pseudo-cycle $f\colon V\to M$ such that $\alpha_f=\alpha$.
Note that if $f(V)\subset f(V^\infty)$ then trivially $\alpha_f=0$.
The well-defined homology class $\alpha_f$ follows from combining Theorem \[prop cycle1\] with Lemma \[lm covering\].
Suppose now that $\alpha\in H_k^{\mathsf{sing}}(M;\bbz)$ is a $k$-cycle given by a smooth singular chain. By pairwise identifying and sufficiently smoothing the $k-1$-dimensional faces of the $k$-simplexes involved in $\alpha$ the cycle-property of $\alpha$ implies that we obtain a $k$-dimensional manifold $V$, not necessarily compact, with a smooth structure, such that the singular chain gives a map $f\colon V\to M$ meeting the pseudo-cycle condition, since $f(V^\infty)$ is covered by the images of the faces of codimension $2$ and higher. Two cohomologous singular chains lead to cobordant pseudo-cycles in the sense of Theorem \[prop cycle1\] (b).
Hence, from now on we can represent integral cycles in singular homology by pseudo-cycles.
The Explicit Isomorphism {#sc expl}
========================
The first part consists of showing that each Morse cycle leads to a well-defined pseudo-cycle, and that the associated singular homology class does not depend on any uncanonical choices involved.
Pseudo-cycles from Morse cycles
-------------------------------
Let $(f,g)$ be a Morse-Smale pair and consider the associated homology $H_\ast(f,g)$. The idea of defining the homomorphism into singular homology is to construct a $k$-dimensional pseudo-cycle $E\colon
Z(a)\to M$ for a given Morse cycle $\{a\}=\{\sum_{x\in\operatorname{Crit}_k f}
a_x\,x\in\}\in H_k(f,g)$. This is essentially based on considering the unstable manifolds $W^u(x)$ from (\[eq unstable\]) with multiplicity $a_x\in\bbz$ and their evaluation maps $E_x\colon W^u(x)\to M$. In order to obtain a well-defined pseudo-cycle we have to carry out a suitable identification on the $1$-codimensional strata of a suitable compactification of $W^u(x)$.
Let $x\in\operatorname{Crit}_k f$ and $y\in\operatorname{Crit}_{k-1}f$ such that $\widehat M_{x,y}$ is a nonempty finite set. We say that a sequence $(w_n)\subset W^u(x)$ is [*weakly convergent*]{} towards a simply broken trajectory, $$w_n\rightharpoonup (\hat u,v)\in \widehat M_{x,y}\times W^u(y),$$ if $w_n\to v$ in $C^\infty_{\text{loc}}((-\infty,0],M)$ and there exists a reparametrization sequence $\tau_n\to-\infty$ such that $\tau_n\ast w_n\to u$ in $C^\infty$ on compact subsets of $\bbr$ for a representative $u$ of the unparameterized trajectory $\hat u$. Note that, in particular, $w_n(0)\to v(0)$.
(600,600)(0,0) (0,0) (50,590)[(0,0)\[lb\][$x$]{}]{} (420,375)[(0,0)\[lb\][$y$]{}]{} (250,500)[(0,0)\[lb\][$\hat u$]{}]{} (415,180)[(0,0)\[lb\][$v$]{}]{} (90,300)[(0,0)\[lb\][$w_n$]{}]{} (70,0)[(0,0)\[lb\][$w_n(0)$]{}]{} (420,0)[(0,0)\[lb\][$v(0)$]{}]{}
The following result is completely analogous to the gluing results developed in [@Sch-Morse], Section 2.5. There, the gluing operation has been constructed for trajectory spaces $M_{y,z}$ instead of $W^u(y)$, but the case of unstable manifolds is handled exactly the same. It provides us with the suitable description of strata of the weak compactification of $W^u(x)$.
\[prop gluing1\] Given an open subset $V\subset W^u(y)$ with compact closure there exists a constant $\rho_V>0$ and a smooth map $$\begin{aligned}
\#^V \colon &\widehat M_{x,y}\times V\times
[\rho_V,\infty) \to W^u(x),\\
&(\hat u,v,\rho) \mapsto \hat u\#_\rho v,\end{aligned}$$ such that
- $\#^V$ is an embedding,
- $\#^V(\hat u,\cdot,\cdot)$ is orientation preserving exactly if $\tau(\hat u)=+1$,
- $\hat u\#_\rho v\rightharpoonup (\hat u,v)$ for $\rho\to\infty$, and for any $w_n\rightharpoonup(\hat u,v)$ there exists an $n_o$ such that for all $n\geqslant n_o$ $w_n=\hat
u_n\#_{\rho_n} v_n$ for unique $(\hat u_n,v_n,\rho_n)$, and
- the evaluation maps $E_x\colon W^u(x)\to M$ and $E_y\colon W^u(y)\to M$ extend to $$\bar E_x\colon W^u(x) \cup_{\#^V}
\widehat M_{x,y}\times V\times [\rho_V,\infty)\to M\,.$$ such that $\bar E_x(\hat u,v,\rho)=E_x(\hat u\#_\rho
v)$ for $\rho\in[\rho_v,\infty)$ and $\bar E_x(\hat
u,v,\rho)=E_y(v)$ for $\rho=\infty$.
Let us define $\overline{W}{}^u(x)$ to be the disjoint union $$\label{eq boundary}
\overline{W}{}^u(x) = W^u(x) \cup \bigcup_{\mu(y)=\mu(x)-1}
\widehat M_{x,y}\times W^u(y)$$ equipped with the topology generated by
- the open subsets of $W^u(x)$,
- the neighborhoods of $(\hat u,v)\in\widehat M_{x,y}\times
W^u(y)$ of the form $$\#^V(\{\hat u\}\times V\times (\rho,\infty)) \cup \{\hat
u\}\times V,\quad \rho\geqslant\rho_V,$$ for $V\subset W^u(y)$ open with compact closure.
This provides a Hausdorff topology and we obtain
\[prop gluing2\] $\overline{W}{}^u(x)$ is an oriented manifold with boundary oriented by $\widehat M_{x,y}\times W^u(y)$ and $\bar
E_x\colon\overline{W}{}^u(x)\to M$ is a smooth embedding.
The proof is given below together with the proof of Lemma \[prop cycle\].
Consider now a Morse-cycle $$a\in Z_k(f,g),\quad a=\sum_x a_xx\,.$$ Given $l\in\bbn$ let us denote by $l\cdot\overline{W}{}^u(x)$ the disjoint union of $l$ copies of $\overline{W}{}^u(x)$, that is, the topological sum. If $l\in\bbz$, $l<0$, we replace $\overline{W}{}^u(x)$ by $\overline{W}{}^u(x)^\ast$, that is, with the orientation reversed. Thus, we associate to $a$ the topological sum $$\label{eq topsum}
a\mapsto \amalg_x a_x\cdot\overline{W}{}^u(x)$$ which is a $k$-dimensional oriented manifold with oriented boundary and it consists of $\sum_x |a_x|$ connected components. Observe that this manifold with boundary is not compact in general.
We denote by $\Delta a$ the following finite set of connecting unparametrized trajectories of relative index $1$, $$\Delta a = \bigcup \{\, a_x\hat u\,|\,
\hat u\in\widehat M_{x,y},\,x\in\operatorname{Crit}_k f,\,
y\in\operatorname{Crit}_{k-1}f\,\}$$ where $a_x\hat u$ is the disjoint union of $|a_x|$ copies of $\{\hat
u\}$. Each $\hat u$ carries the sign $\tau(\hat u)\in\{\pm1\}$ and we assign to every $\gamma\in a_x\hat u$ the new sign $\sigma(\gamma)=\operatorname{sgn}(a_x)\cdot
\tau(\hat u)$. Computing $$\partial a = \sum_x \sum_{\mu(y)=\mu(x)-1}
\sum_{\hat u\in\widehat M_{x,y}}
a_x\tau(\hat u)\,y$$ we immediately obtain
\[lm 1-1\] If $a=\sum_x a_x x$ is a Morse-cycle there exists an equivalence relation $\sim_{\Delta a}$ on $\Delta a$ such that for each $\gamma\in\Delta a$ there exists a unique $\gamma'\not=\gamma$ with $\sigma(\gamma')=-\sigma(\gamma)$, so that $\gamma\sim_{\Delta a}\gamma'$ and $\gamma(+\infty)=\gamma'(+\infty)\in\operatorname{Crit}_{k-1}f$ for $\gamma,\gamma'$ viewed as flow trajectories.
Since $\Delta a$ is an index set for the components of the $k-1$-dimensional manifold from (\[eq topsum\]), $\partial(\amalg_x a_x\overline{W}{}^u(x))$ such that $\sigma(\gamma)$ corresponds to the boundary orientation, we obtain the equivalence relation for points $\{\gamma\}\times\{v\}\in a_x \widehat M_{x,y}\times
W^u(y)$, $$\{\gamma\}\times \{v\} \sim_a \{\gamma'\}\times\{v'\}
\quad\stackrel{\mathsf{def}}{\Longleftrightarrow}\quad
\gamma \sim_{\Delta a} \gamma',\; v=v'\,.$$ We define $$\label{eq cycle}
Z(a) = \amalg_x a_x\overline{W}{}^u(x) \big/ \sim_a\,.$$
(900,500)(0,0) (0,0) (190,400)[(0,0)\[lb\][$x$]{}]{} (190,200)[(0,0)\[lb\][$W^u(x)$]{}]{} (730,400)[(0,0)\[lb\][$x'$]{}]{} (730,200)[(0,0)\[lb\][$W^u(x')$]{}]{} (460,260)[(0,0)\[lb\][$y$]{}]{} (305,300)[(0,0)\[lb\][$\gamma$]{}]{} (605,300)[(0,0)\[lb\][$\gamma'$]{}]{} (400,150)[(0,0)\[lb\][$v$]{}]{}
One easily sees that $Z(a)$ is a topological Hausdorff space and clearly the evaluation maps $\bar E_x$ yields $$E\colon Z(a)\to M,\quad
[\gamma,v] \mapsto v(0)\,.$$ In fact, we obtain
\[prop cycle\] The space $Z(a)$ carries the structure of a $k$-dimensional manifold without boundary and $E\colon Z(a)\to M$ is a smooth map.
Let us consider $x,x'\in\operatorname{Crit}_k f$ and $y\in\operatorname{Crit}_{k-1}f$ with $\hat
u\in\widehat M_{x,y}$ and $\hat u'\in\widehat M_{x',y}$ such that $\hat
u\sim_{\Delta a}\hat u'$, in particular, $\tau(\hat u)=-\tau(\hat u')$. Let $v_o\in W^u(y)$ and $V,V'\subset W^u(y)$ be two relatively compact neighborhoods of $v_o$.
In view of Lemma \[prop gluing2\] we consider the following local coordinates at $(\hat u,v_o)\in\partial\overline{W}{}^u(x)$, respectively for Lemma \[prop cycle\] $[(\hat
u,v_o)]_{\sim_a}\in Z(a)=\amalg_x a_x\overline{W}{}^u(x)/\sim_a$, $$\begin{gathered}
\#^V_{\hat u,\hat u'}\colon V\times [0,\epsilon) \to \overline
{W}^u(x),\\
(v,t) \mapsto \begin{cases}
\hat u \#_{-\frac{1}{t}} v, & t<0,\\
(\hat u,v_o) \sim_a (\hat u',v_o), &t=0,\\
\hat u' \#_{\frac{1}{t}} v, & t>0,
\end{cases}\end{gathered}$$ where $\#$ is the gluing map from Proposition \[prop gluing1\] and $\epsilon>0$ is small enough depending on the compact set $\operatorname{cl}(V)$. Thus, we have to show that
- $(\#^{V'}_{\hat u,\hat u'}){}^{-1}\circ \#^V_{\hat u,\hat u'}
\colon U \times (-\epsilon_o,\epsilon_o) \to (V\cap V')
\times \bbr$ is smooth for $U\subset V\cap V'$ and $\epsilon_o<\min(\epsilon,\epsilon')$ sufficiently small, and that
- $E\circ\#^V_{\hat u,\hat u'}\colon V\times
(-\epsilon,\epsilon)\to M$ is smooth at $(v,0)$.
Let us recall the definition of $\hat u\#_\rho v$ from [@Sch-Morse]. Let $\beta^-\colon\bbr\to[0,1]$ be a cut-off function with $$\beta^-(s)=\begin{cases}
1, & s\leqslant -1,\\
0, & s\geqslant 0,
\end{cases}$$ and $\beta^+(s)=\beta^-(-s)$. We write $$\beta^\pm_\rho(s)=\beta^\pm(s+\rho), \quad
\hat u_\rho(s)=\hat u(s+\rho)\,.$$ For every $v\in V$ and $\rho\geqslant \rho_o$ large enough we define $w=w(\hat u,v,\rho)$ by $$w=\begin{cases}
\hat u(s+2\rho), &s\leqslant -\rho-1,\\
\exp_y\big( \beta^-_\rho \exp_y^{-1}\circ\hat u_{2\rho}
+\beta^+_\rho \exp_y^{-1}\circ v \big)(s),
& |s+\rho|<1,\\
v(s), &s\geqslant -\rho+1\,.
\end{cases}$$ In particular, $w(\rho)=y$. One can find a $\rho_V>0$ and a bundle $\pi\colon L^\bot\to V\times [\rho_V,\infty)$ with $L_{(v,\rho)}^\bot
\subset C^\infty(w^\ast TM)$ such that there exists a unique section $\gamma\colon V\times [\rho_V,\infty) \to L^\bot$ providing $$(\hat u\#_\rho v)(s) =\exp_{w(s)} (\gamma(v,\rho)(s)),
\quad \hat u\#_\rho v\in W^u(x)\,.$$ The bundle $L^\bot$ can be completed fiberwise in terms of a Sobolev space yielding a smooth bundle such that $\gamma$ is a smooth section. (Details can be found in [@Sch-Morse].) Moreover, there is an exponential estimate for the correction term $\gamma(v,\rho)$ between $w(v,\rho)$ and $\hat u\#_\rho v$. Namely, there exists a $\sigma>0$ such that $$\label{eq exponential1}
\sup_{s\in\bbr} |\gamma(v,\rho)(s)| \leqslant
c \operatorname{e}^{-\sigma\rho}$$ for some $c>0$ uniformly for $v\in V$. Moreover, also the covariant derivatives of $\gamma$ with respect to $v$ and $\rho$ satisfy such an exponential estimate as $\rho\to\infty$. This is due to the fact that $w(v,\rho)\rightharpoonup (\hat u,v)$ as $\rho\to\infty$ and that the gradient flow trajectories $\hat u$ and $v$ converge exponentially fast towards $y$, $$d(\hat u(s),y),\, d(v(-s),y) \leqslant
c\operatorname{e}^{-\sigma s} \quad\text{as }s\to\infty\,.$$ The construction of $L^\bot$ and $\gamma$ in [@Sch-Morse] is such that $L^\bot\to V\times [\rho_V,\infty)$ and $L^\bot\to V'\times
[\rho_{V'},\infty)$ coincide over $V\cap V'\times
[\max(\rho_V,\rho_{V'}),\infty)$. One obtains a unique smooth gluing map $$\#^{V\cup V'}_{\hat u}\colon (V\cup V')\times
[\max(\rho_V,\rho_{V'}),\infty)\to W^u(x)$$ extending $\#^V$ and $\#^{V'}$ and assertion (A) follows.
Let us consider now the coordinate chart $\phi_V(v,t)=\#^V_{\hat
u,\hat u'}(v,t)$ with the exponential estimate for the correction term $\gamma(v,\pm\frac{1}{t})$, $$\label{eq exponential2}\textstyle
\| \nabla^\alpha\gamma(v,\pm\frac{1}{t})\|_\infty \leqslant
c_\alpha \operatorname{e}^{-|\frac{1}{t}|}\,$$ where $\nabla^\alpha$ are the covariant derivatives of the section $\gamma$ with respect to the variables $v$ and $\rho$. We obtain for the evaluation map $E\colon Z(a)\to M$, $$\begin{aligned}
E \circ \phi_V(v,t) &=\begin{cases}
E_x(\hat u'\#_{(-\frac{1}{t})} v), & t<0,\\
E_y(v), & t=0,\\
E_x(\hat u \#_{\frac{1}{t}} v), & t>0,
\end{cases}\\
&= \begin{cases}
\exp_{v(0)} \big( \gamma(\hat u',v,-\frac{1}{t})(0)\big),
& t<0,\\
v(0), & t=0,\\
\exp_{v(0)} \big( \gamma(\hat u,v,\frac{1}{t})(0) \big),
& t>0\,.
\end{cases}\end{aligned}$$ Thus, the smoothness of $E\circ\phi_V$ follows from (\[eq exponential2\]) and the standard identities for the covariant derivatives of $\exp\colon TM\to M$ at $0_p\in T_p M$.
The next step is to analyze the map $E\colon Z(a) \to M$ with respect to the end of $Z(a)$, because in general $Z(a)$ is not compact. If it is compact, we immediately obtain the well-defined integral homology class $E_\ast([Z(a)])\in H_k(M;\bbz)$ associated to the Morse cycle $a$ of degree $k$.
\[prop pseudo-cycle\] The evaluation map $E\colon Z(a) \to M$ associated to a Morse cycle $a\in Z_k(f,g)$ is a $k$-dimensional pseudo-cycle.
Consider a point $p\in M$ such that $$p\in E(Z(a)^\infty)=\bigcap_{K\underset{\text{cpt.}}{\subset}Z(a)}
\overline{ E(Z(a)\setminus K)}\,.$$ That is, there exists a sequence $(\gamma_n)\subset Z(a)$ such that $\gamma_n(0)\to p$ in $M$ but $(\gamma_n)$ contains no convergent subsequence in $Z(a)$. We can assume that every $\gamma_n$ corresponds to an element in $W^u(x)$ for some $x\in \operatorname{Crit}_kf$ such that $a_x\not=0$ for $a=\sum_x a_x x$. The compactness result for the space of negative gradient flow trajectories provides a convergent subsequence $$\gamma_{n_k} \stackrel{C^\infty_{\text{loc}}}
{\longrightarrow} \gamma \in W^u(z),$$ with $\mu(z)\leqslant \mu(x)$. Since $(\gamma_{n_k})$ does not converge in $$Z(a)=\amalg_x a_x \overline{W}{}^u(x) \big/\sim\,,$$ we obtain $\mu(z) \leqslant \mu(x) -2$. This shows that $E(Z(a)^\infty)$ is covered by the images of the evaluation maps $$\label{eq cover}
E(Z(a)^\infty) \subset
\bigcup\limits_{\mu(z)\leqslant k-2}
\operatorname{im}E_z \subset M\,.$$ Thus, $E\colon Z(a)\to M$ is a pseudo-cycle.
Given a Morse-cycle $a\in Z_k(f,g)$, we denote the by Lemma \[prop pseudo-cycle\] and Theorem \[cor pseudo-cycle\] uniquely determined homology class by $$[a]\in H_k(M;\bbz)\,.$$ Moreover, the map $a\mapsto [a]$ is linear by construction.
\[prop cycle2\] If $a\in Z_k(f,g)$ is a boundary, i.e. $a=\partial b$ for some $b\in
C_{k+1}(f,g)$, then $[a]=0$. That is, the homomorphism $$\label{eq homom}
\Phi_{f,g}\colon H_k(C_\ast(f,g),\partial) \to H_k(M;\bbz),\quad
\{a\} \mapsto [a]$$ is well-defined.
Consider $a=\sum_x a_x x\in Z_k(f,g)$ and $b=\sum_z b_z z\in
C_{k+1}(f,g)$ such that $\partial b=a$. Similar to the construction of $Z(a)$ we now set $$W = \amalg_{z\in \operatorname{Crit}_{k+1}f}
b_z\cdot \overline{W}{}^u(z), \quad
\overline{W}{}^u(z)= W^u(z) \cup \bigcup\limits_{x\in\operatorname{Crit}_k f}
\widehat M_{z,x}\times W^u(x)\,,$$ as in (\[eq boundary\]). We can obtain the boundary of the manifold $\overline{W}{}^u(z)$ as $$\partial\overline{W}{}^u(z) = \amalg_{x\in\operatorname{Crit}_kf}
n(z,x)\cdot\overline{W}{}^u(x)\,,$$ such that by setting $$U=\amalg_{\mu(x)=k} a_x\cdot W^u(x)$$ we obtain the smooth $(k+1)$-dimensional manifold $W$ with boundary $\partial W=U$. Note that $U\subset Z(a)$ is a $k$-dimensional open submanifold with $$E(Z(a)^\infty) \subset E(U^\infty)$$ and, analogously to (\[eq cover\]), $E(U^\infty)$ and $E(W^\infty)$ are covered by the at most $(k-1)$-dimensional submanifolds $$E(W^\infty) \cup E(U^\infty) \subset
\bigcup_{\mu(y)\leqslant k-1} \operatorname{im}E_y\,.$$ Altogether, we have $E(W^\infty)\cap E(U)=\emptyset$ and $H_k(E(W^\infty);\bbz)=0$ so that Theorem \[prop cycle1\] (b) is applicable.
In order to obtain a homomorphism $\Phi\colon
H^{\text{Morse}}_\ast(M)\to H^{\text{sing}}_\ast(M)$ we have to show that the linear maps $\Phi_{f,g}$ are compatible with the canonical isomorphisms from Theorem \[thm continuation\] $$\Phi_{10}\colon H_\ast(f_0,g_0)\stackrel{\cong}{\longrightarrow}
H_\ast(f_1,g_1)$$ from Theorem \[thm continuation\]. For this purpose we first present an alternative construction of a pseudo-cycle associated to a $(f,g)$-Morse cycle representing the same singular homology class.
Let us consider a smooth $1$-parameter family $(f_s,g_s)$ of functions and Riemannian metrics with $-\infty<s\leqslant 0$ such that for some $R>0$ $$(f_s,g_s)=(f,g),\quad \text{for all}\; s<-R\,.$$ The pair $(f,g)$ is Morse-Smale as above. We now redefine for $x\in\operatorname{Crit}f$ $$W^u(x) = \{\,\gamma\colon (-\infty,0]\to M\,|\,
\dot\gamma(s)+\nabla_{g_s}f_s(\gamma(s))=0,\,
\gamma(-\infty)=x\,\}\,.$$ All statements about the prior unstable manifolds remain valid for this non-autonomous flow. Observe that we have the following weak compactness result: Given any sequence $(\gamma_n)\subset W^u(x)$ which contains no convergent subsequence, there exists a subsequence $(n_k)$ and a reparametrization sequence $\tau_k\to-\infty$ such that $$\gamma_{n_k}\stackrel{C^\infty_{\text{loc}}}{\longrightarrow}
\gamma\in W^u(y)\quad\text{and}\quad
\tau_k\ast\gamma_{n_k}\stackrel
{C^\infty_{\text{loc}}}{\longrightarrow} u\in M_{x',y'}(f,g)\,.$$ for some $x,x',y',y\in\operatorname{Crit}f$ with $\mu(x)\geqslant
\mu(x')>\mu(y')\geqslant\mu(y)$. Again, in general, $\gamma_{n_k}$ can converge weakly towards a multiply broken trajectory. If $\mu(y)=\mu(x)-1$ we have $x'=x$ and $y'=y$.
Carrying out the same constructions as before, now based on the deformed unstable manifolds associated to $(f_s,g_s)$, we obtain pseudo-cycles $$\tilde E\colon \tilde Z(a) \to M$$ associated to Morse-cycles $a\in Z_k(f,g)$ thus leading to a homomorphism $$\label{eq defhomom}
\widetilde\Phi_{f,g}\colon H_\ast(C_\ast(f,g),\partial) \to
H_\ast(M;\bbz)\,.$$
\[prop homotopy1\] The homomorphisms $\Phi_{f,g}$ and $\widetilde\Phi_{f,g}$ are identical.
We have to show that the pseudo-cycles $E\colon Z(a)\to M$ and $\tilde
E\colon \tilde Z(a)\to M$ can be related by a suitable pseudo-cycle cobordism such that Theorem \[prop cycle1\] (b) applies.
Since asymptotically constant families $(f_s,g_s)_{s\in(-\infty,0]}$ as above form a convex set we can consider the continuous path $$(f_s,g_s)_\lambda = (1-\lambda)(f,g) + \lambda
(f_s,g_s),\quad \lambda\in I=[0,1]\,.$$ Given $x\in\operatorname{Crit}f$, the space $$W^u_I(x) = \{\,(\lambda,\gamma)\,|\,
\lambda\in I,\,\gamma\in W^u_\lambda(x)\,\}$$ is a smooth manifold of dimension $\mu(x)+1$, where $W^u_\lambda(x)$ is the unstable manifold of $x$ associated to the pair $(f_s,g_s)_\lambda$. Its boundary is the disjoint union of $W^u_1(x)$ and $W^u_0(x)^\ast$, i.e. the latter with reversed orientation. Analyzing the non-compactness of $W^u_I(x)$, we consider a sequence $(\lambda_n,\gamma_n)$ which contains no convergent subsequence. There exists a subsequence $(n_k)$ such that $\lambda_{n_k}\to\lambda$ and either $$\gamma_{n_k}\rightharpoonup (u,\gamma)\in \widehat
M_{x,y}(f,g)\times W^u_\lambda(x)\,,$$ for $\mu(y)=\mu(x)-1$, or $\gamma_{n_k}$ converges in $C^\infty_{\text{loc}}$ towards a $\gamma\in W^u_\lambda(z)$ with $\mu(z)\leqslant\mu(x)-2$.
Moreover, we can prove a $\lambda$-parameterized version of the gluing result in Lemma \[prop gluing1\] yielding a gluing map $$\#^V\colon \widehat M_{x,y}\times V\times [\rho_V,\infty)
\to W^u_I(x)$$ for every relatively compact, open subset $V\subset W^u_I(y)$ and $\mu(y)=\mu(x)-1$. This allows us to build $\overline{W}_I^u(x)$ as in (\[eq boundary\]) and to construct a smooth manifold $Z_I(a)$ together with a smooth map $$E\colon Z_I(a)\to M,\quad
\partial Z_I(a)=\tilde Z(a) - Z(a),$$ which extends the given maps $\tilde E$ and $E$ on the boundary. Since $$E(Z_I(a)^\infty) \subset \bigcup_{\mu(z)\leqslant\mu(x)-2}
\operatorname{im}E_z$$ for $E_z\colon W^u_I(z)\to M$ with $\dim W^u_I(z)\leqslant \mu(x)-1$, we meet the conditions of Theorem \[prop cycle1\] (b).
In view of (\[eq Morse homology\]), we now show that the pseudo-cycle homomorphisms $\Phi_i=\Phi_{(f^i,g^i)}$ are compatible with the canonical isomorphisms $\Phi_{ij}$,
\[prop natural\] The homomorphisms $\Phi_i\colon H_\ast(f^i,g^i) \to H_\ast(M;\bbz)$ are compatible with $(\Phi_{ij})$, that is, $$\Phi_1 \circ \Phi_{10} = \Phi_0$$ for all Morse-Smale pairs $(f^0,g^0)$ and $(f^1,g^1)$.
We have to compare the pseudo-cycles $$E^0\colon Z^0(a) \to M \quad\text{and}\quad
E^1\colon Z^1(\Phi_{10}(a)) \to M$$ for any Morse cycle $a\in Z_k(f^0,g^0)$. That is, we have to show that the $E^i$ can be extended to a suitable cobordism $E\colon W\to M$ such that, again, Theorem \[prop cycle1\] (b) applies.
Let us consider the space similar to $W^u_I(x)$ in the proof of Lemma \[prop homotopy1\], $$W_{\bbr_+}(x)=\{\,(\lambda,\gamma)\,|\,
\lambda\in[0,\infty),\,\gamma\in
W^u(x;f_{s+\lambda},g_{s+\lambda})\,\}$$ for $x\in \operatorname{Crit}f^0$. (If $[0,\infty)$ is replaced by a compact interval we are in the situation of Lemma \[prop homotopy1\].) Now we have to deal with additional non-compactness for $\lambda_n\to\infty$. Let $(\lambda_n,\gamma_n)\subset W_{\bbr_+}(x)$ be such that $\lambda_n\to\infty$. Then, there exists a subsequence $(n_k)$ such that $$\gamma_{n_k}\stackrel{C^\infty_{\text{loc}}}
{\longrightarrow} \gamma\in W^u(x';f^1,g^1)$$ for some $x'\in \operatorname{Crit}f^1$. Necessarily, $\mu(x')\leqslant
\mu(x)$. If both critical points $x$ and $x'$ have equal Morse index then, up to choosing a subsequence, $$(-\lambda_{n_k})\ast\gamma_{n_k} \stackrel
{C^\infty_{\text{loc}}}{\longrightarrow}
u\in M_{x,x'}(f_s,g_s)\,.$$ In that case we denote this weak convergence again by $$(\lambda_{n_k},\gamma_{n_k})\rightharpoonup
(u,\gamma)\,.$$ For the converse, we have a gluing theorem analogous to Lemma \[prop gluing1\]:
Let $\mu(x)=\mu(x')$. Given $V\subset W^u(x')$, an open and relatively compact subset, there exists a $\lambda_V>0$ and a smooth map $$\#^V\colon M_{x,x'}(f_s,g_s)\times V\times
[\lambda_V,\infty) \to W_{\bbr_+}(x)\,,$$ such that the corresponding properties (a)–(d) as in Lemma \[prop gluing1\] hold true. Extending the construction from the proof of Lemma \[prop homotopy1\] based on the $\lambda$-parametrized gluing, we now glue in boundary manifolds to $W_{\bbr_+}$ such that $$\begin{aligned}
\overline{W}_{\bbr_+}(x)= W_{\bbr_+}(x) \,&\cup
\bigcup_{\mu(y)=\mu(x)-1} \big(M_{x,y}(f^0,g^0)\times
W_{\bbr_+}(y)\big)\\
& \cup \bigcup_{\mu(x')=\mu(x)} \big(M_{x,x'}(f_s,g_s)\times
W^u(x';f^1,g^1)\big)\,.\end{aligned}$$ Note that it is not necessary to glue in the codimension-$2$ manifolds $M_{x,y}(f^0,g^0)\times M_{y,y'}(f_s,g_s)\times W^u(y';f^1,g^1)$ for $\mu(y')=\mu(x)-1$. Building the quotient manifold $$\bar Z(a) = \amalg_{x\in\operatorname{Crit}_kf^0}
a_x\cdot \overline{W}_{\bbr_+}(x) \big/ \sim$$ analogously as above, we obtain a smooth $(k+1)$-dimensional manifold with boundary $$\partial \bar Z(a) = Z^o(\Phi_{10}(a)) - Z(a)$$ where $Z^o(\Phi_{10}(a))$ is the open subset $$\bigcup_{x\in\operatorname{Crit}f^0} a_x\cdot
M_{x,x'}\times W^u(x') \subset Z(\Phi_{10}(a))$$ with complementary strata of codimension at least $1$. The evaluation maps $E(\gamma)=\gamma(0)$ extend from the boundary manifolds to $\bar
Z(a)$ and it is straightforward to verify that the conditions for Theorem \[prop cycle1\] (b) are satisfied.
Summing up, we obtain the well-defined homomorphism $$\label{eq morphism}
\Phi\colon H^{\text{Morse}}_\ast(M;\bbz)\to
H^{\text{sing}}_\ast(M;\bbz)\,.$$
### Remarks on compatibility with other equivalences for Morse homology {#ssc other}
In view of the axiomatic approach to Morse homology adopted in [@Sch-Morse], it is straightforward, based on Lemma \[prop natural\], to verify that the homomorphism $\Phi$ is natural. This means it respects functoriality with respect to closed embeddings, w.r.t. changes of Morse functions, and it is compatible with the relative version of Morse homology. Thus, we can refer to the uniqueness result from [@Sch-Morse], mentioned in \[ssc axiom\], in order to conclude that $\Phi$ is in fact the unique, natural isomorphism between Morse homology and singular homology.
Let us also remark that in case of a self-indexing Morse function $f$, i.e. $$\mu(x)=f(x), \;\forall x\in\operatorname{Crit}f,$$ we have the obvious isomorphism $$\label{eq self-index}
\Gamma_k\colon C_k(f)\stackrel{\cong}{\longrightarrow}
H^{\text{sing}}_k(M^k,M^{k-1};\bbz),$$ for $M^a=\{\,p\in M\,|\,f(p)\leqslant a\,\}$, $a\in\bbr$. A classical proof for the equivalence of Morse homology and singular homology[^7] is to show $$\label{eq compat1}
\Gamma_{k-1}\circ \partial(f,g)=\partial_\ast\circ\Gamma_k$$ for the boundary operator in the long exact sequence associated to the decomposition $(M^k,M^{k-1})_{k=0,\ldots,n}$, $$\label{eq compat2}
H_k(M^k,M^{k-1})\stackrel{\partial_\ast}{\longrightarrow}H_{k-1}(M^{k-1},M^{k-2})\,.$$ Obviously, we have for $j\colon H_k(M^k)\to H_k(M^k,M^{k-1})$, $$\label{eq compat3}
j\circ\Phi_{fg}(a)=\Gamma(a),\;\forall a\in Z_k(f,g)$$ so that the induced isomorphism $\Gamma\colon
H_\ast(f,g)\stackrel{\cong}{\longrightarrow}H^{\text{sing}}_\ast(M;\bbz)$ and $\Phi_{f,g}$ are identical.
The Inverse Homomorphism
------------------------
Although it is already clear that the homomorphism $$\Phi\colon H^{\text{Morse}}_\ast(M;\bbz)\to H^{\text{sing}}_\ast(M;\bbz)$$ is a natural isomorphism, let us nevertheless construct its inverse $\Psi=\Phi^{-1}$ [*explicitly*]{} along the same lines as used for $\Phi$.
The main idea is to define an intersection number for pseudo-cycles and stable manifolds $W^s(x)$ for a generic Morse-Smale pair $(f,g)$. Recall from [@Sch-Morse] the construction of a Banach manifold $\Gcal$ of smooth Riemannian metrics, $L^2$-dense in the space of all smooth Riemannian metrics. We consider a metric $g$ to be generic with respect to a certain property, if we can find a residual set $\Rcal\subseteq \Gcal$ of metrics with that property. The first step is
\[thm psistable\] Let $\chi\colon V^k\to M$ be a smooth map of a $k$-dimensional manifold into $M$ and $f$ a Morse function on $M$ such that $\chi(V)\cap\operatorname{Crit}f=\emptyset$ if $k<n$ or $\operatorname{rank}D\chi(p)=n$ for all $p\in\chi^{-1}(\operatorname{Crit}f)$ if $k=n$. Then there exists a residual set $\Rcal\subseteq\Gcal$ such that $$\Mcal_{\chi;x}(f,g)=\{\,(p,\gamma)\in V\times
W^s(x)\,|\,\dot\gamma+\nabla_g f(\gamma)=0,\;
\gamma(0)=\chi(p)\,\}$$ is a smooth manifold of dimension $$\dim \Mcal_{\chi;x}(f,g)=k-\mu(x)$$ for all $x\in\operatorname{Crit}f$ and $g\in\Rcal$. In particular, it is empty if $k<\mu(x)$.
Also, as will be clear from the proof, if $V^k$, $M$ and $W^s(x)$ are oriented, the intersection manifold $\Mcal_{\chi;x}(f,g)$ inherits a well-defined orientation which is a number $\pm1\in\bbz_2$ if $k=\mu(x)$.
The main ingredient of this transversality theorem is the following
\[lm submersion\] The universal stable manifold $$W^s_{\text{univ}}\{\,(\gamma,g)\in
C^\infty([0,\infty),M)\times\Gcal\,|
\dot\gamma+\nabla_g f(\gamma)=0,\,\gamma(+\infty)=x\,\}$$ for $x\in\operatorname{Crit}f$, $f$ a Morse function, admits a submersion $$E\colon W^s_{\text{univ}}(x)\to M,\quad E(\gamma,g)=\gamma(0),$$ away from the critical point $\gamma\equiv x$. It is also a submersion everywhere if $\mu(x)=0$.
Let us recall some analytic constructions from [@Sch-Morse]. The space $$\Hcal^{1,2}_x=H^{1,2}_x([0,\infty),M)$$ is the $H^{1,2}$-Sobolev completion of the space of smooth curves $\gamma\colon[0,\infty)\to M$ with sufficiently fast convergence toward $x\in M$ as $t\to\infty$. It is in fact a Hilbert manifold. The tangent space to the Banach manifold of smooth Riemannian metrics on $M$ is $$T_g\Gcal=\{\,h\in C^\infty_\epsilon(\operatorname{End}(TM))\,|\,h\,\text{symmetric
w.r.t. }g_o\,\}$$ for some fixed Riemannian metric $g_o$. The function space $C^\infty_\epsilon$ is an $L^2$-dense subspace of $C^\infty$ with a Banach space norm. Let us now consider the smooth map $$\begin{gathered}
F\colon \Hcal^{1,2}_x\times\Gcal\to L^2(\Hcal^{1,2}_x{}^\ast TM),\\
F(\gamma,g)=\dot\gamma+(\nabla_g f)\circ\gamma,\end{gathered}$$ where the right hand side space is a Banach space bundle over the manifold $\Hcal^{1,2}_x$ with fiber $L^2(\gamma^\ast TM)$ of $L^2$-vector fields along the curve $\gamma$. Choosing a Riemannian connection $\nabla$ on $TM$ we obtain the linearization of $F$ as $$\begin{aligned}
DF(\gamma,g)(\xi,h)&= DF_1(\gamma,g)(\xi)+ DF_2(\gamma,g)(h),\\
DF_1(\gamma,g)(\xi)&=\nabla_t\xi+(\nabla_\xi\nabla_g
f)\circ\gamma,\\
DF_2(\gamma,g)(h)&=h(\gamma)\cdot\nabla_g f(\gamma)\,.\end{aligned}$$ Observe that $h(\gamma)$ is an endomorphism of the pull-back bundle $\gamma^\ast TM$. Hence, any variation of $h(\gamma(t))$ as a function of time $t$ can be achieved through a variation of $h$ over $M$ if $\gamma$ is injective. Altogether we obtain the tangent space of the universal stable manifold as $$\label{eq tangent}
T_{(\gamma,g)}W^s_{\text{univ}}(x)=\{\,(\xi,h)\,|\,DF(\gamma,g)(\xi,h)=0\,\},$$ because $0$ is a regular value for $F$, as it will be clear below. Given $\gamma(0)=p\in M$ and $(\gamma,g)\in W^s_{\text{univ}}(x)$ we have to show that for each $v\in T_p M$ there exist $(\xi,h)\in
T_{(\gamma,g)}W^s_{\text{univ}}(x)$ such that $\xi(0)=v$, if either
1. $\gamma(0)\not\in\operatorname{Crit}f$, i.e. $\gamma(0)\not=x$, or
2. $\gamma(0)=x$ and $\mu(x)=0$.
In the latter case (b) we have $\gamma\equiv\operatorname{const}=x$ with $$DF_1(x,g)(\xi)=\dot\xi+\operatorname{Hess}f(x)\cdot\xi,\quad DF_2(x,g)(h)=0,$$ where the Hessian at $x\in\operatorname{Crit}_0 f$ is positive definite. This implies $$\ker DF(x,g)=T_xM\times T_g\Gcal,$$ and hence the submersion property of $E$.
In case (a) let us simplify the operator $DF(\gamma,g)$ by using coordinates with respect to an orthonormal parallel frame of $\gamma^\ast TM$. We obtain the operator, $$\label{eq parallel}\begin{split}
D&\colon H^{1,2}([0,\infty),\bbr^n)\times T_g\Gcal\to L^2([0,\infty),\bbr^n)\\
D&(\xi,h)=\dot\xi+A(t)\xi+h\cdot X,
\end{split}$$ where $A\colon [0,\infty)\to S(n,\bbr)$ is a smooth path in the space of symmetric $n\times n$-matrices with $A(\infty)=\operatorname{Hess}f(x)$ and $X\colon [0,\infty)\to\bbr^n$ with $X(t)\not=0$ for all $t\in[0,\infty)$. We shall now prove that for all $\eta\in L^2([0,\infty),\bbr^n)$ and $v\in\bbr^n$ there exist $\xi\in H^{1,2}([0,\infty),\bbr^n)$ and $h\in
C^\infty_o([0,\infty),S(n))$ such that $D(\xi,h)=\eta$ and $\xi(0)=v$. This concludes the proof of (a) in view of the fact that each such $h$ arises from an $h\in T_g\Gcal$ since $\gamma$ is injective if $\gamma(0)\not=x$.
Suppose that there exist $\eta$ and $v$ such that $$\label{eq cokern}
\langle D(\xi,h),\eta\rangle_{L^2}+\langle
\xi(0),v\rangle_{\bbr^n}=0\quad\text{for all }\xi,h\,.$$ This implies that $\eta\in H^{1,2}([0,\infty),\bbr^n)$ and $\dot\eta-A^t(t)\eta=0$ and therefore $\eta\equiv 0$ if $\eta(0)=0$. Moreover, (\[eq cokern\]) implies that $\langle hX,\eta\rangle=0$ for all $h$ and we have $X(t)\not=0$. If $\eta(0)\not=0$ we can find[^8] $h(t)$ with support in $[0,\epsilon)$ such that $\langle
hX,\eta\rangle\not=0$ contradicting (\[eq cokern\]). Hence we obtain $\eta\equiv 0$ and by (\[eq cokern\]) $\langle
\xi(0),v\rangle_{\bbr^n}=0$ for all $\xi$ which implies $v=0$. Since the cokernel of $D$ in $L^2$ is finite-dimensional it follows that $D$ is surjective.
The proof of Theorem \[thm psistable\] now follows from the parameter version of the Sard-Smale theorem. There exists a residual subset $\Rcal\subseteq\Gcal$ such that for each $g\in\Rcal$ the map $$(\chi,E)\colon V^k\times W^s_g(x)\to M\times M$$ intersects the diagonal $\triangle=\{\,(p,p)\,|\,p\in M\}$ transversely. For such a generic $g$, $$\Mcal_{\chi;x}(f,g)=(\chi,E)^{-1}(\triangle)$$ is a smooth manifold of dimension $k+(n-\mu)-n$.
Note that for $k=\mu(x)$ intersections $(p,\gamma)\in \Mcal_{\chi;x}(f,g)$ for a regular $g$ can only occur if $\operatorname{rank}D\chi(p)=k$. Therefore, it is obvious how the solution space $\Mcal_{\chi;x}$ inherits its orientation from an orientation of $V^k$, $M$ and $W^s(x)$.
The main consequence of the intersection theorem \[thm psistable\] is the compactness result
\[cor intersect\] For each $k$-dimensional pseudo-cycle $\chi\colon V^k\to M$ with $\overline{\chi(V)}\cap\operatorname{Crit}f=\emptyset$ if $k<n$, and $\operatorname{rank}D\chi(p)=n$ for all $p\in\chi^{-1}(\operatorname{Crit}f)$ and $\chi(V^\infty)\cap\operatorname{Crit}f=\emptyset$ if $k=n$, there is a residual set of metrics $\Rcal$ such that the intersection set $\Mcal_{\chi;x}(f,g)$ is finite for all $x\in\operatorname{Crit}f$ with $\mu(x)=k$ and $g\in\Rcal$.
Consider a sequence $(p_n,\gamma_n)\subseteq\Mcal_{\chi;x}(f,g)$. After choosing a suitable subsequence we have $$\label{eq intersect1}
\chi(p_n)\to x_o\in \overline{\chi(V)},\quad
\gamma_n\stackrel{C^\infty_{\text{loc}}}{\longrightarrow}\gamma_o\in
W^s(x'),\;\mu(x')\geqslant \mu(x),\;\gamma_o(0)=x_o\,.$$ In the case that $x_o\in \chi(V^\infty)$, we use that $\chi(V^\infty)$ can be covered by a map $\tilde\chi\colon \tilde V^{k-2}\to M$, so that $(p_o,\gamma_o)\in\Mcal_{\tilde\chi;x'}(f,g)$, $\tilde\chi(p_o)=x_o$. Since the intersection of residual sets is residual it follows from Theorem \[thm psistable\] that for a generic $g$ $\Mcal_{\tilde\chi;x'}$ has to be empty by dimensional reasons. Thus $x_o\in \chi(V^\infty)$ can be excluded.
Sharpening the convergence result (\[eq intersect1\]) we can deduce weak convergence towards a broken trajectory $$\label{eq intersect2}
\gamma_n\rightharpoonup (\gamma_o,u_1,\ldots,u_r),\quad
\gamma_o\in W^s(x'),\;u_1\in\Mcal_{x',x_1},\ldots,\;
u_r\in\Mcal_{x_{r-1},x}\,.$$ For such multiply broken trajectories we must have $\mu(x_{i-1})>\mu(x_i)$. Hence, if $k=\mu(x)$ we cannot have $x'\not=x$ and $\Mcal_{\chi;x}(f,g)$ must be compact and hence finite.
Applying the concept of coherent orientations we can now associate to each intersection $(p,\gamma)\in\Mcal_{\chi;x}(f,g)$ a sign $\tau(p,\gamma)$. Given a $k$-dimensional pseudo-cycle $\chi\colon
V^k\to M$ representing a singular cycle $\alpha=\alpha_\chi\in
H^{\text{sing}}_k(M)$ with $k<n$ we can find a Morse function $f$ such that $\operatorname{Crit}f\cap\overline{\chi(V)}=\emptyset$. If $k=n$, after possibly homotoping $\chi$ to a suitable cobordant pseudo-cycle, we can find a Morse function $f$ such that we have only $p\in\chi^{-1}(\operatorname{Crit}f)$ with $\operatorname{rank}D\chi(p)=n$. We then define in view of Theorem \[thm psistable\] for a generic $g$ $$\label{eq psi}
\Psi(\chi)=\sum_{x\in\operatorname{Crit}_k f}
\#_{\text{alg}}\Mcal_{\chi;x}(f,g)\,x\,\in C_k(f,g)\,.$$
\[thm well-defined\] The chain $\Psi(\chi)\in C_k(f,g)$ is a Morse-cycle, and given two cobordant pseudo-cycles $\chi,\chi'$, the associated Morse-cycles are cohomologous, $\Psi(\chi)-\Psi(\chi')=\partial(f,g)b$ for some $b\in C_{k+1}(f,g)$.
Computing $$\begin{gathered}
\partial(f,g)\sum_{x\in\operatorname{Crit}_k f} \#_{\text{alg}}\Mcal_{\chi;x}(f,g)\,x=
\sum_{\mu(y)=k-1} n(\chi;y)\,y,\\
n(\chi;y)=\sum_{\mu(x)=k}\#_{\text{alg}}\Mcal_{\chi;x}(f,g)\#_{\text{alg}}\Mcal_{x;y}(f,g),\end{gathered}$$ we have to show that $$\label{eq well1}
n(\chi;y)=0\,.$$ This follows readily from the $1$-dimensional compactness result for $\Mcal_{\chi;y}$ analogous to (\[eq intersect1\]) and (\[eq intersect2\]) and the corresponding gluing operation. Namely, since $\dim\chi-\mu(y)=1$, non-compactness of $\Mcal_{\chi;y}(f,g)$ for generic $g$ can only occur in terms of simply broken trajectories in the limit. But exactly as for the proof of the fundamental fact $\partial(f,g)^2=0$, the corresponding gluing result completely analogous to Lemma \[prop gluing1\] shows that the oriented number of boundary components of $\Mcal_{\chi;y}(f,g)$ equals $n(\chi;y)$ and has to vanish since each component of $\Mcal_{\chi;y}$ is diffeomorphic to an interval. This proves (\[eq well1\]).
Given a pseudo-cycle cobordism $F\colon W^{k+1}\to M$ in the sense of Theorem \[prop cycle1\] (b), i.e. $\partial F=\chi-\chi'$, we can define the $1$-dimensional manifold $\Mcal_{F;x}(f,g)$ for $x\in\operatorname{Crit}_k f$, and generic $g$. The same compactness-gluing argument as before shows $$\partial(f,g)\sum_{x\in\operatorname{Crit}_k
f}\#_{\text{alg}}\Mcal_{F;x}(f,g)\,x=\Psi(\chi)-\Psi(\chi')\,.$$
The thus well-defined homomorphism $$\Psi_{f,g}\colon H^{\text{sing}}_\ast(M;\bbz)\to H_\ast(f,g)$$ is compatible with the canonical isomorphism $$\Phi_{10}\colon H_\ast(f^0,g^0)\to H_\ast(f^1,g^1)\,.$$ We have
\[cor isomorphism\] Considering the isomorphism $\Phi_{10}$ for generic Morse-Smale pairs, it holds $$\Phi_{10}\circ\Psi_{f^0,g^0}=\Psi_{f^1,g^1},$$ and we have for the well-defined homomorphism $$\Psi\colon H^{\text{sing}}_\ast(M;\bbz)\to
H^{\text{Morse}}_\ast(M;\bbz)$$ the identity $\Psi\circ\Phi=\operatorname{id}_{H^{\text{Morse}}}$.
The proof of the compatibility $\Phi_{10}\circ\Psi_{f^0,g^0}=\Psi_{f^1,g^1}$ can be carried out exactly analogous to that for Lemma \[prop natural\] using the argument from the proofs of Corollaries \[cor intersect\] and \[thm well-defined\].
Consider now the $k$-dimensional pseudo-cycle $E\colon Z(a)\to M$ associated to a Morse-cycle $a\in Z_k(f^0,g^0)$. Let $y\in\operatorname{Crit}_k f^1$. Then in view of Theorem \[thm psistable\] for a Morse-Smale pair $(f^1,g^1)$ with generic $g^1$ we have intersections $(p,\gamma)\in\Mcal_{E;y}(f^1,g^1)$ only for $p\in Z(a)$ on the $k$-dimensional strata which are exactly the unstable manifolds $W^u(x,f^0,g^0)$ in $a=\sum_{x\in\operatorname{Crit}_k f^0}a_x\,x$. We therefore have $$\Mcal_{E;y}(f^1,g^1)=\{\,(\gamma^-,\gamma^+)\in W^u(x,f^0,g^0)\times
W^s(y;f^1,g^1)\,|\, \gamma^-(0)=\gamma^+(0)\,\}\,.$$ Using a homotopy operator as before we can show easily that this is homologically equivalent to the definition of the operator $\Phi_{10}\colon
C_\ast(f^0,g^0)\to C_\ast(f^1,g^1)$, i.e. $$\label{eq iso1}
\Psi_{f^1,g^1}\circ\Phi_{f^0,g^0}=\Phi_{10}\colon
H_\ast(f^0,g^0)\stackrel{\cong}{\longrightarrow}H_\ast(f^1,g^1)\,.$$ This proves $\Psi\circ\Phi=\operatorname{id}_{H^{\text{Morse}}}$.
Using the fact that $H^{\text{Morse}}_\ast(M)\cong
H^{\text{sing}}_\ast(M)$, it follows immediately that the left-inverse $\Psi$ is the inverse of $\Phi$. Thus we have explicit constructions of both isomorphisms in terms of Morse-pseudo-cycle equivalences.
[Mas78]{}
C.C. Conley, *Isolated invariant sets and the [M]{}orse index*, CBMS Reg. Conf. Series in Math., vol. 38, A.M.S., Providence, R.I., 1978.
S.K. Donaldson and P.B. Kronheimer, *The geometry of four-manifolds*, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1990.
A. Floer, *Witten’s complex and infinite dimensional [M]{}orse theory*, J. Diff. Geom. **30** (1989), 207–221.
J.M. Franks, *Morse-smale flows and homotopy theory*, Topology **18** (1979), 199–215.
W. S. Massey, *Homology and cohomology theory*, Monographs and textbooks in pure and applied mathematics, vol. 46, Marcel Dekker, New York, 1978.
J.W. Milnor, *Lectures on the h-cobordism theorem*, Math. Notes, Princeton Univ. Press, Princeton, 1965.
D. McDuff and D. Salamon, *[$J$]{}-holomorphic curves and quantum cohomology*, University Lecture Series, vol. 6, A.M.S., Providence, R.I., 1994.
S. Piunikhin, D. Salamon, and M. Schwarz, *Symplectic [F]{}loer-[D]{}onaldson theory and quantum cohomology*, Contact and Symplectic Geometry (Cambridge 1994) (C.B. Thomas, ed.), Publ. Newton Inst., vol. 8, Cambridge University Press, 1996, pp. 171–200.
M. Schwarz, *An explicit isomorphism between [F]{}loer homology and quantum cohomology*, in preparation.
M. Schwarz, *Morse homology*, Progress in Mathematics, vol. 111, Birkhäuser, Basel, 1993.
M. Schwarz, *A quantum cup-length estimate for symplectic fixed points*, Invent. math. **133** (1998), 353–397.
P. Seidel, *[$\pi_1$]{} of symplectic automorphism groups and invertibles in quantum homology rings*, Geom. funct. anal. **7** (1997), 1046–1095.
E. Witten, *Supersymmmetry and [M]{}orse theory*, J. Diff. Geom. **17** (1982), 661–692.
[^1]: The author was supported in part by NSF Grant \# DMS 9626430.
[^2]: If $M$ is not orientable, choose homology coefficients in $\bbz_2$.
[^3]: i.e. proper and bounded below. In [@Sch-Morse], this property is called coerciveness.
[^4]: This poses no restriction for our application to Morse homology because we consider only cycles lying in the compact sublevel sets $M^a=\{\,p\in M\,|\,f(p)\leqslant a\,\}$ of an exhausting function.
[^5]: Alternatively, we could also use Borel-Moore homology with specified type of supports.
[^6]: compare [@Don-Kron], Section 9.2.3.
[^7]: used in [@Mil-hcobordism]
[^8]: Compare (2.38) in the proof of Proposition 2.30 in [@Sch-Morse]
|
---
abstract: 'Hans Duistermaat was scheduled to lecture in the 2010 School on Poisson Geometry at IMPA, but passed away suddenly. This is a record of a talk I gave at the 2010 Conference on Poisson Geometry (the week after the School) to share some of my memories of him and to give a brief assessment of his impact on the subject.'
address: 'Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA'
author:
- Reyer Sjamaar
date: '2011-10-25'
title: 'Hans Duistermaat’s contributions to Poisson geometry'
---
Johannes Jisse (Hans) Duistermaat (1942–2010) earned his doctorate in 1968 at the University of Utrecht under the direction of Hans Freudenthal. After holding a postdoctoral position at the University of Lund, a professorship at the University of Nijmegen, and a visiting position at the Courant Institute, he returned to Utrecht in 1975 to take over Freudenthal’s chair after the latter’s retirement. He held this chair until his own retirement in 2007. He continued to play a role in the life of the Utrecht Mathematical Institute and kept up his mathematical activities until he was struck down in March 2010 by a case of pneumonia contracted while on chemotherapy for cancer.
I got to know Hans Duistermaat as an undergraduate when I took his freshman analysis course at Utrecht. At that time he taught the course from lecture notes that were in style and content close to Dieudonné’s book [@dieudonne;foundations]. Although this was a smashing success with some students, I think Hans realized he had to reduce the potency so as not to leave behind quite so many of us, and over the years the lecture notes grew into his and Joop Kolk’s still very substantial undergraduate textbook [@duistermaat-kolk;multidimensional-real]. Anyway, I quickly became hooked and realized that I wanted one day to become his graduate student. This came to pass and I finished in 1990 my thesis work on a combination of two of Hans’ favourite topics, Lie groups and symplectic geometry.
The online Mathematics Genealogy Project (accessed 19 December 2010) lists twenty-four PhD students under Duistermaat’s name. What the website does not tell you is that his true adviser was not Freudenthal, but the applied mathematician G. Braun, who died one year before Hans’ thesis work was finished and about whom I have been able to find little information.
The subject of our conference, Poisson geometry, is now firmly ensconced on both sides of the Atlantic as well as on both sides of the equator, as Alan Weinstein observed this week, and Henrique Bursztyn has kindly asked me to speak about Hans Duistermaat’s impact on the field. In a narrow sense Duistermaat contributed very little to Poisson geometry. The subject dearest to his heart was differential equations, although he had an unusual geometric intuition. As far as I know (thanks to Rui Loja Fernandes), the notion of a *Poisson manifold* appears just once in his written work, namely in a book [@duistermaat;discrete-integrable] on discrete dynamical systems on elliptic surfaces, which he finished not long before his death and which has just been published. But Poisson *brackets* can be found in most of his papers, and the fact is that he has contributed many original ideas to the area.
### Bispectral problem for Schrödinger equations {#bispectral-problem-for-schrödinger-equations .unnumbered}
For instance, his paper [@duistermaat-grunbaum;differential-spectral] with Alberto Grünbaum continues to be influential in the literature on integrable systems and noncommutative algebraic geometry. It contains the solution of the at first rather strange-sounding bispectral problem, for what potentials $V(x)$ do the solutions $f(x,\lambda)$ of the equation $-f''+Vf=\lambda f$ satisfy a differential equation in the spectral parameter $\lambda$? A comment on the bibliography of this paper: most mathematicians disregard the classics, but Hans was never afraid to go back to the sources. He was widely read in the older literature on analysis and differential geometry, and used it to great effect in his own writings.
### Resonances {#resonances .unnumbered}
To mention a lesser-cited paper [@duistermaat;nonintegrability], let me remind you of Duistermaat’s insight how resonances in a Hamiltonian system may preclude complete integrability, as explained to us earlier this week by Nguyen Tien Zung.
### Lie III {#lie-iii .unnumbered}
Sometimes the flow of ideas was remarkably indirect. His and Joop Kolk’s book on Lie groups [@duistermaat-kolk;lie-groups;;2000] (which, despite having appeared in Springer’s Universitext series, is not exactly an elementary graduate text) is much closer to Lie’s original point of view pertaining to differential equations than modern treatments such as Bourbaki [@bourbaki;groupes-algebres], which are more algebraic in spirit. Nevertheless the book is notable for several innovations, particularly its proof of Lie’s third fundamental theorem in global form, which I think deserves to become the standard argument and which runs in outline as follows.
The thing to be proved is that for every finite-dimensional real Lie algebra ${{\mathfrak}{g}}$ there exists a simply connected finite-dimensional real Lie group whose Lie algebra is isomorphic to ${{\mathfrak}{g}}$. Choose a norm on ${{\mathfrak}{g}}$. Then the space $P({{\mathfrak}{g}})$ of continuous paths $[0,1]\to{{\mathfrak}{g}}$ equipped with the supremum norm is a Banach space. For each $\gamma$ in $P({{\mathfrak}{g}})$ let $A_\gamma$ be the continuously differentiable path of linear endomorphisms of ${{\mathfrak}{g}}$ determined by the linear initial-value problem $$A_\gamma'(t)={{\operatorname{\mathrm{ad}}}}(\gamma(t))\circ A_\gamma(t),\qquad
A_\gamma(0)={{\operatorname{\mathrm{id}}}}_{{\mathfrak}{g}}.
$$ Duistermaat and Kolk prove that the multiplication law $$(\gamma\cdot\delta)(t)=\gamma(t)+A_\gamma(t)\delta(t)$$ turns $P({{\mathfrak}{g}})$ into a Banach Lie group. Next they introduce a subset $P({{\mathfrak}{g}})_0$, which consists of all paths $\gamma$ that can be connected to the constant path $0$ by a family of paths $\gamma_s$ which is continuously differentiable with respect to $s$ and has the property that $$\int_0^1A_{\gamma_s}(t)^{-1}\frac{\partial\gamma_s}{\partial
s}(t)\,dt=0
$$ for $0\le s\le1$. The subset $P({{\mathfrak}{g}})_0$ is a closed connected normal Banach Lie subgroup of $P({{\mathfrak}{g}})$ of finite codimension, and the quotient $P({{\mathfrak}{g}})/P({{\mathfrak}{g}})_0$ is a simply connected Lie group with Lie algebra ${{\mathfrak}{g}}$!
One of the many virtues of this proof is that it is manifestly functorial: a Lie algebra homomorphism ${{\mathfrak}{g}}\to{{\mathfrak}{h}}$ induces a continuous linear map on the path spaces $P({{\mathfrak}{g}})\to P({{\mathfrak}{h}})$, which is a homomorphism of Banach Lie groups and maps the subgroup $P({{\mathfrak}{g}})_0$ to $P({{\mathfrak}{h}})_0$, and therefore descends to a Lie group homomorphism.
For the details I refer you to the book; also be sure to read the historical and bibliographical notes at the end of the chapter. The global form of Lie’s third theorem appears to be due to É. Cartan [@cartan;troisieme-theoreme-fondamental], whose first proof was based on the Levi decomposition. A later version [@cartan;topologie-groupes-lie] goes approximately as follows: to build a simply connected group $G$ corresponding to ${{\mathfrak}{g}}$, start with the universal covering group $G_0$ of the adjoint group of ${{\mathfrak}{g}}$, and then construct $G$ as a central extension of $G_0$ by the centre of ${{\mathfrak}{g}}$ (viewed as an abelian Lie group). The extension is obtained from a cocycle $G_0\times G_0\to{\mathfrak}{z}({{\mathfrak}{g}})$, which Cartan finds by integrating the infinitesimal cocycle that corresponds to the Lie algebra extension ${\mathfrak}{z}({{\mathfrak}{g}})\to{{\mathfrak}{g}}\to{{\mathfrak}{g}}_0$. This argument has a natural interpretation in the language of differentiable group cohomology, as was shown by Van Est [@vanest;group-cohomology-lie-algebra]. Much earlier Lie himself suggested a different proof: he surmised that the Lie algebra ${{\mathfrak}{g}}$ ought to be linear and that a group with Lie algebra ${{\mathfrak}{g}}$ can therefore be realized as a subgroup of an appropriate general linear group. This line of argument was justified a few years after Cartan by Ado [@ado;representation-lie-algebras].
The point of this for Poisson geometry is that a few years after publication the Duistermaat-Kolk proof became at Alan Weinstein’s suggestion a central feature of Marius Crainic and Rui Loja Fernandes’ resolution of two longstanding problems in differential geometry: the integrability of a Lie algebroid to a Lie groupoid [@crainic-fernandes;integrability-lie], and the integrability of a Poisson manifold to a symplectic groupoid [@crainic-fernandes;integrability-poisson]. Curiously, a very different work of Duistermaat, which I will get to later, also impinges on these integrability problems. My Cornell colleague Leonard Gross has a paper in preparation that adapts the Duistermaat-Kolk argument to certain infinite-dimensional situations.
Let me now discuss in a bit more detail four of Hans Duistermaat’s papers that are of obvious relevance to the topics of this conference, namely those on the spectrum of elliptic operators [@duistermaat-guillemin;spectrum-bicharacteristics], global action-angle variables [@duistermaat;global-action-angle], the quantum-mechanical spherical pendulum [@duistermaat-cushman;quantum-spherical], and the Duistermaat-Heckman theorem [@duistermaat-heckman;variation].
The spectrum of positive elliptic operators and periodic bicharacteristics
==========================================================================
This paper [@duistermaat-guillemin;spectrum-bicharacteristics], coauthored with Victor Guillemin, is Hans Duistermaat’s most cited work according to MathSciNet. It is perhaps also his technically most accomplished paper. The authors consider a compact $n$-dimensional manifold $X$ and a scalar elliptic pseudodifferential operator $P$ of order $1$ on $X$ which is positive selfadjoint. The spectrum of this operator is a discrete set $$0\le\lambda_0\le\lambda_1\le\cdots\le\lambda_j\le\cdots{\longrightarrow}\infty.
$$ By placing a Dirac measure at each eigenvalue we obtain the spectral distribution $\sigma_P=\sum_{j=0}^\infty\delta_{\lambda_j}$. The principal symbol $p$ of $P$ defines a Hamiltonian vector field $H_p$ on the punctured cotangent bundle $T^*X\setminus X$.
The main example to keep in mind is that of the Laplacian $\Delta$ defined with respect to a Riemannian metric on $X$. For a suitable constant $c$ the operator $c-\Delta$ is positive selfadjoint, so the spectral theorem enables us to define a positive square root $P=\sqrt{c-\Delta}$, which is pseudodifferential of order $1$ and whose principal symbol is given by $p(x,\xi)={\lVert\xi\rVert}$ for $x\in X$ and $\xi\in T_xX$. The Hamiltonian flow of $p$ is the geodesic spray of $X$ (at least on the unit sphere bundle).
The operator $P$ is a quantization of the classical observable $p$. As explained for example in [@guillemin;lectures-spectral-elliptic], the classical analogue of an eigenfunction of $P$ is a periodic trajectory of the Hamiltonian vector field $H_p$ and the classical analogue of the eigenvalue is the energy of the trajectory. The periods of $H_p$ form a “lattice” which is dual to the set of eigenvalues of $P$. The purpose of the paper is to make this analogy precise.
Here I must interrupt myself to state that Hans never spoke to me in such terms. Many of us conceive of mathematics as a system of grandiose functorial schemes and profound analogies or correspondences suggested by the mysteries of nature. Hans’ mind worked differently and the word “quantization” never crossed his lips except in jest. Once he told me that on a visit to Moscow early in his career Gelfand asked him what were his chief mathematical goals in life, and he had no idea what to say.
What moved Hans Duistermaat, as far as I can see, was a gregarious and competitive spirit that took him from one collaboration to the next and from one mathematical problem to the next. These are the same qualities that made him a keen chess player, strong enough to have once played former world champion A. Karpov to a draw in a simultaneous match. An early manifestation of this spirit was his eager participation in the sport of kite flying during his boyhood in the Dutch East Indies, now the Republic of Indonesia. The local variant of the entertainment, which Wikipedia tells me is known as kite fighting, required coating the flying line with glass and abrasives for the purpose of ruining one’s playmates’ equipment. Victor Guillemin relates in his acceptance notice for the 2003 Steele Prize how Hans warned him, not for nothing, against getting involved with a *duistere maat* (murky companion).
Let us turn back to microlocal analysis and look at $$\hat{\sigma}_P(t)={\mathscr}{F}\sigma_P=\sum_{j=0}^\infty e^{-i\lambda_jt},
$$ the Fourier transform of the spectral distribution $\sigma_P$. This can be seen as the distributional trace of the unitary operator $e^{-itP}$, which is a Fourier integral operator. A preliminary result says that $\hat{\sigma}_P$ is a tempered distribution, and therefore so is $\sigma_P$. In particular the eigenvalue counting function $$N_P(\lambda)=\sharp\{\,j\mid\lambda_j\le\lambda\,\}
$$ does not grow faster than a power of $\lambda$, which foreshadows Weyl’s law. The next result is a first hint at the connection between periods and eigenvalues.
$\hat{\sigma}_P$ is $C^\infty$ outside the set of periods of periodic trajectories of $H_p$.
In other words, the singular support of $\hat{\sigma}_P$ is contained in the set of periods of the Hamiltonian $p$. This is suggestive of the Poisson summation formula, where one Fourier transforms a sum of delta functions supported on a lattice and finds a sum of delta functions supported on the dual lattice. The results of Duistermaat and Guillemin describe the singularities of $\hat{\sigma}_P$ and can be viewed as a generalization of this elementary fact.
Every orbit is periodic of period $0$, so one expects $\hat{\sigma}_P$ to have a big singularity at $t=0$. To focus on this singularity take a smooth function $\chi$ such that $\hat{\chi}={\mathscr}{F}\chi$ is a bump function equal to $1$ in a small neighbourhood of $0$. Then $$\hat{\chi}(t)\hat{\sigma}_P(t)=\sum_{j=0}^\infty
e^{-i\lambda_jt}\hat{\chi}(t)
={\mathscr}{F}\biggl(\sum_j\chi(\lambda-\lambda_j)\biggr).
$$
We have an asymptotic expansion $$\sum_j\chi(\lambda-\lambda_j)\sim\frac1{(2\pi)^n}\sum_{k=0}^\infty
c_k\lambda^{n-1-k}
$$ as $\lambda\to\infty$. The constants $c_k$ are independent of $\chi$. The leading coefficient is $$c_0={{\operatorname{\mathrm{vol}}}}\{(x,\xi)\in T^*X\mid p(x,\xi)=1\}.
$$
This yields all sorts of information about the spacing of the eigenvalues, for example the following version of Weyl’s law, which says that the volume of phase space is asymptotically proportional to the number of eigenvalues.
We have an asymptotic expansion $$N_P(\lambda)=\frac{a}{(2\pi)^n}\lambda^n+O(\lambda^{n-1})
$$ as $\lambda\to\infty$, where $a={{\operatorname{\mathrm{vol}}}}\{(x,\xi)\mid p(x,\xi)\le1\}$.
A further analysis leads to a “residue formula”, which describes the poles of $\hat{\sigma}_P$ at nonzero periods.
Let $T\ne0$. Assume that all periodic orbits of $H_p$ of period $T$ are isolated and nondegenerate. Then $$\lim_{t\to
T}(t-T)\hat{\sigma}_P(t)=\sum_\gamma\frac{T_{0,\gamma}}{2\pi}
\frac{i^{m_\gamma}}{{\bigl\lvert{{\operatorname{\mathrm{det}}}}(I-d\Pi_\gamma)\bigr\rvert}^{1/2}}.
$$
There is a close resemblance between this formula and the Lefschetz formula for elliptic complexes of Atiyah and Bott [@atiyah-bott;lefschetz-fixed-elliptic-complex]. The sum on the right is over all closed orbits $\gamma$ of period $T$; $T_{0,\gamma}$ is the primitive period of $\gamma$; and $\Pi_\gamma$ is the Poincaré return map of $\gamma$. Nondegeneracy of $\gamma$ means that ${{\operatorname{\mathrm{det}}}}(I-d\Pi_\gamma)\ne0$. The integer $m_\gamma$ is a Maslov index. For $P=\sqrt{c-\Delta}$ it is the Morse index of the geodesic $\gamma$ for the Euler-Lagrange functional. Eckhard Meinrenken showed in an early paper [@meinrenken;conley-zehnder] that for general $P$ the number $m_\gamma$ can be interpreted as a Conley-Zehnder index.
Duistermaat and Guillemin did not do this work in isolation. Some of the most important prior mathematical work on the subject is that of Weyl [@weyl;verteilungsgesetz], which was inspired by Planck’s model of black-body radiation, and Hörmander [@hormander;spectral-elliptic;acta]. Roughly contemporaneous work includes that of Gutzwiller [@gutzwiller;periodic-quantization], Colin de Verdière [@colin-de-verdiere;spectre-laplacien], and Chazarain [@chazarain;formule-de-poisson]. See [@arendt;weyl] for a historical survey that takes in a good deal of the physics literature. For later developments the reader can consult the Fourier volume in honour of Colin de Verdière, particularly Colin’s own contribution [@colin-de-verdiere;spectrum-laplace] to that volume.
On global action-angle variables
================================
I will give a slightly anachronistic account of Duistermaat’s paper [@duistermaat;global-action-angle] on monodromy in integrable systems, which takes into consideration later work of Dazord and Delzant [@dazord-delzant;actions-angles].
Let $B$ be a connected $n$-manifold. A *Lagrangian fibre bundle* over $B$ is a triple ${\mathscr}{L}=(M,\omega,\pi)$, where $(M,\omega)$ is a symplectic $2n$-manifold and $\pi\colon M\to B$ a surjective submersion with Lagrangian fibres. To keep things simple we will assume that the fibres of $\pi$ are compact and connected.
The standard way to obtain such a bundle is to start with an integrable Hamiltonian system and throw out the singularities of the energy-momentum map. The simplest Lagrangian fibre bundle over a given base $B$ is as follows.
Let $p\colon B\to{{{\text{\bfR}}}}^n$ be a local diffeomorphism. Let ${{{\text{\bfT}}}}$ be the circle ${{{\text{\bfR}}}}/{{{\text{\bfZ}}}}$. The *angle form* on ${{{\text{\bfT}}}}$ is $dq$, where $q$ is the coordinate on ${{{\text{\bfR}}}}$. Let $M$ be the product $B\times{{{\text{\bfT}}}}^n$, equipped with the symplectic form $\omega=\sum_{j=1}^ndp_j\wedge
dq_j$. Let $\pi\colon M\to B$ be the projection onto the first factor. The functions $p_j$ are the *action variables* and the (multivalued) functions $q_j$ are the *angle variables*. The map $p\circ\pi$ is a momentum map for the translation action of ${{{\text{\bfT}}}}^n$ on the second factor of $M$.
An *isomorphism* of Lagrangian fibre bundles over $B$ is given by a symplectomorphism of the total spaces that induces the identity map on the base. There is an equally obvious notion of *localization*, that is restriction of a Lagrangian fibre bundle to an open subset of the base. The Liouville-Mineur-Arnold theorem states that every Lagrangian fibre bundle admits local action-angle variables, i.e. is locally isomorphic to $B\times{{{\text{\bfT}}}}^n$. The problem solved by Duistermaat, and before him in special cases by Nehoro[š]{}ev [@nehorosev;action-angle], is when a Lagrangian fibre bundle over $B$ admits global action-angle variables, i.e. is globally isomorphic to $B\times{{{\text{\bfT}}}}^n$.
A Lagrangian fibre bundle admits global action-angle variables if and only if two invariants, $\mu(P)$ (the affine monodromy) and $\lambda({\mathscr}{L})$ (the Lagrangian class), vanish.
(This is not quite the formulation given by Duistermaat. The Lagrangian class was introduced later by Dazord and Delzant. The quantity called monodromy by Duistermaat is what I will call here the linear monodromy, which ignores the translational part of the affine monodromy. See below for a full discussion.)
Just as interesting as this theorem is the fact that many commonplace integrable systems do *not* admit global action-angle variables, for instance Huygens’ spherical pendulum, which Duistermaat analyses in detail. Let me now explain the two invariants.
Monodromy {#monodromy .unnumbered}
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Let ${\mathscr}{L}=(M,\omega,\pi)$ be a Lagrangian fibre bundle over $B$. The map $TM\to T^*M$ given by $v\mapsto\iota(v)\omega$ is a bundle isomorphism, and we denote its inverse by $\omega^\sharp\colon T^*M\to
TM$. Let $m\in M$ and put $b=\pi(m)\in B$. Given a covector $\alpha\in T_b^*B$, the projection and the symplectic form produce a tangent vector $v_m(\alpha)$, $$T_b^*B\overset{\pi^*}{\longrightarrow}T_m^*M\overset{\omega^\sharp}{\longrightarrow}T_mM,\qquad\alpha\longmapsto\pi^*(\alpha)\longmapsto
\omega^\sharp\pi^*(\alpha)=v_m(\alpha).
$$ Since we can write $\alpha=d_bf$ for a suitable function $f$, we see that $v_m(\alpha)$ is the value at $m$ of the Hamiltonian vector field $H_{\pi^*f}$, and therefore is tangent to the fibre $\pi^{-1}(b)$. The fibre being compact, the vector field $v(\alpha)$ is complete, and we denote by ${\varphi}_b(\alpha)\colon\pi^{-1}(b)\to\pi^{-1}(b)$ its time $1$ flow. The map $${\varphi}_b\colon T_b^*B\times\pi^{-1}(b){\longrightarrow}\pi^{-1}(b)$$ defined by ${\varphi}_b(\alpha,m)={\varphi}_b(\alpha)(m)$ is an action of the abelian Lie group $T_b^*B\cong{{{\text{\bfR}}}}^n$ on $\pi^{-1}(b)$. The map $\alpha\mapsto v_m(\alpha)$ is an isomorphism $T_b^*B\to
T_m(\pi^{-1}(b))$, so, the fibre $\pi^{-1}(b)$ being connected, we conclude that the action ${\varphi}_b$ is transitive and locally free. The kernel of the action $P_b\cong{{{\text{\bfZ}}}}^n$ is the *period lattice* at $b$. Collecting these fibrewise actions gives us an action $${\varphi}\colon T^*B\times_BM{\longrightarrow}M$$ of the bundle of Lie groups $T^*B\to B$ on the bundle $M\to B$. The kernel of this bundle action is the bundle of free abelian groups $P=\coprod_bP_b$ over $B$, called the *period bundle*. The fibrewise quotient $$T=T^*B/P$$ is a bundle over $B$ with general fibre the torus ${{{\text{\bfT}}}}^n$ and structure group ${{\operatorname{\mathrm{Aut}}}}({{{\text{\bfT}}}}^n)\cong{{{\text{\bfGL}}}}(n,{{{\text{\bfZ}}}})$. The quotient action $${\varphi}_T\colon T\times_BM{\longrightarrow}M,$$ which we will write as ${\varphi}_T(t,m)=t\cdot m$, makes $M$ a *$T$-torsor*, a principal homogeneous space for the torus bundle $T$ in the sense that the map $T\times_BM\to M\times_BM$ defined by $(t,m)\mapsto(m,t\cdot m)$ is a diffeomorphism.
The $T^*B$-action defines for each $1$-form on the base $\alpha\in\Omega^1(B)$ a diffeomorphism ${\varphi}(\alpha)$ from $M$ to itself which induces the identity on $B$. This diffeomorphism transforms the symplectic form as follows.
${\varphi}(\alpha)^*\omega=\omega+\pi^*d\alpha$ for every $\alpha\in\Omega^1(B)$.
Recall that a Lagrangian section of a cotangent bundle is the same as a closed $1$-form. Since sections of $P$ induce the identity map on $M$, the lemma tells us therefore that $P$ is a Lagrangian submanifold of $T^*B$.
This has various desirable consequences. First of all, applying the lemma to the translation action of $T^*B$ on itself we conclude that the standard symplectic form is preserved by the $P$-action and so descends to a symplectic form $\omega_T$ on $T$. Thus the Lie group bundle $T$ itself is a Lagrangian fibre bundle over $B$.
More importantly, we see that on any sufficiently small open subset $U$ of the base there exists a coordinate system $p=(p_1,p_2,{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},p_n)$ such that $${\mathscr}{F}(p)=(dp_1,dp_2,{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},dp_n)$$ is a frame of the local system $P|U$. These preferred coordinate systems determine what following recent usage I will call a *tropical affine structure* on $B$, that is an atlas with values in the pseudogroup defined by the *tropical affine group* ${{{\text{\bfG}}}}={{{\text{\bfGL}}}}(n,{{{\text{\bfZ}}}})\ltimes{{{\text{\bfR}}}}^n$. (See e.g.[@gross;tropical-mirror Chapter 1].) Conversely, this atlas determines the Lagrangian lattice bundle $P$.
Analytic continuation of the coordinate system $p$ along a loop $\gamma$ in $B$ based at $b\in U$ gives a new coordinate system $p'$ at $b$, which is related to $p$ by a transformation $g_\gamma\in{{{\text{\bfG}}}}$. The corresponding local frames ${\mathscr}{F}(p)$ and ${\mathscr}{F}(p')$ of $P$ are related by the linear part $g_{0,\gamma}\in{{{\text{\bfG}}}}_0={{{\text{\bfGL}}}}(n,{{{\text{\bfZ}}}})$ of the affine transformation $g_\gamma$. The map $\gamma\mapsto g_\gamma$ induces a homomorphism from $\pi_1(B,b)$ to ${{{\text{\bfG}}}}$. The conjugacy class of this homomorphism, $$\mu(P)\in{{\operatorname{\mathrm{Hom}}}}(\pi_1(B),{{{\text{\bfG}}}})/{{\operatorname{\mathrm{Ad}}}}({{{\text{\bfG}}}})\cong H^1(B,{{{\text{\bfG}}}}),
$$ is the *affine monodromy* of $P$. (Here $H^1(B,{{{\text{\bfG}}}})$ denotes the cohomology set of $B$ with coefficients in the group ${{{\text{\bfG}}}}$ equipped with the discrete topology.) The conjugacy class defined by the map $\gamma\mapsto g_{0,\gamma}$, $$\mu_0(P)\in{{\operatorname{\mathrm{Hom}}}}(\pi_1(B),{{{\text{\bfG}}}}_0)/{{\operatorname{\mathrm{Ad}}}}({{{\text{\bfG}}}}_0)\cong H^1(B,{{{\text{\bfG}}}}_0),
$$ is the *linear monodromy*, which determines the isomorphism class of the local system $P$. The monodromy depends only on the affine structure of $B$, not on $M$ or its symplectic structure.
The linear monodromy $\mu_0(P)$ is trivial if and only if the local system $P$ is trivial. In that case $T\cong B\times{{{\text{\bfT}}}}^n$ is isomorphic to a trivial bundle of Lie groups, $M$ is a principal ${{{\text{\bfT}}}}^n$-bundle over $B$, and $P$ has a global frame of closed $1$-forms $(\alpha_1,\alpha_2,{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},\alpha_n)$. We can then find a covering $f\colon\tilde{B}\to B$ of the base and a local diffeomorphism $\tilde{p}\colon\tilde{B}\to{{{\text{\bfR}}}}^n$ such that $f^*\alpha_j=d\tilde{p}_j$. The full monodromy $\mu(P)$ is trivial if and only if $P$ is trivial and the $\alpha_j$ are exact. If that happens we can define global single-valued action variables $p\colon
B\to{{{\text{\bfR}}}}^n$, and $p\circ\pi$ is a momentum map for the ${{{\text{\bfT}}}}^n$-action on $M$.
Chern class and Lagrangian class {#chern-class-and-lagrangian-class .unnumbered}
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The existence of global angle variables on a Lagrangian fibre bundle ${\mathscr}{L}=(M,\omega,\pi)$ is tantamount to the existence of a global Lagrangian section of $\pi\colon M\to B$.
First let us consider plain smooth sections of $\pi$. We need to introduce a few sheaves of abelian groups on the base space $B$. There is $\Omega^k$, the sheaf of smooth $k$-forms, and its subsheaf ${\mathscr}{Z}^k$ of closed $k$-forms. Then there is the sheaf of smooth sections of $T$, which we will call ${\mathscr}{T}$, and the sheaf of locally constant sections of $P$, which we will call ${\mathscr}{P}$. Let $\{U_i\}_{i\in I}$ be an open cover of $B$ and suppose that we have local smooth sections $s_i\colon U_i\to M$ of $\pi$. Since $M$ is a $T$-torsor, over each intersection $U_{ij}=U_i\cap U_j$ we have a unique section $t_{ij}\in{\mathscr}{T}(U_{ij})$ such that $s_i=t_{ij}\cdot
s_j$. The tuple $t=(t_{ij})$ is a Čech $1$-cocycle and defines an element $[t]\in H^1(B,{\mathscr}{T})$.
Since $T$ is the quotient bundle $T^*B/P$, on the level of sheaves we have a short exact sequence $$0{\longrightarrow}{\mathscr}{P}{\longrightarrow}\Omega^1{\longrightarrow}{\mathscr}{T}{\longrightarrow}0.
$$ The sheaf $\Omega^1$ is fine, so the long exact cohomology sequence gives canonical isomorphisms $$H^k(B,{\mathscr}{T})\cong H^{k+1}(B,{\mathscr}{P})$$ for all $k\ge0$. The image $c(M)\in H^2(B,{\mathscr}{P})$ of $[t]$ is the *Chern class* of the $T$-torsor $M$ and it is the obstruction to the existence of a global section of $\pi$. It is independent of the symplectic structure on $M$.
Since $P$ is Lagrangian, the sheaf ${\mathscr}{P}$ is a subsheaf of ${\mathscr}{Z}^1$, and therefore the exterior derivative $d\colon\Omega^1\to{\mathscr}{Z}^2$ descends to a morphism $$d_P\colon{\mathscr}{T}{\longrightarrow}{\mathscr}{Z}^2.$$ A section $t$ of $T$ is *closed* if $d_Pt=0$. If the open sets $U_i$ are small enough, we can choose the local sections $s_i$ to be Lagrangian, which implies that the transition functions $t_{ij}$ are closed. Thus the $t_{ij}$ are sections of the subsheaf ${\mathscr}{K}=\ker(d_P)$ of ${\mathscr}{T}$, and the corresponding cohomology class lives in $H^1(B,{\mathscr}{K})$. This is the *Lagrangian class* $\lambda({\mathscr}{L})$, which is implicit in the paper of Dazord and Delzant but was named by Zung [@zung;topology-integrable-hamiltonian], and it is the obstruction to the existence of a global *Lagrangian* section of $\pi$. Given a Lagrangian section $s$, the map $T^*B\to M$ defined by $(b,\alpha)\mapsto{\varphi}(\alpha)(s(b))$ identifies the Lagrangian fibre bundle $T$ with ${\mathscr}{L}$.
Therefore the vanishing of the Lagrangian class $\lambda({\mathscr}{L})$ is equivalent to ${\mathscr}{L}$ being isomorphic as a Lagrangian fibre bundle to $T$. The vanishing of both $\lambda({\mathscr}{L})$ and the affine monodromy $\mu(P)$ is equivalent to the existence of global action-angle variables. This is the version of Duistermaat’s theorem established by Dazord and Delzant (who, by the way, also considered the case of less than fully integrable systems).
Symplectic torsors {#symplectic-torsors .unnumbered}
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Dazord and Delzant went on to prove that the Lagrangian class completely classify all Lagrangian fibre bundles on a tropical affine manifold. Let us widen our view a little by fixing a tropical affine manifold $B$ with period bundle $P$ and torus bundle $T=T^*B/P$, and examining arbitrary $T$-torsors over $B$. Any such torsor $\pi\colon
M\to B$ has a well-defined Chern class $c(M)\in H^2(B,{\mathscr}{P})$, where as before ${\mathscr}{P}$ is the locally constant sheaf of sections of $P$. In fact, just as for principal bundles the cohomology group $H^2(B,{\mathscr}{P})$ classifies $T$-torsors up to isomorphism. (If the linear monodromy $\mu_0(P)$ vanishes, then $P$ is the constant local system ${{{\text{\bfZ}}}}^n$, a $T$-torsor is an ordinary principal ${{{\text{\bfT}}}}^n$-bundle, and the Chern class is the ordinary Chern class in $H^2(B,{{{\text{\bfZ}}}}^n)$.)
Let us think about all possible symplectic forms $\omega$ on $M$ which vanish on the fibres of $\pi$, so making ${\mathscr}{L}=(M,\omega,\pi)$ into a Lagrangian fibre bundle. We will call ${\mathscr}{L}$ a *symplectic $T$-torsor* with *total space* $M$. As before we regard two symplectic $T$-torsors as isomorphic if the total spaces are symplectomorphic via a diffeomorphism that fixes the base $B$. The collection of isomorphism classes $[{\mathscr}{L}]$ is an analogue of the Picard group of an algebraic variety and we will denote it by ${{{\text{\bfPic}}}}(B,P)$.
The set ${{{\text{\bfPic}}}}(B,P)$ is equipped with two algebraic operations. The *opposite* of ${\mathscr}{L}=(M,\omega,\pi)$ is $-{\mathscr}{L}=(M,-\omega,\pi)$. (Negating the symplectic form has the effect of reversing the $T$-action, i.e. composing it with the automorphism $t\mapsto t^{-1}$ of $T$.) Given two symplectic $T$-torsors ${\mathscr}{L}_1=(M_1,\omega_1,\pi_1)$ and ${\mathscr}{L}_2=(M_2,\omega_2,\pi_2)$, define $M$ to be the $T$-torsor $(M_1\times_BM_2)/T^-$, where $T^-$ is the antidiagonal subbundle $\{(t,t^{-1})\mid t\in T\}$ of $T\times_BT$. It is a theorem of Ping Xu [@xu;morita-groupoid] that the form $\omega_1+\omega_2$ on $M_1\times_BM_2$ descends to a symplectic form $\omega$ on $M$ which makes ${\mathscr}{L}=(M,\omega,\pi)$ into a symplectic $T$-torsor. We call ${\mathscr}{L}$ the *sum* of ${\mathscr}{L}_1$ and ${\mathscr}{L}_2$. The operation $[{\mathscr}{L}_1]+[{\mathscr}{L}_2]=[{\mathscr}{L}_1+{\mathscr}{L}_2]$ turns ${{{\text{\bfPic}}}}(B,P)$ into an abelian group. The zero element is $[T]$ and the opposite of $[{\mathscr}{L}]$ is $[-{\mathscr}{L}]$.
Can we explicitly describe the Picard group ${{{\text{\bfPic}}}}(B,P)$? The Poincaré lemma implies that $$0{\longrightarrow}{\mathscr}{Z}^k{\longrightarrow}\Omega^k\overset{d}{\longrightarrow}{\mathscr}{Z}^{k+1}{\longrightarrow}0
$$ is a short exact sequence of sheaves. To begin with, this gives us isomorphisms $$H^l(B,{\mathscr}{Z}^k)\cong H^{k+l}(B,{{{\text{\bfR}}}}),$$ because $\Omega^k$ is fine. Furthermore, taking $k=1$ and dividing the first two terms by ${\mathscr}{P}$ we get the short exact sequence $$0{\longrightarrow}{\mathscr}{Z}^1/{\mathscr}{P}{\longrightarrow}{\mathscr}{T}\overset{d_P}{\longrightarrow}{\mathscr}{Z}^2{\longrightarrow}0.
$$ This identifies the kernel ${\mathscr}{K}=\ker(d_P)$ with ${\mathscr}{Z}^1/{\mathscr}{P}$ and yields a long exact sequence $$\begin{gathered}
0{\longrightarrow}H^0(B,{\mathscr}{K}){\longrightarrow}H^0(B,{\mathscr}{T})\overset{d_{P,*}}{\longrightarrow}H^0(B,{\mathscr}{Z}^2)\overset{\partial}{\longrightarrow}H^1(B,{\mathscr}{K}){\longrightarrow}H^1(B,{\mathscr}{T})
\\
\overset{d_{P,*}}{\longrightarrow}H^1(B,{\mathscr}{Z}^2)\overset{\partial} {\longrightarrow}H^2(B,{\mathscr}{K}){\longrightarrow}\cdots
$$ Substituting $H^k(B,{\mathscr}{Z}^2)\cong H^{k+2}(B,{{{\text{\bfR}}}})$ and $H^k(B,{\mathscr}{T})\cong H^{k+1}(B,{\mathscr}{P})$, and noticing that $H^k(B,{\mathscr}{K})\to H^k(B,{\mathscr}{T})\cong H^{k+1}(B,{\mathscr}{P})$ is the connecting homomorphism $\delta$ for the short exact sequence $$0{\longrightarrow}{\mathscr}{P}{\longrightarrow}{\mathscr}{Z}^1{\longrightarrow}{\mathscr}{K}{\longrightarrow}0,
$$ we obtain the long exact sequence that we want, $$\begin{gathered}
0{\longrightarrow}H^0(B,{\mathscr}{K})\overset{\delta}{\longrightarrow}H^1(B,{\mathscr}{P})\overset{d_{P,*}}{\longrightarrow}H^2(B,{{{\text{\bfR}}}})\overset{\partial}{\longrightarrow}H^1(B,{\mathscr}{K})\overset{\delta}{\longrightarrow}H^2(B,{\mathscr}{P})
\\
\overset{d_{P,*}}{\longrightarrow}H^3(B,{{{\text{\bfR}}}})\overset{\partial}{\longrightarrow}H^2(B,{\mathscr}{K})\overset{\delta}{\longrightarrow}\cdots
$$ If a $T$-torsor $M$ admits a symplectic form $\omega$ vanishing on the fibres, then $\delta$ maps the Lagrangian class $\lambda(M,\omega,\pi)$ to the Chern class $c(M)$, and therefore $d_{P,*}c(M)=0$. So we see that $d_{P,*}c(M)=0$ is a necessary condition for $M$ to be the total space of a symplectic $T$-torsor. Dazord and Delzant show that this condition is actually sufficient, and that every $\lambda\in H^1(B,{\mathscr}{K})$ satisfying $\delta\lambda=c(M)$ is the Lagrangian class of a unique isomorphism class of Lagrangian fibre bundles with total space $M$. The conclusion is as follows.
Let $B$ be a tropical affine manifold with period bundle $P$. Let $T$ be the torus bundle $T^*B/P$, let ${\mathscr}{T}$ be the sheaf of smooth sections of $T$, and let ${\mathscr}{K}$ be the kernel of the sheaf homomorphism $d_P\colon{\mathscr}{T}\to{\mathscr}{Z}^2$.
1. The map ${{{\text{\bfPic}}}}(B,P)\to H^1(B,{\mathscr}{K})$ defined by $[{\mathscr}{L}]\mapsto\lambda({\mathscr}{L})$ is a group isomorphism.
2. We have a short exact sequence $$0{\longrightarrow}H^2(B,{{{\text{\bfR}}}})/d_{P,*}H^1(B,{\mathscr}{P}){\longrightarrow}{{{\text{\bfPic}}}}(B,P){\longrightarrow}\ker(d_{P,*}){\longrightarrow}0.
$$
This theorem gives us two different descriptions of the identity component of the Picard group, namely ${{{\text{\bfPic}}}}^0(B,P)$ is equal to the group of symplectic torsors of “degree” (i.e. Chern class) $0$, and $${{{\text{\bfPic}}}}^0(B,P)\cong H^2(B,{{{\text{\bfR}}}})/d_{P,*}H^1(B,{\mathscr}{P}).$$ The “Néron-Severi group” (i.e. component group) ${{{\text{\bfPic}}}}(B,P)/{{{\text{\bfPic}}}}^0(B,P)$ is isomorphic to the subgroup $\ker(d_{P,*})$ of $H^2(B,{\mathscr}{P})$. If the base $B$ is of finite type, the Picard group is finite-dimensional and the Néron-Severi group is finitely generated.
Suppose that we are given a $T$-torsor $\pi\colon M\to B$ with Chern class $c\in H^2(B,{\mathscr}{P})$ and let us denote by ${{{\text{\bfPic}}}}(M,B,P)\cong\delta^{-1}(c)$ the collection of isomorphism classes of symplectic $T$-torsors with total space $M$. The theorem tells us that ${{{\text{\bfPic}}}}(M,B,P)$ is nonempty if and only if $d_{P,*}c=0$ and that the group ${{{\text{\bfPic}}}}^0(B,P)$ acts simply transitively on ${{{\text{\bfPic}}}}(M,B,P)$. The ${{{\text{\bfPic}}}}^0(B,P)$-action is the *gauge action* given by the formula $[\sigma]\cdot[M,\omega,\pi]=[M,\omega+\pi^*\sigma,\pi]$, where $\sigma\in Z^2(B)$ is a de Rham representative of a class in $H^2(B,{{{\text{\bfR}}}})$.
Zung has obtained a version of these results for certain *singular* Lagrangian fibrations.
Twisted symplectic torsors {#twisted-symplectic-torsors .unnumbered}
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It is instructive to go one step further in the long exact sequence and ask what happens if $d_{P,*}c(M)$ is nonzero. This leads to a “nonholonomic” version of the Duistermaat-Dazord-Delzant theorems. I will outline the results and publish the proofs elsewhere. We define a *twisted Lagrangian fibre bundle* ${\mathscr}{L}=(M,\omega,\pi)$ over a base manifold $B$ in the same way as a Lagrangian fibre bundle, except that we relax the requirement that $\omega$ be closed to the requirement that $d\omega$ be *basic* in the sense that $d\omega=\pi^*\eta$ for a $3$-form $\eta$ on $B$. Thus $\omega$ is an almost symplectic form. The form $\eta=\eta({\mathscr}{L})$ is closed and uniquely determined by $\omega$. It is an isomorphism invariant and we refer to it as the *twisting form* of the twisted Lagrangian fibre bundle ${\mathscr}{L}$. The pair $(\omega,\eta)$ is a cocycle in the relative de Rham complex of the projection $\pi$.
It turns out that, just as in the Lagrangian case, the cotangent bundle $T^*B$ acts on the total space $M$ of a twisted Lagrangian fibre bundle ${\mathscr}{L}$ and that the kernel of the action is a bundle of lattices $P$, which is Lagrangian with respect to the standard symplectic form on $T^*B$. So again $B$ is a tropical affine manifold and $M$ is a torsor for the torus bundle $T=T^*B/P$.
We now fix the tropical affine manifold $(B,P)$ and look at any twisted Lagrangian bundle ${\mathscr}{L}=(M,\omega,\pi)$ which is at the same time a $T$-torsor. We assume that the almost symplectic form $\omega$ is *compatible* with the $T$-action in the sense that the $T^*B$-action on $M$ induced by $\omega$ has kernel $P$. We call such an ${\mathscr}{L}$ a *twisted symplectic $T$-torsor* and set ourselves the task of classifying up to isomorphism all twisted symplectic $T$-torsors. We denote the set of isomorphism classes by ${{{\text{\bfTPic}}}}(B,P)$.
The first observation is that this set is an abelian group in the same way as the ordinary Picard group. We will refer to ${{{\text{\bfTPic}}}}(B,P)$ as the *twisted* Picard group of the tropical affine manifold $(B,P)$.
The next observation is that every $T$-torsor $M$ possesses a compatible almost symplectic form and that the extent to which it is not closed is measured by the class $d_{P,*}c(M)$.
Every $T$-torsor $\pi\colon M\to B$ possesses a compatible almost symplectic form $\omega$. Its twisting form $\eta\in Z^3(B)$ satisfies $[\eta]=d_{P,*}c(M)$.
The Dazord-Delzant theorem generalizes as follows.
We have an exact sequence $$0{\longrightarrow}\Omega^2(B)/d_PH^0(B,{\mathscr}{T}){\longrightarrow}{{{\text{\bfTPic}}}}(B,P){\longrightarrow}H^2(B,{\mathscr}{P}){\longrightarrow}0.
$$
So the moduli space of twisted Lagrangian fibre bundles is typically infinite-dimensional. These degrees of freedom can be taken away by introducing a coarser form of gauge equivalence, namely by letting an arbitrary $2$-form $\sigma\in\Omega^2(B)$ on the base act on a twisted Lagrangian fibre bundle ${\mathscr}{L}=(M,\omega,\pi)$ by the formula $\sigma\cdot{\mathscr}{L}=(M,\omega+\pi^*\sigma,\pi)$. This action changes the twisting form by the exact $3$-form $d\beta$. It follows from the theorem that ${{{\text{\bfTPic}}}}(B,P)/\Omega^2(B)$ is isomorphic to $H^2(B,{\mathscr}{P})$, in other words every $T$-torsor has a compatible almost symplectic structure which is unique up to coarse gauge equivalence.
A twisted symplectic $T$-torsor does not have a well-defined Lagrangian class, but the *difference* ${\mathscr}{L}_1-{\mathscr}{L}_2={\mathscr}{L}_1+-{\mathscr}{L}_2$ of two twisted symplectic $T$-torsors that have the same twisting forms, $\eta({\mathscr}{L}_1)=\eta({\mathscr}{L}_2)$, is a symplectic $T$-torsor and therefore has a well-defined Lagrangian class. It follows that if we fix a closed $3$-form $\eta\in Z^3(B)$ the set of isomorphism classes of twisted symplectic $T$-torsors with twisting form $\eta$ is a principal homogeneous space of ${{{\text{\bfPic}}}}(B,P)$. If in addition we fix a class $c\in H^2(B,{\mathscr}{P})$ satisfying $d_{P,*}c=[\eta]$, then the set of isomorphism classes of twisted symplectic $T$-torsors with Chern class $c$ and twisting form $\eta$ is a principal homogeneous space of ${{{\text{\bfPic}}}}^0(B,P)$.
Groupoids and realizations {#groupoids-and-realizations .unnumbered}
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At the end of my talk Alan Weinstein pointed out that Duistermaat’s study of global action-angle variables provided one of the incentives for him to formulate the symplectic groupoid program [@coste-dazord-weinstein;groupoides], [@weinstein;symplectic-groupoids-poisson-manifolds]. In the language of that program a Lagrangian fibre bundle is nothing but a *realization* of the base manifold $B$ equipped with the zero Poisson structure, and the torus bundle $T$ is a symplectic groupoid over $B$ (with source and target maps being equal) which *integrates* this Poisson manifold.
Every manifold $B$ with zero Poisson structure is obviously integrable and the associated source-simply connected symplectic groupoid is just the cotangent bundle $T^*B$. What makes the tropical affine case special is the existence of a *proper* symplectic groupoid $T$ which integrates the trivial Poisson structure. The Duistermaat-Dazord-Delzant theorems then amount to a classification of all realizations of $B$ which are free and fibre-transitive under $T$. The group ${{{\text{\bfPic}}}}(B,P)$ is referred to as the *static* Picard group of the groupoid $T$ in [@bursztyn-weinstein;picard-poisson]. (The full, noncommutative, Picard group is the semidirect product of ${{{\text{\bfPic}}}}(B,P)$ with the group of tropical affine automorphisms of $B$.) The twisted case also fits into this framework, as one can see from the papers [@bursztyn-crainic-weinstein-zhu;integration-twisted] and [@cattaneo-xu;integration-twisted].
The quantum-mechanical spherical pendulum {#the-quantum-mechanical-spherical-pendulum .unnumbered}
-----------------------------------------
Having spent far more time on action-angle variables than I intended, let me be very brief about the quantum-mechanical picture. A treatment of quantum monodromy in the spherical pendulum was given by Richard Cushman and Hans Duistermaat [@duistermaat-cushman;quantum-spherical]. A different interpretation was given soon afterwards by Victor Guillemin and Alejandro Uribe [@guillemin-uribe;monodromy-quantum]. Let me quote from Hans’ review of the latter paper in the *Mathematical Reviews*, which he starts by explaining his own approach:
If one considers the Schrödinger operator $E=-(\hbar^2/2)\Delta+V$, where $\Delta$ is the Laplace operator on the $2$-dimensional standard sphere $S$ in ${{{\text{\bfR}}}}^3$ and the potential $V$ is the vertical coordinate function, then the rotational symmetry around the vertical axis yields an operator $L=i\hbar(x_1\partial/\partial x_2-x_2\partial/\partial x_1)$ which commutes with $S$ \[*sic*\].
Replacing $i\hbar\partial/\partial x_j$ by the conjugate variable $p_j$, we get principal symbols $e$ and $l$ of $E$ and $L$, respectively, which Poisson commute and define an integrable Hamiltonian system on the phase space $T^*S$, the cotangent bundle of $S$. One has straightforward generalizations to $n$ commuting operators $E_1$,…, $E_n$ with principal symbols $e_1$,…, $e_n$ on $n$-dimensional manifolds $M$.
Because the operators $E_j$ commute, one has common eigenfunctions $\psi_k$, $k=1$, $2$,…, with eigenvalues ${\varepsilon}_{j,k}$ ($E_j\psi_k={\varepsilon}_{j,k}\cdot\psi_k$). The rule for finding the $n$-dimensional spectrum $({\varepsilon}_{1,k},{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},{\varepsilon}_{n,k})\in{{{\text{\bfR}}}}^n$ for $k=1$, $2$,…, asymptotically for $\hbar\downarrow0$ and near a regular value of the mapping $(e_1,\cdots,e_n)$ is as follows. One constructs locally so-called action variables, which are functions $(a_1,{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},a_n)$ of the $(e_1,{\relax\ifmmode\ldots\else$\,\ldots\,$\fi},e_n)$, in such a way that $(\partial a_i/\partial e_j)$ is invertible and the Hamiltonian flows of the $a_j$ are periodic with period $2\pi$. Then the $n$-dimensional spectrum is given asymptotically by $a^{-1}({{{\text{\bfZ}}}}^n+\alpha)$, where ${{{\text{\bfZ}}}}^n$ is the integer lattice, $\alpha$ is a Maslov shift, and $a$ is the vector of action variables given above. This means that the actions, in particular the nonexistence of global action variables, can be read off from the asymptotics of the spectrum.
He proceeds to explain the different approach taken by Guillemin and Uribe. Although it has some very convincing illustrations, Cushman and Duistermaat’s paper is little more than an announcement and there does not seem to exist a more comprehensive version. Some ten years after its appearance experimental evidence of quantum monodromy was found and finally Vũ Ngc San wrote two papers [@vu;quantum-monodromy-integrable-systems], [@vu;quantum-monodromy-bohr-sommerfeld] clarifying and elaborating on Cushman and Duistermaat’s ideas.
Duistermaat-Heckman
===================
Of all Hans Duistermaat’s accomplishments the best known to differential geometers is probably the Duistermaat-Heckman theorem [@duistermaat-heckman;variation]. This is so familiar to most of the audience that I passed it over in my talk, but in this written version I can’t resist making some remarks about it. Recall that in its simplest form the theorem states that $$\int_M\exp(\omega-tf)=\int_X\frac{\exp(\omega-tf)}{{{{\text{\bfe}}}}(X,t)}.
$$ Here $(M,\omega)$ is a compact symplectic manifold, $t$ is a complex parameter, $f$ is a periodic Hamiltonian, $X$ is the critical manifold of $f$, and ${{{\text{\bfe}}}}(X,t)$ is the equivariant Euler class of the normal bundle of $X$ in $M$. The integral on the left is to be interpreted as the integral of $e^{-tf}\omega^n/n!$, where $2n=\dim M$. This is precisely the Fourier-Laplace transform of the measure $f_*(\omega^n/n!)$ obtained by pushing forward the Liouville measure $\omega^n/n!$ to the real line. The critical manifold $X$ usually consists of connected components of various dimensions, so the integral on the right is to be read as a sum of integrals, one for each component.
The theorem contains as a special case Archimedes’ result that the surface area of a sphere is equal to that of the circumscribed cylinder, an illustration of which, according to Cicero [@cicero;disputations Liber V, §§64–66], adorned the Syracusan’s tomb. A modern antecedent of the theorem is Bott’s residue formula for holomorphic vector fields [@bott;residue]. Soon after publication three interesting alternative proofs appeared, one based on the localization principle in equivariant cohomology by Atiyah and Bott [@atiyah-bott;moment-map-equivariant-cohomology] and Berline and Vergne [@berline-vergne;zeros] (see also [@berline-getzler-vergne;heat-kernels] and [@guillemin-sternberg;supersymmetry-equivariant]), one based on partial action-angle variables by Dazord and Delzant [@dazord-delzant;actions-angles], and one based on the coisotropic embedding theorem by Guillemin and Sternberg [@guillemin-sternberg;birational].
The index theorem {#the-index-theorem .unnumbered}
-----------------
My favourite meta-application of the Duistermaat-Heckman theorem is Atiyah’s heuristic derivation [@atiyah;circular-symmetry] of the Atiyah-Singer index theorem for the Dirac operator suggested by ideas of Witten [@witten;supersymmetry-morse]. Let $X$ be a compact Riemannian manifold and let $M=C^\infty(S^1,X)$ be the loop space of $X$. A tangent vector to $M$ at a loop $\gamma$ is a vector field along $\gamma$, i.e. a section of $\gamma^*(TX)$. The loop space has a Riemannian structure: the inner product of two tangent vectors $s_1$, $s_2\in T_\gamma M$ is defined to be the integral $\int_{S^1}(s_1(\theta),s_2(\theta))\,d\theta$. The circle $S^1$ acts on $M$ by spinning the loops, and we let $\alpha$ be the $1$-form on $M$ dual to the infinitesimal generator of this action. Then $\omega=d\alpha$ is a presymplectic structure on $M$; it degenerates for example at the closed geodesics of $X$.
Despite this degeneracy the circle action is generated by a Hamiltonian, namely the energy function $E\colon M\to{{{\text{\bfR}}}}$ given by $E(\gamma)=\frac12\int_{S^1}{\lVertd\gamma\rVert}^2$. The Duistermaat-Heckman theorem tells us to integrate the functional $e^{-tE}$ times a “Liouville” volume form on $M$. The “Riemannian” volume form on $M$ is the Wiener measure $d\gamma$, and just as in the finite-dimensional case we must multiply this by the (regularized) Pfaffian of the skew symmetric endomorphism of $TM$ defined by the presymplectic form. This Pfaffian exists if the manifold $X$ has a ${{{\text{\bfSpin}}}}$-structure, and the Duistermaat-Heckman integral $\int_Me^{-tE(\gamma)}{{\operatorname{\mathrm{Pf}}}}(\omega)\,d\gamma$ is seen to be the index of the associated Dirac operator ${\textit{\dh}}$. The Duistermaat-Heckman theorem then says that this integral localizes to the fixed point set $M^{S^1}$, which is a copy of $X$. By calculating the weights of the action on the normal bundle of $X$ in $M$ one arrives at the A-roof genus and thus concludes that ${{\operatorname{\mathrm{index}}}}({\textit{\dh}})=\hat{A}(X)$.
Nitta’s theorem {#nittas-theorem .unnumbered}
---------------
Is there a generalization of the Duistermaat-Heckman theorem to Poisson manifolds? In general this seems too much to ask for, but a reasonable compromise was found by Yasufumi Nitta [@nitta;duistermaat-heckman], [@nitta;reduction-calabi-yau]. I state his result in a more general form obtained by (my student and Hans’ grand-student) Yi Lin [@lin;equivariant-twisted]. Let $(M,\rho)$ be a compact *generalized Calabi-Yau manifold*, that is a $2n$-dimensional manifold equipped with a (possibly twisted) generalized complex structure defined by a pure spinor $\rho\in\Omega^*(M,{{{\text{\bfC}}}})$ with the property that the $2n$-form $\nu=(\rho,\bar{\rho})$ is a volume form. (Such manifolds are discussed in more detail in Gil Cavalcanti’s lecture notes in these proceedings.) Let $T$ be a torus acting on $M$ in a Hamiltonian fashion with generalized moment map $\Phi\colon M\to{{\mathfrak}{t}}^*$. Then the pushforward measure $\Phi_*(\nu)$ on ${{\mathfrak}{t}}^*$ is equal to a piecewise polynomial function times Lebesgue measure. For a symplectic manifold $(M,\omega)$ we have $\rho=e^{i\omega}$ and $\nu=\omega^n/n!$, and so this assertion is exactly the Fourier transformed version of the classical Duistermaat-Heckman theorem.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
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---
abstract: 'A difficulty with previous treatments of the gravitational self-force is that an explicit formula for the force is available only in a particular gauge (Lorenz gauge), where the force in other gauges must be found through a transformation law. For a class of gauges satisfying a “parity condition” ensuring that the Hamiltonian center of mass of the particle is well-defined, I show that the gravitational self-force is always given by the angle-average of the bare gravitational force. To derive this result I replace the computational strategy of previous work with a new approach, wherein the form of the force is first fixed up to a gauge-invariant piece by simple manipulations, and then that piece is determined by working in a gauge designed specifically to simplify the computation. This offers significant computational savings over the Lorenz gauge, since the Hadamard expansion is avoided entirely and the metric perturbation takes a very simple form. I also show that the rest mass of the particle does not evolve due to first-order self-force effects. Finally, I consider the “mode sum regularization” scheme for computing the self-force in black hole background spacetimes, and use the angle-average form of the force to show that the same mode-by-mode subtraction may be performed in all gauges satisfying the parity condition. This helps provide a practical foundation for the computation of self-forces in the Kerr background.'
author:
- 'Samuel E. Gralla'
title: 'Gauge and Averaging in Gravitational Self-force'
---
The leading-order deviation from geodesic motion proportional to the mass of a body—interpreted as the force due to the body’s own gravitational field—is known as the gravitational self-force. A recurring source of difficulty in both the theoretical treatment and practical computation of the self-force has been the choice of *gauge* in which the metric perturbation and force are expressed. In particular, early treatments [@mino-sasaki-tanaka; @quinn-wald; @detweiler-whiting; @poisson-review] required a specific gauge choice—Lorenz gauge—even to define the perturbed trajectory (via a point particle hypothesis coupled with “Lorenz gauge relaxation” [@quinn-wald; @gralla-wald] to allow non-geodesic motion), and a proposed extension of the results to other gauges [@barack-ori-gauge] restricted to gauge vectors that are continuous at the particle, even though the metric perturbation is singular. At a theoretical level, this elevates a particular gauge to fundamental status, to the exclusion of other gauges that seem equally nice, such as any gauge where the point particle $1/r$ singularity corresponds to linearized Schwarzschild in Cartesian Schwarzschild coordinates, as opposed to the Cartesian isotropic coordinates that correspond to Lorenz gauge.[^1] And at a practical level, one has excluded the standard gauges of black hole perturbation theory [@barack-ori-gauge].
Previous work [@gralla-wald] (hereafter paper I) eliminated the fundamental status of the Lorenz gauge by giving a definition of perturbed motion holding for any gauge where the particle is represented by a $1/r$ singularity. However, in this work we still relied on the Lorenz gauge for our computations and, more importantly, for the *expression* of the final result, as a formula holding in the Lorenz gauge together with a generalized transformation law. At a theoretical level, we still have a preferred role for the Lorenz gauge; and at a practical level, the results suggest that the computation of a self-force in an alternative gauge must always proceed through Lorenz gauge, eliminating much of the appeal of working in alternative gauges in the first place.
In this paper I will identify a class of gauges based on the requirement that the center of mass as defined by Regge and Teitelboim [@regge-teitelboim] is well-defined (in the “near zone”), and show that the force in any such gauge is given by the angle-average of the bare force in that gauge. To derive this result I adopt the assumptions of paper I but take a new computational approach, wherein the form of the force in any gauge is fixed up to a gauge-invariant piece by simple manipulations, and then that piece is determined by working in a gauge chosen specifically to make the computation as simple as possible. This approach avoids much of the computational complexity of previous work (eliminating the Hadamard expansion entirely and significantly reducing the calculation needed thereafter), while organizing the computation so that the final equation automatically takes a gauge-independent form.
The precise results are as follows. We define the (lowest-order) mass $M$, spin $S_{ab}$, and center of mass deviation $Z^a$ of the particle as tensors on a timelike worldline $\gamma$ (four velocity $u^a$) in a vacuum background metric $g_{ab}$.[^2] Then we find that that $\gamma$ is a geodesic, that the mass and spin are constant (parallel propagated), and that the deviation $Z^a$ satisfies $$\label{eom}
u^b \nabla_b (u^c \nabla_c Z^a) = \frac{1}{4\pi} \lim_{r\rightarrow 0}\int F^a d\Omega + R_{bcd}^{\ \ \ a} u^b Z^c u^d + \frac{1}{2M} R_{bcd}^{\ \ \ a} S^{bc} u^d,$$ where $F^a$ is the “bare gravitational force”, $$\label{gravforce}
F^a = - \left(g^{ab}+u^a u^b\right)\left( \nabla_d h_{cd} - \frac{1}{2}\nabla_b h_{cd} \right)u^c u^d,$$ and $h_{\mu \nu}$ is the metric perturbation of a point particle, $$\label{pp}
G^{(1)}_{ab}[h] = 8 \pi M \int_\gamma \delta_4 (x,z(\tau)) u_a u_b d\tau,$$ which must be expressed in a “parity-regular” gauge, where the singular part of the spatial metric is even-parity on the sphere, $C_{ij}(t,-\vec{n})=C_{ij}(t,\vec{n})$ in equation . The bare force $F^a$ is familiar from the perturbed geodesic equation, and here is defined only off of $\gamma$ (and only locally, where $u^a$ is extended off $\gamma$ by parallel transport along spacelike geodesics orthogonal to $u^a$) since the metric perturbation is divergent. The integral in is defined by using the exponential map based on $\gamma$ to associate a flat metric, in terms of which the integral is over a fixed 2-sphere of spatial distance $r$ in the hyperplane orthogonal to $u^a$, with $n^a$ its unit normal and $d\Omega$ its area element, and the integration is done component-wise under the exponential map. I also define the perturbed mass of the particle and show that it is constant in time.
The first term on the right hand side of equation is proportional to the metric perturbation and therefore corresponds to the gravitational self-force. We see that the force in any parity-regular gauge is given by the angle-average of the bare force in that gauge, so that the self-force may be viewed as the net gravitational force on the particle. If the Lorenz gauge is adopted and the Hadamard form for the metric perturbation is computed (choosing the retarded solution with no incoming radiation) and plugged in, then this term reduces to the standard “tail integral” expression for the self-force (e.g., [@poisson-review]). The second term corresponds to the geodesic deviation equation and reflects the particle’s desire to move on a new geodesic once it has been displaced from the original. The third term is the Papapetrou spin force. If the parity condition is violated, equation does not hold, and the equation of motion takes a complicated form involving an explicit gauge transformation to a parity-regular gauge, equation .
A practical technique for computing self-forces in black hole background spacetimes is known as “mode sum regularization” ([@barack-ori-modesum] and many other references). In this approach one numerically solves for the spherical harmonic modes of the metric perturbation (the sum over which diverges at the particle), and performs a mode-by-mode subtraction that regularizes the sum in such a way that the correct self-force is computed. While extensive work has determined the form of the subtraction in the Lorenz gauge, only restricted results are available in other gauges. Taking advantage of a connection between mode decompositions and averaging (and hence self-force), I give a simple argument that the same subtraction may be performed in all parity-regular gauges. Since both the Lorenz gauge and a modified radiation gauge of Kerr [@k] are in this class, this result helps provide a practical foundation for the computation of self-force effects in the Kerr spacetime.
I use the conventions of Wald [@wald]. Greek indices label tensor components, while early-alphabet Latin indices $a,b,\dots$ are abstract indices. When coordinates $t,x^i$ are used, the time and space components are denoted by $0$ and mid-alphabet Latin indicies $i,j,\dots$, respectively.
Review of Formalism {#sec:review}
===================
The central idea of paper I is to introduce a mathematically precise formulation of the notion of the “near-zone” of a body, and to use the requirement of a sensible near-zone to demand a sensible perturbation family. Given a family of metrics $g_{ab}(\lambda)$ in coordinates ($t, x^i$), we define a scaled metric $\bar{g}_{ab} \equiv \lambda^{-2} g_{ab}$ and scaled coordinates $\bar{t} \equiv \lambda^{-1}(t-t_0)$ and $\bar{x}^i \equiv \lambda^{-1} x^i$. Denoting the scaled metric in scaled coordinates by $\bar{g}_{\bar{\mu} \bar{\nu}}$, consider the $\lambda \rightarrow 0$ limit, $\bar{g}^{(0)}_{\bar{\mu} \bar{\nu}} \equiv \bar{g}_{\bar{\mu} \bar{\nu}}|_{\lambda=0}$. This limit effectively “zooms in” on the spacetime point ($t = t_0, x^i = 0$), and will recover the near zone of a body if the one-parameter-family contains a body whose radius and mass shrink down linearly with $\lambda$ to the worldline $x^i = 0$ (denoted by $\gamma$). By demanding its existence (and associated conditions), we automatically consider bodies of small size and mass. Note that the components of the original and scaled metrics are related simply by “plugging in the new coordinates,” $$\label{plugin}
\bar{g}_{\bar{\mu} \bar{\nu}}(\lambda; t_0; \bar{t},\bar{x}^i)
= g_{\mu \nu} (\lambda; t=t_0 + \lambda \bar{t}, x^i=\lambda \bar{x}^i).$$ This equation relates components of the scaled metric in scaled coordinates, $\bar{g}_{\bar{\mu} \bar{\nu}}$, to corresponding components of the original metric in the original coordinates, $g_{\mu \nu}$.
One may perform perturbation theory in either the original (“far-zone”) picture or the scaled (“near-zone”) picture. The far-zone background and perturbations will be denoted by $g^{(n)}_{\mu \nu} \equiv (1/n!) (\partial_\lambda)^n g_{\mu \nu}|_{\lambda=0}$ (I also write $h_{\mu \nu}=g^{(1)}_{\mu \nu}$), while the near-zone background and perturbations will be denoted by $\bar{g}^{(n)}_{\bar{\mu} \bar{\nu}} \equiv (1/n!) (\partial_\lambda)^n \bar{g}_{\bar{\mu} \bar{\nu}}|_{\lambda=0}$. Our assumptions together with the choice of Fermi coordinates (e.g., [@poisson-review]) about $\gamma$ for the far-zone background metric constrain the far-zone quantities to take the form $$\begin{aligned}
g^{(0)}_{\mu \nu} & = \eta_{\mu \nu} + B_{\mu i \nu j}(t)x^i x^j + O(r^3) \label{g0} \\
h_{\mu \nu} = g^{(1)}_{\mu \nu} & = \frac{C_{\mu \nu}(t,\vec{n})}{r} + D_{\mu \nu}(t,\vec{n}) + r E_{\mu \nu}(t,\vec{n})+ O(r^2) \label{g1}
\\
g^{(2)}_{\mu \nu} & = \frac{F_{\mu \nu}(t,\vec{n})}{r^2} + \frac{H_{\mu \nu}(t,\vec{n})}{r} + K_{\mu \nu}(t,\vec{n}) + O(r),\label{g2}\end{aligned}$$ where we have defined $r=\sqrt{\delta_{ij}x^i x^j}$ and $n^i=x^i/r$. Here $B_{\mu i \nu j}$ is related to the Riemann curvature on the background worldline $x^i=0$ (see e.g. [@poisson-review] for the exact expression), whereas $C_{\mu \nu},\dots,K_{\mu \nu}$ are unspecified smooth functions on $\mathbb{R} \times S^2$ (denoted by arguments $(t,\vec{n})$). The lack of a linear term in equation is a consequence of $\gamma$ being geodesic, as shown in paper I. We also showed that the metric perturbation has effective point particle source, equation . In light of equation the near-zone perturbation series takes the form $$\begin{aligned}
\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}} & = \eta_{\mu \nu} + \frac{C_{\mu \nu}(t_0,\vec{n})}{\bar{r}} + \frac{F_{\mu \nu}(t_0,\vec{n})}{\bar{r}^2} + O\left( \frac{1}{\bar{r}^3} \right) \label{gb0} \\
\bar{g}^{(1)}_{\bar{\mu} \bar{\nu}} & = D_{\mu \nu} + \frac{H_{\mu \nu}}{\bar{r}} + O\left( \frac{1}{\bar{r}^2} \right) + \bar{t} \left[ \frac{\dot{C}_{\mu \nu}}{\bar{r}} + \frac{\dot{F}_{\mu \nu}}{\bar{r}^2} + O \left( \frac{1}{\bar{r}^3} \right)
\right] \label{gb1} \\
\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}} & = B_{\mu i \nu j}\bar{x}^i \bar{x}^j + \bar{r} E_{\mu \nu} + K_{\mu \nu} + O\left( \frac{1}{\bar{r}} \right) \nonumber \\ & + \bar{t} \left[ \dot{D}_{\mu \nu} + \frac{\dot{H}_{\mu \nu}}{\bar{r}} + O\left( \frac{1}{\bar{r}^2} \right) \right] + \bar{t}^2 \left[ \frac{\ddot{C}_{\mu \nu}}{\bar{r}} + \frac{\ddot{F}_{\mu \nu}}{\bar{r}^2} + O\left( \frac{1}{\bar{r}^3} \right) \right], \label{gb2}\end{aligned}$$ where an overdot denotes a $t$-derivative, and the order symbols $O(1/\bar{r}^n)$ refer to $\bar{t}$-independent functions. The dependence on $(t_0,\vec{n})$ is suppressed in equations and for readability. Note that the lack of growing-in-$\bar{r}$ terms in equation is inherited from the absence of a linear term in , which is a consequence of $\gamma$ being geodesic (and choosing Fermi coordinates). The (stationary and asymptotically flat) near-zone background metric $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ represents the local state of the body at time $t_0$, and as such its properties should characterize those of the body. In particular, the multipole moments of $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ should correspond to those of the body, at lowest non-trivial perturbative order. To the (far-zone) perturbative order considered in this paper, only the monopole and dipole moments can play a role. We therefore define $$\begin{aligned}
M(t_0) & \equiv \frac{-1}{8\pi} \lim_{\bar{r} \rightarrow \infty} \int n^i \partial_i \bar{g}^{(0)}_{00} \bar{r}^2 d\Omega \label{Mdef} \\
D^i(t_0) & \equiv \frac{3}{8\pi} \lim_{\bar{r} \rightarrow \infty} \int \bar{g}^{(0)}_{00} n^i \bar{r}^2 d\Omega \label{Ddef} \\
S_{ij}(t_0) & \equiv \frac{3}{8\pi} \lim_{\bar{r} \rightarrow \infty} \int \bar{g}^{(0)}_{0[i} n^{\ }_{j]} \bar{r}^2 d\Omega, \label{Sdef}\end{aligned}$$ and refer to $M$, $D^i$, and $S_{ij}$ as the (lowest-order) mass, mass dipole, and spin (current dipole) of the body.[^3] In equations -, the bars on the indices of coordinate components of the near-zone metric have been dropped, and the integrals are taken on fixed coordinate two-spheres with respect to the “flat” volume element $d\Omega$. It is well known (and easily checked) that the mass dipole may be set to zero by translating the coordinates, $\bar{x}^i \rightarrow \bar{x}^i - Z^i$, with $$\label{Zdef}
Z^i(t_0) \equiv D^i(t_0)/M(t_0).$$ This gives the new coordinates the interpretation of being mass centered, so that $Z^i$ represents center of mass position of the body in the original coordinates. Since the above translation corresponds to a first-order gauge transformation in the far zone (recall $\bar{x}^i=x^i/\lambda$), we identify $Z^i$ with the first-order deviation of the center of mass position from its background position $\gamma$. While the mass, spin, and deviation are defined in the near zone “at spatial infinity” (as a function of the coordinate time $t_0$ along $\gamma$ at which the near-zone limit is taken), they may equally well be viewed as tensors defined in the far-zone “at the spatial origin”, i.e., as tensor fields on $\gamma$. It was shown in paper I that the mass and spin do not evolve with time $t_0$ (i.e., that they are parallel propagated tensor fields on $\gamma$), while a Lorenz-gauge equation of motion (together with a gauge transformation law) was derived for the deviation vector.
The analysis of this paper will require transformation properties of $Z^i$ not only under ordinary translations, but also under transformations of the form $\delta \bar{x}^i = \alpha^i(\vec{n}) + O(1/\bar{r})$, i.e., under angle-dependent translations, or supertranslations. Using the fact that $\bar{g}_{00}=-1+2M/\bar{r} + O(1/\bar{r}^2)$ for all metrics of the form , it is easy to check that $Z^i$ transforms as $$\label{changeZ}
\delta Z^i = \frac{3}{4\pi} \int \alpha^j n_j n^i d\Omega.$$ Since the angle-average of a vector picks out an $\ell=1$ part, we may restrict consideration to $\alpha^i$ of the form $\alpha^i = B^j n_j n^i + C^i$ for constants $B^i$ and $C^i$. The associated change in $Z^i$ is then simply $B^i+ C^i$. Thus $Z^i$ changes under supertranslations with $B^i \neq 0$, in addition to its change under ordinary translations $\alpha^i=C^i$.
Coordinate Choices and Perturbed Mass {#sec:dM}
=====================================
The mass, spin, and deviation of the body were defined at lowest-order in the near zone, so that perturbative corrections to these quantities would naturally be defined at first and higher orders in the near-zone. However, while the background metric is stationary and asymptotically flat (so that its multipole moments are well-defined), the $n$’th-order perturbation may have growth in $n$ combined powers of $\bar{t}$ and $\bar{r}$ (c.f. paper I and equations (\[gb0\]-\[gb1\])). It turns out, however, that at first order the situation is more under control. As already noted, the choice of Fermi coordinates in the far-zone background—together with the fact that $\gamma$ is geodesic—eliminates growing-in-$\bar{r}$ terms from the first near-zone perturbation, so that the perturbation is asymptotically flat. Furthermore, as I now show, the constancy of the mass and spin similarly allows one to eliminate all $\bar{t}/\bar{r}$ and $\bar{t}/\bar{r}^2$ terms, so that the perturbation is stationary to $O(1/\bar{r}^2)$. To see this, recall that $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ is a stationary and asymptotically flat metric, so that, introducing $\delta_{\mu \nu}=\textrm{diag}(1,1,1,1)$ and $t^\alpha = (1,0,0,0)$, it may (at each $t_0$) be put in the standard form [@MTW], $$\label{gb0form}
\bar{g}^{(0)}_{\bar{\mu} \bar{\nu}} = \eta_{\mu \nu} + \frac{2M}{\bar{r}}\delta_{\mu \nu} + \frac{M^2}{2 \bar{r}^2} (3 \eta_{\mu \nu}- t_\mu t_\nu) - \frac{4 n^i t_{(\mu} S_{\nu)i}}{\bar{r}^2} + O\left(\frac{1}{\bar{r}^3}\right),$$ where $S_{i0}=0$ for all $t_0$ (implying $D^i=0$, so that the coordinates are mass centered and “track” the motion of the body), while a priori $M$ and $S_{ij}$ may depend on $t_0$, reflecting evolution of the mass and spin. However, since the $M$ and $S_{ij}$ in do correspond to those defined in and , we know that these quantities are in fact independent of time $t_0$, as shown in paper I and remarked above. Thus the near-zone background metric is identical (through order $O(1/\bar{r}^2)$) at all $t_0$, which implies by the form of that the near-zone perturbation is independent of $\bar{t}$ through $O(1/\bar{r}^2)$, becoming simply $$\label{gb1form}
\bar{g}^{(1)}_{\bar{\mu} \bar{\nu}} = D_{\mu \nu}(\vec{n}) + H_{\mu \nu}(\vec{n})\frac{1}{\bar{r}} + O\left(\frac{1}{\bar{r}^2}\right) + \bar{t} \ O\left(\frac{1}{\bar{r}^3}\right).$$ Further simplification can now be made. The perturbation $\bar{g}^{(1)}_{\bar{\mu}\bar{\nu}}$ must satisfy the (vacuum) linearized Einstein equation about the near-zone background $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$. The quantity $D_{\mu \nu}$ appears at order $O(1/\bar{r}^2)$ (and higher) in the linearized Einstein tensor, while $H_{\mu \nu}$ appears first at $O(1/\bar{r}^3)$. However, at these orders only the first two terms in the background will appear, so that the background is effectively Schwarzschild. Therefore, when considering only the explicitly displayed terms in the perturbation , we may use well known results [@regge-wheeler; @zerilli] for perturbations of the Schwarzschild spacetime. In particular, since stationary perturbations scaling as $\bar{r}^0$ and $\bar{r}^{-1}$ (as in ) are known to be $\ell=0$ (spherically symmetric) up to gauge, the perturbation in is simply a change in mass, and a gauge may be chosen so that $$\label{gb1form2}
\bar{g}^{(1)}_{\bar{\mu} \bar{\nu}} = \frac{2\ \delta M(t_0)}{\bar{r}}\delta_{\mu \nu} + O\left(\frac{1}{\bar{r}^2}\right) + \bar{t} \ O\left(\frac{1}{\bar{r}^3}\right),$$ where $\delta M$ is an arbitrary constant (which here may a priori depend on $t_0$), which I refer to as the *perturbed mass* of the body. An explicit formula may be given in analogy with , $$\delta M(t_0) = \frac{-1}{8\pi} \lim_{\bar{r} \rightarrow \infty} \int n^i \partial_i \bar{g}^{(1)}_{00} \bar{r}^2 d\Omega \label{dMdef},$$ with the caveat that this equation holds only in coordinates where $D_{\mu \nu}=\dot{C}_{\mu \nu}=0$. If one wishes to compute the mass in other coordinates (such as when the Lorenz gauge is used, and $D_{\mu \nu}$ is equal to the value of the “tail integral”), a more complicated expression (which must implicitly involve transforming to appropriate coordinates) must be derived. However, since the mass is an intrinsic property of a spacetime, there is no need to consider such coordinates in defining $\delta M$ and determining its evolution.[^4] Note that since the perturbation is still stationary at $O(1/\bar{r}^2)$, it may be possible to define perturbed spin and deviation by an analogous procedure. However, these quantities appear at one order in $\lambda$ higher than pursued in this paper, and are not considered here.
I now show that the perturbed mass does not evolve with time. With our previous coordinate choices, the second-order near-zone perturbation takes the from $$\label{gb2form}
\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}} = B_{\mu i \nu j}\bar{x}^i \bar{x}^j + \bar{r} E_{\mu \nu} + K_{\mu \nu} + \frac{\bar{t}}{\bar{r}}2 \dot{\delta M} \delta_{\mu \nu} + O(1/\bar{r}) + \bar{t} \ O(1/\bar{r}^2) + \bar{t}^2 \ O(1/\bar{r}^3).$$ This perturbation satisfies the (vacuum) linearized Einstein equation about $\bar{g}_{\bar{\mu}\bar{\nu}}^{(0)}$ (equation ) with effective sources constructed from $\bar{g}_{\bar{\mu}\bar{\nu}}^{(1)}$ (equation ). However, it is easy to see that these effective sources are $O(1/\bar{r}^4)$, $\bar{t} O(1/\bar{r}^6)$, and $\bar{t}^2 O(1/\bar{r}^8)$, while the error in the linearized Einstein tensor introduced by including only the explicitly displayed terms in the second-order perturbation is $O(1/\bar{r}^3)$, $\bar{t}O(1/\bar{r}^4)$, and $\bar{t}^2 O(1/\bar{r}^5)$. Thus the effective source terms may be ignored, and denoting the linearized Einstein tensor of $\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}}$ about $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ by $\mathcal{G}_{\bar{\mu} \bar{\nu}}$, we have $\mathcal{G}_{\bar{\mu} \bar{\nu}}=0$ to the appropriate order, i.e., $$\mathcal{G}_{\bar{\mu} \bar{\nu}} = O(1/\bar{r}^3)+\bar{t}O(1/\bar{r}^4)+\bar{t}^2O(1/\bar{r}^5).$$
To determine the mass evolution $\dot{\delta M}$ we focus on the $\ell=0$ part of $\mathcal{G}_{\bar{\mu} \bar{\nu}}$. While the background $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ is not spherically symmetric due to the presence of the spin term, it is easy to see that to the relevant order in $\mathcal{G}_{\bar{\mu} \bar{\nu}}$ the spin term only contributes in product with the “B” term in the perturbation . However, since far-zone background spacetime is assumed to be vacuum, its Riemann tensor may be decomposed into two rank-two symmetric, trace-free spatial tensors (the “electric” and “magnetic” parts, e.g. [@poisson-review]), showing that the “B” term is pure $\ell=2$. Since the spin term is $\ell=1$, the combination can therefore make no contribution to the $\ell=0$ part of $\mathcal{G}_{\bar{\mu} \bar{\nu}}$ at the relevant order. Instead, the $\ell=0$ part is completely determined by the remaining terms in the perturbation, which “see” only the Schwarzschild metric. In particular only the $\ell=0$ parts of these terms may contribute to the $\ell=0$ part of $\mathcal{G}_{\bar{\mu} \bar{\nu}}$, and we conclude that the $\ell=0$ part of $\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}}$ must be a perturbation of the Schwarzschild spacetime to the relevant order. We may now use Zerilli’s result [@zerilli] that $\ell=0$ perturbations of Scwarzschild simply shift the mass, being writable as a $1/\bar{r}$ term plus a gauge transformation. In particular, taking a time derivative we must have $$\label{fancyG}
\partial_0 \stackrel{\textit{\tiny M}}{\nabla}_{(\mu} \xi_{\nu)} = \stackrel{\textit{\tiny M}}{\nabla}_{(\mu} \partial_0 \xi_{\nu)} = \frac{2 \dot{\delta M}}{\bar{r}} \delta_{\mu \nu} + O(1/\bar{r}^2),$$ where $\stackrel{\textit{\tiny M}}{\nabla}$ is the derivative operator compatible with Schwarzschild, and $\xi^\mu$ is a vector field. However, since the mass peturbation is not pure gauge, we immediately have a contradiction unless $\dot{\delta M}=0$.
Incorporating this result, we may now summarize the coordinate choices made in this section as $$\begin{aligned}
\bar{g}^{(0)}_{\bar{\mu} \bar{\nu}} & = \eta_{\mu \nu} + \frac{2M}{\bar{r}}\delta_{\mu \nu} + \frac{M^2}{2 \bar{r}^2} (3 \eta_{\mu \nu}- t_\mu t_\nu) - \frac{4 n^i t_{(\mu} S_{\nu)i}}{\bar{r}^2} + O\left(\frac{1}{\bar{r}^3}\right) \label{gb0gauged}\\
\bar{g}^{(1)}_{\bar{\mu} \bar{\nu}} & = \frac{2\delta M}{\bar{r}}\delta_{\mu \nu} + O\left(\frac{1}{\bar{r}^2}\right) + \bar{t} \ O\left(\frac{1}{\bar{r}^3}\right) \label{gb1gauged} \\
\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}} & = B_{\mu i \nu j}\bar{x}^i \bar{x}^j + \bar{r} E_{\mu \nu} + K_{\mu \nu} + O\left(\frac{1}{\bar{r}}\right) + \bar{t} \ O\left(\frac{1}{\bar{r}^2}\right) + \bar{t}^2 \ O\left(\frac{1}{\bar{r}^3}\right),\label{gb2gauged}\end{aligned}$$ with $S_{0i}=0$. By fixing the mass and spin terms to a standard form at all $t_0$ and choosing the mass dipole to vanish for all $t_0$, the metric form has been made very simple, and all non-stationarity has been eliminated from the relevant orders in $\bar{r}$. These properties make this gauge much simpler to use than the Lorenz gauge used in [@gralla-wald] and elsewhere. Rewritten in the far-zone, the perturbation series in these coordinates becomes $$\begin{aligned}
g_{\mu \nu}^{(1)} & = \frac{2 M \delta_{\mu \nu}}{r} + E_{\mu \nu} r + O(r^2) \label{g1gauged} \\
g_{\mu \nu}^{(2)} & = \frac{M^2}{2 r^2} (3 \eta_{\mu \nu}- t_\mu t_\nu) - \frac{4 n^i t_{(\mu} S_{\nu)i}}{r^2} + \frac{2\delta M}{r}\delta_{\mu \nu} + K_{\mu \nu} + O(r), \label{g2gauged}\end{aligned}$$ with the background $g^{(0)}_{\mu \nu}$ still given by .
Parity Condition
================
The definition of center of mass adopted in section \[sec:review\] is based on the dipole moment of the time-time component of a stationary, asysmptotically flat metric. An alternative definition of center of mass is given by equation (5.13) of Regge and Teitelboim [@regge-teitelboim], derived as the conserved quantity canonically conjugate to the asymptotic boost symmetry of asymptotically flat general relativity.[^5] Like the Hamiltonian notion of mass (normally referred to as the ADM mass), this “Hamiltonian center of mass” involves only the spatial metric, and is more general in that it may be applied to time-dependent spacetimes in addition to the stationary spacetimes we consider. However, unlike the Hamiltonian notion of mass, the center of mass comes with an additional restriction: In order to ensure the existence of the integral defining the center of mass, Regge and Teitelboim impose a “parity condition” that the monopole $(1/r)$ part of the spatial metric be even parity, $C_{ij}(\vec{n})=C_{ij}(-\vec{n})$ in .[^6] This restriction is not necessary to define the mass dipole, which is finite for any metric of the form .
Since the general metric form satisfies the parity condition, we see that the parity condition is simply a coordinate condition in the context of stationary, asymptotically flat spacetimes. Rotations and translations will automatically preserve the parity condition, while a supertranslation $\delta x^i = \alpha^i(\vec{n}) + O(1/\bar{r})$ must now satisfy $\alpha^i = c^i + \Sigma^i(\vec{n})$, with $c^i$ constant and $\Sigma^i$ odd parity, $\Sigma^i(-\vec{n})=-\Sigma^i(\vec{n})$. This form provides a natural split between the “pure translation” part $c^i$ and a “pure supertranslation” part $\Sigma^i$, which is odd-parity. It is easily checked from equation that the mass dipole center of mass changes by $c^i$ under this transformation, so that the parity condition removes its supertranslation dependence. In this case the transformation properties of the mass dipole center of mass agree with those established by Regge and Teitelboim for their center of mass; and since both notions give zero on the metric , the two definitions are equivalent in our context. Therefore, the question of which definition to use is simply the question of whether to impose the parity condition.
One may take one of two alternative viewpoints on this matter. First, since the formal Hamiltonian analysis yields a center of mass formula that diverges (in general) in coordinates that violate the parity condition, one may argue that such coordinates are “too irregular” to admit a notion of center of mass, even if the mass dipole formula is finite. Alternatively, one may view the mass dipole formula as providing an extension of the Regge Teitelboim center of mass to a larger class of coordinates within the stationary case. In any case, the parity condition adds a number of simplifying properties in the context of the present work: It eliminates the supertranslation dependence of the mass dipole, it allows the equations of motion to be expressed purely in terms of the local spacetime metric (see discussion in appendix \[sec:eom-noparity\]), and it makes the mode-sum regularization scheme gauge invariant.
Equation of Motion {#sec:eom}
==================
In the gauge of section \[sec:dM\], the equation of motion for the deviation is simply $Z^i(t)=0$. In principle, therefore, giving the change in $Z^i$ under a change in gauge provides the complete description of motion. However, the more useful description of motion in other gauges, equation , may be derived as follows. Begin with gauge transformations $x'^\mu = x^\mu + \xi^\mu$ of the form[^7] $$\xi^\mu = \alpha^\mu(t,\vec{n}) + O(r), \label{xi} \\$$ where $\alpha^\mu$ is a smooth functions of its arguments. It is easy to check that such transformations preserve the form of equations (\[g0\]-\[g1\]), and thus are allowed under our assumptions. As described in [@gralla-wald-gaugenote], we furthermore believe (but have not proven) that expressibility in the form of equation is a necessary condition on an allowed transformation, except in the case of certain trivial one-parameter-families, where additional $\log r$ terms are allowed. Thus we believe (but have not proven) that such transformations correspond precisely the coordinate choices allowed by our formalism (not including the parity condition) at first order in $\lambda$ for non-trivial families of solutions. In order to preserve the parity condition $C_{ij}(t,-\vec{n})=C_{ij}(t,\vec{n})$, we must restrict the form of $\alpha^\mu$ so that $$\xi^i = c^i(t) + \Sigma^i(t,\vec{n}) + O(r), \label{xi-parity} \\$$ with $\Sigma^i$ odd-parity, $\Sigma^i(t,-\vec{n})=-\Sigma^i(t,\vec{n})$. I will refer to gauge transformations of the above form as parity-regular transformations, and I will define parity-regular gauges as those that are related to the gauge of section \[sec:dM\] by a parity-regular transformation. If equation gives the general transformation preserving our assumptions (as we believe), then it follows that parity-regular gauges are the general class allowed by our assumptions plus the parity condition $C_{ij}(t,-\vec{n})=C_{ij}(t,\vec{n})$. It is easy to check that the Lorenz gauge is parity-regular. The analysis of [@barack-ori-gauge] gives some evidence that the Regge-Wheeler gauge conditions of Schwarzschild are compatible with parity-regularity. In [@k] it was checked that a modified radiation gauge of Kerr is parity-regular.
Under a change of gauge of the form , the near-zone background coordinates change by $\bar{x}'^i=\bar{x}^i + c^i(t_0) + \Sigma^i(t_0,\vec{n}) + O(1/\bar{r})$. Using equation , we see that the deviation $Z^i(t_0)$ changes by $c^i(t_0)$, as noted in the previous section. We may express this in terms of the gauge vector $\xi^\mu$ by taking an angle-average over a small constant-$r$ sphere,[^8] $$\label{deltaZ}
\delta Z^i = \langle \xi^i \rangle_{r \rightarrow 0} \equiv \frac{1}{4\pi} \lim_{r\rightarrow 0} \int \xi^i d\Omega.$$ Equation gives the change in deviation due to a parity-regular transformation made on any perturbation of the assumed form .
The key manipulation now follows. Consider the second time derivative, $\delta \ddot{Z}^i = \partial_0 \partial_0 \delta Z^i$. We have $$\begin{aligned}
\delta \ddot{Z}_i & = \langle \partial_0 \partial_0 \xi_i \rangle_{r \rightarrow 0} \nonumber \\
& = \langle \nabla_0 \nabla_0 \xi_i + R_{0j0k}x^k \partial_j \xi_i + \nabla_0 \nabla_i \xi_0 - \nabla_i \nabla_0 \xi_0 + R_{i00}^{\ \ \ j}\xi_j \rangle_{r \rightarrow 0} \nonumber \\
& = \langle -( \nabla_0 \delta h_{0i} - \frac{1}{2} \nabla_i \delta h_{00}) \rangle_{r \rightarrow 0} - R_{0i0j} \langle{ \xi^j \rangle}_{r\rightarrow 0} \nonumber \\
& = \delta \langle F_i \rangle_{r \rightarrow 0} - R_{0i0j} \delta Z^j\label{dZdd}\end{aligned}$$ In the second line we have rewritten in terms of covariant derivatives ($R_{0k0l}x^l \partial_k \xi_i$ is a Christoffel term), as well as added zero in the form of the Ricci identity. However, since $\partial_j \xi_i$ is even parity to leading order and the Riemann tensor is smooth, the first Riemann term vanishes by virtue of the parity condition. Noting that the remaining derivatives of $\xi^i$ appear only in symmetrized form, in the third line we reexpress in terms of the change in the metric perturbation, $\delta h_{\mu \nu} = -2 \nabla_{(\mu}\xi_{\nu)}$, finding precisely the gravitational force form of equation . We also separate off the remaining Riemann term, which takes a geodesic deviation form. In the last line we use equations and , where the finiteness[^9] of $\langle F_i \rangle_{r \rightarrow 0}$ allows us to pull the $\delta$ out of the angle-average. We may now “drop the $\delta$’s” to obtain $$\label{Zdd}
\ddot{Z}^i = \langle F^i \rangle_{r \rightarrow 0} - R_{0j0}^{\ \ \ i} Z^j + \mathcal{A}^i,$$ where $\mathcal{A}^i$ is the constant of integration—an unknown gauge-invariant acceleration. Thus by simple manipulations we have fixed the form of the equation of motion up to a gauge-invariant piece, and may now work in any convenient (parity-regular) gauge to determine $\mathcal{A}^i$.
Since the gauge of section \[sec:dM\] has $Z^i(t)=0$ it immediately eliminates two terms in , giving simply $$\label{Finsimplegauge}
\mathcal{A}^i = - \langle F^i \rangle_{r \rightarrow 0}$$ in this gauge. The interpretation of the gauge is that the gravitational self-force is exactly opposite to the gauge-invariant force, so that the total force is zero, and there is no deviation from geodesic motion ($Z^i(t)=0$). Since the angle-average of a three-vector picks out an $\ell=1$, electric parity part, we need consider only the $\ell=1$, electric parity part of the bare force $F^i$ in order to compute $\mathcal{A}^i$ from equation . Since only the “E” term in the perturbation contributes, we may focus on the $\ell=1$, electric parity part of $E_{\mu \nu}$. To do so, we return to second-order near-zone perturbation theory.
As remarked above in the derivation of the constancy of the perturbed mass, the relevant terms in the second-order metric perturbation $\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}}$ satisfy the linearized Einstein equation off of the $\bar{g}^{(0)}_{\bar{\mu}\bar{\nu}}$ to the relevant order, equation . As further remarked, the $\ell=1$ spin term in the background appears only in combination with the $\ell=2$ “B” term in the perturbation. While there is no contribution to the $\ell=0$ mode relevant for the perturbed mass, there can be a contribution to the $\ell=1$ mode relevant for the deviation, which significantly complicates the analysis (see appendix \[sec:spin\]). However, if the spin is assumed to be zero from the outset, then, as in the mass evolution case, the metric perturbation “sees” only Schwarzschild to the relevant order, and we can make use of Zerilli’s [@zerilli] analysis of perturbations of Schwarzschild. In particular, Zerilli showed that $\ell=1$, electric parity perturbations are pure gauge, so that by a gauge transformation we may eliminate the $\ell=1$, electric parity part of $E_{\mu \nu}$ entirely, whence it immediately follows from that $\mathcal{A}^i=0$. Thus the equation of motion in the spinless case may be derived with very little effort, involving only the few lines of algebra of equation . If the spin is not assumed zero, more algebra is required (though still significantly less than needed when the Lorenz gauge is used). This case is treated in appendix \[sec:spin\], leading to $$\label{AequalsS}
M \mathcal{A}_i = \frac{1}{2}S^{kl}R_{kl0i},$$ showing that $\mathcal{A}^i$ is simply the acceleration due to the Papapetrou spin force. We have thus justified the final equation of motion, which appears in covariant form in .
Mode-Sum regularization {#sec:modesum}
=======================
The computation of gravitational self-forces on black hole background spacetimes is an important problem for gravitational-wave astronomy of extreme mass-ratio inspirals (e.g., [@barack-review]). The angle-average formula suggests a straightforward way of proceeding: first numerically compute the metric perturbations of a point particle in any parity-regular gauge, and then perform an average to determine the force in that gauge. However, while simple in principle, such a procedure may be difficult to carry out in practice, due to the singular nature of the quantity being averaged. Instead, to achieve an accurate result it is likely preferable to use an alternative technique, such as that of *mode sum regularization*, first introduced in [@barack-ori-modesum] and widely employed thereafter. This method takes advantage of the fact that numerical calculations in black hole spacetimes usually employ a spherical (or spheroidal) harmonic decomposition, which in particular has the property that the individual modes of the bare force are finite at the particle. One then performs a finite subtraction on each mode, which is designed so that the resulting sum converges to the correct self-force. While extensive work has determined the form of this subtraction (in terms of “regularization parameters”) for arbitrary orbits of Schwarzschild and Kerr in the Lorenz gauge, regularization in alternative gauges has been by contrast neglected, with the first results, for circular orbits in Schwarzschild in a modified radiation gauge, appearing only recently [@k]. In this section I show that the mode sum regularization scheme is gauge-invariant under the parity condition, in the sense that the same subtraction may be employed to determine the force in any parity-regular gauge. Combined with the Lorenz gauge results of [@barack-ori-kerr], this provides a complete regularization prescription for parity-regular gauges, such as the radiation gauge of [@k].
Let $(\tilde{t},\tilde{r},\hat{\theta},\hat{\phi})$ be Boyer-Lindquist coordinates for the Kerr spacetime. For a given point along the worldline $\gamma$ where we wish to compute the self-force, we may rotate and time-translate the coordinates so that the particle is located at $\tilde{t}=\tilde{\phi}=0$, while taking the remaining coordinate positions to be $\tilde{r}=r_0$ and $\hat{\theta}=\theta_0$. Despite the lack of a full rotational symmetry we (following [@barack-ori-modesum]) nevertheless also perform an additional rotation in the $\hat{\theta}$ direction, so that the particle is located at the pole of the new coordinates. More precisely, define new angular coordinates $(\tilde{\theta},\tilde{\phi})$ by $$\begin{aligned}
\cos \hat{\theta} & = - \sin \tilde{\theta} \cos \tilde{\phi} \cos \theta_0 + \cos \tilde{\theta} \cos \theta_0 \label{theta-hat} \\
\tan \hat{\phi} & = \frac{\sin \tilde{\theta} \sin \tilde{\phi}}{\sin \tilde{\theta} \cos \tilde{\phi} \cos \theta_0 + \cos \tilde{\theta} \sin \theta_0} \label{phi-hat}\end{aligned}$$ to obtain “rotated Boyer-Lindquist” coordinates $\tilde{t},\tilde{r},\tilde{\theta},\tilde{\phi}$ in which the particle position is given by $\tilde{t}=0,\tilde{r}=r_0,\tilde{\theta}=0$. In these coordinates the metric components are smooth everywhere except for the pole $\theta=0$, where they acquire non-trivial direction-dependent limits. Below we will need the lowest-order relationship between the spatial Fermi coordinates $x^i$ and the rotated Boyer-Lindquist coordinates, restricted to the sphere. A straightforward computation gives this to be $$\label{fermi-tilde}
x^i|_{\tilde{t}=0, \tilde{r}=r_0} = \alpha^i \ \tilde{\theta} \cos \tilde{\phi} + \beta^i \ \tilde{\theta} \sin \tilde{\phi} + O(\tilde{\theta}^2),$$ where $\alpha^i$ and $\beta^i$ are constants independent of $\tilde{\theta}$ and $\tilde{\phi}$ (dependent on the Boyer-Lindquist position $\theta_0$, the three-velocity, and the mass and spin parameters of the Kerr metric), and where the $O(\tilde{\theta}^2)$ term may depend on $\tilde{\phi}$.
The advantage of placing the particle at the pole is the simplification of the spherical harmonic description by the elimination of modes with non-zero $m$ when the series is evaluated at the particle. Let a subscript $\ell$ denote the $\ell$’th term in the expansion evaluated at the pole/particle, $A_{\ell} = \int A Y_{\ell 0} d\tilde{\Omega}$ for integrable functions $A(\tilde{\theta},\tilde{\phi})$, and suitably generalized for distributions. When viewed in light of our angle-average result, the mode sum regularization prescription amounts to finding some $S_\ell^i$ such that $$\label{modesum}
\langle F^i \rangle_{r \rightarrow 0} = \sum_{\ell=0}^\infty \left( F^i_{\ell} - S^i_{\ell} \right).$$
![A diagram illustrating the geometrical setup of the mode sum regularization argument. The particle is at the pole of rotated Boyer-Lindquist coordinates. The mode decomposition is taken relative to the background coordinate sphere, while the self-force is given by an average over the local inertial sphere. For the change in bare force, a general theorem relates the mode sum at the particle/pole to the average over the polar circle, which agrees with the average over the local inertial sphere when the parity condition is satisfied.[]{data-label="fig:setup"}](setup)
This formula relates an average over an infinitesimal sphere surrounding the particle to a spherical harmonic decomposition on a finite sphere surrounding the black hole, evaluated at the particle (see figure \[fig:setup\]). Such a connection between mode sums and local averaging is familiar from ordinary Fourier series, where, if a function is of bounded variation, its series converges to the two-sided average at a discontinuity. For spherical harmonic expansions, an analogous result (e.g. section III22b of [@sansone]) states that if the average of a function over latitude lines is of bounded variation (as a function of latitude), then its spherical harmonic series evaluated at the pole (“Laplace series”) converges to the average on an infinitesimal latitude line surrounding the pole. (This result is easily understood at a formal level, by noting that $Y_{\ell 0}$ is independent of $\tilde{\phi}$, so that the formula for $A_{\ell}$ takes an average over $\tilde{\phi}$.) The theorem does not apply to the bare force $F^i$ (which is divergent), but it does apply to the *change* in bare force under a change in gauge, $\delta F^i$. In particular, a simple calculation (which reverses the calculations internal to the average in ) gives $$\delta F^i = \partial_0 \partial_0 \xi^i + R^{\ i}_{0 \ 0j}\xi^j - R^{\ k}_{0 \ 0l} x^l \partial_k \xi^i + O(r) \label{Fchange}$$ for any transformation of the form . If the transformation is parity-regular, equation , we see that $\delta F^i$ has a Fermi coordinate expansion of the form $$\label{Fchange-parity}
\delta F^i = \mathcal{C}^i(t) + \mathcal{S}^i(t,\vec{n}) + O(r),$$ where $\mathcal{S}^i$ is odd-parity, $\mathcal{S}^i(t,-\vec{n})=-\mathcal{S}^i(t,\vec{n})$.[^10] Restricting to the background coordinate sphere $\tilde{t}=0,\tilde{r}=r_0$ and expanding in $\tilde{\theta}$ at fixed $\tilde{\phi}$, we have $$\begin{aligned}
\delta F^i|_{\tilde{t}=0,\tilde{r}=\tilde{r}_0}(\tilde{\theta},\tilde{\phi}) & = \mathcal{C}^i(0) + \mathcal{S}^i(0,\vec{n}|_{\tilde{t}=0,\tilde{r}=\tilde{r}_0}(\tilde{\theta},\tilde{\phi})) + O(\tilde{\theta}) \label{starthere}\\
& = \mathcal{C}^i(0) + \mathcal{S}^i(0,\vec{n}|_{\tilde{t}=0,\tilde{r}=\tilde{r}_0}(\tilde{\theta}=0,\tilde{\phi})) + O(\tilde{\theta}) \label{gethere}\\
& = \mathcal{C}^i(0) + \mathcal{S}^i(0,\vec{n}_0(\tilde{\phi})) + O(\tilde{\theta}),\label{deltaFexp}\end{aligned}$$ where the $O(\tilde{\theta})$ terms may depend on $\tilde{\phi}$ and we have defined $$\label{n0}
\vec{n}_0(\tilde{\phi}) = \lim_{\tilde{\theta}\rightarrow 0} \vec{n}(\tilde{t}=0,\tilde{r}=\tilde{r}_0,\tilde{\theta},\tilde{\phi}).$$ In moving from equation to equation we have used the fact that the restriction of $\vec{n}$ to the background coordinate sphere is smooth in $\tilde{\theta}$ at fixed $\tilde{\phi}$ (see equation and recall $n^i=x^i/r$), as well as the fact that $\mathcal{S}^i$ is smooth in $\vec{n}$. Equation shows that the restriction of $\delta F^i$ to the background coordinate sphere is continuous (in fact smooth, by our assumptions) in $\tilde{\theta}$ at fixed $\tilde{\phi}$. In particular its average over $\tilde{\phi}$ is of bounded variation, and by the theorem we have $$\begin{aligned}
\label{poleavg}
\sum_{\ell} (\delta F^i)_\ell & = \lim_{\tilde{\theta}\rightarrow 0} \frac{1}{2 \pi} \int \delta F^i|_{\tilde{t}=0,\tilde{r}=r_0}(\tilde{\theta},\tilde{\phi}) d\tilde{\phi} \\
& = \mathcal{C}^i(0) + \frac{1}{2 \pi} \int \mathcal{S}^i(0,\vec{n}_0(\tilde{\phi})) d\tilde{\phi},\end{aligned}$$ where in the second line we have plugged in the form of equation . Since $\mathcal{S}^i$ is odd parity, the second term on the right-hand-side will vanish if $\vec{n}_0$ is odd under $\tilde{\phi} \rightarrow \tilde{\phi} + \pi$. This property is expected from the geometry of the setup (figure \[fig:setup\]), and is easily confirmed from equations and . Thus the term involving $\mathcal{S}^i$ vanishes, so that the Laplace series for $\delta F^i$ in fact converges to $\mathcal{C}^i$. However, the angle-average that computes the self-force also returns $\mathcal{C}^i$ on the form . Therefore, when the parity condition is satisfied the averages agree, and we have simply $$\label{sumavg}
\sum_\ell (\delta F^i)_\ell = \langle \delta F^i \rangle_{r \rightarrow 0},$$ showing that the Laplace series for the change in bare force $\delta F^i$ in fact converges to its local inertial angle-average, i.e., to the change in self-force it effects. This means in particular that no extra $S^i_\ell$ must be subtracted in the new gauge, since merely the process of decomposing $\delta F^i$ into modes and resumming returns its contribution to the self-force. To see this explicitly, let $F^i_\textrm{old}$ denote the bare force in a gauge that satisfies the parity condition, and write $$\begin{aligned}
\sum_\ell\left( F^i_\ell - S^i_\ell \right) & = \sum_\ell\left[ (F^i_{\textrm{old}})_\ell + (\delta F^i)_\ell - S_{\ell}^i \right] \nonumber \\
& = \sum_\ell\left[ (F^i_{\textrm{old}})_\ell - S_{\ell}^i \right] + \langle \delta F^i \rangle_{r \rightarrow 0} \nonumber \\
& = \langle F^i_{\textrm{old}}+\delta F^i \rangle_{r \rightarrow 0}. \label{showedit} \end{aligned}$$ In writing the second line we have used , and in writing the third line we have used equation . This shows that equation holds in the new gauge if it held in the old, i.e., that $S^i_\ell$ is a correct piece to subtract in any parity-regular gauge.
We note that previous work has organized the subtraction so that one first subtracts an $\hat{S}^i_\ell$ of the form $\hat{S}^i_\ell = A^i (\ell + 1/2) + B^i + C^i/(\ell + 1/2)$, then sums over modes (the result is now finite), and finally adds in a finite residual $D^i$ to get the correct self-force. (In terms of our $S^i_\ell$, $D^i$ is a “finite piece” $D^i \equiv \sum_\ell (S_\ell^i - \hat{S}_\ell^i)$.[^11]) The ($\ell$-independent) $A,B,C,D$ are the “regularization parameters” for the particular orbit and spacetime, which by the results of this section do not depend on the choice of (parity-regular) gauge.
I gratefully acknowledge Robert Wald for many helpful conversations and Laszlo Szabados for helpful correspondence. This research was supported in part by NSF grant PHY08-54807 to the University of Chicago.
Derivation in the case of non-zero spin {#sec:spin}
=======================================
When the spin is non-zero, we may not rely on Zerilli’s results for Schwarzschild [@zerilli], and the analysis of second-order Einstein equation, equation , becomes more complicated. In this case it pays to systematically consider the contributions to the linearized Einstein tensor $\mathcal{G}_{\bar{\mu}\bar{\nu}}$ from the various terms in the background and perturbation . Since all terms are stationary to the relevant orders, no $\bar{t}$-dependence will appear, and we may count orders in $1/\bar{r}$. At leading order $O(1)$ in $\mathcal{G}_{\bar{\mu}\bar{\nu}}$, the only contributions are from the “B” term in the perturbation and the flat “$\eta$” term in the background. Since the “B” term is (by the Fermi coordinate construction) a perturbation of flat spacetime, the linearized Einstein equation is automatically satisfied and we learn no new information. At next order $O(1/\bar{r})$ in $\mathcal{G}_{\bar{\mu}\bar{\nu}}$, both the “E” term in the perturbation and the mass term $2M \delta_{\mu \nu}/\bar{r}$ in the background can now contribute. Expanding the background metric in powers of $1/\bar{r}$, the linearized Einstein equation at $O(1/\bar{r})$ may be written $$\label{moo}
G^{(1)}_{\eta}[E_{\mu \nu} \bar{r}] + 2 G^{(2)}_{\eta}[2M/\bar{r}\delta_{\mu \nu},B_{\mu i \nu j}\bar{x}^i \bar{x}^j] = 0,$$ where $G^{(1)}_{\eta}$ and $G^{(2)}_{\eta}$ are the first and second-order Einstein tensors (respectively) off of flat spacetime. However, since the “B” term has no $\ell=1$, electric parity part while the $M$ term is spherically symmetric, the second term in has no $\ell=1$, electric parity part. Therefore the $\ell=1$, electric parity part of the “E” term satisfies the vacuum linearized Einstein equation about flat spacetime, $$\label{oink}
G^{(1)}_{\eta}[(E_{\mu \nu} \bar{r})_{\ell=1,+}] = 0,$$ where the subscript “$\ell=1,+$” indicates the $\ell=1$, electric parity part. Since $\ell=1$, electric parity perturbations of flat spacetime that scale linearly with $\bar{r}$ are pure gauge (e.g., [@zhang]), all solutions to equation may be written $(E_{\mu \nu} \bar{r})_{\ell=1,+}=\partial_{(\mu} \mathcal{E}_{\nu)}$, where $\mathcal{E}_\mu$ is $\ell=1$ and electric parity. If we make a (second-order near-zone) gauge transformation generated by $\mathcal{E}$, then we may set the $\ell=1$, electric parity part of $E_{\mu \nu}$ to zero. However, one can check that in general the required gauge vector $\mathcal{E}_\mu$ is $\bar{t}$-dependent (despite the “E” term being stationary), so that this gauge transformation would introduce $\bar{t}$-dependence at higher orders, in violation of our previous choices that eliminated such dependence. (In particular, terms of order $\bar{t}/\bar{r}$ and $\bar{t}^2/\bar{r}^2$ would appear in $\bar{g}^{(2)}_{\bar{\mu}\bar{\nu}}$, contradicting the previous choices $\dot{H}_{\mu \nu}=\ddot{C}_{\mu \nu}=0$—see equations and .) Without invalidating these choices we may only make gauge transformations whose gauge vector is $\bar{t}$-independent. One may check that this allows us to put $E_{\mu \nu}$ in the form $$\label{cluck}
(E_{\mu \nu} \bar{r})_{\ell=1,+} = -2 a_i \bar{x}^i t_\mu t_\nu,$$ where $a_i$ is an arbitrary spatial vector, named since in this form $(E_{\mu \nu} \bar{r})_{\ell=1,+}$ is an “acceleration perturbation” familiar from Fermi coordinates about an accelerated worldline. Making such a gauge transformation (and “absorbing” its effects at $O(1)$ into the arbitrary tensor $K_{\mu \nu}$), equation becomes simply $$\label{Finsimplegauge2}
\mathcal{A}^i = a^i.$$
We have now made coordinate choices that eliminate all $\bar{t}$-dependence and reduce the relevant $\ell=1$, electric parity part of $E_{\mu \nu}$ into a simple form with one unknown, the gauge-invariant acceleration $\mathcal{A}^i=a^i$. It remains to use the linearized Einstein equation at order $O(1/\bar{r}^2)$ to determine $\mathcal{A}^i$. Again expanding the background in powers of $1/\bar{r}$, at this order the $\ell=1$, electric parity parts of the second-order Einstein equation may be written as $$\left( G^{(1)}_{\eta}\left[K_{\mu \nu}\right] + 2 G^{(2)}_{\eta}\left[\frac{2M \delta_{\mu \nu}}{\bar{r}}, -2\mathcal{A}_i \bar{x}^i t_{\mu} t_{\nu}\right] + 2 G^{(2)}_{\eta}\left[\frac{-4 n^i t_{(\mu} S_{\nu)i}}{\bar{r}^2},B_{\mu i \nu j}\bar{x}^i \bar{x}^j\right] \right)_{\ell=1,+}=0,$$ where terms that can give no $\ell=1$, electric part have not been displayed. This equation gives relationships between $K_{\mu \nu}$, $\mathcal{A}_i$, $M$, $S_{ij}$, and $R_{\mu i \nu j}$. To determine a relationship not involving the unknown tensor $K_{\mu \nu}$, first write out the linearized Einstein tensor about flat spacetime for stationary perturbations, and note that there is a particular $\ell=1$ part[^12] that vanishes for all $K_{\mu \nu}$. Then computing this particular $\ell=1$ part of the remaining two terms will determine $\mathcal{A}_i$ in terms of the mass, spin, and Riemann tensor. Performing this calculation yields $M \mathcal{A}_i = \frac{1}{2}S^{kl}R_{kl0i}$, as claimed in equation .
Equation of motion in parity-irregular gauges {#sec:eom-noparity}
=============================================
I now consider the form of the equations of motion in parity-irregular gauges, adopting the mass dipole definition of center of mass. Under the general gauge transformations , we have from equation that[^13] $$\delta Z^i = \frac{3}{4\pi} \lim_{r\rightarrow 0}\int \xi^j n_j n^i = 3 \langle \xi^j n_j n^i \rangle_{r \rightarrow 0} \label{Zchange}.$$ Beginning with the equation of motion in a parity-regular gauge, we may now derive an equation for $Z^i$ in a new gauge, $$\begin{aligned}
\ddot{Z}^i - & \langle F^i \rangle_{r \rightarrow 0} + R_{0j0}^{\ \ \ i} Z^j - M^{-1} S^{kl} R_{kl0i} = \nonumber \\
& \langle ( -\partial_0 \partial_0 \xi_j (\delta^i_{\ j} - 3n^i n_j) + R_{0j0k}x^k \partial^j \xi^i - R_{0i0}^{\ \ \ j}\xi_k (\delta^k_{\ j} - 3 n^k n_j) ) \rangle_{r \rightarrow 0}, \label{eom-noparity}\end{aligned}$$ where equation has been used. If the parity condition is satisfied, and $\xi^i = c^i + \Sigma^i(\vec{n})+O(r)$ with $\Sigma^i$ odd-parity, then the right hand side vanishes and the equation of motion retains the original form, depending only on the local spacetime metric (at zeroth and first order). However, if the parity condition is not satisfied, then the right hand side does not in general vanish,[^14] and the equation of motion for $Z^i$ takes a complicated form involving the gauge transformation to some reference gauge. Another way to see the difficulty is to repeat the calculations of for a general gauge transformation, giving $$\delta \ddot{Z}^i = 3 \langle ( \delta F_j + R_{0k0l} x^l \partial_k \xi_j - R_{0j0k}\xi^k ) n^j n^i \rangle_{r\rightarrow 0}.$$ Without the parity condition the Riemann terms do not simplify into the geodesic deviation form $R_{0i0j}\delta Z^j$. It appears that no expression in terms of just $\delta Z^i$ and $\delta h_{\mu \nu}$ is possible, so that the equation for $Z^i$ in parity-irregular gauges must involve a gauge vector explicitly.
[99]{} Y. Mino, M. Sasaki, and T. Tanaka, *Phys. Rev. D*, **55**, 3457-3476, (1997) T.C. Quinn and R.M. Wald, *Phys. Rev. D*, **56**, 3381-3394, (1997) S. Detweiler and B.F. Whiting, *Phys. Rev. D*, **67**, 024025, (2003) E. Poisson, *Liv. Rev. Rel.* **7** 6 (2004) S. Gralla and R. Wald, *Class. Quantum Grav.* **25** 205009 (2008) L. Barack and A. Ori, *Phys. Rev. D*, **64**, 124003 (2001) T. Regge and C. Teitelboim, *Ann. Phys.* **88** 286 (1974) L. Barack and A. Ori, *Phys. Rev. D* **61** 061502(R) (2000) T. Keidl, A. Shah, J. Friedman, D. Kim and L. Price, *Phys. Rev. D* **82** 124012 (2010) R. Wald 1984 *General Relativity* (Chicago, IL: University of Chicago Press) R. Beig and N. OMurchadha, *Ann. Phys.* **174** 463 (1987) C. Misner, K. Thorne and J. Wheeler 1973 *Gravitation* (San Francisco, CA: Freeman) S. Gralla and R. Wald, arXiv:1104.5205 F. Zerilli, *Phys. Rev. D* **2** 2141 (1970) T. Regge and J. Wheeler, *Phys. Rev.* **108** 1063–1069 (1957) X.H. Zhang, *Phys. Rev. D* **34** 991 (1986) A. Pound, *Phys. Rev. D* **81** 024023 (2010) T. Quinn and R. Wald, *Phys. Rev. D* **60** 064009 (1999) L. Barack, *Class. Quant. Grav.* **26** 213001 (2009) L. Barack and A. Ori, *Phys. Rev. Lett.* **90** 111101 (2003) G. Sansone 1977 *Orthogonal Functions* (Huntington, New York: Robert E. Krieger Publishing)
[^1]: The (discontinuous) gauge vector that changes the singularity from isotropic to Schwarzschild type is given by $\xi^i=n^i=x^i/r$ in Fermi normal coordinates.
[^2]: The spin is antisymmetric and satisfies $u^a S_{ab}=0$. The deviation satisfies $Z^a u_a=0$.
[^3]: The mass dipole may also be thought of as the time-space component of the spin tensor, $S^{0i}$. However, we will work with a spin tensor that is orthogonal to $\gamma$, $S^{0i}=0$, defining a separate mass dipole.
[^4]: A definition of perturbed mass was given in [@pound], which appears to correspond to equation applied in the Lorenz gauge. This definition would not be sensible within our framework. The mass defined in [@pound] was found to evolve with time. The conclusion that the perturbed mass evolves with time appears to be at odds with the analysis of [@quinn-wald-energy], where it was found that energy conservation is satisfied under the assumption of no change in mass.
[^5]: We note that a later formula due to Beig and O’Murchadha [@beig-omurchadha] is equivalent given the parity condition.
[^6]: Regge and Teitelboim also impose a parity condition on the extrinsic curvature. However, this condition is not needed for the center of mass and plays no role in our analysis.
[^7]: In the appendix of paper I an opposite sign convention, $x'^\mu = x^\mu - \xi^\mu$, was used for the definition of the gauge vector.
[^8]: We could equivalently express $\delta Z^i$ as an average over a circle or over two antipodal points, since these averages all agree for a “constant plus odd parity” function. The entire derivation of the equation of motion could then proceed unchanged, so that the self-force in fact may equivalently be written as the average of the bare force over a (constant geodesic distance) sphere, circle, or pair of points.
[^9]: The angle average of $F^i$ is manifestly finite in the gauge of section \[sec:dM\], and, by reversing the calculations of , may be easily seen to transform finitely (see also ).
[^10]: The parity condition is not required to show the applicability of the theorem, but will be necessary for the later analysis of this section.
[^11]: Remarkably, it has been found (by lengthy computation in the Lorenz gauge) that $D^i=0$ in every circumstance, so that the subtraction of $A^i (\ell + 1/2) + B^i + C^i/(\ell + 1/2)$ in fact returns the correct force. This surprising relationship between the large-$\ell$ expansion of a point particle metric perturbation (which uniquely determines $A,B,C$) and the physical self-force has thus far defied a more fundamental explanation.
[^12]: In the notation of [@gralla-wald], this part is $(2/3) R^D_i + R^E_i$, involving only spatial trace-free components of the linearized Ricci tensor.
[^13]: As already noted in section \[sec:eom\], the sign convention for $\xi^\mu$ used in this paper differs from that used in the appendix of paper I.
[^14]: For example, with $\xi^i = B^j n_j n^i$ with $B^i$ constant, the right hand side of evaluates to $(16/15)R_{0i0j}B^j$.
|
---
abstract: 'We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy$_1$ process with the role of space and time interchanged. The predictions, which contain no free parameter, are numerically well confirmed for space-time statistics in the stationary state. In particular the spatial fluctuations of the interface computed numerically agree well with those given by the GOE edge distribution of Tracy and Widom.'
author:
- 'G.T. Barkema[^1]'
- 'P.L. Ferrari[^2]'
- 'J.L. Lebowitz[^3]'
- 'H. Spohn[^4]'
date: '15. September 2014'
title: KPZ universality class and the anchored Toom interface
---
Introduction {#sec1}
============
Toom [@T80] studied a family of probabilistic cellular automata on $\mathbb{Z}^2$ which have a unique stationary state at high noise level and (at least) two stationary states for low noise. Most remarkably, the low noise states are stable against small changes in the update rules [@BG85]. This is in stark contrast to models satisfying the condition of detailed balance. For example the two-dimensional (2D) ferromagnetic Ising model with Glauber spin flip dynamics at sufficiently low temperatures and zero external magnetic field, $h=0$, has two equilibrium phases with non-zero spontaneous magnetization. But by a small change of $h$ uniqueness is regained [@HS77].
We consider the 2D Toom model with NEC (North East Center) majority rule. The system consists of Ising spins ($S_{i,j} = \pm 1$) located on a square lattice which evolve in discrete time. (We use magnetic language only for convenience. In physical realizations $S_{i,j}$ is a two-valued order parameter field). At each time step, all spins $S_{i,j}$ are updated independently according to the rule $$\label{1.1a}
S_{i,j}(t+1) =
\begin{cases}
\mathrm{sign}\big(S_{i,j+1}(t) + S_{i+1,j}(t) +S_{i,j}(t)\big) & \quad \mathrm{with\,\, probability\,\,} 1 - p - q\,,\\
+1 & \quad \mathrm{with\,\, probability\,\,} p\,,\\
-1 & \quad \mathrm{with\,\, probability\,\,} q\,.
\end{cases}$$ For $p=q=0$ we have a deterministic evolution: each updated spin becomes equal to the majority of itself and of its northern and eastern neighbors. Non-zero $p,q$ represents the effect of a noise which favors the $+$ sign with probability $p$ and the $-$ sign with probability $q$. It was proved by Toom that for low enough noise ($p,q$ sufficiently small) the automaton has at least two translation invariant stationary states, such that the spins are predominantly $+$ or $-$, respectively. The probability with which one is obtained depends on the initial conditions.
To investigate the spatial coexistence of the two phases, specific boundary conditions were introduced in [@DLLS91; @BBLS96]. More concretely, the Toom model restricted to the third quadrant was studied with the boundary conditions $S_{i,0} = 1$ and $S_{0,j} = -1$ for all $i,j<0$ and all $t$. Since the information is traveling southwest, in the long time limit a steady state is reached, for which the upper part is in one phase and the lower half in the other one. The phases are bordered by an interface which fluctuates but has a definite slope, depending on $p,q$, on the macroscopic scale. Of interest are steady state static and dynamical fluctuations of this non-equilibrium interface. Since both pure phases have already a nontrivial intrinsic structure, to analyse properties of the interface seems to be a difficult enterprise. In [@DLLS91; @BBLS96] a low noise approximation is used for which the interface is governed by an autonomous stochastic dynamics in continuous time, see Figure \[Fig1\]. The interface can be represented by a spin configuration on the semi-infinite lattice $\mathbb{Z}_+$. Such spin configurations inherit then a dynamics in which spins are randomly exchanged. It is this Toom spin exchange model described below which is the focus of our contribution. For more information we refer to [@DLLS91; @BBLS96].
\[cc\][$N$]{} \[cc\][$E$]{} \[cc\][$W$]{} \[cc\][$S$]{} \[cc\][$1$]{} \[cc\][$\lambda$]{} \[cc\][$n$]{} \[cc\][$\;\;M_n$]{} \[cc\][$-$]{} \[cc\][$+$]{} ![Representation of the Toom interface model. The black/white dots are the spin values $+/-$ in the Toom spin exchange model.[]{data-label="Fig1"}](FigToom "fig:"){height="6cm"}
Toom spin exchange model {#toom-spin-exchange-model .unnumbered}
------------------------
We consider the 1D lattice $\mathbb{Z}$ and spin configurations $\{\sigma_j, j \in \mathbb{Z}, \sigma_j = \pm1\}$. A exchanges with the closest to the right at rate $\lambda$ and, correspondingly, a exchanges with the closest $+$ spin to the right at rate $1$. $\lambda \in [0,1]$ is an asymmetry parameter. The Bernoulli measures are stationary under this dynamics and we label them by their average magnetization, $\mu = \langle \sigma_0\rangle_\mu$. On a finite ring of $N$ sites the dynamics is correspondingly defined, replacing right by clockwise. As can be seen from Figure \[Fig1\], the interface is enforced by a hard wall at $0$, that is, spin configurations are restricted to the half lattice $\mathbb{Z}_+ = \{1,2,\ldots \}$, but the dynamics remains unaltered. The Toom spin model on the half lattice has an unusual independence property. If one considers the dynamics of the subsystem $\{\sigma_1(t),\ldots,\sigma_L(t)\}$, then it evolves as a continuous time Markov chain. However the magnetization is no longer conserved. If, for some $j$, the entire block $[j,\ldots,L]$ has spin $+$, then $\sigma_j(t)$ flips to $- \sigma_j(t)$ with rate $\lambda$ and correspondingly for a block of touching the right border the flip is done with rate 1. As a consequence, a unique limiting probability measure is approached as $t \to \infty$. In our approximation, the height of anchored interface of the Toom automaton is just the magnetization of the Toom spin model, $$\label{1.1}
M_n(t) = \sum_{j=1}^n \sigma_j(t)\,.$$ The argument $t$ is omitted in case the $n$-dependence at fixed $t$ is considered. Averages in the steady state are denoted by $\langle \cdot \rangle$. Note that by time stationarity and time correlations such as $\langle M_n(t) M_{n'}(t')\rangle$ depend only on $t - t'$. At $\lambda = 1$ the interface is along the diagonal and fluctuates symmetrically, $\langle M_n \rangle = 0$, while for $0 < \lambda < 1$ the interface becomes asymmetric.
Based on theoretical and numerical evidence, in [@DLLS91] it was concluded that, for large $n$, $$\label{1.2}
\langle M_n^2 \rangle - \langle M_n \rangle^ 2\simeq n^{1/2} \quad\mathrm{for} \,\,\lambda = 1$$ with possibly logarithmic corrections, while $$\label{1.3}
\langle M_n^2 \rangle - \langle M_n \rangle^ 2\simeq n^{2/3} \quad\mathrm{for} \,\,0 < \lambda < 1 \,.$$ Most remarkably, using the then just being developed multi-spin coding techniques, the full probability density function (pdf) for $M_n$ was recorded, see [@BBLS96], Fig. 3. For $\lambda = 1$, the pdf is well fitted by a Gaussian, in agreement with the prediction of the collective variable approximation (CVA) [@DLLS91]. The variance differed however by a logarithmic correction from the $\sqrt{n}$ prediction, in the scaling limit, given by the CVA. For $\lambda = \tfrac{1}{4}$, the scaling function obtained through the CVA was used as a fit to the numerical data. This is given by $\mathrm{Ai}(x)^4$ with $\mathrm{Ai}$ the standard Airy function. Somewhat *ad hoc*, the left tail of $\mathrm{Ai}$ was cut at its first zero. Looking eighteen years later at the same figure, with the hindsight of the much improved understanding of the KPZ universality class, it is a safe guess that in fact a Tracy-Widom distribution from random matrix theory is displayed. Apparently the fluctuations of the anchored Toom interface share the same fate as the length of the longest increasing subsequence of random permutations. Without knowing, Odlyzko [@OR99] observed the GUE Tracy-Widom distribution. We refer to [@HL14] for a more complete account of the history. For us Fig. 3 of [@BBLS96] is a compelling motivation to return to the fluctuations of the anchored Toom interface and to understand better how they fit into the KPZ universality class.
In this note, we will provide numerical and theoretical evidence that in fact $$\label{1.4}
M_n \simeq \mu_0 n + (\Gamma n )^{1/3}\tfrac{1}{2} \xi_{\mathrm{GOE}}$$ for large $n$ and $0 < \lambda < 1$. Here the coefficients $\mu_0,\Gamma$ depend on $\lambda$ and are computed explicitly. The random amplitude $\xi_{\mathrm{GOE}}$ is GOE Tracy-Widom distributed. The general form of (\[1.4\]) is familiar from other models in the KPZ universality class. To have fluctuations governed by the GOE edge distribution came as a complete surprise and has not been anticipated before. To be on the safe side we also investigate the covariance $\langle M_n(t) M_{n}(0)\rangle - \langle M_n(0)\rangle^2$ and compare it with the prediction coming from the covariance of the Airy$_{1}$ process. Besides running multi-spin coding on more modern machines, we present a much improved analysis on interchanging the role of space and time for the interface dynamics.
Mesoscopic description of the Toom interface {#sec2}
============================================
To study the fluctuations of the Toom interface, it is convenient to start from a mesoscopic description of the height $$\label{2.3}
h(x,t) \simeq M_n(t)\,,$$ where $x$ stands for the continuum approximation of $n$. Firstly note that on $\mathbb{Z}$ the Toom spin model conserves the magnetization and thus has a one-parameter family of stationary states labeled by the average magnetization, $\mu$. In the steady state the spins are independent and the spin current is given by $$\label{2.1}
J(\mu,\lambda) = 2 \Big( \lambda \frac{1+\mu}{1-\mu} - \frac{1-\mu}{1+\mu}\Big)\,,$$ see [@DLLS91]. For the anchored Toom interface we expect (and have checked numerically) that in small segments very far away from the origin the spins will be independent, so that, $\langle \sigma _i\sigma_{i+j} \rangle - \langle \sigma _i\rangle \langle\sigma_{i+j}\rangle \to 0$ as $i \to \infty$ at fixed $j \neq 0$. To have a stationary state for the semi-infinite system thus requires $J =0$. Using (\[2.1\]) this has the unique solution $$\label{2.2}
\mu_0 = \frac{1-\sqrt{\lambda}}{1 +\sqrt{\lambda}}\,,$$ which determines the asymptotic magnetization. If $h$ is slowly varying on the scale of the lattice, then locally it will maintain a definite slope $u = \partial_x h$. The local slope is conserved, hence governed by the conservation law $$\label{2.3a}
\partial_t u + \partial_x J(u,\lambda) = 0\,,$$ which should be viewed as the Euler equation for the magnetization of the spin model. Equivalently, there is a Hamilton-Jacobi type equation for $h$, $$\label{2.3c}
\partial_t h + J(\partial_x h,\lambda) = 0\,,$$ with $h(x) = \mu_0x$ as stationary solution.
To describe on a mesoscopic scale the statistical properties of the Toom interface, we follow the common practice to add noise to the deterministic equation (\[2.3c\]), see for example [@LL]. More concretely to the spin current in (\[2.3a\]) we add the dissipative term $-\tfrac{1}{2}D\partial_x u$, $D$ the diffusion constant, and, since local exchanges are essentially uncorrelated, the space-time white noise $\kappa W(x,t)$. $W(x,t)$ is normalized and $\kappa$ is the noise strength. Since the deviation from the constant slope profile $\mu_0 n$ will be studied, in fact it suffices, by power counting, to keep the current $J(\mu,\lambda)$ up to second order relative to $\mu_0$ as $$\label{2.3b}
J(\mu -\mu_0, \lambda ) = v(\lambda) (\mu -\mu_0)+\tfrac{1}{2}G(\lambda)(\mu -\mu_0)^2 + \mathcal{O}((\mu -\mu_0)^3)\,,$$ where $$\label{2.5}
v(\lambda) = 2(1+ \sqrt{\lambda})^2\,,\quad G(\lambda) = (1+ \sqrt{\lambda})^3(1 - \sqrt{\lambda})\frac{1}{\sqrt{\lambda}}\,.$$ Thereby one obtains that on a mesoscopic scale the fluctuating height $h(x,t)$ is governed by $$\label{2.6}
\partial_t h = - v\partial_x h - \tfrac{1}{2}G(\partial_x h)^2 + \tfrac{1}{2}D\partial_x^2h + \kappa W(x,t)$$ for $t \geq 0$. For the Toom interface the height is pinned at the origin which leads to the restriction $x \geq 0$ and the boundary condition $$\label{2.2c}
h(0,t) = 0\,.$$ The coefficients $v,G$ depend on $\lambda$. If $G =0$, as is the case for $\lambda = 1$, the second order expansion does not suffice. Fourth order is irrelevant, but the third order term generates logarithmic corrections [@P], which are the theoretical reason behind the already mentioned logarithmic corrections for the interface variance at $\lambda = 1$.
Eq. (\[2.6\]) is the much studied one-dimensional KPZ equation [@KPZ] with two important differences. Firstly the height function is over the half-line, being pinned at the origin, and secondly there is the outward drift $v(\lambda)$, which cannot be removed because of this boundary condition. At such brevity our reasoning may look *ad hoc*. But the scheme should be viewed as a particular case of the KPZ scaling theory [@KHM; @S], which has been confirmed through extensive Monte Carlo simulations for related models, for example see [@BS].
For magnetization $\mu$ the spin susceptibility $A$ equals, for independent spins, $\langle \sigma_0^2\rangle_\mu - \langle \sigma_0\rangle_\mu^2 = 1 -\mu^2$ and at $\mu_0$ is given by, $$\label{2.7}
A = 4 \sqrt{\lambda} (1+ \sqrt{\lambda})^{-2}\,.$$ To connect with the parameters of Eq. (\[2.6\]), one checks that on $\mathbb{R}$ the steady state has the slope statistics given by spatial white noise with variance $\kappa^2/D$. Therefore we identify as $$\label{2.6a}
A = \kappa^2/D\,.$$ In the scaling regime only $A$ will appear, which is unambiguously defined by (\[2.7\]) in terms of the spin model, while $D$ and $\kappa$ separately are regarded as phenomenological coefficients. As for $M_n(t)$, our focus is the stationary process determined by (\[2.6\]), (\[2.2c\]).
Interchanging the role of space and time {#sec2a}
========================================
Considering Eq. (\[2.6\]), it would be of advantage to interchange $x$ and $t$, because then the boundary value $h(0,t) = 0$ turns into an initial condition, which is more accessible. From the perspective of stochastic partial differential equation, such an interchange looks impossible. But once we write the Cole-Hopf solution of (\[2.6\]), for example see [@CH], our scenario becomes fairly plausible.
The Cole-Hopf transformation is defined by $$\label{2.8}
Z(x,t) = \mathrm{e}^{(G/D)h(x,t)}\,,$$ which satisfies $$\label{2.9}
\partial_tZ = \tfrac{1}{2} D\partial_x^2 Z -v\partial_x Z - (G\kappa/D) W Z$$ on $\mathbb{R}_+$ with boundary condition $Z(0,t) = 1$ and some initial condition $Z_0(x)$. The first two terms generate a Brownian motion with constant drift, which is used in the Feynman-Kac discretization to formally integrate (\[2.9\]). Let $b(t)$ be a Brownian motion with $b(0) = 0$ and variance variance $ \mathbb{E}(b(t)^2) = Dt$, $ D > 0$, $ \mathbb{E}$ denoting the expectation for $b(t)$. In the usual parlance $b(t)$ is called a directed polymer, since it moves forward in the time direction. Furthermore let $T$ be the largest $s$ such that $x + b(t-s) -v(t - s) =0$, i.e. $T$ is the first time of hitting of $0$ for a Brownian motion with drift starting at $x$. Then Eq. (\[2.9\]) integrates to $$\label{2.10}
Z(x,t) = \mathbb{E} \Big( \mathrm{e}^{-(G\kappa/D) \int_{(t - T) \vee 0}^t ds W(x + b(t-s) -v(t-s),s)}\big(Z_0(b(t) -vt)\mathbbm{1}_{[t \leq T]}
+ \mathbbm{1}_{[t > T]}\big)\Big)\,.$$ For large $t$, the path $\{x + b(t-s) - v(t-s), 0 \leq s \leq t\}$ will hit 0 before $s = 0$ with a probability close to one. Hence the contribution from the term with $Z_0$ will vanish, the particular initial conditions are forgotten, and $$\label{2.11}
\lim_{t \to \infty} Z(x,t) = Z_{\infty}(x) = \mathbb{E}\Big( \mathrm{e} ^{-(G\kappa/D)\int^0_{-T} ds W(x +b(-s) +vs,s)}
\Big)\,.$$ Going back to (\[2.8\]), $(D/G)\log Z_{\infty}(x)$ defines the stationary measure for Eqs. (\[2.6\]), (\[2.2c\]). The stationary process for all $t \in \mathbb{R}$ is obtained by shifting $W$ in $t$ as $$\label{2.12}
Z_{\mathrm{st}}(x,t) = \mathbb{E}\Big( \mathrm{e} ^{-(G\kappa/D)\int^0_{-T} ds W(x + b(-s) +vs,s+t)}
\Big)\,.$$
To understand the interchange between $x$ and $t$, at least in principle, we discretize (\[2.12\]) by replacing $\mathbb{R}_+ \times
\mathbb{R}$ by $\mathbb{Z}_+\times\mathbb{Z}$. Then the continuum directed polymer $b(t) -vt$ is replaced by its discrete cousin, namely a random walk $\omega$ with down-left paths only. The walk starts at $\vec{j}_0$, $\vec{j} =(j_1,j_2)$. The transitions are $\omega_n$ to $\omega_n - (1,0)$ with probability $p$ and $\omega_n$ to $\omega_n - (0,1)$ with probability $q$, $p+q = 1$. $T$ is the time of first hitting the line $\{j_2 = 1\}$. $W(x,t)$ is replaced by a collection of independent standard Gaussian random variables $\{W(j_1,j_2),j_1\in\mathbb{Z}, j_2 \in \mathbb{Z}_+\}$. The integral in the exponent of (\[2.12\]) now turns into the sum over $W(j_1,j_2)$ along $\omega$ until the boundary is reached. Since the path $\omega$ is decreasing, it can be viewed with either $j_1$ or $j_2$ as time axis. In the first version, the continuum limit equals $-(q/p)t + \sqrt{q}\,b(t)$ and in the second version $-(p/q)t + \sqrt{p}\,b(t)$.
Eq. (\[2.12\]) corresponds to the first version. Instead we now take $j_2$ as time axis and consider the continuum version of the partition function as in Eq. (\[2.12\]). Then the directed polymer is parametrized as $u \mapsto t + \tilde{b}(x-u) - \tilde{v}(x - u)$. $\tilde{b}(u)$ is a Brownian motion with $\tilde{b}(0) = 0$ and variance $\tilde{\mathbb{E}}(\tilde{b}(u)^2) = \tilde{D} u $ and the transformed drift is $\tilde{v} = v^{-1}$. With these conventions the partition function reads $$\label{2.12a}
\tilde{Z}_{\mathrm{st}}(x,t) = \tilde{\mathbb{E}} \Big( \mathrm{e} ^{(\tilde{G}\tilde{\kappa}/\tilde{D})\int_0^{x} du W(u, t + \tilde{b}(x-u) -\tilde{v}(x-u))}
\Big)\,,$$ where $x>0$ and $t\in\mathbb{R}$. By defining $\tilde{h} = (\tilde{D}/\tilde{G})\log \tilde{Z}_{\mathrm{st}}$, one arrives at $$\label{2.20a}
\partial_x \tilde{h} = - \tilde{v}\partial_t \tilde{h} - \tfrac{1}{2} \tilde{G} (\partial_t \tilde{h})^2
+ \tfrac{1}{2}\tilde{D} \partial_t^2\tilde h + \tilde{\kappa} W$$ with initial condition $$\label{2.13a}
\tilde{h}(0,t) =0\,.$$
There is no good reason for having a strict identity between $h$ and $\tilde{h}$. But one would expect both to have the same asymptotic behavior, *provided* one appropriately adjusts $\tilde{G}$ and $\tilde{A} = \tilde {\kappa}^2/\tilde{D}$. The argument given is not specific enough for finding out the correctly transformed coefficients. For this purpose we return to the Toom spin model on $\mathbb{Z}$ and first consider the macroscopic height evolution. Then, as in (\[2.3a\]), $$\label{2.13}
\partial_t h +J(\partial_x h) = 0\,,$$ where $J(\partial_x h) = J(\partial_x h, \lambda)$. Since $J$ is monotone, it is invertible and $$\label{2.14}
\partial_x h + \tilde{J}(\partial_t h) = 0\,,\quad J(\tilde{J}(u)) = u\,,$$ and, expanding in $\partial_t h$, $$\label{2.15}
\partial_x h = - v^{-1}\partial_t h + \tfrac{1}{2} G v^{-3}(\partial_t h)^2 + \mathcal{O}((\partial_t h )^3)\,.$$ We conclude that $$\label{2.16h}
v\tilde{v} = 1\,, \quad G = -\tilde{G} v^3\,.$$
As a second task we have to find out the transformed susceptibility $\tilde{A}$. For this purpose we consider the stationary Toom spin model, $\sigma_j(t)$, on $\mathbb{Z}\times \mathbb{R}$ with average magnetization $\mu$. Since the steady state is Bernoulli, one already knows that $$\label{2.17}
\sum_{j\in\mathbb{Z}} \big(\langle \sigma_j(0)\sigma_0(0)\rangle _\mu - \mu^2\big) = 1 - \mu^2 = A\,.$$ $\tilde{A}$ is the corresponding susceptibility in the $t$-direction, which is defined by $$\label{2.16a}
\int_{-\infty}^{\infty} dt \big(\langle \sigma_0(t)\sigma_0(0)\rangle _\mu - \mu^2\big) = \tilde{A} \,.$$ The computation of $\tilde{A}$ requires dynamical correlations, which looks like a difficult task. Help comes from the very special correlation structure which holds for a large class of 1D spin models with exchange dynamics. While for some models such structure can be checked from the exact solution [@PrSp04; @FeSp06], for the Toom spin model it is an assumption. But there is no good reason why the Toom spin model should behave exceptionally. We consider correlations between $(0,0)$ and $(j,t)$. There then is a special direction, determined through the speed of propagation of small disturbances, $v(\lambda)$. Along $(v(\lambda)s,s)$ the line integral as in (\[2.16a\]) vanishes, while in all other directions it converges to a strictly positive value.
For the complete argument it is convenient to first define the height function for the Toom spin model by $$\label{2.18}
\mathsf{h}(j,t) = \begin{cases}
\sum_ {i = 1}^j (\sigma_i(t) - \mu)
& \quad \mathrm{for\,\, } j> 0\,,\\
\mathcal{J}_{(0,1)}([0,t] ) - Jt& \quad \mathrm{for\,\, } j=0\,,\\
\sum_ {i = j}^{-1} (\sigma_i(t) -\mu) & \quad \mathrm{for\,\, } j< 0\,.
\end{cases}$$ Here $ \mathcal{J}_{(0,1)}([0,t] )$ is the actual time-integrated spin current across the bond $(0,1)$ up to time $t$ implying the convention $\mathsf{h}(0,0) = 0$. By definition the spin susceptibility along the $j$-axis is given by $$\label{2.19}
\langle (\mathit{\mathsf{h}}(j,0) - \mathsf{h}(0,0) )^2\rangle = A j$$ for large $j$, $j > 0$. Correspondingly in the $t$-direction $$\label{2.20}
\langle (\mathsf{h}(0,t) - \mathsf{h}(0,0) )^2\rangle = \tilde{A} t$$ for large $t$, $t > 0$. In the direction of the propagation speed, the height fluctuations of are suppressed, $$\label{2.21}
\langle (\mathsf{h}(vt,t) - \mathsf{h}(0,0) )^2\rangle = \mathcal{O}(t^{2/3})\,.$$ We set $X= \mathsf{h}(vt,t) - \mathsf{h}(0,t)$ and $Y=\mathsf{h}(0,t)- \mathsf{h}(0,0)$. Using the general bound $ |\langle X^2 \rangle - \langle Y^2 \rangle| \leq
\langle (X+Y)^2\rangle^{1/2} (2 \langle X^2 \rangle + 2 \langle Y^2 \rangle )^{1/2}$ and stationarity, $$\label{2.22}
\mathsf{h}(vt,t) - \mathsf{h}(0,t) = \mathsf{h}(vt,0) - \mathsf{h}(0,0)$$ in distribution, one concludes that $$\label{2.23}
\lim_{t\to \infty} t^{-1} \langle (\mathsf{h}(vt,0) - \mathsf{h}(0,0) )^2\rangle
= Av = \tilde{A} = \lim_{t\to \infty} t^{-1} \langle (\mathsf{h}(0,t) - \mathsf{h}(0,0) )^2\rangle\,.$$
Our argument only used that along a particular direction the height fluctuations are subdiffusive. While such a property is expected to hold for a large class of spin exchange dynamics, it has been proved only for a few models, in particular for the asymmetric simple exclusion process (ASEP) [@LY04; @BS10]. Here particles hop to the right with rate $p$ and to the left with rate $q$, $p+q = 1$, provided the target site is empty. As for the Toom spin model the invariant measures are Bernoulli, say with density $\rho$. Then $A = \rho (1 - \rho)$ and $v = (p-q)(1 -2\rho)$. The identity (\[2.23\]) states that $\langle \big(\mathcal{J}_{(0,1)}([0,t] ) - \rho(1- \rho)t\big)^2\rangle = \tilde{A} t $ for large $t$ with $\tilde{A} = |(p-q)(1 - 2\rho)|\rho (1 - \rho)$. In fact, this identity is proved in [@FF94], including the corresponding central limit theorem.
Asymptotic properties {#sec3}
=====================
As argued in the previous section, on a large space time scale the stationary process $M_n(t) - \mu_0 n$ is approximated by $\tilde{h}(x,t)$ governed by Eq. (\[2.20a\]) with initial conditions $\tilde{h}(0,t) = 0$, which is known as KPZ equation with flat initial conditions. Available are a replica solution [@CL13] and proofs for a few discrete models in the KPZ universality class [@F04; @Sas07; @BFPS07; @BFP07; @BFP08; @FSW14]. We summarize the findings, which then immediately yields the predictions for the anchored Toom interface. The non-universal parameters are , $\tilde{A} = 8 \sqrt{\lambda}$, and $\tilde{G} = -2^{-3}(1+ \sqrt{\lambda})^{-3}(1 - \sqrt{\lambda})\frac{1}{\sqrt{\lambda}}$. Following [@TS12] we introduce $$\label{3.1}
\tilde{\Gamma} = |\tilde{G}|\tilde{A}^2 = 8\sqrt{\lambda}(1 - \sqrt{\lambda})(1+ \sqrt{\lambda})^{-3}\,.$$ Then, for large $x$, $$\label{3.2}
\tilde{h}(x,0) \simeq \tilde{v} x +
(\tilde{\Gamma} x)^{1/3} \tfrac{1}{2} \xi_{\mathrm{GOE}}\,,$$ where the random amplitude $\xi_{\mathrm{GOE}}$ is GOE Tracy-Widom distributed. More precisely $\xi_{\mathrm{GOE}}$ has the distribution function $$\label{3.3}
\mathbb{P}(\xi_{\mathrm{GOE}} \leq s) = F_1(s)\,,\quad F_1(2s) = \det(\mathbbm{1} - K)_{L^2((s,\infty))}\,.$$ The integral kernel of $K$ reads $K(u,u') = \mathrm{Ai}(u+u')$, see [@FS06] for this particular representation of $F_1$. As a consequence, for large $n$, $M_n - \mu_0n $ is predicted to have the distribution function $$\label{3.4}
\mathbb{P}(M_n - \mu_0 n \leq s) \simeq F_1(2(\tilde{\Gamma} n)^{-1/3}s)\,.$$
$\frac12 \xi_{\rm GOE}$ has mean $-0.6033$, variance $0.408$, and decays rapidly at infinity as for the right tail and $\exp[-|s|^3/6]$ for the left tail. The GOE Tracy-Widom distribution was originally derived in the context of random matrices [@TW94]. One considers the Gaussian orthogonal ensemble of real symmetric $N\times N$ matrices, $H$, with probability density $$\label{3.5}
Z^{-1} \exp\left(- \tfrac{1}{4N}\mathrm{tr}H^2\right) dH,$$ where $dH=\prod_{1\leq i\leq j\leq N}dH_{i,j}$. Let $\lambda_N$ be the largest eigenvalue of $H$. Then, for large $N$, $$\label{3.6}
\lambda_N \simeq 2N + N^{1/3} \xi_{\mathrm{GOE}} \,.$$
Next we consider $t \mapsto \tilde{h}(x,t)$ as a stationary stochastic process in $t$. It is correlated over times of order $(\tilde{\Gamma} x)^{2/3}$. In fact after an appropriate scaling $\tilde{h}(x,t)$ converges to a stochastic process known as Airy$_1$. In formulas $$\label{3.7}
\lim_{x \to\infty} (\tilde{\Gamma} x )^{-1/3}\big(\tilde{h}(x, 2\tilde{A}^{-1} (\tilde{\Gamma} x )^{2/3}t) -\tilde{v}x\big)= \mathcal{A}_1(t)\,.$$ For the joint distribution of $\mathcal{A}_1(t_1),\ldots,\mathcal{A}_1(t_n)$, $t_1 <\ldots< t_n$, one has a determinantal formula. In particular for two times $t_1,t_2$ $$\label{3.8}
\mathbb{P}( \mathcal{A}_1(t_1) \leq s_1, \mathcal{A}_1(t_2) \leq s_2) = \det (\mathbbm{1} - \mathsf{K})_{L^2(\mathbb{R}\times\{1,2\})}.$$ $\mathsf{K}$ is a operator with kernel given by $$\mathsf{K}(x,i;x',j)=\mathbbm{1}(x > s_i) K_1(t_i,x;t_j,x')\mathbbm{1}(x' > s_j),$$ with $$\label{3.8a}
\begin{aligned}
K_1(t,x;t',x') =& \mathrm{Ai}(x'+x+(t'-t)^2)\exp \big((t'-t)(s'+s)+\tfrac{2}{3}(t' - t)^3\big)\\
&-\,\frac{1}{\sqrt{4 \pi (t'-t)}}\exp\left( -\frac{(x'-x)^2}{4(t'-t)}\right) \mathbbm{1}(t'>t)\,.
\end{aligned}$$ From the expression (\[3.8\]) one obtains the covariance $$\label{3.9}
g_1(t) = \langle \mathcal{A}_1(0) \mathcal{A}_1(t)\rangle
- \langle \mathcal{A}_1(0)\rangle^2\,.$$ To actually compute $g_1$, one uses a matrix approximation of the operators in (\[3.8\]) by evaluating the kernels at judiciously chosen base points [@Bo08], for which the determinants are then readily obtained by a standard numerical routine. The limit in (\[3.7\]) implies that, for large $x$, $$\label{3.10}
\langle \tilde{h}(x,0) \tilde{h}(x,t)\rangle
- \langle \tilde{h}(x,0)\rangle^2 \simeq (\Gamma x)^{2/3}g_1(\tilde{A}t/2(\tilde{\Gamma} x)^{2/3})\,.$$ Returning to the Toom interface one arrives at the result that, for large $n$, $$\label{3.11}
\langle (M_n(t) - \mu_0 n)(M_n(0) - \mu_0 n) \rangle
- \langle( M_n(0) - \mu_0 n)\rangle^2 \simeq (\tilde{\Gamma} n)^{2/3} g_1(\tilde{A}t/2(\tilde{\Gamma} n)^{2/3})\,.$$ Based on KPZ scaling theory, (\[3.4\]) and (\[3.11\]) are our predictions for the fluctuations of the Toom interface. They will be tested numerically in the following section.
Numerical studies {#sec4}
=================
The Toom spin model lends itself well for an efficient simulation technique, often referred to as multispin coding [@Newman99], which was used already in [@BBLS96] and is used also in this study. The basic idea is that the time-consuming part of the algorithm is written down as a sequence of single-bit operations, but the computer then acts on 64-bit words, thereby performing 64 simulations simultaneously. Most of the computational effort is invested into selecting a random site, flipping the spin value at that site, and then walking along the array of spins until an opposite spin is encountered, which is then also flipped. A piece of code in the programming language C which achieves this is:
i=random()*n;
first=spin[i];
todo=randword()|first;
spin[i]^=todo;
for (j=i+1;(j<n)&&(todo!=0);j++)
{
flip=todo&(first^spin[j]);
spin[j]^=flip;
todo&= (~flip);
}
In this example code, the introduction of a random pattern [**randword()**]{} in the third line introduces a bias; the density of 1s in this random pattern should equal $\lambda$.
For the actual simulations, we start from a random spin distribution, that is, the initial spins are independent Bernoulli random variables with parameter $1/2$, and then evolve the system over $n^2/2$ units of time to achieve the steady state. Next, in one set of simulations, we keep evolving the system, and make a histogram of $M_n(k)$ for $k=0,n,\dots,10^7n$, where $M_n(k)$ is the magnetization after $k$ units of time. This data are used to determine the distribution function of $M_n-\mu_0 n$.
In another set of simulations we obtain an estimate of $\left< (M_n(0)-M_n(t))^2\right>$ by averaging $(M_n(i)-M_n(i+j))^2$ for $i=0,n+T,\dots, 10^3 (n+T)$ and $j=0,1,\dots,T$ in which $T=2n^{2/3}$ is the longest time difference over which we measure the correlation.
We have made simulations for $\lambda=1/8$, a value at which the convergence with increasing system size is relatively fast, for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$. We import the data sets in Mathematica and rescale them according to the theoretical predictions of (\[3.4\]) and (\[3.11\]). First we consider the scaling of the magnetization as $$M_n^{\rm resc}=\frac{M_n-\mu_0 n}{(\tilde \Gamma n)^{1/3}}$$ and compare its density with the one of $\frac12 \xi_{\rm GOE}$ (the data for $\xi_{\rm GOE}$ are taken from [@PSKPZ]), see Figures \[FigDensity\] and \[FigDensityDiff\]. The agreement is remarkable and at first approximation one only sees a (non-random) shift of the distributions to the right, which goes to zero as $n^{-1/3}$ as observed previously in other models in the KPZ universality class, see [@TS10; @TS12; @FF11; @Tak12].
Secondly, we focus at the covariance. Since our simulation is in steady state, we can derive the covariance from $\left< (M_n(0)-M_n(t))^2\right>$ simply by the relation $${\rm Cov}(M_n(0),M_n(t)) = {\rm Var}(M_n(0))-\tfrac12 \left< (M_n(0)-M_n(t))^2\right>.$$ The value of ${\rm Var}(M_n(0))$ can be be obtained using the first set of data or by making an average over the region of times $t$ where $\left< (M_n(0)-M_n(t))^2\right>$ is constant. We used the latter approach, since it turns out to be less sensitive to long-lived correlations in the total magnetization, associated with the system’s state close to the origin. The estimate of the variance has been made by averaging the values of $\tfrac12 \left< (M_n(0)-M_n(t))^2\right>$ for times $t\in [n^{2/3},2 n^{2/3}]$. In that region the theoretical prediction gives that the covariance (of the rescaled process) is about $10^{-6}$, which is much below the statistical noise that is about $10^{-3}$.
![Densities of $M_n^{\rm resc}$ for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$ compared with the theoretical prediction, that is, the density of $\frac12 \xi_{\rm GOE}$. The insert at the top figure is the log-log plot of the function $n\mapsto \langle M_n^{\rm resc}\rangle- \tfrac12\langle \xi_{\rm GOE}\rangle$. The line has slope $-1/3$. The arrow indicates the shift of the curves as $n$ increases.[]{data-label="FigDensity"}](FigDensity "fig:")![Densities of $M_n^{\rm resc}$ for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$ compared with the theoretical prediction, that is, the density of $\frac12 \xi_{\rm GOE}$. The insert at the top figure is the log-log plot of the function $n\mapsto \langle M_n^{\rm resc}\rangle- \tfrac12\langle \xi_{\rm GOE}\rangle$. The line has slope $-1/3$. The arrow indicates the shift of the curves as $n$ increases.[]{data-label="FigDensity"}](FigDensityLog "fig:")
![Difference of the densities of $M_n^{\rm resc}$ for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$ and the theoretical prediction.[]{data-label="FigDensityDiff"}](FigDensityDiffColors)
We considered the scaled process according to (\[3.11\]), namely $$M_n^{\rm resc}(t)=\frac{M_n(2 t (\tilde \Gamma n)^{2/3}/\tilde A)-\mu_0 n}{(\tilde \Gamma n)^{1/3}}.$$ Using the approach described above, we determine the covariance of $M_n^{\rm resc}$ and plot it against the covariance $g_1(t)$ of the Airy$_1$ process, see Figure \[FigCov\].
![Densities of ${\rm Cov}(M_n^{\rm resc}(0),M_n^{\rm resc}(t))$, $t\in [0,1.5]$, for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$ compared with the theoretical prediction $g_1(t)$.[]{data-label="FigCov"}](FigCovColors "fig:")![Densities of ${\rm Cov}(M_n^{\rm resc}(0),M_n^{\rm resc}(t))$, $t\in [0,1.5]$, for $n=10^4$, $2\times 10^4$, $5\times 10^4$, and $10^5$ compared with the theoretical prediction $g_1(t)$.[]{data-label="FigCov"}](FigCovLogColors "fig:")
Mean Variance Skewness Kurtosis
-------------- ------------------- ----------------- ------------------ ----------------------------
$n=10\,000$ $-0.5198; -14\%$ $0.4335; 6.3\%$ $0.2657; -9.4\%$ $3.154; -0.33\%$
$n=20\,000$ $-0.5344; -11\%$ $0.4239; 3.9\%$ $0.2757; -5.9\%$ $3.159; -0.18\%$
$n=50\,000$ $-0.5496; -9.0\%$ $0.4162; 2.0\%$ $0.2820; -3.8\%$ $3.152; -0.40\%$
$n=100\,000$ $-0.5612; -7.0\%$ $0.4116; 0.9\%$ $0.2897; -1.2\%$ $3.168; \phantom{+}0.09\%$
$n=\infty$ $-0.6033$ $0.4080$ $0.2931$ $3.165$
: Mean, Variance, Skewness, and Kurtosis of $M_n^{\rm resc}$ and their relative difference with the asymptotic values.[]{data-label="Table"}
The precision in the agreement between theory and Monte Carlo data can be tested also through recording the higher order statistics, see Table \[Table\]. One expects that generically the $\ell$-th cumulant approaches its asymptotic value as $n^{-\ell/3}$. In particular the mean should have the slowest decay, consistent with our data.
Conclusions {#sec5}
===========
Using improved computer resources we have identified the distribution sampled in [@BBLS96], Fig. 3, as the GOE Tracy-Widom edge distribution. In our figures there is no free scaling parameter. All model-dependent parameters are computed from a sophisticated version of the KPZ scaling theory. One might wonder whether similar properties hold for other 1D spin models with short range spin exchange dynamics. Such a model would have a spin current $J(\mu)$ depending on the average magnetization $\mu$. We crucially used that $J(\mu) =0$ has a unique solution, $\mu_0$, with $|\mu_0| <1$. The case of multiple solutions has not been considered yet. Furthermore we needed $J'(\mu_0) > 0$ corresponding to the right half lattice. If in addition $J''(\mu_0) \neq 0$, the same properties as discussed in our note are predicted. If $J''(\mu_0) = 0$ but $J'''(\mu_0) \neq 0$, the variance grows as $\sqrt{n}$ with logarithmic corrections. In principle also $J'''(\mu_0)$ could vanish. Then the asymptotics should behave exactly as $\sqrt{n}$. The Toom spin model is singled out because it appears naturally from an underlying cellular automaton.
### Acknowledgements {#acknowledgements .unnumbered}
We thank Michael Prähofer, Jeremy Quastel, and Gene Speer for very helpful comments. The work was mostly done when the three last authors stayed at the Institute for Advanced Study, Princeton. We are grateful for the support. The work of JLL was supported by NSF Grant DMR-1104501. PLF was supported by the German Research Foundation via the SFB 1060–B04 project.
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[^1]: Institute for Theoretical Physics, Universiteit Utrecht, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands, and Instituut-Lorentz, Universiteit Leiden, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands. E-mail: [G.T.Barkema@uu.nl]{}
[^2]: Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany. E-mail: [ferrari@uni-bonn.de]{}
[^3]: Departments of Mathematics and Physics, Rutgers University, NJ 08854-8019, USA and Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA. E-mail: [lebowitz@math.rutgers.edu]{}
[^4]: Zentrum Mathematik, TU München, Boltzmannstrasse 3, D-85747 Garching, Germany and Institute for Advanced Study, Einstein Drive, Princeton NJ 08540, USA. E-mail: [spohn@ma.tum.de]{}
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address: |
$^{1}$Theoretische Physik - ETH-Honggerberg CH-8093, Switzerland\
$^{2}$Departamento de Física Teórica I, Universidad Complutense. 28040-Madrid, Spain\
author:
- 'Germán Sierra$^{1}$[@ger] and Miguel A. Martín-Delgado$^{2}$'
title: 'A Rotating-Valence-Bond scenario for the 2D Antiferromagnetic Heisenberg Model'
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50000
Despite of some initial controversies there is by now sufficient theoretical and experimental evidence for the existence of antiferromagnetic long range order (AF LRO) in the 2d spin 1/2 Heisenberg antiferromagnet [@manousakis] (and references therein). This property has been observed in parent compounds of hight-${T_c}$ materials such as $ La_2 Cu O_4$ [@manousakis]. From a theoretical point of view this means that the strong quantum fluctuations implied by the low dimensionality and the spin 1/2 do not destroy completely the Neel order, as it happens [@bethe] in 1d. Though there is no a satisfactory physical explanation of this fact, which may be important regarding the interplay between antiferromagnetism and superconductivity upon doping. The RVB scenario originally proposed by Anderson [@anderson; @kivelson], while yielding an appealing picture of the ground state, does not explain the presence of AF LRO. This type of order may however be incorporated a posteriori in long range RVB ansatzs of factorized form [@liang-doucot-anderson], with predictions similar to the ones obtained using Quantum Monte Carlo methods [@carlson] and variational plus Lanczos techniques [@hebb-rice]. A class of physical systems where the RVB approach may be actually realized is in spin ladders with an even number of chains [@white-noack-scalapino; @dagotto-rice]. The previous works leave still room to investigate in more depth the interplay between the RVB scenario, or more generally “valence bond scenarios”, and the AF order present in the 2d AF-magnets, described by the AF Hamiltonian $H=J \sum _{\langle i,j \rangle} {\bf S}_i \cdot {\bf S}_j$.
In this letter we shall study a new possible effect of the antiferromagnetism on the dynamics of bonds, namely bond-rotation. Let us recall that the basic mechanism considered by the RVB picture is the resonance between parallel neighbour bonds, which causes a substantial reduction of the ground state energy [@anderson-fazekas]. In addition to this effect we shall explore the possibility that the bonds also rotate around their ends under the influence of the AF background. To test this idea we propose a variational ground state in which the bonds rotate but do not resonate among themselves. This simplified ansatz allows us to decouple the rotation of bonds of other phenomena and in particular the resonance. A more realistic ansatz would include both rotation and resonance of bonds, but we shall not deal with this issue here. The analysis of the pure rotational-valence-bond ansatz leads us to a value of the staggered magnetization closer to the exact numerical result of the AFH model than to the Neel magnetization while we find the opposite result for the ground state energy. We interpret these results in terms of the previous picture of a bond scenario compatible with AF LRO and not containing the resonating mechanism, which is responsable for the lowering of the energy. An important ingredient of our construction is the use of real space renormalization group techniques, which allows us to obtain exact analytical results for any value of the spin $S$ of the model ($S$ is integer or half-integer and in the discussion above $S=1/2$). The physical reason for this is that every cluster of 5 spins which share a rotating bond, behave as an effective spin $3S$, coupled with its neighbours through a Heisenberg interaction with an effective coupling constant. Hence the effective spin renormalizes to infinity, which allows us to compare the staggered magnetization and energy of the ansatz with the semiclassical spin-wave $1/S$ expansions of these quantities for the ground state of the AF-magnet. This comparison confirms the above picture of the rotating-valence-bond state as a state close to the exact ground state of the AF Heisenberg model in staggered magnetization but not in energy.
Let us begin our approach by considering the cluster of 5 spins 1/2 of Fig.\[fig1\] a). The configuration showed in Fig.\[fig1\] a) is the exact ground state of the Ising piece of the Heisenberg Hamiltonian, given by $H_{z} = J\sum_{i=1,\dots,4} S^z_0 S^z_i $ , where $S_0^z $ and $S_i^z$ are the third component of the spin operators at the center and the $i^{th}$ position off the center respectively. As soon as the “transverse” Hamiltonian $H_{xy} = J \sum_{i=1,\dots,4} S^x_0 S^x + S^y_0 S^y_i $ is switched on, the down-spin in the middle starts to move around the cluster, and a valence bond between the center and the remaining sites is formed in a s-wave ($l=0$) symmetric state as shown in Fig.\[fig1\] b). Other rotational states with $l\neq 0$ may appear corresponding to excitations ($l$ being the orbital angula momentum of the bond). An alternative description of this state is given by first combining the 4 spins sourrounding the center into a spin 2 irrep, which in turn is combined with the spin 1/2 at the center yielding a spin 3/2 irrep with energy $e_0 = - 3 J/2$. If instead of the spin 1/2 at each site there is a spin S the previous analysis can be easily generalized as follows: the ground state of the AFH Hamiltonian of the 5-cluster has total spin 3S and is obtained by first combining all the surrounding spins into a spin 4S, which in turn becomes a spin 3S after multiplication with the spin S at the center. In a certain sense this state can be viewed as the formation of bonds between the center and its four neighbours. After applying several steps of the real-space RG, as we shall see below, new bonds are generated between sites at longer distances apart. Thus our valence-bond scenario is a type of long range valence bond state.
To study the AFH model in the entire square lattice we begin by first tesselating this plane using the cluster of Fig.\[fig1\] a) as the fundamental cell (see Fig. \[fig2\]). Notice that the centers of the 5-cluster form a new square lattice with lattice spacing $a' =
\sqrt{5} a$. Given this tesselation we can apply the standard RG method of replacing clusters of spins by an effective spin [@jullienlibro; @jaitisi]. This method has been applied for the 1d AFH model by Rabin [@rabin] for clusters or blocks with 3 sites, obtaining a ground state energy with an error of $12\%$. The effective spin of every 3-block in 1d has spin 1/2. In our case, as we have discussed above, the effective spin of the 5-blocks have spin 3S and the energy per block equal to $e_0
= - J S( 4S +1)$. The effective spins $S^\prime=3S$ interact by means of an effective Hamitonian which to first order in perturbation theory can be derived if we know the renormalization of the spin operators ${\bf S}_{\alpha} \rightarrow \xi_{\alpha}
{\bf S}^\prime ,
\alpha = 0, 1, \dots, 4$.
The [*renormalization spin factor*]{} $\xi_{\alpha}$ can be shown to be given by the sum $\xi_{\alpha}=\case{1}{3S} \; \sum_{m_0,m_1,\ldots,m_4} m_{\alpha}
(C^{3S}_{m_0,m_1,\ldots,m_4})^2$ subject to the constraint $\sum_{\alpha=0}^4 m_{\alpha}=3S$. $ C^{3S}_{m_0,m_1,...}$ is the CG coefficient which describes the ground state of spin 3S in terms of the 5 original spins S, whose expression is a product of 4 standard CG coefficients. The $\xi_{\alpha}$ satisfy the [*sum rule*]{} $\sum_{\alpha=0}^4 \xi_{\alpha}=1$. We arrive at the following result,
$$\begin{aligned}
\xi_{\alpha}(S) = {1\over3S} {6S+1\over 8S+1}
{[(2S)!]^5\over[(8S)!]^2}\cr \times \sum_{m_1,\ldots,m_4} \;
m_{\alpha}
{(4S-\sum_1^4 m_i)! \; [(4S+\sum_1^4 m_i)!]^2 \over \prod_1^4 (S-m_i)!
\; (S+m_i)! \; [-2S+\sum_1^4 m_i]!} \label{3}\end{aligned}$$
where if $\alpha=0$ then $m_0=3S-\sum_1^4 m_i$. It follows that the renormalization factors for the four external spins in the 5-block are all equal $\xi_1=\xi_2=\xi_3=\xi_4\equiv \xi (S)$, while that of the central spin $\xi_0$ is determined by the sum rule. Amazingly enough the sum (\[3\]) can be performed in a close manner yielding,
$$\xi (S) = {1\over3} \; {S+\case{1}{4}\over
S+\case{1}{3}} \label{4}$$
For spin $S=\case{1}{2}$ one obtains $\xi (\case{1}{2})=\case{3}{10}$. Moreover, Eq. (\[4\]) correctly reproduces the classical limit $\lim_{S\rightarrow \infty}
\xi(S)=\case{1}{3}$ (recall $S=S^{\text{old}}=\case{1}{3}
S^\prime=\xi_{\text{cl}} S^\prime$). Notice also that the value for $S=\case{1}{2}$ is already close to the classical value.
The RG-equations for the spin operators $\bf{S}_i$ $i=1,2,3,4$ allows us to compute the renormalized Hamiltonian $H^\prime$ which turns out to be of the same form as the original AFH Hamiltonian. In fact, we arrive at the following RG-equations,
\[5\] $$\begin{aligned}
H^\prime(N,S,J) =
-J S (4S+1){N\over5} \cr +
H(\case{N}{5},3 S,3\xi^2(S) J) \label{5a}\end{aligned}$$ $$N^\prime = {N\over5},\; S^\prime = 3
S, \; J^\prime = 3 \xi^2(S) J \label{5b}$$
where the first contribution in Eq. (\[5a\]) comes from the energy of the blocks. As $3 \xi^2(S)<1$, the flow equation (\[5b\]) implies that the coupling constant flows to zero $J^{(n)}
\stackrel{n\rightarrow \infty}{\rightarrow} 0$ which means that the AFH model remains [*massless*]{} for arbitrary value of the spin $S$. This fact allows us to compute the density of energy $e_{\infty}(S)$ (per site) as the following series,
\[6\] $$\begin{aligned}
e_{\infty}(S) =
-\case{1}{5} \sum_{n=0}^{\infty} \case{1}{5^n} J^{(n)} S^{(n)} (4
\times 3^n S + 1) \label{6a} \end{aligned}$$ $$S^{(n+1)} = 3 S^{(n)}, \; J^{(n+1)} = 3 \xi^2(S^{(n)}) J^{(n)}
\label{6b}$$
Using eqs. (4) and (6) we can compute the ground state energy of our variational RG state for any value of the spin S. In particular for S=1/2 we get the value $e_{\infty}$ = -0.5464. This value has to be compared with the “exact” numerical result -0.6692, which is obtained using Green-function Monte Carlo methods [@carlson]. We observe two facts: i) the error of the computation is $18\%$, which is surprisingly bigger than in 1d where [@rabin] it amounts to a $12\%$ and ii) the result is closer to the Neel state energy than to the exact ground state energy of the AFH model. This last observation is confirmed by the semiclassical expansion of $e_{\infty}$ in the spin S, which we compare with the standard formulas of Anderson and Kubo obtained using the spin wave methods [@swt],
$$\begin{aligned}
& e_{\infty} = - 2 S ( S + \frac{0.2545}{S} + \cdots
) & \label{6c} \\ & e_{\infty}^{sw} = -2 S ( S + 0.158 +
\frac{0.0062}{S} +\cdots ) & \nonumber \end{aligned}$$
Observe that the lowest order correction to the Neel energy is absent in our case.
Yet another explanation for the difference between our energy and the numerical value is that the rotating-valence-bond state as depicted in Fig.\[fig2\] does not have parallel adjacent bonds, which excludes the resonance among themselves.
In order to have a better insight into the physics of the model it is convenient to compute the staggered magnetization $M\equiv \langle
\case{1}{N} \sum_j (-1)^j S^z_j \rangle$. We have been able to obtain a closed formula for arbitrary spin $S$ which is capable of analytical study. To this purpose, we use the RG-equality for V.E.V. $\langle
\psi_0| {\cal O} |\psi_0 \rangle= \langle \psi^\prime_0| {\cal
O}^\prime |\psi^\prime_0 \rangle$ for renormalized observables ${\cal
O}^\prime$ in the ground state and divide the sum in $M$ into 5-block contributions. With the help of the renormalization spin factors we arrive at the RG-equation for the staggered magnetization,
$$M_N (S) = {8 \xi (S) - 1\over5} M_{N/5} (3 S)
\label{7}$$
The explicit knowledge of $\xi (S)$ (\[3\]) allows us to solve this RG-equation for the staggered magnetization in the thermodynamic limit $N\rightarrow \infty$. In fact, as we know by now that the Hamiltonian renormalizes to its classical limit, we have $\lim_{S\rightarrow \infty} M(S)=S$. Defining $M(S) \equiv S f(S)$, Eq. (\[7\]) amounts to solving the equation $f(S)={S+1/5\over S+1/3}
f(3 S)$ subject to the boundary condition $f(\infty)=1$. Thus, we obtain the following formula for the staggered magnetization for arbitrary spin,
$$M(S) = S \; \prod_{n=0}^{\infty} {S+\case{1}{5} 3^{-n}
\over S+\case{1}{3} 3^{-n}} \label{8}$$
This is a nice formula in several regards. For spin $S=\case{1}{2}$ we get $M(\case{1}{2})=0.373$ to be compared with $0.34\pm0.01$ obtained with Green-function Monte Carlo methods [@carlson] and Variational Monte Carlo plus Lanczos algorithm [@hebb-rice]. It amounts to a $7\%$ error. Other approximate methods employed so far lead to values of $M(\case{1}{2})$ such as, e.g., spin wave theory plus $1/S$-expansion gives [@swt] 0.303, spin wave theory plus perturbation theory gives [@huse] 0.313, etc. Our value is close to the one found[@parrinello] with pertubation theory around the Ising model to order 4 which is 0.371. Another interesting feature of our formula (\[8\]) is that it allows us to make a $1/S$-expansion yielding the result,
$$M(S) = S - 0.2 + 0.06 {1\over S} + O(1/S^2)
\label{9}$$
which is to be compared with the spin wave theory result $M(S)=S - 0.198 + O(1/S^2)$ showing excellent agreement for the first two terms while discrepancies start from the term $1/S$ onwards.
The above results confirm the picture given at the beginning of this letter that the rotating-valence-bond state represents a suggestive proposal to combine the valence-bond and antiferromagnetic scenarios, but that resonance phenomena must be taken into account before claiming that rotation effects take places among the bonds of the 2d-AFH model. As was mentioned above, the rotating-valence-bond state so far introduced in this letter does not allow for resonace phenomena. However, we can imagine that a bond moves its center of rotation in the direction of a neighbour bond by hopping (see Fig.\[fig3\]). In this case expect that the two bonds will form a resonant pair. Conversely, we can imagine that a pair of resonant bonds may break into two rotating neighbour bonds as depicted in Fig.\[fig3\]. These intuitive considerations suggest that a resonating-rotating-valence-bond scenario may give a satisfactory picture of the ground state of the 2d AFH model.
G.S. gratefully acknowledges the hospitality of the Theoretical Physics Institute at ETH (specially T.M. Rice) while part of this work was carried out.
Work partially supported in part by the Swiss National Science Foundation and by the Spanish Fund DGICYT, Ref. PR95-284 (G.S.) and by CICYT under contract AEN93-0776 (M.A.M.-D.) .
On leave of absence from Instituto de Matemáticas y Física Fundamental, C.S.I.C., Serrano 123, 28006-Madrid, Spain.
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abstract: 'Complex black-box predictive models may have high accuracy, but opacity causes problems like lack of trust, lack of stability, sensitivity to concept drift. On the other hand, interpretable models require more work related to feature engineering, which is very time consuming. Can we train interpretable and accurate models, without timeless feature engineering? In this article, we show a method that uses elastic black-boxes as surrogate models to create a simpler, less opaque, yet still accurate and interpretable glass-box models. New models are created on newly engineered features extracted/learned with the help of a surrogate model. We show applications of this method for model level explanations and possible extensions for instance level explanations. We also present an example implementation in Python and benchmark this method on a number of tabular data sets.'
author:
- |
Alicja Gosiewska\
Faculty of Mathematics and Information Science\
Warsaw University of Technology\
`a.gosiewska@mini.pw.edu.pl`\
Aleksandra Gacek\
Faculty of Mathematics and Information Science\
Warsaw University of Technology\
`a.gacek@student.mini.pw.edu.pl`\
Piotr Lubon\
Faculty of Mathematics and Information Science\
Warsaw University of Technology\
`lubonp@student.mini.pw.edu.pl`\
Przemyslaw Biecek\
Faculty of Mathematics, Informatics and Mechanics\
University of Warsaw\
Faculty of Mathematics and Information Science\
Warsaw University of Technology\
`przemyslaw.biecek@gmail.com`\
bibliography:
- 'safe.bib'
title: 'SAFE ML: Surrogate Assisted Feature Extraction for Model Learning'
---
Motivation
==========
Questions of trust in machine learning models became crucial issues in recent years. Complex predictive models have various applications in different areas [@PALIWAL20092; @KOUROU20158] and an increasing number of people use machine learning solutions in everyday life. Hence, it is important to ensure that predictions of these models are reliable. There are four requirements whose fulfillment is essential to ensure that predictive model is trustworthy and accessible: (1) high model performance, (2) auditability, (3) interpretability, and (4) automaticity.
\(1) High model performance means that a model rarely makes wrong predictions or the prediction error is small on average. Usually, this can be achieved by using complex, so-called black-box models, such as, boosting trees [@DBLP:journals/corr/ChenG16] or deep neutral networks [@Goodfellow-et-al-2016]. The opposite of black-boxes are glass-boxes. They are simple, interpretable models, such as linear regression, logistic regression, decision trees, regression trees, and decision rules.
Model performance ensures only a part of information about model’s quality. Model’s (2) auditability guarantees that the model can be verified with respect to different criteria. They are, for example, stability, fairness, and sensitivity to a concept drift. There are tools that allow to audit black-box models [@gosiewska2018auditor], yet simple glass-boxes offer more extended range of diagnostic methods [@Harrell:2006:RMS:1196963].
The third requirement is an (3) interpretability, which became an important topic in recent years [@ONeil]. Machine learning models influence people’s lives, in particular, they are used by financial, medical, and security institutions. Models have an impact on whether we get a loan [@HUANG2007847], what type of treatment we receive [@doi:10.1177/117693510600200030], or even whether we are searched by the police [@4053200]. Therefore, models reasoning should be transparent and accessible. There is an ongoing debate about the right to explanation, what does it mean and how it can be achieved [@DBLP:journals/corr/abs-1711-00399; @Edwards_Veale_2018].
The (4) automaticity of machine learning methods is spreading rapidly. Due to the increasing computational power, it becomes easier and easier to obtain more precise models, usually in an automatic manner. There are automated frameworks for AutoML like autokeras, auto-sklearn, TPOT [@jin2018efficient; @NIPS2015_5872; @Olson2016] that allow one to train a model even without any statistical knowledge or even programming skills. Yet, machine learning specialists can also take an advantage of automated methods of modeling. Such methods reduce time needed to train the model, therefore human effort can be directed towards more creative and sophisticated tasks than testing wide range of parameters and models.
People usually choose automatically fitted black-box models that achieve high performance at the cost of auditability and interpretability. As a response to this problem, the methodology for explaining predictions of black-box models, so called post-hoc interpretability, is under active development. There are several approaches to explaining the global behavior of black-boxes. Model can be reduced to simple if-then rules [@MAGIX] or decision trees [@proc-jsm-2018]. However, these explanations are simplifications of models and may be inaccurate. As a consequence, they may be misleading or even harmful. Hence, in many applications it is better to train a transparent, interpretable model than apply explanations to a complex model [@2017arXiv171006169T; @pleseStop]. Therefore, automated methods of obtaining interpretable models, while maintaining the predictive capabilities of a complex model, are extremely important.
In this article, we present a method for Surrogate Assisted Feature Extraction for Model Learning (SAFE ML). This method uses a surrogate model to assist feature engineering and lead to training accurate and transparent glass-box model. In this approach, surrogate model should be accurate to produce best feature transformation, yet it does not have to be interpretable. Based on the new features, the transparent glass-box model is trained. In many cases the high accuracy of black-box models comes from good data representation and this is something than can be next extracted from the model.
The SAFE ML method is flexible and model agnostic, any class of models may be used as a surrogate model and as a glass-box model. Therefore, surrogate model may be selected to fit the data as best as possible, while glass-box model one can be selected according to the particular task or abilities of the end-users to interpreting models.
An advantage of this methodology is that the final glass-box model has a performance close to the surrogate model. By changing the representation of the data, SAFE ML allows to gain interpretability with minimal or no reduction of model performance.
The SAFE ML method can be used as a step in training a model with AutoML methods. We can use AutoML to fit elastic and complex model, then use SAFE to obtain a transparent model.
The paper is organized as follows. Section \[SAFE\_algorithm\] provides a description of the SAFE algorithm. Section \[SAFE\_application\] contains illustrations and benchmarks for the SAFE method for regression and classification problems. Extensions for instance-level approaches and interactions are discussed in Section \[extension\]. Conclusions are in Section \[discussion\].
Description of the SAFE Algorithm {#SAFE_algorithm}
=================================
{width="\textwidth"}
The SAFE ML algorithm uses a complex model as a surrogate. New binary features are created on the basis of surrogate predictions. These new features are used to train a simple refined model.
Illustration of the SAFE ML method is presented in Figure \[fig:safeDiagram\]. In the Algorithm \[alg:SAFEdescription\] we describe how data transformations are extracted from the surrogate model while in Algorithm \[alg:SAFElearning\] we show how to train a new refined model based on transformed features. Below, we explain details of the terminology being used in algorithms. Let $x_1, x_2, ..., x_p$ be features in the surrogate model $M$. A subset of all features except $x_i$ we denote as $x_{-i}$.
**The partial dependence profile** [@PDP] is defined as $$f_i(x_i) = \mathbb{E}_{x_{-i}}[ M(x_i, x_{-i}) ],$$ and calculated as $$\hat f_i(x_i) = \frac{1}{n} \sum_{j=1}^{n} M(x_{i}^j, x_{-i}^j),$$ where $n$ is the number of observations and $x_i^j$ is a value of the $i$-th feature for the $j$-th instance. Partial dependence function describes the expected output condition on a selected variable. The visualization of this function is Partial Dependence Plot [@RJ-2017-016], an example plot is presented in Step 1 in Figure \[fig:safeDiagram\].
**The change point method** [@DBLP:journals/corr/abs-1801-00718] is used to identify times when the probability distribution of a time series changes. **The hierarchical clustering** [@Rokach2005] is an algorithm that groups observations into clusters. It involves creating a hierarchy of clusters that have a predetermined ordering. Step 2 in corresponds to both change point method and hierarchical clustering.
data $X_{n \times p}$, surrogate model $M$, regularization penalty $\lambda$. Let $x_i$ be $i$-th feature. Calculate partial dependence profile $f_i(x)$ for feature $x_i$. Approximate $f_i(x)$ with interpretable features $x^*_{i}$, for example, use the change point method to discretize the variable with regularization penalty $\lambda_i$. Save transformation $t_i(x)$ that transforms $x_i$ into $x_i^*$. Calculate model responses for each observation with imputed each possible value of $x_i$. Merge levels of $f_i(x)$ with similar model responses, for example use the hierarchical clustering with number of clusters $\lambda_i$. Save transformation $t_i(x)$ that transforms $x_i$ into $x_i^*$. Sets of transformations $T^* = \{t_1, ..., t_p\}$ may be used to create new data $X^*$ from features $x_i^* = t_i(x_i)$.
data $X^{new}_{m \times p}$, set of transformations $T^*$ derived from surrogate model $M$. Transform dataset $X$ into $X^{*, new} = T^*(X^{new})$. Create transparent model $M^{new}$ based on $X^{*, new}$.
Application and Benchmarks {#SAFE_application}
==========================
In this section, we perform SAFE ML on selected data sets for regression and classification problems. A summary discussion of the results is conducted at the end of this section.
Examples are generated with scikit-learn models [@scikit-learn] and SafeTransformer. SafeTransformer is a Python library that implements SAFE ML method.
Code that generates artificial data sets and performs SAFE ML method and can be found in the GitHub repository: <https://github.com/agosiewska/SAFE_examples>.
Classification - Artificial Data Set {#subsection_classification_artificial}
------------------------------------
We compare performance of naïve logistic regression, surrogate xgboost, and refined logistic regression. Here naïve regression means that we fill vanilla regression model without any feature engineering.
This example is performed on the artificial data set SIMULD2 for binary classification. SIMULD2 consists of 500 observations and three variables. Variable $y$ is a binary target. Variable $X1$ is continuous, uniform distributed at range from $-5$ to $5$ with normally distributed noise. Variable $X2$ is categorical with 40 levels.
As can be seen in Table \[tab:class\_results\], refined logistic regression performs better than the other two models. Refined logistic regression achieves even better accuracy and AUC than xgboost model, while being a more transparent model.
It may be surprising that the refined model is better than the surrogate one, however there are some reasons for that. Elastic models are better to capture non-linear relations but at the price of larger variance for parameter estimation. In some cases the refined models will work on better features and will have less parameters to train, thus it can outperform the surrogate model.
0.15in
Model Accuracy AUC
----------------------------- ----------- ----------- -- --
Naïve logistic regression 0.736 0.897
Surrogate xgboost 0.960 0.982
Refined logistic regression **0.976** **0.989**
: Results of the SAFE method for models trained on the SIMULD2 data set. SAFE was performed with penalty equals $0.42625$.[]{data-label="tab:class_results"}
-0.1in
Partial Dependence Plot in Figure \[fig:class\_pdp\] shows the relationship between variable $X1$ and output of the xgboost model. This pattern is close to real association, which is a step function with discontinuities in $-3$ and $2.5$. This relationship could not be caught by logistic regression. However, in Figure \[fig:class\_pdp\], we can see that SAFE ML method divided $X1$ variable into three binary variables. This make it possible for refined logistic regression to capture the non-linearity.
Variable $X2$ consists of 40 levels, yet process of generating target variable y distinguishes between variables in three groups. When examining how SAFE ML has grouped variables, one can see that groups almost match up with real dependencies. This caused that instead of one variable of 40 levels, the new model was trained on 3 binary variables. This means that transformed features better reflected the real relationships.
0.2in
![An expected response of the model conditioned on the variable X1. Black vertical lines marks points of the discretization calculated with SAFE ML.[]{data-label="fig:class_pdp"}](fig/clasification_PDP_X2.pdf){width="60.00000%"}
-0.2in
Regression - Boston Housing {#subsection_regression_boston}
---------------------------
Second example is performed on Boston Housing data set [@HARRISON197881]. Boston Housing consists of 506 rows and 14 columns. The target variable is medv (median value of owner-occupied homes).
We compare performances of naïve linear regression, surrogate xgboost, and refined linear regression.
As described in Section \[SAFE\_algorithm\], feature extraction in SAFE ML algorithm depends on a choice of a regularization penalty $\lambda$. Figure \[boston\_results\] shows performances of models as functions of penalty. Mean Square Errors (MSE) of the refined linear regression models are, in general, close to MSE of surrogate model. Thus, the use of a simpler model did not negatively affect the performance. At the same time, we gained transparency.
0.2in
![Dependence between SAFE ML method’s penalty and MSE for refined model. []{data-label="boston_results"}](fig/boston_results.pdf){width="60.00000%"}
-0.2in
Partial Dependence Plot for xgboost model and variable ZN is presented in Figure \[boston\_pdp\]. Flexible boosting model captured the non-linear relationship between variable ZN and target medv. As a result, SAFE ML method divided ZN variable into two binary features to improve performance of refined model.
0.2in
![Partial Dependence Plot (PDP) of the gradient boosting model and ZN variable . Black vertical line indicates variable split generated with SAFE ML method.[]{data-label="boston_pdp"}](fig/boston_PDP_ZN.pdf){width="60.00000%"}
-0.2in
Benchmark on a Number of Tabular Data Sets {#large_benchmark}
------------------------------------------
In this section we benchmark the SAFE ML method on a number of tabular data sets for regression and classification problems. We compare performances of three groups of models: simple models trained without SAFE ML feature transformation, complex surrogate models, and refined interpretable models.
### Benchmark for Classification {#benchmark_classification}
We train classification models on six different data sets. They are two simulated data sets, Titanic from Kaggle, Blood Transfusion Service Center [@Yeh2009], Teaching Assistant Evaluation form UCI Machine Learning Repository [@Dua:2017], and Pima Indian Diabetes [@johannes1988using].
We use Accuracy and AUC metrics to evaluate models. Logistic regression and classification trees trained without any feature extraction are baselines. Complex xgboost models are surrogates required to perform SAFE ML algorithm. Parameters of surrogate models differ between data sets. Refined models are logistic regression models and classification trees. To chose best penalty for SAFE ML transformations, for each surrogate model we examined 25 equally spaced penalties in the range from $0.01$ to $10$. The criterion was performance of a refined model.
Results of benchmarking are in Table \[table\_classification\]. For 22 out of 24 cases, refined model surpasses baseline model. In more than half cases, refined model outperforms baseline and surrogate model.
0.15in
[lcccr]{}\
Data set & BASE. & SURR. & REF.\
SIMULD1 (A) & 0.833 & **0.980** & 0.963\
SIMULD2 (A) & 0.897 & 0.982 & **0.989**\
Titanic & 0.861 & **0.896** & 0.870\
Blood Transfusion & 0.670 & **0.679** & 0.668\
Teaching Evaluation & 0.725 & **0.838** & 0.821\
Pima Indian Diabetes & 0.814 & 0.822 & **0.838**\
\
Data set & BASE. & SURR. & REF.\
SIMULD1 (A) & 0.744 & 0.888 & **0.912**\
SIMULD2 (A) & 0.736 & 0.960 & **0.976**\
Titanic & 0.798 & **0.834** & **0.834**\
Blood Transfusion & 0.749 & **0.754** & 0.668\
Teaching Evaluation & 0.842 & 0.842 & **0.868**\
Pima Indian Diabetes & 0.745 & 0.734 & **0.771**\
\
Data set & BASE. & SURR. & REF.\
SIMULD1 (A) & 0.877 & **0.980** & 0.972\
SIMULD2 (A) & 0.928 & 0 982 & **0.983**\
Titanic & 0.777 & **0.896** & 0.878\
Blood Transfusion & 0.598 & 0.667 & **0.683**\
Teaching Evaluation & 0.763 & 0.817 & **0.842**\
Pima Indian Diabetes & 0.665 & **0.822** & 0.767\
\
Data set & BASE. & SURR. & REF.\
SIMULD1 (A) & 0.896 & 0.888 & **0.912**\
SIMULD2 (A) & 0.928 & 0.96 & **0.976**\
Titanic & 0.794 & 0.834 & **0.839**\
Blood Transfusion & 0.738 & **0.775** & 0.759\
Teaching Evaluation & 0.842 & 0.842 & **0.895**\
Pima Indian Diabetes & 0.688 & 0.734 & **0.760**\
-0.1in
### Benchmark for regression
In this section, we examine performance of the SAFE ML method on 5 data sets for regression problems. They are Energy Efficiency and Yacht Hydrodynamics form UCI Machine Learning Repository [@Dua:2017], Boston Housing [@HARRISON197881], Warsaw Apartments [@DALEX], and Real Estates [@Yeh:2018:BRE:3198938.3199153].
Base and refined models are linear regression models. We use xgboosts as a surrogate model, xgboost parameters differ between data sets. To chose best penalty for SAFE ML transformations we examine 25 equally spaced penalties in the range from $0.01$ to $10$ and MSE criterion.
Results are presented in Table \[table\_regression\]. For all data sets, baseline models outperform base models. For 3 out of 5 data sets, refined linear model achieves better performance than xgboost model.
0.15in
Data set BASE. SURR. REF.
----------------------- ------- ------------ ----------- --
Warsaw Apartments (A) 1 7.12 **64.99**
Real Estates 1 1.02 **1.38**
Boston Housing 1 1.27 **1.32**
Energy efficiency 1 **43.09** 8.88
Yacht Hydrodynamics 1 **267.75** 105.17
: Performances of models trained on five data sets for regression problem. Artificial data sets are marked by (A). Baseline models (BASE.) and refined models (REF.) are linear regression models. Surrogate models (SURR.) are xgboost models. Performance metric is MSE, values in columns are scaled to MSE for baseline model.[]{data-label="table_regression"}
-0.1in
Benchmark Summary {#application_summary}
-----------------
In more than half of the cases, refined model had outperformed surrogate model. In majority of the rest examples performance differences between surrogate and refined models were minimal.
Refined models are simple, with a small number of parameters, therefore one could conclude that refined models generalize data better than complex models. However, it is worth noting that the refined models generalize relationships that were captured by surrogate models. Thus, without a complex model as a surrogate, it would not have been possible. With SAFE ML method, transferring knowledge about relationships to a simple model is automatic and do not require detailed investigation of the complex model.
Even if black-box model gains better results, it is still worth considering applying transparent glass-box model. As we have seen in previous examples, performance of surrogate and refined model were, in general, close to each other. The advantage of a simpler model is that we gain transparency, interpretability and auditability.
Future extensions of the SAFE ML method {#extension}
=======================================
Instance Level Problems
-----------------------
In previous sections, we showed how to use complex surrogate models to extract global, interpretable features. SAFE ML method could be also extended to instance level feature extraction. A complex model can capture local relationships between variables. Therefore, we may consider several local, interpretable models, instead of one global model.
There are several approaches to obtain locality, we can subset data set, reweight original data, or simulate instances from the original data distribution. One of the examples of local model approximations is LIME (Local interpretable model-agnostic explanations) [@lime]. It is a method for generating local models that approximate the predictions of the underlying complex model. Local models are simple, such as, LASSO regression. Since LASSO is a method for selecting variables, while applying LIME we perform also a feature extraction. However, this method is not capable of extracting new interpretable features.
An extension of the LIME that includes extraction of interpretable features is localModel (Local Explanations of Machine Learning Models for Tabular Data) [@localModel]. Local interpretable features are created by discretization of numerical features due to the splits of the decision tree. Categorical features are discretized by merging levels using the marginal relationship between the feature and the model response. Locality is obtained by generating a random number of interpretable inputs around the explained instance. Then, LASSO regression model is fitted to new features and original model’s responses.
The idea behind localModel is similiar to SAFE ML. However, localModel is used to make statements about predictions and behaviour of the underlying black-box model, while the idea of SAFE is to create new refined model to make its own predictions.
Interactions Extractions {#interactions}
------------------------
SAFE ML algorithm is used for transforming single features. One can consider extending this approach of interactions. There are methods of capturing interactions from random forest [@randomForestExplainer] or xgboost [@xgboostExplainer]. This can be used for extraction of new features which contain information about interactions between variables.
Discussion
==========
In this article, we presented SAFE ML algorithm that uses surrogate model to feature transformations. New features are then used to train refined glass-box model.
We benchmarked SAFE ML for regression and classification problems. The results confirmed that SAFE ML algorithm produces features that can be further used to fit accurate and transparent model. We also justified the advantage of refined models over surrogate black-boxes.
We also discussed possible extensions of SAFE ML to instance level problems. In addition, we see the possibility of extending the SAFE ML method to include interaction extraction.
Benchmarking Methodology
------------------------
Benchmarks in Section \[SAFE\_application\] were based on a single split into training and test data sets. In further research, benchmarks could include k-fold cross-validation technique. However, while applying cross-validation, it would be necessary to take into account values of penalty. In Section \[SAFE\_application\] we were selecting a penalty on the basis of the model performance on a test data. The use of cross-validation will cause that values of penalty for each fold will be different. Thus it will not be possible to point the best penalty.
Conclusions
-----------
The SAFE ML method allows us to fulfill four requirements of trustworthy predictive model, stated in Section \[motivation\]. One can choose a final refined model, accordingly to the simplicity and transparency, therefore statement (3) about interpretability is accomplished. Simple models, such as, linear regression and logistic regression are extensively described from a mathematical point of view. As a result, there are many methods to diagnose such models. Therefore, requirement of the (2) auditability is also fulfilled. In Section \[SAFE\_application\] we showed that performances of refined models are close to performance of complex surrogate models. Therefore, SAFE ML method allows to gain (1) high model performance. In Section \[SAFE\_application\] we also argued that SAFE ML algorithm allows automatic feature transformation for the purpose of fitting refined model. This approach allows you to omit examining a complex model. Thus (4) automaticity is also accomplished.
Similar Nomenclature
--------------------
The phrase *surrogate model* is occasionally referred to an interpretable glass-box model that approximates predictions of a black-box model [@h2o_mli_booklet]. The surrogate model in this sense mimics most of the properties of the model under consideration, and is used to makes statements about the black-box model and not about the real world. However, there is no unambiguous nomenclature for this kind of problem. Models that mimic black-boxes are called also proxy models, shadow models, metamodels, response surface models, emulators [@molnar; @proc-jsm-2018].
Therefore, our meaning of the term *surrogate model* is not a duplication the meaning of the existing phrase. In this article, we refer *surrogate model* to a complex model that supports training interpretable model.
Software and Code {#software}
-----------------
Benchmarks from Section \[SAFE\_application\] were generated with SafeTransformer Python library available at (<https://github.com/olagacek/SAFE>). Code that generates benchmarks is availible on Github: (<https://github.com/agosiewska/SAFE_examples>).
Acknowledgements
================
Alicja Gosiewska was financially supported by the grant of Polish Centre for Research and Development POIR.01.01.01-00-0328/17. Przemyslaw Biecek was financially supported by the grant NCN Opus grant 2017/27/B/ST6/01307.
|
---
abstract: 'We present detailed investigations of the magnetic properties of an Fe monolayer on W and Ta $(110)$ surfaces based on the *ab initio* screened Korringa–Kohn–Rostoker method. By calculating tensorial exchange coupling coefficients, the ground states of the systems are determined using atomistic spin dynamics simulations. Different types of ground states are found in the systems as a function of relaxation of the Fe layer. In case of W$(110)$ substrate this is reflected in a reorientation of the easy axis from in-plane to out-of-plane. For Ta$(110)$ a switching appears from the ferromagnetic state to a cycloidal spin spiral state, then to another spin spiral state with a larger wave vector and, for large relaxations, a rotation of the normal vector of the spin spiral is found. Classical Monte Carlo simulations indicate temperature-induced transitions between the different magnetic phases observed in the Fe/Ta$(110)$ system. These phase transitions are analyzed both quantitatively and qualitatively by finite-temperature spin wave theory.'
author:
- Levente Rózsa
- László Udvardi
- László Szunyogh
- 'István A. Szabó'
title: 'Magnetic phase diagram of an Fe monolayer on W(110) and Ta(110) surfaces based on *ab initio* calculations'
---
Introduction
============
The Dzyaloshinsky–Moriya interaction[@Dzyaloshinsky; @Moriya] between local magnetic moments has a great impact in spintronics applications through the formation of chiral spin structures like magnetic skyrmions[@Heinze; @Romming] and chiral domain walls,[@Chen] while it may also lead to an asymmetry in the magnon spectrum of ferromagnetic thin films, as was shown theoretically[@Udvardi] and examined in spin-polarized electron energy loss spectroscopy experiments[@Zakeri; @Zakeri2] for Fe/W$(110)$. Spin-polarized scanning tunneling microscopy experiments enabled the real-space observation of spin spiral orderings at low temperatures in several ultrathin films such as Mn monolayer on W$(110)$,[@Bode] Pd/Fe double-layer on Ir$(111)$,[@Romming] Cr monolayer on W$(110)$[@Santos] and Fe double-layer on W$(110)$.[@Kubetzka; @Meckler]
A double-layer of Fe on W$(110)$ shows unusual phase transitions when the temperature is increased. While the monolayer is ferromagnetic up to $T_{c}\approx 230\,\textrm{K}$,[@Elmers] in the double-layer the low-temperature spin spiral phase disappears at around $200\,\textrm{K}$,[@Bergmann] developing an in-plane ferromagnetic state as in the case of the monolayer, which persists up to $T_{c}\approx 450\,\textrm{K}$.[@Elmers] This is in agreement with the asymmetry of the spin wave spectrum found in Ref. \[\] at $T\approx 300\,\textrm{K}$, since the spectrum around a cycloidal spin spiral ground state would be symmetric if the Dzyaloshinsky–Moriya interaction were perpendicular to the plane of the spiral.[@Michael; @Michael2] Using the experimentally obtained wavelength of the low-temperature spiral state it was possible to find micromagnetic exchange (spin stiffness), Dzyaloshinsky–Moriya and anisotropy parameters describing this type of order.[@Meckler; @Meckler2] However, both micromagnetic[@Heide; @Zimmermann] and atomistic[@Bergqvist] *ab initio* calculations indicated a ferromagnetic ground state in the system. For an Fe monolayer on W$(110)$, theoretical calculations[@Qian; @Nakamura; @Bergman] agree with experiments[@Elmers2] in determining an in-plane ferromagnetic ground state. For a Mn monolayer on W$(110)$, Ref. \[\] provided consistent experimental and theoretical descriptions of the spiral ground state. *Ab initio* calculations[@Zimmermann] and experiments[@Santos] are also in agreement about the spiral ground state of Cr monolayer on W$(110)$.
Various types of magnetic ground state configurations were found by *ab initio* calculations in an Fe monolayer on the $(100)$ surface of W$_{1-x}$Ta$_x$ ($0 \le x \le 1$) alloys[@Ferriani; @Ondracek] as a function of Ta concentration $x$, ranging from an antiferromagnetic state on pure W to a ferromagnetic state on pure Ta. Both W and Ta have bcc lattice structure but the lattice constant of Ta is about $4.3\%$ larger than that of W ($a_{\rm Ta}=3.301\,$ Å and $a_{\rm W}=3.165\,$ Å ). This difference was taken into account by calculating the lattice constant of the alloy, but the relaxation of the Fe layer towards the top substrate layer was kept fixed during the calculations at the value determined for Fe/W$(100)$, although Fe should have a larger inward relaxation in the case of Ta with the larger lattice constant. For different relaxations, Fe on Ta$(100)$ may have either ferromagnetic or antiferromagnetic ground state as shown in Ref. \[\].
In this paper we examine the magnetic ground state of an Fe monolayer on W and Ta $(110)$ surfaces as a function of the relaxation of the Fe layer with respect to the top substrate layer. The electronic structure calculations were performed by using the relativistic screened Korringa-Kohn-Rostoker method.[@Szunyogh2] For the determination of the magnetic ground state we mapped the spin system onto a generalized classical Heisenberg model, where the parameters are taken from the relativistic generalization[@Udvardi2] of the method of infinitesimal rotations introduced by Liechtenstein *et al.*[@Liechtenstein] The ground state of the system was found by atomistic spin dynamics simulations based on the Landau-Lifshitz-Gilbert[@Landau; @Gilbert] equations. These results are described in Sec. \[sec2\].
Besides changing the relaxation, thermal fluctuations may also induce transitions between the different types of ordered states found in these systems. Classical Monte Carlo simulations were performed using the previously obtained spin model to find these transitions. In one of the transitions found in Fe monolayer on Ta$(110)$ the increasing temperature drives the system from the ferromagnetic ground state into a non-collinear spin spiral state. Most likely, this transition is driven by the Dzyaloshinsky–Moriya interactions and the easy-axis anisotropy in the system. Such a transition was already studied in Refs. \[\] and \[\] using a Ginzburg–Landau model, which is, however, unsuitable for employing Heisenberg model parameters obtained from *ab initio* calculations.
Instead of relying on a continuum model, we used spin wave expansion to describe the transition between the different ordered states. This method was found to be a powerful tool[@Yosida; @Miwa] for explaining a transition from a low-temperature ferromagnetic to a high-temperature helical state in bulk Dy. In the present work we incorporated the Dzyaloshinsky–Moriya interaction into such an analysis, which was unnecessary in bulk systems with an inversion center, but it plays an important role in case of ultrathin films. We also used the spin wave expansion technique to handle higher order terms (magnon-magnon interactions) perturbatively, since perturbation theory makes it possible to estimate the temperature where the system reaches the paramagnetic state. This method was originally used to calculate the Curie temperature in a simple cubic lattice described by a ferromagnetic Heisenberg model.[@Bloch] By using a simplified model Hamiltonian consistent with the different types of ground states found in an Fe monolayer on Ta$(110)$, in Sec. \[sec3\] we present a detailed analysis of the temperature-induced magnetic phase transitions and relate the results to those obtained from Monte Carlo simulations.
Magnetic states and phase transitions in an Fe monolayer on W and Ta $(110)$ surfaces \[sec2\]
==============================================================================================
*Ab initio* calculation of collinear magnetic states\[sec2A\]
-------------------------------------------------------------
For the *ab initio* calculations we used the relativistic screened Korringa-Kohn-Rostoker method,[@Szunyogh; @Zeller; @Szunyogh2] using the local spin density approximation and the atomic sphere approximation. First we performed calculations for W and Ta bulk with the lattice constants $a_{\textrm{W}}=3.165$Å and $a_{\textrm{Ta}}=3.301$Å, respectively. The layered systems considered for the deposited Fe monolayers comprised eight layers of bulk atoms, one layer of Fe and three layers of empty spheres, sandwiched between the semi-infinite bulk calculated in the previous step and a semi-infinite vacuum. Theoretical calculations using the full-potential linearized augmented plane-wave method give relaxation values between $12-13\%$ for an Fe monolayer on W$(110)$,[@Qian2; @Qian3; @Bergman; @Huang] while the experimental values are in the range of $7-13\%$.[@Albrecht; @Tober; @Meyerheim] On Ta$(110)$ Fe should have an even larger relaxation due to the larger lattice constant. Therefore the calculations were performed for different values of the distance between the Fe monolayer and the top bulk monolayer, adjusting the Wigner–Seitz radius of the atomic spheres related to the Fe atoms correspondingly. Both for W and Ta, the relative relaxation with respect to the ideal distance between bcc$(110)$ atomic layers was changed between $10\%$ and $17\%$. All the atomic layers but the Fe layer were kept at the ideal lattice geometry since calculations[@Qian2; @Qian3; @Huang] indicate that the W-W relaxations are below $1\%$ even between the topmost W monolayers. We determined the potential and the exchange-correlation magnetic field self-consistently, serving as an input to the evaluation of the exchange coefficients, see Sec. \[sec2B\].
The spin and orbital magnetic moments obtained from the *ab initio* calculations are listed in Table \[table1\]. The sum of the spin and orbital moments in the Fe layer on W(110) for $13\%$ inward relaxation compares within $10\%$ to the total magnetic moments given in the literature.[@Qian2; @Qian3; @Huang] The induced moments in the topmost W layer are antiparallel to the Fe moments, in agreement with Refs. \[\] and \[\], but they are parallel in the next two W layers. It is worth noting that the spin and orbital moments are parallel for the W atoms although the W $d$–shell is less than half-filled, which indicates a violation of Hund’s third rule, cf. Ref. \[\]. Apparently, this is not the case for Ta. It is also notable that the induced moments of the Ta atoms are larger than those of the corresponding W atoms. Ref. \[\] agrees with our calculation inasmuch as increasing the relaxation decreases the magnetic moments of the Fe atoms, most likely due to the increased hybridisation between the Fe and the substrate layers.
[ccccccccc]{}
&\
& &\
relaxation & Fe1 & W1 & W2 & W3 & Fe1 & W1 & W2 & W3\
10% & 2.355 & -0.164 & 0.007 & 0.003 & 0.180 & -0.027 & 0.001 & -0.001\
13% & 2.244 & -0.164 & 0.012 & 0.003 & 0.169 & -0.018 & 0.005 & 0.000\
15% & 2.181 & -0.161 & 0.017 & 0.004 & 0.162 & -0.014 & 0.007 & 0.001\
17% & 2.122 & -0.156 & 0.022 & 0.004 & 0.156 & -0.011 & 0.010 & 0.002\
\
&\
& &\
relaxation & Fe1 & Ta1 & Ta2 & Ta3 & Fe1 & Ta1 & Ta2 & Ta3\
10% & 2.587 & -0.278 & -0.037 & -0.027 & 0.100 & 0.031 & 0.005 & 0.003\
13% & 2.520 & -0.310 & -0.045 & -0.030 & 0.097 & 0.036 & 0.006 & 0.003\
15% & 2.466 & -0.333 & -0.046 & -0.027 & 0.094 & 0.040 & 0.006 & 0.002\
17% & 2.406 & -0.358 & -0.044 & -0.023 & 0.090 & 0.044 & 0.006 & 0.001\
Calculated exchange interactions\[sec2B\]
-----------------------------------------
Using the self-consistent potentials obtained before, the relativistic torque method[@Udvardi2] was employed to map the energy of the magnetic system onto a generalized Heisenberg model, $$H=\frac{1}{2}\sum_{\substack {i,j \\ (i \ne j)} } J_{ij}^{\alpha\beta}S_{i}^{\alpha}S_{j}^{\beta}+\sum_{i}K_{i}^{\alpha\beta}S_{i}^{\alpha}S_{i}^{\beta},\label{eqn1}$$ where $i,j$ and $\alpha,\beta$ label lattice sites and Cartesian indices, respectively, ${S}^\alpha_{i}$ are the components of the unit vector representing the orientation of the spin at lattice site $i$, while $J_{ij}^{\alpha\beta}$ and $K_{i}^{\alpha\beta}$ stand for the matrix elements of the exchange coupling tensors and of the second-order on-site anisotropy energy tensors. The relativistic torque method relies on the magnetic force theorem and requires the calculation of coupling coefficients around different collinear reference states for at least three linearly independent magnetization directions, since for a given direction, only those components of the $\boldsymbol{J}_{ij}$ tensors can be obtained which lie in the plane perpendicular to the magnetization. In particular, we considered the magnetization directions $[1\overline{1}0]$, $[001]$, and $[110]$. The spins in a given layer must be ferromagnetically aligned, but the antiferromagnetic ordering between the different layers was taken into account. To perform the necessary integrations, $16$ energy points were taken along a semicircle contour in the upper complex semiplane, and from $204$ up to $6653$ $\boldsymbol{k}$-points were sampled in the Brillouin zone, gradually increasing for energies approaching the Fermi level.
![(color online) Calculated isotropic exchange interactions $J_{ij}$ obtained from the relativistic torque method, for (a) W(110) and (b) Ta(110) surfaces and different values of relaxations of the Fe layer.[]{data-label="figJij"}](fig1a.png "fig:"){width="\columnwidth"} ![(color online) Calculated isotropic exchange interactions $J_{ij}$ obtained from the relativistic torque method, for (a) W(110) and (b) Ta(110) surfaces and different values of relaxations of the Fe layer.[]{data-label="figJij"}](fig1b.png "fig:"){width="\columnwidth"}
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
The isotropic part of the exchange tensors between the Fe atoms, $$\begin{aligned}
J_{ij}=\frac{1}{3}\sum_{\alpha}J_{ij}^{\alpha\alpha},\end{aligned}$$ is shown in Fig. \[figJij\], for W and Ta surfaces and different relaxation values. Note that with the sign convention of Eq. (\[eqn1\]), $J_{ij}<0$ and $J_{ij}>0$ indicate ferromagnetic and antiferromagnetic couplings, respectively. In case of W(110) surface, the nearest-neighbor ferromagnetic coupling is fairly insensitive to the relaxation, while the next-nearest-neighbor coupling (at the distance of one lattice constant) is antiferromagnetic for lower relaxations, but becomes ferromagnetic above $15\%$ relaxation. For Ta(110) surface, the exchange couplings for the two nearest neighbors are ferromagnetic for all considered values of relaxations. The weaker nearest-neighbor interaction decreases and the next-nearest-neighbor interaction increases in size with increasing relaxation. Also notable is the increasingly antiferromagnetic character of some further (third and fifth) neighbor couplings with increasing relaxation, which will give rise to the formation of a short wavelength spin spiral along the $[1\overline{1}0]$ direction in Fe/Ta$(110)$, see Sec. \[sec2C\]. In particular, this might happen since the strong ferromagnetic coupling between the next-nearest neighbors does not play a role in the formation of the spiral state since it only couples spins along the $[001]$ direction (see coupling $J_{2}$ in Fig. \[fig4\]).
The antisymmetric part of the exchange tensors between the Fe atoms is shown in Fig. \[figDij\] in terms of the components of the Dzyaloshinsky–Moriya vectors, $$\begin{aligned}
D_{ij}^{\alpha}=\frac{1}{2}\sum_{\beta,\gamma}\varepsilon^{\alpha\beta\gamma}J_{ij}^{\beta\gamma}.\end{aligned}$$ According to the symmetry rules set up by Moriya,[@Moriya2] all the Dzyaloshinsky–Moriya vectors lie in the (110) plane. Note that the $x$ and $y$ directions correspond to the $[1\overline{1}0]$ (long) axis and to the $[001]$ (short) axis, respectively. The components of the Dzyaloshinsky–Moriya vectors are only drawn for neighbors with $R_{ij}^{x} \ge 0 $ and $R_{ij}^{y} \ge 0$. The components for the related neighbors can be obtained by symmetry: $(-D_{ij}^{x},D_{ij}^{y})$ for $(R_{ij}^{x},-R_{ij}^{y})$, $(D_{ij}^{x},-D_{ij}^{y})$ for $(-R_{ij}^{x},R_{ij}^{y})$ and $(-D_{ij}^{x},-D_{ij}^{y})$ for $(-R_{ij}^{x},-R_{ij}^{y})$. $D_{ij}^{x}$ is, therefore, only finite between atoms which have a finite distance along the $[001]$ ($y$) direction; for example, the atoms at $\sqrt{2}a$ distance are located along the $[1\overline{1}0]$ ($x$) axis, thus $D_{ij}^{x}=0$. Similarly, $D_{ij}^{y}$ is only finite if $R_{ij}^{x}\neq 0$. The Dzyaloshinsky-Moriya interactions are comparable in magnitude to the isotropic exchange interactions and they also show oscillating behavior.
The presence of the Dzyaloshinsky-Moriya interactions may stabilize spin spiral states and the sign of the components of the Dzyaloshinsky-Moriya vectors determines the chirality of the spin spiral. Let $\boldsymbol{q}$ be the wave vector of the spiral, $\boldsymbol{n}$ the normal vector of the monolayer pointing outwards from the substrate, and introduce the vector $\boldsymbol{\chi}=\boldsymbol{S}_{i}\times\boldsymbol{S}_{j}$ such that $\boldsymbol{q}(\boldsymbol{R}_{j}-\boldsymbol{R}_{i}) >0 $, where $\boldsymbol{R}_{i}$ and $\boldsymbol{R}_{j}$ are the position vectors of neighboring spins in the lattice. Note that for cycloidal spin spirals the direction of $\boldsymbol{\chi}$ is independent of the choice of the lattice sites $i$ and $j$. Following Refs. \[\] and \[\], a cycloidal spin spiral is called right-rotating when the vectors $\left(\boldsymbol{q},\boldsymbol{\chi},\boldsymbol{n}\right)$ form a right-handed system. If they form a left-handed system, the spin spiral is called left-rotating. With our sign convention and only taking into account the largest Dzyaloshinsky–Moriya interactions in both directions, in the case of W substrate the $D_{ij}^{x}$ component prefers a right-rotating spiral along the $[001]$ direction and the $D_{ij}^{y}$ component prefers a left-rotating spiral along the $[1\overline{1}0]$ direction. This is in agreement with the results in Ref. \[\] and the chirality of the spin spiral state along the $[001]$ direction in double-layer Fe on W(110).[@Meckler] For Ta substrate, the sign of the largest $D_{ij}^{x}$ vector component is flipped compared to the case of W substrate. This means that the Dzyaloshinsky–Moriya interactions prefer left-rotating spirals in an Fe monolayer on Ta(110) along both the $[001]$ and $[1\overline{1}0]$ directions.
Ground states obtained from spin dynamics simulations \[sec2C\]
---------------------------------------------------------------
After obtaining the coupling coefficients from collinear configurations, we performed atomistic spin dynamics simulations to find the ground states of the systems. These are based on the numerical solution of the Landau-Lifshitz-Gilbert equations, $$\partial_{t}\boldsymbol{S}_{i}=-\gamma'\boldsymbol{M}_{i}-\alpha \gamma' \boldsymbol{S}_{i}\times\boldsymbol{M}_{i},\label{eqn2}$$ with $\gamma'=\frac{1}{1+\alpha^{2}}\frac{ge}{2m}$ the gyromagnetic coefficient ($g$ the g-factor, $e$ the magnitude of charge and $m$ the mass of the electron) and $\alpha$ the dimensionless Gilbert damping factor. The torque $\boldsymbol{M}_{i}$ acting on the spin vector $\boldsymbol{S}_{i}$ is defined as $$\boldsymbol{M}_{i}=\boldsymbol{S}_{i}\times\left(-\frac{1}{m_{i}}\frac{\partial H}{\partial \boldsymbol{S}_{i}}\right),\label{eqn3}$$ and $m_{i}$ is the magnitude of the magnetic moment of the atom at site $i$, associated with the spin magnetic moment from the *ab initio* calculations in Sec. \[sec2A\], while $H$ is the spin Hamiltonian in Eq. (\[eqn1\]).
We also calculated the exchange couplings between the Fe atoms and the atoms in the topmost bulk layer which had the largest induced moment, see Table \[table1\]. However, we found that including these couplings did not change the ground state considerably, they just give rise to an antiparallel alignment of the induced moments with respect to the neighboring Fe moments. This implies that for the considered systems only the stable Fe moments are relevant to be included into the Hamiltonian (\[eqn1\]). This feature is essential since the quasiclassical description (\[eqn1\])-(\[eqn2\]) is shown to be a reliable description for the rigid moments,[@Antropov] but it is probably not valid for the induced moments.
 Energies per Fe spin of an Fe monolayer (a) on W$(110)$ and (b) on Ta(110) for different magnetic states as a function of the relaxation of the Fe layer obtained from spin dynamics simulations for a system consisting of $N=64\times 64$ atoms with periodic boundary conditions. The energy of the ground state (GS) is highlighted by blue solid line and the types of the ground state magnetic orderings are displayed for the whole range of relaxations. For the explanation of the different spin spiral states (SS I, SS II, SS III) see the text.](fig3a.png "fig:"){width="\columnwidth"}  Energies per Fe spin of an Fe monolayer (a) on W$(110)$ and (b) on Ta(110) for different magnetic states as a function of the relaxation of the Fe layer obtained from spin dynamics simulations for a system consisting of $N=64\times 64$ atoms with periodic boundary conditions. The energy of the ground state (GS) is highlighted by blue solid line and the types of the ground state magnetic orderings are displayed for the whole range of relaxations. For the explanation of the different spin spiral states (SS I, SS II, SS III) see the text.](fig3b.png "fig:"){width="\columnwidth"}
Starting the spin dynamics simulations from a random initial configuration, the system will generally converge to a metastable equilibrium state, that is to a local energy minimum. However, this configuration may not be the ground state – the global energy minimum –, therefore the determination of the ground state may require multiple runs. It was found that a random initial state often leads to a spin spiral state, even if it has slightly higher energy than the ferromagnetic state. Furthermore, the obtained equilibrium states may contain skyrmion-like local excitations which are stable with respect to the dynamics of the system, but represent a positive energy correction compared to the ground state.
{width="2.0\columnwidth"}
![(color online) Spin spiral energies per spin relative to the energy of the ferromagnetic state, calculated from the Heisenberg model parameters in the spin spiral configuration Eq. (\[eqnspinconf\]) for wave vectors along the $[1\overline{1}0]$ direction, $q_x$ (given in units of $\frac{2\pi}{\sqrt{2}a}$). The points at which the spin spiral energies are calculated in Fig. \[fig-gs\] are denoted by squares for SS I and circles for SS II. The inset shows a magnified view of the range $0 \le q_{x} \le 0.15$.[]{data-label="figJq"}](fig5.png){width="\columnwidth"}
The energies obtained from the spin dynamics simulations with the Hamiltonian (\[eqn1\]) are shown in Fig. \[fig-gs\](a) as a function of the relaxation of the Fe layer in case of W(110). The ground state energy of the system is compared to the energies of the ferromagnetic alignments along the main crystallographic directions $[1\overline{1}0]$, $[001]$ and $[110]$. The ground state of the Fe monolayer on W$(110)$ was found to be ferromagnetic for all relaxations, however, a reorientation transition occurs at around $15\%$ relaxation of the Fe layer from the in-plane $[1\overline{1}0]$ direction to the out-of-plane $[110]$ direction. The in-plane easy axis at the experimentally observed relaxation value $13\%$ is in agreement with the experiments.[@Elmers2] It is worth noting that a double-layer of Fe on W$(110)$ has an out-of-plane easy axis,[@Kubetzka; @Slezak] similarly to the case here for large relaxation.
In Fig. \[fig-gs\](b) the energies of the ferromagnetic states and also of different spin spiral states are shown for the Fe monolayer on Ta$(110)$. The energies of the cycloidal spiral states SS I and SS II were calculated in the homogeneous left-rotating spin spiral configuration, $$\begin{aligned}
\boldsymbol{S}_{i}=\left(-\sin\left(\boldsymbol{q}\boldsymbol{R}_{i}\right),0,\cos\left(\boldsymbol{q}\boldsymbol{R}_{i}\right)\right), \label{eqnspinconf}\end{aligned}$$ where the different spin components correspond to the directions $\left(x,y,z\right)=\left([1\overline{1}0],[001],[110]\right)$. The normal vector and rotational sense of the spirals chosen in Eq. (\[eqnspinconf\]) are consistent with the obtained ground states shown in Fig. \[fig-spinconf\]. The spiral energies were calculated for $\boldsymbol{q}$ values in the whole Brillouin zone, but only the $\boldsymbol{q}$ vectors along the $[1\overline{1}0]$ direction, denoted by $q_{x}$, showed complex behavior, see Fig. \[figJq\]. In Fig. \[figJq\], the energy difference between the spin spiral states and the ferromagnetic state along the $[110]$ direction does not go to $0$ as $\boldsymbol{q}\rightarrow\boldsymbol{0}$ due to the anisotropy in the system.
Fig. \[fig-gs\](b) indicates phase transitions at $10.5\%$ relaxation from the ferromagnetic state with out-of-plane easy axis (FM) to the SS I spin spiral state, at $13.8\%$ relaxation between the SS I and SS II states, and at $14.5\%$ relaxation between the SS II and SS III states. All the spiral states have a wave vector parallel to the $[1\overline{1}0]$ direction, and all the spins in the spiral are confined to a plane. For the SS I and the SS II states, the spins are located in the $[110]-[1\overline{1}0]$ plane, forming a left-rotating cycloidal spin spiral as in a Mn monolayer on W$(110)$,[@Bode] although it is clear from Fig. \[fig-gs\](b) that the $[1\overline{1}0]$ direction is the hard axis since the ferromagnetic state along this direction has the highest energy. The plane of the spiral is thus clearly a consequence of the Dzyaloshinsky–Moriya interaction in the system which prefers spin spiral states oriented perpendicular to the Dzyaloshinsky–Moriya vector. For a spin spiral along the $[1\overline{1}0]$ direction, only the $[001]$ component of the Dzyaloshinsky–Moriya interaction plays a role in the ground state energy, leading to the cycloidal spiral state resembling a Néel domain wall.
The SS I state has a small wave number, the value of which increases continuously with increasing relaxation (see the squares in Fig. \[figJq\]), but jumps to the much larger wave number of the SS II spin spiral at relaxation 13.8%. The presence of spin spiral energy minima at different wave vectors and the transition between these minima is a consequence of the frustrated isotropic exchange interactions around these relaxations, see Fig. \[figJij\](b). The SS III state has similar wave number to the SS II state, however, the anisotropy is strong enough to rotate the plane of the spiral out from the $[110]-[1\overline{1}0]$ plane, that is the normal vector $[001]$ changes to a general direction in the $[1\overline{1}0]-[001]$ plane. The ground state energies obtained from the spin dynamics simulations in Fig. \[fig-gs\](b) are somewhat lower than the spin spiral energies presented in Fig. \[figJq\], since due to the anisotropy the spiral can gain energy by being deformed with respect to the perfect sinusoidal shape.[@Meckler2] This difference is the largest for the SS III state, but in that case this is also a consequence of the rotation of the normal vector of the spin spiral.
Phase transitions at finite temperature using Monte Carlo simulations\[sec2D\]
------------------------------------------------------------------------------
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
We examined the phase transitions in the systems also for fixed relaxations as a function of temperature, using classical Monte Carlo simulations with Metropolis dynamics. These phase transitions were expected to occur for relaxation values close to the transition points. The order parameter of the simulations was defined as $$m^{2}\left(\boldsymbol{q}\right)=\sum_{\alpha=x,y,z}m_{\alpha}^{2}\left(\boldsymbol{q}\right)
,\label{eqnop}$$ with $$m_{\alpha}^{2}\left(\boldsymbol{q}\right)=\left\langle\left|\frac{1}{N}\sum_{i}\textrm{e}^{-\textrm{i}\boldsymbol{q}\boldsymbol{R}_{i}}S_{i}^{\alpha}\right|^{2}\right\rangle,\label{eqnopcomp}$$ where $\langle \; \rangle$ denotes thermal average. As discussed in Sec. \[sec2C\], the shape of the spiral state will differ from a perfect sinusoidal shape due to the anisotropy in the system. Therefore the order parameter for wave vector ${\boldsymbol{q}}$ does not perfectly fit this anharmonic spiral with the same wave vector due to the appearance of higher Fourier harmonics, but still it gives a good approximation to characterize the ordering.[@Rocio]
The temperature dependence of the order parameters is shown in Fig. \[figMC\]. For the Fe monolayer on W$(110)$, see Fig. \[figMC\](a), no reorientation transition occurred in the system, although the relaxation value of 15% was close to the transition point. Similarly, no temperature-induced reorientation was found on the other side of the phase boundary, at $15.2\%$ relaxation. The paramagnetic state was reached at $T_{\textrm{c}}\approx 350\,\textrm{K}$, somewhat higher than the experimentally determined critical temperature, $T_{c}\approx 230\,\textrm{K}$.[@Elmers]
In case of the Ta substrate several types of temperature-induced transitions happened between the different ordered phases before reaching the paramagnetic phase, if the chosen relaxation value was close to the phase boundaries shown in Fig. \[fig-gs\](b). The SS I phase turned out to be the most stable one against thermal fluctuations: systems with ferromagnetic ground state at $10\%$ relaxation or with a SS II ground state at $13.8\%$ relaxation turned into the SS I state, in both cases at around $130\,\textrm{K}$, as indicated by a change in the wave number of the order parameter in Fig. \[figMC\](b) and Fig. \[figMC\](c), respectively. Moreover, in case of the FM-SS I phase transition a continuous increase of the wave number can be inferred from Fig. \[figMC\](b) above the critical temperature of the phase transition. For the case of a SS III ground state at $15\%$ relaxation, Fig. \[figMC\](d) shows that the $m_{z}^{2}$ component decreases with the temperature similarly to the order parameter $m^{2}$ in Figs. \[figMC\](b)-(c). However, $m_{x}^{2}$ initially increases with the temperature, which is accompanied by a more pronounced decrease of $m_{y}^{2}$. This indicates that the normal vector of the spin spiral rotates towards the $y=[001]$ axis and at about $80\,\textrm{K}$ a phase transition to the SS II state occurs. The paramagnetic state was reached at $T_{\textrm{c}}\approx 140-220\,\textrm{K}$ in the case of Ta substrate depending on the relaxation.
Description of the phase transitions in Fe/Ta$(110)$ based on spin wave expansion\[sec3\]
=========================================================================================
In this Section, the temperature-induced phase transitions in the Fe monolayer on Ta(110) surface will be discussed in terms of spin wave expansion. Keeping the same global coordinate system as in Sec. \[sec2C\], $\left(x,y,z\right)=\left([1\overline{1}0],[001],[110]\right)$, we will use a simplified model Hamiltonian, $$\begin{aligned}
H=&&\frac{1}{2}\sum_{\substack{i,j \\ (i \ne j)}}
J_{ij}\boldsymbol{S}_{i}\boldsymbol{S}_{j}+\frac{1}{2}\sum_{\substack{i,j \\ (i \ne j)}}\boldsymbol{D}_{ij}\left(\boldsymbol{S}_{i}\times\boldsymbol{S}_{j}\right)\nonumber
\\
&&+\sum_{i}\left[K_{x}\left(S_{i}^{x}\right)^{2}+K_{z}\left(S_{i}^{z}\right)^{2}\right], \label{eqn29}\end{aligned}$$ where $J_{ij}=J_{ji}$, $\boldsymbol{D}_{ij}=\left(0,D_{ij},0\right)$ with $D_{ij} >0$ for $R^{x}_{ij}>0$ and $D_{ij}=-D_{ji}$, $K_{z} < 0$ and $K_{x} > 0$, that is $z$ is the easy axis and $x$ is the hard axis. We choose the parameters such that the above Hamiltonian reproduces the different phases found in Sec. \[sec2C\]. Since the spin spirals have a wave vector parallel to the $x$ axis, only such parameters are relevant which influence the ordering along this direction. These are the effective exchange couplings denoted by $J_{1},J_{2},J_{3},J_{7},J_{11}$ and a Dzyaloshinsky–Moriya vector between the nearest neighbors $\boldsymbol{D}_{1}$ parallel to the $y$ axis (see Fig. \[fig4\]). The isotropic couplings are summed up along the $y$ axis: for example, $J_{3}$ represents the coupling between the spin at site $0$ and all the atoms which have the same $x$ coordinate as the third neighbors. This is because the contributions of these Fe-Fe pairs add up in the energy of the spin spirals with wave vectors along the $x$ axis. The anisotropy constants are chosen in agreement with the energies of the ferromagnetic states along the different axes in Fig. \[fig-gs\](b).
![(color online) Sketch of the lattice and the model parameters considered in Eq. (\[eqn29\]) for an Fe monolayer on Ta$(110)$. $J_{j}$ denote effective exchange couplings between the spin at site $0$ and its neighbors (see text). Equivalent neighbors are formed by mirroring on the $xz$ and $yz$ planes: there are four neighbors of types $1$ and $7$, as well as two neighbors of types $2$, $3$ and $11$. Only the nearest-neighbor Dzyaloshinsky–Moriya vector $\boldsymbol{D}_{1}$ is taken into account, and it transforms as an axial vector.[]{data-label="fig4"}](fig7.png){width="\columnwidth"}
Within the spin wave expansion, the energy of the spin system is expanded around a stable equilibrium state using small spin deviations with respect to this state. To lowest order, the Hamiltonian can be written as $$\begin{aligned}
H_{0}=E_{0}+\sum_{k}\omega_{k}a^{*}_{k}a_{k},\label{eqn17}\end{aligned}$$ where $E_{0}$ is the energy of the equilibrium state, the $a_{k}$ variables are the classical equivalents of bosonic spin wave annihilation operators and the spin wave energies, $\omega_{k}\ge 0$, stand for the energy corrections due to the spin excitations represented by $a_{k}$.
For $K_{x}=K_{z}=0$, a homogeneous cycloidal spiral state in the $xz$ plane with wave vector $\boldsymbol{q}_{0}$ along the $x$ axis, $\boldsymbol{S}_{i}=\left(-\sin\left(\boldsymbol{q}_{0}\boldsymbol{R}_{i}\right),0,\cos\left(\boldsymbol{q}_{0}\boldsymbol{R}_{i}\right)\right)$, is either a stable or an unstable equilibrium state of the system. The energy per atom of the spin spiral is given by $$\begin{aligned}
\frac{E_{0}\left(\boldsymbol{q}_{0}\right)}{N}
=&&\frac{1}{2}J(\boldsymbol{q}_{0})-\frac{1}{2}\textrm{i}D(\boldsymbol{q}_{0}),\label{eqn33}\end{aligned}$$ with $$\begin{aligned}
J(\boldsymbol{q})=&&\sum_{j (\ne i)}
J_{ij}\textrm{e}^{-\textrm{i}\boldsymbol{q}\left(\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right)},
\\
D(\boldsymbol{q})=&&\sum_{j (\ne i)}
D_{ij}\textrm{e}^{-\textrm{i}\boldsymbol{q}\left(\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right)}.\end{aligned}$$ The spin wave spectrum around a homogeneous cycloidal spiral state with wave vector $\boldsymbol{q}_{0}$ is given by[@Michael; @Michael2] $$\begin{aligned}
\omega_{\boldsymbol{q};\boldsymbol{q}_{0}}=&&\sqrt{C_{+}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)C_{-}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)}, \label{eqn32}\end{aligned}$$ with $$\begin{aligned}
C_{+}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)=&&\frac{1}{2}\left[J\left(\boldsymbol{q}-\boldsymbol{q}_{0}\right)+J\left(\boldsymbol{q}+\boldsymbol{q}_{0}\right)\right]\nonumber
\\
&&-\frac{1}{2}\left[\textrm{i}D\left(\boldsymbol{q}+\boldsymbol{q}_{0}\right)-\textrm{i}D\left(\boldsymbol{q}-\boldsymbol{q}_{0}\right)\right]\nonumber
\\
&&-J\left(\boldsymbol{q}_{0}\right)+\textrm{i}D\left(\boldsymbol{q}_{0}\right),\label{eqn30}
\\
C_{-}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)=&&J\left(\boldsymbol{q}\right)-J\left(\boldsymbol{q}_{0}\right)+\textrm{i}D\left(\boldsymbol{q}_{0}\right),\end{aligned}$$ where the excitations are indexed with the Fourier transformation wave vectors $\boldsymbol{q}$. The equilibrium state is stable if both $C_{+}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)$ and $C_{-}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)$ are non-negative for every $\boldsymbol{q}$, which leads to real and non-negative spin wave frequencies.[@Kaplan] The condition $C_{+}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)\ge 0$ generally holds true if the wave vector $\boldsymbol{q}_{0}$ is close, but not necessarily equal, to the value for which Eq. (\[eqn33\]) is minimized. Without Dzyaloshinsky–Moriya interactions, $C_{-}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)\ge 0$ only holds if $J\left(\boldsymbol{q}_{0}\right)$ is the global minimum of $J\left(\boldsymbol{q}\right)$. However, the presence of the Dzyaloshinsky–Moriya interaction stabilizes several spiral states with different $\boldsymbol{q}_{0}$ values by achieving $C_{-}\left(\boldsymbol{q};\boldsymbol{q}_{0}\right)\ge 0$, even ones which do not minimize Eq. (\[eqn33\]). This leads to the appearance of metastable states for which the spin wave expansion (\[eqn17\]) applies. The presence of the anisotropy may also stabilize these spiral states, either by introducing a hard axis perpendicular to the spiral plane ($K_{y}>0$ in our model, cf. Ref. \[\]) or by introducing an easy axis in the plane of the spiral ($K_{z}<0$, cf. Ref. \[\]).
At finite temperatures, the free energy per atom of a system described by the spin-wave Hamiltonian (\[eqn17\]) can be expressed as $$\begin{aligned}
\frac{F}{N}=\frac{E_{0}}{N}+\frac{k_{\textrm{B}}T}{N}\sum_{k}\ln\omega_{k}+C\left(T\right),\label{eqn20}\end{aligned}$$ where $C\left(T\right)$ does not depend on the parameters of the equilibrium state $E_{0}$ and $\omega_{k}$. This expression can describe a transition between two different stable equilibrium states specified by parameters $E_{0},\omega_{k}$ and $E'_{0},\omega'_{k}$. If $E_{0}<E'_{0}$ and the relation $\sum_{k}\ln\omega_{k}>\sum_{k}\ln\omega'_{k}$ applies, then the system will switch from the first state to the second one at the temperature $$\begin{aligned}
k_{\textrm{B}}T_{\textrm{trans}}=\frac{E'_{0}-E_{0}}{\sum_{k}\ln\omega_{k}-\sum_{k}\ln\omega'_{k}}.\label{eqn23}\end{aligned}$$ The quantum version of this method was applied in Refs. \[\] and \[\] to describe the transition from a ferromagnetic to a spin spiral state in Dy. It should be noted that this method only gives numerically good transition temperatures if the temperature itself is small, since the spin wave expansion for the free energy (\[eqn20\]) becomes less accurate as the temperature is increased.
A way of including a perturbative correction in the calculations is by writing the free energy as $$\begin{aligned}
F=&&E_{0}+\sum_{k}\omega_{k}n_{k}+\frac{1}{2}\sum_{k,k'}P_{kk'}n_{k}n_{k'}\nonumber
\\
&&-k_{\textrm{B}}T\sum_{k}\ln{n_{k}},\label{eqn27}\end{aligned}$$ where $P_{kk'}$ is a symmetric matrix representing higher order corrections to the energy (\[eqn17\]) and $n_{k}$ is the occupation number of the spin wave with energy $\omega_{k}$. Minimizing (\[eqn27\]) with respect to $n_{k}$ leads to self-consistent equations which have real nonnegative solutions only for $T<T_{\textrm{c}}$, giving an estimate of the transition temperature into the paramagnetic phase. This method was originally applied in Ref. \[\] to find the Curie temperature of a Heisenberg ferromagnet on a simple cubic lattice.
The FM-SS I transition\[sec3A\]
-------------------------------
Based on the *ab initio* calculations, we chose different sets of model parameters which are close to the transition points, and employed the spin wave expansion described above to obtain the possible phase transitions as a function of temperature. The calculations were compared to Monte Carlo simulations using the Metropolis algorithm. For $10-11\%$ relaxations the spin spiral energy in Fig. \[figJq\] had a single minimum, which we reproduced by choosing a nearest-neighbor ferromagnetic coupling $J_{1}=-2.0\,\textrm{mRyd}$ and a Dzyaloshinsky–Moriya parameter of $D_{1}=0.4\,\textrm{mRyd}$. $K_{z}=-0.22\,\textrm{mRyd}$ was used to move the energy of the ferromagnetic state below the minimum of the spin spiral energy curve.
As pointed out in Sec. \[sec2C\], the equilibrium spin spiral states of the system are no longer perfect sinusoidal waves, since the spins will prefer the $z$ direction over the $x$ axis. After finding this exact equilibrium state numerically, the spin wave expansion was first performed on the basis of Eq. (\[eqn20\]), that is for a system of free spin waves. The results are given in Table \[table2\] for a lattice size of $N=128\times 64$. The energies of the equilibrium states increase with increasing wave vector, with the ferromagnetic state ($q=0$) being the ground state. Although the size of the lattice influences the allowed wave vectors in the Brillouin zone, the ground state remains ferromagnetic even in the continuum limit[@Dzyaloshinsky2; @Izyumov] corresponding to an infinite lattice, if the anisotropy is large enough. The free energy correction per spin due to free spin waves ($\sum_{k} \ln \omega_{k} $) decreases when the wave number of the spin spiral increases, leading to the expected transition from the ferromagnetic to the spiral state with increasing temperature. After this transition, the wave number of the equilibrium spin spiral gradually increases. This change is continuous in the continuum limit, therefore the spiral orderings with different wave vectors do not actually represent different phases.
------------------------------------------------------- ---------------- -------------------------- -------------------------------------------- ------------------------------------------
$q \left(\frac{2\pi}{\sqrt{2}a_{\textrm{Ta}}}\right)$ $\lambda$ (nm) $\frac{E_{0}}{N}$ (mRyd) $\frac{F_{\textrm{SW}}}{Nk_{\textrm{B}}T}$ $T_{\textrm{trans}}^{\textrm{free}}$ (K)
$0.000000$ $\infty$ $-4.2200$ $1.9630$ $0.0$
$0.015625$ $29.88$ $-4.2176$ $1.9531$ $38.2$
$0.031250$ $14.94$ $-4.2152$ $1.9438$ $40.7$
$0.046875$ $9.96$ $-4.2124$ $1.9353$ $51.9$
$0.062500$ $7.47$ $-4.2072$ $1.9282$ $115.5$
$0.078125$ $5.98$ $-4.1966$ $1.9215$ $249.4$
$0.093750$ $4.98$ $-4.1786$ $1.9138$ $368.6$
------------------------------------------------------- ---------------- -------------------------- -------------------------------------------- ------------------------------------------
: Energy ($E_0/N$), free energy correction ($F_{\textrm{SW}}/Nk_{\textrm{B}}T=\sum_{k}\ln\omega_{k}/N$) per spin and transition temperature ($T_{\rm trans}^{\textrm{free}}$) as defined in Eq. (\[eqn23\]) for different wave numbers ($q$) and corresponding wavelengths ($\lambda$) for the FM-SS I transition, calculated for a lattice of $N=128\times 64$ atoms with periodic boundary conditions.[]{data-label="table2"}
------------------------------------------------------- ---------------- -------------------------- ------------------------------------- ----------------------
$q \left(\frac{2\pi}{\sqrt{2}a_{\textrm{Ta}}}\right)$ $\lambda$ (nm) $T_{\textrm{trans}}$ (K) $T^{\rm free}_{\textrm{trans}}$ (K) $T_{\textrm{c}}$ (K)
$0.000000$ $\infty$ $0.0$ $0.0$ $201.8$
$0.031250$ $14.94$ $49.4$ $38.4$ $223.9$
$0.062500$ $7.47$ $88.5$ $91.1$ $261.7$
------------------------------------------------------- ---------------- -------------------------- ------------------------------------- ----------------------
: Transition temperatures for different wave numbers ($q$) and corresponding wavelengths ($\lambda$), calculated for a lattice size of $N=64\times 32$ with periodic boundary conditions. $T_{\textrm{trans}}$ and $T^{\rm free}_{\textrm{trans}}$ indicate the temperature where the SS I spiral with the given wave number becomes the global minimum of the free energy derived from the perturbation theory, Eq. (\[eqn27\]), and for free spin waves, Eq. (\[eqn20\]), respectively. $T_{\textrm{c}}$ is the temperature where the state becomes unstable according to perturbation theory.[]{data-label="table3"}
Including perturbation corrections in the calculations on the basis of Eq. (\[eqn27\]) makes it possible to give an approximation for $T_{\textrm{c}}$, where the equilibrium state loses its stability and becomes paramagnetic. This gives an upper bound for the transition temperatures, $T_{\textrm{trans}}$. The results are summarized in Table \[table3\]. It is worth noting that although the ferromagnetic state remains metastable for a wide temperature range in the SS I phase, there is a temperature region where only the spiral state is stable and the ferromagnetic state becomes paramagnetic, in agreement with the prediction of Ref. \[\]. Including the perturbative correction also modifies the transition temperature $T_{\textrm{trans}}$ compared to the non-interacting case. The transition temperature from the ferromagnetic state to the first spiral state is significantly increased for the interacting case, as can be inferred from Fig. \[fig-fdiff\] and Table \[table3\]. Interestingly, the transition temperature between the spin spiral states with different wave vectors is hardly affected by the perturbation correction. Note that the $T_{\textrm{trans}}^{\textrm{free}}$ transition temperatures are slightly different in Tables \[table2\] and \[table3\] because of the different lattice sizes used in the calculations. The reason for this is that the lattice size influences not only the allowed $q$ values, but also the spin wave energies.
![(color online) Temperature dependence of free energy differences $\Delta F/N$ between different states, including the ferromagnetic state and spin spirals at different wave numbers. The line at $\Delta F/N=0$ is a guide to the eye, identifying the transition temperatures. The differences obtained with perturbation theory, Eq. (\[eqn27\]), are compared to the linear functions of the free spin wave theory, Eq. (\[eqn20\]), for a lattice size of $N=64 \times 32$. The wave numbers are given in units of $\frac{2\pi}{\sqrt{2}a}$.[]{data-label="fig-fdiff"}](fig8.png){width="\columnwidth"}
![(color online) The order parameter defined in Eq. (\[eqnop\]) for different wave numbers obtained from Monte Carlo simulations as a function of (a) increasing and (b) decreasing temperature, describing the FM-SS I transition, for a lattice size of $N=128\times 64$. The wave numbers are given in units of $\frac{2\pi}{\sqrt{2}a}$.[]{data-label="fig-mc"}](fig9a.png "fig:"){width="\columnwidth"} ![(color online) The order parameter defined in Eq. (\[eqnop\]) for different wave numbers obtained from Monte Carlo simulations as a function of (a) increasing and (b) decreasing temperature, describing the FM-SS I transition, for a lattice size of $N=128\times 64$. The wave numbers are given in units of $\frac{2\pi}{\sqrt{2}a}$.[]{data-label="fig-mc"}](fig9b.png "fig:"){width="\columnwidth"}
Fig. \[fig-mc\] shows the results of Monte Carlo simulations for the same model system. As clear from Fig. \[fig-mc\](a), the simulation results are in good agreement with the spin wave calculations: starting from a ferromagnetic ground state, the system will turn into a spiral state with gradually increasing wave vector until the temperature becomes high enough to remove all kinds of magnetic order from the system. The values for $k_{\textrm{B}}T_{\textrm{trans}}$ are somewhat inaccurate (compare Tables \[table2\]-\[table3\] with Fig. \[fig-mc\]), mainly because the transitions apparently show hysteresis. The lower wave vector states will remain metastable at higher temperatures than the point where the free energy minimum moves to a different wave vector (see Fig. \[fig-fdiff\]). This is even more pronounced in Fig. \[fig-mc\](b), where the simulation was performed for decreasing temperature, starting from a random initial state. Although the $q=0.062500$ state is not the ground state, the system freezes into this metastable state in this case. On the other hand, the transition point to the paramagnetic state $T_{\textrm{c}}$ is well approximated by the perturbation theory: for the $q=0.062500$ spiral state, it predicts $T_{\textrm{c}}=261.7\,\textrm{K}$, while the critical temperature from the simulation is around $220\,\textrm{K}$. For comparison, the random phase approximation[@Tyablikov] gives $T_{\textrm{c}}=271.1\,\textrm{K}$ for the critical temperature of the ferromagnetic state. The same kind of transition was obtained using the *ab initio* coupling coefficients instead of the model parameters, compare Fig. \[figMC\](b) with Fig. \[fig-mc\](a).
The SS II-SS I transition
-------------------------
The SS II-SS I transition can be examined using the same methods as in the previous case. The main difference is that the energy of the spin spiral must have two different minima, both corresponding to spiral orderings, that is $\boldsymbol{q}_{1}, \boldsymbol{q}_{2} \ne \boldsymbol{0}$ (see Fig. \[figJq\]). This requires at least four different coupling coefficients in the spin model (\[eqn29\]) along the $x$ axis, illustrated in Fig. \[fig4\]. For the model calculations we chose $J_{1}=-2.0\,\textrm{mRyd}$, $J_{3}=2.58\,\textrm{mRyd}$, $J_{7}=-1.0\,\textrm{mRyd}$ and $J_{11}=0.8\,\textrm{mRyd}$, which could reproduce the shape of the curves in Fig. \[figJq\], with a slightly lower minimum at high wave number $q=0.593750$ and a somewhat higher one at $q=0.156250$. We also considered the same Dzyaloshinsky–Moriya interaction between the nearest neighbors as in the previous case, $D_{1}=0.4\,\textrm{mRyd}$, since this is necessary to stabilize both spiral states at zero temperature, see Eq. (\[eqn32\]) and the subsequent discussion. We omitted the anisotropy terms needed to make the ferromagnetic state energetically favorable in Sec. \[sec3A\], since they are irrelevant for the current discussion. The energies and free energy corrections are given in Table \[table4\], for lattice size of $N=128\times 64$.
------------------------------------------------------- ---------------- -------------------------- -------------------------------------------- -------------------------------------
$q \left(\frac{2\pi}{\sqrt{2}a_{\textrm{Ta}}}\right)$ $\lambda$ (nm) $\frac{E_{0}}{N}$ (mRyd) $\frac{F_{\textrm{SW}}}{Nk_{\textrm{B}}T}$ $T_{\rm trans}^{\textrm{free}}$ (K)
$0.593750$ $0.79$ $-2.9894$ $1.4779$ $0.0$
$0.156250$ $2.99$ $-2.9736$ $1.4168$ $40.8$
------------------------------------------------------- ---------------- -------------------------- -------------------------------------------- -------------------------------------
: Energy ($E_0/N$), free energy correction ($F_{\textrm{SW}}/Nk_{\textrm{B}}T=\sum_{k}\ln\omega_{k}/N$) per spin and transition temperature ($T_{\rm trans}^{\textrm{free}}$) values as in Table \[table2\] for different wave numbers (or wavelengths) for the SS II-SS I transition, for a lattice size of $N=128\times 64$ with periodic boundary conditions.[]{data-label="table4"}
![The order parameter for different wave numbers as a function of temperature obtained from Monte Carlo simulations for the SS II-SS I transition, for a lattice size of $N=128\times 64$. The wave numbers are given in units of $\frac{2\pi}{\sqrt{2}a}$.[]{data-label="fig9"}](fig10.png){width="\columnwidth"}
The spin wave calculations indicate that starting from a high wave vector ground state, the system may indeed switch to a low wave vector ordering. This is in agreement with the Monte Carlo simulation results with the same parameter set, shown in Fig. \[fig9\], as well as simulations performed with the *ab initio* coupling coefficients, see Fig. \[figMC\](c). The spin wave expansion again underestimates the transition temperature as in the case of the FM-SS I transition.
The SS III-SS II transition
---------------------------
The third type of transition found in the Fe monolayer on Ta$(110)$ surface corresponds to the case when the wave vector of the spiral remains fixed, but the spiral normal vector rotates from the $y$ axis (the cycloidal state) towards a direction in the $xy$ plane. For modelling this transition we supposed that the wave vector $\boldsymbol{q}_{0}$ of the spiral state is determined by the isotropic exchange couplings, while the Dzyaloshinsky–Moriya interaction and the anisotropy terms were taken into account as a perturbation. For the anisotropy we chose $K_{x}>0$ and $K_{z}=0$, since *ab initio* calculations indicated that at 15% relaxation the ferromagnetic states along the $y$ and $z$ axes have almost the same energy, while the $x$ axis is a hard axis (see Fig. \[fig-gs\](b)). The angle between the $xz$ plane and the plane of the spin spiral will be denoted by $\varphi$. In this case, the energy contribution per spin from the Dzyaloshinsky–Moriya interaction and the anisotropy terms can be expressed as $$\begin{aligned}
\frac{\Delta E}{N}=-\frac{1}{2}\textrm{i}D\left(\boldsymbol{q}_{0}\right)\cos\varphi+\frac{1}{2}K_{x}\cos^{2}\varphi. \label{eqn39}\end{aligned}$$
Differentiating (\[eqn39\]) with respect to $\varphi$ leads to the stationary points
$$\begin{aligned}
\sin\varphi^{(1)}=&&0,
\\
\cos\varphi^{(2)}=&&\frac{\textrm{i}D\left(\boldsymbol{q}_{0}\right)}{2K_{x}}. \label{eqn41}\end{aligned}$$
Substituting the solutions into (\[eqn39\]) gives $$\begin{aligned}
\frac{\Delta E^{(1)}}{N}=&&\mp\frac{1}{2}\textrm{i}D\left(\boldsymbol{q}_{0}\right)+\frac{1}{2}K_{x}, \label{e1}
\\
\frac{\Delta E^{(2)}}{N}=&&-\frac{\left(\textrm{i}D\left(\boldsymbol{q}_{0}\right)\right)^{2}}{8K_{x}}, \label{e2}\end{aligned}$$ implying that whenever the second stationary point exists, $$\begin{aligned}
\left|\frac{\textrm{i}D\left(\boldsymbol{q}_{0}\right)}{2K_{x}}\right|<1, \label{cond}\end{aligned}$$ it will correspond to the energy minimum. This describes the rotation of the spiral normal vector away from the $y$ axis when the Dzyaloshinsky–Moriya interaction is weak compared to the anisotropy. Calculating the spin wave spectrum reveals that only one of the states is stable for any value of $D$ and $K_{x}$, therefore the spin wave expansion is not suitable for describing this type of transition.
For the present simulations the exchange parameters $J_{1}=-2.0\,\textrm{mRyd}$, $J_{3}=3.0\,\textrm{mRyd}$ and $J_{7}=-1.0\,\textrm{mRyd}$ were chosen which lead to a spin spiral along the $x$ axis with a wave number $q=0.546875$ ($\lambda=0.85\,\textrm{nm}$). We took $D_{1}=0.05\,\textrm{mRyd}$ between the nearest neighbors and $K_{x}=0.2\,\textrm{mRyd}$, and found that these values did not influence the shape of the spiral considerably, but confined the spins to a plane with a normal vector lying in the $xy$ plane, as shown in Fig. \[fig-spinconf\](d). We also used a ferromagnetic coupling between the neighbors in the $y$ direction, $J_{2}=-2.0\,\textrm{mRyd}$, which does not influence the spiral state but removes the possible domain walls from the system along the $y$ axis. These domain walls occur because Eq. (\[eqn41\]) has two solutions $\pm\varphi^{(2)}$ with the same energy, therefore if the spins are weakly coupled along the $y$ direction, $\varphi^{(2)}$ and $-\varphi^{(2)}$ domains may be simultaneously present in the system.
![(color online) Different components of the order parameter Eq. (\[eqnopcomp\]) as a function of temperature, for the SS III-SS II transition. The wave number of the spin spiral was $q=0.546875 \frac{2\pi}{\sqrt{2}a}$ and a lattice size of $N=128\times 64$ was used.[]{data-label="fig10"}](fig11.png "fig:"){width="\columnwidth"} -10pt
The SS III-SS II transition is shown in Fig. \[fig10\], in agreement with the simulations performed with *ab initio* coupling coefficients, see Fig. \[figMC\](d). By increasing the temperature, the plane of the normal vector of the spiral rotates towards the $y$ axis, which is the one preferred by the Dzyaloshinsky–Moriya interaction over the $x$ direction preferred by the anisotropy. This indicates that with increasing temperature the magnitude of the effective Dzyaloshinsky–Moriya contribution to the free energy decreases slower than the anisotropy contribution.
Summary and conclusions
=======================
We examined the phase diagram of an Fe monolayer on the $(110)$ surfaces of W and Ta as a function of the relaxation of the Fe layer and the temperature. We used the relativistic screened Korringa-Kohn-Rostoker method to determine the single-particle potential of the systems within the local density approximation of density functional theory. In terms of the relativistic torque method, we calculated the tensorial coupling coefficients which appear in the generalized Heisenberg model describing the spin system, Eq. (\[eqn1\]). Based on this spin model, we determined the magnetic ground state from spin dynamics simulations, and we performed Monte Carlo simulations to explore the magnetic phase transitions at finite temperature.
In case of W substrate the obtained magnetic moments and ground states were in good agreement with previous calculations[@Qian2; @Qian3; @Huang] and with experiments.[@Elmers2] The ground state was ferromagnetic with an easy axis along the $[1\overline{1}0]$ direction for relaxations smaller than $15\%$, including the experimentally and theoretically determined relaxation values around $12\%-13\%$. For larger relaxations, the system remained ferromagnetic, but the easy axis turned into the out-of-plane $[110]$ direction. For fixed relaxations, we found no thermally induced transition between these two states.
In case of Ta substrate four different phases were identified in the considered relaxation range, see Fig. \[fig-gs\](b), with transitions occurring at $10.5\%$, $13.8\%$ and $14.5\%$ relaxation values. At low relaxations the ground state was ferromagnetic with an easy axis along $[110]$. The next two phases, denoted by SS I and SS II, correspond to cycloidal spin spirals with wave vectors along the $[1\overline{1}0]$ direction and normal vector along the $[001]$ axis, the SS II state having a significantly larger wave number. The SS II and SS III spin spirals had similar wave vectors but the normal vector of the spiral left the $[001]$ axis in the SS III state.
Choosing the relaxation value close to one of the transition points, different types of transitions were obtained between these states at finite temperature. These possible phase transitions were described theoretically using spin wave expansion and compared to Monte Carlo simulations performed on model systems. We found that starting from a ferromagnetic ground state, the system may turn into a spin spiral state at finite temperature before becoming paramagnetic. Although the appearance of the spin spiral state as a consequence of the Dzyaloshinsky–Moriya interaction is a well-known effect in two-dimensional systems such as Mn monolayer[@Bode] or Fe double-layer[@Meckler] on W$(110)$, there was no such transition observed as a function of temperature. However, *ab initio* calculations[@Heide; @Zimmermann; @Bergqvist] indicated a ferromagnetic ground state for Fe double-layer on W$(110)$, suggesting that this system is probably very close to such a ferromagnetic-spin spiral transition. We have also shown that the high wave vector SS II state may turn into the low wave vector SS I spiral by increasing the temperature, while in the case of the SS III-SS II transition the normal vector of the spin spiral rotated from a general in-plane direction towards the $[001]$ direction.
For all three phase transitions, the simulations performed on model systems and using the *ab initio* coupling coefficients gave results which were in agreement with the predictions based on spin wave expansion. Compared to the Monte Carlo simulations, the spin wave expansion gave good approximations for the temperature $T_{\textrm{c}}$ where any magnetic order disappears and somewhat underestimated the transition temperature $T_{\textrm{trans}}$ between the ordered states. The latter difference is also a consequence of the metastability of the states, indicating that conventional Monte Carlo simulations are not well-suited for finding the actual transition temperature.
Given the wide variety of possible ground states in a relatively narrow range of relaxations, our present work might motivate experiments to determine the actual magnetic ground state of an Fe monolayer on Ta$(110)$. It may even be possible to find one of the thermally induced transitions described here. On the other hand, the spin wave expansion method may also be applied for the finite-temperature description of other stable equilibrium configurations such as the skyrmion lattice structure found in ultrathin magnetic films.[@Heinze; @Romming]
The authors thank Eszter Simon, László Ujfalusi and Bernd Zimmermann for enlightening discussions. Financial support was provided by the Hungarian Scientific Research Fund under project No. K84078 and by the European Union under FP7 Contract No. NMP3-SL-2012-281043 FEMTOSPIN. The work of LS and IAS was also supported by the project TÁMOP-4.2.2.A-11/1/KONV-2012-0036 co-financed by the European Union and the European Social Fund.
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|
---
address: Celestijnenlaan 200B
author:
- Emmanuel
bibliography:
- 'biblio.bib'
date: April 2015
title: Motivic Integration and Logarithmic Geometry
---
|
---
abstract: 'We consider the gravitationally induced particle production from the quantum vacuum which is defined by a free, massless and minimally coupled scalar field during the formation of a gauge cosmic string. Previous discussions of this topic estimate the power output per unit length along the string to be of the order of $10^{68}$ ergs/sec/cm in the s-channel. We find that this production may be completely suppressed. A similar result is also expected to hold for the number of produced photons.'
---
[**Vanishing of Gravitational Particle Production**]{}
[ **in the**]{}
[**Formation of Cosmic Strings**]{}
[Iver Brevik]{}[^1]\
[Bjørn Jensen]{}[^2][^3]\
[**PACS**]{} number(s): 11.27.+d, 98.80.Cq
Introduction
============
When a long and straight gauge cosmic string is formed, it is expected that the resulting spacetime geometry outside the string core to a very high degree of approximation can be described by a flat space with a conical deficit angle. Previous studies indicate that the formation of such strings is accompanied by an intense burst of particles and radiation which are released from the quantum vacuum. The power output per unit length along a string during the formation process may be very large as to be of the order of $10^{68}$ ergs/sec/cm for a free, massless and minimally coupled scalar field [@Parker]. The power output is strongly suppressed when one considers higher angular momentum quantum numbers. A similar estimate for the production of photons was deduced in [@Brevik]. The purpose of this communication is to show that the energy produced during the formation of a gauge cosmic string may be significantly lower than what previous estimates have indicated. It is found in particular that there is no energy production in the zero-angular momentum sector when one considers the free, massless and minimally coupled scalar-field sector to leading order in the string tension. We also expect a similar conclusion to hold for the number of produced photons.
A String Geometry
=================
Let us first briefly review some properties of the geometry which is induced by a gauge cosmic string. It is in general believed that the spacetime geometry outside a long and straight gauge cosmic string can be approximated by the spacetime found outside a corresponding fundamental string, since the typical radius of a cosmic string is typically of the order of $10^{-30}$ cm.. Hence, the cosmic string action can be approximated by the fundamental string action $$S=-\frac{T}{2}\int d^2\sigma\sqrt{-h}\partial_\mu X^A\partial_\nu X^Bh^{\mu\nu}g_{AB}\,
,$$ where Greek letters denote world-sheet indices, while capital Latin ones denote target-space coordinates. $h$ denotes the world-sheet geometry, $g$ the target space geometry, and $T\, (>0)$ is the string tension. It was found in [@Jensen1; @Jensen2] that such a string which is static, straight and infinitely long will, in the [*Weyl gauge*]{} and using cylinder coordinates, give rise to the geometry $$ds^2=a_0^{-1}(-dt^2+dr^2+dz^2)+a_0r^2d\phi^2$$ where $a_0=1-4G\mu_0$, and $\mu_0(=T)$ is the proper energy density in the string. In the following we will use both $\mu_0$ and $T$ to denote the proper energy density in the string. It is at the outset put no restrictions on the value of $\mu_0$ so that $\mu_0$ parametrises a [*single*]{} and infinitely large family of geometries. The string source is assumed to be positioned at $r=0$, such that the region $r>0$ is a flat vacuum region. The coordinates are all assumed to be independent of each other, and of $\mu_0$. The world-sheet geometry is related to the target space geometry by $$h_{\mu\nu}=\partial_\mu X^A\partial_\nu X^B g_{AB}\, .$$ By adjusting the $t$ and $z$ coordinates to coincide with the timelike and spacelike coordinates in the string world-sheet we have that $$h_{\mu\nu}=a_0^{-1}\eta_{\mu\nu}\, .$$ Clearly, when $\mu_0>(4G)^{-1}$ we see that the world-sheet geometry changes sign, while the orientation of the induced geometry in the string world-sheet is unchanged. It is correspondingly straightforward to see that the target space geometry changes signature when $\mu_0$ becomes greater that $(4G)^{-1}$. The geometry in eq.(2) can be brought in the form $$ds^2=-dT^2+dR^2+dZ^2+a_0^2R^2d\phi^2$$ with the use of the rescalings $T=a_0^{-1/2}t$, $R=a_0^{-1/2}r$ and $Z=a_0^{-1/2}z$ [*provided that*]{} $\mu_0<(4G)^{-1}$. This last expression for the string geometry can be derived directly from the Levi-Civita form of the line-element [@Hiscock; @Jensen2].
The action in eq.(1) is not invariant under target space diffeomorphisms. Using eq.(5) we find that the string action takes the form $$S=-\frac{T}{2}\int d^2\sigma\partial_\mu X^A\partial_\nu X^B\eta^{\mu\nu}\eta_{AB}\, ,$$ where $\eta$ denotes the corresponding Minkowski geometry, while using the geometry in eq.(2) we find that $$S=-\frac{T}{2a_0}\int d^2\sigma\partial_\mu X^A\partial_\nu X^B\eta^{\mu\nu}\eta_{AB}\,
.$$ Hence, in the geometry eq.(2) we can perceive the string to carry a renormalised tension $T_{\mbox{\small ren}}$which is given by [^4] [^5] $$T_{\mbox{\small ren}}\equiv\frac{T}{1-4GT}\Rightarrow
T=\frac{T_{\mbox{\small ren}}}{1+4GT_{\mbox{\small ren}}}\, .$$ Since $\mu_0$ (or $T$) denotes the canonical energy measure of the string relative to the geometry in eq.(5), it is more correct to formulate the geometry in eq.(2) in terms of the energy measure of the string relative to the coordinate system which is used there, i.e. in terms of $T_{\mbox{\small ren}}$. The string geometry then takes the form $$ds^2=b_0(-dt^2+dr^2+dz^2)+b_0^{-1}r^2d\phi^2\, ,$$ where $b_0\equiv 1+4GT_{\mbox{\small ren}}=(1-4G\mu_0)^{-1}\geq 1$.
In [@Parker], and in a large number of consecutive studies (see [@Brevik; @Pullin; @Mendel], e.g.), the induced string geometry was taken in the form $$ds^2=-dt^2+dr^2+dz^2+a_0^2r^2d\phi^2\, .$$ We have used the [*same*]{} coordinates in eq.(10) as in the other equations above. [^6] We can do this since it is assumed (implicitly) in all the references above that the coordinates used in eq.(10)
[**(1)**]{} cover the complete spacetime manifold ($r>0$)
[**(2)**]{} do not depend on $\mu_0$.
The singularity structure carried by the manifold described by eq.(10) coincide with the singularity structure in the manifold described in eq.(2) at $r=0$ (a curvature singularity), but the singularity structures differ significantly at the point $\mu_0=(4G)^{-1}$ in the $\mu_0$-parameter space where the geometry in eq.(2) changes signature and $T_{\mbox{\small ren}}\rightarrow\infty$. Hence, since eq.(2) is derived with no particular restriction on the value of $\mu_0$ eq.(2) and eq.(10) describe two [*different*]{} (families of) manifolds. However, at each [*fixed*]{} point $\mu_0$ in the $\mu_0$-parameter space, and such that $0\leq\mu_0<(4G)^{-1}$ (which is the physically interesting regime), these spaces can be brought into manifestly corresponding forms via the rescalings following eq.(5). These spaces can also be so related when $\mu_0>(4G)^{-1}$. However, in this case we must in addition to a set of rescalings also change the signature of either the geometry in eq.(10), or the geometry in eq.(2). Note that $a_0$ is the same in both eq.(2) and eq.(10) since it is a function of the [*proper*]{} energy density in the source. One way to give a presice description of the relation between the geometry in eq.(2) and the geometry in eq.(10) is to observe [@Jensen1; @Jensen2] that eq.(2) can be written in the form $$ds^2=a_0^{-1}(-dt^2+dr^2+dz^2+a_0^2r^2d\phi^2)\, .$$ Hence, [*under the assumptions in [**(1)**]{} and [**(2)**]{} above*]{}, it follows that the geometry which is used in a majority of previous works on gauge cosmic string theory is conformally related to the geometry in eq.(2), i.e. one can go from eq.(2) and to the form in eq.(10) by multiplication with an overall constant scale factor.
Gravitational Particle Creation
===============================
Even though a cosmic string may have the impressing proper mass per unit length $\mu_0\sim 10^{22}$ g/cm $\sim 10^{43}$ erg/cm, the deviation from the Minkowski space which the string induces in $dt=dz=0$ hyperplanes is only of the (less impressing) order $G\mu_0\sim 10^{-6}$. However, it has been argued in earlier studies that even though a static string barely distorts spacetime, the energy production $W$ during the formation of the string may be huge, and at least of the order of $W\sim 10^{20}$ erg/cm during a formation time which was taken to be of the order $\Delta t\sim 10^{-35}$ sec.. In these studies a variety of different scalar fields as well as Maxwell fields were studied. Hence, physical particles released from the quantum vacuum due to the creation of a string may represent a potentially significant source of entropy in the early universe [@Parker]. We will now turn to a reassessment of these production estimates.
In the computation of the energy released in the creation of a string we will use the instantaneous approximation, which was pioneered in this context in [@Parker]. In this approach one assumes that the creation of the string happens instantaneously. We will follow [@Parker] in that we will assume that the initial spacetime is Minkowski space, but we will describe the final spacetime by eq.(9) (or equivalently eq.(11)) and not by eq.(10). The actual time-dependent problem can thus be captured by writing the relevant spacetime geometry as in eq.(9), but with $T_{\mbox{\small ren}}$ replaced by a time-dependent function. In our case this time-dependent function should be the Heaviside step-function $\Theta$, i.e. $$T_{\mbox{\small ren}}\rightarrow T_{\mbox{\small ren}}(t)=T_{\mbox{\small ren}}\Theta
(t-t_0)
\Rightarrow b_0\rightarrow b(t)\, .$$ The moment of creation of the string is at $t=t_0$. In explicit calculations we will set $t_0=0$ without loss of any generality.
We will direct our attention to a massless and minimally coupled scalar field $\Phi (x)$ configuration with the density $${\cal L}=\sqrt{-g}\partial_A\Phi (x)\partial^A\Phi (x)\, .$$ Since we always have that $\sqrt{-g}g^{AA}=1\, ;\, A\neq\phi$, it follows that the corresponding field equation reduces to the form $$(\Box_3+\frac{b^2(t)}{r}\partial^2_\phi)\Phi (x)=0\, ,$$ where $\Box_3$ is the d’Alembertian in a 2+1-dimensional Minkowski geometry for [*all times*]{} $t$. The reader should be aware that the form of eq.(14) does not depend on the particular substitution in eq.(12), but is valid for [*any*]{} general function $b(t)$. From this equation of motion it follows in particular that the continuity condition across $t=t_0$ in the time direction simply reduces to $$(\partial_t\Phi (x))|_{t_0^{-}}=(\partial_t \Phi (x))|_{t_0^{+}}\, .$$ This fact will be of crucial importance, and will represent one main reason, for why our findings differ from those stemming from similar previous excursions into this subject.
In the following we will confine the quantum field to the interior of a straight cylinder centered at the origin $r=0$, with the constant coordinate radius $r=R$. The top and buttom of the cylinder are assumed to be at the fixed coordinate positions $z=0$ and $z=L$, respectively. The cylinder thus defines a [*co-moving*]{} volume, since the proper volume changes in the transition from Minkowski spacetime and to the situation when a string is present. Note that the proper area of a cross section of the cylinder defined by $dt=dz=0$ always equals $\pi R^2$. The change in the proper volume of the cylinder is thus solely induced by a “stretching” of the cylinder in the $z$-direction.
In the region $t<t_0$ we decompose the scalar field operator according to $$\Phi (\vec{x},t)=\sum_{j}( a_jf_j(\vec{x},t)+a^\dagger_jf^*_j(\vec{x},t))\, ,$$ where $j=(n,m,s); n,m=0,\pm 1,\pm 2,..., s=1,2,3,...$, and the annihilation and the creation operators $a_j$ and $a^\dagger _j$ satisfy the usual canonical commutation relations. The mode functions $f_j$, which constitute a complete set with respect to the canonical symplectic form, are given by $$f_{n,m,s}(\vec{x},t)=N_1e^{-i\omega_st}e^{ikz}e^{im\phi}J_{|m|}((\omega_s^2-k^2)^{1/2}r)
\, ,$$ where the normalisation factor $N_1$ is given by $$\begin{aligned}
&&N_1=(2\omega_s V_1)^{-1/2}((\partial_rJ_{|m|} ((\omega_s^2-k^2)^{1/2}r)|_{r=R})^{-1}\,
,\\
&&V_1\equiv\pi LR^2\,\, ,\,\, k=\frac{2\pi n}{L}\, .\end{aligned}$$ The $\omega_s$’s are determined from the equation $J_{|m|}((\omega_s^2-k^2)^{1/2}R)=0$. $s$ does therefore denote a radial quantum number. The canonical vacuum state is defined as $a_j|0\rangle_{\mbox{\small in}}=0$. When $t>t_0$ we similarly expand the quantum field as $$\Phi (\vec{x},t)=\sum_{j}(b_jg_j(\vec{x},t)+b^\dagger _jg^*_j(\vec{x},t))\, .$$ The mode-functions $g_j$ are given by $$g_j(\vec{x},t)=N_2e^{-iW_st}e^{ikz}e^{im\phi}J_{\nu_m}((W_s^2-k^2)^{1/2}r)$$ with $\nu_m\equiv b_0 |m|$. The $W_s$’s are determined from $J_{\nu_m}((W_s^2-k^2)^{1/2}R)=0$. The canonical symplectic form is defined by $$(\psi_n,\psi_m)=i\int_{\Sigma}d^3x\sqrt{g_3}N^A\psi_n{\mathop{\partial}
\limits^{\leftrightarrow}} _{_A}\psi^*_m\, ,$$ where $\vec{N}$ is defined as a future pointing unit vector, $\vec{N}^2=-1$, which is everywhere orthogonal to the spacelike hypersurface $\Sigma$. $d^3x\equiv drdzd\phi$ and $g_3$ is the determinant of the induced 3-geometry in $\Sigma$. We choose the normal vector field to be the canonical one, i.e. we choose $$\vec{N}|_{t_0^-}=\partial_t\,\, ,\,\, \vec{N}|_{t_0^+}=b_0^{-1/2}\partial_t\, .$$ With this normalisation we find that $$(g_j,g_j)=i\int_{\Sigma} d^3x r(g_j
\mathop{\partial_t}\limits^{\leftrightarrow}
g^*_j)|_{t_0^+}\, .$$ It follows that $N_2$ has the same form as $N_1$ except that $\omega\rightarrow W$ and $|m|\rightarrow \nu_m$, of course.
On the spacelike hypersurface $\Sigma$ of instantaneous creation of the string we will have the following general relation between $f$-modes and $g$-modes $$f_j(\vec{x},t_0^-)=\sum_{j'}(\alpha_{j';j}g_{j'}(\vec{x},t_0^+)
+\beta_{j';j}g^*_{j'}(\vec{x},t_0^+))\, .$$ The total number of produced particles $|\beta |^2$ as they are defined in the $t>t_0$ region relative to the incoming vacuum $|0\rangle_{\mbox{\small in}}$ is formally given by $$|\beta |^2\equiv\sum_{j'j}|\beta_{j';j}|^2\equiv
\sum_{j'j}\, _{\mbox{\small in}}\langle 0|N_{j'j}|0\rangle_{\mbox{\small in}}\, ,$$ where $N_{j'j}\equiv b^\dagger _{j'}b_j$. From eq.(25) we then easely deduce that $$\begin{aligned}
\beta_{j';j}&=&i\int_\Sigma d^3x(f_j|_{t_0^-}(\sqrt{g_3}\vec{N}g^*_{j'})|_{t_0^+}-
g_{j'}|_{t_0^+}(\sqrt{g_3}\vec{N}f^*_j)|_{t_0^-})\\
&=&i\int_\Sigma d^3x(f_j|_{t_0^-}(\partial_t g^*_{j'})|_{t_0^+}-
g_{j'}|_{t_0^+}(\partial_tf_j^*)|_{t_0^-}))\, .\end{aligned}$$ Since we have $$f_{n,0,s}|_{t_0^-}=g_{n,0,s}|_{t_0^+}\,\, ,\,\,
\partial_tf_{n,0,s}|_{t_0^-}=\partial_tg_{n,0,s}|_{t_0^+}\, ,$$ we can immediately conclude that $$\beta_{n,0,s;n,0,s}=0 \, .$$ However, even though the diagonal elements in the scattering matrix $\beta_{s';s}$ vanish, the off-diagonal elements may not. The explicit form for $\beta_{j';j}$ is $$\frac{\beta_{j';j}}{\beta^{(10)}_{j';j}}=b_0^{1/2}(\frac{2(s'-s)-|m|+|m'|b_0}{(2s'+\frac
{3}{2})
+(|m'|-|m|-2s-\frac{3}{2})b_0})\, ,$$ where $\beta^{(10)}_{j';j}$ represents the resulting production estimate if we had used the geometry in eq.(10) in order to compute the particle production. We identify $\beta^{(10)}_{j';j}$ with the corresponding expression in [@Parker]. In the derivation of this expression (which is straightforward and will therefore not be reproduced here) we assumed that $R$ is very large in order to utilise the asymptotic properties of the Bessel functions. We also put $n=n'=0$, since the inclusion of these quantum numbers does not provide us with any additional insights. In the physically interesting regime we can effectively set $m=m'=0$ [@Parker], so that $$\frac{\beta_{s';s}}{\beta^{(10)}_{s';s}}=(\frac{2(s'-s)\sqrt{1-4G\mu_0}}
{((2s'+\frac{3}{2})(1-4G\mu_0)-(2s+\frac{3}{2}))})\, .$$ Clearly, $\beta_{s;s}=0$, while $\beta_{s';s\neq s'}\neq 0$. However, this off-diagonal production is sub-leading. Indeed, from [@Parker] one finds that to leading order in $4G\mu_0$($<<1$) $$\beta^{(10)}_{s';s}\sim 2G\mu_0\delta_{s',s}\, .$$ Hence, the potentially [*physically significant*]{} production of scalar particles from the quantum vacuum, due to the creation of a physically realistic cosmic string, is completely suppressed in the zero angular momentum sector.
Conclusion
==========
In previous studies [@Parker; @Brevik; @Pullin; @Mendel] (e.g.) one made the replacement $a_0\rightarrow a(t)$ in eq.(10) in order to approximate the description of cosmic string creation. Clearly, the resulting geometric structure is not simply related to the corresponding geometry in eq.(2). From this point of view it is perhapse not surprising that our results differ significantly from the results of these previous studies. The conformal form of the string geometry in eq.(11) were explored and partially utilised in [@Jensen1; @Jensen2] in order to understand the properties of the usual form of the string geometry in eq.(10). In [@Jensen3] this form of the string geometry was also used in order to extract the qualitative properties of the amount of particles produced during string creation compared to the estimate one computes directly from eq.(10). When the conformal scalar field sector is considered along with the associated conformal vacuum structure [@Birrell], it was shown in [@Jensen3] that the total number of particles produced in the appropriately generalised version of eq.(11) ($|\beta^{(11)}|^2$) is less than the corresponding amount produced in eq.(10) ($|\beta^{(10)}|^2$), in the instantaneous approximation. These quantities were found to be related by [@Jensen3] $$|\beta^{(11)}|^2=(1-4G\mu_0)|\beta^{(10)}|^2\, .$$ Clearly, the conformal vacuum is a very special configuration, and is probably of greater theoretical interest than of physical significance. However, the findings in [@Jensen3] are indeed along the lines implied by the findings in this paper, since $|\beta^{(11)}|^2<|\beta^{(10)}|^2$ to leading order in $4G\mu_0$. Indeed, the $(1-4G\mu_0)$ factor in eq.(34) is also manifestly present in the squared version of eq.(32). However, the differences between eq.(34) and eq.(32) do also illustrate that the question of whether any physically significant particle production occur in string creation or not, is very sensitive to the nature of the coupling of the scalar field to gravity.
Acknowledgements
================
BJ thanks NORDITA for a travelling grant, and NORDITA, the Norwegian University of Science and Technology and the Niels Bohr Institute for hospitality during the time when this work was carried out.
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[^1]: e-mail:iver.h.brevik@mtf.ntnu.no
[^2]: e-mail:bjensen@gluon.uio.no
[^3]: On leave of absence from Institute of Physics, University of Oslo, Norway.
[^4]: One can also come to this conclusion without the explicit use of the string action by a very simple and straightforward calculation of the energy density in the string in the coordinate system in eq.(2), when $T$ is defined as the proper energy density in the string.
[^5]: Note that this string tension renormalisation is of a purely classical and non-perturbative nature, and does therefore represent an additional string tension renormalisation mechanism in addition to the perturbative ones of quantum mechanical origin (see [@Damour], and the references therein).
[^6]: The reason for identifying the coordinates in eq.(2) and eq.(10) is to be able to relate the number of particles released in string formation when calculated in the geometry in eq.(2) and eq.(10), since quantum field theory is [*not*]{} generally covariant.
|
---
abstract: 'Given a collection of images, humans are able to discover landmarks of the depicted objects by modeling the shared geometric structure across instances. This idea of *“geometric equivariance”* has been widely used for unsupervised discovery of object landmark representations. In this paper, we develop a simple and effective approach based on contrastive learning of *“invariant”* representations. We show that when a deep network is trained to be invariant to geometric and photometric transformations, representations from its intermediate layers are highly predictive of object landmarks. Furthermore, by stacking representations across layers in a “*hypercolumn*” their effectiveness can be improved. Our approach is motivated by the phenomenon of the gradual emergence of invariance in the representation hierarchy of a deep network. We also present a unified view of existing equivariant and invariant representation learning approaches through the lens of contrastive learning, shedding light on the nature of invariances learned. Experiments on standard benchmarks for landmark discovery, as well as a challenging one we propose, show that the proposed approach surpasses prior state-of-the-art. [[^1]]{}'
author:
- |
Zezhou Cheng Jong-Chyi Su Subhransu Maji\
University of Massachusetts Amherst\
[{zezhoucheng, jcsu, smaji}@cs.umass.edu]{}
bibliography:
- 'reference.bib'
title: Unsupervised Discovery of Object Landmarks via Contrastive Learning
---
The project is supported in part by Grants \#1661259 and \#1749833 from the National Science Foundation of United States. Our experiments were performed on the University of Massachusetts, Amherst GPU cluster obtained under the Collaborative Fund managed by the Massachusetts Technology Collaborative. We would also like to thank Erik Learned-Miller, Dan Sheldon and Rui Wang for discussion and feedback on the draft.
[^1]: Project webpage: <https://people.cs.umass.edu/~zezhoucheng/contrastive_landmark>
|
---
abstract: |
We give a polynomial time algorithm that finds the maximum weight stable set in a graph that does not contain an induced path on seven vertices or a bull (the graph with vertices $a, b, c, d, e$ and edges $ab, bc, cd, be, ce$). With the same arguments with also give a polynomial algorithm for any graph that does not contain $S_{1,2,3}$ or a bull.
*Keywords*: maximum weight stable set problem, polynomial algorithm, ($P_7$, bull)-free, ($S_{1,2,3}$, bull)-free
author:
- 'Frédéric Maffray[^1]'
- 'Lucas Pastor[^2]'
title: 'Maximum Weight Stable Set in ($P_7$, bull)-free graphs and ($S_{1,2,3}$, bull)-free graphs'
---
Introduction
============
In a graph $G$, a *stable set* (or *independent set*) is a subset of pairwise non-adjacent vertices. The *Maximum Stable Set problem* (shortened as MSS) is the problem of finding a stable set of maximum cardinality. In the weighted version, let $w : V(G)
\rightarrow \mathbb{N}$ be the weight function over the set of vertices. The weight of any subset of vertices is defined as the sum of the weight of all its elements. The *Maximum Weight Stable Set problem* (shortened as MWSS) is the problem of finding a stable set of maximum weight. It is known that MSS and MWSS are NP-hard in general [@GarJoh].
Given a set of graphs $\mathcal{F}$, a graph $G$ is *$\mathcal{F}$-free* if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{F}$. If $\mathcal{F}$ is composed of only one element $F$, we say that $G$ is $F$-free. On the other hand, we say that $G$ *contains* $F$ when $F$ is isomorphic to an induced subgraph of $G$. For any integer $k$, we let $P_k$, $C_k$ and $K_k$ denote respectively the chordless path on $k$ vertices, the chordless cycle on $k$ vertices, and the complete graph on $k$ vertices. The *claw* is the graph with four vertices $a,x,y,z$ and three edges $ax,ay,az$. Let $S_{i, j, k}$ be the graph obtained from a claw by subdividing its edges into respectively $i$, $j$ and $k$ edges. Let us say that a graph is *special* if every component of the graph is a path or an $S_{i,j,k}$ for any $i,j,k$.
- Alekseev [@A] proved that MSS remains NP-hard in the class of $\mathcal{F}$-free graphs whenever $\mathcal{F}$ is a finite set of graphs such that no member of $\mathcal{F}$ is special.
- Several authors [@FOS; @M; @NT; @NS; @S] proved that MWSS can be solved in polynomial time for claw-free graphs ($S_{1, 1, 1}$-free graphs).
- Lozin and Milanič [@LM] proved that MWSS can be solved in polynomial time for fork-free graphs ($S_{1, 1, 2}$-free graphs).
- Lokshtanov, Vatshelle and Villager [@LVV] proved that MWSS can be solved in polynomial time for $P_5$-free graphs ($S_{0, 2, 2}$-free graphs).
The results above settle the complexity of MWSS in $F$-free graph whenever $F$ is a connected special graph on at most five vertices. Therefore the new frontier to explore is when the forbidden induced subgraph has six or more vertices. There are several results on the existence of a polynomial time algorithm for MWSS in subclasses of $P_6$-free graphs [@K; @KM; @MP; @RM1999; @RM2009; @RM2013]. Mosca [@RM2008] proved that MWSS is solvable in polynomial for the class of ($P_7$, banner)-free graphs. Brandstädt and Mosca [@BM] proved that there exists a polynomial time algorithm for the MWSS problem in the class of ($P_7$, $K_3$)-free graphs. The *bull* is the graph with vertices $a, b, c, d, e$ and edges $ab, bc, cd, be,
ce$ (see Figure \[fig:bull\]). Our main results are the following two theorems.
\[thm:P7\] The Maximum Weight Stable Set problem can be solved in polynomial time in the class of ($P_7$, bull)-free graphs.
\[thm:S123\] The Maximum Weight Stable Set problem can be solved in polynomial time in the class of ($S_{1,2,3}$, bull)-free graphs.
Theorem \[thm:P7\] generalizes the main results in [@BM] and [@MP], and Theorem \[thm:S123\] generalizes the main results in [@KM2] and [@MP].
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Our paper is organised as follows. In the rest of this section we recall some definitions, notations and well known results. In Section 2 we develop a structural description that we can use to solve the MWSS efficiently. In Section \[sec:P7\], thanks to the detailed structure, we show how to solve the MWSS in polynomial time in the class of ($P_7$, bull)-free graphs. In Section \[sec:S123\], we show how to solve the MWSS in polynomial time in the class of ($S_{1,2,3}$, bull)-free graphs.
Let $G$ be a graph. For any vertex $v \in V(G)$, we denote by $N(v) =
\{u \in V(G) \mid uv \in E(G)\}$ the *neighborhood* of $v$. For any $S \subseteq V(G)$ we denote by $G[S]$ the *induced subgraph* of $G$ with vertex-set $S$. For any $X\subseteq V(G)$, we may write $G\setminus X$ instead of $G[V(G)\setminus X]$. For any $S\subseteq
V(G)$ and $x\in V(G)$, we let $N_S(x)$ stand for $N(x)\cap S$. For two sets $K, S \subseteq V(G)$, we say that $K$ is *complete* to $S$ if every vertex of $K$ is adjacent to every vertex of $S$, and we say that $K$ is *anticomplete* to $S$ if no vertex of $K$ is adjacent to any vertex of $S$. A *homogeneous set* is a set $S
\subseteq V(G)$ such that every vertex in $V(G) \setminus S$ is either complete to $S$ or anticomplete to $S$. A homogeneous set is *proper* if it contains at least two vertices and is different from $V(G)$. A graph is *prime* if it has no proper homogeneous set.
A *hole* in a graph is any induced cycle on at least four vertices. An *antihole* is the complement of a hole. A graph $G$ is perfect if, for every induced subgraph $G'$ of $G$, the chromatic number of $G'$ is equal to the maximum clique size in $G'$. The Strong Perfect Graph Theorem [@CRST] establishes that a graph is perfect if and only if it contains no odd hole and no odd antihole.
In a series of papers [@Ch-bull1; @Ch-bull23] Chudnovsky established a decomposition theorem for all bull-free graphs. Based on this decomposition, Thomassé, Trotignon and Vušković [@TTV] proved that the MWSS problem is fixed-parameter tractable in the class of bull-free graphs. It might be that these results could be adapted so as to yield an alternate proof of Theorems \[thm:P7\] and \[thm:S123\]. However we are able to avoid using the rather complex machinery of [@TTV] and [@Ch-bull1; @Ch-bull23]. Our proof is based on conceptually simple ideas derived from [@BM] and is self-contained.
Structural description
======================
A class of graphs is *hereditary* if, for every graph $G$ in the class, every induced subgraph of $G$ is also in the class. For example, for any set $\mathcal{F}$ of graphs, the class of $\mathcal{F}$-free graphs is hereditary. We will use the following theorem of Lozin and Milanič [@LM].
\[thm:LM\] Let $\cal{G}$ be a hereditary class of graphs. Suppose that there is a constant $c \geq 1$ such that the MWSS problem can be solved in time $O(|V(G)|^c)$ for every prime graph $G$ in $\cal{G}$. Then the MWSS problem can be solved in time $O(|V(G)|^c + |E(G)|)$ for every graph $G$ in $\cal{G}$.
The classes of ($P_7$, bull)-free graphs and ($S_{1,2,3}$, bull)-free graphs are hereditary. Hence, in order to prove Theorems \[thm:P7\] and \[thm:S123\] it suffices to prove them for prime graphs.
In a graph $G$, let $H$ be a subgraph of $G$. For each $k>0$, a *$k$-neighbor* of $H$ is any vertex in $V(G)\setminus V(H)$ that has exactly $k$ neighbors in $H$. The following two lemmas are straightforward and we omit their proof.
\[lem:c5n\] Let $G$ be a bull-free graph. Let $C$ be an induced $C_5$ in $G$, with vertices $c_1, \ldots, c_5$ and edges $c_ic_{i+1}$ for each $i$ modulo $5$. Then:
- Any $2$-neighbor of $C$ is adjacent to $c_i$ and $c_{i+2}$ for some $i$.
- Any $3$-neighbor of $C$ is adjacent to $c_i$, $c_{i+1}$ and $c_{i+2}$ for some $i$.
- If a non-neighbor of $C$ is adjacent to a $k$-neighbor of $C$, then $k\in\{1,2,5\}$.
\[lem:c7n\] Let $G$ be a bull-free graph. Let $C$ be an induced $C_7$ in $G$, with vertices $c_1, \ldots, c_7$ and edges $c_ic_{i+1}$ for each $i$ modulo $7$. Then:
- Any $2$-neighbor of $C$ is adjacent to $c_i$ and either $c_{i+2}$ or $c_{i+3}$ for some $i$.
- Any $3$-neighbor of $C$ is adjacent to either to $c_i$, $c_{i+1}$ and $c_{i+2}$ or to $c_i$, $c_{i+2}$ and $c_{i+4}$ for some $i$.
- $C$ has no $k$-neighbor for any $k\in\{4,5,6\}$.
For any integer $k\ge 5$, a *$k$-wheel* is a graph that consists of a $C_k$ plus a vertex (called the center) adjacent to all vertices of the cycle. The following lemma was proved for $k\ge 7$ in [@RSb]; actually the same proof holds for all $k\ge 6$ as observed in [@DM; @FMP].
\[lem:wheel\] A prime bull-free graph contains no $k$-wheel for any $k\ge 6$.
Since the bull is a self-complementary graph, the lemma also says that a prime bull-free graph does not contain the complementary graph of a $k$-wheel with $k\ge 6$ (a *$k$-antiwheel*).
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An *umbrella* is a graph that consists of a $5$-wheel plus a vertex adjacent to the center of the $5$-wheel only (see Figure \[fig:4graphs\]).
\[lem:umbr\] A prime bull-free graph contains no umbrella.
A *parasol* is a graph that consists of a $P_5$, plus a sixth vertex adjacent to all vertices of the $P_5$, plus a seventh vertex adjacent to the sixth vertex only (see Figure \[fig:4graphs\]).
\[lem:parasol\] A prime bull-free graph contains no parasol.
Let $G$ be a prime bull-free graph, and suppose that it contains a parasol, with vertices $p_1, \ldots, p_5, x,y$ and edges $p_ip_{i+1}$ for $i=1,2,3,4$, and $xp_j$ for $j=1,\ldots,5$ and $xy$. Let $P=\{p_1, \ldots, p_5\}$. Let $A$ be the set of vertices that are complete to $P$, and let $Z$ be the set of vertices that are anticomplete to $P$. Let: $$\begin{aligned}
A' &=& \{a\in A\mid a \mbox{ has a neighbor in } Z\}. \\
A'' &=& \{a\in A\setminus A'\mid a \mbox{ has a non-neighbor in }
A'\}.\end{aligned}$$ Note that $y\in Z$ and $x\in A'$, so $A'\neq\emptyset$, and that $A''$ is anticomplete to $Z$, by the definition of $A'$. Let $H$ be the component of $G\setminus (A'\cup A'')$ that contains $P$. We claim that: $$\label{apvh}
\mbox{$A'\cup A''$ is complete to $V(H)$.}$$ Proof: Suppose on the contrary that there exist non-adjacent vertices $a,u$ with $a\in A'\cup A''$ and $u\in V(H)$. We use the following notation. If $a\in A'$, let $z$ be a neighbor of $a$ in $Z$. If $a\in A''$, let $b$ be a non-neighbor of $a$ in $A'$, and let $z$ be a neighbor of $b$ in $Z$; in that case we know that $a$ is not adjacent to $z$, since $a\notin A'$. By the definition of $H$, there is a path $u_0$-$\cdots$-$u_\ell$ in $H$ with $u_0\in P$ and $u_\ell=u$, and $\ell\ge 0$. We know that $a$ is adjacent to $u_0$ by the definition of $A$, so $\ell\ge 1$. We choose $u$ that minimizes $\ell$, so the path $u_0$-$\cdots$-$u_\ell$ is chordless, and $a$ is complete to $\{u_0,\ldots,u_{\ell-1}\}$, and if $\ell\ge 2$ then $u_2,\ldots,u_\ell\in Z$.\
Suppose that $\ell=1$. Suppose that $u_1\in A$. By the definition of $H$ we have $u_1\in A\setminus(A'\cup A'')$, so $u_1$ is not adjacent to $z$ and is complete to $A'$, and so $a\notin A'$, hence $a\in A''$, and $u_1$ is adjacent to $b$. Then $\{z,b,u_1,u_0,a\}$ induces a bull, a contradiction. Hence $u_1\notin A$. So there is an integer $i\in\{1,2,3,4\}$ such that $u_1$ has a neighbor and a non-neighbor in $\{p_i,p_{i+1}\}$. Suppose that $u_1$ is not adjacent to $z$. If $a\in A'$, then $\{z,a,p_i,p_{i+1},u_1\}$ induces a bull. If $a\in
A''$, then $u_1$ is adjacent to $b$, for otherwise $\{z,b,p_i,p_{i+1},u_1\}$ induces a bull; but then $\{z,b,u_1,p,a\}$ induces a bull (for $p\in\{p_i,p_{i+1}\}\cap N(u_1)$). Hence $u_1$ is adjacent to $z$. It follows that there is no integer $j$ such that $\{u_1, p_j, p_{j+1}\}$ induces a triangle, for otherwise there is an integer $k$ such that $\{z,u_1,p_k,p_{k+1},p_{k+2}\}$ induces a bull. If we can take $i=1$, then $u_1$ is adjacent to $p_4$, for otherwise $\{u_1, p_1,p_2,a,p_4\}$ induces a bull; and similarly $u_1$ is adjacent to $p_5$; but then $\{u_1, p_4, p_5\}$ induces a triangle, a contradiction. Hence $u_1$ is either complete or anticomplete to $\{p_1,p_2\}$, and actually it is anticomplete to that set since $\{u_1,p_1,p_2\}$ does not induce a triangle. Likewise $u_1$ is anticomplete to $\{p_4,p_5\}$. Hence $u_1$ is adjacent to $p_3$. But then $\{u_1,p_3,p_2,a,p_5\}$ induces a bull, a contradiction.\
Therefore $\ell\ge 2$. We have $u_1\notin A$, for otherwise we would have $u_1\in A'$ because $u_2\in Z$. Since $u_1\notin A$ and the graph $\overline{P_5}$ is connected, there are non-adjacent vertices $p,q\in P$ such that $u_1$ is adjacent to $p$ and not to $q$. We may assume up to relabeling that $u_0=p$. Then $\{u_\ell, u_{\ell-1},
u_{\ell-2}, a, q\}$ induces a bull, a contradiction. Thus (\[apvh\]) holds.
Let $R=V(G)\setminus (A'\cup A''\cup V(H))$. By the definition of $H$, there is no edge between $V(H)$ and $R$. By (\[apvh\]), $V(H)$ is complete to $A'\cup A''$. Hence $V(H)$ is a homogeneous set, and it is proper because $P\subseteq V(H)$ and $A'\neq \emptyset$.
Let $G_1$ be the graph with vertices $p_1,\ldots,p_5,d,a$ such that $p_1$-$p_2$-$p_3$-$p_4$-$p_5$-$p_1$ is a $C_5$, $d$ is adjacent to $p_5$, $a$ is adjacent to $p_5,p_1,p_2$, and there is no other edge. Let $G_2$ be the graph with vertices $p_1,\ldots,p_5,d,a$ such that $p_1$-$p_2$-$p_3$-$p_4$-$p_5$-$p_1$ is a $C_5$, $d$ is adjacent to $p_5$, $a$ is adjacent to $p_1,p_2,p_3$, and there is no other edge. See Figure \[fig:4graphs\].
\[lem:nog12\] A prime bull-free graph $G$ contains no $G_1$ and no $G_2$.
First suppose that $G$ contains a $G_1$, with the same notation as above. Let $X=\{x\in V(G)\mid$ $xp_5,xp_2\in E(G)$ and $xd,xp_3,xp_4\notin E(G)\}$ (so $a,p_1\in X$), and let $Y$ be the vertex-set of the component of $G[X]$ that contains $a$ and $p_1$. Since $G$ is prime, $Y$ is not a homogeneous set, so there are adjacent vertices $y,z\in Y$ and a vertex $b\in V(G)\setminus Y$ such that $by\in E(G)$ and $bz\notin E(G)$. Suppose that $bp_5\notin
E(G)$. Then $bd\in E(G)$, for otherwise $\{b,y,z,p_5,d\}$ induces a bull; and similarly $bp_4\in E(G)$. If $bp_2\notin E(G)$, then $bp_3\in E(G)$, for otherwise $\{b,y,z,p_2,p_3\}$ induces a bull; but then $\{p_2,p_3,b,p_4,p_5\}$ induces a bull; so $bp_2\in E(G)$. Then $bp_3\in E(G)$, for otherwise $\{d,b,y,p_2,p_3\}$ induces a bull; but then $\{d,b,p_3,p_2,z\}$ induces a bull. Hence $bp_5\in E(G)$. Suppose that $bp_2\notin E(G)$. Then $bd\in E(G)$, for otherwise $\{p_2,y,b,p_5,d\}$ induces a bull; and $bp_4\in E(G)$, for otherwise $\{p_2,y,b,p_5,p_4\}$ induces a bull; and $bp_3\in E(G)$, for otherwise $\{z,p_5,b,p_4,p_3\}$ induces a bull; but then $\{d,b,p_4,p_3,p_2\}$ induces a bull. Hence $bp_2\in E(G)$. If $bp_3\in E(G)$, then $bp_4\in E(G)$, for otherwise $\{z,p_2,b,p_3,p_4\}$ induces a bull, and $bd\in E(G)$, for otherwise $\{d,p_5,p_4,b,p_2\}$ induces a bull; but then $\{z,p_5,d,b,p_3\}$ induces a bull. Hence $bp_3\notin E(G)$. Then $bp_4\notin E(G)$, for otherwise $\{p_3,p_4,b,p_5,z\}$ induces a bull, and $bd\notin E(G)$, for otherwise $\{d,b,y,p_2,p_3\}$ induces a bull. But now we see that $b\in Y$, a contradiction.
Now suppose that $G$ contains a $G_2$, with the same notation as above. Let $X=\{x\in V(G)\mid$ $xp_1,xp_3\in E(G)$ and $xd,xp_5,xp_4\notin E(G)\}$ (so $a,p_2\in X$), and let $Y$ be the vertex-set of the component of $G[X]$ that contains $a$ and $p_2$. Since $Y$ is not a homogeneous set, there is a vertex $b \in V(G)
\setminus Y$ and two adjacent vertices $x, y \in Y$ such that $b$ is adjacent to $x$ and not adjacent to $y$. If $bp_4\notin E(G)$, then $bp_3\in E(G)$, for otherwise $\{b,x,y,p_3,p_4\}$ induces a bull, and $bp_1\in E(G)$, for otherwise $\{p_1,x,b,p_3,p_4\}$ induces a bull, and $bp_5\notin E(G)$, for otherwise $\{y,p_1,b,p_5,p_4\}$ induces a bull, and $bd\notin E(G)$, for otherwise $\{d,b,x,p_3,p_4\}$ induces a bull; but then we see that $b\in Y$, a contradiction. Hence $bp_4\in
E(G)$. If $bp_5\in E(G)$, then $bd\in E(G)$, for otherwise $\{x, b,
p_4, p_5, d\}$ induces a bull, and $bp_3\notin E(G)$, for otherwise $\{y,p_3,p_4,b,d\}$ induces a bull, and $bp_1\in E(G)$, for otherwise $\{p_3,p_4,b,p_5,p_1\}$ induces a bull; but then $\{d,b,p_1,x,p_3\}$ induces a bull. Hence $bp_5\notin E(G)$. Then $bp_3\notin E(G)$, for otherwise $\{y,p_3,b,p_4,p_5\}$ induces a bull, and $bp_1\in E(G)$, for otherwise $\{b,x,y,p_1,p_5\}$ induces a bull; but then $\{p_5,
p_1,b,x,p_3\}$ induces a bull, a contradiction.
$(P_7,\mbox{bull})$-free graphs {#sec:P7}
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Before giving the proof of Theorem \[thm:P7\] we need another lemma.
\[lem:c7p\] Let $G$ be a connected $(P_7,\mbox{bull})$-free graph. Assume that $G$ contains a $C_7$ but no $C_5$ and no $7$-wheel. Then $V(G)$ can be partitioned into seven non-empty sets $A_1,\ldots,A_7$ such that for each $i\in\{1,\ldots,7\}$ $(\bmod~7)$ the set $A_i$ is complete to $A_{i-1}\cup A_{i+1}$ and anticomplete to $A_{i-3}\cup A_{i-2}\cup
A_{i+2}\cup A_{i+3}$.
Since $G$ contains a $C_7$, there exist seven pairwise disjoint and non-empty sets $A_1,\ldots,A_7\subset V(G)$ such that for each $i\in\{1,\ldots,7\}$ $(\bmod~7)$ the set $A_i$ is complete to $A_{i-1}\cup A_{i+1}$ and anticomplete to $A_{i-3}\cup A_{i-2}\cup
A_{i+2}\cup A_{i+3}$. We choose these sets so as to maximize their union $U=A_1\cup\cdots\cup A_7$. Hence we need only prove that $V(G)=U$, so suppose the contrary. Since $G$ is connected, there is a vertex $x$ in $V(G)\setminus U$ that has a neighbor in $U$. For each $i\in\{1,\ldots,7\}$ pick a vertex $c_i\in A_i$ so that $x$ has a neighbor in the cycle $C$ induced by $\{c_1, \ldots, c_7\}$. So $x$ is a $k$-neighbor of $C$ for some $k>0$. Since $G$ contains no $7$-wheel, and by Lemma \[lem:c7n\], we have $k\in\{1,2,3\}$. If $k=1$, say $x$ is adjacent to $c_1$, then $x$-$c_1$-$c_2$-$c_3$-$c_4$-$c_5$-$c_6$ is an induced $P_7$. If $k=2$ and $x$ is adjacent to $c_i$ and $c_{i+3}$ for some $i$, then $\{x,
c_i, c_{i+1}, c_{i+2}, c_{i+3}\}$ induces a $C_5$. If $k=3$ and $x$ is adjacent to $c_i$, $c_{i+2}$ and $c_{i+4}$ for some $i$, then $\{x,
c_i, c_{i-1}, c_{i-2}, c_{i-3}\}$ induces a $C_5$. Therefore, by Lemma \[lem:c7n\], it must be that $N_C(x)$ is equal to either $\{c_{i-1}, c_{i+1}\}$ or $\{c_{i-1}, c_i, c_{i+1}\}$ for some $i$, say $i=7$. Pick any $c'\in A_1\setminus \{c_1\}$ and let $C'$ be the cycle induced by $(V(C)\setminus\{c_1\})\cup\{c'\}$. Then by the same arguments applied to $C'$ and $x$, we deduce that $x$ is adjacent to $c'$. So $x$ is complete to $A_1$, and similarly $x$ is complete to $A_6$. Likewise, Lemma \[lem:c7n\] and the fact that $G$ is $C_5$-free implies that $x$ has no neighbor in $A_2\cup A_3\cup
A_4\cup A_5$. But now the sets $A_1,\ldots,A_6,A_7\cup\{x\}$ contradict the maximality of $U$. So $V(G)=U$ and the lemma holds.
Now we can prove the main result of this section.
[**Proof of Theorem \[thm:P7\].**]{} Let $G$ be a $(P_7,\mbox{bull})$-free graph, and let $w$ be a weight function on the vertex set of $G$. By Theorem \[thm:LM\], we may assume that $G$ is prime. By Lemmas \[lem:wheel\]—\[lem:nog12\], $G$ contains no $k$-wheel and no $k$-antiwheel for any $k\ge 6$, no umbrella, no parasol, no $G_1$ and no $G_2$. To find the maximum weight stable set in $G$ it is sufficient to compute, for every vertex $c$ of $G$, a maximum weight stable set containing $c$, and to choose the best set over all $c$. So let $c$ be any vertex in $G$. The maximum weight of a stable set that contains $c$ is equal to $w(c)+\sum_{K} \alpha_w(K)$, where the sum is over all components $K$ of $G\setminus(\{c\}\cup N(c))$ (the non-neighborhood of $c$) and $\alpha_w(K)$ is the maximum weight of any stable set in $K$. So let $K$ be an arbitrary component of $G\setminus(\{c\}\cup N(c))$. If $K$ is perfect, we can use the algorithm from [@Penev] to find a maximum weight stable set in $K$. Therefore let us assume that $K$ is not perfect. We note that $K$ contains no antihole of length at least $6$, for otherwise the union of such a subgraph with $c$ forms an antiwheel. Hence, by the Strong Perfect Graph Theorem [@CRST], and since $G$ is $P_7$-free, $K$ contains a $C_5$ or a $C_7$.
Since $G$ is prime it is connected, so there is a neighbor $d$ of $c$ that has a neighbor in $K$. Let $H=N_K(d)$ and $Z=V(K)\setminus H$. We claim that every $C_5$ in $K$ contains at most two vertices from $H$, and if it contains two they are non-adjacent. Indeed, in the opposite case, there is a $C_5$ in $K$ with vertices $v_1,\ldots,v_5$ and edges $v_iv_{i+1}$ ($\bmod~5$) such that $v_1,v_2\in H$. Then $v_3\in H$, for otherwise $\{c,d,v_1,v_2,v_3\}$ induces a bull; and similarly $v_4,v_5\in H$; but then $\{v_1,\ldots,v_5,d,c\}$ induces an umbrella, which contradicts Lemma \[lem:umbr\]. So the claim is established. Henceforth, for $q\in\{0,1,2\}$ we say that a $C_5$ in $K$ is of type $q$ if it contains exactly $q$ vertices from $H$. So every $C_5$ in $K$ is of type $0$, $1$ or $2$, and if it is of type $2$ its two vertices from $H$ are non-adjacent. Our proof follows the pattern from [@BM], but in some parts we will use different arguments.
[**Case 1: $K$ contains a $C_7$ and no $C_5$.**]{}
Since $K$ is connected and contains no $7$-wheel, Lemma \[lem:c7p\] implies that $V(K)$ can be partitioned into seven non-empty sets $A_1,\ldots,A_7$ such that for each $i\in\{1,\ldots,7\}$ ($\bmod 7$) the set $A_i$ is complete to $A_{i-1}\cup A_{i+1}$ and anticomplete to $A_{i-3}\cup A_{i-2}\cup A_{i+2}\cup A_{i+3}$. Clearly we have $\alpha_w(K)=\max_{i\in\{1,\ldots,7\}} \{\alpha_w(G[A_i])
+\alpha_w(G[A_{i+2}]) +\alpha_w(G[A_{i+4}])\}$, so we need only compute $\alpha_w(G[A_i])$ for each $i\in\{1,\ldots,7\}$. For each $i$ pick a vertex $a_i\in A_i$. The graph $G[A_i]$ contains no $C_5$, no $P_5$ and no $\overline{P_5}$, for otherwise adding $a_{i+1}$ and either $a_{i+2}$ or $a_{i+3}$ to such a subgraph we obtain an umbrella or a parasol in $G$ or $\overline{G}$, which contradicts Lemmas \[lem:umbr\] and \[lem:parasol\]. By results from [@CHMW] and [@HoangPO], MWSS can be solved in time $O(n^3)$ in graphs with no $C_5$, $P_5$ and $\overline{P_5}$. Hence, since the $A_i$’s are pairwise disjoint, MWSS can be solved in time $O(|V(K)|^3)$ in $K$.
[**Case 2: $K$ contains a $C_5$ of type $2$ and no $C_5$ of type $1$ or $0$.**]{}
For adjacent vertices $u,v$ in $Z$ we say that the edge $uv$ is *red* if there exists a $P_4$ $h'$-$u$-$v$-$h''$ for some $h',h''\in H$. For every vertex $h$ in $H$ we define its *score*, $sc(h)$, as the number of red edges that contain a neighbor of $h$. Let $h$ be a vertex of maximum score in $H$. $$\label{xmax0}
\longbox{Suppose that $K\setminus N(h)$ contains a $C_5$ of type~$2$
$t$-$h_1$-$a$-$b$-$h_2$-$t$, with $h_1,h_2\in H$ and $a,b,t\in Z$.
Then $hh_1, hh_2\notin E(G)$, and $Z$ contains vertices
$y_1,z_1,y_2,z_2$ such that $y_1z_1, y_2z_2, hy_1,hy_2\in E(G)$,
$hz_1,hz_2, h_1y_1,h_1z_1, h_2y_2,h_2z_2\notin E(G)$, and, up to
symmetry, $\{y_1,y_2\}$ is complete to~$a$ and anticomplete to~$b$,
and $\{z_1,z_2\}$ is anticomplete to $a$, and $bz_2\in E(G)$.}$$ Proof: Clearly $h\notin\{h_1,h_2\}$. Note that $ab$ is a red edge. There must be a red edge $y_1z_1$ (with $y_1,z_1\in Z$) that is counted in $sc(h)$ and not in $sc(h_1)$, for otherwise we have $sc(h_1)\ge sc(h)+1$ (because of $ab$), which contradicts the choice of $h$. So $h_1$ has no neighbor in $\{y_1,z_1\}$. We may assume that $hy_1\in E(G)$. Let $h'$-$y_1$-$z_1$-$h''$ be a $P_4$ with $h',h''\in H$. If $hz_1\in E(G)$, then $hh'\notin E(G)$, for otherwise $\{c,d,h',h,z_1\}$ induces a bull; and similarly $hh''\notin
E(G)$; but then $\{h',y_1,h,z_1,h''\}$ induces a bull. Hence $hz_1\notin E(G)$. Clearly $a\notin\{y_1,z_1\}$. If $a$ has no neighbor in $\{y_1,z_1\}$, then $b$ has a neighbor in $\{y_1,z_1\}$, for otherwise $b$-$a$-$h_1$-$d$-$h$-$y_1$-$z_1$ is an induced $P_7$; and $b$ is adjacent to both $y_1,z_1$, for otherwise $\{c,d,h_1,a,b,y_1,z_1\}$ induces a $P_7$; but then $\{h,y_1,z_1,b,a\}$ induces a bull, a contradiction. So $a$ has a neighbor in $\{y_1,z_1\}$. If $a$ is adjacent to both $y_1,z_1$, then $\{h,y_1,z_1,a,h_1\}$ induces a bull. So $a$ has exactly one neighbor in $\{y_1,z_1\}$, which leads to the following two cases:\
— (i) $ay_1\in E(G)$ and $az_1\notin E(G)$. Then also $y_1b\notin
E(G)$, for otherwise $\{h,y_1,b,a,h_1\}$ induces a bull.\
— (ii) $az_1\in E(G)$ and $ay_1\notin E(G)$. Then also $z_1b\notin
E(G)$, for otherwise either $\{h_1,a,z_1,b,h_2\}$ induces a bull (if $z_1h_2\notin E(G)$), or $\{c,d,h_1,a,z_1,b,h_2\}$ induces a $G_2$ (if $z_1h_2\in E(G)$), which contradicts Lemma \[lem:nog12\]. Moreover, $y_1b\in E(G)$, for otherwise $c$-$d$-$h$-$y_1$-$z_1$-$a$-$b$ is an induced $P_7$.\
Similarly, there is a red edge $y_2z_2$ (with $y_2,z_2\in Z$) that is counted in $sc(h)$ and not in $sc(h_2)$, so $h_2$ has no neighbor in $\{y_2,z_2\}$. We may assume that $hy_2\in E(G)$, and by the same argument as above we have $hz_2\notin E(G)$ and either:\
— (iii) $by_2\in E(G)$, $bz_2\notin E(G)$, and $y_2a\notin E(G)$, or\
— (iv) $bz_2\in E(G)$, $by_2\notin E(G)$, $z_2a\notin E(G)$, and $y_2a\in E(G)$.\
Now if either (i) and (iii) occur, or (ii) and (iv) occur, then either $\{d,h,y_1,y_2,a\}$ induces a bull (if $y_1y_2\in E(G)$) or $\{h,y_1,y_2,a,b\}$ induces a $C_5$ of type $1$ (if $y_1y_2\notin
E(G)$), a contradiction. Therefore we may assume, up to symmetry, that (i) and (iv) occur. Thus (\[xmax0\]) holds.
Now we claim that: $$\label{xmax}
\longbox{If $K\setminus N(h)$ contains a $C_5$ of type~$2$, with the
same notation as in (\ref{xmax0}), then $K\setminus (N(h)\cup N(a))$
contains no $C_5$ of type~$2$.}$$ Proof: Let $y_1,z_1,y_2,z_2$ be vertices of $Z$ as in (\[xmax0\]). Suppose that $K\setminus (N(h)\cup N(a))$ contains a $C_5$ of type $2$ $t'$-$h_3$-$a'$-$b'$-$h_4$-$t'$, with $h_3,h_4\in H$ and $t',a',b'\in
Z$. By the analogue of (\[xmax0\]) there exist vertices $y_4,z_4$ in $Z$ such that $y_4z_4, hy_4\in E(G)$, $hz_4, h_4y_4,h_4z_4\notin
E(G)$, and, up to symmetry, $y_4a', z_4b'\in E(G)$ and $y_4b',z_4a'\notin E(G)$. We have $y_4a\notin E(G)$, for otherwise $c$-$d$-$h_4$-$b'$-$a'$-$y_4$-$a$ is an induced $P_7$; and $y_4y_1\notin E(G)$, for otherwise $\{d,h,y_4,y_1,a\}$ induces a bull; and $y_4b\notin E(G)$, for otherwise $\{h,y_1,a,b,y_4\}$ induces a $C_5$ of type $1$. Then $ba'\notin E(G)$, for otherwise $c$-$d$-$h$-$y_4$-$a'$-$b$-$a$ is an induced $P_7$. If $y_1b'\in
E(G)$, then $y_1z_4\in E(G)$, for otherwise $\{h,y_1,b',z_4,y_4\}$ induces a $C_5$ of type $1$, and $y_1h_4\in E(G)$, for otherwise $\{h,y_1,z_4,b',h_4\}$ induces a bull; but then $\{d,h_4,b',y_1,a\}$ induces a bull. So $y_1b'\notin E(G)$. Then $bb'\notin E(G)$, for otherwise $c$-$d$-$h$-$y_1$-$a$-$b$-$b'$ is an induced $P_7$. Then $a'y_1\in E(G)$, for otherwise $b$-$a$-$y_1$-$h$-$y_4$-$a'$-$b'$ is an induced $P_7$. Then $h_3y_1\notin E(G)$, for otherwise $\{d,h_3,a',y_1,a\}$ induces a bull, and $h_3b\notin E(G)$, for otherwise $\{h_3,a',y_1,a,b\}$ induces a $C_5$ of type $1$. But then $c$-$d$-$h_3$-$a'$-$y_1$-$a$-$b$ is an induced $P_7$, a contradiction. Thus (\[xmax\]) holds.
[**Case 3: $K$ contains a $C_5$ of type $0$ or $1$.**]{}
We will prove that: $$\label{case2}
\longbox{There is a vertex $x\in V(K)$ such that $K\setminus N(x)$
contains no $C_5$ of type~$0$ or~$1$.}$$
We first make some remarks about the $C_5$’s of type $1$ and make a few more claims. Let $H_1=\{h\in H\mid$ $h$ lies in a $C_5$ of type $1\}$. $$\label{a1234}
\longbox{Let $h\in H_1$, and let $C = h$-$p_1$-$p_2$-$p_3$-$p_4$-$h$
be any $C_5$ of type~$1$ that contains~$h$. Let $a$ be any vertex in
$Z$. Then either $N_C(a)$ is a stable set, or $N_C(a)=
\{p_1,p_2,p_3,p_4\}$.}$$ Proof: Suppose that $N_C(a)$ is not a stable set. If $a$ is adjacent to $h$ and one of $p_1,p_4$, say $ap_1\in E(G)$, then $ap_2\in E(G)$, for otherwise $\{d,h,a,p_1,p_2\}$ induces a bull, and $ap_3\notin
E(G)$, for otherwise $\{d,h,p_1,a,p_3\}$ induces a bull, and $ap_4\notin E(G)$, for otherwise $\{d,h,p_4,a,p_2\}$ induces a bull. But then $\{p_1,p_2,p_3,p_4,h,d,a\}$ induces a $G_1$, which contradicts Lemma \[lem:nog12\]. Now suppose that $ah\notin E(G)$. If $a$ is adjacent to $p_2$ and $p_3$, then $a$ also has a neighbor in $\{p_1,p_4\}$, for otherwise $\{p_1,p_2,a,p_3,p_4\}$ induces a bull. So in any case, up to symmetry, we may assume that $a$ is adjacent to $p_1$ and $p_2$. Then $ap_3\in E(G)$, for otherwise $\{h,p_1,a,p_2,p_3\}$ induces a bull, and $ap_4\in E(G)$, for otherwise $\{p_1,p_2,p_3,p_4,h,d,a\}$ induces a $G_2$, which contradicts Lemma \[lem:nog12\]. Thus (\[a1234\]) holds.
$$\label{match}
\longbox{Let $h\in H_1$, and let $C = h$-$t$-$u$-$v$-$w$-$h$ be any
$C_5$ of type~$1$ that contains~$h$. Suppose that
$C'=h'$-$t'$-$u'$-$v'$-$w'$-$h'$ is a $C_5$ of type~$1$ in which $h$
has no neighbor, with $h'\in H$. Then either $N_{C'}(t)=\{u',w'\}$
and $N_{C'}(w)=\{t',v'\}$, or vice-versa.}$$
Proof: Clearly $h\neq h'$. Let $Y=\{t,u,v,w\}$ and $Y'=\{t',u',v',w'\}$. Suppose that $\{t,w\}$ is anticomplete to $Y'$. Then $h'w\in E(G)$, for otherwise $w$-$h$-$d$-$h'$-$w'$-$v'$-$u'$ is an induced $P_7$, and similarly $h't\in E(G)$. If $h'u\in E(G)$, then $ut'\notin E(G)$ (by (\[a1234\]) applied to $C'$ and $u$), but then $\{h,t,u,h',t'\}$ induces a bull. So $h'u\notin E(G)$, and similarly $h'v\notin E(G)$. Then one of $u,v$, say $u$, has a neighbor in $Y'$, for otherwise $u$-$v$-$w$-$h'$-$w'$-$v'$-$u'$ is an induced $P_7$; moreover $u$ is complete to $Y'$, for otherwise $c,d,h,t,u$ plus two vertices from $Y'$ induce a $P_7$. Then $v$ has no neighbor $y'\in
Y'$, for otherwise $\{t,u,y',v,w\}$ induces a bull; but then $\{h',t',u',u,v\}$ induces a bull. So $\{t,w\}$ is not anticomplete to $Y'$, and we may assume up to symmetry that $w$ has a neighbor in $Y'$.\
We have $|N_{Y'}(w)|\ge 2$ and $N_{Y'}(w)\neq\{t',w'\}$, for otherwise $c,d,h,w$ plus three vertices from $Y'$ induce a $P_7$; and $w$ is not complete to $Y'$, for otherwise, by (\[a1234\]), $\{h,w,v',w',h'\}$ induces a bull. Hence, by (\[a1234\]) and up to symmetry, we have $N_{C'}(w)=\{t',v'\}$. Since $t'h\notin E(G)$, we have $t'v\notin
E(G)$, for otherwise, by (\[a1234\]), $\{h,w,v,t',h'\}$ induces a bull. If also $t$ has a neighbor in $Y'$, then by the same argument as with $w$ we have either (i) $N_{C'}(t)=\{u',w'\}$ or (ii) $N_{C'}(t)=\{t',v'\}$. In case (i) we obtain the desired result, so assume that (ii) holds. By (\[a1234\]), $t'u\notin E(G)$. Then $h'$ has a neighbor in $\{u,v\}$, for otherwise $c$-$d$-$h'$-$t'$-$w$-$v$-$u$ is an induced $P_7$; say $h'u\in E(G)$. Then $h'v\notin E(G)$, for otherwise $\{t,u,h',v,w\}$ induces a bull. Then $v'$ has neighbor in $\{u,v\}$, for otherwise $c$-$d$-$h'$-$u$-$v$-$w$-$v'$ is an induced $P_7$; and by (\[a1234\]) we have $N_C(v')=Y$. But then $\{h,t,v',u,h'\}$ induces a bull, a contradiction. So we may assume that $t$ has no neighbor in $Y'$. Then $th'\in E(G)$, for otherwise $t$-$h$-$d$-$h'$-$t'$-$u'$-$v'$ is an induced $P_7$; and $uh'\notin
E(G)$, for otherwise by (\[a1234\]), $N_C(h') = Y$, which would imply $N_{C'}(w) \neq \{v', t'\}$; and $vh'\in E(G)$, for otherwise $c$-$d$-$h'$-$t$-$u$-$v$-$w$ is an induced $P_7$. By (\[a1234\]) we have $|N_{Y'}(v)|\le 1$ and $N_{Y'}(v)\subset\{u',v'\}$. We have $vv'
\notin E(G)$, for otherwise $\{h, w, v, v', u'\}$ induces a bull, so we have $vu'\in E(G)$, for otherwise $c$-$d$-$h'$-$v$-$w$-$v'$-$u'$ is an induced $P_7$. Then $uu'\notin E(G)$ by (\[a1234\]) (since $wu'\notin E(G)$). But then $c$-$d$-$h$-$t$-$u$-$v$-$u'$ is an induced $P_7$. Thus (\[match\]) holds.
Now we deal with $C_5$’s of type $0$. Clearly any such $C_5$ lies in a component of $G[Z]$, and any such component has a neighbor in $H$ since $G$ is connected. $$\label{z0}
\longbox{Let $T$ be any component of $G[Z]$ that contains a $C_5$, let
$C$ be any $C_5$ in $T$, and let $h$ be any vertex in $H$ that has
a neighbor in $T$. Then $h$ is a $2$-neighbor of $C$.}$$ Proof: There is a shortest path $p_0$-$p_1$-$p_2$-$\cdots$-$p_r$ such that $p_0 = c$, $p_1 = d$, $p_2 = h$ and $p_r$ has a neighbor in $C$, and $r \geq 2$. By Lemma \[lem:c5n\], $p_r$ is either a $1$-neighbor, a $2$-neighbor or a $5$-neighbor of $C$. If $p_r$ is a $5$-neighbor, then $V(C)\cup\{p_r,p_{r-1}\}$ induces an umbrella, which contradicts Lemma \[lem:umbr\]. If $p_r$ is a $1$-neighbor of $C$, then $p_{r-2}, p_{r-1}, p_r$ and four vertices of $C$ induce a $P_7$. So $p_r$ is a $2$-neighbor of $C$. Now if $r\ge 3$, then $p_{r-3}, p_{r-2}, p_{r-1}, p_r$ and three vertices of $C$ induce a $P_7$. So $r=2$, and (\[z0\]) holds.
$$\label{z01}
\mbox{At most one component of $G[Z]$ contains a $C_5$.}$$
Proof: Suppose that two components $T$ and $T'$ of $G[Z]$ contain a $C_5$. Let $C$ a $C_5$ in $T$, with vertices $c_1, \ldots, c_5$ and edges $c_ic_{i+1}$ ($\bmod~5$), and let $C'$ a $C_5$ in $T'$, with vertices $c'_1, \ldots, c'_5$ and edges $c'_ic'_{i+1}$ ($\bmod~5$). Pick any $h \in H$ that has a neighbor in $T$, and pick any $h'$ in $H$ that has a neighbor in $T'$. By (\[z0\]) and Lemma \[lem:c5n\] we may assume that $N_C(h)=\{c_1,c_4\}$ and $N_{C'}(h')=\{c'_1,c'_4\}$. If $h$ has a neighbor in $T'$, then, by (\[z0\]) and Lemma \[lem:c5n\], we have $N_{C'}(h)
=\{c'_j,c'_{j+2}\}$ for some $j$. But then $c_3$-$c_2$-$c_1$-$h$-$c'_j$-$c'_{j-1}$-$c'_{j-2}$ is an induced $P_7$. So $h$ has no neighbor in $T'$, and similarly $h'$ has no neighbor in $T$. Then either $c_3$-$c_2$-$c_1$-$h$-$d$-$h'$-$c'_1$ or $c_3$-$c_2$-$c_1$-$h$-$h'$-$c'_1$-$c'_2$ is an induced $P_7$. So (\[z01\]) holds.
$$\label{hinz0}
\longbox{If a component $T$ of $G[Z]$ contains a $C_5$, and $h$ is any
vertex in $H$ that has a neighbor in $T$, then $K\setminus N(h)$ has
no $C_5$ of type~$0$ or~$1$.}$$
Proof: By (\[z0\]) and (\[z01\]), $K\setminus N(h)$ has no $C_5$ of type $0$. So suppose that there is a $C_5$ of type $1$ $C'=h'$-$t'$-$u'$-$v'$-$w'$-$h'$ (with $h'\in H$) in which $h$ has no neighbor. Let $C$ be a $C_5$ in $T$, with vertices $c_1,\ldots,c_5$ and edges $c_ic_{i+1}$ ($\bmod~5$). By (\[z0\]) and Lemma \[lem:c5n\], we may assume that $N_C(h)=\{c_1,c_4\}$. Let $C_h=h$-$c_1$-$c_2$-$c_3$-$c_4$-$h$; so $C_h$ is a $C_5$ of type $1$. By (\[match\]) and up to symmetry, we have $N_{C'}(c_1)=\{t',v'\}$ and $N_{C'}(c_4)=\{u',w'\}$, and $t',u',v',w'\in T$. Then $c_5$ has a neighbor in $\{u',v'\}$, for otherwise $\{c_1,v',u',c_4,c_5\}$ induces a $C_5$ of type $0$ in which $h'$ has at most one neighbor, contradicting (\[z0\]). If $c_5u' \in E(G)$, then $c_5v' \in E(G)$, for otherwise $\{h, c_4, c_5, u', v'\}$ induces a bull. If $c_5v' \in
E(G)$, then $c_5u' \in E(G)$, for otherwise $\{h, c_1, c_5, v', u'\}$ induces a bull. In both cases, by (\[a1234\]), $c_5$ is complete to $\{t', u', v', w'\}$. But then $\{h,c_1,t',c_5,w'\}$ induces a bull. Thus (\[hinz0\]) holds.
$$\label{u}
\longbox{Suppose that there is no $C_5$ of type~$0$. Pick any $h\in
H_1$, and suppose that there is a $C_5$ of type~$1$
$C'=h'$-$b_2$-$u$-$v$-$a_2$-$h'$ in which $h$ has no neighbor. Then
$K\setminus N(u)$ has no $C_5$ of type~$1$.}$$
Proof: Let $h$-$a_1$-$v'$-$u'$-$b_1$-$h$ be any $C_5$ of type $1$ that contains $h$. By (\[match\]), we may assume that $N_{C'}(a_1)=\{b_2,v\}$ and $N_{C'}(b_1)=\{a_2,u\}$. Let $C=h$-$a_1$-$v$-$u$-$b_1$-$h$; then $C$ is a $C_5$ of type $1$ in which $h'$ has no neighbor, so $h$ and $h'$ play symmetric roles. Let $C_{a_1}= h$-$a_1$-$b_2$-$u$-$b_1$-$h$ and $C_{a_2}=
h'$-$a_2$-$b_1$-$u$-$b_2$-$h'$. Suppose that there is a $C_5$ of type $1$ $C''=h''$-$t''$-$u''$-$v''$-$w''$-$h''$ in which $u$ has no neighbor. Let $X=\{a_1,b_1,a_2,b_2,u,v\}$ and $Y''=\{t'',u'',v'',w''\}$.\
We observe that $G[X\cup Y'']$ is bipartite: indeed in the opposite case, and since $K$ contains no $C_5$ of type $0$ and no $C_7$, there is a triangle in $G[X\cup Y'']$, and so there is either (i) a vertex $y''\in Y''$ with two adjacent neighbors in $X$, or (ii) a vertex $x\in X$ with two adjacent neighbors in $Y''$. In case (i), by (\[a1234\]) applied to $y''$ and the cycles $C, C', C_{a_1},
C_{a_2}$, we see that $y''$ is complete to $X$, which is not possible since $uy''\notin E(G)$. So suppose we have case (ii). By (\[a1234\]) we have $N_{C''}(x)=Y''$. Clearly $x\neq u$. Moreover, $x\notin\{b_1,b_2,v\}$, for otherwise $\{u,x,v'',w'',h''\}$ induces a bull. So, up to symmetry, $x=a_1$. By case (i) we have $v''b_2,w''b_2\notin E(G)$; but then $\{h'',w'',v'',a_1,b_2\}$ induces a bull. So $G[X\cup Y'']$ is bipartite. Let $A,B$ be a bipartition of $X\cup Y''$ in two stable sets. Up to symmetry we may assume that $A=\{a_1,a_2,u,u'',w''\}$ and $B=\{b_1,b_2,v,t'',v''\}$.\
Note that $h''$ has a neighbor in $C$, for otherwise (\[match\]) is contradicted (since $u$ has no neighbor in $\{t'',w''\}$), and similarly $h''$ has a neighbor in $C'$, in $C_{a_1}$ and in $C_{a_2}$. Suppose that $h''a_1\in E(G)$. Then $h''b_2\notin E(G)$, for otherwise $\{d,h'',a_1,b_2,u\}$ induces a bull, and $h''b_1\in E(G)$, for otherwise $c$-$d$-$h''$-$a_1$-$b_2$-$u$-$b_1$ is an induced $P_7$, and $h''a_2\notin E(G)$, for otherwise $\{d,h'',a_2,b_1,u\}$ induces a bull, and $h''h'\notin E(G)$, for otherwise $\{c,d,h'',h',a_2\}$ induces a bull. By (\[a1234\]), $h''$ is not adjacent to $v$. But then $h''$ has no neighbor in $C'$, a contradiction. So $h''a_1\notin
E(G)$, and similarly $h''a_2\notin E(G)$. So $h''\notin\{h,h'\}$; moreover $h''h\notin E(G)$, for otherwise $\{c,d,h'',h,a_1\}$ induces a bull, and similarly $h''h'\notin E(G)$. Then $h''$ has a neighbor in $\{b_1,b_2\}$, say $h''b_1\in E(G)$, for otherwise $h''$ has no neighbor in $C_{a_1}$; and then $h''b_2\in E(G)$, for otherwise $c$-$d$-$h''$-$b_1$-$u$-$b_2$-$a_1$ is an induced $P_7$, and $h''v\in
E(G)$, for otherwise $c$-$d$-$h''$-$b_1$-$a_2$-$v$-$a_1$ is an induced $P_7$. So $N_X(h'')=\{b_1,b_2,v\}$. By (\[a1234\]), $b_1,b_2$ and $v$ have no neighbor in $\{t'',w''\}$; and since $B$ is a stable set they are not adjacent to $v''$.\
Suppose that $b_1u''\in E(G)$. Then $a_1v''\notin E(G)$, for otherwise $c$-$d$-$h''$-$b_1$-$u''$-$v''$-$a_1$ is an induced $P_7$, and $hu''\notin E(G)$, for otherwise $\{d,h,u'',b_1,u\}$ induces a bull. Then $h$ has exactly one neighbor in $\{v'',w''\}$, for otherwise either $c$-$d$-$h$-$b_1$-$u''$-$v''$-$w''$ is an induced $P_7$ or $\{d,h,w'',v'',u''\}$ induces a bull. However, if $hw''\in
E(G)$, then $b_2u''\in E(G)$, for otherwise $u''$-$v''$-$w''$-$h$-$a_1$-$b_2$-$u$ is an induced $P_7$, and then $c$-$d$-$h$-$w''$-$v''$-$u''$-$b_2$ is an induced $P_7$; while if $hv''\in E(G)$, then $u''v\notin E(G)$, for otherwise $c$-$d$-$h$-$v''$-$u''$-$v$-$u$ is an induced $P_7$, and then $u''$-$v''$-$h$-$d$-$h''$-$v$-$u$ is an induced $P_7$, a contradiction. Hence $b_1u''\notin E(G)$ and, by symmetry, $b_1$ and $b_2$ have no neighbor in $Y''$.\
If $vu''\in E(G)$, then $hu''\in E(G)$, for otherwise $c$-$d$-$h$-$b_1$-$u$-$v$-$u''$ is an induced $P_7$, but then $c$-$d$-$h$-$u''$-$v$-$u$-$b_2$ is an induced $P_7$. So $vu''\notin
E(G)$, and so $v$ has no neighbor in $Y''$. Then $a_1v''\notin E(G)$, for otherwise $c$-$d$-$h''$-$v$-$a_1$-$v''$-$u''$ is an induced $P_7$; and $a_1t''\notin E(G)$, for otherwise $b_1$-$u$-$v$-$a_1$-$t''$-$u''$-$v''$ is an induced $P_7$. Hence, by symmetry, $a_1$ and $a_2$ have no neighbor in $Y''$. Now $h$ has a neighbor in $\{t'',u'',v'',w''\}$, for otherwise $a_1$-$h$-$d$-$h''$-$t''$-$u''$-$v''$ is an induced $P_7$. If $h$ has two adjacent neighbors in $Y''$, then $h$ is complete to $Y''$, for otherwise $d,h$ plus three consecutive vertices of $Y''$ induce a bull; but then $\{h'',t'',u'',h,a_1\}$ induces a bull. So we may assume that $N_{C''}(h)=\{t'',v''\}$, for otherwise $u,v,a_1,h$ and three consecutive vertices in $Y''$ induce a $P_7$. But then $u''$-$v''$-$h$-$d$-$h''$-$v$-$u$ is an induced $P_7$, a contradiction. Thus (\[u\]) holds.
Now, (\[case2\]) follows from (\[hinz0\]) and (\[u\]). This completes the proof in Case 3.
To conclude, we give the general outline of the algorithm to solve MWSS in $K$. For each type $q\in\{0,1,2\}$, we find a vertex $x$ such that $K \setminus N(x)$ contains no $C_5$ of type $q$. We then solve the MWSS in $K \setminus N(x)$ and in $K \setminus \{x\}$. Since every maximum weight stable set of $K$ either contains $x$ or not, the best of these two solutions is a solution for the MWSS in $K$. We repeat this until there are no more $C_5$’s of this type. More formally:
**(I)** Suppose that $K$ contains no $C_5$. If $K$ also contains no $C_7$, then $K$ is perfect, so we can solve the MWSS in $K$ by using the algorithm from [@Penev]. If $K$ contains a $C_7$, then MWSS can be solved in time $O(|K|^3)$ as explained in Case 1 of the proof.
**(II)** Suppose that $K$ contains a $C_5$ of type $2$ and no $C_5$ of type $0$ or $1$. Let $h$ be a vertex of maximum score as in Case 2 of the proof. Then MWSS in $K$ can be solved by successively solving the MWSS in (a) $G[K \setminus N(h)]$ and in (b) $G[K \setminus \{h\}]$.
Step (a) can be done as follows: If $G[K \setminus N(h)]$ contains no $C_5$, then we are in (I). If $G[K \setminus N(h)]$ contains a $C_5$ (of type $2$), then by (\[xmax\]) there is a vertex $a$ in this $C_5$ such that $G[K \setminus (N(h)\cup N(a))]$ contains no $C_5$. Hence we solve MWSS in (a1) $G[K \setminus (N(h)\cup N(a))]$ and in (a2) $G[K \setminus (N(h)\cup \{a\})]$. Step (a1) can be done in polynomial time by referring to (I). Step (a2) can be computed by recursively calling Step (a). The number of recursive calls is bounded by $|Z|$.
Step (b) can be computed by recursively calling [(II)]{}. After a number of calls there is no longer any $C_5$ of type $2$, so we are in (I). The number of recursive calls is bounded by $|H|$.
**(III)** Suppose that $K$ contains a $C_5$ of type $1$ and no $C_5$ of type $0$. Let $u$ be a vertex such that $K \setminus
N(u)$ has no $C_5$ of type $1$, as in Claim (\[u\]). Then MWSS in $K$ can be solved by successively solving the MWSS in (a) $G[K
\setminus N(u)]$ and in (b) $G[K \setminus \{u\}]$. Step (a) can be done in polynomial time by referring to (II) or (I). Step (b) can be computed by recursively calling [(III)]{}. After a number of calls there is no longer any $C_5$ of type $1$, so we are in [(II)]{} or (I). The number of recursive calls is bounded by $|K|$.
**(IV)** Suppose that $K$ contains a $C_5$ of type $0$. Let $T$ be the component of $G[Z]$ (unique by Claim (\[z01\])) that contains a $C_5$. Let $H_0=\{h\in H\mid$ $h$ has a neighbor in $T\}$. Let $h$ be any vertex in $H_0$. By (\[hinz0\]) we know that $G[K
\setminus N(h)]$ contains no $C_5$ of type $0$ or $1$. Then the MWSS in $K$ can be solved by successively solving the MWSS in (a) $G[K
\setminus N(h)]$ and in (b) $G[K \setminus \{h\}]$. Step (a) can be computed in polynomial time by calling [(II)]{} or (I). Step (b) can be computed by recursively calling [(IV)]{}. The number of recursive calls is equal to $|H_0|$. At the end of this step, the component $T$ becomes isolated because we have removed all vertices of $H_0$, but we still need to solve MWSS in $T$. This can be done as follows. Consider any vertex $h\in H_0$. By Claim (\[z0\]) every $C_5$ in $T$ contains exactly two vertices from $N(h_0)\cap V(T)$, and these two vertices are not adjacent. Hence MWSS can be solved in $T$ using the same technique as in (II) and the analogue of Claim (\[xmax\]).
The total number of recursive calls is in $O(n)$ since there are three different cycle types. For each computation of MWSS in $K$, we end up calling the algorithm in [@Penev] which runs in $O(n^6)$. Furthermore, at each step we need to compute the list of all the cycles of length $5$, which takes $O(n^5)$, but this is additive. We need to run all the previous steps on every connected component $K$ of the non-neighborhood of a fixed vertex of $V(G)$, there are at most $n$ such components. Finally, we repeat this for every vertex in $V(G)$, so the overall complexity of our algorithm is $O(n^{9})$. This completes the proof of Theorem \[thm:P7\].
One may wonder whether Claims (\[xmax\]) and (\[case2\]) could be subsumed by the following single claim: There is a vertex $x$ in $K$ such that $K\setminus N(x)$ contains no $C_5$ of any type. Here is an example showing that such a claim does not hold. Let $Z$ have six vertices $c_1,\ldots,c_5$ and $z$, such that $c_1,\ldots,c_5$ induce a $C_5$ with edges $c_ic_{i+1}$ ($i\bmod 5$), and $z$ has no neighbor in this $C_5$. Let $H$ have five vertices $h_1,\ldots,h_5$ such that for each $i$ we have $N_Z(h_i)=\{c_{i-1},c_{i+1},z\}$. Let $V(G)=\{c,d,h_1,\ldots,h_5,c_1,\ldots,c_5,z\}$. It is a routine matter to check that $G$ is $(P_7$, $K_3$)-free and that $K\setminus
N(x)$ contains a $C_5$ for every vertex $x\in K$.
$(S_{1,2,3}$, bull)-free graphs {#sec:S123}
===============================
[**Proof of Theorem \[thm:S123\].**]{} Let $G$ be a $(S_{1,2,3},\mbox{bull})$-free graph, and let $w$ be a weight function on the vertex set of $G$. We proceed as in the proof of Theorem \[thm:P7\]. By Theorem \[thm:LM\], we may assume that $G$ is prime, and by Lemmas \[lem:wheel\]—\[lem:nog12\], $G$ contains no wheel, no antiwheel, no umbrella, no $G_1$ and no $G_2$. Let $c$ be any vertex of $G$, and let $K$ be an arbitrary component of $G\setminus(\{c\}\cup N(c))$. If $K$ is perfect, we can use the algorithm from [@Penev] to find a maximum weight stable set in $K$. Therefore let us assume that $K$ is not perfect. By the Strong Perfect Graph Theorem [@CRST], $K$ contains an odd hole or an odd antihole. In fact $K$ contains no antihole of length at least $6$, for otherwise the union of such a subgraph and $c$ induces an antiwheel. So $K$ contains an odd hole. We observe that: $$\label{ncs}
\longbox{If $C$ is a hole of length at least $5$ in $G$, and $x,y$ are
vertices in $V(G)\setminus V(C)$ such that $xy\in E(G)$ and $x$ has no
neighbor in $C$, then $N_C(y)$ is a stable set.}$$ Proof: Let $C$ have length $\ell\ge 5$. If $y$ has two consecutive neighbors on $C$, then $y$ is complete to $V(C)$, for otherwise $x,y$ and three consecutive vertices of $C$ induce a bull; but then either $V(C)\cup\{y\}$ induces a $k$-wheel, with $k \geq 6$ (if $\ell\ge 6$) or $V(C)\cup\{x,y\}$ induces an umbrella (if $\ell=5$), a contradiction. Thus (\[ncs\]) holds.
Since $G$ is prime it is connected, so there is a neighbor $d$ of $c$ that has a neighbor in $K$. Let $H=N_K(d)$ and $Z=V(K)\setminus H$. Now we claim that: $$\label{c75}
\longbox{$K$ contains no odd hole of length at least $7$. Moreover,
every $C_5$ in $K$ contains one or two vertices of~$H$, and if it
contains two they are not adjacent.}$$ Proof: Suppose that $K$ contains a hole $C$ of odd length $\ell\ge 5$, with vertices $c_1, \ldots, c_\ell$ in order. Since $G$ is prime, it is connected, so there exists a path $p_0$-$p_1$-$\cdots$-$p_k$ with $p_0\in V(C)$, $p_k=c$, and $k\ge 2$, and we choose a shortest such path, so $p_1,\ldots,p_{k}\notin V(C)$ and $p_2,\ldots, p_k$ have no neighbor in $C$. By (\[ncs\]) applied to $p_1,p_2$ and $C$, we know that $N_C(p_1)$ is a stable set. Since $\ell$ is odd, it follows that there is an integer $i\in\{1,\ldots,\ell\}$ such that $p_1c_i\in E(G)$ and $p_1c_{i+1}, p_1c_{i+2}\notin E(G)$, and $p_1c_{i-1}\notin E(G)$, say $i=1$. If $\ell\ge 7$, then $p_1c_4\in E(G)$, for otherwise $\{c_1,c_\ell, p_1,p_2,c_2,c_3,c_4\}$ induces an $S_{1,2,3}$; and then $p_1c_5\notin E(G)$. Then $p_1c_6\in E(G)$, for otherwise $\{p_1,p_2,c_1,c_2,c_4,c_5,c_6\}$ induces an $S_{1,2,3}$. But then $\{p_1,p_2,c_6,c_5,c_1,c_2,c_3\}$ induces an $S_{1,2,3}$, a contradiction. This proves the first sentence of (\[c75\]). Now $\ell=5$. If $k\ge 3$, then $\{c_1,c_5,c_2,c_3,p_1,p_2,p_3\}$ induces an $S_{1,2,3}$. So $k=2$, and so $p_1=d$, and we already know that $N_C(d)$ is equal to $\{c_1\}$ or $\{c_1,c_4\}$. This proves the second sentence of (\[c75\]). Thus (\[c75\]) holds.
For $q\in\{1,2\}$ we say that a $C_5$ in $K$ is *of type $q$* if it contains exactly $q$ vertices from $H$. By (\[c75\]) and the Strong Perfect Graph Theorem, $K$ contains a $C_5$, and every $C_5$ in $K$ is of type $1$ or $2$.
For adjacent vertices $u,v$ in $Z$ we say that the edge $uv$ is *red* if there exists a $P_4$ $h'$-$u$-$v$-$h''$ for some $h',h''\in H$. For every vertex $h$ in $H$ we define its *score*, $sc(h)$, as the number of red edges that contain a neighbor of $h$.
We choose a vertex $h_0\in H$ as follows: if there exists a red edge, let $h_0$ be a vertex of maximum score in $H$; if there is no red edge, let $h_0$ be any vertex in $H$ that has a neighbor in $Z$. We claim that: $$\label{h0}
\mbox{$K\setminus N(h_0)$ contains no $C_5$.}$$ Proof: Suppose on the contrary that $K\setminus N(h_0)$ contains a $C_5$ $C$. First suppose that $C$ is of type $1$. So $C=h$-$t$-$u$-$v$-$w$-$h$ for some $h\in H$ and $t,u,v,w\in Z$. Let $z$ be any neighbor of $h_0$ in $Z$. By (\[ncs\]) applied to $h_0,z$ and $C$, we know that $N_{C}(z)$ is a stable set. Suppose that $zh\in E(G)$. Then $zt, zw\notin E(G)$. Then $z$ has a neighbor in $\{u,v\}$, for otherwise $\{h,z,d,c,t,u,v\}$ induces an $S_{1,2,3}$. So, up to symmetry, $N_{C}(z)=\{h,u\}$. But then $\{u,t,v,w,z,h_0,d\}$ induces an $S_{1,2,3}$. Therefore $z$ is not adjacent to $h$. Now $z$ has a neighbor in $\{t,u\}$, for otherwise $\{d,c,h_0,z,h,t,u\}$ induces an $S_{1,2,3}$, and similarly $z$ has a neighbor in $\{v,w\}$. Since $N_{C}(z)$ is a stable set, we may assume that $N_{C}(z)$ consists of $t$ plus one of $v,w$. Then $\{z,t,v,w,h_0,d,c\}$ induces an $S_{1,2,3}$, a contradiction.
Now supppose that $C$ is of type $2$. So $C=t$-$h_1$-$a$-$b$-$h_2$-$t$, for some $h_1,h_2\in H$ and $a,b,t\in
Z$. Clearly $h_0\notin\{h_1,h_2\}$. Note that $ab$ is a red edge. There must be a red edge $y_1z_1$ (with $y_1,z_1\in Z$) that is counted in $sc(h_0)$ and not in $sc(h_1)$, for otherwise we have $sc(h_1)\ge sc(h_0)+1$ (because of $ab$), which contradicts the choice of $h_0$. So $h_1$ has no neighbor in $\{y_1,z_1\}$. We may assume that $h_0y_1\in E(G)$. Let $h'$-$y_1$-$z_1$-$h''$ be a $P_4$ with $h',h''\in H$. If $h_0z_1\in E(G)$, then $h_0h'\notin E(G)$, for otherwise $\{c,d,h',h_0,z_1\}$ induces a bull; and similarly $h_0h''\notin E(G)$; but then $\{h',y_1,h_0,z_1,h''\}$ induces a bull. Hence $h_0z_1\notin E(G)$. Clearly $a\notin\{y_1,z_1\}$. If $a$ has no neighbor in $\{y_1,z_1\}$, then $\{d,c,h_1,a,h_0,y_1,z_1\}$ induces an $S_{1,2,3}$; while if $a$ is complete to $\{y_1,z_1\}$, then $\{h_0,y_1,z_1,a,h_1\}$ induces a bull. Hence $a$ has exactly one neighbor in $\{y_1,z_1\}$.\
Suppose that $a$ is adjacent to $y_1$ and not to $z_1$. Then $y_1b\notin E(G)$, for otherwise $\{h_0,y_1,b,a,$ $h_1\}$ induces a bull; and $z_1b\in E(G)$, for otherwise $\{y_1,z_1,a,b,h_0,d,$ $c\}$ induces an $S_{1,2,3}$. If $ty_1\in E(G)$, then $h_2y_1\notin E(G)$, for otherwise $\{a,y_1,t,h_2,$ $d\}$ induces a bull; but then $\{y_1,t,z_1,b,h_0,d,c\}$ induces an $S_{1,2,3}$. Hence $ty_1\notin
E(G)$. Then $tz_1\notin E(G)$, for otherwise $\{y_1,a,z_1,t,h_0,d,c\}$ induces an $S_{1,2,3}$. But now $\{h_1,t,d,c,a,y_1,z_1\}$ induces an $S_{1,2,3}$, a contradiction.\
Therefore $a$ is adjacent to $z_1$ and not to $y_1$. If $z_1b\in
E(G)$, then $z_1h_2\in E(G)$, for otherwise $\{h_1,a,z_1,b,h_2\}$ induces a bull; but then $\{c,d,h_1,a,b,z_1,h_2\}$ induces a $G_2$. Hence $z_1b\notin E(G)$. Then $y_1b\in E(G)$, for otherwise $\{a,b,z_1,y_1,h_1,d,c\}$ induces an $S_{1,2,3}$; and $ty_1\in E(G)$, for otherwise $\{h_1,t,d,c,a,b,y_1\}$ induces an $S_{1,2,3}$. But then $\{y_1,t,b,a,h_0,d,c\}$ induces an $S_{1,2,3}$. Thus (\[h0\]) holds.
Finally we give the outline of the algorithm to solve MWSS in $K$.
- Suppose that $K$ contains no $C_5$. Then $K$ is perfect, so we can compute the MWSS in $K$ by using the algorithm from [@Penev].
- Suppose that $K$ contains a $C_5$. Let $h_0$ be a vertex of $H$ as in (\[h0\]). Then MWSS in $K$ can be solved by successively solving MWSS in (a) $K\setminus{N(h_0)}$ and in (b) $K
\setminus \{h_0\}$. Step (a) can be done in polynomial time by referring to (I) since $K\setminus{N(h_0)}$ is perfect by (\[c75\]) and (\[h0\]). Step (b) can be computed by recursively calling (II). The number of recursive calls is bounded by $|H|$.
For each computation of MWSS in $K$, we end up calling the algorithm in [@Penev] which runs in $O(n^6)$. Furthermore, at each step we need to compute the list of all the cycles of length $5$, which takes $O(n^5)$, but this is additive. We need to run all the previous steps on every connected component $K$ of the non-neighborhood of a fixed vertex of $V(G)$, there are at most $n$ such components. Finally, we repeat this for every vertex in $V(G)$, so the overall complexity of our algorithm is $O(n^{8})$. This completes the proof of Theorem \[thm:S123\].
Concluding remarks
==================
The technique used here is essentially that which was developed by Brandstädt and Mosca in [@BM]. The new results that it has enabled us to establish are generalizations of results [@KM2; @MP] that were obtained earlier using different techniques. It is not clear to us if the same ideas can be used to solve MWSS in other classes of graphs, such as ($S_{2,2,2}$, bull)-free graphs or ($P_8$, bull)-free graphs or similar classes (not necessarily subclasses of bull-free graphs). These may be interesting open problems.
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[^1]: CNRS, Laboratoire G-SCOP, University of Grenoble-Alpes, France.
[^2]: Laboratoire G-SCOP, University of Grenoble-Alpes, France. The authors are partially supported by ANR project STINT (reference ANR-13-BS02-0007).
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---
abstract: 'We demonstrate the efficient transverse compression of a 12.5 MeV/c muon beam stopped in a helium gas target featuring a vertical density gradient and strong crossed electric and magnetic fields. The vertical spread of the muon stopping distribution was reduced from 10 to 0.7 mm within 3.5 . The simulation including proper cross sections for low-energy $\mu^+$ - $\text{He}$ elastic collisions and the charge exchange reaction $ \mu^+ + \text {He} \longleftrightarrow \text{Mu} + \text{He}^+ $ describes the measurements well. By combining the transverse compression stage with a previously demonstrated longitudinal compression stage, we can improve the phase space density of a $\mu^+ $ beam by a factor of $ 10^{10} $ with $ 10^{-3} $ efficiency.'
author:
- 'A. Antognini'
- 'N. J. Ayres'
- 'I. Belosevic'
- 'V. Bondar'
- 'A. Eggenberger'
- 'M. Hildebrandt'
- 'R. Iwai'
- 'D. M. Kaplan'
- 'K. S. Khaw'
- 'K. Kirch'
- 'A. Knecht'
- 'A. Papa'
- 'C. Petitjean'
- 'T. J. Phillips'
- 'F. M. Piegsa'
- 'N. Ritjoho'
- 'A. Stoykov'
- 'D. Taqqu'
- 'G. Wichmann'
title: 'Demonstration of Muon-Beam Transverse Phase-Space Compression'
---
Next generation precision experiments with muons and muonium atoms [@Gorringe2015], such as muon $ g-2 $ and EDM measurements [@Iinuma2011; @Adelmann2010; @Crivellin2018], muonium spectroscopy [@Cr], and muonium gravity measurements [@Kirch2014a; @Kaplan2018], require high-intensity muon beams at low energy with small transverse size and energy spread. The standard surface muon beams currently available do not fulfill these requirements. To improve the quality of the muon beam, phase space cooling techniques are needed. However, conventional methods, such as stochastic cooling [@VanderMeer1985] and electron cooling [@Budker1978], are not applicable due to the short muon lifetime of $ 2.2 $ . Alternative beam cooling techniques based on muon energy moderation in materials have been developed [@Muhlbauer1999; @Morenzoni1994], however, they typically suffer from low cooling efficiencies ($<10^{-4} $).
At the Paul Scherrer Institute, we are developing a novel device (muCool) that produces a high-quality muon beam, reducing the full (transverse and longitudinal) phase space of a standard $\mu^{+}$ beam by 10 orders of magnitude with $10^{-3}$ efficiency [@Taqqu2006]. The whole device is placed inside a $ 5 $ magnetic field, pointing in the $ +z $-direction, as sketched in Fig. \[scheme\]. First, a surface muon beam propagating in the $-z $-direction is stopped in a few mbar of helium gas at cryogenic temperature, reducing the muon energy to the eV range. The stopped muons are then guided into a sub-mm spot using a combination of strong electric and magnetic fields and gas density gradients in three stages.
[ 0= =0=0 0]{}
In the first stage (transverse compression), the electric field is perpendicular to the applied magnetic field and at $ 45 $ with respect to the $ x $-axis: $ \vec{E} = (E_x, E_y, 0) $, with $ E_x=E_y\approx 1$ . In vacuum, applying such crossed electric and magnetic fields would prompt the stopped muons to drift in the [$ \hat{E} \times \hat{B}$]{}-direction, performing cycloidal motion with frequency $\omega=eB/m_{\mu}$ (cyclotron frequency), where $ m_{\mu} $ is the muon mass.
However, in the muCool device, the muons also collide with He gas atoms with an average frequency $ \nu_{c} $, which depends on the gas density, elastic $ \mu^+ $-He cross section, and $\mu^+$ - $\text{He}$ relative velocity. These collisions lead to muon energy loss and change of direction. Thus, the muon motion is modified compared to that in vacuum, so that the muon drift velocity also acquires a component in the $ \hat{E} $-direction, making muons drift at an angle $ \theta $ relative to the [$ \hat{E} \times \hat{B}$]{}-direction. The average drift angle $ \theta $ is proportional to the collision frequency $ \nu_{c} $ [@Heylen1980]: $$\label{eq:drift_angle}
\tan\theta= \frac{\nu_c}{\omega}.$$ The reason for such behavior is as follows: for $ \nu_{c}>\omega $, the muon will not complete the full period of the cycloidal motion before the next collision (blue trajectory in Fig. \[scheme\]), resulting in a large drift angle $ \theta $. If $ \nu_{c}<\omega $, the muon will perform several periods of cycloidal motion between collisions, resulting in a smaller drift angle $ \theta $ (green trajectory in Fig. \[scheme\]).
We use this feature to manipulate the direction of the muon drift by changing the collision frequency. The most straightforward way to modify the collision frequency is by changing the gas density. In the transverse compression stage, this is done by having the bottom of the apparatus at $ 6 $ K and the top at $ 19 $ K, thus creating a temperature gradient in the vertical ($ y $-) direction. The gas pressure is chosen so that $ \frac{\nu_c}{\omega}=1 $ at $ y=0 $. Muons at different $ y $-positions (at fixed $ x $-position) experience different densities, resulting in different drift directions: a muon stopped at the bottom of the target (higher density) experiences more collisions with He atoms ($ \frac{\nu_c}{\omega}>1 $) and will thus drift predominately in the $ \hat{E} $-direction (large $ \theta $, upwards), while a muon stopped in the top part (lower density) collides less frequently with He atoms ($ \frac{\nu_c}{\omega}<1 $), resulting in drift predominately in the [$ \hat{E} \times \hat{B}$]{}-direction (small $ \theta $, downwards). The net result is that muons stopped at different $ y $-positions converge towards $ y=0 $, while simultaneously drifting in the $ +x $-direction. Hence, the muon beam spread is reduced in the $ y $-direction to sub-mm size (transverse compression).
After that, the muons are transported to the second stage, which is at room temperature. In this stage, the electric field has a component parallel to the magnetic field and pointing towards $ z=0 $, leading to a muon drift towards the center of the target, thus reducing muon spread in the $ z $-direction (longitudinal compression). The vertical component of the electric field ($ E_y $) at this low density causes an additional drift in the [$ \hat{E} \times \hat{B}$]{}-direction ($ +x $-direction in this case), thus transporting the muons towards the final compression stage. From there, the sub-mm muon beam can be extracted though a small orifice into the vacuum, re-accelerated with pulsed electric fields to keV energies, and extracted out of the magnetic field (see Fig. \[scheme\]). The efficient longitudinal compression of a muon beam, together with an [$ \hat{E} \times \hat{B}$]{}-drift, has already been demonstrated [@Bao2014; @Belosevic2019].
This Letter presents the first demonstration of the muon transverse compression stage of the muCool device. For this demonstration, about $2\cdot10^4~\mu^+$/s at 12.5 MeV/c were injected into the $ 25 $-cm-long target placed in the center of a 5-tesla solenoid. Before entering the target, the muons traversed a 55--thick entrance detector, several thin foils, and a copper aperture, defining the beam injection position.
The gas volume of the transverse target was enclosed by a Kapton foil, folded around triangular PVC end-caps. Kapton and PVC are both electrical and thermal insulators, and are thus capable of sustaining high voltage, while keeping the heat transport between top and bottom walls small. The large thermal conductivity of the single crystal sapphire plates glued to the top and the bottom target walls assured homogeneous temperatures of these walls. The required temperature gradient from 6 to 19 K was produced by heating the top wall with 500 mW power and thermally connecting the bottom wall to a copper cold finger [@Wichmann2016a].
The Kapton foil enclosing the gas volume was lined with metallic electrodes extending along the $ z $-direction. The 45 electric field was defined by applying appropriate voltages to several of these electrodes (see Fig. \[scheme\]), which were connected to the rest of the electrodes via voltage dividers.
Several detectors were placed around the target, A1...A3 in Fig. \[scheme\], to monitor the muon movement by detecting the $\mu^+$-decay positrons. The detectors consisted of plastic scintillator bars with a groove inside which a wavelength-shifting fiber was glued. These 2 m-long wavelength-shifting fibers transported the scintillation light from the cryogenic temperatures in vacuum to room temperature and air, where they were read out by $1.3 \times 1.3$ silicon photomultipliers. The scintillators were mounted inside a collimator to improve their position resolution.
0= =0=0
0
The probability of detecting the decay positrons vs. the muon decay position is shown in Fig. \[fig:trans\_detector\_acc\] for the detectors A1 and A2L. By recording time spectra for each detector (with $ t=0 $ given by the muon entrance detector), we can observe indirectly the muon drift in the transverse target. To demonstrate transverse compression we also need to show that the muon beam size decreases during the drift. This is achieved by comparing the time spectra obtained when the temperature gradient ($ 6 $–$ 19 $ K at $ 8.6 $ mbar) is applied, to the time spectra with negligible temperature gradient ($ 4 $–$ 6 $ K at $ 3.5 $ mbar): in the first case both drift and compression are expected; in the second only drift (“pure drift”). We further increased the contrast between “compression” and “pure-drift” measurements by injecting the muons through either “top hole” or “bottom hole” apertures at $y=\pm4.5$ mm (see Fig. \[scheme\]). If we were to inject the muons through a single large aperture, the majority of the muons would be stopped close to $y=0$, and would thus drift straight towards the tip for both “compression” and “pure drift.” Contrarily, with either of the two smaller apertures, we target only a narrow vertical region of the gas, with distinct density conditions for “compression” and “pure drift.”
0= =0=0
0
Geant4 [@Agostinelli2003] simulations of the muon trajectories for “top-hole” and “bottom-hole” injections, and for “compression” and “pure drift,” are shown in Fig. \[fig:trajectories\], with an applied high voltage of $ \mathrm{HV}=5.0 $ kV. The simulation included the most relevant low-energy $\mu^+$ - $\text{He}$ interactions: elastic collisions and charge exchange. The cross sections for these processes were appropriately scaled from the proton-He cross sections [@Belosevic2020; @Taqqu2006]. Simulations show that for the “compression” case, muons injected through both apertures reach the tip of the target efficiently, while in the “pure drift” case, most of the muons stop in the target walls before reaching the tip.
The measured time spectra of A1 and A2L under “compression” conditions are presented in Fig. \[fig:time spectra\] (red dots), for both “top-” and “bottom-hole” injection. Note that the time spectra were corrected for muon decay by multiplying the counts by $\exp(t/2.2~\mathrm{\mu s})$. The A1 counts increase with time, for both injection positions. This indicates that the muons were gradually moving towards the tip of the target, [i.e.]{}, towards the A1 acceptance region (see Fig. \[fig:trans\_detector\_acc\]). The A2L counts first increase, then decrease with time, suggesting that the muons first entered, then exited the acceptance region of the detector. After a certain time, the number of counts stays constant in both detectors, implying that the muons reached the target walls. For “bottom-hole” injection, the times at which muons reach the detector acceptance regions are delayed by up to 1500 ns compared to the “top-hole” injection. Indeed, muons injected through the “bottom hole” travel through a region of higher gas density compared to that for “top-hole” injection, leading to a slower drift. All these features are consistent with the simulation results of Fig. \[fig:trajectories\] (left).
The “pure-drift” time spectra for “top-hole” injection (black points in Fig. \[fig:time spectra\] (top row)) differ significantly from the “compression” time spectra. The A1 counts remain very low at all times, suggesting that the muons never reached the tip of the target. The A2L counts never decrease after reaching the maximum, implying that muons did not manage to “fly by” the detector, but they stopped in the target walls before leaving the acceptance region of the detector. This is consistent with the simulated trajectories of Fig. \[fig:trajectories\] (right). However, for “bottom-hole” injection, measured “pure-drift” time spectra (black points, Fig. \[fig:time spectra\] (bottom row)) are almost identical to the “compression” time spectra (red points). This is mainly due to detector resolution, which is worse for the “bottom-hole” measurements because muons drift at larger distances from the detectors.
0= =0=0
0
To better compare the measurements with the Geant4 simulations, the muon trajectories of Fig. \[fig:trajectories\] were folded with the detector acceptance of Fig. \[fig:trans\_detector\_acc\] to produce the corresponding time spectra. These time spectra (dashed curves in Fig. \[fig:time spectra\]) were then fitted to the measurements using two fit parameters per time spectrum: a normalization, to account for the uncertainties in the detection and stopping efficiencies, and a flat background, to account for beam-related stops in the walls. To improve the fits, the detector positions were shifted by up to $ 1 $ in the simulation compared to the design value, to account for possible shifts of various parts of the setup during the cool-down, and mechanical uncertainty. Relatively large reduced chi-squareds, especially for the “pure-drift” measurements, point to systematic discrepancies between the simulation and the measurements.
One possible explanation of the discrepancy could be a misalignment (tilt) between the target and the magnetic field axes. Even a small misalignment would shift the initial beam position, particularly affecting the “pure-drift” measurements, as the time and position at which muons crash into the target walls would shift accordingly. In the “compression” measurements, such misalignment is less problematic, because the temperature gradient ensures that muons reach the tip of the target, regardless of their initial position.
The effects of such misalignments were investigated by simulating time spectra for various tilts between the target and the magnetic field axes, and fitting them to the data. The best fit for “compression” and “pure drift” simultaneously was obtained by rotating the target axis from the $(0,0,1)$ direction to $(0.018,0.007,0.9998)$ for the “top hole,” and to $(0.019, 0.005, 0.9998)$ for the “bottom hole,” leading to average shifts of the initial muon stopping distribution by up to $ 3.5 $ in the $ xy $-plane.
Including such a tilt in the simulation improves the fit significantly (solid curves in Fig. \[fig:time spectra\]). Even better agreement is achieved by allowing different tilts for the “compression” and “pure-drift” measurements, which likely points to a mismatch between the assumed and actual initial muon beam momentum distribution. Further possible improvements of the fit would be simultaneous fine-tuning of the tilt, the muon momentum distribution, and the detector positions. However, little further insight would be gained from such a time-consuming optimization, since the simulations presented here already reproduce well the main features of the measured time spectra. Hence, the measurements demonstrate the transverse compression of a muon beam.
Next, we investigate the muon drift versus its energy by varying the electric field strength. As explained above, the muon drift angle $ \theta $ is proportional to the average $\mu^+$ - $\text{He}$ collision frequency $\nu_{c}$, which can be written as $$\nu_{c}=N\sigma_{MT}(E_{CM})\left|\vec{v}_{r}\right|,$$ where $N$ is the helium number density, $\sigma_{MT}(E_{CM})$ is the energy-dependent $\mu^+$ - $\text{He}$ momentum transfer cross section and $\vec{v}_{r}$ is the $\mu^+$ - $\text{He}$ relative velocity.
0= =0=0
0
0= =0=0
0
0= =0=0
0
For muon energies $ \lesssim 1$ eV, $\sigma_{MT}$ is proportional to $1/\sqrt{E_{CM}}\propto 1/\left|\vec{v}_{r}\right|$ [@Mason1979b], so that the collision frequency is independent of the muon energy. For energies larger than $ 1 $ eV, the product $\sigma_{MT}\left|\vec{v}_{r}\right|$ decreases with energy, as shown in Fig. \[fig:cs\*sqrtE vs E\]. Hence, the collision frequency and the muon drift angle $\theta$ decrease with increasing muon energy. Since the muon energy in the He gas increases with electric field strength, the muon drift direction approaches the [$ \hat{E} \times \hat{B}$]{}-direction with increasing electric field. The simulated muon trajectories of Fig. \[fig:trans\_newtop\_comp\_sim\_scan\] for various applied HV confirm this behavior.
The dependence of the muon motion on the electric field strength was studied experimentally for various beam injections and density conditions. In Fig. \[fig:trans\_newtop\_comp\_data\_scan\] we present the measurements for “top-hole” and “compression” configuration, which illustrate all relevant features. We can see that the maximum number of counts in A1 decreases with decreasing HV, indicating that increasingly fewer muons reach the tip of the target. Moreover, for smaller HV the drift is slower, as visible from the shift of the A2L time spectrum maximum towards later times. This is consistent with the simulated trajectories of Fig. \[fig:trans\_newtop\_comp\_sim\_scan\].
The simulated time spectra for each HV and for each detector were fitted independently to the measurements with two free parameters: normalization and flat background. It was not possible to perform a simultaneous fit for all HV values. We indeed observed that the normalization giving the best fit depends systematically on the HV. This might be caused by the lack of a precise definition of the electric field at the target tip, which would affect most significantly the measurements with strong electric fields, where muons actually reach the tip of the target. A mismatch of the initial muon stop distribution between simulations and experiment would also contribute to this systematic effect.
Still, the simulation reproduces correctly the muon drift velocity, which is given by the energy-dependent $\mu^+$ - $\text{He}$ elastic cross section. Indeed, the maxima of the time spectra, occurring when muons arrive in the detector acceptance region, are consistent between measurements and simulations.
To summarize, this Letter presents the first demonstration of transverse compression of a muon beam with the muCool device. One critical aspect of this demonstration was distinguishing between a simple muon drift versus drift with simultaneous reduction of the beam transverse size. Such distinction was accomplished by performing the measurements with and without a vertical temperature gradient, which is needed for the transverse compression, and by injecting the muon beam into different density regions of the target. The muon motion corresponding to realistic experimental conditions was simulated using the Geant4 package including custom low-energy ($ <1 $ keV) muon processes [@Belosevic2020]. Very good agreement between the simulations and measurements was achieved after accounting for small target and detector misalignments. According to the simulations, under these experimental conditions, a muon beam with initial diameter of 10 mm and 830 keV energy with 17 keV spread is transformed within 3.5 into a beam of 0.7 mm size (FWHM) in the $ y $-direction and 5 eV energy with 5 eV spread.
Besides demonstrating the transverse compression, the dependence of the muon drift on the applied electric field strength was explored experimentally and found to be in fair agreement with Geant4 simulations. This validates our modeling of the low-energy $\mu^+$ - $\text{He}$ elastic collisions.
Connecting the transverse compression stage demonstrated in this Letter to the previously demonstrated longitudinal compression stage [@Bao2014; @Belosevic2019] will allow us to realize a high-brightness low-energy muon beam as proposed in [@Taqqu2006], with a phase space compression factor of $ 10^{10} $ and $ 10^{-3} $ efficiency relative to the input beam.
The experimental work was performed at the $ \pi E 1 $ beamline at the PSI proton accelerator HIPA. We thank the machine and beam line groups for providing excellent conditions. We gratefully acknowledge the outstanding support received from the workshops and support groups at ETH Zurich and PSI. Furthermore, we thank F. Kottmann, M. Horisberger, U. Greuter, R. Scheuermann, T. Prokscha, D. Reggiani, K. Deiters, T. Rauber, and F. Barchetti for their help. This work was supported by the SNF grants No. 200020\_159754 and 200020\_172639.
[20]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty [****, ()](https://doi.org/10.1016/j.ppnp.2015.06.001) [****, ()](https://doi.org/10.1088/1742-6596/295/1/012032) [****, ()](https://doi.org/10.1088/0954-3899/37/8/085001) [****, ()](https://doi.org/10.1103/PhysRevD.98.113002) [****, ()](https://doi.org/10.1007/s10751-018-1525-z) [****, ()](https://doi.org/10.1142/S2010194514602580), [****, ()](https://doi.org/10.3390/atoms6020017) [****, ()](https://doi.org/10.1103/RevModPhys.57.689) [****, ()](https://doi.org/10.1070/PU1978v021n04ABEH005537) [****, ()](https://doi.org/10.1023/A:1012624501134) [****, ()](https://doi.org/10.1103/PhysRevLett.72.2793) [****, ()](https://doi.org/10.1103/PhysRevLett.97.194801) [****, ()](https://doi.org/10.1049/ip-a-1.1980.003) [****, ()](https://doi.org/10.1103/PhysRevLett.112.224801) [****, ()](https://doi.org/10.1140/epjc/s10052-019-6932-z) [****, ()](https://doi.org/10.1016/j.nima.2016.01.018) [****, ()](https://doi.org/10.1016/S0168-9002(03)01368-8) **, [Ph.D. thesis](https://doi.org/10.3929/ETHZ-B-000402802), () [**** ()](http://www-cfadc.phy.ornl.gov/elastic/home.html) [****, ()](https://doi.org/10.1088/0022-3700/12/24/023)
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---
abstract: 'We compute the distribution of the decay rates (also referred to as residues) of the eigenstates of a disordered slab from a numerical model. From the results of the numerical simulations, we are able to find simple analytical formulae that describe those results well. This is possible for samples both in the diffusive and in the localised regime. As example of a possible application, we investigate the lasing threshold of random lasers.'
author:
- 'M. Patra'
bibliography:
- 'paper.bib'
title: Decay Rate Distributions of Disordered Slabs and Application to Random Lasers
---
Introduction
============
A very successful approach to describe disordered materials is supplied by random-matrix theory, see Refs. for reviews. While one can put the beginning of random-matrix theory at Wigner’s surmise for describing the scattering spectra of heavy atomic nuclei,[@wigner:56a] its theoretical foundations were laid only much later.[@bohigas:84a] It was very successfully applied to electronic transport in disordered wires and mesoscopic quantum dots, and recently these methods have been adopted to model (quantum) transport of optical radiation in media with spatially fluctuating dielectric constant.[@beenakker:98a; @beenakker:98b; @patra:98a; @patra:99a]
In the theoretical treatment of disordered materials, two particular geometries are of special importance, namely the disordered slab and the chaotic cavity (see Fig. \[figGeometrien\]). The principal difference between the geometries is easily explained: A chaotic cavity is an object in which the dynamics is chaotic due to the shape of the cavity or due to scatterers placed at random positions. The size of the opening is small compared to the total surface area of the cavity. Particles (electrons, photons) are then “trapped” inside the structure for a time that is long enough to ergodically explore the entire cavity. In a disordered slab, particles cannot be trapped that efficiently. They can no longer explore the entire volume ergodically but they still stay long enough to explore the entire cross-section of the sample, thus still making a random-matrix description possible. In order to call this geometry a “wire” or a “waveguide” its length should be much larger than its width. To be able to apply the theory only the much weaker criterion that the length is sufficiently larger than the mean-free path of the medium has to be fulfilled.
Two different aspects are of special important in the theory of disordered media, namely transport properties and resonances. The transport properties (moments of the eigenvalues of the transmission and reflection matrices) are known for the disordered slab in the limit that it is wide,[@brouwer:98a] for the chaotic cavity with an opening that is so small that only one or two modes can propagate through it,[@brouwer:95b; @brouwer:97a] or a chaotic cavity with a wide opening.[@beenakker:98b]
Less is known about the poles of such systems. (The eigenvalues of the Hamiltonian correspond to poles of the scattering matrix, and these show up as resonances in a scattering experiment. Hence the somewhat inconsistent nomenclature.) The beginning of random-matrix theory can be put at the moment when Wigner surmised the eigenvalue distribution for a closed chaotic cavity.[@wigner:56a; @mehta:90; @beenakker:97a] Here we are interested in open systems, where the eigenvalues acquire an imaginary part. (The imaginary part is referred to as residue.) It determines the decay rate of the (quasi-)eigenstate of the system. For chaotic cavities with broken time-reversal symmetry, the decay rate distribution is known analytically for an opening of arbitrary size.[@fyodorov:97a] The distribution for the more important case of preserved time-reversal symmetry[^1] is not known but can be approximated by a cavity with broken symmetry and an opening of half the real size.
Information on the residues of a disordered slab is very limited, and only the scaling behaviour of the large residue-tail in the localised regime was determined recently.[@titov:00a; @terrano:00a] This deficiency is felt especially strong in the random-laser community since the location of the residues gives the lasing threshold of an optical system, and most experimental setups resemble a disordered slab much more than they resemble a cavity. This paper fills this gap by presenting the results of numerical simulations. The quality of the numerical decay rate distributions is good enough that it allows us to arrive at analytical formulae for the distribution function, including its dependence on the parameters of the system. The idea to use high-quality simulations to arrive at formulae is not completely new as the distribution of the scattering strengths of chaotic cavities was found in the same way.[@beenakker:98b]
This paper is organised as follows: First, we introduce the Anderson Hamiltonian used the describe the disordered slab. In Sec. \[secEigenSolver\] we show how the eigenvalues of that Hamiltonian can be computed in a efficient numerical way. Depending on the length of the slab, it can be in either the diffusive or in the localised regime. We will analyse the decay rate distributions for both regimes separately, first in Sec. \[secDiffusive\] for the diffusive and then in Sec. \[secLocalised\] for the localised regime. Until that point all results are completely general and can be applied to electronic and photonic systems. In Sec. \[secThreshold\] we specialise on the lasing threshold in amplifying disordered media. We distinguish between the diffusive and the localised regimes (Secs. \[secThresholdDiffusive\] and \[secThresholdLocalised\]). We conclude in Sec. \[secConclusions\].
Anderson Hamiltonian for a disordered slab {#secModell}
==========================================
We consider a two-dimensional slab of length $L$ and width $N$. The slab is described by an Anderson-type lattice Hamiltonian with spacing $1$. In the Anderson model, transport is modelled by nearest-neighbour hopping between lattice sites. Without loss of generality we can set the hopping rate to $1$. With a spatially varying potential $P(x,y)$ the Hamiltonian becomes
\[Hdef\] $$\begin{aligned}
\mathcal{H}_{(x,y),(x,y)} &= P(x,y) &(y \ne 1, L) \\
\mathcal{H}_{(x,y),(x,y)} &= P(x,y) - i \kappa &(y = 1, L) \label{Hdefkappa} \\
\mathcal{H}_{(x,y),(x+1,y)} &= 1 &(x<W) \\
\mathcal{H}_{(x,y),(x-1,y)} &= 1 &(x>1) \\
\mathcal{H}_{(x,y),(x,y+1)} &= 1 &(y< L) \\
\mathcal{H}_{(x,y),(x,y-1)} &= 1 &(y>1) \end{aligned}$$
All other elements are zero. $x$ runs from $1$ to $N$, and $y$ from $1$ to $L$.
The real part $E$ of the eigenvalues of $\mathcal{H}$ in the limit of large $N$ and $L$ is confined to the interval $[-4;4]$. (If the average of $P(x,y)$ is nonzero, the interval is simply shifted by that average. If $P(x,y)$ is fluctuating as a function of $x$ and $y$ — like it does for a disordered medium — the interval becomes a bit wider.) For electronic systems, $E$ gives the energy of the eigenstate, and Eq. (\[Hdef\]) hence describes a slab with a conduction band of width $8$. For photonic systems, the real part of the eigenvalue gives the eigenfrequency. For both systems, the imaginary part $\gamma$ of the eigenvalue gives the decay rate of the eigenstate. (Actually not $\gamma$ but rather $\gamma/2$ is the decay rate but for the ease of notation we will continue to refer to $\gamma$ simply as the decay rate.)
$\kappa$ in Eq. (\[Hdefkappa\]) quantifies the strength of the coupling between the slab and the outside.[@fyodorov:97a] Using the units introduced above, $\kappa$ has the value $\sin^2 k$ where $k$ is the wavevector at the energy at which particles are injected respectively emitted. This quantity is proportional to the velocity of the particle perpendicular to the interface. In the centre of the band $\sin k=1$ whereas at the edges $\sin k=0$.
If $\kappa$ is chosen to be constant (i.e. not to depend on energy) ideal outcoupling can be described only for one specific value of the energy. We will do this since otherwise solving the Hamiltonian no longer is a standard eigenvalue problem, and set $\kappa\equiv 1$, hence modelling ideal coupling at the centre of the band.[^2] Working at the centre of the conduction bands offers the advantage that the width $N$ of the sample is identical to the number $N$ of propagating modes, and thus allows the describe the largest number of propagating modes for given size of the Hamiltonian (i.e. given numerical work). It is possible to include energy dependent coupling terms[@terraneo:00a] but it should be stressed that a constant $\kappa$ is more efficient and gives completely correct results as long as only eigenvalues near the respective energy are considered. We set $\kappa=1$, meaning ideal coupling at the centre of the conduction band.
It should be stressed that – even though we are modelling a two-dimensional system — the results are valid for three-dimensional systems as long as $L\gtrsim N$. A particle that is injected into the slab ergodically explores the entire cross-section of the sample before being emitted again, and hence loose its memory of which sites are “connected” by hopping terms. The sites can then be re-arranged, e.g. in a three-dimensional structure. Only for very short samples, $L\lesssim N$, this is not possible but for such samples already applying the Anderson model (i.e. only allowing nearest-neighbour hopping) is very questionable. The only “real” restriction that can limit the application of our results to certain photonic three-dimensional systems is that particles can leave the sample only at the front and at the back — and not through the “sides”.
In the formulation of Eq. (\[Hdef\]) the matrix $\mathcal{H}$ has double indices but these are easily removed by considering $\mathcal{H}_{n n'}$ with $n=x+(y-1) N$. (It would not make sense to set $n=y+(x-1) L$ since usually $L\gg
N$, and we want to work with a band matrix that is as small as possible.) This results is a matrix of the form as depicted in Fig. \[figmatrix\]. It is a banded $L N \times L N$ matrix with band width $2 N + 1$. Also within the band most elements are zero (since usually $N\gg 1$). The matrix is symmetric but non-Hermitian as there are complex numbers on the diagonal.
The model (\[Hdef\]) has been widely used since an efficient way to compute the transmission through such a slab is known.[@baranger:91a] The method of recursive Greens functions allows to compute the entire scattering matrix, hence all linear transport properties, in a time of order $\mathcal{O}(L N^3)$ and with only minimal storage requirements $\mathcal{O}(N^2)$. No explicit reference to the Hamiltonian $\mathcal{H}$ is made, so that eigenvalues cannot be computed by this method.
Computing eigenvalues of symmetric complex non-Hermitian banded matrices {#secEigenSolver}
========================================================================
Since the Hamiltonian $\mathcal{H}$ from Eq. (\[Hdef\]) is both banded and sparse one might be tempted to use an eigensolver for sparse matrices to compute the eigenvalues of Eq. (\[Hdef\]). A sparse eigenvalue routine needs to be able to solve the equation $$( \mathcal{H} - \mu {\mathbbm{1}}) \vec{x} = \vec{y}
\label{inviteration}$$ for the unknown vector $\vec{x}$ for arbitrary $\mu$ and $\vec{y}$. In particular, the eigensolver needs to set $\mu$ close to an eigenvalue of $\mathcal{H}$ so that the matrix $\mathcal{H} - \mu {\mathbbm{1}}$ is ill-conditioned. A numerical solution of Eq. (\[inviteration\]) is then difficult and very expensive. Furthermore, only one eigenvalue is found at a time, and control of which eigenvalue the algorithm will converge to is difficult. (Algorithms for sparse matrices always use inverse iteration so that the corresponding eigenvector will be returned without additional effort but the eigenvector is of no use for us.) Using an algorithm for banded matrices is thus the better alternative.
There exist a number of algorithms for real symmetric or complex Hermitian band matrices. Both problems are characterised by real eigenvalues, so that they are conceptually identical. Only one algorithm for computing an eigenvalue (plus the corresponding eigenvector) of a general complex band matrix has been published.[@schrauf:91a] It uses inverse iteration, so it is of limited use here.
We thus had to implement our own eigenvalue solver. The eigenrepresentation of $\mathcal{H}$ in terms of the diagonal matrix $\Lambda = \mathrm{diag}(\lambda_1,\ldots,\lambda_N)$ of the eigenvalues $\lambda_i$ of A and the matrix $U$ of eigenvectors is $$\mathcal{H} = U \Lambda U^{-1} \;.
\label{AewDecomp}$$ We now observe that for symmetric, that includes *complex* symmetric, $\mathcal{H}$ it is always possible to chose eigenvectors such that $U
U^T = {\mathbbm{1}}$. If $U$ would be a real matrix, one would call $U$ orthogonal but since it is complex there is no special name for the property $U U^T = {\mathbbm{1}}$.
Algorithms for diagonalising a real symmetric matrix $A$ implicitly decompose $A$ as $$A = Q D Q^T\;,\qquad Q Q^T = {\mathbbm{1}}$$ with the matrix $D$ of eigenvalues of $A$. It is therefore possible to adapt such an algorithm for our needs. Most algorithms for banded matrices first reduce $A$ to tridiagonal form $A'$ by transformations of the form $A'=Q' A Q'^T$, and we will adopt this strategy. (A matrix is called tridiagonal if only the diagonal and its neighbouring elements are nonzero. If $A$ would be real, the transformation $A\to A'$ would be called a similarity transformation.) For a band matrix this is possible in an efficient way since it is not necessary to compute (and thus store) the full matrices $Q'$, and by annihilating the elements of the matrix $A$ in a clever order, the band structure is kept intact in all steps.[@kaufman:84a; @kaufman:00a]
The reduction of the complex matrix $\mathcal{H}$ to tridiagonal shape is done by straight-forward adaptation of this algorithm from real to complex numbers, where care needs to be taken that the dot product $\vec{x} \cdot \vec{y} = \sum_i x_i y_i$ is used and not the dot product $\vec{x} \cdot \vec{y} = \sum_i \overline{x}_i
y_i$ normally used for complex vectors. (The overbar marks the complex conjugate.)
To compute the eigenvalues of the tridiagonal matrix for the real symmetric or complex Hermitian case, methods that isolate eigenvalues in disjunct intervals are used (“divide and conquer” and similar methods[@golub:89a]). Such methods work for both of these cases as all eigenvalues are real and can thus be ordered. This no longer is possible here as the eigenvalues are complex. We therefore use the QR respectively QL method.[@lapack:99]
For an $K\times K$ banded matrix with band width $W$ the time needed to compute the eigenvalues increases as $\mathcal{O}(K^2 W)$ whereas the storage requirements increase as $\mathcal{O}(K W)$. In terms of the dimensions $L$ and $N$ of the disordered slab, this means that the time increases as $\mathcal{O}(L^2 N^3)$ and the storage space as $\mathcal{O}(L
N^2)$. For given computational resources, both scalings impose an upper limit on the system size that can feasible be treated. For typical values of the ratio $L$ and $N$, and for “realistic” computer equipment, the time limit is reached somewhat earlier than the memory limit.[^3]
With respect to a similar algorithm for full matrices one wins a factor $L$ (usually of order several hundred) for both time and memory by using the banded algorithm, thus allowing to treat system that could not be treated otherwise. Still, the work presented in this paper is a big numerical challenge. To arrive at the results, of the order of 100000 hours of cpu time on fast PC’s were needed.
Numerical simulations
=====================
Disorder is modelled by assigning random values to $P(x,y)$. It is assumed that those random numbers are uniformly distributed in the interval $[-w;w]$ so that $w$ measures the amount of disorder.
We only consider eigenvalues near the centre of the conduction band as the assumption of ideal coupling is only valid there. For numerical reasons it is essential that the centre of the conduction band is at $E=0$, i.e. one is not allowed to add an offset to $P(x,y)$.[^4]
We hence chose a window $[-d;d]$ and only include eigenvalues in the further analyses when their real part is inside that window. If the window is too large, systematic errors are introduced while too small a window leads to bad statistics. As can be seen from Fig. \[windowsizefig\], for of $d=0.1$ the distribution function already agrees with the distribution function for $d=0.01$ but has much better statistics. $d=0.5$ and $d=1.0$ gives significant systematic deviations. For this reason, all results presented in this paper assume a window with $d=0.1$.
The formulation of the model in Sec. \[secModell\] is in terms of generic units. Contact with a microscopic model or an experiment is best made in terms of the mean free path. It can be computed from the length-dependence of the transmission probability $T$ through the sample. In the diffusive regime, $l\lesssim L\ll N l$, it is given by [@beenakker:97a] $$\frac{1}{T} = 1 + \frac{L}{l} \;.$$ The mean free path can be computed by fitting $T(L)$ to this functional form.
The transmission probability has been computed using the method of recursive Green’s functions[@baranger:91a] for variable disorder strength $w$. As Fig. \[freiwegfig\] shows, the numerically computed mean free path $l$ is for the range of $w$ in question in very good approximation given by $$l = \frac{6}{w^{3/2}} \;.
\label{freiwegeq}$$ (Computed for each value of $w$ from $50$ samples with $L=2,4,\ldots,98,100$ and $N=50$.) In the following, we will no longer make explicit reference to $w$ but rather give the more intuitive mean-free path $l$.
Diffusive regime {#secDiffusive}
================
For a sample length $L$ with $l\lesssim L \ll N l$ the sample is said to be in the diffusive regime. It is immediately obvious that the diffusive regime can only be observed in sufficiently wide samples, $N\gg 1$.
For chaotic cavities with broken time-reversal symmetry an analytical result for the decay rate distribution has been given by Fyodorov and Sommers.[@fyodorov:97a] We start from their result and rescale it, $$\mathcal{P}(y) = \frac{1}{y^2 M!} \int_0^{M y} x^M e^{-x} d x
= \frac{1}{y^2} \left[ 1 - e^{-M y} \sum_{k=0}^M \frac{M^k}{k!}
y^k \right] \;.
\label{PyAnsatz}$$ $\mathcal{P}(y)$ is normalised to one and in our scaling is for all $M>1$ peaked near a value of $y$ of order $1$. In the original formulation for a chaotic cavity, $M$ is the number of modes propagating through the opening of the cavity.
In the following we will argue that the decay rate distribution $P(g)$ can be written in the form (\[PyAnsatz\]) as $$P(\gamma)=\frac{1}{\gamma_0}
\mathcal{P}\bigl(\frac{\gamma}{\gamma_0}\bigr)$$ with some scaling factor $\gamma_0$ and some effective number of modes $M\ne N$. In Fig. \[PyGueltigFig\] a comparison between the analytical suggestion and a simulation is given, and the agreement is striking. The horizontal axis has been plotted logarithmically since this results in both the differences between the $P(y)$ for different $N$ becoming easier to recognise and in giving a more prominent place to the small-$\gamma$ tail of $P(\gamma)$. In most applications, including the random laser discussed later in this paper, one is much more interested in small $\gamma$ than in large $\gamma$.
The results of the simulations are fitted “by eye” against the functional form (\[PyAnsatz\]), resulting in one pair of values for $\gamma_0$ and $M$ for each set of parameters. Especially at very small $\gamma$, there are sometimes numerical errors that introduce artifacts into the numerical histogram so that using an automatic fitting algorithm is not feasible. (Usually we computed 500–1500 realisations for each parameter set.) From our simulations, we find that the scaling factor $\gamma_0$ only depends on the length $L$ of the sample and its mean free path $l$ but not on its width $N$, and seems to be given by $$\gamma_0 = \frac{2 l}{L^2} \;.
\label{g0Gleichung}$$ As Fig. \[g0Grafik\] shows, the agreement between the result of the numerical simulations and Eq. (\[g0Gleichung\]) is good, and all major deviations are for small $L$ where universal scaling is expected to be worse than for larger $L$. The model equations set the speed of propagation to $1$ but it is obvious that for some other choice for the propagation speed $c$ one has to change Eq. (\[g0Gleichung\]) to $\gamma_0 = 2 c l / L^2$.
While the determination of $\gamma_0$ is very precise, there is a somewhat larger error involved in determining $M$ by fitting the analytical form to the results of numerical simulations. First, we only fitted against integer $M$, though in principle a generalisation of Eq. (\[PyAnsatz\]) to noninteger $M$ is possible, see Eq. (\[PvertGamma\]). Secondly, if $M \gtrsim 25$, the difference between $P(y)$ for $M$ and for $M+1$ becomes too small to tell with certainty which of these two values describes the numerical result better. Thirdly and finally, even with 500-1500 samples for each set of parameter values, there are still some fluctuations in the numerically computed histogram for the decay rate distribution that in some cases make the decision on the right $M$ a bit difficult. Considering all of this, one should allow for an error of $1$ for $M$, and even of $2$ for $M\gtrsim 25$.
We have computed $M$ for a series of samples with increasing length for three different widths $N$. As Fig. \[Mgrafik\] shows, the effective number $M$ of modes is well approximated by $$M = \frac{N}{1 + L / (6 l)} \;.
\label{Mgleichung}$$ The agreement between this suggested analytical form and the numerical simulation becomes better as the width $N$ of the sample is increased. From the simulations it is obvious that the functional form Eq. (\[Mgleichung\]) is correct but there still is the (small) possibility that the factor $6$ might need to be replaced by a slightly smaller value. To answer this question with certainty, we would need to increase both $L$ and $N$ significantly. Unfortunately, such simulations are outside the present time and memory constraints.
Equations (\[PyAnsatz\]–\[Mgleichung\]) give a good description of the decay rate distribution of a disordered slab in the diffusive regime, provided the slab is sufficiently wide. Since the transversal length scales are set by microscopic quantities (wave length of the light for optical systems, Fermi wave length for electronic systems), all macroscopic objects are “wide”.
Localised regime {#secLocalised}
================
If the length $L$ of a disorder medium is increased, the phenomenon of localisation sets in once $L\gtrsim N l$ (see Ref. for a review). In the localised regime the probability of transmission $T$ through the sample is reduced significantly and decays exponentially with the length $L$ of the sample. The length scale $\xi$ is called the localisation length, and can be computed from an ensemble of disordered slabs by computing the average of the logarithm of the transmission as a function of the length of the samples, hence $$-L/\xi = \langle \ln T(L) \rangle \;.$$ One should note that this is not identical to fitting the transmission to $\langle T(L)
\rangle \propto \exp( -L/\xi )$ since the large sample-to-sample fluctuations of $T(L)$ in the localised regime would give a value for $\xi$ that is off by a factor $4$. The localisation length can also be computed analytically from the mean-free path using the DPMK equation, with the result[@efetov:83b; @efetov:83c; @dorokhov:82a; @dorokhov:82b; @dorokhov:83a; @dorokhov:83b] $$\xi = \frac{N+1}{2} l \;,
\label{XiVorhersage}$$ and agrees well with our numerical results.
It is generally accepted that the distribution of the decay rates $\gamma$ (at least for small $\gamma$) in the localised regime is log-normal, i.e., $\ln
\gamma$ is distributed according to a Gaussian distribution. Recent interest has rather been in the large-$\gamma$ tail which was shown to follow a power-law.[@titov:00a; @terrano:00a] In a log-normal distribution, most of the weight lies in the right tail, so those papers give a sufficient description for most of the eigenmodes. In the context of applications to random lasers we are, however, interested in the small decay rate tail, hence in the log-normal distribution.
To our knowledge, there is only a single paper by Titov and Fyodorov that gives explicit expressions for the parameters of that log-normal distribution.[@titov:00a] However, their analytical results are for a somewhat different system so it is difficult to tell whether they agree or disagree with our findings. We will return to this aspect at the end of this section. First, we want to present the results of our numerical simulations.
Using the log-normal ansatz, the distribution of the decay rates $\gamma$ is $$P(\gamma) = b \exp\left(-\frac{(\log \gamma - \log
\gamma_0)^2}{\sigma^2} \right) \;.
\label{pLokalAnsatz}$$ The numerically computed histograms indeed follow this form, see Fig. \[logbeispiel\], except for the large-$\gamma$ tail — as already mentioned above but this deviation is only seen in a log-log plot.
Fig. (\[figlocmax\]) shows in the left the numerically computed $\gamma_0$ as a function of $L$ for $N=15$. Also displayed is the localisation length $\xi$ computed numerically from the transmission, being in good agreement with the analytical prediction (\[XiVorhersage\]). The quality of the data is good enough to say with confidence that $\gamma_0$ decays exponentially with a length scale that is somewhat larger than $\xi$. Fig. (\[figlocmax\]) shows in the right the value of $\gamma_0$ also for two other values of $N$, and all three cases are well-described by introducing a numerical fitting factor $a$, $$\gamma_0 \propto \exp\left(-\frac{L}{a \xi}\right) \quad
\mathrm{with}\quad a=1.12 \;.
\label{ylocposprop}$$ It is known that working at the centre of the conduction band when in the localised regime can introduce certain artefacts, especially in analytical approaches. Among other, the localisation length at the band centre can differ by approximately $10\,\%$ from the value outside the centre.[@kappus:81a] We have defined $\xi$ based on the transmission through the sample (at an energy corresponding to the band centre), and in transmission resonances at all energies can contribute. A numerical prefactor $a$ that differs by about $10\,\%$ from $1$ thus does not come as a complete surprise.
We still need to compute the proportionality factor appearing in Eq. (\[ylocposprop\]). For this purpose we need to plot the ratio of the numerically computed $\gamma_0$ and the right-hand side of Eq. (\[ylocposprop\]) for different values of $N$. We did this for $L=71.55~l$. Since this is a very expensive operation, we have computed a large number of samples only for $N=10,15,20$ so that their statistics is better than for the other values of $N$. An estimate of the error for these “better” data points has been included in the figure. This allows us to conclude that $$\gamma_0 = \frac{a}{N^2} \exp\left(-\frac{L}{a \xi}\right) \quad
\mathrm{with}\quad a=1.12 \;.
\label{ylocpos}$$ It should be noted that this equation contains two numerical coefficients, and there is no obvious reason why they should be identical. Still, we find that they both are approximately $a=1.12$.
Re-introducing “physical units” into Eq. (\[ylocpos\]) is a bit more difficult than it was for Eq. (\[g0Gleichung\]) where it was obvious that one simply has to multiply by the velocity of propagation $c$. Here one has to multiply by $c/\Delta$ where $\Delta$ is the perpendicular grid spacing. Due to the assumption of one propagating mode per (lateral) grid point made in Sec. \[secModell\], $\Delta$ is not arbitrary but has a well-defined physical meaning. For the electronic case, $\Delta=\pi/k_{\mathrm{F}}$ with $k_{\mathrm{F}}$ the wave vector at the Fermi level, and for the photonic case $\Delta=2\lambda/\pi$ with $\lambda$ the wave length of the light (hence $c/\Delta = 1/(4\nu)$).
\
Determining the width $\sigma$ of the distribution is more difficult since we can only use the left wing of the distribution — the right wing eventually turns into a power-law tail and thus no longer follows a log-normal distribution. Once again, we have accumulated more data for $N=10,15,20$ so that some indication of the error is possible for those three data points.
From our data, we propose the formula $$\sigma = \frac{2}{3} \left(\frac{L}{a \xi}\right)^{2/3} \;,
\label{eqsigma}$$ where $a=1.12$ has the same value as in Eq. (\[ylocpos\]). As Fig. \[figsigma\] shows, there clearly is no disagreement between the numerical data and Eq. (\[eqsigma\]). However, please remember that the $\frac{2}{3}$ should be thought of as a fitting factor that might not be exactly $2/3$ but perhaps rather $0.67$ or some other numerical factor.
Since the distribution is log-normal only for not too large $\gamma$ (remember the power-law tail for large $\gamma$) the normalisation is nontrivial \[$P(\gamma)$ is not normalised to $1$ any longer!\] and cannot be computed from $\gamma_0$ and $\sigma$. The constant $a$ in Eq. (\[pLokalAnsatz\]) is directly equal to the height of the peak of the numerically computed $P(\gamma)$. Since the total area underneath the numerically computed $P(y)$ (and hence its normalisation) is dominated by the large-$\gamma$ tail, $a$ has a relatively large error. Taking all the available data, the most likely value is $$b=\frac{1}{N^2} \exp\left(\frac{L}{a\xi}\right) \;.$$ This value has been determined from a large number of simulations that for space reasons cannot be presented here. Unfortunately the quality of the data is not good enough to decide whether an additional prefactor $a=1.12$ should appear.
At the present it is not possible to tell whether our results agree with the ones put forward by Titov and Fyodorov.[@titov:00a] In particular, they arrive at $$\gamma_0 \propto \exp\left(-\frac{3 L'}{\xi}\right) \;,
\label{EQmisha}$$ whereas our finding (\[ylocpos\]) was $\gamma_0 \propto \exp(-L/1.12\xi)$. There are two obvious differences between the model used by them and the model employed by us. First, for numerical reasons we work at the centre of the conduction band while they work near (but sufficiently far away from) the band edges. This might explain the factor $a=1.12$ that we have to introduce. Secondly and probably more importantly, they consider a system of length $L'$ that is closed at one end whereas our systems have length $L$ and are open at both ends. It is obvious that a half-closed system of length $L'$ corresponds to an open system of length $L>L'$. Eq. (\[EQmisha\]) suggests that those two systems could be mapped into each other by setting $L\approx 3 L'$ but there is no further evidence to support this claim.
Lasing threshold of a random laser {#secThreshold}
==================================
A random laser is a laser where the necessary feedback is not due to mirrors at the ends of the laser but due to random scattering inside the medium.[@wiersma:95a; @wiersma:97a; @beenakker:98b] We model the random laser as a disordered slab containing a dye that is able to amplify the radiation in a certain frequency interval with rate $1/\tau_{\mathrm{a}}$. The lasing threshold is the amplification rate at which the intensity of the emitted radiation diverges in a linear model. If saturation effects are included, the emitted intensity increases abruptly but finitely at crossing the lasing threshold.
The lasing threshold is given by the value of the smallest decay rate of all eigenmodes in the amplification window.[@misirpashaev:98a] (Remember that $\gamma$ actually is twice the decay rate. On the other hand, also $1/\tau_{\mathrm{a}}$ enters the relevant formulae only with a prefactor $1/2$. $\gamma$ thus indeed gives the necessary amplification rate $1/\tau_{\mathrm{a}}$.) This is easily understood since this simply means that in the mode with the smallest decay rate the photons are created faster by amplification than they can leave the sample (=decay). It, however, also follows from a complete quantum mechanical analysis.[@schomerus:00a; @patra:99a]
The distribution of the decay rate has been computed in this paper. A certain number $K$ of modes will be in the frequency window where amplification is possible. The lasing threshold is given by the smallest $\gamma$ of these $K$ modes. In a simple picture that is valid once $K\gg 1$ we can assume that the $K$ different $\gamma$’s are distributed independently according to $P(\gamma)$.[@misirpashaev:98a] The distribution $\tilde{P}(\gamma)$ of the smallest mode and hence of the lasing threshold then becomes $$\tilde{P}(\gamma) = K P(\gamma)\left[ 1-\int_0^{\gamma}
P(\gamma') d\gamma' \right]^{K-1} \;.
\label{eqtreshold1}$$
For $K\ne 1$, the distribution $\tilde{P}(\gamma)$ of the lasing threshold is not longer identical to the distribution $P(\gamma)$ of the decay rate of each individual mode. In particular, not only the precise form of these two distribution will be different, but also the “typical” value of the lasing threshold can be different from the “typical” decay rate $\gamma_0$. Interestingly, for chaotic cavities in the diffusive regime it was found that the latter two quantities differ only insignificantly[@frahm:00a; @schomerus:00a] which might seem counter-intuitive. A slab geometry is more “complicated” in that the scaling $K\propto N$ “tries” to lower the lasing threshold with increasing $N$.
For $K\gg 1$ the distribution $\tilde{P}(\gamma)$ is sharply peaked around its maximum. The position $\gamma_{\mathrm{m}}$ of the maximum is given by the solution of the equation $d \tilde{P}(y_{\mathrm{m}})/d\gamma_{\mathrm{m}}=0$, hence $$0 = \frac{d P(\gamma_{\mathrm{m}})}{d\gamma_{\mathrm{m}}}
\left[ 1-\int_0^{\gamma_{\mathrm{m}}} P(\gamma') d\gamma'
\right]
- (K-1) [ P (\gamma_{\mathrm{m}})]^2 \;.
\label{eqtreshold2}$$ While Eq. (\[eqtreshold1\]) is difficult to compute numerically due to the large exponent $K-1\gg 1$, in Eq. (\[eqtreshold2\]) this exponent no longer appears.
Eq. (\[eqtreshold2\]) depends on $P(\gamma)$ which in turn depends on the dimensions $L$ and $N$ of the system. In assuming that the number of propagating modes is equal to the width $N$ of the sample we already have made the assumption that the width (and hence also the length) is measured in units of $\lambda/2$. (The “$2$” accounts for polarisation.) The total number of modes in the sample thus is $L N$. We assume that a fraction $f$ of them is inside the amplification window of the dye, hence $K=f N L$. For simplicity we neglect complications as the shape of the mode profile. (It is easily incorporated into the numerics and we refrain from doing this just to prevent having to introduce even more parameters.) $f$ depends only on the chemical properties of the dye and not on the dimensions of the sample.
In the following we will show how to compute the most likely lasing threshold for samples in both the diffusive and in the localised regime.
Lasing threshold in the diffusive regime {#secThresholdDiffusive}
----------------------------------------
The change of the lasing threshold with increasing system size is influenced by a subtle interplay between $L$ and $N$ in determining the distribution $P(\gamma)$ and in determining the number $K=f N L$ of total modes.
If $K\gg 1$ the lasing mode has a decay rate in the low-$\gamma$ tail of $P(\gamma)$ (i.e. $\gamma<\gamma_0$ or $y<1$). The weight of this tail is $$\int_0^1 P(y) d y = \frac{M^{M-1}}{(M-1)!} e^{-M} \;,
\label{P0bis1}$$ and goes to zero as $M$ becomes larger. For $M\to\infty$ the tail disappears completely, as is already obvious from the asymptotic form of the distribution, $$P_{M\to\infty}(y)=\left\{ \begin{aligned}
& 0 & (y<1) \\
& 1/y^2 & (y\ge 1)
\end{aligned} \right.$$ With increasing $N$ and hence increasing $M$, the probability that a given mode has a small $y$ thus decreases rapidly. On the other hand, we are interested in the smallest decay rate out of $K$ modes, and $K$ increases linearly with $N$. This are two counter-acting effects, and it is not obvious which of these two is stronger.
The effect of an increase of the system size $L$, on the contrary, is obvious. First, the average decay rate $\gamma_0$ decreases according to Eq. (\[g0Gleichung\]). Secondly, $M$ decreases from Eq. (\[Mgleichung\]), leading to even smaller values for $\gamma$ of the lasing mode.
There have been some analytical attempts to compute the lasing threshold for a chaotic cavity [@frahm:00a; @schomerus:00a]. For large $M$, the small-$y$ tail of Eq. (\[PyAnsatz\]) was approximated by $$P(y) \approx \frac{1}{2 M}
\left[ 1 + \mathrm{erf}\bigl( \sqrt{M/2} [y-1]
\bigr) \right] \;.
\label{Pyapproximation}$$ This allows to arrive at scaling laws of the lasing threshold for variable $M$ at fixed $K$. Unfortunately, the difference between two counter-acting effects of an increase in $N$ are so small that Eq. (\[Pyapproximation\]) is a bit too crude for our needs.
We thus have to revert to a numerical procedure. Eq. (\[PyAnsatz\]) can be rewritten using the incomplete Gamma function $$\Gamma(a,x)=\int_x^{\infty} t^{n-1} e^{-t} d t \;.$$ $\Gamma(a,0)$ reduces to the well-known Gamma function $\Gamma(a)$. For numerical reasons it is advisable to introduce the regularised Gamma function $Q(a,x)=\Gamma(a,x)/\Gamma(a)$. Fast numerical algorithms to compute $Q(a,x)$ exist. \[Please note that in the literature the definitions of the regularised Gamma function sometimes disagree in that our $Q(a,x)$ is denoted as $1-Q(a,x)$.\] Now we can express Eq. (\[PyAnsatz\]) and its derivative and integral as
\[PvertGamma\] $$\begin{aligned}
P(y) &= \frac{1}{y^2} \bigl[1 - Q(M+1, M y)\bigr] \;, \\
\frac{d P(y)}{d y} &= \frac{(M y)^M}{y^2 \Gamma(M)} e^{-M y}
- \frac{2}{y^3}\bigl[1- Q(M+1, M y)\bigr] \;, \\
\int_0^y P(y') d y' &= \frac{1}{y} \bigl[Q(M+1,M y)-1\bigr] + 1
- Q(M,M y) \;,\end{aligned}$$
so that Eq. (\[eqtreshold2\]) can be evaluated efficiently.
Lasing threshold in the localised regime {#secThresholdLocalised}
----------------------------------------
From Eq. (\[pLokalAnsatz\]) we directly arrive at
$$\begin{aligned}
\frac{d P(\gamma)}{d \gamma} &=
-2 b \frac{\ln\gamma-\ln\gamma_0}{\gamma\sigma^2}
\exp\left[
-\frac{(\ln\gamma-\ln\gamma0)^2}{\sigma^2}\right]
\;, \\
\int_0^{\gamma} P(\gamma') d \gamma' &=
\frac{b \sqrt{\pi} \sigma \gamma_0}{2}
e^{\sigma^2/4} \left[1+\mathrm{erf}\left(
\frac{2 \ln( \gamma / \gamma_0)-\sigma^2}{2\sigma}\right)
\right] \;.\end{aligned}$$
A further simplification is not possible, and we did not manage to find suitable approximations. Also for the localised regime we thus are restricted to a numerical evaluation.
Numerical results
-----------------
The lasing threshold is computed numerically from Eq. (\[eqtreshold2\]), using the formulae from Secs. \[secThresholdDiffusive\] and \[secThresholdLocalised\]. Into the formulae presented there, we have to insert the correct dependence of the $\gamma_0$, $M$, $\sigma$, etc. on $L$ and $N$ that was presented earlier in this paper. Despite this complication the numerical calculation is straight forward as Eq. (\[eqtreshold2\]) possesses a single root only. Since this root has a change of sign, it is easily found numerically.
Fig. \[figLaserschwelle\] shows the results for both the diffusive and the localised regimes, for both $f=0.1~l$ and $f=0.001~l$. (The mean-free path appears as a factor since the figure is in units $L/l$ and not $L$.) The formulae found in this paper are valid deep in the diffusive regime respectively deep in the localised regime. Near the cross over, hence near the line $L\approx N l$, this condition is not fulfilled. The numerical values near the diagonal line in Fig. \[figLaserschwelle\] should thus be viewed with caution.
As can be seen from the figure, in the diffusive regime with $N\gg L/l$ the lasing threshold becomes almost independent of the width $N$ of the sample (for sufficiently large $N$), and the most likely value of the lasing threshold is about $$\gamma_{\mathrm{m}} \approx \frac{2 c l}{L^2} \;,
\label{gammaMdiffuse}$$ hence the value given by Eq. (\[g0Gleichung\]). This means that even though $K\gg 1$, $P(y)$ for $y<1$ is already so small that it dominates over the large value of $K$. Differences between this simple approximation and the precise numerical result appear for finite $N$, with the size of this difference depending on $f$. However, for designing experiments it is obvious from the results presented here that the only feasible way to lower the lasing threshold of a random laser in the diffusive regime is increasing its length, not modifying its width.
As Fig. \[figLaserschwelle\] shows, also in the localised regime there is only a small dependence on $f$. This means that in a log-normal distribution the weight of the left tail is so small that unless $K$ is exponentially large $\gamma_{\mathrm{m}}$ cannot become much smaller than the position $\gamma_0$ of the peak of the distribution. The difference to the diffusive regime is that the lasing threshold can be lowered efficiently not only by increasing the length but also decreasing the width $N$ and hence driving the system farther into localisation.
It is no surprise that samples in the localised regime generally have a lower lasing threshold than samples in the diffusive regime. We have shown that also the diffusive samples can have an “acceptably small” lasing threshold as it is trivial to make them very long (since there is no need to care much about their width). For both the diffusive and the localised regime, the typical decay rates of a single mode are comparable to the lasing threshold.
Conclusions {#secConclusions}
===========
We have numerically computed the distributions of the residues (or decay rates) of a disordered slab. The slab has length $L$, mean free path $l$, width respectively cross-sectional area $N$ ($N$ is given as number of propagating channels) and velocity of propagation $c$. We were able to “guess” simple analytical formulae that are able to describe the numerical results well.
For a sample in the diffusive regime ($L\lesssim N l$) we found in Eqs. (\[PyAnsatz\]–\[Mgleichung\])
\[ergebnis1\] $$\begin{aligned}
P(\gamma)&=\frac{L^2}{2 l c}
\mathcal{\mathcal{P}}\bigl(\frac{\gamma L^2}{2 l c}\bigr) \;, \\
\mathcal{P}(y) &= \frac{1}{y^2} \Bigl[1 -
\frac{\Gamma(1+\frac{N}{1 + L/6 l},\frac{N y}{1 + L/6 l})}{
\Gamma(1+\frac{N}{1 + L/6 l})} \Bigr] \;,\end{aligned}$$
where $\Gamma(a,x)$ is the incomplete Gamma function. The agreement between the numerical results and the proposed formulae is good, and there is the possibility that Eq. (\[ergebnis1\]) might become exact in the limit $L/l\gg N \gg 1$. However, there is only numerical and no analytical evidence to back this claim.
For a sample in the localised regime ($L \gtrsim N l$) with localisation length $\xi=(N+1)l/2$ we found in Sec. \[secLocalised\] $$\begin{gathered}
P(\gamma) = \frac{1}{N^2} \exp\left(\frac{L}{a\xi}
-\frac{(\log \gamma - \log
\gamma_0)^2}{\sigma^2} \right) \;, \quad a=1.12 \;, \nonumber\\
\gamma_0 = \frac{a}{N^2} \exp\left(-\frac{L}{a \xi}\right) \;,\quad
\sigma = \frac{2}{3} \left(\frac{L}{a \xi}\right)^{2/3} \;.
\label{ergebnis2}\end{gathered}$$ The quality of the simulations results in the localised regime is somewhat less than in the diffusive regime. For this reason, Eq. (\[ergebnis2\]) should be understood as an approximate fit only, and it very probably differs from the exact relation, especially outside the band centre.
These results can be applied to both electronic and photonic systems. For photonic systems we have shown that under realistic assumptions the lasing threshold of a random laser is close to $\gamma_0$ both in the diffusive and in the localised regime. Eqs. (\[ergebnis1\]) and (\[ergebnis2\]) thus not only give the distribution of the decay rate of each individual mode but also a good estimate of the lasing threshold, i.e. the smallest decay rate of a large number of modes.
Valuable discussions with C.W.J. Beenakker are acknowledged.
[^1]: Optical experiments always preserve time-reversal symmetry unless a magneto-optical effect is included. For electric systems time-reversal symmetry can be broken by applying a large magnetic field to the sample. (Such fields are created routinely in experiments.)
[^2]: It is not possible to have more than ideal coupling. For $\kappa<1$ the loss rates are smaller than for $\kappa=1$, so this is easily identified as “sub-ideal”. For $\kappa>1$ the loss rates split into two separate parts: Most become smaller, as for $\kappa<1$, while a few loss rates become very large, thereby fulfilling the requirement that the average loss rate has to be proportional to $\kappa$. We should note that this somewhat counter-intuitive behaviour is also observed for chaotic cavities.[@fyodorov:97a]
[^3]: On a modern computer a single diagonalisation for a $L=700$, $N=70$ systems takes about $2$ days and uses $256$ Mbytes of memory. While this memory requirement frequently is no problem, the computing time usually is. Remember that the task is to compute the distribution of the decay rates. Hence, many matrices with different realisations of the random potential $P(x,y)$ have to be diagonalised — not just a single matrix. However, the restrictions imposed by time and memory are of the same order of magnitude.
[^4]: The algorithms will return eigenvalues $z'$ that have a very small but finite deviation $|z-z'|$ from their correct value $z$. Since we are primarily interested in the imaginary part of the eigenvalue and want it to be as precise as possible the magnitude of the real part has to be as small as possible.
|
---
author:
- 'Gabriel Török, Martin Urbanec, Karel Adámek & Gabriela Urbancová'
date:
- 'Received / Accepted'
- 'Received: 12 February 2014 / Accepted: 10 March 2014'
title: ' [ Appearance of innermost stable circular orbits of accretion discs around rotating neutron stars ]{}'
---
Introduction {#section:introduction}
============
In Newtonian theory, circular trajectories of test particles orbiting around a spherical central body of mass $M$ are stable to small radial perturbations at any external radii $r$. The same trajectories calculated using a general relativistic description exhibit an instability below the critical radius of the marginally stable cicular orbit $r_{\mathrm{ms}}$ [e.g., @bar-etal:1972]. This radius is frequently considered as the innermost stable circular orbit (ISCO) of an accretion disc that orbits a black hole or a neutron star (NS), $r_{\mathrm{ISCO}}= r_{\mathrm{ms}}$. The ISCO is often assumed as a unique prediction of Einstein’s general relativity. Several decades after the ISCO concept has been introduced, it was argued that ISCO also appears around highly elliptic bodies described by the Newtonian theory [see works of @zdu-gou:2001; @ams-etal:2002; @klu-ros:2013]. In this sense, several phenomena related to rotating oblate neutron stars can be well understood in terms of the interplay between the effects of general relativity and Newtonian theory. Namely, as in the Lense-Thirring or Kerr spacetime, the position of ISCO decreases with growing angular momentum $j$ of the neutron star (given by the NS spin). At the same time, it increases with the increasing influence of the NS quadrupole moment $q$ [which determines the oblateness parameter $\tilde{q}\equiv q/j^2$, see, e.g., @urb-etal:2013].
In this paper we explore the consequences of this interplay on the behaviour of NS compactness. Finally, we focus on low-mass X-ray binary systems (LMXBs) and find astrophysical implications for the distribution of neutron stars with an ISCO (ISCO-NS).
{width="1\hsize"}
Behaviour of ISCO radius and NS compactness {#section:radius}
===========================================
Numerous works have investigated the behaviour of test particle motion, Keplerian frequencies, and the properties of ISCO in the vinicity of a rotating neutron star [e.g. @klu-etal:1990; @shi-sas:1998; @pac-etal:2006; @pac-etal:2012; @bak-etal:2012; @gut-etal:2013]. It has been shown that for geodesic motion, these are well described by formulae derived considering NS spacetime approximated by the Hartle-Thorne metric [@har-tho:1968; @abr-etal:2003; @ber-etal:2005]. Here we explore in more detail the behaviour of function $r_{\mathrm{ISCO}}(j\,,~q)$ calculated by using this approximation.[^1] We find that for $\tilde{q}\gtrsim2.8$, $r_{\mathrm{ISCO}}$ has a minimum when the angular momentum and the quadrupole moment of the central object are related as follows: $$\tilde{q}_{\mathrm{min}}= 1 + \frac{32\left(9\sqrt{6} + 4j_{\mathrm{min}}\right)}{135 j_{\mathrm{min}} \left[4608 \log(3/2)-1865\right]}\,.$$ The smallest allowed ISCO radius is then given by $$r_{\mathrm{min}}= 6 - 2j_{\mathrm{min}}\sqrt{2/3}\,.$$ In the left panel of Figure \[figure:risco\] we illustrate the behaviour of ISCO and the turning points $[r_\mathrm{min},j_\mathrm{min} ]$ for several values of $\tilde{q}$. Apparently, for $\tilde{q}\gtrsim7$, the turning points arise for low angular momentum values, $j\lesssim0.27$.
ISCO calculated for particular NS models
----------------------------------------
So far we have only considered the behaviour of ISCO without focusing on its relation to NS models. Models of neutron stars based on modern equations of state (EoS) have been extensively developed through the use of various numerical methods and codes [see @lat-pra:2001; @lat-pra:2007 for a review]. Results of the published studies indicate that in contrast to the behaviour of $r_{\mathrm{ISCO}}$, the NS radius $R_{\NS}$ very slowly evolves with the NS spin. Taking this into account, one can expect that the non-monotonicity of the ISCO behaviour discussed above may imply a non-monotonicity of the quantity $K\equiv r_{\ISCO}/ R_{\NS}$ (hereafter denoted as compactness factor). Considering concrete NS models we confirm this expectation using the numerical code of [@urb-etal:2013].
In the right panel of Figure \[figure:risco\] we plot the NS compactness factor $K$ calculated for the SKl5 Skyrme EoS [e.g., @rik-etal:2003]. We assume in the figure several values of the neutron star mass $M$ and spin frequency $\nu_{\mathrm{spin}}$, which range from the non-rotating limit to highest frequencies corresponding to the mass-shedding limit. To obtain smooth curves we have computed [$600$]{} NS configurations. One can clearly see that $K$ is not a monotonic function of $\nu_{\mathrm{spin}}$. Its extrema arise for very different values of $\nu_{\mathrm{spin}}$. For a high neutron star mass, ($M\sim2M_{\sun}$), they arise close to the mass-shedding limit, above 1000Hz. For a low neutron star mass, ($M\sim1M_{\sun}$), they arise for low values of $\nu_{\mathrm{spin}}$, below $\nu_{\mathrm{spin}}=200$Hz.
Appearance and disappearance of ISCO and the consequences
=========================================================
Using the right panel of Figure \[figure:risco\] we can directly compare the NS radii and the ISCO radii. Apparently, when $M$ is high, the ISCO is located above the NS surface for almost any $\nu_{\mathrm{spin}}$. On the other hand, when $M$ is low, the ISCO arises for only high values of $\nu_{\mathrm{spin}}$. For an intermediate mass, $M\sim1.6M_{\odot}$, the ISCO only occurs for two short intervals of $\nu_{\mathrm{spin}}$ - one close to the non-rotating limit and the other one at high frequencies.
In the left panel of Figure \[figure:distribution\] we show the map of mass–spin regions where the ISCO arises above the NS surface for Skl5 EoS. This map, which relates to curves drawn in the right panel of Figure \[figure:risco\], is emphasized by the thick line and dark colour. Remarkably, for $\nu_{\mathrm{spin}}\sim800$Hz, there are two intervals of $M$ that allow an ISCO appearance. On the other hand, for $\nu_{\mathrm{spin}}\lesssim700$Hz (or $\nu_{\mathrm{spin}}\gtrsim1200$Hz), there is only one such interval of $M$. For the sake of comparison we also consider three other EoS within the same figure: the SV and GS EoS, which represent two more parametrizations of the Skyrme potential [see @rik-etal:2003 and references therein], and the UBS EoS [@urb-etal:2010:aca]. Apparently, these EoS display a similar behaviour as the Skl5 EoS.
{width="1\hsize"}
Possible double-peaked spin distribution of ISCO-NS
---------------------------------------------------
While it is often speculated that most neutron stars in LMXBs may have ISCO, little clear observational evidence for this claim has been achieved yet, and the mass and spin distribution of these possible ISCO-NS remains puzzling [see, however @bar-etal:2005; @bar-etal:2006; @tor-etal:2010; @tor-etal:2012; @wan-etal:2013]. The mass distribution of neutron stars in LMXBs has hardly been explored so far. The mean mass is currently estimated as $M_0\in(1.4M_\odot-1.8M_\odot)$. More accurate estimates on NS mass distribution are available in double NS binary systems where masses are concentrated in a very narrow peak around 1.4$M_\odot$ [see @Lat:2012:].[^2] Accurate measurements for NS spin frequencies are available in radio pulsars where the distribution of rotational frequencies is narrowed around two peaks - one is of order of Hz, while the other is $\sim$100Hz [@Man-etal:2005:]. Spin measurements are much less frequent in LMXBs. [The spin frequencies inferred from the observed X-ray oscillations seem to be significantly higher than those of radio pulsars, reaching typical values of few hundreds of Hz. Nevertheless, several slowly rotating sources have been discovered as well [e.g., IGR J17480-2446 in globular cluster Terzan X-5 with $\nu_{\mathrm{spin}}=11$Hz, @bor-etal:2010; @mar-str:2010]. An overview of the currently measured spins of more than 20 LMXB NSs along with detailed references can be found in [@lam-bou:2008] and [@pat-wat:2012].]{}
Motivated by a simple analytical analysis and justified by the consideration of NS models assuming modern EoS, our results imply strong restrictions on distribution of the ISCO-NS parameters. This distribution could be very different from the (so far unknown) distribution of all NS in LMXBs. We illustrate these restrictions using an example calculated for the Skl5 EoS and the simplified assumption that all NS follow a two dimensional Gaussian distribution in the mass–spin space. For the distribution of all NS we set mean values $M_0\in\left\{1M_{\sun},~1.4M_{\sun},~2M_{\sun}\right\},$ and $\nu_{\mathrm{spin}}^0=500\mathrm{Hz}$ together with scales $\sigma_M=0.2M_{\sun}$ and $\sigma_{\nu\mathrm{spin}}=300$Hz. The subsets of ISCO-NS were then determined by the map depicted in the left panel of Figure \[figure:distribution\]. We show the spin distribution of these subsets in the right panel of Figure \[figure:distribution\]. We can see that when the neutron stars were distributed around high mass $M_0=2M_{\sun}$, the ISCO-NS spin distribution is approximately the same as the spin distribution of all NS that peak around $\nu_{\mathrm{spin}}^0$. On the other hand, when this mass was low, $M_0=1M_{\sun}$, the ISCO-NS distribution had a peak only around a high value of $\nu_{\mathrm{spin}}$. A different situation occurs for an intermediate value of $M_0=1.4M_{\sun}$, which implies an ISCO-NS distribution divided into two groups of slow and fast rotators.
Discussion and conclusions {#section:conclusions}
==========================
Our analysis clearly revealed the non-monotonicity of the dependence of the compactness factor $K$ on NS spin. The occurence of NS spin interval in which the ISCO appears for two very different ranges of NS mass (and the inferred double-peaked ISCO-NS spin distribution) depends on the detailed behaviour of $K$.
For the four EoS discussed here, this behaviour is rather similar. However, normalization of $K$ and its exact dependence on $\nu_{\mathrm{spin}}$ and $M$ depend on the particular EoS. More detailed investigation that takes into account a large set of EoS needs to be performed to make better astrophysical assessments. Moreover, we assumed here the simplified identity $r_{\mathrm{ISCO}}= r_{\mathrm{ms}}$, but various effects might lead to a decrease or increase of $K$ (e.g. viscosity, pressure forces, or magnetic field influence), and the chosen spacetime description can play some role as well [e.g. @alp-psa:2008; @str-sra:2009; @bak-etal:2010; @bak-etal:2012; @kot-etal:2008]. Despite these uncertainties, which need to be adressed in the subsequent analysis, our findings can be useful in several astrophysical applications. [Clearly, the predictions on the NS parameters would be strong if the ISCO presence or absence in the individual systems were observationally confirmed. Moreover, one can speculate on some consequences for evolution of the spinning-up sources.]{}
[Although more development is required, some implications can be consired immediately. One possibility is to use our results in a study aimed to distinguish between different models of high-frequency quasiperiodic oscillations (HF QPOs). There is no consensus as yet on the HF QPO origin, and various models have been proposed [e.g., @alp-sha:1985; @lam-etal:1985; @mil-etal:1998; @psa-etal:1999; @ste-vie:1999; @abr-klu:2001; @tit-ken:2002; @rez-etal:2003; @pet:2005; @zha:2005; @sra-etal:2007; @stu-etal:2008 and several others]. Among numerous ideas,]{} it was suggested that the strong luminosity variations observed on the kHz time-scales in the neutron star X-ray binaries result from the modulation of accretion flow that heats the boundary layer between the flow and the NS surface [@pac:1987; @hor:2005; @abr-etal:2007]. Motivated by this suggestion, [@urb-etal:2010] assumed the Paczynski modulation mechanism for the specific epicyclic resonance QPO model and investigated the implied restrictions on the NS mass and angular momentum. Taking into account our present findings and the general idea that (any) disc oscillations are responsible for exciting the Paczynski modulation mechanism, one might expect that the spin distribution of the HF QPO sources is double-peaked when NS masses are distributed around the intermediate value $M_0\sim1.4M_{\sun}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to acknowledge the support of the Czech grant GAČR 209/12/P740 and the project CZ.1.07/2.3.00/20.0071 “Synergy”, aimed to foster international collaboration of the Institute of Physics of SU Opava. Furthermore, we acknowledge financial support from the internal grants of SU Opava SGS/11/2013 and SGS/23/2013. We thank the anonymous referee for his or her comments and suggestions that helped to improve the paper.
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[^1]: The appropriate explicit formulae determining the function $r_{\mathrm{ISCO}}$ are too long to be included in this Letter. They are given, e.g., in [@abr-etal:2003] or [@tor-etal:2008].
[^2]: This rather brief description of situation does not include recent observations of the pulsar J0348+0432 with mass $2.01 \pm 0.04 M_\odot$, which currently represents the most massive confirmed neutron star [@Ant-etal:2013:].
|
---
abstract: 'We study rigidity properties of lattices in ${\operatorname{Isom}}({\mathbf{H}}^n)\simeq {\mathrm{SO}}_{n,1}({{\mathbb R}})$, $n\ge 3$, and of surface groups in ${\operatorname{Isom}}({\mathbf{H}}^2)\simeq {\mathrm{SL}}_{2}({{\mathbb R}})$ in the context of *integrable measure equivalence*. The results for lattices in ${\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$, are generalizations of Mostow rigidity; they include a *cocycle* version of strong rigidity and an integrable measure equivalence classification. For surface groups integrable measure equivalence rigidity is obtained via a cocycle version of the Milnor-Wood inequality. The integrability condition appears in certain (co)homological tools pertaining to bounded cohomology. Some of these homological tools are developed in a companion paper [@sobolev].'
address:
- 'The Technion, Haifa'
- 'University of Illinois at Chicago, Chicago'
- 'University of Chicago, Chicago'
author:
- Uri Bader
- Alex Furman
- Roman Sauer
title: Integrable measure equivalence and rigidity of hyperbolic lattices
---
Introduction and statement of the main results {#sec:introduction}
==============================================
Introduction {#sub:discussion_of_results}
------------
*Measure equivalence* is an equivalence relation on groups, introduced by Gromov [@gromov-invariants] as a measure-theoretic counterpart to quasi-isometry of finitely generated groups. It is intimately related to *orbit equivalence* in ergodic theory, to the theory of von Neumann algebras, and to questions in descriptive set theory. The study of rigidity in measure equivalence or orbit equivalence goes back to Zimmer’s paper [@zimmer-csr], which extended Margulis’ superrigidity of higher rank lattices [@Margulis:1974:ICM] to the context of *measurable cocycles* and applied it to problems of orbit equivalence. The study of measure equivalence and related problems has recently experienced a rapid growth, with [@FurmanOEw; @FurmanME; @Gaboriau-cost; @gaboriau-l2; @MonodShalom; @hjorth; @Ioana+Peterson+Popa; @popa-betti; @popa:wI; @popa:wII; @popa-cocycle; @popa-spectralgap; @Kida:thesis; @Kida:ME; @ioana] being only a partial list of important advances. We refer to [@Shalom:2005ECM; @Popa:ICM; @Furman:MGT] for surveys and further references.
One particularly fruitful direction of research in this area has been in obtaining the complete description of groups that are measure equivalent to a given one from a well understood class of groups. This has been achieved for lattices in simple Lie groups of higher rank [@FurmanME], products of hyperbolic-like groups [@MonodShalom], mapping class groups [@Kida:thesis; @Kida:ME; @Kida:OE], and certain amalgams of groups as above [@Kida:amalgamated]. In all these results, the measure equivalence class of one of such groups turns out to be small and to consist of a list of “obvious” examples obtained by simple modifications of the original group. This phenomenon is referred to as *measure equivalence rigidity*. On the other hand, the class of groups measure equivalent to lattices in ${\mathrm{SL}}_2({{\mathbb R}})$ is very rich: it is uncountable, includes wide classes of groups and does not seem to have an explicit description (cf. [@Gaboriau:2005exmps; @bridson]).
In the present paper we obtain measure equivalence rigidity results for lattices in the least rigid family of simple Lie groups ${\operatorname{Isom}}({\mathbf{H}}^n)\simeq {\mathrm{SO}}_{n,1}({{\mathbb R}})$ for $n\ge 2$, including surface groups, albeit within a more restricted category of *integrable measure equivalence*, hereafter also called *${\mathrm{L}}^1$-measure equivalence* or just *${\mathrm{L}}^1$-ME*. Let us briefly state the classification result, before giving the precise definitions and stating more detailed results.
\[T:ME-rigidity2and3\] Let $\Gamma$ be a lattice in $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 2$; in the case $n=2$ assume that $\Gamma$ is cocompact. Then the class of all finitely generated groups that are ${\mathrm{L}}^1$-measure equivalent to $\Gamma$ consists of those $\Lambda$, which admit a short exact sequence $\{1\}\to F\to\Lambda\to\bar\Lambda\to\{1\}$ where $F$ is finite and $\bar\Lambda$ is a lattice in $G$; in the case $n=2$, $\bar\Lambda$ is cocompact in $G={\operatorname{Isom}}({\mathbf{H}}^2)$.
The integrability assumption is necessary for the validity of the rigidity results for cocompact lattices in ${\operatorname{Isom}}({\mathbf{H}}^2)\cong{\mathrm{PGL}}_2({{\mathbb R}})$. It remains possible, however, that the ${\mathrm{L}}^1$-integrability assumption is superfluous for lattices in ${\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$. We also note that a result of Fisher and Hitchman [@fisher] can be used to obtain ${\mathrm{L}}^2$-ME rigidity results similar to Theorem \[T:ME-rigidity2and3\] for the family of rank one Lie groups ${\operatorname{Isom}}({\mathbf{H}}_{{{\mathbb H}}}^n)\simeq{\rm Sp}_{n,1}({{\mathbb R}})$ and ${\operatorname{Isom}}({\mathbf{H}}_{{{\mathbb O}}}^2)\simeq F_{4(-20)}$[^1]; here we do not know either whether the ${\mathrm{L}}^2$-integrability assumption is necessary.
The proof of Theorem \[T:ME-rigidity2and3\] for the case $n\ge 3$ proceeds through a cocycle version of Mostow’s strong rigidity theorem stated in Theorems \[T:SOn1-1-taut\] and \[T:SOn1-L1-cocycle\]. This cocycle version relates to the original Mostow’s strong rigidity theorem in the same way in which Zimmer’s cocycle superrigidity theorem relates to the original Margulis’ superrigidity for higher rank lattices. Our proof of the cocycle version of Mostow rigidity, which is inspired by Gromov-Thurston’s proof of Mostow rigidity using simplicial volume [@thurston] and Burger-Iozzi’s proof for dimension $3$ [@burger+iozzi2], heavily uses bounded cohomology and other homological methods. A major part of the relevant homological technique (like *Sobolev homology*), which applies to general Gromov hyperbolic groups, is developed in the companion paper [@sobolev]; *in fact, Theorem \[thm:main result about induction in cohomology\] taken from [@sobolev] is the only place in this paper where we require the integrability assumption.*
Theorem \[T:ME-rigidity2and3\] and the more detailed Theorem \[T:ME-rigidity\] are deduced from the strong rigidity for integrable cocycles (Theorem \[T:SOn1-1-taut\]) using a general method described in Theorem \[T:reconstruction\]. The latter extends and streamlines the approach developed in [@FurmanME], and further used in [@MonodShalom] and in [@Kida:ME].
The proof of Theorem \[T:ME-rigidity2and3\] for surfaces uses a cocycle version of the fact that an abstract isomorphism between uniform lattices in ${\mathrm{PGL}}_2({{\mathbb R}})$ is realized by conjugation in ${\operatorname{Homeo}}(S^1)$. The proof of this generalization uses homological methods mentioned above and a cocycle version of the Milnor-Wood-Ghys phenomenon (Theorem \[thm:taut relative homeo\]), in which an integrable ME-cocycle between surface groups is conjugate to the identity map in ${\operatorname{Homeo}}(S^1)$. In the case of surfaces in Theorem \[T:ME-rigidity2and3\], this result is used together with Theorem \[T:reconstruction\] to construct a representation $\rho:\Lambda\to {\operatorname{Homeo}}(S^1)$. Additional arguments (Lemma \[L:furstenberg\] and Theorem \[T:intoPGL2\]) are then needed to deduce that $\rho(\Lambda)$ is a uniform lattice in ${\mathrm{PGL}}_2({{\mathbb R}})$.
Let us now make precise definitions and describe in more detail the main results.
Basic notions
-------------
### **Measure equivalence of locally compact groups** {#ssub:measure_equivalence_of_locally_compact_groups}
We recall the central notion of *measure equivalence* which was suggested by Gromov [@gromov-invariants]\*[0.5.E]{}. It will be convenient to work with general unimodular, locally compact second countable (lcsc) groups rather than just countable ones.
\[D:ME\] Let $G$, $H$ be unimodular lcsc groups with Haar measures $m_G$ and $m_H$. A $(G,H)$-*coupling* is a Lebesgue measure space $(\Omega,m)$ with a measurable, measure-preserving action of $G\times H$ such that there exist finite measure spaces $(X,\mu)$, $(Y,\nu)$ and equivariant measure space isomorphisms $$\label{e:ij-fd}
\begin{aligned}
&i:(G,m_G)\times (Y,\nu)\xrightarrow{\cong} (\Omega,m)\quad &\text{so that}\quad &g:i(g',y)\mapsto i(gg',y),\\
&j:(H,m_H)\times (X,\mu)\xrightarrow{\cong} (\Omega,m)\quad &\text{so that}\quad &h: j(h',x)\mapsto j(hh',x),
\end{aligned}$$ for $g,g'\in G$ and $h,h'\in H$. Groups which admit such a coupling are said to be *measure equivalent* (abbreviated ME).
In the case where $G$ and $H$ are countable groups, the condition on the commuting actions $G{\curvearrowright}(\Omega,m)$ and $H{\curvearrowright}(\Omega,m)$ is that they admit finite $m$-measure Borel fundamental domains $X,Y\subset \Omega$ with $\mu=m|_X$ and $\nu=m|_Y$ being the restrictions.
As the name suggest, measure equivalence is an equivalence relation between unimodular lcsc groups. For reflexivity, consider the $G\times G$-action on $(G,m_G)$, $(g_1,g_2):g\mapsto g_1 g g_2^{-1}$. We refer to this as the *tautological self coupling* of $G$. The symmetry of the equivalence relation is obvious. For transitivity and more details we refer to Appendix \[subs:composition\].
\[exmp:lattices\] Let $\Gamma_1,\Gamma_2$ be lattices in a lcsc group $G$[^2]. Then $\Gamma_1$ and $\Gamma_2$ are measure equivalent, with $(G,m_G)$ serving as a natural $(\Gamma_1,\Gamma_2)$-coupling when equipped with the action $(\gamma_1,\gamma_2):g\mapsto \gamma_1 g \gamma_2^{-1}$ for $\gamma_i\in\Gamma_i$. In fact, any lattice $\Gamma<G$ is measure equivalent to $G$, with $(G,m_G)$ serving as a natural $(G,\Gamma)$-coupling when equipped with the action $(g,\gamma):g'\mapsto g g' \gamma^{-1}$.
### **Taut groups** {#ssub:subsubsection_name}
We now introduce the following key notion of taut couplings and taut groups.
\[D:M-rigidity\] A $(G,G)$-coupling $(\Omega,m)$ is *taut* if it has the tautological coupling as a factor uniquely; in other words if it admits a up to null sets unique measurable map $\Phi:\Omega\to G$ so that for $m$-a.e. $\omega\in\Omega$ and all $g_1,g_2\in G$[^3] $$\Phi((g_1,g_2)\omega)=g_1\Phi(\omega)g_2^{-1}.$$ Such a $G\times G$-equivariant map $\Omega\to G$ will be called a *tautening map*. A unimodular lcsc group $G$ is *taut* if every $(G,G)$-coupling is taut.
The requirement of uniqueness for tautening maps in the definition of taut groups is equivalent to the property that the group in question is *strongly ICC* (see Definition \[def:strongy ICC definition\]). This property is rather common; in particular it is satisfied by all center free semi-simple Lie groups and all ICC countable groups, i.e. countable groups with infinite conjugacy classes. On the other hand the existence of tautening maps for $(G,G)$-coupling is hard to obtain; in particular taut groups necessarily satisfy Mostow’s strong rigidity property.
Let $G$ be a taut unimodular lcsc group. If $\tau:\Gamma_1\xrightarrow{\cong}\Gamma_2$ is an isomorphism of two lattices $\Gamma_1$ and $\Gamma_2$ in $G$, then there exists a unique $g\in G$ so that $\Gamma_2=g^{-1} \Gamma_1 g$ and $\tau(\gamma_1)=g^{-1}\gamma g$ for $\gamma\in\Gamma_1$.
The lemma follows from considering the tautness of the measure equivalence $(G,G)$-coupling given by the $G\times G$-homogeneous space $G\times G/\Delta_\tau$, where $\Delta_\tau$ is the graph of the isomorphism $\tau:\Gamma_1\to\Gamma_2$; see Lemma \[L:taut-MR\] for details.
The phenomenon, that any isomorphism between lattices in $G$ is realized by an inner conjugation in $G$, known as *strong rigidity* or *Mostow rigidity*, holds for all simple Lie groups[^4] $G\not\simeq{\mathrm{SL}}_2({{\mathbb R}})$. More precisely, if $X$ is an irreducible non-compact, non-Euclidean symmetric space with the exception of the hyperbolic plane ${\mathbf{H}}^2$, then $G={\operatorname{Isom}}(X)$ is Mostow rigid. Mostow proved this remarkable rigidity property for uniform lattices [@Mostow:1973book]. It was then extended to the non-uniform cases by Prasad [@prasad] (${\rm rk}(X)=1$) and by Margulis [@Margulis-Mostow] (${\rm rk}(X)\ge 2$).
In the higher rank case, more precisely, if $X$ is a symmetric space without compact and Euclidean factors with ${\rm rk}(X)\ge 2$, Margulis proved a stronger rigidity property, which became known as *superrigidity* [@Margulis:1974:ICM]. Margulis’ superrigidity for lattices in higher rank, was extended by Zimmer in the cocycle superrigidity theorem [@zimmer-csr]. Zimmer’s cocycle superrigidity was used in [@FurmanME] to show that higher rank simple Lie groups $G$ are *taut* (the use of term tautness in this context is new). In [@MonodShalom] Monod and Shalom proved another case of cocycle superrigidity and proved a version of tautness property for certain products $G=\Gamma_1\times\cdots\times\Gamma_n$ with $n\ge 2$. In [@Kida:ME; @Kida:thesis] Kida proved that mapping class groups are taut. Kida’s result may be viewed as a cocycle generalization of Ivanov’s theorem [@ivanov].
### **Measurable cocycles** {#ssub:measurable_cocycles}
Let us elaborate on this connection between tautness and *rigidity* of *measurable cocycles*. Recall that a *cocycle* over a group action $G{\curvearrowright}X$ to another group $H$ is a map $c:G\times X\to H$ such that for all $g_1,g_2\in G$ $$c(g_2g_1,x)=c(g_2,g_1 x)\cdot c(g_1,x).$$ Cocycles that are independent of the space variable are precisely homomorphisms $G\to H$. One can conjugate a cocycle $c:G\times X\to H$ by a map $f:X\to H$ to produce a new cocycle $c^f:G\times X\to H$ given by $$c^f(g,x)=f(g.x)^{-1}c(g,x)f(x).$$ In our context, $G$ is a lcsc group, $H$ is lcsc or, more generally, a Polish group, and $G{\curvearrowright}(X,\mu)$ is a measurable measure-preserving action on a Lebesgue finite measure space. In this context all maps, including the cocycle $c$, are assumed to be $\mu$-measurable, and all equations should hold $\mu$-a.e.; we then say that $c$ is a *measurable cocycle*.
Let $(\Omega,m)$ be a $(G,H)$-coupling and $H\times X\xrightarrow{j}\Omega\xrightarrow{i^{-1}} G\times Y$ be as in (\[e:ij-fd\]). Since the actions $G{\curvearrowright}\Omega$ and $H{\curvearrowright}\Omega$ commute, $G$ acts on the space of $H$-orbits in $\Omega$, which is naturally identified with $X$. This $G$-action preserves the finite measure $\mu$. Similarly, we get the measure preserving $H$-action on $(Y,\nu)$. These actions will be denoted by a dot, $g:x\mapsto g.x$, $h:y\mapsto h.y$, to distinguish them from the $G\times H$ action on $\Omega$. Observe that in $\Omega$ one has $$g:j(h,x)\mapsto j(h h_1^{-1},g.x)$$ where $h_1\in H$ depends only on $g\in G$ and $x\in X$, and therefore may be denoted by $\alpha(g,x)$. One easily checks that the map $$\alpha:G\times X\to H$$ that was just defined, is a measurable cocycle. Similarly, one obtains a measurable cocycle $\beta:H\times Y\to G$. These cocycles depend on the choice of the measure isomorphisms in (\[e:ij-fd\]), but different measure isomorphisms produce conjugate cocycles. Identifying $(\Omega,m)$ with $(H,m_H)\times (X,\mu)$, the action $G\times H$ takes the form $$\label{e:me-via-coc}
(g,h):j(h',x)\mapsto j(h h'\alpha(g,x)^{-1},g.x).$$ Similarly, cocycle $\beta:H\times Y\to G$ describes the $G\times H$-action on $(\Omega,m)$ when identified with $(G,m_G)\times(Y,\nu)$. In general, we call a measurable coycle $G\times X\to H$ that arises from a $(G,H)$-coupling as above an *ME-cocycle*.
The connection between tautness and cocycle rigidity is in the observation (see Lemma \[L:coc-taut\]) that a $(G,G)$-coupling $(\Omega,m)$ is taut iff the ME-cocycle $\alpha:G\times X\to G$ is conjugate to the identity isomorphism $$\alpha(g,x)=f(g.x)^{-1}g f(x)$$ by a unique measurable $f:X\to G$. Hence one might say that $G$ is *taut* iff it satisfies a *cocycle version* of Mostow rigidity.
### **Integrability conditions** {#ssub:integrability_conditions}
Our first main result – Theorem \[T:SOn1-1-taut\] below – shows that $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge3$, are $1$-*taut* groups, i.e. all *integrable* $(G,G)$-couplings are taut. We shall now define these terms more precisely.
A *norm* on a group $G$ is a map $|\cdot|:G\to [0,\infty)$ so that $|gh|\le |g|+|h|$ and ${{\left\lvert g^{-1}\right\rvert}}={{\left\lvert g\right\rvert}}$ for all $g,h\in G$. A norm on a lcsc group is *proper* if it is measurable and the balls with respect to this norm are pre-compact. Two norms $|\cdot |$ and $|\cdot|'$ are equivalent if there are $a,b>0$ such that $|g|'\le a\cdot|g|+b$ and $|g|\le a \cdot |g|'+b$ for every $g\in G$. On a compactly generated group[^5] $G$ with compact generating symmetric set $K$ the function ${{\left\lvert g\right\rvert}}_K=\min\{n\in{{\mathbb N}}\mid g\in K^n\}$ defines a proper norm, whose equivalence class does not depend on the chosen $K$. Unless stated otherwise, we mean a norm in this equivalence class when referring to a proper norm on a compactly generated group.
\[D:Lp-ME\] Let $H$ be a compactly generated group with a proper norm ${{\left\lvert \cdot\right\rvert}}$ and $G$ be a lcsc group. Let $p\in[1,\infty]$. A measurable cocycle $c:G\times X\to H$ is ${\mathrm{L}}^p$-*integrable* if for a.e. $g\in G$ $$\int_X |c(g,x)|^p\,d\mu(x)<\infty.$$ For $p=0$ we require that the essential supremum of $|c(g,-)|$ is finite for a.e. $g\in G$. If $p=1$, we also say that $c$ is *integrable*. If $p=0$, we say that $c$ is *bounded*.
The integrability condition is independent of the choice of a norm within a class of equivalent norms. ${\mathrm{L}}^p$-integrability implies ${\mathrm{L}}^q$-integrability whenever $1\le q\le p$. In the Appendix \[sub:Lp-integrability\] we show that, if $G$ is also compactly generated, the ${\mathrm{L}}^p$-integrability of $c$ implies that the above integral is uniformly bounded on compact subsets of $G$.
A $(G,H)$-coupling $(\Omega,m)$ of compactly generated, unimodular, lcsc groups is ${\mathrm{L}}^p$-*integrable*, if there exist measure isomorphisms as in (\[e:ij-fd\]) so that the corresponding ME-cocycles $G\times X\to H$ and $H\times Y\to G$ are ${\mathrm{L}}^p$-integrable. If $p=1$ we just say that $(\Omega, m)$ is *integrable*. Groups $G$ and $H$ that admit an ${\mathrm{L}}^p$-integrable $(G,H)$-coupling are said to be ${\mathrm{L}}^p$-*measure equivalent*.
For each $p\in[1,\infty]$, being ${\mathrm{L}}^p$-measure equivalent is an equivalence relation on compactly generated, unimodular, lcsc groups (see Lemma \[L:composition-of-lp-coc\]). Furthermore, ${\mathrm{L}}^p$-measure equivalence implies ${\mathrm{L}}^{q}$-measure equivalence if $1\le q\le p$. So among the ${\mathrm{L}}^p$-measure equivalence relations, ${\mathrm{L}}^\infty$-measure equivalence is the strongest and ${\mathrm{L}}^1$-measure equivalence is the weakest one; all being subrelations of the (unrestricted) measure equivalence.
A lcsc group $G$ is $p$-*taut* if every ${\mathrm{L}}^p$-integrable $(G,G)$-coupling is taut.
The definition of ${\mathrm{L}}^p$-integrability for couplings is motivated by the older definition of *${\mathrm{L}}^p$-integrability for lattices*, which e.g. plays an important role in [@shalom] and which we recall here. Let $\Gamma<G$ be a lattice, where $\Gamma$ is a finitely generated and $G$ is a compactly generated group. Then $\Gamma$ is ${\mathrm{L}}^p$-*integrable* if $(G,m_G)$ is an ${\mathrm{L}}^p$-coupling. Equivalently, if there exists a Borel cross-section $s:G/\Gamma\to G$ of the projection, so that the cocycle $c:G\times G/\Gamma\to\Gamma$, $c(g,x)=s(g.x)^{-1}gs(x)$ is ${\mathrm{L}}^p$-integrable. Note also that ${\mathrm{L}}^\infty$-integrable lattices are precisely the uniform ones.
Statement of the main results {#sub:statement_of_the_main_results}
-----------------------------
\[T:SOn1-1-taut\] The groups $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$, are $1$-taut.
This result has an equivalent formulation in terms of cocycles.
\[T:SOn1-L1-cocycle\] Let $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$, $G{\curvearrowright}(X,\mu)$ be a probability measure preserving action, and $c:G\times X\to G$ be an integrable ME-cocycle. Then there is a measurable map $f:X\to G$, which is unique up to null sets, such that for $\mu$-a.e. $x\in X$ and every $g\in G$ we have $$c(g,x)=f(g.x)^{-1}\,g\,f(x).$$
Note that this result generalizes Mostow-Prasad rigidity for lattices in these groups. This follows from the fact that any $1$-taut group satisfies Mostow rigidity for $L^1$-integrable lattices, and the fact, due to Shalom, that all lattices in groups $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$, are ${\mathrm{L}}^1$-integrable.
\[thm:Shalom and lattices\] All lattices in simple Lie groups not locally isomorphic to ${\operatorname{Isom}}({\mathbf{H}}^2)\simeq{\mathrm{PSL}}_2({{\mathbb R}})$, ${\operatorname{Isom}}({\mathbf{H}}^3)\simeq{\mathrm{PSL}}_2({{\mathbb C}})$, are ${\mathrm{L}}^2$-integrable, hence also ${\mathrm{L}}^1$-integrable. Further, lattices in ${\operatorname{Isom}}({\mathbf{H}}^3)$ are ${\mathrm{L}}^1$-integrable.
The second assertion is not stated in this form in [@shalom]\*[Theorem 3.6]{} but the proof therein shows exactly that. In fact, for lattices in ${\operatorname{Isom}}({\mathbf{H}}^n)$ Shalom shows ${\mathrm{L}}^{n-1-\epsilon}$-integrability.
Lattices in $G={\operatorname{Isom}}({\mathbf{H}}^2)\cong{\mathrm{PGL}}_2({{\mathbb R}})$, such as surface groups, admit a rich space of deformations – the Teichmüller space. In particular, these groups do not satisfy Mostow rigidity, and therefore are not taut (they are not even $\infty$-taut). However, it is well known viewing $G={\operatorname{Isom}}({\mathbf{H}}^2)\cong{\mathrm{PGL}}_2({{\mathbb R}})$ as acting on the circle $S^1\cong \partial{\mathbf{H}}^2\cong {{{\mathbb R}}\operatorname{P}}^1$, any abstract isomorphism $\tau:\Gamma\to\Gamma'$ between cocompact lattices $\Gamma,\Gamma'<G$ can be realized by a conjugation in ${\operatorname{Homeo}}(S^1)$, that is, $$\exists f\in {\operatorname{Homeo}}(S^1)~\forall \gamma\in\Gamma\colon\pi\circ\tau(\gamma)=f^{-1}\circ \pi(\gamma)\circ f,$$ where $\pi:G\to {\operatorname{Homeo}}(S^1)$ is the imbedding as above. Motivated by this observation we generalize the notion of tautness as follows.
\[D:tautrel\] Let $G$ be a unimodular lcsc group, ${\mathcal{G}}$ a Polish group, $\pi:G\to{\mathcal{G}}$ a continuous homomorphism. A $(G,G)$-coupling is *taut relative* to $\pi:G\to{\mathcal{G}}$ if there exists a up to null sets unique measurable map $\Phi:\Omega\to {\mathcal{G}}$ such that for $m$-a.e. $\omega\in\Omega$ and all $g_1, g_2\in G$ $$\Phi((g_1,g_2)\omega)=\pi(g_1)\Phi(\omega)\pi(g_2)^{-1}.$$ We say that $G$ is *taut* (resp. $p$-*taut*) *relative to* $\pi:G\to{\mathcal{G}}$ if all (resp. all ${\mathrm{L}}^p$-integrable) $(G,G)$-couplings are taut relative to $\pi:G\to{\mathcal{G}}$.
Observe that $G$ is taut iff it is taut relative to itself. Note also that if $\Gamma<G$ is a lattice, then $G$ is taut iff $\Gamma$ is taut relative to the inclusion $\Gamma<G$; and $G$ is taut relative to $\pi:G\to{\mathcal{G}}$ iff $\Gamma$ is taut relative to $\pi|_\Gamma:\Gamma\to{\mathcal{G}}$ (Proposition \[P:taut-lattice\]). If $\Gamma<G$ is ${\mathrm{L}}^p$-integrable, then these equivalences apply to $p$-tautness.
\[thm:taut relative homeo\] The group $G={\operatorname{Isom}}({\mathbf{H}}^2)\cong{\mathrm{PGL}}_2({{\mathbb R}})$ is $1$-taut relative to the natural embedding $G<{\operatorname{Homeo}}(S^1)$. Cocompact lattices $\Gamma<G$ are $1$-taut relative to the embedding $\Gamma<G<{\operatorname{Homeo}}(S^1)$.
We skip the obvious equivalent cocycle reformulation of this result.
\[R:circle-taut\]
1. The ${\mathrm{L}}^1$-assumption cannot be dropped from Theorem \[thm:taut relative homeo\]. Indeed, the free group $\mathbf{F}_2$ can be realized as a lattice in ${\mathrm{PSL}}_2({{\mathbb R}})$, but most automorphisms of $\mathbf{F}_2$ cannot be realized by homeomorphisms of the circle.
2. Realizing isomorphisms between surface groups in ${\operatorname{Homeo}}(S^1)$, one obtains somewhat regular maps: they are Hölder continuous and quasi-symmetric. We do not know (and do not expect) Theorem \[thm:taut relative homeo\] to hold with ${\operatorname{Homeo}}(S^1)$ being replaced by the corresponding subgroups.
We now state the ${\mathrm{L}}^1$-ME rigidity result which is deduced from Theorem \[T:SOn1-1-taut\], focusing on the case of countable, finitely generated groups.
\[T:ME-rigidity\] Let $G={\operatorname{Isom}}({\mathbf{H}}^n)$ with $n\ge 3$, and $\Gamma<G$ be a lattice. Let $\Lambda$ be a finitely generated group, and let $(\Omega,m)$ be an integrable $(\Gamma,\Lambda)$-coupling. Then
1. there exists a short exact sequence $$1\to F\to \Lambda\to\bar\Lambda\to 1$$ where $F$ is finite and $\bar{\Lambda}$ is a lattice in $G$,
2. and a measurable map $\Phi:\Omega\to G$ so that for $m$-a.e. $\omega\in\Omega$ and every $\gamma\in\Gamma$ and every $\lambda\in\Lambda$ $$\Phi((\gamma,\lambda)\omega)=\gamma \Phi(\omega) \bar\lambda^{-1}.$$
Moreover, if $\Gamma\times\Lambda{\curvearrowright}(\Omega,m)$ is ergodic, then
1. either the push-forward measure $\Phi_\ast m$ is a positive multiple of the Haar measure $m_G$ or $m_{G^0}$;
2. or, one may assume that $\Gamma$ and $\bar\Lambda$ share a subgroup of finite index and $\Phi_*m$ is a positive multiple of the counting measure on the double coset $\Gamma e\bar\Lambda\subset G$.
This result is completely analogous to the higher rank case considered in [@FurmanME], except for the ${\mathrm{L}}^1$-assumption. We do not know whether Theorem \[T:ME-rigidity\] remains valid in the broader ME category, that is, without the ${\mathrm{L}}^1$-condition, but should point out that if the ${\mathrm{L}}^1$ condition can be removed from Theorem \[T:SOn1-1-taut\] then it can also be removed from Theorem \[T:ME-rigidity\].
Theorem \[T:ME-rigidity\] can also be stated in the broader context of unimodular lcsc groups, in which case the ${\mathrm{L}}^1$-measure equivalence rigidity states that a compactly generated unimodular lcsc group $H$ that is ${\mathrm{L}}^1$-measure equivalent to $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$, admits a short exact sequence $1\to K\to H\to \bar{H}\to 1$ where $K$ is compact and $\bar{H}$ is either $G$, or its index two subgroup $G^0$, or is a lattice in $G$.
Measure equivalence rigidity results have natural consequences for *(stable, or weak) orbit equivalence* of essentially free probability measure-preserving group actions (cf. [@FurmanOEw; @MonodShalom; @Kida:OE; @popa-cocycle]). Two probability measure preserving actions $\Gamma{\curvearrowright}(X,\mu)$, $\Lambda{\curvearrowright}(Y,\nu)$ are *weakly*, or *stably*, *orbit equivalent* if there exist measurable maps $p:X\to Y$, $q:Y\to X$ with $p_*\mu\ll \nu$, $q_*\nu\ll \mu$ so that a.e. $$p(\Gamma.x)\subset \Lambda.p(x),\quad q(\Lambda.y)\subset \Gamma.q(y),\qquad
q\circ p(x)\in\Gamma.x,\quad p\circ q(y)\in \Lambda.y.$$ (see [@FurmanOEw]\*[§2]{} for other equivalent definitions). If $\Gamma_1,\Gamma_2$ are lattices in some lcsc group $G$, then $\Gamma_1{\curvearrowright}G/\Gamma_2$ and $\Gamma_2{\curvearrowright}G/\Gamma_1$ are stably orbit equivalent via $p(x)=s_1(x)^{-1}$, $q(y)=s_2(y)^{-1}$, where $s_i:G/\Gamma_i\to G$ are measurable cross-sections. Moreover, given any (essentially) free, ergodic, probability measure preserving (p.m.p.) action $\Gamma_1{\curvearrowright}(X_1,\mu_1)$ and $\Gamma_1$-equivariant quotient map $\pi_1:X_1\to G/\Gamma_2$, there exists a canonically defined free, ergodic p.m.p. action $\Gamma_2{\curvearrowright}(X_2,\mu_2)$ with equivariant quotient $\pi_2:X_2\to G/\Gamma_1$ so that $\Gamma_i{\curvearrowright}(X_i,\mu_i)$ are stably orbit equivalent in a way compatible to $\pi_i:X_i\to G/\Gamma_{3-i}$ [@FurmanOEw]\*[Theorem C]{}.
We shall now introduce integrability conditions on weak orbit equivalence, assuming $\Gamma$ and $\Lambda$ are finitely generated groups. Let $|\cdot|_\Gamma$, $|\cdot|_\Lambda$ denote some word metrics on $\Gamma$, $\Lambda$ respectively, and let $\Gamma{\curvearrowright}(X,\mu)$ be an essentially free action. Define an extended metric $d_\Gamma:X\times X\to [0,\infty]$ on $X$ by setting $d_\Gamma(x_1,x_2)=|\gamma |_\Gamma$ if $\gamma. x_1=x_2$ and set $d_\Gamma(x_1,x_2)=\infty$ otherwise. Let $d_\Lambda$ denote the extended metric on $Y$, defined in a similar fashion. We say that $\Gamma{\curvearrowright}(X,\mu)$ and $\Lambda{\curvearrowright}(Y,\nu)$ are ${\mathrm{L}}^p$*-weakly/stably orbit equivalent*, if there exists maps $p:X\to Y$, $q:Y\to X$ as above, and such that for every $\gamma\in \Gamma$, $\lambda\in\Lambda$ $$\bigl(x\mapsto d_\Lambda(p(\gamma. x),p(x))\bigr)\in {\mathrm{L}}^p(X,\mu),\qquad
\bigl(x\mapsto d_\Gamma(q(\lambda. y),q(y))\bigr)\in {\mathrm{L}}^p(Y,\nu).$$ Note that the last condition is independent of the choice of word metrics.
The following result[^6] is deduced from Theorem \[T:ME-rigidity\] in essentially the same way Theorems A and C in [@FurmanOEw] are deduced from the corresponding measure equivalence rigidity theorem in [@FurmanME]. The only additional observation is that the constructions respect the integrability conditions.
\[T:L1OE-rigidity\] Let $G={\operatorname{Isom}}({\mathbf{H}}^n)$ where $n\ge 3$, and $\Gamma<G$ be a lattice. Assume that there is a finitely generated group $\Lambda$ and essentially free, ergodic, p.m.p actions $\Gamma{\curvearrowright}(X,\mu)$ and $\Lambda{\curvearrowright}(Y,\nu)$, which admit a stable ${\mathrm{L}}^1$-orbit equivalence $p:X,\to Y$, $q:y\to X$ as above. Then either one the following two cases occurs:
Virtual isomorphism
: There exists a short exact sequence $1\to F\to \Lambda\to\bar\Lambda\to 1$, where $F$ is a finite group and $\bar\Lambda<G$ is a lattice with $\Delta=\Gamma\cap\bar\Lambda$ having finite index in both $\Gamma$ and $\bar\Lambda$, and an essentially free ergodic p.m.p action $\Delta{\curvearrowright}(Z,\zeta)$ so that $\Gamma{\curvearrowright}(X,\mu)$ is isomorphic to the induced action $\Gamma{\curvearrowright}\Gamma\times_\Delta (Z,\zeta)$, and the quotient action $\bar\Lambda{\curvearrowright}(\bar{Y},\bar\nu)=(Y,\nu)/F$ is isomorphic to the induced action $\bar\Lambda{\curvearrowright}\bar\Lambda\times_\Delta (Z,\zeta)$, or
Standard quotients
: There exists a short exact sequence $1\to F\to \Lambda\to\bar\Lambda\to 1$, where $F$ is a finite group and $\bar\Lambda<G$ is a lattice, and for $G'=G$ or $G'=G^0$ (only if $\bar\Lambda, \Gamma<G^0$), and equivariant measure space quotient maps $$\qquad\pi:(X,\mu)\to (G'/\bar\Lambda,m_{G'/\bar\Lambda}),\qquad
\sigma:(Y,\nu)\to (G'/\Gamma,m_{G'/\Gamma})$$ with $\pi(\gamma.x)=\gamma.\pi(x)$, $\sigma(\lambda.y)=\bar\lambda.\sigma(y)$. Moreover, the action $\bar\Lambda{\curvearrowright}(\bar Y,\bar\nu)=(Y,\nu)/F$ is isomorphic to the canonical action associated to $\Gamma{\curvearrowright}(X,\mu)$ and the quotient map $\pi:X\to G'/\bar\Lambda$.
The family of rank one simple real Lie groups ${\operatorname{Isom}}({\mathbf{H}}^n)$ is the least rigid one among simple Lie groups. As higher rank simple Lie groups are rigid with respect to measure equivalence, one wonders about the remaining families of simple real Lie groups: ${\operatorname{Isom}}({\mathbf{H}}^n_{{\mathbb C}})\simeq{\mathrm{SU}}_{n,1}({{\mathbb R}})$, ${\operatorname{Isom}}({\mathbf{H}}^n_{{\mathbb H}})\simeq{\mathrm{Sp}}_{n,1}({{\mathbb R}})$, and the exceptional group ${\operatorname{Isom}}({\mathbf{H}}^2_{{{\mathbb O}}})\simeq F_{4(-20)}$. The question of measure equivalence rigidity (or ${\mathrm{L}}^p$-measure equivalence rigidity) for the former family remains open, but the latter groups are rigid with regard to ${\mathrm{L}}^2$-measure equivalence. Indeed, recently, using harmonic maps techniques (after Corlette [@corlette] and Corlette-Zimmer [@corlette+zimmer]), Fisher and Hitchman [@fisher] proved an ${\mathrm{L}}^2$-cocycle superrigidity result for isometries of quaternionic hyperbolic space ${\mathbf{H}}^n_{{{\mathbb H}}}$ and the Cayley plane ${\mathbf{H}}^2_{{{\mathbb O}}}$. This theorem can be used to deduce the following.
The rank one Lie groups ${\operatorname{Isom}}({\mathbf{H}}^n_{{{\mathbb H}}})\simeq{\mathrm{Sp}}_{n,1}({{\mathbb R}})$ and ${\operatorname{Isom}}({\mathbf{H}}^2_{{{\mathbb O}}})\simeq F_{4(-20)}$ are $2$-taut.
Using this result as an input to the general machinery described above one obtains:
The conclusions of Theorems \[T:ME-rigidity\] and \[T:L1OE-rigidity\] hold for all lattices in ${\operatorname{Isom}}({\mathbf{H}}^n_{{{\mathbb H}}})$ and ${\operatorname{Isom}}({\mathbf{H}}^2_{{{\mathbb O}}})$ provided the ${\mathrm{L}}^1$-conditions are replaced by ${\mathrm{L}}^2$-ones.
Organization of the paper
-------------------------
The rigidity properties of general taut groups, including Theorem \[T:reconstruction\], are proved in Section \[sec:measure\_equivalence\]. The generalizations of Mostow rigidity, Theorem \[T:SOn1-1-taut\], and cocycle version of Milnor-Wood-Ghys phenomenon, Theorem \[thm:taut relative homeo\], are proved in Section \[sec:mostow\_rigidity\_for\_maximal\_cocycles\] using the homological methods. In Section \[sec:proofs\_of\_the\_main\_results\] the remaining results stated in the introduction are deduced, including the ${\mathrm{L}}^1$-measure equivalence rigidity of surface groups (Theorem \[T:ME-rigidity2and3\] case $n=2$). General facts about measure equivalence which are used throughout the paper are collected in the Appendix \[sec:appendix\_measure\_equivalence\].
Acknowledgements
----------------
Uri Bader and Alex Furman were supported in part by the BSF grant 2008267. Uri Bader was also supported in part by the ISF grant number 704/08 and the GIF grant 2191-1826.6/2007. Alex Furman was also supported in part by the NSF grants DMS 0604611 and 0905977. Roman Sauer acknowledges support from the *Deutsche Forschungsgemeinschaft*, made through grant SA 1661/1-2.
Measure equivalence rigidity for taut groups {#sec:measure_equivalence}
============================================
This section contains general tools related to the notion of taut couplings and taut groups. The results of this section apply to general unimodular lcsc groups, including countable groups, and are not specific to ${\operatorname{Isom}}({\mathbf{H}}^n)$ or semi-simple Lie groups. Whenever we refer to ${\mathrm{L}}^p$-integrability conditions, we assume that the groups are also compactly generated. We shall rely on some basic facts about measure equivalence which are collected in Appendix \[sec:appendix\_measure\_equivalence\].
The basic tool is the following:
\[T:reconstruction\] Let $G$ be a unimodular lcsc group that is taut (resp. $p$-taut). Any unimodular lcsc group $H$ that is measure equivalent (resp. ${\mathrm{L}}^p$-measure equivalent) to $G$ admits a short exact sequence with continuous homomorphisms $$1\to K\to H\to \bar{H}\to 1,$$ where $K$ is compact and $\bar{H}$ is a closed subgroup in $G$ such that $G/\bar{H}$ carries a $G$-invariant Borel probability measure.
Theorem \[T:split+hom\] below contains a more general statement.
The strong ICC property and strongly proximal actions {#sub:the_strong_icc_property_and_strongly_proximal_actions}
-----------------------------------------------------
We need to introduce a notion of *strongly ICC* group $G$ and, more generally, the notion of a group ${\mathcal{G}}$ being strongly ICC relative to a subgroup ${\mathcal{G}}_0<{\mathcal{G}}$.
\[def:strongy ICC definition\] A Polish group ${\mathcal{G}}$ is *strongly ICC relative* to ${\mathcal{G}}_0<{\mathcal{G}}$ if ${\mathcal{G}}\setminus\{e\}$ does not support any Borel probability measure that is invariant under the conjugation action of ${\mathcal{G}}_0$ on ${\mathcal{G}}\setminus\{e\}$. A Polish group ${\mathcal{G}}$ is *strongly ICC* if it is strongly ICC relative to itself.
The key properties of this notion are discussed in the appendix \[sub:strong\_icc\_property\].
\[exa:examples of strongly icc groups\] The basic examples of the strong ICC property include the following:
1. A countable group $\Gamma$ is strongly ICC iff it is an ICC[^7] group.
2. Center-free semi-simple Lie groups $G$ without compact factors are examples of non-discrete strongly ICC groups. In fact, such groups are strongly ICC relative to any unbounded Zariski dense subgroup (cf. [@locally-compact Proof of Theorem 2.3]).
3. The Polish group ${\operatorname{Homeo}}(S^1)$ is strongly ICC relative to ${\mathrm{PGL}}_2({{\mathbb R}})$ (see Lemma \[L:sICCinHomeo\]).
Let $M$ be a compact metrizable space. Recall that a continuous action $G{\curvearrowright}M$ is *minimal and strongly proximal* if the following equivalent conditions hold:
1. for every Borel probability measure $\nu\in{\operatorname{Prob}}(M)$ and every non-empty open subset $V\subset M$ one has $$\sup_{g\in G} g_*\nu(V)= 1.$$
2. for every $\nu\in{\operatorname{Prob}}(M)$ the convex hull of the $G$-orbit $g_*\nu$ is dense in ${\operatorname{Prob}}(M)$ in the weak-\* topology.
Recall that this condition is satisfied by the standard action of $G={\mathrm{PGL}}_2({{\mathbb R}})$ and its lattices on the circle. More generally, any connected semi-simple center free group $G$ without compact factors $G$ acts on $M=G/Q$ where $Q$ is a parabolic in a minimal and strongly proximal fashion [@Margulis:book]\*[Theorem 3.7 on p. 205]{}.
\[L:sICCinHomeo\] Let $M$ be a compact metrizable space and $G<{\operatorname{Homeo}}(M)$ be a subgroup which acts minimally and strongly proximally on $M$. Then ${\operatorname{Homeo}}(M)$ is strongly ICC relative to $G$.
Let $\mu$ be a probability measure on ${\operatorname{Homeo}}(M)$. The set of $\mu$-*stationary* probability measures on $M$ $${\operatorname{Prob}}_\mu(M)=\Bigl\{ \nu\in{\operatorname{Prob}}(M) \mid \nu=\mu*\nu=\int f_*\nu\,d\mu(f) \Bigr\}$$ is a non-empty convex closed (hence compact) subset of ${\operatorname{Prob}}(M)$, with respect to the weak-\* topology. Suppose $\mu$ is invariant under conjugations by $g\in G$. Since $$g_*(\mu*\nu)=\mu^g*(g_*\nu)=\mu*(g_*\nu)$$ it follows that ${\operatorname{Prob}}_\mu(M)$ is a $G$-invariant set. Minimality and strong proximality of the $G$-action implies that ${\operatorname{Prob}}_\mu(M)={\operatorname{Prob}}(M)$. In particular, every Dirac measure $\nu_x$ is $\mu$-stationary; hence $\mu\{ f \mid f(x)=x\}=1$. It follows that $\mu=\delta_e$.
Tautness and the passage to self couplings {#sub:tautness_and_the_passage_to_self_couplings}
------------------------------------------
\[T:split+hom\] Let $G$, $H$ be unimodular lcsc groups. Let $(\Omega,m)$ be a $(G,H)$-coupling. Let ${\mathcal{G}}$ be a Polish group and $\pi:G\to{\mathcal{G}}$ a continuous homomorphism. Assume that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$ and the $(G,G)$-coupling $\Omega\times_H{{\Omega}^*}$ is taut relative to $\pi$.
Then there exists a continuous homomorphism $\rho:H\to {\mathcal{G}}$ and a measurable map $\Psi:\Omega\to {\mathcal{G}}$ so that a.e.: $$\Psi((g,h)\omega) = \pi(g)\Psi(\omega)\rho(h)^{-1}\qquad (g\in G,\ h\in H)$$ and the unique tautening map $\Phi:\Omega\times_H{{\Omega}^*}\to {\mathcal{G}}$ is given by $$\Phi([\omega_1,\omega_2]) = \Psi(\omega_1)\cdot\Psi(\omega_2)^{-1}.$$ The pair $(\Psi,\rho)$ is unique up to conjugations $(\Psi^{x}, \rho^{x})$ by $x\in{\mathcal{G}}$, where $$\Psi^{x}(\omega)=\Psi(\omega)x^{-1},\qquad \rho^{x}(h)=x\rho(h)x^{-1}.$$ If, in addition, $\pi:G\to{\mathcal{G}}$ has compact kernel and closed image $\bar{G}=\pi(G)$, then the same applies to $\rho:H\to{\mathcal{G}}$, and there exists a Borel measure $\bar{m}$ on ${\mathcal{G}}$, which is invariant under $$(g,h): x\mapsto \pi(g)x\rho(h)$$ and descends to finite measures on $\pi(G){\backslash}{\mathcal{G}}$ and ${\mathcal{G}}/\rho(H)$. In other words, $({\mathcal{G}},\bar{m})$ is a $(\pi(G),\rho(H))$-coupling which is a quotient of $({\operatorname{Ker}}(\pi)\times{\operatorname{Ker}}(\rho)){\backslash}(\Omega,m)$.
We shall first construct a homomorphism $\rho:H\to {\mathcal{G}}$ and the $G\times H$-equivariant map $\Psi:\Omega\to {\mathcal{G}}$. Consider the space $\Omega^3=\Omega\times\Omega\times\Omega$ and the three maps $p_{1,2},p_{2,3}$, $p_{1,3}$, where $$p_{i,j}\colon\Omega^3\ {{\buildrel{}\over\longrightarrow}}\ \Omega^2\ {{\buildrel{}\over\longrightarrow}}\ \Omega \times_H {{\Omega}^*}$$ is the projection to the $i$-th and $j$-th factor followed by the natural projection. Consider the $G^3\times H$-action on $\Omega^3$: $$(g_1,g_2,g_3,h):(\omega_1,\omega_2,\omega_3)\mapsto ((g_1,h)\omega_1,(g_2,h)\omega_2,(g_3,h)\omega_3).$$ For $i\in\{1,2,3\}$ denote by $G_i$ the corresponding $G$-factor in $G^3$. For $i,j\in \{1,2,3\}$ with $i\neq j$ the group $G_i\times G_j<G_1\times G_2\times G_3$ acts on $\Omega\times_H {{\Omega}^*}$ and on ${\mathcal{G}}$ by $$(g_i,g_j):\,[\omega_1,\omega_2]\mapsto [g_i\omega_1,g_j\omega_2],
\qquad (g_i,g_j):x\mapsto \pi(g_i)\,x\,\pi(g_j)^{-1}\qquad(x\in{\mathcal{G}})$$ respectively. Let $\{i,j,k\}=\{1,2,3\}$. The map $p_{i,j}:\Omega^3\to \Omega\times_H{{\Omega}^*}$ is $G_k\times H$-invariant and $G_i\times G_j$-equivariant. This is also true of the maps $$F_{i,j}=\Phi\circ p_{i,j}\colon\Omega^3 \xrightarrow{p_{i,j}} \Omega\times_H{{\Omega}^*}
\xrightarrow{\Phi} {\mathcal{G}},$$ where $\Phi:\Omega\times_H{{\Omega}^*}\to {\mathcal{G}}$ is the tautening map. For $\{i,j,k\}=\{1,2,3\}$, the three maps $F_{i,j}$, $F_{j,i}^{-1}$ and $F_{i,k}\cdot F_{k,j}$ are all $G_k\times H$-invariant, hence factor through the natural map $$\Omega^3\rightarrow\Sigma_k=(G_k\times H){\backslash}\Omega^3.$$ By an obvious variation on the argument in Appendix \[ssub:composition\_of\_couplings\] one verifies that $\Sigma_k$ is a $(G_i, G_j)$-coupling. The three maps $F_{i,j}$, $F_{j,i}^{-1}$ and $F_{i,k}\cdot F_{k,j}$ are $G_i\times G_j$-equivariant. Since ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$, there is at most one $G_i\times G_j$-equivariant measurable map $\Sigma_k\to {\mathcal{G}}$ according to Lemma \[L:sICC2uniq\]. Therefore, we get $m^3$-a.e. identities $$\label{e:Fij}
F_{i,j}=F_{j,i}^{-1}=F_{i,k}\cdot F_{k,j}.$$ Denote by $\bar{\Phi}:\Omega^2\to {\mathcal{G}}$ the composition $\Omega^2{{\buildrel{}\over\longrightarrow}} \Omega\times_H {{\Omega}^*}{{\buildrel{\Phi}\over\longrightarrow}} {\mathcal{G}}$. By Fubini’s theorem, (\[e:Fij\]) implies that for $m$-a.e. $\omega_2\in \Omega$, for $m\times m$-a.e. $(\omega_1,\omega_3)$ $$\bar{\Phi}(\omega_1,\omega_3)
=\bar{\Phi}(\omega_1,\omega_2)\cdot \bar\Phi(\omega_2,\omega_3)
=\bar{\Phi}(\omega_1,\omega_2)\cdot\bar{\Phi}(\omega_3,\omega_2)^{-1}.$$ Fix such a generic $\omega_2\in \Omega$ and define $\Psi:\Omega\to {\mathcal{G}}$ by $\Psi(\omega)=\bar\Phi(\omega,\omega_2)$. Then for a.e. $[\omega,\omega']\in \Omega\times_H\Omega$ $$\label{e:PhiPsi}
\Phi([\omega,\omega'])=\bar{\Phi}(\omega,\omega')=\Psi(\omega)\cdot\Psi(\omega')^{-1}.$$ We proceed to construct a representation $\rho:H\to {\mathcal{G}}$. Equation (\[e:PhiPsi\]) implies that for every $h\in H$ and for a.e $\omega,\omega'\in \Omega$: $$\Psi(h \omega)\Psi(h \omega')^{-1}=\bar{\Phi}(h \omega,h \omega')
=\bar{\Phi}(\omega,\omega')=\Psi(\omega)\Psi(\omega')^{-1},$$ and in particular, we get $$\Psi(h\omega)^{-1}\Psi(\omega)=\Psi(h\omega')^{-1}\Psi(\omega').$$ Observe that the left hand side is independent of $\omega'\in\Omega$, while the right hand side is independent of $\omega\in\Omega$. Hence both are $m$-a.e. constant, and we denote by $\rho(h)\in{\mathcal{G}}$ the constant value. Being coboundaries the above expressions are cocycles; being independent of the space variable they give a homomorphism $\rho:H\to {\mathcal{G}}$. To see this explicitly, for $h,h'\in H$ we compute using $m$-a.e. $\omega\in\Omega$: $$\begin{aligned}
\rho(hh')&=\Psi(hh' \omega)^{-1}\Psi(\omega)\\
&=\Psi(hh' \omega)^{-1}\Psi(h' \omega) \Psi(h' \omega)^{-1}\Psi(\omega)\\
&=\rho(h)\rho(h').\end{aligned}$$ Since the homomorphism $\rho$ is measurable, it is also continuous [@zimmer-book Theorem B.3 on p. 198]. By definition of $\rho$ we have for $h\in H$ and $m$-a.e $\omega\in\Omega$: $$\label{e:psi-h-eq}
\Psi(h \omega)=\Psi(\omega)\rho(h)^{-1}.$$ Since $\Psi(\omega)=\bar\Phi(\omega,\omega_2)$, it also follows that for $g\in G$ and $m$-a.e. $\omega\in \Omega$ $$\label{e:psi-g-eq}
\Psi(g \omega)=\pi(g)\Psi(\omega).$$ Consider the collection of all pairs $(\Psi,\rho)$ satisfying (\[e:psi-h-eq\]) and (\[e:psi-g-eq\]). Clearly, ${\mathcal{G}}$ acts on this set by $x:(\Psi,\rho)\mapsto (\Psi^{x},\rho^{x})=(\Psi\cdot x,x^{-1} \rho x)$; and we claim that this action is transitive. Let $(\Psi_i,\rho_i)$, $i=1,2$, be two such pairs in the above set. Then $$\tilde\Phi_i(\omega,\omega')=\Psi_i(\omega)\Psi_i(\omega')^{-1}\qquad (i=1,2)$$ are $G\times G$-equivariant measurable maps $\Omega\times\Omega\to{\mathcal{G}}$, which are invariant under $H$. Hence they descend to $G\times G$-equivariant maps $\Phi_i:\Omega\times_H{{\Omega}^*}\to{\mathcal{G}}$. The assumption that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$, implies a.e. identities $\Phi_1=\Phi_2$, $\tilde\Phi_1=\tilde\Phi_2$. Hence for a.e. $\omega,\omega'$ $$\Psi_1(\omega)^{-1}\Psi_2(\omega)=\Psi_1(\omega')^{-1}\Psi_2(\omega').$$ Since the left hand side depends only on $\omega$, while the right hand side only on $\omega'$, it follows that both sides are a.e. constant $x\in{\mathcal{G}}$. This gives $\Psi_1=\Psi_2^x$. The a.e. identity $$\Psi_1(\omega)\rho_1(h)=\Psi_1(h^{-1}\omega)=\Psi_2(h^{-1}\omega)x
=\Psi_2(\omega)\rho_2(h)x=\Psi_1(\omega)x^{-1}\rho_2(h)x$$ implies $\rho_1=\rho_2^x$. This completes the proof of the first part of the theorem.
Next, we assume that ${\operatorname{Ker}}(\pi)$ is compact and $\pi(G)$ is closed in ${\mathcal{G}}$, and will show that the kernel $K={\operatorname{Ker}}(\rho)$ is compact, the image $\bar{H}=\rho(H)$ is closed in ${\mathcal{G}}$, and that ${\mathcal{G}}/\bar{H}$, $\pi(G){\backslash}{\mathcal{G}}$ carry finite measures. These properties will be deduced from the assumption on $\pi$ and the existence of the measurable map $\Psi:\Omega\to {\mathcal{G}}$ satisfying (\[e:psi-h-eq\]) and (\[e:psi-g-eq\]). We need the next lemma, which says that $\Omega$ has measure space isomorphisms as in (\[e:ij-fd\]) with special properties.
\[lem:convenient fundamental domains\] Let $\rho:H\to {\mathcal{G}}$ and $\Psi:\Omega\to {\mathcal{G}}$ be as above. Then there exist measure space isomorphisms $i:G\times Y\cong \Omega$ and $j:H\times X\cong \Omega$ as in (\[e:ij-fd\]) that satisfy in addition $$\Psi(i(g,y))=\pi(g),\qquad \Psi(j(h,x))=\rho(h).$$
We start from some measure space isomorphisms $i_0:G\times Y\cong \Omega$ and $j_0:H\times X\cong \Omega$ as in (\[e:ij-fd\]) and will replace them by $$i(g,y)=i_0(gg_y,y),\qquad j(h,x)=j_0(hh_x,x)$$ for some appropriately chosen measurable maps $Y\to G$, $y\mapsto g_y$ and $X\to H$, $x\mapsto h_x$. The conditions (\[e:ij-fd\]) remain valid after any such alteration.
Let us construct $y\mapsto g_y$ with the required property; the map $x\mapsto h_x$ can be constructed in a similar manner. By (\[e:psi-g-eq\]) for $m_G\times\nu$-a.e. $(g_1,y)\in G\times Y$ the value $$\pi(g)^{-1}\Psi\circ i_0(gg_1,y)$$ is $m_G$-a.e. independent of $g$; denote it by $f(g_1,y)\in {\mathcal{G}}$. Fix $g_1\in G$ for which $$\Psi\circ i_0(gg_1,y)=\pi(g) f(g_1,y)$$ holds for $m_G$-a.e. $g\in G$ and $\nu$-a.e. $y\in Y$. There exists a Borel cross section ${\mathcal{G}}\to G$ to $\pi:G\to{\mathcal{G}}$. Using such, we get a measurable choice for $g_y$ so that $$\pi(g_y)=f(g_1,y)^{-1}\pi(g_1).$$ Setting $i(g,y)=i_0(g g_y,y)$, we get $m_G\times\nu$-a.e. that $\Psi\circ i (g,y)=\pi(g)$.
\[L:cover\] Given a neighborhood of the identity $V\subset H$ and a compact subset $Q\subset {\mathcal{G}}$, the set $\rho^{-1}(Q)$ can be covered by finitely many translates of $V$: $$\rho^{-1}(Q)\subset h_1 V\cup\cdots\cup h_N V.$$
Since $\pi:G\to{\mathcal{G}}$ is assumed to be continuous, having closed image and compact kernel, for any compact $Q\subset{\mathcal{G}}$ the preimage $\pi^{-1}(Q)\subset G$ is also compact. Let $W\subset H$ be an open neighborhood of identity so that $W\cdot W^{-1}\subset V$; we may assume $W$ has compact closure in $H$. Then $\pi^{-1}(Q)\cdot W$ is precompact. Hence there is an open precompact set $U\subset G$ with $\pi^{-1}(Q)\cdot W\subset U$. Consider subsets $A=j(W\times X)$, and $B=i(U\times Y)$ of $\Omega$, where $i$ and $j$ are as in the previous lemma. Then $$m(A)=m_H(W)\cdot \nu(Y) >0,\qquad m(B)=m_G(U)\cdot \mu(X) <\infty.$$ Let $\{h_1,\dots,h_n\}\subset \rho^{-1}(Q)$ be such that $h_kW\cap h_l W=\emptyset$ for $k\ne l\in\{1,\dots,n\}$. Then the sets $h_k A=j(h_k W\times X)$ are also pairwise disjoint and have $m(h_k A)=m(A)>0$ for $1\le k\le n$. Since $$\Psi(h_k A)=\rho(h_k W)=\rho(h_k)\rho(W)\subset Q\cdot\rho(W)\subset \rho(U),$$ it follows that $h_kA\subset B$ for every $1\le k\le n$. Hence $n\le m(B)/m(A)$. Choosing a *maximal* such set $\{h_1,\dots,h_N\}$, we obtain the desired cover.
Lemma \[L:cover\] implies that the closed subgroup $K={\operatorname{Ker}}(\rho)$ is compact. More generally, it implies that the continuous homomorphism $\rho:H\to {\mathcal{G}}$ is proper, that is, preimages of compact sets are compact. Therefore $\bar{H}=\rho(H)$ is closed in ${\mathcal{G}}$.
We push forward the measure $m$ to a measure $\bar{m}$ on ${\mathcal{G}}$ via the map $\Psi:\Omega\to {\mathcal{G}}$. The measure $\bar{m}$ is invariant under the action $x\mapsto \pi(g)\,x\,\rho(h)$. Since $\bar{H}=\rho(H)\cong H/\ker(\rho)$ is closed in ${\mathcal{G}}$, the space ${\mathcal{G}}/\bar{H}$ is Hausdorff. As ${\operatorname{Ker}}(\rho)$ and ${\operatorname{Ker}}(\pi)$ are compact normal subgroups in $H$ and $G$, respectively, the map $\Psi:\Omega\to {\mathcal{G}}$ factors through $$(\Omega,m){{\buildrel{}\over\longrightarrow}} (\Omega',m')=({\operatorname{Ker}}(\pi)\times {\operatorname{Ker}}(\rho)){\backslash}(\Omega,m)\xrightarrow{\Psi'}{\mathcal{G}}.$$ Let $\bar{G}=G/{\operatorname{Ker}}(\pi)$. Starting from measure isomorphisms as in Lemma \[lem:convenient fundamental domains\], we obtain equivariant measure isomorphisms $(\Omega',m')\cong (\bar{H}\times X,m_{\bar{H}}\times\mu)$ and $(\Omega',m')\cong (\bar{G}\times Y,m_{\bar{G}}\times\nu)$. In particular, $(\Omega',m')$ is a $(\bar{G},\bar{H})$-coupling. The measure $\bar{m}$ on ${\mathcal{G}}$ descends to the $\bar{G}$-invariant finite measure on ${\mathcal{G}}/\bar{H}$ obtained by pushing forward $\mu$. Similarly, $\bar{m}$ descends to the $\bar{H}$-invariant finite measure on $\bar{G}{\backslash}{\mathcal{G}}$ obtained by pushing forward $\nu$. This completes the proof of Theorem \[T:split+hom\].
Theorem \[T:reconstruction\] immediately follows from Theorem \[T:split+hom\]. In case of ${\mathrm{L}}^p$-conditions, one observes that if $(\Omega,m)$ is an ${\mathrm{L}}^p$-integrable $(G,H)$-coupling, then $\Omega\times_H{{\Omega}^*}$ is an ${\mathrm{L}}^p$-integrable $(G,G)$-coupling (Lemma \[lem:composition of lp couplings\]); so it is taut under the assumption that $G$ is $p$-taut.
Lattices in taut groups {#sub:lattices_in_taut_groups}
-----------------------
\[P:taut-lattice\] Let $G$ be a unimodular lcsc group, ${\mathcal{G}}$ a Polish group, $\pi:G\to{\mathcal{G}}$ a continuous homomorphism. Assume that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$.
Then $G$ is taut (resp. $p$-taut) relative to $\pi:G\to{\mathcal{G}}$ iff $\Gamma$ is taut (resp. $p$-taut) relative to $\pi|_\Gamma:\Gamma\to{\mathcal{G}}$.
In particular, $G$ is taut iff any/all lattices in $G$ are taut relative to the inclusion $\Gamma<G$.
For the proof of this proposition we shall need the following.
\[L:induction-taut\] Let $G$ be a unimodular lcsc group, ${\mathcal{G}}$ a Polish group, $\pi:G\to{\mathcal{G}}$ a continuous homomorphism, and $\Gamma_1,\Gamma_2<G$ lattices. Let $(\Omega,m)$ be a $(\Gamma_1,\Gamma_2)$-coupling, and assume that the $(G,G)$-coupling $\bar\Omega=G\times_{\Gamma_1}\Omega\times_{\Gamma_2} G$ is taut relative to $\pi:G\to{\mathcal{G}}$. Then there exists a $\Gamma_1\times\Gamma_2$-equivariant map $\Omega\to {\mathcal{G}}$.
It is convenient to have a concrete model for $\bar{\Omega}$. Choose Borel cross-sections $\sigma_i$ from $X_i=G/\Gamma_i$ to $G$, and form the cocycles $c_i:G\times X_i\to \Gamma_i$ by $$c_i(g,x)=\sigma_i(g.x)^{-1}g\sigma_i(x),\qquad (i=1,2).$$ Then, suppressing the obvious measure from the notations, $\bar{\Omega}$ identifies with $X_1\times X_2\times\Omega$, while the $G\times G$-action is given by $$(g_1,g_2): (x_1,x_2,\omega)\mapsto (g_1.x_1,g_2.x_2, (\gamma_1,\gamma_2)\omega)
\qquad\text{where}\qquad \gamma_i=c_i(g_i,x_i).$$ By the assumption there exists a measurable map $\bar\Phi:\bar\Omega\to {\mathcal{G}}$ so that $$\bar\Phi((g_1,g_2)(x_1,x_2,\omega))
=\pi(g_1)\cdot \bar\Phi(x_1,x_2,\omega)\cdot \pi(g_2)^{-1}\qquad (g_1,g_2\in G)$$ for a.e. $(x_1,x_2,\omega)\in\bar\Omega$. Fix a generic pair $(x_1,x_2)\in X_1\times X_2$, denote $h_i=\sigma_i(x_i)$ and consider $g_i=\gamma_i^{h_i}$ ($=h_i\gamma_i h_i^{-1}$), where $\gamma_i\in\Gamma_i$ for $i\in\{1,2\}$. Then $g_i.x_i=x_i$, $c_i(g_i,x_i)=\gamma_i$ and the map $\Phi':\Omega\to{\mathcal{G}}$ defined by $\Phi'(\omega)=\bar\Phi(x_1,x_2,\omega)$ satisfies $m$-a.e. $$\begin{aligned}
\Phi'((\gamma_1,\gamma_2)\omega)=\bar\Phi((g_1,g_2)(x_1,x_2,\omega))
&=\pi(g_1)\cdot \Phi'(\omega)\cdot \pi(g_2)^{-1}\\
&=\pi(\gamma_1^{h_1})\cdot \Phi'(\omega)\cdot \pi(\gamma_2^{h_2})^{-1}.\end{aligned}$$ Thus $\Phi(\omega)=\pi(h_1)^{-1} \Phi'(\omega) \pi(h_2)$ is a $\Gamma_1\times\Gamma_2$-equivariant measurable map $\Omega\to {\mathcal{G}}$, as required.
Assuming that $G$ is taut relative to $\pi:G\to{\mathcal{G}}$ and $\Gamma<G$ is a lattice, we shall show that $\Gamma$ is taut relative to $\pi|_\Gamma:\Gamma\to{\mathcal{G}}$.
Let $(\Omega,m)$ be a $(\Gamma,\Gamma)$-coupling. Then the $(G,G)$-coupling $\bar\Omega=G\times_{\Gamma}\Omega\times_{\Gamma} G$ is taut relative to ${\mathcal{G}}$, and by Lemma \[L:induction-taut\], $\Omega$ admits a $\Gamma\times\Gamma$-tautening map $\Phi:\Omega\to{\mathcal{G}}$. Since ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)<{\mathcal{G}}$, the map $\Phi:\Omega\to {\mathcal{G}}$ is unique as a $\Gamma\times\Gamma$-equivariant map (Lemma \[L:sICC2uniq\].(\[i:2lattice\])). This shows that $\Gamma$ is taut relative to ${\mathcal{G}}$.
Observe, that if $G$ is assumed to be only $p$-taut, while $\Gamma<G$ to be ${\mathrm{L}}^p$-integrable, then the preceding argument for the existence of $\Gamma\times\Gamma$-tautening map for a ${\mathrm{L}}^p$-integrable $(\Gamma,\Gamma)$-coupling $\Omega$ still applies. Indeed, the composed coupling $\bar\Omega=G\times_{\Gamma}\Omega\times_{\Gamma} G$ is then ${\mathrm{L}}^p$-integrable and therefore admits a $G\times G$-tautening map $\bar{\Phi}:\bar\Omega\to{\mathcal{G}}$, leading to a $\Gamma\times\Gamma$-tautening map $\Phi:\Omega\to {\mathcal{G}}$.
Next assume that $\Gamma<G$ is a lattice and $\Gamma$ is taut (resp. $p$-taut) relative to $\pi|_\Gamma:\Gamma\to{\mathcal{G}}$. Let $(\Omega,m)$ be a $(G,G)$-coupling (resp. a ${\mathrm{L}}^p$-integrable one). Then $(\Omega,m)$ is also a $(\Gamma,\Gamma)$-coupling (resp. a ${\mathrm{L}}^p$-integrable one). Since $\Gamma$ is assumed to be taut (resp. $p$-taut) there is a $\Gamma\times\Gamma$-equivariant map $\Phi:\Omega\to{\mathcal{G}}$. As ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$ it follows from (\[i:extn\]) in Lemma \[L:sICC2uniq\] that $\Phi:\Omega\to {\mathcal{G}}$ is automatically $G\times G$-equivariant. The uniqueness of tautening maps follows from the strong ICC assumption.
The explicit assumption that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$ is superfluous. If no integrability assumptions are imposed, the strong ICC follows from the tautness assumption by Lemma \[L:uniq2sICC\]. However, if one assumes merely $p$-tautness, the above lemma yields strong ICC property for a restricted class of measures; and the argument that this is sufficient becomes unjustifiably technical in this case.
The isometry group of hyperbolic space is $1$-taut {#sec:mostow_rigidity_for_maximal_cocycles}
==================================================
Proofs of Theorems \[T:SOn1-1-taut\] and \[thm:taut relative homeo\] {#sub:proof_of_theorem_BC}
--------------------------------------------------------------------
We prove Theorems \[T:SOn1-1-taut\] and \[thm:taut relative homeo\] relying on the results of Subsections \[sub:the\_cohomological\_induction\_map\_for\_integrable\_me\_couplings\]–\[sub:preserving maximal simplices\]. Throughout, let $G={\operatorname{Isom}}({\mathbf{H}}^n)$ be the isometry group of hyperbolic $n$-space. We assume that $n\ge 2$. Let $${\mathcal{G}}=\begin{cases}
G&\text{ if $n\ge 3$,}\\
{\operatorname{Homeo}}(S^1)&\text{ if $n=2$.}
\end{cases}$$ Further, we define $$G\xrightarrow{\pi}{\mathcal{G}}=\begin{cases}
G\xrightarrow{{\operatorname{id}}} G&\text{ if $n\ge 3$,}\\
\text{standard action of $G$ on $\partial{\mathbf{H}}^2\cong S^1$}&\text{ if $n=2$.}
\end{cases}$$ Theorems \[T:SOn1-1-taut\] and \[thm:taut relative homeo\] state that $G$ is $1$-taut relative to $\pi\colon G\to{\mathcal{G}}$.
### **Reduction to cocycles of lattices and ergodicity** {#ssub:reduction_to_cocycles_of_lattices}
Let $\Gamma<G^0$ be a torsion-free uniform lattice in the connected component $G^0\subset G$ of the identity. By Proposition \[P:taut-lattice\] it suffices to show that $\Gamma$ is $1$-taut relative to $\pi=\pi\vert_\Gamma\colon\Gamma\to{\mathcal{G}}$. Let $(\Omega,m)$ be an integrable $(\Gamma,\Gamma)$-coupling. It is sometimes convenient to denote the left copy of $\Gamma$ by $\Gamma_l$ and the right copy by $\Gamma_r$.
By Lemma \[L:coc-taut\] the $(\Gamma,\Gamma)$-coupling $\Omega$ is taut relative to $\pi$ if and only if there is an essentially unique measurable map $f\colon X\to{\mathcal{G}}$ such that a.e. $$\label{eq:conjugation}
\pi\circ\alpha(\gamma,x)=f(\gamma.x)\pi(\gamma) f(x)^{-1},$$ that is, the cocycle $\pi\circ\alpha$ is conjugate to the constant cocycle $\pi$. By [@fisher-nonergodic]\*[Corollary 3.6]{}[^8] it is sufficient to prove on a.e. ergodic component of $X$, each of which corresponds to an ergodic component in the ergodic decomposition $(\Omega,m_t)$ of the coupling $\Omega$ [@FurmanME]\*[Lemma 2.2]{} where $m=\int m_td\eta(t)$ and $\eta$ some probability measure. Let $\alpha\colon\Gamma_r\times X\to\Gamma_l$ be an integrable ME-cocycle associated to a $\Gamma_l$-fundamental domain $X\subset\Omega$. Let ${{\left\lvert \_\right\rvert}}\colon\Gamma\to{{\mathbb N}}$ be the length function associated to some word-metric on $\Gamma$. Then the integrability of $(\Omega,m)$ means that for every $\gamma\in\Gamma$ $$\int\int_X{{\left\lvert \alpha(\gamma,x)\right\rvert}}dm_t(x)d\eta(t)=\int_X{{\left\lvert \alpha(\gamma,x)\right\rvert}}dm(x)<\infty,$$ which yields that $\int_X{{\left\lvert \alpha(\gamma,x)\right\rvert}}dm_t(x)<\infty$ for $\eta$-a.e. $t$. Hence $(\Omega,m_t)$ is integrable for $\eta$-a.e. $t$. Thus we may assume for the rest of the proof that $(\Omega,m)$ is an ergodic and integrable $(\Gamma,\Gamma)$-coupling. By [@sobolev]\*[Corollary 1.11]{}, the coupling index of $\Omega$ is $1$.
### **Volume cocycle** {#ssub:volume_cocycle}
We identify the boundary at infinity $\partial{\mathbf{H}}^n$ with $B=G/P$ and endow it with the measure class of the push-forward of the Haar measure on $G$. In the functorial theory of bounded cohomology as developed by Burger-Monod [@burger+monod; @monod-book], the measurable map $${\operatorname{dvol}}_b\colon B^{n+1}\rightarrow{{\mathbb R}}$$ that assigns to $(b_0,\dots,b_n)$ the oriented volume of the geodesic, ideal simplex with vertices $b_0,\dots,b_n$ is a $\Gamma$-invariant (even $G^0$-invariant) cocycle and defines an element ${\operatorname{dvol}}_b\in{\mathrm{H}_\mathrm{b}}^n(\Gamma,{{\mathbb R}})$ (Theorem \[thm:boundary resolutions by Burger-Monod\]). The forgetful map (comparison map) from bounded cohomology to ordinary cohomology is denoted by $${\operatorname{comp}}^\bullet\colon{\mathrm{H}_\mathrm{b}}^\bullet(\Gamma,{{\mathbb R}})\to{\mathrm{H}}^\bullet(\Gamma,{{\mathbb R}}).$$ By Theorem \[thm:bounded volume cocycle to volume coycle\] the bounded cocycle ${\operatorname{dvol}}_b$ is a lift of the volume cocycle ${\operatorname{dvol}}\in{\mathrm{H}}^n(\Gamma,{{\mathbb R}})\cong {\mathrm{H}}^n(\Gamma{\backslash}{\mathbf{H}}^n,{{\mathbb R}})$ of the $n$-dimensional closed manifold $\Gamma{\backslash}{\mathbf{H}}^n$, that is, $${\operatorname{comp}}^n({\operatorname{dvol}}_b)={\operatorname{dvol}}.$$
### **A higher-dimensional Milnor-Wood inequality** {#ssub:milnor-wood}
To show the existence of $f$ as in (\[eq:conjugation\]), we consider the induction homomorphism $${\mathrm{H}_\mathrm{b}}^\bullet(\Omega)\colon {\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_l,{\mathrm{L}}^\infty(\Gamma_r{\backslash}\Omega))\to
{\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_r,{\mathrm{L}}^\infty(\Gamma_l{\backslash}\Omega))$$ in bounded cohomology associated to $\Omega$ (see Subsection \[sub:the\_cohomological\_induction\_map\_for\_integrable\_me\_couplings\]). Let $$\begin{aligned}
{\mathrm{H}_\mathrm{b}}^\bullet(j^\bullet)&\colon {\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_l,{{\mathbb R}})\to{\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_l,{\mathrm{L}}^\infty(\Gamma_r{\backslash}\Omega))\\
{\mathrm{H}_\mathrm{b}}^\bullet({\mathrm{I}}^\bullet)&\colon {\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_r,{\mathrm{L}}^\infty(\Gamma_l{\backslash}\Omega))\to {\mathrm{H}_\mathrm{b}}^\bullet(\Gamma_r,{{\mathbb R}})\end{aligned}$$ be the homomorphisms induced by inclusion of constant functions in the coefficients and by integration in the coefficients, respectively. Inspired by the classical Euler number of a surface representation we define:
\[eq:Euler number\] Let $[\Gamma]\in {\mathrm{H}}_n(\Gamma,{{\mathbb R}})$ be the homological fundamental class of the manifold $\Gamma{\backslash}{\mathbf{H}}^n$. The *Euler number* ${\operatorname{eu}}(\Omega)$ of $\Omega$ is the evaluation of the cohomology class ${\operatorname{comp}}^n\circ{\mathrm{H}_\mathrm{b}}^n({\mathrm{I}}^\bullet)\circ{\mathrm{H}_\mathrm{b}}^n(\Omega)\circ{\mathrm{H}_\mathrm{b}}^n(j^\bullet)({\operatorname{dvol}}_b)$ against the fundamental class $[\Gamma]$ $${\operatorname{eu}}(\Omega)=\bigl\langle {\operatorname{comp}}^n\circ{\mathrm{H}_\mathrm{b}}^n({\mathrm{I}}^\bullet)\circ{\mathrm{H}_\mathrm{b}}^n(\Omega)\circ{\mathrm{H}_\mathrm{b}}^n(j^\bullet)({\operatorname{dvol}}_b), [\Gamma]\bigr\rangle.$$
For any $(\Gamma,\Gamma)$-coupling $\Omega$ (without assuming integrability) we prove the following *higher-dimensional Milnor-Wood inequality* in Theorem \[thm:evaluation by fundamental class\]: $$\label{eq:Milnor Wood}
{{\left\lvert {\operatorname{eu}}(\Omega)\right\rvert}}\le{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n).$$
### **Maximality of the Euler number provided $\Omega$ is integrable** {#ssub:maximality_of_the_euler_number_provided_omega_is_integrable}
Next we appeal to the following general result from our companion paper [@sobolev], which relies on the integrability of the coupling. In fact, this is the only place in the proof where we use the integrability.
\[thm:main result about induction in cohomology\] Let $M$ and $N$ be closed, oriented, negatively curved manifolds of dimension $n$. Let $(\Omega,\mu)$ be an ergodic, integrable ME-coupling $(\Omega,\mu)$ of the fundamental groups $\Gamma=\pi_1(M)$ and $\Lambda=\pi_1(N)$ with coupling index $c=\frac{\mu({\Lambda}{\backslash}\Omega)}{\mu({\Gamma}{\backslash}\Omega)}$. Suppose that $x_\Gamma^b\in{\mathrm{H}_\mathrm{b}}^n(\Gamma,{{\mathbb R}})$ is an element that maps to the cohomological fundamental class $x_G\in{\mathrm{H}}^n(\Gamma,{{\mathbb R}})\cong {\mathrm{H}}^n(M,{{\mathbb R}})$ of $M$ under the comparision map. Define $x_\Lambda\in{\mathrm{H}}^n(\Lambda,{{\mathbb R}})$ analogously. Then the composition $$\begin{gathered}
\label{eq:composition induction cohomological}
{\mathrm{H}_\mathrm{b}}^n(\Gamma, {{\mathbb R}})\xrightarrow{{\mathrm{H}_\mathrm{b}}^n(j^\bullet)} {\mathrm{H}_\mathrm{b}}^n(\Gamma,{\mathrm{L}}^\infty(\Lambda{\backslash}\Omega,{{\mathbb R}}))
\xrightarrow{{\mathrm{H}_\mathrm{b}}^n(\Omega)}{\mathrm{H}_\mathrm{b}}^n(\Lambda,{\mathrm{L}}^\infty(\Gamma{\backslash}\Omega,{{\mathbb R}}))\\
\xrightarrow{{\mathrm{H}_\mathrm{b}}^n({\mathrm{I}}^\bullet)} {\mathrm{H}_\mathrm{b}}^n(\Lambda,{{\mathbb R}})\xrightarrow{{\operatorname{comp}}^n} {\mathrm{H}}^n(\Lambda,{{\mathbb R}})
\end{gathered}$$ sends $x_\Gamma^b$ to $\pm c\cdot x_\Lambda$.
We apply this theorem to $M=N=\Gamma{\backslash}{\mathbf{H}}^n$ and $\Lambda=\Gamma$. In our case the coupling index is $1$. Therefore the bounded class ${\operatorname{dvol}}_b$ is mapped to $\pm{\operatorname{dvol}}$ under . In other words, the Euler class of $\Omega$ is maximal: $$\label{eq:eval against fundamental class}
{\operatorname{eu}}(\Omega)=\pm{\operatorname{vol}}\bigl(\Gamma{\backslash}{\mathbf{H}}^n\bigr).$$
### **Boundary maps** {#ssub:boundary_maps}
Next we want to express in terms of the *boundary map* associated to the cocycle $\alpha$. Boundary theory, in the sense of Furstenberg [@Furstenberg] (see [@burger-mozes]\*[Corollary 3.2]{}, or [@MonodShalom-cocycle]\*[Proposition 3.3]{} for a detailed argument applying to our situation), yields the existence of an essentially unique measurable map, called *boundary map* or *Furstenberg map*, $$\label{eq:boundary map}
\phi:X\times B\to B
\qquad\text{satisfying}\qquad
\phi(\gamma x,\gamma b)=\alpha(\gamma, x)\phi(x,b)$$ for every $\gamma\in\Gamma$ and a.e. $(x,b)\in X\times B$.
To deal with some measurability issues we need the following construction. For a standard Borel probability space $(S,\nu)$ and a Polish space $W$ we consider the set ${\mathrm{F}}(S,W)$ of measurable functions $S\to W$, where two functions are identified if they agree on $\nu$-conull set. One can endow ${\mathrm{F}}(S,W)$ with the topology of convergence in measure. The Borel algebra of this topology turns ${\mathrm{F}}(S,W)$ into a standard Borel space [@fisher-nonergodic]\*[Section 2A]{}. Two different Polish topologies on $W$ with the same Borel algebra give rise to the same standard Borel space ${\mathrm{F}}(S,W)$ [@fisher-nonergodic]\*[Remark 2.5]{}.
In our situation, the map $\phi$ gives rise to a measurable map $f:X\to{\mathrm{F}}(B,B)$ defined for almost every $x\in X$ by $f(x)= \phi(x,\_)$ [@fisher-nonergodic]\*[Corollary 2.9]{}. We also write $f(x)=\phi_x$.
Theorem \[thm:evaluation by fundamental class\] below, allows us to express the Euler class ${\operatorname{eu}}(\Omega)$ in terms of the boundary map $\phi$, and interpret the equality in as $$\int_X\int_{G^0/\Gamma}
{\operatorname{vol}}\bigl(\phi_x(gz_0),\dots,\phi_x(gz_n)\bigr)\,dg\,d\mu(x)=\pm v_{\mathrm{max}}.$$ where $v_{\mathrm{max}}$ is the volume of a positively oriented ideal maximal simplex $(z_0,\dots,z_n)$ in $B^{n+1}$ and the quotient $G^0/\Gamma$ carries the normalized Haar measure. Since the integrand is at most $v_{\mathrm{max}}$ for a.e. $x\in X$, we conclude: Either the ideal simplex $(\phi_x(gb_0),\dots,\phi_x(gb_n))$ is non-degenerate and positively oriented for a.e. $g\in G$ and a.e. $x\in X$, or $(\phi_x(gb_0),\dots,\phi_x(gb_n))$ is non-degenerate and negatively oriented for a.e. $g\in G$ and a.e. $x\in X$.
For $n=2$ any non-degenerate ideal triangle has oriented volume $\pm v_{\mathrm{max}}$; hence either for a.e. $x\in X$ the map $\phi_x:S^1\to S^1$ preserves the cyclic order of a.e. triple of points on the circle $B=\partial{\mathbf{H}}^2=S^1$, or for a.e. $x\in X$ the map $\phi_x$ reverses the orientation of a.e. triple on the circle.
In the $n\ge 3$ case it follows that $(\phi_x(gb_0),\dots,\phi_x(gb_n))$ is a maximal, hence regular, ideal simplex for a.e. $g\in G$ and a.e. $x\in X$.
### **Conclusion** {#ssub:conclusion}
Firstly consider the case $n\ge 3$. By a general fact about standard Borel spaces the measurable injection $j\colon G\rightarrow F(B,B)$ given by the action of $G$ on $B$ by Moebius transformations is a Borel isomorphism onto its image and the image is measurable in ${\mathrm{F}}(B,B)$ [@kechris]\*[Corollary 15.2 on p. 89]{}. Lemma \[lem:key lemma in mostow rigidity\] yields that the image of $f:X\to{\mathrm{F}}(B,B)$ is contained in $j(G)$, thus $f$ can be regarded as a measurable map $X\to G$. Equation (\[eq:boundary map\]) for $\phi$ implies that $f$ satisfies equation (\[eq:conjugation\]), which concludes the proof of Theorem \[T:SOn1-1-taut\].
Next let $n=2$. Here $B=S^1$. By *loc. cit.* the measurable injective map $j\colon{\operatorname{Homeo}}(S^1)\rightarrow{\mathrm{F}}(S^1,S^1)$ is a Borel isomorphism of ${\operatorname{Homeo}}(S^1)$ onto its measurable image. By Proposition \[prop:orientation cocycle maximal\] the image of $f:X\to {\mathrm{F}}(S^1,S^1)$ is contained in the image of $j$. Thus $f$ can be regarded as a measurable map to ${\operatorname{Homeo}}(S^1)$. Again, we conclude that $f$ satisfies (\[eq:conjugation\]), which finishes the proof of Theorem \[thm:taut relative homeo\].
The cohomological induction map {#sub:the_cohomological_induction_map_for_integrable_me_couplings}
-------------------------------
The following *cohomological induction map* associated to an ME-coupling was introduced by Monod and Shalom in [@MonodShalom].
\[prop:induction from Monod-Shalom\] Let $(\Omega,m)$ be a $(\Gamma,\Lambda)$-coupling. Let $Y\subset $ be a measurable fundamental domain for the $\Gamma$-action. Let $\chi\colon\Omega\to\Gamma$ be the measurable $\Gamma$-equivariant map uniquely defined by $\chi(\omega)^{-1}\omega\in Y$ for $\omega\in\Omega$. The maps $$\begin{gathered}
{\mathrm{C}_\mathrm{b}}^\bullet(\chi)\colon
{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma,{\mathrm{L}}^\infty(\Omega))
\to{\mathrm{C}_\mathrm{b}}^\bullet(\Lambda,{\mathrm{L}}^\infty(\Omega))\\
{\mathrm{C}_\mathrm{b}}^k(\chi)(f)(\lambda_0,\dots,\lambda_k)(y)
= f\bigl(\chi(\lambda_0^{-1}y)),\dots,\chi(\lambda_k^{-1}y)\bigr)(y)
\end{gathered}$$ defines a $\Gamma\times\Lambda$-equivariant chain morphism with regard to the following actions: The $\Gamma\times\Lambda$-action on ${\mathrm{C}_\mathrm{b}}^\bullet(\Gamma,{\mathrm{L}}^\infty(\Omega))\cong {\mathrm{L}}^\infty(\Gamma^{\bullet+1}\times\Omega)$ is induced by $\Gamma$ acting diagonally on $\Gamma^{\bullet+1}\times\Omega$ and by $\Lambda$ acting only on $\Omega$. The $\Gamma\times\Lambda$-action on ${\mathrm{C}_\mathrm{b}}^\bullet(\Lambda,{\mathrm{L}}^\infty(\Omega)\cong{\mathrm{L}}^\infty(\Lambda^{\bullet+1}\times\Omega)$ is induced by $\Lambda$ acting diagonally on $\Lambda^{\bullet+1}\times\Omega$ and by $\Gamma$ acting only on $\Omega$.
The chain map ${\mathrm{C}_\mathrm{b}}^\bullet(\chi)$ induces, after taking $\Gamma\times\Lambda$-invariants and identifying ${\mathrm{L}}^\infty(\Gamma{\backslash}\Omega)$ with ${\mathrm{L}}^\infty(\Omega)^\Gamma$ and similary for $\Lambda$, an isometric isomorphism $${\mathrm{H}_\mathrm{b}}^\bullet(\chi) \colon {\mathrm{H}_\mathrm{b}}^\bullet(\Gamma,{\mathrm{L}}^\infty(\Lambda{\backslash}\Omega))
\xrightarrow{\cong}{\mathrm{H}_\mathrm{b}}^\bullet(\Lambda,{\mathrm{L}}^\infty(\Gamma{\backslash}\Omega)).$$ in cohomology. This map does not depend on the choice of $Y$, or equivalently $\chi$, and will be denoted by ${\mathrm{H}_\mathrm{b}}^\bullet(\Omega)$. We call ${\mathrm{H}_\mathrm{b}}^\bullet(\Omega)$ the *cohomological induction map associated to $\Omega$*.
Apart from the fact that the isomorphism is isometric, this is exactly Proposition 4.6 in [@MonodShalom] (with $S=\Omega$ and $E={{\mathbb R}}$). The proof therein relies on [@monod-book]\*[Theorem 7.5.3 in §7]{}, which also yields the isometry statement.
\[prop:induction reciprocity\] Retain the setting of the previous proposition. Let $\alpha\colon\Lambda\times Y\to\Gamma$ be the corresponding ME-cocycle. Let $B_\Gamma$ and $B_\Lambda$ be standard Borel spaces endowed with probability Borel measures and measure-class preserving Borel actions of $\Gamma$ and $\Lambda$, respectively. Assume the action on $B_\Lambda$ is amenable in the sense of Zimmer. Let $\phi\colon B_\Lambda\times \Gamma{\backslash}\Omega\to B_\Gamma$ be a measurable $\alpha$-equivariant map (upon identifying $Y$ with $\Gamma{\backslash}\Omega$). Then the chain morphism $$\begin{gathered}
{\mathrm{C}_\mathrm{b}}^\bullet(\phi)\colon{\mathcal{B}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})\to{{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_\Lambda^{\bullet+1}, {\mathrm{L}}^\infty(\Omega))\\
{\mathrm{C}_\mathrm{b}}^k(\phi)(f)(\dots,b_i,\dots)(\omega)=f\bigl(\dots,\chi(\omega)\phi(b_i,[\omega]),\dots\bigr).
\end{gathered}$$ is $\Gamma\times\Lambda$-equivariant with regard to the following actions: The action on ${\mathcal{B}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})$ is induced from $\Gamma$ acting diagonally $B^{\bullet+1}$ and $\Lambda$ acting trivially. The action on ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_\Lambda^{\bullet+1}, {\mathrm{L}}^\infty(\Omega))\cong {\mathrm{L}}^\infty(B_\Lambda^{\bullet+1}\times\Omega)$ is induced from $\Lambda$ acting diagonally on $B_\Lambda^{\bullet+1}\times\Omega$ and from $\Gamma$ acting only on $\Omega$.
Further, every $\Gamma\times\Lambda$-chain morphism from ${\mathcal{B}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})$ to ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_\Lambda^{\bullet+1}, {\mathrm{L}}^\infty(\Omega))$ that induces the same homomorphism on ${\mathrm{H}_\mathrm{b}}^0$ as ${\mathrm{C}_\mathrm{b}}^\bullet(\phi)$ is equivariantly chain homotopic to ${\mathrm{C}_\mathrm{b}}^\bullet(\phi)$.
Firstly, we show equivariance of ${\mathrm{C}_\mathrm{b}}^\bullet(\phi)$. By definition we have $${\mathrm{C}_\mathrm{b}}^\bullet(\phi)((\gamma,\lambda)f)(\dots,b_i,\dots)(\omega) =
f\bigl(\dots,\gamma^{-1}\chi(\omega)\phi(b_i,[\omega]),\dots\bigr).$$ By definition, $\Gamma$-equivariance of $\chi$, and $\alpha$-equivariance of $\phi$ we have $${\mathrm{C}_\mathrm{b}}^\bullet(\phi)(f)\bigl(\dots,\lambda^{-1}b_i,\dots)(\gamma^{-1}\lambda^{-1}\omega\bigr)=
f\bigl(\dots,\gamma^{-1}\chi(\lambda^{-1}\omega)\alpha(\lambda^{-1},[\omega])\phi(b_i,[\omega]),\dots\bigr).$$ It remains to check that $$\chi(\lambda^{-1}\omega)\alpha(\lambda^{-1}, [\omega])=\chi(\omega).$$ Since both sides are $\Gamma$-equivariant, we may assume that $\omega\in Y$, i.e., $\chi(\omega)=1$. In this case it follows from the defining properties of $\chi$ and $\alpha$.
Next we prove the uniqueness up to equivariant chain homotopy. By Proposition \[prop:B-complex strong resolution\] the complex ${\mathcal{B}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})$ is a strong resolution of the trivial $\Gamma\times\Lambda$-module ${{\mathbb R}}$. The $\Gamma\times\Lambda$-action on $B_\Lambda^{\bullet+1}\times\Omega$ is amenable if the $\Lambda$-action on $B_\Lambda^{\bullet+1}\times\Gamma{\backslash}\Omega$ is amenable [@adams+elliot]\*[Corollary C]{}. The latter action is amenable since the $\Lambda$-action on $B_\Lambda$ is amenable and because of [@zimmer-book]\*[Proposition 4.3.4 on p. 79]{}. By Theorem \[thm:boundary resolutions by Burger-Monod\] ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_\Lambda^{\bullet+1}, {\mathrm{L}}^\infty(\Omega))
\cong {\mathrm{L}}^\infty(B_\Lambda^{\bullet+1}\times\Omega)$ is a relatively injective, strong resolution of the trivial $\Gamma\times\Lambda$-module, and Theorem \[thm:main homological theorem\] yields uniqueness up to equivariant homotopy.
The map ${\mathrm{C}_\mathrm{b}}^\bullet(\phi)$ cannot be defined on ${\mathrm{L}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})$ since we do not assume that $\phi$ preserves the measure class. The idea to work with the complex ${\mathcal{B}}^\infty(B_\Gamma^{\bullet+1},{{\mathbb R}})$ to circumvent this problem in the context of boundary maps is due to Burger and Iozzi [@burger+iozzi].
The Euler number in terms of boundary maps {#sub:the_cohomological_induction_map_and_boundary_maps}
------------------------------------------
In the Burger-Monod approach to bounded cohomology one can realize bounded cocycles in the bounded cohomology of $\Gamma$ as cocycles on the boundary $B$. However, it is not immediately clear how the evaluation of a bounded $n$-cocycle realized on $B$ at the fundamental class of $\Gamma{\backslash}{\mathbf{H}}^n$ can be explicitly computed since the fundamental class is not defined in terms of the boundary. Lemma \[lem:image of fund class under Patterson-Sullivan map\] below achieves just that. The two important ingredients that go into its proof are Thurston’s measure homology and the cohomological Poisson transform ${\mathrm{PT}}^\bullet:
{\mathrm{L}}^\infty(B^{\bullet+1},{{\mathbb R}})\to{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma, {{\mathbb R}})$ (see Definition \[def:poisson transform\]).
\[def:visual measure\] For $z\in{\mathbf{H}}^n$ let $\nu_z$ be the *visual measure* at $z$ on the boundary $B=\partial{\mathbf{H}}^n$ at infinity, that is, $\nu_z$ is the push-forward of the Lebesgue measure on the unit tangent sphere $\mathrm{T}^1_z{\mathbf{H}}^n$ under the homeomorphism $\mathrm{T}^1_z{\mathbf{H}}^n\to\partial{\mathbf{H}}^n$ given by the exponential map. For a $(k+1)$-tuple $\sigma=(z_0,\ldots, z_k)$ of points in ${\mathbf{H}}^n$ we denote the product of the $\nu_{z_i}$ on $B^{k+1}$ by $\nu_\sigma$.
The measure $\nu_z$ is the unique Borel probability measure on $B$ that is invariant with respect to the stabilizer of $z$. All visual measures are in the same measure class. Moreover, we have $\nu_{g z}=g_\ast\nu_z=\nu_z(g^{-1}\_)$ for every $g\in G$.
\[lem:image of fund class under Patterson-Sullivan map\] Let $\Gamma\subset G^0$ be a torsion-free and uniform lattice. Let $\sigma_0=(z_0,\dots,z_n)$ be a positively oriented geodesic simplex in ${\mathbf{H}}^n$. Let $f\in {\mathrm{L}}^\infty(B^{n+1},{{\mathbb R}})^\Gamma$ be an alternating cocycle. Then $$\bigl\langle {\operatorname{comp}}^n\circ\,{\mathrm{H}_\mathrm{b}}^n({\mathrm{PT}}^\bullet)(f), [\Gamma]\bigr\rangle =
\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{{\operatorname{vol}}(\sigma_0)}\int_{\Gamma{\backslash}G^0}\int_{B^{n+1}}
f(gb_0,\dots,gb_n)\,d\nu_{\sigma_0} dg.$$
We need Thurston’s description of singular homology by *measure cycles* [@thurston]: Let $M$ be a topological space. We equip the space ${\mathcal{S}}_k(M)={\operatorname{Map}}(\Delta^k,M)$ of continuous maps from the standard $k$-simplex to $M$ with the compact-open topology. The group ${\mathrm{C}^\mathrm{m}}_k(M)$ is the vector space of all signed, compactly supported Borel measures on ${\mathcal{S}}_k(M)$ with finite total variation. The usual face maps $\partial_i: {\mathcal{S}}_k(M)\to{\mathcal{S}}_{k-1}(M)$ are measurable, and the maps ${\mathrm{C}^\mathrm{m}}_k(M)\rightarrow{\mathrm{C}^\mathrm{m}}_{k-1}(M)$ that send $\mu$ to $\sum_{i=0}^k
(-1)^i(\partial_i)_{\ast}\mu$ turn ${\mathrm{C}^\mathrm{m}}_\bullet(M)$ into a chain complex. The map $$ {\mathrm{D}}_\bullet\colon {\mathrm{C}}_\bullet(M)\rightarrow{\mathrm{C}^\mathrm{m}}_\bullet(M),~\sigma\mapsto\delta_\sigma$$ that maps a singular simplex $\sigma$ to the point measure concentrated at $\sigma$ is a chain map that induces an (isometric) homology isomorphism provided $M$ is homeomorphic to a CW-complex [@loeh; @zastrow]. We will consider the case $M=\Gamma{\backslash}{\mathbf{H}}^n$ next.
Fix a basepoint $x\in{\mathbf{H}}^n$. Consider the $\Gamma$-equivariant chain homomorphism $j_k\colon{\mathrm{C}}_k(\Gamma)\rightarrow{\mathrm{C}}_k({\mathbf{H}}^n)$ that maps $(\gamma_0,\dots,\gamma_k)$ to the geodesic simplex with vertices $(\gamma_0 x,\dots,\gamma_k x)$. Let ${\mathcal{B}}^\infty\bigl({\mathcal{S}}_\bullet({\mathbf{H}}^n),{{\mathbb R}}\bigr)\subset {\mathrm{C}}^\bullet({\mathbf{H}}^n)$ be the subcomplex of bounded measurable singular cochains on ${\mathbf{H}}^n$. The Poisson transform[^9] factorizes as $${\mathrm{L}}^\infty(B^{\bullet+1},{{\mathbb R}})\xrightarrow{{\mathrm{P}}^\bullet}{\mathcal{B}}^\infty
\bigl({\mathcal{S}}_\bullet({\mathbf{H}}^n),{{\mathbb R}}\bigr)\xrightarrow{{\mathrm{R}}^\bullet}{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma)$$ where ${\mathrm{P}}^k(l)(\sigma)=\int_{B^{k+1}}l(b_0,\dots,b_k)d\nu_{\sigma}$ and ${\mathrm{R}}^k(f)=f\circ j_k$. For every $k\ge 0$ there is a Borel section $s_k:{\mathcal{S}}_k(\Gamma{\backslash}{\mathbf{H}}^n)\rightarrow{\mathcal{S}}_k({\mathbf{H}}^n)$ of the projection [@loeh]\*[Theorem 4.1]{}. The following pairing is independent of the choice of $s_k$ and descends to cohomology: $$\begin{gathered}
\langle\_,\_\rangle_m\colon{\mathcal{B}}^\infty\bigl({\mathcal{S}}_\bullet({\mathbf{H}}^n),{{\mathbb R}}\bigr)^\Gamma\otimes {\mathrm{C}^\mathrm{m}}_\bullet(\Gamma{\backslash}{\mathbf{H}}^n)\rightarrow{{\mathbb R}}\\
\langle l,\mu\rangle_m=
\int_{{\mathcal{S}}_\bullet(\Gamma{\backslash}{\mathbf{H}}^n)}l\bigl(s_\bullet(\sigma)\bigr)d\mu(\sigma)
\end{gathered}$$ One sees directly from the definitions that for every $x\in{\mathrm{H}}_n(\Gamma)$ $$\begin{aligned}
\label{eq:fundamental cycles and adjunction}
\bigl\langle {\operatorname{comp}}^n\circ\, {\mathrm{H}}^n\bigl({\mathrm{PT}}^\bullet)(f),x\bigr\rangle &=
\bigl\langle {\operatorname{comp}}^n\circ\,{\mathrm{H}}^n\bigl({\mathrm{R}}^\bullet)\circ{\mathrm{H}}^n({\mathrm{P}}^\bullet)(f), x\bigr\rangle\\
&=\bigl\langle {\mathrm{H}}^n({\mathrm{P}}^\bullet)(f), {\mathrm{H}}_n({\mathrm{D}}_\bullet\circ j_\bullet)(x)\bigr\rangle_m. \notag
\end{aligned}$$ For any positively oriented geodesic $n$-simplex $\sigma$, let ${\operatorname{sm}}(\sigma)$ denote the push-forward of the normalized Haar measure under the measurable map $$\Gamma{\backslash}G^0\rightarrow {\operatorname{Map}}(\Delta^n, \Gamma{\backslash}{\mathbf{H}}^n),~g\mapsto
{\operatorname{pr}}(g\sigma).$$ Let $\rho\in G$ be the orientation reversing isometry that maps $(z_0,z_1,\dots,z_n)$ to $(z_1,z_0,\dots,z_n)$. By [@ratcliffe]\*[Theorem 11.5.4 on p. 551]{} the image ${\mathrm{H}}_n({\mathrm{D}}_\bullet\circ j_\bullet)([\Gamma])$ of the fundamental class $[\Gamma]\in{\mathrm{H}}_n(\Gamma)\cong {\mathrm{H}}_n(M)$ of $M$ is represented by the measure[^10] $$\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{2{\operatorname{vol}}(\sigma_0)}\bigl({\operatorname{sm}}(\sigma_0)-
{\operatorname{sm}}(\rho\circ\sigma_0)\bigr).$$ In combination with and the fact that $f$ is alternating, this yields the assertion.
The next theorem is well known to experts, and we only prove it for the lack of a good reference. Although it can be seen as a special case of Theorem \[thm:evaluation by fundamental class\] we separate the proofs. The proofs of Theorems \[thm:bounded volume cocycle to volume coycle\] and \[thm:evaluation by fundamental class\] are given at the end of the subsection.
\[thm:bounded volume cocycle to volume coycle\] Let $\Gamma\subset G^0$ be a torsion-free and uniform lattice. Then $$\bigl\langle{\operatorname{comp}}^n({\operatorname{dvol}}_b), [\Gamma]\bigr\rangle={\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n).$$ Equivalently, this means that ${\operatorname{comp}}^n({\operatorname{dvol}}_b)={\operatorname{dvol}}$.
We view the following theorem as a higher-dimensional cocycle analog of the Milnor-Wood inequality for homomorphisms of a surface group into ${\operatorname{Homeo}}_+(S^1)$.
\[thm:evaluation by fundamental class\] Let $(\Omega,m)$ be a $(\Gamma,\Gamma)$-coupling of a torsion-free and uniform lattice $\Gamma\subset G^0$. Let $\phi\colon X\times B\to B$ be the $\alpha$-equivariant boundary map from , where $\alpha\colon\Gamma_r\times X\to\Gamma_l$ is a ME-cocycle for $\Omega$. If $\sigma=(z_0,\dots,z_n)$ with $z_i\in B$ is a positively oriented ideal regular simplex, then $${\operatorname{eu}}(\Omega)=
\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{v_{\mathrm{max}}}\int_X\int_{\Gamma{\backslash}G^0}
{\operatorname{vol}}\bigl(\phi_x(gz_0),\dots,\phi_x(gz_n)\bigr)\,dg\,d\mu(x).$$ In particular, we have the inequality ${{\left\lvert {\operatorname{eu}}(\Omega)\right\rvert}}\le{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)$.
We shall need the upcoming, auxiliary Lemmas \[lem:evaluation by fund class - first step: visual measures\] and \[lem:Lebesgue\_differentiation\] before we conclude the proof of the preceding theorem at the end of this subsection. We retain the setting of Theorem \[thm:evaluation by fundamental class\] for the rest of this subsection.
\[lem:evaluation by fund class - first step: visual measures\] If $\sigma=(z_0,\dots,z_n)$ with $z_i\in{\mathbf{H}}^n$ is a positively oriented geodesic simplex, then $${\operatorname{eu}}(\Omega)=
\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{{\operatorname{vol}}(\sigma)}
\int_X\int_{G^0/\Gamma}\int_{B^{n+1}}
{\operatorname{vol}}\bigl(\phi_x(gb_0),\dots,\phi_x(gb_n)\bigr)d\nu_\sigma\,dg\,d\mu(x).$$
Consider the diagram below. The unlabelled maps are the obvious ones, sending a function to its equivalence class up to null sets and inclusion of constant functions. For better readability, we denote the copy of $B$ on which $\Gamma_l$ acts by $B_l$; similarly for $B_r$. All the maps are $\Gamma_l\times\Gamma_r$-equivariant chain morphisms as explained now. On ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_l^{\bullet+1},{{\mathbb R}})$ and ${\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_l,{{\mathbb R}})$ we have the usual $\Gamma_l$-actions and the trivial $\Gamma_r$-actions. The lower Poisson transform is then clearly $\Gamma_l\times\Gamma_r$-equivariant. The actions on the domain and target of the maps ${\mathrm{C}_\mathrm{b}}^\bullet(\chi)$ and ${\mathrm{C}_\mathrm{b}}^\bullet(\phi)$ are defined in Propositions \[prop:induction from Monod-Shalom\] and \[prop:induction reciprocity\], and is proven there that these maps are $\Gamma_l\times\Gamma_r$-equivariant. The Poisson transform in the upper row, which is $\Gamma_r$-equivariant, is also $\Gamma_l$-equivariant, since $\Gamma_l$ acts only by its natural action on $\Omega$. $$\xymatrix{
{\mathcal{B}}^\infty(B_l^{\bullet+1},{{\mathbb R}})\ar[d]\ar[r]^-{{\mathrm{C}_\mathrm{b}}^\bullet(\phi)} & {{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_r^{\bullet+1},{\mathrm{L}}^\infty(\Omega))\ar[r]^{{\mathrm{PT}}^\bullet} & {\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_r,{\mathrm{L}}^\infty(\Omega))\\
{{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_l^{\bullet+1},{{\mathbb R}})\ar[r]^{{\mathrm{PT}}^\bullet} & {\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_l,{{\mathbb R}})\ar[r]&{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_l,{\mathrm{L}}^\infty(\Omega))\ar[u]^{{\mathrm{C}_\mathrm{b}}^\bullet(\chi)}
}$$ Using Proposition \[prop:induction reciprocity\] again, one sees that the diagram commutes up to equivariant chain homotopy.
The volume cocycle ${\operatorname{dvol}}_b$, which we defined as a cocycle in ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B^{n+1},{{\mathbb R}})$, is everywhere defined and everywhere $\Gamma$-invariant and strictly satisfies the coycle condition; hence it lifts to a cocycle in ${\mathcal{B}}^\infty(B^{n+1},{{\mathbb R}})$ which we denote by ${\operatorname{dvol}}_{\mathrm{strict}}$. The commutativity of the diagram up to equivariant chain homotopy yields that $$\label{eq: another euler}
{\operatorname{eu}}(\Omega)=\bigl\langle{\operatorname{comp}}^n\circ\,{\mathrm{H}_\mathrm{b}}^n({\mathrm{I}}^\bullet)\circ
{\mathrm{H}_\mathrm{b}}^n({\mathrm{PT}}^\bullet)\circ{\mathrm{H}_\mathrm{b}}^n(\phi)({\operatorname{dvol}}_{\rm strict}),[\Gamma]\bigr\rangle.$$ The Poisson transform in the upper row after taking $\Gamma_l$-invariants followed by integration in the coefficients $${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(B_r^{\bullet+1},{\mathrm{L}}^\infty(\Gamma_l{\backslash}\Omega))\xrightarrow{{\mathrm{PT}}^\bullet}
{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_r,{\mathrm{L}}^\infty(\Gamma_l{\backslash}\Omega))\to
{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma_r,{{\mathbb R}})$$ is the same as first integrating the coefficients followed by the Poisson transform with trivial coefficients. With this fact and in mind, we invoke Lemma \[lem:image of fund class under Patterson-Sullivan map\] to conclude the proof.
\[lem:Lebesgue\_differentiation\] Fix points $o\in {\mathbf{H}}^n$ and $b_0\in \partial{\mathbf{H}}^n$. Denote by $d=d_o$ the visual metric on $\partial{\mathbf{H}}^n$ associated with $o$. Let $\{z^{(k)}\}_{k=1}^\infty$ be a sequence in ${\mathbf{H}}^n$ converging radially to $b_0$. Let $\phi\colon B\to B$ be a measurable map. For every $\epsilon>0$ and for a.e. $g\in G$ we have $$\lim_{k\to\infty} \nu_{z^{(k)}}\left\{b\in B \mid
d(\phi(gb),\phi(gb_0))>\epsilon\right\} = 0.$$
For the domain of $\phi$, it is convenient to represent $\partial{\mathbf{H}}^n$ as the boundary $\hat{{\mathbb R}}^n=\{(x_1,\dots,
x_n,0)\mid x_i\in{{\mathbb R}}\}\cup\{\infty\}$ of the upper half space model $${\mathbf{H}}^n=\{(x_1,\dots,x_{n+1})\mid x_{n+1}>0\}\subset{{\mathbb R}}^{n+1}.$$ We may assume that $o=(0,\dots,0,1)$ and $b_0=0\in{{\mathbb R}}^{n}\subset\hat{{\mathbb R}}^n$. The points $z^{(k)}$ lie on the line $l$ between $o$ and $b_0$. The subgroup of $G$ consisting of reflections along hyperplanes containing $l$ and perpendicular to $\{x_{n+1}=0\}$ leaves the measures $\nu_{z^{(k)}}$ invariant, i.e. each $\nu_{z^{(k)}}$ is ${\rm O}(n)$-invariant. Since the probability measure $\nu_{z^{(k)}}$ is in the Lebesgue measure class, the Radon-Nikodym theorem, combined with the ${\rm O}(n)$-invariance, yields the existence of a measurable functions $h_k:[0,\infty)\to[0,\infty)$ such that for any bounded measurable function $l$ $$\int l\,d\nu_{z^{(k)}}=\int_0^\infty
\left(\frac{1}{{\operatorname{vol}}(B(0,r))}\int_{B(0,r)}l(y)\,dy\right) h_k(r)\,dr$$ holds[^11] and $$\int_0^\infty h_k(r)\,dr=1.$$ Since the $\nu_{z^{(k)}}$ weakly converge to the Dirac measure at $0\in{{\mathbb R}}^n$, we have for every $r_0>0$ $$\label{eq:convergence}
\lim_{k\to\infty}\int_{r_0}^\infty h_k(r)\,dr=0.$$ For the target of $\phi$, we represent $B=\partial {\mathbf{H}}^n$ as the boundary $S^{n-1}\subset{{\mathbb R}}^n$ of the Poincare disk model. The visual metric is then just the standard metric of the unit sphere. Considering coordinates in the target, it suffices to prove that every measurable function $f:\hat{{\mathbb R}}^n\to [-1,1]$ satisfies $$ \lim_{k\to\infty} \int_{\hat{{\mathbb R}}^n}{{\left\lvert f(gx)-f(g0)\right\rvert}}d\nu_{z^{(k)}}(x) = 0.$$ for a.e. $g\in G$. By the Lebesgue differentiation theorem the set $L_f$ of points $x\in {{\mathbb R}}^n$ with the property $$\label{eq:lebesgue diff}
\lim_{r\to 0}\frac{1}{{\operatorname{vol}}(B(0,r))} \int_{B(x,r)} |f(y)-f(x)|\,dy =0$$ is conull in ${{\mathbb R}}^n$. The subset of elements $g\in G$ such that $g0\in L_f$ and $g0\ne \infty$ is conull with respect to the Haar measure. From now on we fix such an element $g\in G$. By compactness there is $L>0$ such that the diffeomorphism of $\hat{{\mathbb R}}^n$ given by $g$ has Lipschitz constant at most $L$ and its Jacobian satisfies ${{\left\lvert {\rm Jac}(g)\right\rvert}}>1/L$ everywhere on ${{\mathbb R}}^n\subset\hat{{\mathbb R}}^n$. Let $\epsilon>0$. According to choose $r_0>0$ such that for all $r<r_0$ $$\label{eq: L estimate}
\frac{L}{{\operatorname{vol}}(B(0,r))}\int_{B(g0,Lr)}{{\left\lvert f(y)-f(g0)\right\rvert}}dy<\frac{\epsilon}{2}.$$ According to choose $k_0\in{{\mathbb N}}$ such that $$\int_{r_0}^\infty h_k(r)\,dr<\frac{\epsilon}{4}$$ for every $k> k_0$. So we obtain that $$\begin{aligned}
\int_{\hat{{\mathbb R}}^n}{{\left\lvert f(gx)-f(g0)\right\rvert}}d\nu_{z^{(k)}}&<
\int_0^{r_0}\frac{1}{{\operatorname{vol}}(B(0,r))}\int_{B(0,r)}{{\left\lvert f(gx)-f(g0)\right\rvert}}dx\; h_k(r)dr+\frac{\epsilon}{2}\\
&\le \int_0^{r_0}\frac{L}{{\operatorname{vol}}(B(0,r))}\int_{gB(0,r)}{{\left\lvert f(y)-f(g0)\right\rvert}}dy\; h_k(r)dr+\frac{\epsilon}{2}
\end{aligned}$$ for $k>k_0$. Because of $gB(0,r)\subset B(g0,Lr)$ and we obtain that for $k>k_0$ $$\int_{\hat{{\mathbb R}}^n}{{\left\lvert f(gx)-f(g0)\right\rvert}}d\nu_{z^{(k)}}<\epsilon.
\qedhere$$
We start with the proof of Theorem \[thm:evaluation by fundamental class\]. For every $i\in\{0,\dots,n\}$ we pick a sequence $(z_i^{(k)})_{k\in{{\mathbb N}}}$ on the geodesic ray from a basepoint $o$ to $z_i$ converging to $z_i$. Let $\sigma_k$ be the geodesic simplex spanned by the vertices $z_0^{(k)},\dots, z_n^{(k)}$. By Lemma \[lem:evaluation by fund class - first step: visual measures\], $$ {\operatorname{eu}}(\Omega)=
\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{{\operatorname{vol}}(\sigma_k)}
\int_X\int_{\Gamma{\backslash}G^0}\int_{B^{n+1}}
{\operatorname{vol}}\bigl(\phi_x(gb_0),\dots,\phi_x(gb_n)\bigr)
\,d\nu_{\sigma_k}\,dg\,d\mu.$$ We now let $k$ go to $\infty$. Note that the left hand side does not depend on $k$. First of all, the volumes ${\operatorname{vol}}(\sigma_k)$ converge to ${\operatorname{vol}}(\sigma)=v_{\mathrm{max}}$. By Lemma \[lem:Lebesgue\_differentiation\], $$\lim_{k\to \infty}\nu_{\sigma_k}
\bigl\{(b_0,\dots,b_n)\mid d(\phi_x(gz_i),\phi_x(gb_i))<\epsilon\bigr\}= 1$$ for every $\epsilon>0$ and a.e. $(x,g)\in X\times G$. Since the volume is continuous on $B^{n+1}$ for $n\ge 3$ [@ratcliffe]\*[Theorem 11.4.2 on p. 541]{} and constant on non-degenerate ideal simplices for $n=2$, this implies that $$\lim_{k\to\infty}\int_{B^{n+1}}
{\operatorname{vol}}\bigl(\phi_x(gb_0),\dots,\phi_x(gb_n)\bigr)
\,d\nu_{\sigma_k}\,dg=
{\operatorname{vol}}\bigl(\phi_x(g z_0),\dots,\phi_x(g z_n)\bigr),$$ for a.e. $(x,g)\in X\times G$, which finally yields Theorem \[thm:evaluation by fundamental class\] by the dominated convergence theorem. The proof of Theorem \[thm:bounded volume cocycle to volume coycle\] is even easier since it does not require Lemma \[lem:Lebesgue\_differentiation\]. One obtains from Lemma \[lem:image of fund class under Patterson-Sullivan map\] that $$\begin{gathered}
\langle {\operatorname{comp}}^n({\operatorname{dvol}}_b), [\Gamma]\rangle=
\frac{{\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)}{{\operatorname{vol}}(\sigma_k)}
\int_X\int_{\Gamma{\backslash}G^0}\int_{B^{n+1}}
{\operatorname{vol}}\bigl(gb_0,\dots,gb_n\bigr)
\,d\nu_{\sigma_k}\,dg\,d\mu\end{gathered}$$ which converges for $k\to\infty$ to ${\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)$ by continuity of ${\operatorname{vol}}\colon B^{n+1}\to{{\mathbb R}}$ and the weak convergence of $\nu_{z_i^{(k)}}$ to the point measure at $z_i$ for every $i\in\{0,\dots,n\}$.
Order-preserving measurable self maps of the circle {#sub:order_preserving_measurable_self_maps_of_the_circle}
---------------------------------------------------
Consider the function $c$, called the *orientation cocycle*, which is defined on triples of points on the circle $S^1=\partial{\mathbf{H}}^2$ by $$c(b_0,b_1,b_3)=v_{\mathrm{max}}^{-1}\cdot {\operatorname{vol}}(b_0,b_1,b_2).$$ It takes values in $\{-1,0,1\}$ with $c (b_0,b_1,b_2)=1$ if the triple $(b_0,b_1,b_2)$ consists of distinct points in the positive orientation/cyclic order, $c=-1$ if the cyclic order is reversed, and $c=0$ if the triple is degenerate. Let $\nu$ denote a probability measure in the Lebesgue class, and suppose that $\phi:(S^1,\nu)\to S^1$ is a measurable map so that for $\nu^3$-a.e. $(b_0,b_1,b_2)$: $$c(\phi(b_0), \phi(b_1), \phi(b_2))=c(b_0,b_1,b_2).$$ It follows from [@iozzi]\*[Proposition 5.5]{} that the following conditions on such measurable orientation preserving $\phi:(S^1,\nu)\to S^1$ are equivalent:
1. The push-forward measure $\phi_*\nu$ has full support;
2. $\phi$ agrees a.e. with a homeomorphism $f\in {\operatorname{Homeo}}(S^1)$.
Let $\alpha:\Gamma\times X\to\Gamma$ be the ME-cocycle associated with an ergodic $(\Gamma,\Gamma)$-coupling $(\Omega,m)$ and an identification $i:\Gamma\times X\to\Omega$. Let $\phi_x\colon (S^1,\nu)\to S^1$, $x\in X$, be the boundary map associated to $\alpha$ as in Subsection \[ssub:boundary\_maps\].
\[prop:orientation cocycle maximal\] If the orientation cocycle is preserved by $\phi_x$ a.e., that is, $$c(\phi_x(b_0),\phi_x(b_1),\phi_x(b_2))=c(b_0,b_1,b_2)\quad\text{$\nu^3$-a.e}$$ for a.e. $x\in X$, then the map $\phi_x$ agree a.e. with a homeomorphism $f_x\in {\operatorname{Homeo}}(S^1)$ for a.e. $x\in X$.
We have to prove that the measurable family of open sets $U_x=S^1\setminus {\operatorname{supp}}(\phi_x\nu)$ satisfies a.e. $U_x=\emptyset$. The fact that $\nu$ is $\Gamma$-quasi-invariant and the identity $$\phi_{\gamma.x}(\gamma b)=\alpha(\gamma,x)\phi_x(b)$$ imply the following *a priori* equivariance of $\{U_x \mid x\in X\}$ $$\label{e:open-inv-set}
U_{\gamma.x}=\alpha(\gamma,x)\,U_x.$$ Since $U_x\neq S^1$ for every $x\in X$, the proposition is implied by the following general lemma and the fact that the action of $G={\mathrm{PSL}}_2({{\mathbb R}})$ and of its lattices on the circle $S^1$ is minimal and strongly proximal [@furstenberg-bourbaki]\*[Propositions 4.2 and 4.4]{}.
\[L:furstenberg\] Let $M$ be a compact metrizable space, and let $\Gamma{\curvearrowright}M$ act minimally and strongly proximally. Let $\{ U_x \mid x\in X\}$ be a measurable family of open subsets of $M$ satisfying (\[e:open-inv-set\]) for a ME-cocycle $\alpha:\Gamma\times X\to\Gamma$ over an ergodic coupling. Then either $U_x=\emptyset$ or $U_x=M$ for a.e. $x\in X$.
We first reduce the question to the trivial cocycle. To distinguish the two copies of $\Gamma$ acting on $\Omega$ denote them by $\Gamma_1$ and $\Gamma_2$. Let $i:\Gamma_2\times X\cong \Omega$ be a measure space isomorphism as in (\[e:ij-fd\]); in particular $$(g_1,g_2): i(\gamma,x)\mapsto i(g_2 \gamma \alpha(g_1,x)^{-1},\ g_1.x)\qquad(g_i\in\Gamma_i).$$ Consider the measurable family $\{O_\omega\}$ of open subsets of $M$ indexed by $\omega\in\Omega$, defined by $O_{i(g,x)}=g U_x$. Then for $\omega=i(\gamma,x)$ and $g_i\in \Gamma_i$ we have $$O_{(g_1,g_2)\omega}=g_2\gamma\alpha(g_1,x)^{-1}U_{g_1.x}=g_2 \gamma U_x=g_2 O_\omega.$$ Note that $\omega\to O_\omega$ is invariant under the action of $\Gamma_1$. Therefore it descends to a measurable family of open sets $\{V_y\}$ indexed by $y\in Y\cong \Omega/\Gamma_1$, and satisfying a.e. on $Y$ $$V_{\gamma.y}=\gamma V_y\qquad (\gamma\in\Gamma).$$ The claim about $\{U_x \mid x\in X\}$ is clearly equivalent to the similar claim about $\{V_y \mid y\in Y\}$. By ergodicity, it suffices to reach a contradiction from the assumption that $V_y\neq \emptyset, M$ for a.e. $y\in Y$.
Denote by $\mu$ the $\Gamma$-invariant and ergodic probability measure on $Y$. Since $M$ has a countable base for its topology, while $\mu(\{ y \mid V_y\neq \emptyset\})=1$, it follows that there exists a non-empty open set $W\subset M$ for which the set $$A=\{ y\in Y \mid W\subset V_y\}$$ has $\mu(A)>0$. Since $M\setminus V_y\neq \emptyset$ for $\mu$-a.e. $y\in Y$, there exists a measurable map $s:Y\to M$ with $s(y)\notin V_y$ a.e. Let $\sigma\in {\operatorname{Prob}}(M)$ denote the distribution of $s(y)$, i.e., $\sigma(E)=\mu\{ y\in Y \mid s(y)\in E\}$. Then for any $\gamma\in \Gamma$ $$\begin{aligned}
\sigma(\gamma^{-1}W)&=&\mu\{ y\in Y \mid s(y) \in \gamma^{-1}W \} \\
&\le& \mu(Y\setminus \gamma^{-1}A)+\mu\{ y\in \gamma^{-1}A \mid s(y)\in \gamma^{-1} V_{\gamma.y}=V_y\} \\
&=&1-\mu(\gamma^{-1}A)=1-\mu(A).\end{aligned}$$ This contradicts the assumption that the action $\Gamma{\curvearrowright}M$ is minimal and strongly proximal.
Preserving maximal simplices of the boundary {#sub:preserving maximal simplices}
--------------------------------------------
Recall that a geodesic simplex in $\bar{\mathbf{H}}^n={\mathbf{H}}^n\cup\partial {\mathbf{H}}^{n}$ is called *regular* if any permutation of its vertices can be realized by an element in ${\operatorname{Isom}}({\mathbf{H}}^n)$. The set of ordered $(n+1)$-tuples on the boundary $B$ that form the vertex set of an ideal regular simplex is denoted by $\Sigma^{\rm reg}$. The set $\Sigma^{\rm reg}$ is a disjoint union $\Sigma^{\rm reg}=\Sigma^{\rm reg}_+\cup \Sigma^{\rm reg}_-$ of two subsets that correspond to the positively and negatively oriented ideal regular $n$-simplices, respectively.
\[lem:key facts from Mostow\]
1. The diagonal $G$-action on $\Sigma^{\rm reg}$ is simply transitive. The diagonal $G^0$-action on $\Sigma^{\rm reg}_-$ and $\Sigma^{\rm reg}_+$ are simply transitive, respectively.
2. An ideal simplex has non-oriented volume $v_{\mathrm{max}}$ if and only if it is regular.
3. Let $n\ge 3$. Let $\sigma,\sigma'$ be two regular ideal simplices having a common face of codimension one. Let $\rho$ be the reflection along the hyperspace spanned by this face. Then $\sigma=\rho(\sigma')$.
\(i) See the proof of [@ratcliffe]\*[Theorem 11.6.4 on p. 568]{}.\
(ii) The statement is trivial for $n=2$, as all non-degenerate ideal triangles in $\bar{\mathbf{H}}^2$ are regular, and $G$ acts simply transitively on them. The case $n=3$ is due to Milnor, and Haagerup and Munkholm [@haagerup] proved the general case $n\ge 3$.\
(iii) This is a key feature distinguishing the $n\ge 3$ case from the $n=2$ case where Mostow rigidity fails. See [@ratcliffe]\*[Lemma 13 on p. 567]{}.
We shall need the following lemma, which in dimension $n=3$ is due to Thurston [@thurston]\*[p. 133/134]{}. Recall that $B=\partial {\mathbf{H}}^n$ is considered equipped with the Lebesgue measure class. We consider the natural measure $m_{\Sigma^{\rm reg}_+}$ on $\Sigma^{\rm reg}_+$ corresponding to the Haar measure on $G^0$ under the simply transitive action of $G^0$ on $\Sigma^{\rm reg}_+$.
\[lem:key lemma in mostow rigidity\] Let $n\ge 3$ and $\phi:B\to B$ be a Borel map such that $\phi^{n+1}=\phi\times\cdots\times\phi$ maps a.e. point in $\Sigma^{\rm reg}_+$ into $\Sigma^{\rm reg}_+$. Then there exists a unique $g_0\in G^0={\operatorname{Isom}}_+({\mathbf{H}}^n)$ with $\phi(b)=g_0 b$ for a.e. $b\in B$.
Fix a regular ideal simplex $\sigma=(b_0,\dots,b_n)\in\Sigma^{\rm reg}_+$, and identify $G^0$ with $\Sigma^{\rm reg}_+$ via $g\mapsto g\sigma$. Then there is a Borel map $f:G^0\to G^0$ such that for a.e. $g\in G^0$ $$\label{e:def-f}
(\phi(gb_0),\dots,\phi(gb_n))=(f(g)b_0,\dots,f(g)b_n).$$ Interchanging $b_0$, $b_1$ identifies $\Sigma^{\rm reg}_+$ with $\Sigma^{\rm reg}_-$, and allows to extend $f$ to a measurable map $G\to G$ satisfying (\[e:def-f\]) for a.e. $g\in G$. Let $\rho_0,\dots,\rho_n\in G$ denote the reflections in the codimension one faces of $\sigma$. Then Lemma \[lem:key facts from Mostow\] (iii) implies that $$f(g\rho)=f(g)\rho\qquad\text{for a.e.}\qquad g\in G$$ for $\rho$ in $\{\rho_0,\dots,\rho_n\}$. It follows that the same applies to each $\rho$ in the countable group $R<G$ generated by $\rho_0,\dots,\rho_n$. We claim that there exists $g_0\in G$ so that $f(g)=g_0 g$ for a.e. $g\in G$, which implies that $\phi(b)=g_0 b$ also holds a.e. on $B$.
The case $n=3$ is due to Thurston ([@thurston]\*[p. 133/134]{}). So hereafter we focus on $n>3$, and will show that in this case the group $R$ is dense in $G$ (for $n=2,3$ it forms a lattice in $G$). Consequently the $R$-action on $G$ is ergodic with respect to the Haar measure. Since $g\mapsto f(g)g^{-1}$ is a measurable $R$-invariant map on $G$, it follows that it is a.e. a constant $g_0\in G^0$, i.e., $f(g)=g_0 g$ a.e. proving the lemma.
It remains to show that for $n>3$, $R$ is dense in $G$. Not being able to find a convenient reference for this fact, we include the proof here.
For $i\in\{0,\dots,n\}$ denote by $P_i<G$ the stabilizer of $b_i\in\partial{\mathbf{H}}^n$, and let $U_i<P_i$ denote its unipotent radical. We shall show that $U_i$ is contained in the closure $\overline{R\cap P_i}<P_i$ (in fact, $\overline{R\cap P_i}=P_i$ but we shall not need this). Since unipotent radicals of any two opposite parabolics, say $U_0$ and $U_1$, generate the whole connected simple Lie group $G^0$, this would show $G^0<\bar{R}<G$. Since $R$ is not contained in $G^0$, it follows that $\bar{R}=G$ as claimed.
Let $f_i:\partial {\mathbf{H}}^n\to {\mathbf{E}}^{n-1}\cup\{\infty\}$ denote the stereographic projection taking $b_i$ to the point at infinity. Then $f_i P_i f_i^{-1}$ is the group of similarities ${\operatorname{Isom}}({\mathbf{E}}^{n-1})\rtimes{{\mathbb R}}^\times_+$ of the Euclidean space ${\mathbf{E}}^{n-1}$. We claim that the subgroup of translations ${{\mathbb R}}^{n-1}\cong U_i<P_i$ is contained in the closure of $R_i=R\cap P_i$. To simplify notations we assume $i=0$. The set of all $n$-tuples $(z_1,\dots,z_n)$ in ${\mathbf{E}}^{n-1}$ for which $(b_0,f_0^{-1}(z_1),\dots,\dots,f_0^{-1}(z_n))$ is a regular ideal simplex in $\bar{{\mathbf{H}}}^{n}$ is precisely the set of all regular Euclidean simplices in ${\mathbf{E}}^{n-1}$ [@ratcliffe]\*[Lemma 3 on p. 519]{}. So conjugation by $f_0$ maps the group $R_0=R\cap P_0$ to the subgroup of ${\operatorname{Isom}}({\mathbf{E}}^{n-1})$ generated by the reflections in the faces of the Euclidean simplex $\Delta=(z_1,\dots,z_n)$, where $z_i=f_0(b_i)$. For $1\le j<k\le n$ denote by $r_{jk}$ the composition of the reflections in the $j$th and $k$th faces of $\Delta$; it is a rotation leaving fixed the co-dimension two affine hyperplane $L_{jk}$ containing $\{z_i\mid i\neq j,k\}$. The angle of this rotation is $2\theta_n$, where $\theta_n$ is the dihedral angle of the simplex $\Delta$. One can easily check that $\cos(\theta_n)=-1/(n-1)$, using the fact that the unit normals $v_i$ to the faces of $\Delta$ satisfy $v_1+\dots+v_n=0$ and $\langle v_i, v_j\rangle=\cos(\theta_n)$ for all $1\le i<j\le n$. Thus $w=\exp(\theta_n\sqrt{-1})$ satisfies $w+1/w=-2/(n-1)$. Equivalently, $w$ is a root of $$p_n(z)=(n-1)z^2+2z+(n-1).$$ This condition on $w$ implies that $\theta_n$ is not a rational multiple of $\pi$. Indeed, otherwise, $w$ is a root of unit, and therefore is a root of some cyclotomic polynomial $$c_m(z)=\prod_{k\in\{1..m-1\mid \gcd(k,m)=1\}}(z-e^{\frac{2\pi k i}{m}})$$ whose degree is Euler’s totient function $\deg(c_m)=\phi(m)$. The cyclotomic polynomials are irreducible over ${{\mathbb Q}}$. So $p_n(z)$ and $c_m(z)$ share a root only if they are proportional, which in particular implies $\phi(m)=2$. The latter happens only for $m=3$, $m=4$ and $m=6$; corresponding to $c_3(z)=z^2+z+1$, $c_4(z)=z^2+1$, and $c_6(z)=z^2-z+1$. The only proportionality between these polynomials is $p_3(z)=2 c_2(z)$; and it is ruled out by the assumption $n>3$.
Thus the image of $R_0$ in ${\operatorname{Isom}}({\mathbf{E}}^{n-1})$ is not discrete. Let $$\pi:\overline{R\cap P_0}\to {\operatorname{Isom}}({\mathbf{E}}^{n-1})\to {\rm O}({{\mathbb R}}^{n-1})$$ denote the homomorphism defined by taking the linear part. Then $\pi(r_{jk})$ is an irrational rotation in ${\rm O}({{\mathbb R}}^{n-1})$ leaving invariant the linear subspace parallel to $L_{jk}$. The closure of the subgroup generated by this rotation is a subgroup $C_{jk}<{\rm O}({{\mathbb R}}^{n-1})$, isomorphic to ${\rm SO}(2)$. The group $K<{\rm O}({{\mathbb R}}^{n-1})$ generated by all such $C_{jk}$ acts irreducibly on ${{\mathbb R}}^{n-1}$, because there is no subspace orthogonal to all $L_{jk}$. Since $\overline{R\cap P_0}$ is not compact (otherwise there would be a point in ${\mathbf{E}}^{n-1}$ fixed by all reflections in faces of $\Delta$), the epimorphism $\pi:\overline{R\cap P_0}\to K$ has a non-trivial kernel $V<{{\mathbb R}}^{n-1}$, which is invariant under $K$. As the latter group acts irreducibly, $V={{\mathbb R}}^{n-1}$ or, equivalently, $U_0<\overline{R\cap P_0}$. This completes the proof of the lemma.
Proofs of the main results {#sec:proofs_of_the_main_results}
==========================
In this section we use the results of Section \[sec:measure\_equivalence\] and Theorems \[T:SOn1-1-taut\] and \[thm:taut relative homeo\] to prove the remaining results stated in the introduction.
Measure equivalence rigidity: Theorem \[T:ME-rigidity\] {#sub:measure_equivalence_rigidity_for_nge3_}
-------------------------------------------------------
Let $G={\operatorname{Isom}}({\mathbf{H}}^n)$, $n\ge 3$. Let $\Gamma<G$ be a lattice, and $\Lambda$ a finitely generated group which admits an integrable $(\Gamma,\Lambda)$-coupling $(\Omega,m)$. By Lemma \[lem:composition of lp couplings\] the $(\Gamma,\Gamma)$-coupling $\Omega\times_\Lambda{{\Omega}^*}$ is integrable.
By Theorem \[T:SOn1-1-taut\] and Proposition \[P:taut-lattice\] the lattice $\Gamma$ is $1$-taut relative to the inclusion $\Gamma<G$. Hence the coupling $\Omega\times_\Lambda{{\Omega}^*}$ is taut. By Example \[exa:examples of strongly icc groups\] the group $G$ is strongly ICC relative to $\Gamma<G$. Applying Theorem \[T:split+hom\] we obtain a continuous homomorphism $\rho:\Lambda\to G$ with finite kernel $F$, image $\bar{\Lambda}=\rho(\Lambda)$ being discrete in $G$, and a measurable ${\operatorname{id}}_\Gamma\times\rho$-equivariant map $\Psi:\Omega\to G$.
To complete the proof of statement (1) of Theorem \[T:ME-rigidity\] and case $n\ge 3$ of Theorem \[T:ME-rigidity2and3\] it remains to show that $\bar\Lambda$ is not merely discrete, but is actually a lattice in $G$. This can be deduced from the application of Ratner’s theorem below which is needed for the precise description of the push-forward measure $\Psi_*m$ on $G$ as stated in part (2) of Theorem \[T:ME-rigidity\]. Let us also give the following direct argument which relies only on the strong ICC property of $G$.
Consider the composition $(G,\Lambda)$-coupling $\widetilde\Omega=G\times_\Gamma\Omega$, and the $(G,G)$-coupling $\widetilde\Omega\times_\Lambda{{\widetilde\Omega}^*}$. Since $\Gamma$ is an integrable lattice in $G$ (Theorem \[thm:Shalom and lattices\]) by Lemma \[lem:composition of lp couplings\] both $\widetilde\Omega$ and $\widetilde\Omega\times_\Lambda{{\widetilde\Omega}^*}$ are integrable couplings. Theorem \[T:SOn1-1-taut\] provides a unique tautening map $$\widetilde\Phi:\widetilde\Omega\times_\Lambda{{\widetilde\Omega}^*}\to G.$$ Applying Theorem \[T:reconstruction\] (a special case of Theorem \[T:split+hom\] with ${\mathcal{G}}=G$), we obtain a homomorphism $\widetilde\rho:\Lambda\to G$ with finite kernel and image being a lattice in $G$. There is also a ${\rm Id}_G\times \widetilde\rho$-equivariant measurable map $$\widetilde\Psi:\widetilde\Omega=G\times_\Gamma\Omega \to G.$$ We claim that $\rho, \widetilde\rho:\Lambda\to G$ are conjugate representations. To see this observe that since $G$ is strongly ICC, there is only one tautening map $\widetilde\Omega\times_\Lambda{{\widetilde\Omega}^*}\to G$. This implies the a.e. identity $$\widetilde\Psi([g_1,\omega_1])\widetilde\Psi([g_2,\omega_2])^{-1}=g_1 \Psi(\omega_1)\Psi(\omega_2)^{-1} g_2^{-1}.$$ Equivalently, we have a.e. identity $$\Psi(\omega_1)^{-1}g_1^{-1}\widetilde\Psi([g_1,\omega_1])
=\Psi(\omega_2)^{-1}g_2^{-1}\widetilde\Psi([g_2,\omega_2]).$$ Hence the value of both sides are a.e. equal to a constant $g_0\in G$. It follows that for a.e. $g\in G$ and $\omega\in \Omega$ $$g^{-1}\widetilde\Psi([g,\omega])=\Psi(\omega)g_0.$$ Finally, the fact that $\Psi$, $\widetilde\Psi$ are $\rho$-, $\widetilde\rho$- equivariant respectively, implies: $$\widetilde\rho(\lambda)=g_0 \rho(\lambda) g_0^{-1}\qquad(\lambda\in\Lambda).$$ In particular, $\bar\Lambda=g_0^{-1}\widetilde\rho(\Lambda)g_0$ is a lattice in $G$.
We proceed with the proof of statement (2): given the ${\rm Id}_\Gamma\times\rho$-equivariant measurable map $\Psi:\Omega\to G$ we shall describe the pushforward $\Psi_*m$ on $G$. (We shall use the discreteness of $\bar\Lambda=\rho(\Lambda)$, but the fact that it is a lattice will not be needed; in fact, it will follow from the application of Ratner’s theorem.) Recall that the measure $\Psi_*m$ is invariant under the action $x\mapsto \gamma x \rho(\lambda)^{-1}$, and descends to a finite $\Gamma$-invariant measure $\mu$ on $G/\bar{\Lambda}$ and to a finite $\bar{\Lambda}$-invariant measure $\nu$ on $\Gamma{\backslash}G$. Assuming $m$ was $\Gamma\times\Lambda$-ergodic, $\mu$ and $\nu$ are ergodic under the $\Gamma$- and $\bar\Lambda$-action, respectively. One can now apply Ratner’s theorem [@ratnerICM] to describe $\mu$, and thereby $\Psi_*m$, as in [@FurmanME]\*[Lemma 4.6]{}. For the reader’s convenience we sketch the arguments.
Let $\Lambda^0=\bar{\Lambda}\cap G^0$; so either $\Lambda^0=\bar{\Lambda}$ or $[\bar{\Lambda}:\Lambda^0]= 2$. In the first case we set $\mu'=\mu$, in the latter case let $\mu'$ denote the $2$-to-$1$ lift of $\mu$ to $G/\Lambda^0$. Let $\Gamma^0=\Gamma\cap G^0$, and let $\mu^0$ be an ergodic component of $\mu'$ supported on $G^0/\Lambda^0$. We consider the homogeneous space $Z=G^0/\Gamma^0\times G^0/\Lambda^0$ which is endowed with the following probability measure $$\tilde\mu^0=\int_{G^0/\Gamma^0} \delta_{g\Gamma^0}\times g_*\mu^0\,dm_{G^0/\Gamma^0}.$$ Observe that $\tilde\mu^0$ well defined because $\mu^0$ is $\Gamma^0$-invariant. Moreover, $\tilde\mu^0$ is invariant and ergodic for the action of the diagonal $\Delta(G^0)\subset G^0\times G^0$ on $Z$. Since $G^0$ is a connected group generated by unipotent elements, Ratner’s theorem shows that $\tilde\mu^0$ is *homogeneous*. This means that there is a connected Lie subgroup $L<G^0\times G^0$ containing $\Delta(G^0)$ and a point $z\in Z$ such that the stabilizer $L_z$ of $z$ is a lattice in $L$ and $\tilde\mu^0$ is the push-forward of the normalized Haar measure $m_{L/L_z}$ to the $L$-orbit $Lz\subset Z$. Since $G^0$ is a simple group, there are only two possibilities for $L$: either (i) $L=G^0\times G^0$ or (ii) $L=\Delta(G^0)$.
In case (i), $\tilde\mu^0$ is the Haar measure on $G^0/\Gamma^0\times G^0/\Lambda^0$, and $\mu^0$ is the Haar measure on $G^0/\Lambda^0$. (In particular, $\Lambda^0$ is a lattice in $G^0$, and $\Lambda$ is a lattice in $G$). The original measure $\mu$ may be either the $G$-invariant measure $m_{G/\bar{\Lambda}}$, or a $G^0$-invariant measure on $G/\bar{\Lambda}$. In the latter case, by possibly multiplying $\Phi$ and conjugating $\rho$ with some $x\in G\setminus G^0$, we may assume that $\mu$ is the $G^0$-invariant probability measure on $G^0/\bar\Lambda$.
In case (ii), the fact that $L_z$ is lattice in $L=\Delta(G^0)$, implies that $\mu^0$ and the original measure $\mu$ are atomic. Since $\Gamma$ acts ergodically on $(G/\bar{\Lambda},\mu)$, this atomic measure is necessarily supported and equidistributed on a finite $\Gamma$-orbit of some $g_0\bar\Lambda\in G/\bar{\Lambda}$. It follows that $\Gamma\cap g_0^{-1}\bar\Lambda g_0$ has finite index in $\Gamma$. (This also implies that $\bar\Lambda$ is a lattice in $G$). Upon multiplying $\Psi$ and conjugating $\rho$ by $g_0\in G$, we may assume that $\Phi_*m$ is equidistributed on the double coset $\Gamma e \bar\Lambda$ and that $\Gamma$, $\bar\Lambda$ are commensurable lattices. This completes the proof of Theorem \[T:ME-rigidity\].
Convergence actions on the circle: case $n=2$ of Theorem \[T:ME-rigidity2and3\] {#sub:actions_on_the_circle}
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Let $\Gamma$ be a uniform lattice in $G={\operatorname{Isom}}({\mathbf{H}}^2)\cong {\mathrm{PGL}}_2({{\mathbb R}})$. The group $G$ is a subgroup of ${\operatorname{Homeo}}(S^1)$ by the natural action of ${\mathrm{PGL}}_2({{\mathbb R}})$ on $S^1\cong{{{\mathbb R}}\operatorname{P}}^1$. Consider a compactly generated unimodular group $H$ that is ${\mathrm{L}}^1$-measure equivalent to $\Gamma$. We will prove a more general statement than in Theorem \[T:ME-rigidity2and3\], which is formulated for discrete $H=\Lambda$. Since $\Gamma$ is uniform, hence integrable in $G$, we can induce any integrable $(\Gamma,H)$-coupling to an integrable $(G,H)$-coupling (Lemma \[lem:composition of lp couplings\]). Let $(\Omega,m)$ be an integrable $(G,H)$-coupling $(\Omega,m)$.
From Theorem \[T:split+hom\] we obtain a continuous homomorphism $\rho:H\to{\operatorname{Homeo}}(S^1)$ with compact kernel and closed image $\bar{H}<{\operatorname{Homeo}}(S^1)$ and, by pushing forward $m$, a measure $\bar{m}$ on ${\operatorname{Homeo}}(S^1)$ that is invariant under all bilateral translations on $f\mapsto g f \rho(h)^{-1}$ with $g\in G$ and $h\in{\operatorname{Homeo}}(S^1)$ and descends to a finite $\bar H$-invariant measure $\mu$ on $G{\backslash}{\operatorname{Homeo}}(S^1)$ and a finite $G$-invariant measure $\nu$ on ${\operatorname{Homeo}}(S^1)/\bar{H}$.
The next step is to show that $\bar{H}$ can be conjugated into $G$. To this end, we shall use the existence of the finite $\bar{H}$-invariant measure $\mu$ on $G{\backslash}{\operatorname{Homeo}}(S^1)$, which may be normalized to a probability measure. We need the following theorem which we prove relying on the deep work by Gabai [@Gabai] and Casson-Jungreis [@Casson] on the determination of convergence groups as Fuchsian groups.
\[T:intoPGL2\] Let $\mu$ be a Borel probability measure on $G{\backslash}{\operatorname{Homeo}}(S^1)$. Then the stabilizer $H_\mu=\{ f\in {\operatorname{Homeo}}(S^1) \mid f_*\mu=\mu\}$ for the action by the right translations is conjugate to a closed subgroup of $G$.
We fix a metric $d$ on the circle, say $d(x,y)=\measuredangle(x,y)$. Let ${\operatorname{Trp}}\subset S^1\times S^1\times S^1$ be the space of distinct triples on the circle. The group ${\operatorname{Homeo}}(S^1)$ acts diagonally on ${\operatorname{Trp}}$. We denote elements in ${\operatorname{Trp}}$ by bold letters $\mathbf x\in{\operatorname{Trp}}$; the coordinates of $\mathbf x\in{\operatorname{Trp}}$ or $\mathbf y\in{\operatorname{Trp}}$ will be denoted by $x_i$ or $y_i$ where $i\in\{1,2,3\}$, respectively. For $f\in{\operatorname{Homeo}}(S^1)$ we write $f(\mathbf x)$ for $(f(x_1),f(x_2),f(x_3))$. We equip ${\operatorname{Trp}}$ with the metric, also denoted by $d$, given by $$d(\mathbf{x},\mathbf{y})=\max_{i\in\{1,2,3\}}d(x_i,y_i).$$ The following lemma will eventually allow us to apply the work of Gabai-Casson-Jungreis.
\[L:Hmu\] For every compact subset $K\subset {\operatorname{Trp}}$ and every $\epsilon>0$ there is $\delta>0$ so that for all $h,h'\in H_\mu$ and $\mathbf{y}\in K\cap h^{-1}K$ and $\mathbf{y}'\in K\cap h'^{-1}K$ one has the implication: $$d(\mathbf{y},\mathbf{y}')<\delta~\text{ and }~d(h(\mathbf{y}),h'(\mathbf{y}'))<\delta
~\Longrightarrow~
\sup_{x\in S^1} d(h(x),h'(x))<\epsilon$$
For an arbitrary triple $\mathbf{z}\in {\operatorname{Trp}}$ and $x\in S^1\setminus\{z_3\}$ consider the real valued *cross-ratio* $$ [x,z_1;z_2,z_3]=\frac{(x-z_1)(z_2-z_3)}{(x-z_3)(z_2-z_1)}.$$ In this formula we view the circle as the one-point compactification of the real line. Denote by $[z_1,z_2]_{z_3}$ the circle arc from $z_1$ to $z_2$ not including $z_3$. As a function in the first variable, $[\_,z_1;z_2,z_3]$ is a monotone homeomorphism between the closed arc $[z_1,z_2]_{z_3}$ and the interval $[0,1]$. For $f\in{\operatorname{Homeo}}(S^1)$ and $\mathbf{z}\in{\operatorname{Trp}}$ we define the function $$F_{\mathbf{z},f}:[z_1,z_2]_{z_3}\to[0,1],~~ F_{\mathbf{z},f}(x)=[f(x),f(z_1);f(z_2),f(z_3)].$$ Since the cross-ratio is invariant under $G$ [@ratcliffe]\*[Theorem 4.3.1 on p. 116]{}, we have $F_{\mathbf{z},gf}(x)=F_{\mathbf{z},f}(x)$ for any $g\in G$. Hence we may and will use the notation $F_{\mathbf{z},Gf}(x)$. We now average $F_{\mathbf{z},Gf}(x)$ with regard to the measure $\mu$ and obtain the function $\bar{F}_{\mathbf{z}}:[z_1,z_2]_{z_3}\to[0,1]$ with $$\bar{F}_{\mathbf{z}}(x)=\!\int\limits_{G{\backslash}{\operatorname{Homeo}}(S^1)} \!\!\!F_{\mathbf{z},Gf}(x)\,d\mu(Gf).$$ The $H_\mu$-invariance of $\mu$ implies that $$\label{eq: H_mu equivariance}
\bar{F}_{h(\mathbf{z})}(h(x))=\bar{F}_{\mathbf{z}}(x)$$ for every $h\in H_\mu$ and every $x\in [z_1,z_2]_{z_3}$. Let us introduce the following notation: Whenever $K\subset{\operatorname{Trp}}$ is a subset, we denote by $\widetilde K$ the subset $$\widetilde{K}=\bigl\{ (x,\mathbf{z}) \mid \mathbf{z}\in K,~ x\in [z_1,z_2]_{z_3}\bigr\}
\subset S^1\times S^1\times S^1\times S^1.$$ Next let us establish the following continuity properties:
1. \[i:barFinv\] For every compact $K\subset{\operatorname{Trp}}$ and every $\epsilon>0$ there is $\eta>0$ such that:
$\displaystyle
\forall (s,\mathbf{z}),(t,\mathbf z)\in\widetilde{K}\colon |\bar{F}_{\mathbf{z}}(t)-\bar{F}_{\mathbf{z}}(s)|<\eta
~\Rightarrow~ d(t,s)<\frac{\epsilon}{5}$
2. \[i:equiF\] For every compact $K\subset{\operatorname{Trp}}$ and every $\eta>0$ there is $\delta>0$ such that:
$\displaystyle
\forall (t,\mathbf{y}),(t,\mathbf{z})\in \widetilde{K}\colon
d(\mathbf{y}, \mathbf{z})<\delta
~\Rightarrow~
|\bar{F}_{\mathbf{y}}(t)-\bar{F}_{\mathbf{z}}(t)|<\frac{\eta}{2}
$
*Proof of (\[i:barFinv\]):* Let $K\subset{\operatorname{Trp}}$ be compact and $\epsilon>0$. Let $f\in{\operatorname{Homeo}}(S^1)$. The family of homeomorphisms $\bar{F}_{\mathbf{z},Gf}\colon [z_1,z_2]_{z_3}\to[0,1]$ depends continuously on $\mathbf{z}\in{\operatorname{Trp}}$. The inverses of these functions are equicontinuous when $\mathbf{z}$ ranges in a compact subset. Hence there exists $\theta(Gf)>0$ such that for every $\mathbf{z}\in K$ and all $t,s\in [z_1,z_2]_{z_3}$ we have the implication $${{\left\lvert F_{\mathbf{z},Gf}(t)-F_{\mathbf{z},Gf}(s)\right\rvert}}<\theta(Gf)
~\Rightarrow~d(t,s)<\frac{\epsilon}{5}.$$ The set $G{\backslash}{\operatorname{Homeo}}(S^1)$ is the union of an increasing sequence of measurable sets $$A_n=\bigl\{Gf\in G{\backslash}H \mid \theta(Gf)>\frac{1}{n} \bigr\}.$$ Fix $n$ large enough so that $\mu(A_n)>1/2$. We claim that $\eta=(2n)^{-1}$ satisfies (\[i:barFinv\]). Suppose that $\mathbf{z}\in K$ and $t,s\in [z_1,z_2]_{z_3}$ satisfy $d(t,s)>\epsilon/5$. Up to exchanging $t$ and $s$, we may assume that $[s,z_1;z_2,z_3]\ge [t,z_1;z_2,z_3]$. Then $F_{\mathbf{z},Gf}(s)\ge F_{\mathbf{z},Gf}(t)$ for all $f\in{\operatorname{Homeo}}(S^1)$, and $$\bar{F}_{\mathbf{z}}(s)-\bar{F}_{\mathbf{z}}(t)\ge
\int_{A_n} (F_{\mathbf{z},Gf}(s)-F_{\mathbf{z},Gf}(t))\,d\mu>\mu(A_n)\cdot \frac{1}{n}>\eta.$$ *Proof of (\[i:equiF\]):* Let $K\subset{\operatorname{Trp}}$ be compact, and let $\eta>0$. Let $f\in{\operatorname{Homeo}}(S^1)$. Since $\widetilde K$ is compact, $\bar{F}_\mathbf{z}(x)$ as a function on $\widetilde K$ is equicontinuous. Hence there is $\delta(Gf)>0$ such that for all $(x,\mathbf y)\in \widetilde K$ and $(x,\mathbf z)\in \widetilde K$ with $d(\mathbf{y}, \mathbf{z})<\delta(Gf)$ we have $${{\left\lvert F_{\mathbf{y},Gf}(x)-F_{\mathbf{z},Gf}(x)\right\rvert}}<\frac{\eta}{2}.$$ The set $G{\backslash}{\operatorname{Homeo}}(S^1)$ is the union of an increasing sequence of measurable sets $$B_n=\bigl\{Gf\in G{\backslash}H \mid \delta(Gf)>\frac{1}{n}\bigr\}.$$ We choose $n\in{{\mathbb N}}$ with $\mu(B_n)>1-\eta/2$ and set $\delta=n^{-1}$. Then for $(x,\mathbf y)\in \widetilde K$ and $(x,\mathbf z)\in \widetilde K$ with $d(\mathbf{y}, \mathbf{z})<\delta$ we have $$|\bar{F}_{\mathbf{y}}(x)-\bar{F}_{\mathbf{z}}(x)|\le
\int_{B_n}|F_{\mathbf{y},Gf}(x)-F_{\mathbf{z},Gf}(x)|\,d\mu(Gf) + \frac{\eta}{2}
<\eta,$$ proving (\[i:equiF\]).
We can now complete the proof of the lemma. Let $K\subset {\operatorname{Trp}}$ be a compact subset. Let $\epsilon>0$. We can choose $r>0$ such that $$K\subset\bigl\{\mathbf{x}\in{\operatorname{Trp}}\mid d(x_1, x_2), d(x_2,x_3), d(x_3,x_1)\ge r\bigl\}.$$ For the given $\epsilon$ and $K$ let $\eta>0$ be as in (\[i:barFinv\]). For the given $\epsilon$ and $K$ and this $\eta$ let $\delta>0$ be as in (\[i:equiF\]). We may also assume that $$\delta<\frac{\epsilon}{5}<\frac{r}{3}.$$ Consider $h,h'\in H_\mu$ and $\mathbf{y}, \mathbf{y}'\in K$ where $\mathbf{z}=h(\mathbf{y})$, $\mathbf{z}'=h'(\mathbf{y}')$ are also in $KÅ$, and assume that $d(\mathbf{y}, \mathbf{y}')<\delta$ and $d(\mathbf{z}, \mathbf{z}')<\delta$. There are several possibilities for the cyclic order of the points $\{y_1, y_1', y_2, y_2', y_3, y_3'\}$, but since the pairs $\{y_i, y_i'\}$ of corresponding points in the triples $\mathbf{y}, \mathbf{y}'$ are closer ($d(y_i,y_i')<\delta<r/3$) than the separation between the points in the triples ($d(y_i,y_j), d(y_i', y_j')\ge r$), these points define a partition of the circle into three long arcs $L_{ij}$ separated by three short arcs $S_k$ (possibly degenerating into points) in the following cyclic order $$S^1=L_{12} \cup S_2 \cup L_{23} \cup S_3 \cup L_{31} \cup S_1.$$ The end points of the arc $S_i$ are $\{y_i, y_i'\}$; and if $(i,j,k)=(1,2,3)$ up to a cyclic permutation, then $$L_{ij}=[y_i,y_j]_{y_k}\cap[y_i',y_j']_{y_k'}.$$ Note that for any $x\in L_{ij}$ we have $$h(x), h'(x)\in [z_i, z_j]_{z_k}\cap [z_i', z_j']_{z_k'}.$$ Using (\[i:equiF\]) and we obtain $$\begin{aligned}
|\bar{F}_{\mathbf{z}}(h(x))-\bar{F}_{\mathbf{z}}(h'(x))| &\le
|\bar{F}_{\mathbf{z}}(h(x))-\bar{F}_{\mathbf{z}'}(h'(x))|
+|\bar{F}_{\mathbf{z}'}(h'(x))-\bar{F}_{\mathbf{z}}(h'(x))|\\
&\le |\bar{F}_{\mathbf{z}}(h(x))-\bar{F}_{\mathbf{z}'}(h'(x))|+\frac{\eta}{2}\\
&=|\bar{F}_{\mathbf{y}}(x)-\bar{F}_{\mathbf{y}'}(x)|+\frac{\eta}{2}
<\eta.\end{aligned}$$ By (\[i:barFinv\]) it follows that $d(h(x),h'(x))<\epsilon/5$ for every $x\in L_{12}\cup L_{23}\cup L_{31}$. It remains to consider points $x\in S_i$, $i=1,2,3$, which can be controlled via the behavior of the endpoints $y_i, y_i'$ of the short arc $S_i$.
First observe that the image $h(S_i)$ of $S_i$ is the short arc defined by $h(y_i), h(y_i')$. Indeed, on one hand the two points are close: $$d(h(y_i), h(y_i'))\le d(h(y_i), h'(y_i'))+d(h'(y_i'),h(y_i'))<\delta+\frac{\epsilon}{5}<\frac{2}{5}\epsilon.$$ On the other hand, the compliment $S^1\setminus S_i$ of $S_i$ contains a point $y_j$ with $j\in \{1,2,3\}\setminus \{i\}$; therefore $h(y_j)\notin h(S_i)$. Since $h(\mathbf{y})\in K$ we have $$d(h(y_i),h(y_j))\ge r>2\epsilon/5.$$ Hence $h(S_i)$ is the short arc defined by $2\epsilon/3$-close points $h(y_i), h(y_i')$, implying $$d(h(x), h(y_i))<\frac{2}{5}\epsilon\qquad (x\in S_i).$$ Similarly, $h'(S_i)$ is the short arc defined by $2\epsilon/5$-close points $h'(y_i), h'(y_i')$, and $$d(h'(x), h'(y_i))<\frac{2}{5}\epsilon\qquad (x\in S_i).$$ Since $y_i\in L_{ij}$, $d(h(y_i), h'(y_i))<\epsilon/5$. Therefore for any $x\in S_i$ $$d(h(x),h'(x))\le d(h(x), h(y_i))+d(h(y_i),h'(y_i))+d(h'(x), h'(y_i))<\epsilon.$$ This completes the proof of the lemma.
We claim that $H_\mu<{\operatorname{Homeo}}(S^1)$ is a *convergence group*, i.e., for any compact subset $K\subset {\operatorname{Trp}}$ the set $$H(\mu,K)=\{ h \in H_\mu \mid h^{-1} K\cap K\ne\emptyset \}$$ is compact. In particular, the Polish group $H_\mu$ is locally compact. Let us fix a compact subset $K\subset{\operatorname{Trp}}$. Since $H(\mu,K)$ is a closed subset in the Polish group ${\operatorname{Homeo}}(S^1)$, it suffices to show that any sequence $\{h_n\}_{n=1}^\infty$ in $H(\mu,K)$ contains a Cauchy subsequence. Choose triples $\mathbf{y}_n\in h_n^{-1} K\cap K$. Upon passing to a subsequence, we may assume that the points $\mathbf y_n$ converge to some $\mathbf y\in K$ and the points $\mathbf z_n=h_{n}(\mathbf{y}_{n})$ converge to some $\mathbf z\in K$. Let $\epsilon>0$. For the given $\epsilon$ and $K$ let $\delta>0$ be as in Lemma \[L:Hmu\]. Choose $N\in \mathbf{N}$ be large enough to ensure that $d(\mathbf{y}_{n},\mathbf{y}_m)<\delta$ and $d(\mathbf{z}_{n}, \mathbf{z}_m)<\delta$ for all $n,m>N$. It follows from Lemma \[L:Hmu\] that $h_n$ and $h_m$ are $\epsilon$-close whenever $n,m>N$. This proves that $H_\mu$ is a convergence group on the circle.
Finally, it follows that $H_\mu$ is conjugate to a closed subgroup of $G$. For discrete groups this is a well known results of Gabai [@Gabai] and Casson – Jungreis [@Casson]. The case of non-discrete convergence group $H_\mu<{\operatorname{Homeo}}(S^1)$ can be argued more directly. The closed convergence group $H_\mu$ is a locally compact subgroup of ${\operatorname{Homeo}}(S^1)$; the classification of all such groups is well known, and the only ones with convergence property are conjugate to ${\mathrm{PGL}}_2({{\mathbb R}})$ .
We return to the proof of Theorem \[T:ME-rigidity2and3\] in case of $n=2$. Starting from an integrable $(G,H)$-coupling $(\Omega,m)$ between $G={\mathrm{PGL}}_2({{\mathbb R}})$ and an unknown compactly generated unimodular group $H$ a continuous representation $\rho:H\to {\operatorname{Homeo}}(S^1)$ with compact kernel and closed image was constructed. Theorem \[T:intoPGL2\] implies that, up to conjugation, we may assume that $$\bar{H}=\rho(H)<G={\mathrm{PGL}}_2({{\mathbb R}}).$$ Since $\bar{H}$ is measure equivalent to $G={\mathrm{PGL}}_2({{\mathbb R}})$, it is non-amenable.
Case (1): $\bar{H}<G={\mathrm{PGL}}_2({{\mathbb R}})$ is non-discrete. (This does not occur in the original formulation of Theorem \[T:ME-rigidity2and3\], but is included in the broader context of lcsc $H$ adapted in this proof). There are only two non-discrete non-amenable closed subgroups of $G$: the whole group $G$ and its index two subgroup $G^0={\mathrm{PSL}}_2({{\mathbb R}})$. Both of these groups may appear as $\bar{H}$; in fact, direct products of the form $H\cong G\times K$ or $H\cong G^0\times K$ with compact $K$ and certain almost direct products $G'\times K'/C$ as in [@locally-compact]\*[Theorem A]{} give rise to an integrable measure equivalence between $H$ and $G$ (cf. [@locally-compact]\*[Theorem C]{}).
Case (2): $\bar{H}$ is discrete. We claim that such $\bar{H}$ is a cocompact lattice in $G$. Indeed, every finitely generated discrete non-amenable subgroup of $G$ is either cocompact or is virtually a free group ${{\mathbb F}}_2$. The latter possibility is ruled out by the following.
The free group ${{\mathbb F}}_2$ is not ${\mathrm{L}}^1$-measure equivalent to $G$.
Note that these groups are measure equivalent since ${{\mathbb F}}_2$ forms a lattice in $G$.
Assuming ${{\mathbb F}}_2$ is ${\mathrm{L}}^1$-measure equivalent to $G$, one can construct an integrable measure equivalence between $G$ and the automorphism group $H={\rm Aut}({\rm Tree}_4)$ of the $4$-regular tree, which contains ${{\mathbb F}}_2$ as a cocompact lattice. By Theorems \[thm:taut relative homeo\] and \[T:split+hom\] this would yield a continuous homomorphism $H\to {\operatorname{Homeo}}(S^1)$ with closed image. This leads to a contradiction, because $H$ is totally disconnected and virtually simple [@tits-trees]\*[Théorème 4.5]{}, while ${\operatorname{Homeo}}(S^1)$ has no non-discrete totally disconnected subgroups [@ghys]\*[Theorem 4.7 on p. 345]{}.
Measure equivalence {#sec:appendix_measure_equivalence}
===================
The appendix contains some general facts related to measure equivalence (Definition \[D:ME\]), the strong ICC property (Definition \[def:strongy ICC definition\]), and the notions of taut couplings and groups (Definition \[D:M-rigidity\]).
The category of couplings {#subs:composition}
-------------------------
Measure equivalence is an equivalence relation on unimodular lcsc groups. Let us describe explicitly the constructions which show reflexivity, symmetry and transitivity of measure equivalence.
### **Tautological coupling** {#ssub:tautological_coupling}
The tautological coupling is the $(G\times G)$-coupling $(G,m_G)$ given by $(g_1,g_2):g\mapsto g_1 g g_2^{-1}$. It demonstrates reflexivity of measure equivalence.
### **Duality** {#ssub:duality}
Symmetry is implied by the following: Given a $(G,H)$-coupling $(\Omega,m)$ the dual $({{\Omega}^*},{{m}^*})$ is the $(H,G)$-coupling ${{\Omega}^*}$ with the same underlying measure space $(\Omega,m)$ and the $H\times G$-action $(h,g):{{\omega}^*}\mapsto (g,h){{\omega}^*}$.
### **Composition of couplings** {#ssub:composition_of_couplings}
Compositions defined below shows that measure equivalence is a transitive relation. Let $G_1,H, G_2$ be unimodular lcsc groups, and $(\Omega_i,m_i)$ be a $(G_i,H)$-coupling for $i\in\{1,2\}$. We describe the $(G_1,G_2)$-coupling $\Omega_1\times_H{{\Omega}^*}_2$ modeled on the space of $H$-orbits on $(\Omega_1\times\Omega_2,m_1\times m_2)$ with respect to the diagonal $H$-action. Consider measure isomorphisms for $(\Omega_i,mi)$ as in : For $i\in\{1,2\}$ there are finite measure spaces $(X_i,\mu_i)$ and $(Y_i,\nu_i)$, measure-preserving actions $G_i{\curvearrowright}(X_i,\mu_i)$ and $H{\curvearrowright}(Y_i,\nu_i)$, measurable cocycles $\alpha_i:G_i\times X_i\to H$ amd $\beta_i:H\times Y_i\to G_i$, and measure space isomorphisms $G_i\times Y_i\cong \Omega_i\cong H\times X_i$ with respect to which the $G_i\times H$-actions are given by $$\begin{aligned}
(g_i,h)\ :\ & (h',x)\mapsto (h h' \alpha_i(g_i,x)^{-1},\, g_i.x),\\
(g_i,h)\ :\ & (g',y)\mapsto (g_i g' \beta(h,y)^{-1},\,h.y).\end{aligned}$$ The space $\Omega_1\times_H{{\Omega}^*}_2$ with its natural $G_1\times G_2$-action is equivariantly isomorphic to $(X_1\times X_2\times H,\mu_1\times\mu_2\times m_H)$ endowed with the $G_1\times G_2$-action $$(g_1,g_2): (x_1,x_2,h)\mapsto (g_1.x_1,\, g_2.x_2,\, \alpha_1(g_1,x_1) h \alpha_2(g_2,x_2)^{-1}).$$ To see that it is a $(G_1,G_2)$-coupling, we identify this space with $Z\times G_1$ equipped with the action $$(g_1,g_2): (g',z)\mapsto (g_1 g' c(g_2,z)^{-1}, g_2.z)\qquad(g'\in G_1,\ z\in Z)$$ where $Z=X_2\times Y_1$, while the action $G_2{\curvearrowright}Z$ and the cocycle $c:G_2\times Z\to G_1$ are given by $$\label{e:composition-coc}
\begin{aligned}
&g_2:(x,y)\mapsto (g_2.x,\alpha_2(g_2,x).y),\\
&c(g_2,(x,y))=\beta_1(\alpha_2(g_2,x), y).
\end{aligned}$$ Similarly, $\Omega_1\times_H{{\Omega}^*}_2\cong W\times G_2$, for $W\cong X_1\times Y_2$.
### **Morphisms** {#ssub:morphisms}
Let $(\Omega_i,m_i)$, $i\in\{1,2\}$, be two $(G,H)$-couplings. Let $F:\Omega_1\to\Omega_2$ be a measurable map such that for $m_1$-a.e. $\omega\in\Omega_1$ and every $g\in G$ and every $h\in H$ $$F((g,h)\omega)=(g,h)F(\omega).$$ Such maps are called *quotient maps* or *morphisms*.
### **Compact kernels** {#ssub:compact kernels}
Let $(\Omega,m)$ be a $(G,H)$-coupling, and let $$\{1\}\to K\to G\to \bar{G}\to\{1\}$$ be a short exact sequence where $K$ is compact. Then the natural quotient space $(\bar\Omega,\bar{m})=(\Omega,m)/K$ is a $(\bar{G},H)$-coupling, and the natural map $F:\Omega\to\bar{\Omega}$, $F:\omega\mapsto K\omega$, is equivariant in the sense of $F((g,h)\omega)=(\bar{g},h) F(\omega)$. This may be considered as an *isomorphism of couplings up to compact kernel*.
### **Passage to lattices** {#ssub:passage to lattices}
Let $(\Omega,m)$ be a $(G,H)$-coupling, and let $\Gamma<G$ be a lattice. By restricting the $G\times H$-action on $(\Omega,m)$ to $\Gamma\times H$ we obtain a $(\Gamma,H)$-coupling. Formally, this follows by considering $(G,m_G)$ as a $\Gamma\times G$-coupling and considering the composition $G\times_G \Omega$ as $\Omega$ with the $\Gamma\times H$-action.
${\mathrm{L}}^p$-integrability conditions {#sub:Lp-integrability}
-----------------------------------------
Let $G$ and $H$ be compactly generated unimodular lcsc groups equipped with proper norms $|\cdot |_G$ and $|\cdot |_H$. Let $c:G\times X\to H$ be a measurable cocycle, and fix some $p\in [1,\infty)$. For $g\in G$ we define $$\|g\|_{c,p}=\Bigl(\int_X |c(g,x)|^p_H\,d\mu(x)\Bigr)^{1/p}.$$ For $p=\infty$ we use the essential supremum. Assume that $\|g\|_{c,p}<\infty$ for a.e. $g\in G$. We claim that there are constants $a,A>0$ so that for every $g\in G$ $$\label{e:sublinear}
\|g\|_{c,p}\le A\cdot |g|_G+a.$$ Hence $c$ is ${\mathrm{L}}^p$-integrable in the sense of Definition \[D:Lp-ME\]. The key observation here is that $\|-\|_{c,p}$ is subadditive. Indeed, by the cocycle identity, subadditivity of the norm $|-|_H$, and the Minkowski inequality, for any $g_1,g_2\in G$ we get $$\begin{aligned}
\|g_2 g_1\|_{c,p}&\le\Bigl(\int_X
\bigl(|c(g_2,g_1. x)|_H+
|c(g_1, x)|_H\bigr)^p\,d\mu(x)
\Bigr)^{1/p}\\
&\le \Bigl(\int_X |c(g_2,-)|^p_H\,d\mu\Bigr)^{1/p}
+\Bigl(\int_X |c(g_1,-)|^p_H\,d\mu\Bigr)^{1/p}\\
&= \|g_2\|_{c,p}+\|g_1\|_{c,p}.\end{aligned}$$ For $t>0$ denote $E_t=\{g\in G : \|g\|_{c,p}<t\}$. We have $E_t\cdot E_s\subseteq E_{s+t}$ for any $t,s>0$. Fix $t$ large enough so that $m_G(E_t)>0$. By [@doran+fell]\*[Corollary 12.4 on p. 235]{}, $E_{2t}\supseteq E_t\cdot E_t$ has a non-empty interior. Hence any compact subset of $G$ can be covered by finitely many translates of $E_{2t}$. The subadditivity implies that $\|g\|_{c,p}$ is bounded on compact sets. This gives (\[e:sublinear\]).
\[L:composition-of-lp-coc\] Let $G$,$H$,$L$ be compactly generated groups, $G{\curvearrowright}(X,\mu)$, $H{\curvearrowright}(Y,\nu)$ be finite measure-preserving actions, and $\alpha:G\times X\to H$ and $\beta:H\times Y\to L$ be ${\mathrm{L}}^p$-integrable cocycles for some $1\le p\le \infty$. Consider $Z=X\times Y$ and $G{\curvearrowright}Z$ by $g:(x,y)\mapsto (g.x, \alpha(g,x).y)$. Then the cocycle $\gamma:G\times Z\to L$ given by $$\gamma(g,(x,y))=\beta(\alpha(g,x),y).$$ is ${\mathrm{L}}^p$-integrable.
For $p=\infty$ the claim is obvious. Assume $p<\infty$. Let $A,a,B,b$ be constants such that $\|h\|_{\beta,p}\le B\cdot |h|_H+b$ and $\|g\|_{\alpha,p}\le A\cdot |g|_G+a$. Then $$\begin{aligned}
\|g\|_{\gamma,p}^p &= \int_{X\times Y} |\beta(\alpha(g,x),y)|_{\mathrm{L}}^p\,d\mu(x)\,d\nu(y)\\
&\le \int_X (B\cdot |\alpha(g,x)|_H+b)^p\,d\mu(x)\\
&\le \max(B,b)^p\cdot \|g\|_{\alpha,p}^p\le (C\cdot |g|_G+c)^p\end{aligned}$$ for appropriate constants $c>0$ and $C>0$.
\[lem:composition of lp couplings\] Let $G_1,H,G_2$ be compactly generated unimodular lcsc groups. For $i\in\{1,2\}$ let $(\Omega_i,m_i)$ be an ${\mathrm{L}}^p$-integrable $(G_i,H)$-coupling. Then $\Omega_1\times_H {{\Omega}^*}_2$ is an ${\mathrm{L}}^p$-integrable $(G_1,G_2)$-coupling.
This follows from Lemma \[L:composition-of-lp-coc\] using the explicit description (\[e:composition-coc\]) of the cocycles for $\Omega_1\times_H {{\Omega}^*}_2$.
We conclude that for each $1\le p\le \infty$, ${\mathrm{L}}^p$-measure equivalence is an equivalence relation between compactly generated unimodular lcsc groups.
Tautening maps {#sub:equivariant_tautening_maps}
--------------
\[L:taut-MR\] Let $G$ be a lcsc group, $\Gamma$ a countable group and $j_1,j_2:\Gamma\to G$ be homomorphisms with $\Gamma_i=j_i(\Gamma)$ being lattices in $G$. Assume that $G$ is taut (resp. $p$-taut and $\Gamma_i$ are ${\mathrm{L}}^p$-integrable). Then there exists $g\in G$ so that $$j_2(\gamma)=g\,j_1(\gamma)\,g^{-1}\qquad (\gamma\in\Gamma).$$ If $\pi:G\to{\mathcal{G}}$ is a continuous homomorphism into a Polish group and $G$ is taut relative to $\pi:G\to{\mathcal{G}}$ (resp. $G$ is $p$-taut relative to $\pi:G\to{\mathcal{G}}$ and $\Gamma_i$ are ${\mathrm{L}}^p$-integrable) then there exists $y\in {\mathcal{G}}$ with $$\pi(j_2(\gamma))=y \pi(j_1(\gamma))y^{-1}\qquad (\gamma\in\Gamma).$$
We prove the more general second statement. The group $$\Delta=\{(j_1(\gamma),j_2(\gamma))\in G\times G \mid \gamma\in \Gamma\}$$ is a closed discrete subgroup in $G\times G$. The homogeneous $G\times G$-space $\Omega=G\times G/\Delta$ equipped with the $G\times G$-invariant measure is easily seen to be a $(G,G)$-coupling. It will be ${\mathrm{L}}^p$-integrable if $\Gamma_1$ and $\Gamma_2$ are ${\mathrm{L}}^p$-integrable lattices. Let $\Phi:\Omega\to {\mathcal{G}}$ be the tautening map. There are $a,b\in G$ and $x\in{\mathcal{G}}$ such that for all $g_1,g_2\in G$ $$\Phi((g_1a,g_2b)\Delta_f)=\pi(g_1) x \pi(g_2)^{-1}.$$ Since $(a,b)$ and $(j_1(\gamma)^a \,a,\,j_2(\gamma)^b\,b)$ are in the same $\Delta$-coset, where $g^h=hgh^{-1}$, we get for all $g_1,g_2\in G$ and every $\gamma\in \Gamma$ $$\pi(g_1) x \pi(g_2)^{-1}=\pi(g_1) \pi(j_1(\gamma)^a) x\pi(j_2(\gamma)^b)^{-1} \pi(g_2)^{-1}.$$ This implies that $j_1$ and $j_2$ are conjugate homomorphisms.
The following lemma relates tautening maps $\Phi:\Omega\to G$ and cocycle rigidity for ME-cocycles.
\[L:coc-taut\] Let $G$ be a unimodular lcsc group, ${\mathcal{G}}$ be a Polish group, $\pi:G\to{\mathcal{G}}$ a continuous homomorphism. Let $(\Omega,m)$ be a $(G,G)$-coupling and $\alpha:G\times X\to G$, $\beta:G\times Y\to G$ be the corresponding ME-cocycles. Then $\Omega$ is taut relative to $\pi$ iff the ${\mathcal{G}}$-valued cocycle $\pi\circ\alpha$ is conjugate to $\pi$, that is, $$\pi\circ\alpha(g,x)=f(g.x)^{-1} \pi(g) f(x)$$ for a unique measurable map $f:X\to {\mathcal{G}}$. This is also equivalent to $\pi\circ\beta$ being uniquely conjugate to $\pi:G\to{\mathcal{G}}$.
Let $\alpha:G\times X\to G$ be the ME-cocycle associated to a measure space isomorphism $i:(G,m_G)\times (X,\mu)\to (\Omega,m)$ as in (\[e:ij-fd\]). In particular, $$(g_1,g_2): i(g,x)\mapsto i(g_2 g\alpha(g_1,x)^{-1},\,g_1.x).$$ We shall now establish a $1$-to-$1$ correspondence between Borel maps $f:X\to {\mathcal{G}}$ with $$\pi\circ\alpha(g,x)=f(g.x)^{-1}\pi(g) f(x)$$ and tautening maps $\Phi:\Omega\to{\mathcal{G}}$. Given $f$ as above one verifies that $$\Phi:\Omega\to {\mathcal{G}},\qquad \Phi(i(g,x))=f(x)\pi(g)^{-1}$$ is $G\times G$-equivariant.
For the converse direction, suppose $\Phi:\Omega\to G$ is a tautening map. Thus, $$g_1 \Phi(g_0,x) g_2^{-1}=\Phi((g_1,g_2)(g_0,x))=\Phi(g_2 g_0 \alpha(g_1,x)^{-1},\, g_1.x).$$ For $\mu$-a.e. $x\in X$ and a.e. $g\in G$ the value of $\Phi(g,x)g$ is constant $f(x)$. The above identity implies the required identity $\alpha(g,x)=f(g.x)^{-1}g f(x)$.
Strong ICC property {#sub:strong_icc_property}
-------------------
\[L:uniq2sICC\] Let $G$ be a unimodular lcsc group, ${\mathcal{G}}$ a Polish group, $\pi:G\to{\mathcal{G}}$ a continuous homomorphism. Suppose that ${\mathcal{G}}$ is not strongly ICC relative to $\pi(G)$. Then there is a $(G,G)$-coupling $(\Omega,m)$ with two distinct tautening maps to ${\mathcal{G}}$.
Let $\mu$ be a Borel probability measure on ${\mathcal{G}}$ invariant under conjugations by $\pi(G)$. Consider $\Omega=G\times {\mathcal{G}}$ with the measure $m=m_G\times\mu$ where $m_G$ denotes the Haar measure, and measure-preserving $G\times G$-action $$(g_1,g_2): (g,x)\mapsto (g_1 g g_2^{-1},\, \pi(g_2)\, x\, \pi(g_2)^{-1}).$$ This is clearly a $(G,G)$-coupling and the following measurable maps $\Phi_i:\Omega\to {\mathcal{G}}$, $i\in \{1,2\}$, are $G\times G$-equivariant: $\Phi_1(g,x)=\pi(g)$ and $\Phi_2(g,x)=\pi(g)\cdot x$. Note that $\Phi_1=\Phi_2$ on a conull set iff $\mu=\delta_e$.
\[L:sICC2uniq\] Let $G$ be a unimodular lcsc group and ${\mathcal{G}}$ a Polish group. Assume that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$. Let $(\Omega,m)$ be a $(G,G)$-coupling. Then:
1. \[i:unique\] There is at most one tautening map $\Phi\colon\Omega\to {\mathcal{G}}$.
2. \[i:descent\] Let $F:(\Omega,m)\to (\Omega_0,m_0)$ be a morphism of $(G,G)$-couplings and suppose that there exists a tautening map $\Phi:\Omega\to{\mathcal{G}}$. Then it descends to $\Omega_0$, i.e., $\Phi=\Phi_0\circ F$ for a unique tautening map $\Phi_0:\Omega_0\to{\mathcal{G}}$.
3. \[i:2lattice\] If $\Gamma_1,\Gamma_2<G$ are lattices, then $\Phi:\Omega\to {\mathcal{G}}$ is unique as a $\Gamma_1\times\Gamma_2$-equivariant map.
4. \[i:extn\] If $\Gamma_1,\Gamma_2<G$ are lattices, and $(\Omega,m)$ admits a $\Gamma_1\times\Gamma_2$-equivariant map $\Phi:\Omega\to{\mathcal{G}}$, then $\Phi$ is $G\times G$-equivariant.
5. \[i:Dirac\] If $\eta:\Omega\to {\operatorname{Prob}}({\mathcal{G}})$, $\omega\mapsto \eta_\omega$, is a measurable $G\times G$-equivariant map to the space of Borel probability measures on ${\mathcal{G}}$ endowed with the weak topology, then it takes values in Dirac measures: We have $\eta_\omega=\delta_{\Phi(\omega)}$, where $\Phi:\Omega\to {\mathcal{G}}$ is the unique tautening map.
We start from the last claim and deduce the other ones from it.
[(\[i:Dirac\])]{}. Given an equivariant map $\eta:\Omega\to {\operatorname{Prob}}({\mathcal{G}})$ consider the convolution $$\nu_\omega=\check\eta_\omega * \eta_\omega,$$ namely the image of $\eta_\omega\times\eta_\omega$ under the map $(a,b)\mapsto a^{-1}\cdot b$. Then $$\nu_{(g,h)\omega}=\nu_\omega^{\pi(g)}\qquad (g,h\in G),$$ where the latter denotes the push-forward of $\nu_\omega$ under the conjugation $$a\mapsto a^{\pi(g)}=\pi(g)^{-1}a \pi(g).$$ In particular, the map $\omega\mapsto \nu_\omega$ is invariant under the action of the second $G$-factor. Therefore $\nu_\omega$ descends to a measurable map $\tilde\nu:\Omega/G\to {\operatorname{Prob}}({\mathcal{G}})$, satisfying $$\tilde\nu_{g.x}=\tilde\nu_x^{\pi(g)}\qquad (x\in X=\Omega/G,\ g\in G).$$ Here we identify $\Omega/G$ with a finite measure space $(X,\mu)$ as in . Consider the center of mass $$\bar\nu=\mu(X)^{-1}\int_{X}\tilde\nu_x \,d\mu(x).$$ It is a probability measure on ${\mathcal{G}}$, which is invariant under conjugations. By the strong ICC property relative to $\pi(G)$ we get $\bar\nu=\delta_e$. Since $\delta_e$ is an extremal point of ${\operatorname{Prob}}({\mathcal{G}})$, it follows that $m$-a.e. $\nu(\omega)=\delta_e$. This implies that $\eta_\omega=\delta_{\Phi(\omega)}$ for some measurable $\Phi:\Omega\to {\mathcal{G}}$. The latter is automatically $G\times G$-equivariant.
[(\[i:unique\])]{}. If $\Phi_1,\Phi_2:\Omega\to {\mathcal{G}}$ are tautening maps, then $\eta_\omega=\frac{1}{2}(\delta_{\Phi_1(\omega)}+\delta_{\Phi_2(\omega)})$ is an equivariant map $\Omega\to{\operatorname{Prob}}({\mathcal{G}})$. By (\[i:Dirac\]) it takes values in Dirac measures, which is equivalent to the $m$-a.e. equality $\Phi_1=\Phi_2$.
[(\[i:descent\])]{}. Disintegration of $m$ with respect to $m_0$ gives a $G\times G$-equivariant measurable map $\Omega_0\to\mathcal{M}(\Omega)$, $\omega\mapsto m_{\omega_0}$, to the space of finite measures on $\Omega$. Then the map $\eta:\Omega_0\to{\operatorname{Prob}}({\mathcal{G}})$, given by $$\eta_{\omega_0}=\|m_{\omega_0}\|^{-1}\cdot \Phi_* (m_{\omega_0}).$$ is $G\times G$-equivariant. Hence by (\[i:Dirac\]), $\eta_{\omega_0}=\delta_{\Phi_0(\omega_0)}$ for the unique tautening map $\Phi_0:\Omega_0\to{\mathcal{G}}$. The relation $\Phi=\Phi_0\circ F$ follows from the fact that Dirac measures are extremal.
[(\[i:2lattice\])]{} follows from (\[i:extn\]) and (\[i:unique\]).
[(\[i:extn\])]{}. The claim is equivalent to: For $m$-a.e. $\omega\in\Omega$ the map $F_\omega\colon G\times G\to {\mathcal{G}}$ with $$F_\omega(g_1,g_2)=\pi(g_1)^{-1}\,\Phi((g_1,g_2)\omega)\,\pi(g_2)$$ is $m_G\times m_G$-a.e. constant $\Phi_0(\omega)$. Note that the family $\{F_\omega\}$ has the following equivariance property: For $g_1,g_2,h_1,h_2\in G$ we have $$\begin{aligned}
F_{(h_1,h_2)\omega}(g_1,g_2)&=&\pi(g_1)^{-1}\Phi((g_1h_1,g_2h_2)\omega)\pi(g_2)\\
&=&\pi(h)_1^{-1} F_\omega(g_1h_1, g_2h_2) \pi(h_2).\end{aligned}$$ Since $\Phi$ is $\Gamma_1\times\Gamma_2$-equivariant, for $m$-a.e. $\omega\in\Omega$ the map $F_\omega$ descends to $G/\Gamma_1\times G/\Gamma_2$. Let $\eta_\omega\in{\operatorname{Prob}}({\mathcal{G}})$ denote the distribution of $F_\omega(\cdot,\cdot)$ over the probability space $G/\Gamma_1\times G/\Gamma_2$, that is, for a Borel subset $E\subset{\mathcal{G}}$ $$\eta_\omega(E)=m_{G/\Gamma_1}\times m_{G/\Gamma_2}\{ (g_1,g_2) \mid F_\omega(g_1,g_2)\in E\}.$$ Since this measure is invariant under translations by $G\times G$, it follows that $\eta_\omega$ is a $G\times G$-equivariant maps $\Omega\to {\operatorname{Prob}}({\mathcal{G}})$. By (\[i:Dirac\]) one has $\eta_\omega=\delta_{f(\omega)}$ for some measurable $G\times G$-equivariant map $f:\Omega\to {\mathcal{G}}$. Hence $F_\omega(g_1,g_2)=f(\omega)$ for a.e. $g_1,g_2\in G$; it follows that $$\label{e:phi-g1g2}
\Phi((g_1,g_2)\omega)=\pi(g_1) \Phi(\omega) \pi(g_2)^{-1}$$ holds for $m_G\times m_G\times m$-a.e. $(g_1,g_2,\omega)$.
Let $\pi:G\to{\mathcal{G}}$ be as above and assume that ${\mathcal{G}}$ is strongly ICC relative to $\pi(G)$. Then the collection of all $(G,G)$-couplings which are taut relative to $\pi:G\to{\mathcal{G}}$ is closed under the operations of taking the dual, compositions, quotients and extensions.
The uniqueness of tautening maps follow from the relative strong ICC property (Lemma \[L:sICC2uniq\].(\[i:unique\])). Hence we focus on the existence of such maps.
Let $\Phi:\Omega\to{\mathcal{G}}$ be a tautening map. Then $\Psi({{\omega}^*})=\Phi(\omega)^{-1}$ is a tautening map ${{\Omega}^*}\to{\mathcal{G}}$.
Let $\Phi_i:\Omega_i\to{\mathcal{G}}$, $i=1,2$, be tautening maps. Then $\Psi([\omega_1,\omega_2])=\Phi(\omega_1)\cdot\Phi(\omega_2)$ is a tautening map $\Omega_1\times_G\Omega_2\to{\mathcal{G}}$.
If $F:(\Omega_1,m_1)\to(\Omega_2,m_2)$ is a quotient map and $\Phi_1:\Omega_1\to{\mathcal{G}}$ is a tautening map, then, by Lemma \[L:sICC2uniq\].(\[i:descent\]), $\Phi_1$ factors as $\Phi_1=\Phi_2\circ F$ for a tautening map $\Phi_2:\Omega_2\to{\mathcal{G}}$. On the other hand, given a tautening map $\Phi_2:\Omega_2\to{\mathcal{G}}$, the map $\Phi_1=\Phi_2\circ F$ is tautening for $\Omega_1$.
Bounded cohomology {#sec:cohomological tools}
==================
Our background references for bounded cohomology, especially for the functorial approach to it, are [@burger+monod; @monod-book]. We summarize what we need from Burger-Monod’s theory of bounded cohomology. Since we restrict to discrete groups, some results we quote from this theory are already go back to Ivanov [@ivanov].
Banach modules {#sub:banach_modules}
--------------
All Banach spaces are over the field ${{\mathbb R}}$ of real numbers. By the dual of a Banach space we understand the normed topological dual. The dual of a Banach space $E$ is denoted by $E^\ast$. Let $\Gamma$ be a discrete and countable group. A *Banach $\Gamma$-module* is a Banach space $E$ endowed with a group homomorphism $\pi$ from $\Gamma$ into the group of isometric linear automorphisms of $E$. We use the module notation $\gamma\cdot
e=\pi(\gamma)(e)$ or just $\gamma e=\pi(\gamma)(e)$ for $\gamma\in\Gamma$ and $e\in E$ whenever the action is clear from the context. The submodule of $\Gamma$-invariant elements is denoted by $E^\Gamma$. Note that $E^\Gamma\subset E$ is closed.
If $E$ and $F$ are Banach $\Gamma$-modules, a *$\Gamma$-morphism* $E\rightarrow F$ is a $\Gamma$-equivariant continuous linear map. The space ${\mathcal{B}}(E,F)$ of continuous, linear maps $E\rightarrow F$ is endowed with a natural Banach $\Gamma$-module structure via $$\label{eq:G action on space of bounded maps}
(\gamma\cdot f)(e)=\gamma f(\gamma^{-1}e).$$ The *contragredient* Banach $\Gamma$-module structure $E^\sharp$ associated to $E$ is by definition ${\mathcal{B}}(E,{{\mathbb R}})=E^\ast$ with the $\Gamma$-action . A *coefficient $\Gamma$-module* is a Banach $\Gamma$-module $E$ contragredient to some separable continuous Banach $\Gamma$-module denoted by $E^\flat$. The choice of $E^\flat$ is part of the data. The specific choice of $E^\flat$ defines a topology on $E$. The only examples that appear in this paper are $E={\mathrm{L}}^\infty(X,\mu)$ with $E^\flat=L^1(X,\mu)$ and $E=E^\flat={{\mathbb R}}$.
For a coefficient $\Gamma$-module $E$ let ${\mathrm{C}_\mathrm{b}}^k(\Gamma,E)$ be the Banach $\Gamma$-module ${\mathrm{L}}^\infty(\Gamma^{k+1},E)$ consisting of bounded maps from $\Gamma^{k+1}$ to $E$ endowed with the supremum norm and the $\Gamma$-action: $$\label{eq:action on cochains}
(\gamma\cdot f)(\gamma_0,\dots,\gamma_k)=\gamma\cdot f(\gamma^{-1}\gamma_0,\dots,\gamma^{-1}\gamma_k).$$
For a coefficient $\Gamma$-module $E$ and a standard Borel $\Gamma$-space $S$ with quasi-invariant measure let ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(S, E)$ be the space of -measurable essentially bounded maps from $S$ to $E$, where maps are identified if they only differ on a null set. The space ${{\mathrm{L}}_{\mathrm{w}\ast}}^\infty(S, E)$ is endowed with the essential supremum norm and the $\Gamma$-action . For a measurable space $X$ the Banach space ${\mathcal{B}}^\infty(X, E)$ is the space of -measurable bounded maps from $X$ to $E$ endowed with supremum norm [@burger+iozzi]\*[Section 2]{} and the $\Gamma$-action .
Injective resolutions {#sub:bounded_cohomology}
---------------------
Let $\Gamma$ be a discrete group and $E$ be a Banach $\Gamma$-module. The sequence of Banach $\Gamma$-modules ${\mathrm{C}_\mathrm{b}}^k(\Gamma, E)$, $k\ge
0$, becomes a chain complex of Banach $\Gamma$-modules via the standard homogeneous coboundary operator $$\label{eq:homogeneous differential}
d(f)(\gamma_0,\dots,\gamma_k)=\sum_{i\ge 0}^k(-1)^i
f(\gamma_0,\dots, \hat{\gamma_i},\dots,\gamma_k).$$
The *bounded cohomology* ${\mathrm{H}_\mathrm{b}}^\bullet(\Gamma,E)$ of $\Gamma$ with coefficients $E$ is the cohomology of the complex of $\Gamma$-invariants ${\mathrm{C}_\mathrm{b}}^\bullet(\Gamma, E)^\Gamma$. The bounded cohomology ${\mathrm{H}_\mathrm{b}}^\bullet(\Gamma,E)$ inherits a semi-norm from ${\mathrm{C}_\mathrm{b}}^\bullet(\Gamma,
E)$: The *(semi-)norm* of an element $x\in {\mathrm{H}_\mathrm{b}}^k(\Gamma, E)$ is the infimum of the norms of all cocycles in the cohomology class $x$.
Next we briefly recall the functorial approach to bounded cohomology as introduced by Ivanov [@ivanov] for discrete groups and further developed by Burger-Monod [@burger+monod; @monod-book]. We refer for the definition of *relative injectivity* of a Banach $\Gamma$-module to [@monod-book]\*[Definition 4.1.2 on p. 32]{}. A *strong resolution* $E^\bullet$ of $E$ is a resolution, i.e. an acyclic complex, $$0\to E\to E^0\to E^1\to E^2\to\dots$$ of Banach $\Gamma$-modules that has chain contraction which is contracting with respect to the Banach norms. The key to the functorial definition of bounded cohomology are the following two theorems:
\[thm:main homological theorem\] Let $E$ and $F$ be Banach $\Gamma$-modules. Let $E^\bullet$ be a strong resolution of $E$. Let $F^\bullet$ be a resolution $F$ by relatively injective Banach $\Gamma$-modules. Then any $\Gamma$-morphism $E\to F$ extends to a $\Gamma$-morphism of resolutions $E^\bullet\to F^\bullet$ which is unique up to $\Gamma$-homotopy. Hence $E\to F$ induces functorially continuous linear maps ${\mathrm{H}}^\bullet({E^\bullet}^\Gamma)\to{\mathrm{H}}^\bullet({F^\bullet}^\Gamma)$.
\[thm:the definining complex is strong injective resolution\] Let $E$ be a Banach $\Gamma$-module. The complex $E\to{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma,E)$ with $E\to{\mathrm{C}_\mathrm{b}}^0(\Gamma,E)$ being the inclusion of constant functions is a strong, relatively injective resolution.
For a coefficient $\Gamma$-module, a measurable space $X$ with measurable $\Gamma$-action, and a standard Borel $\Gamma$-space $S$ with quasi-invariant measure we obtain chain complexes ${\mathcal{B}}^\infty(X^{\bullet+1},E)$ and ${{\mathrm{L}}_{\mathrm{w}\ast}}(S^{\bullet+1},
E)$ of Banach $\Gamma$-modules via the standard homogeneous coboundary operators (similar as in ).
The following result is important for expressing induced maps in bounded cohomology in terms of boundary maps [@burger+iozzi].
\[prop:B-complex strong resolution\] Let $E$ be a coefficient $\Gamma$-module. Let $X$ be a measurable space with measurable $\Gamma$-action. The complex $E\to{\mathcal{B}}^\infty(X^{\bullet+1},E)$ with $E\to{\mathcal{B}}^\infty(X,E)$ being the inclusion of constant functions is a strong resolution of $E$.
The next theorem is one of the main results of the functorial approach to bounded cohomology by Burger-Monod:
\[thm:boundary resolutions by Burger-Monod\] Let $S$ be a regular $\Gamma$-space and be $E$ a coefficient $\Gamma$-module. Then $E\to{{\mathrm{L}}_{\mathrm{w}\ast}}(S^{\bullet+1}, E)$ with $E\to{{\mathrm{L}}_{\mathrm{w}\ast}}(S^{\bullet+1},E)$ being the inclusion of constant functions is a strong resolution. If, in addition, $S$ is amenable in the sense of Zimmer [@monod-book]\*[Definition 5.3.1]{}, then each ${{\mathrm{L}}_{\mathrm{w}\ast}}(S^{k+1},E)$ is relatively injective, and according to Theorem \[thm:main homological theorem\] the cohomology groups ${\mathrm{H}}^\bullet({{\mathrm{L}}_{\mathrm{w}\ast}}(S^{\bullet+1},E)^\Gamma)$ are canonically isomorphic to ${\mathrm{H}_\mathrm{b}}^\bullet(\Gamma,E)$.
\[def:poisson transform\] Let $S$ be a standard Borel $\Gamma$-space with a quasi-invariant probability measure $\mu$. Let $E$ be a coefficient $\Gamma$-module. The *Poisson transform* ${\mathrm{PT}}^\bullet:
{{\mathrm{L}}_{\mathrm{w}\ast}}(S^{\bullet+1},E)\to{\mathrm{C}_\mathrm{b}}^\bullet(\Gamma, E)$ is the $\Gamma$-morphism of chain complexes defined by $${\mathrm{PT}}^k(f)(\gamma_0,\dots,\gamma_k)= \int_{S^{k+1}}
f(\gamma_0 s_0,\dots,\gamma_k s_k)d\mu(s_0)\dots d\mu(s_k).$$
If $S$ is amenable, then the Poisson transform induces a canonical isomorphism in cohomology (Theorem \[thm:boundary resolutions by Burger-Monod\]). By the same theorem this isomorphism does not depend on the choice of $\mu$ within a given measure class.
[^1]: Here $\simeq$ means *locally isomorphic*.
[^2]: Any lcsc group containing a lattice is necessarily unimodular.
[^3]: If one only requires equivariance for almost all $g_1,g_2\in G$ one can always modify $\Phi$ on a null set to get an everywhere equivariant map [@zimmer-book]\*[Appendix B]{}.
[^4]: For the formulation of Mostow rigidity above we have to assume that $G$ has trivial center.
[^5]: Every connected lcsc group is compactly generated [@stroppel]\*[Corollary 6.12 on p. 58]{}.
[^6]: The formulation of the virtual isomorphism case in terms of induced actions is due to Kida [@Kida:OE].
[^7]: has Infinite Conjugacy Classes.
[^8]: The target ${\mathcal{G}}$ is assumed to be locally compact in this reference but the proof therein works the same for a Polish group ${\mathcal{G}}$.
[^9]: Here the Poisson transform is defined in terms of $\nu_x$. Since the visual measures are all in the same measure class, the Poisson transform in cohomology does not depend on the choice of $x$ (see the remark after Definition \[def:poisson transform\]).
[^10]: The reader should note that in *loc. cit.* the Haar measure is normalized by ${\operatorname{vol}}(\Gamma{\backslash}{\mathbf{H}}^n)$ whereas we normalize it by $1$.
[^11]: ${\operatorname{vol}}(B(0,r))$ is here the Lebesgue measure of the Euclidean ball of radius $r$ around $0\in{{\mathbb R}}^n$.
|
---
abstract: 'We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous “Sullivan dictionary", this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.'
address:
- |
CNRS\
Département de Mathématique d’Orsay\
Bâtiment 425, Université de Paris Sud, 91405 Orsay cedex, France.
- |
LAMA\
Université Paris-Est Marne-la-Vallée\
5 boulevard Descartes\
77454 Champs sur Marne\
France
author:
- Bertrand Deroin
- Romain Dujardin
title: 'Complex projective structures: Lyapunov exponent, degree and harmonic measure'
---
Introduction {#introduction .unnumbered}
============
Our purpose in this paper is to introduce several new objects associated with a $\mathbb{CP}^1$-structure on a Riemann surface $X$ of finite type, and study their relationships. In the non-compact case, we assume that the projective structure is “parabolic at the cusps", in a sense that will be made precise below.
To such a projective structure $\sigma$, we associate:
a [*Lyapunov exponent*]{} $\chi(\sigma)$, which was constructed in [@Bers1];
a [*degree*]{} $\deg(\sigma)$ which is simply the (normalized) asymptotic covering degree of the developing map $\widetilde X{\rightarrow}{{\mathbb{P}^1}}$ ($\widetilde X$ the universal cover of $X$);
a family of [*harmonic measures*]{} $(\nu_x)_{x\in \widetilde X}$ on ${{\mathbb{P}^1}}$, which generalize the traditional harmonic measures on the limit sets of Kleinian groups (throughout the paper, ${{\mathbb{P}^1}}$ stands for ${\mathbb{C}}{{\mathbb{P}^1}}$).
We will show that the Lyapunov exponent and the degree are related by a simple formula, and give estimates for the Hausdorff dimension of the harmonic measures in terms of $\chi$. We give several applications of these ideas to the study of the space $P(X)$ of (parabolic) projective structures on $X$, in particular revealing new aspects of the famous Sullivan dictionary between rational and Möbius dynamics on ${{\mathbb{P}^1}}$.
Before proceeding to a detailed presentation of these results, let us present the main characters of the story.
Parabolic $\mathbb P^1$-structures {#ss:parabolic type PS}
----------------------------------
Let us fix a Riemann surface $X$ of finite type (genus $g$ with $n$ punctures) with negative Euler characteristic $\mathrm{eu(X)}=2-2g-n$, as well as a universal cover $c : \widetilde{X}\rightarrow X$. Throughout this paper, by definition the (unmarked) fundamental group $\pi_1(X)$ is the deck group of this covering. Recall that the automorphism group of the Riemann sphere is the group $\text{PSL} (2,\mathbb C)$ acting on $\mathbb P^1 = \mathbb C \cup {\left\{\infty\right\}} $ by the formula $ {\left(\begin{smallmatrix}
a & b \\ c& d \end{smallmatrix}\right)} \cdot z = \frac{az + b }{cz + d}$. By definition a [*Kleinian group*]{} is a discrete subgroup of ${\mathrm{PSL}(2,\mathbb C)}$.
It is convenient to define a ${{\mathbb{P}^1}}$-structure on $X$ in terms of the so-called development-holonomy pair $(\mathsf{dev},\mathsf{hol})$. Consider a non constant locally injective meromorphic map $\mathsf{dev} : \widetilde{X} \rightarrow \mathbb P ^1 $, satisfying the equivariance property $\mathsf{dev}\circ \gamma = \mathsf{hol}(\gamma)\circ \mathsf{dev} $, where $\mathsf{hol}$ is a representation $\pi_1(X){\rightarrow}\text{PSL} (2,\mathbb C)$. If $A\in {\mathrm{PSL}(2,\mathbb C)}$, the pairs $(\mathsf{dev},\mathsf{hol})$ and $(A\circ \mathsf{dev},A \circ \mathsf{hol}\circ A^{-1})$ will be declared as equivalent. By definition, a *$\mathbb P^1 $-structure* is an equivalence class of such pairs. We refer here to the survey paper by Dumas, see [@dumas] for a comprehensive treatment of this notion.
When the surface $X$ is not compact (hence by assumption it is biholomorphic to a compact Riemann surface punctured at a finite set), we restrict ourselves to the subclass of *parabolic* $\mathbb P^1$-structures. Such a structure has the following well-defined local model around the punctures: each puncture has a neighborhood which is [*projectively*]{} equivalent to the quotient of the upper half plane by the translation $z\mapsto z+1$. For instance, the canonical projective structure $\sigma_{\rm Fuchs}$ induced by uniformization (i.e. viewing $X$ as a quotient of ${\mathbb{H}}$ under a Fuchsian group) is of this type. More generally, the proof of the Ahlfors finiteness theorem (see [@ahlfors Lemma 1]) shows that if $\Gamma$ is any torsion free Kleinian group, and ${\Omega}$ is the orbit of a discontinuity component, then the induced projective structure on ${\Omega}/\Gamma$ is parabolic. These have been known as [*covering projective structures*]{}, because the developing map is a covering onto its image in this case [@kra1; @kra2]. An important example is given by quasi-Fuchsian deformations of the canonical structure $\sigma_{\rm Fuchs}$. There are many other examples of parabolic $\mathbb P^1$-structures. For instance surgery operations such as grafting (see Hejhal’s original construction in [@hejhal; @grafting]) may produce a parabolic $\mathbb P^1$-structure with holonomy a Kleinian group that is not of covering type. Such projective structures are usually called *exotic*. There are yet other examples of $\mathbb P^1$-structures: a remarkable theorem of Gallo, Kapovich and Marden [@gkm] asserts that (when $X$ is compact) a representation $\pi_1(X) \rightarrow \text{PSL}(2,\mathbb C)$ is the holonomy of a $\mathbb P^1$-structure on $X$ (for some Riemann surface structure on $X$) if and only if it is non elementary and it lifts to a representation with values in $\text{SL}(2,\mathbb C)$. In particular, there exist $\mathbb P^1$-structures with holonomy a non discrete (or even dense) subgroup of $\text{PSL} (2,\mathbb C)$.
It is also of interest to deal with the case of [*branched ${{\mathbb{P}^1}}$-structures*]{}, where the local injectivity assumption on the developing map is dropped. Examples include conical metrics of constant curvature equal to $1,0$ or $-1$, with conical points of angle multiple of $2\pi$. It turns out that some of the results in the paper carry over to this setting. The necessary adaptations will be explained in Appendix \[sec:appendix\].
The degree
----------
Let ${\mathbb{H}}$ denote the upper half-plane ${\left\{\tau, \Im(\tau)>0\right\}}$, and $g_P = \frac{2|d\tau|}{\Im \tau}$ be the *Poincaré* (or [*hyperbolic*]{}) [*metric*]{} on ${\mathbb{H}}$, that is the unique complete conformal metric of curvature $-1$, which is invariant under $\mathrm{Aut}(\mathbb H)\simeq \mathrm{PSL}(2, {\mathbb{R}})$. Taking the pull-back of this metric under any biholomorphism between $\widetilde{X}$ and $\mathbb H$ endows $\widetilde{X}$ with a complete conformal metric of curvature $-1$ invariant under $\pi_1(X)$, therefore this metric descends to $X$. It is well known that when $X$ is of finite type, the hyperbolic metric has finite volume.
Recall that a representation $\pi_1(X) \rightarrow \text{PSL} (2, \mathbb C)$ is if it does not preserve any probability measure on the Riemann sphere. The holonomy of a parabolic projective structure is always non elementary: see [@gkm Theorem 11.6.1, p. 695] for the compact case, and [@cdfg Lemma 10] for a proof in the case of the fourth punctured sphere, which readily extends to all punctured surfaces.
If $\sigma$ is a parabolic projective structure, we want to define $\delta(\sigma)$ as a nonnegative number counting the average asymptotic covering degree of $\mathsf{dev}_\sigma : \widetilde X {\rightarrow}{{\mathbb{P}^1}}$. For any $x\in \widetilde{X}$ we denote by $B(x,R)$ the ball centered at $x$ of radius $R$ in the Poincaré metric, and by $\operatorname{vol}$ the hyperbolic volume.
\[defprop:degree\] Let $X$ be a Riemann surface of finite type and $\sigma$ be a parabolic projective structure on $X$. Choose a developing map $\mathsf{dev}: \widetilde{X} \rightarrow \mathbb P^1$. Let $(x_n)$ be a sequence of points in $\widetilde{X}$ whose projections $c (x_n)$ stay in a compact subset of $X$, $(R_n)$ be a sequence of radii tending to infinity, and $(z_n)$ be an arbitrary sequence in ${{\mathbb{P}^1}}$. Then the limit $$\label{eq:limit} \delta = \lim_{n\rightarrow \infty} \frac{ \#B(x_n,R_n)\cap {\mathsf{dev}}^{-1}(z_n)}{\mathrm{vol}(B(x_n,R_n))}$$ exists, and does not depend on the chosen sequences $(x_n)$, $(R_n)$ nor on the developing map $\mathsf{dev}$. The number $\mathrm{deg}(\sigma)= \mathrm{vol} (X) \delta$ is by definition the [*degree*]{} of the projective structure.
The existence of the limit in is not obvious, in particular due to the possibility of boundary effects. The proof ultimately relies on a result of Bonatti and Gómez-Mont [@bgm] and will be carried out in §\[ss:existence of degree\]. Observe that this result is reminiscent from Nevanlinna theory, though the information we obtain is much more precise.We can actually derive the asymptotics of the Nevanlinna theoretic counting function $N(r,{\mathsf{dev}},z)$ and characteristic $T(r, {\mathsf{dev}})$ of the developing map (see [@nevanlinna]) and show that these quantities are governed by the degree. Namely, for every $z\in \mathbb P^1$ an easy computation shows that $$\label{eq:nevanlinna characteristic} N(r,{\mathsf{dev}},z) \underset{r\rightarrow \infty}\sim T(r, {\mathsf{dev}}) \underset{r\rightarrow 1}\sim 2\pi \delta \log\big(\frac{1}{1-r}\big).$$ Besides, Nevanlinna theory is known to have connections with Brownian motion, see [@carne]. In this paper we will explore this relationship from a different point of view.
The reason for introducing the normalized invariant $\mathrm{deg}(\sigma)$ is that $\delta(\sigma)$ is invariant under finite coverings, hence does not behave like a degree. We also show that projective structures with vanishing degree are exactly the covering projective structures (Proposition \[p:vanishing\]).
The Lyapunov exponent {#ss:lyapunov}
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The second invariant, the *Lyapunov exponent* of a parabolic projective structure was defined in our previous work [@Bers1]. It depends only on the holonomy $\mathsf{hol}_\sigma$ of the structure, and also on the induced Riemann surface structure on $X$. Fix a basepoint $\star\in X$, in particular an identification between the covering group $\pi_1(X)$ and the usual fundamental group $\pi_1(X,\star)$. As $X$ is endowed with its Poincaré metric, Brownian motion on $X$ is well-defined. Throughout the paper, [*Brownian motion*]{} will refer to the stochastic process with continuous time whose infinitesimal generator is the hyperbolic laplacian (instead of ${\frac{1}{2}}\Delta$, which is another usual convention). Let $W_\star$ be the Wiener measure on the set of continuous paths $\omega: [0,\infty) \rightarrow X$ starting at $\omega(0)= \star$.
\[def:lyap\] Let $X$ and $\sigma$ be as above. Define a family of loops as follows: for $t>0$, consider a Brownian path ${\omega}$ issued from $\star$, and concatenate ${\omega}{ \arrowvert_{[0, t]}}$ with a shortest geodesic joining ${\omega}(t)$ and $\star$, thus obtaining a closed loop $\widetilde {\omega}_t$. Then for $W_\star$ a.e. Brownian path ${\omega}$ the limit $$\label{eq:deflyap}
\chi (\sigma) = \lim_{t{\rightarrow}\infty} {\frac{1}{t}}\log {\left\Vert\mathsf{hol}{\left(\widetilde {\omega}_t\right)}\right\Vert}$$ exists and does not depend on ${\omega}$. This number is by definition the Lyapunov exponent of $\sigma$.
Here ${\left\Vert\cdot\right\Vert}$ is any matrix norm on ${\mathrm{PSL}(2,\mathbb C)}$. The existence of the limit in was established in [@Bers1 Def-Prop. 2.1]. As expected it is a consequence of the subadditive ergodic theorem. With notation as in [@Bers1], $\chi(\sigma) = \chi_{\rm Brown} (\mathsf{hol})$. Another way to define $\chi (\sigma)$ goes as follows (see [@Bers1 Rmk 3.7]: identify $\pi_1(X)$ with a Fuchsian group $\Gamma$ and independently random elements $\gamma_n \in \Gamma\cap B_{{\mathbb{H}}}(0, R_n)$, relative to the counting measure. Here $(R_n)$ is a sequence tending to infinity as fast as, say $n^\alpha$ for $\alpha>0$. Then almost surely $${\frac{1}{d_{\mathbb{H}}(0, \gamma_n(0)) }}\log {\left\Vert\mathsf{hol}(\gamma_n)\right\Vert} \underset{n{\rightarrow}\infty}\longrightarrow \chi(\sigma).$$
The harmonic measures {#ss:harmonic measure}
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The third object that we associate to a $\mathbb P^1$-structure on $X$ is a family of *harmonic measures* $\{ \nu_x \} _{x\in \widetilde{X}}$ on the Riemann sphere, indexed by $\widetilde X$. It can be defined in several ways. The following appealing presentation was introduced by Hussenot in his PhD thesis [@hussenot]:
\[defprop:harmonic measure\] Let $X$ be a Riemann surface of finite type and $\sigma$ be a parabolic projective structure on $X$. Choose a representing pair $(\mathsf{dev}, \mathsf{hol})$. Then for every $x\in \widetilde X$, and $W_x$ a.e. Brownian path starting at $\omega(0)=x$, there exists a point $\mathrm e(\omega)$ on ${{\mathbb{P}^1}}$ defined by the property that $$\frac{1}{t} \int _0 ^t \mathsf{dev}_*{\left(\delta _{\omega (s)}\right)} ds \underset{t\rightarrow +\infty}\longrightarrow \delta_{\mathrm e(\omega)} .$$ The distribution of the point $\mathrm e(\omega )$ subject to the condition that $\omega(0)= x$ is the measure $\nu_x$.
For covering ${{\mathbb{P}^1}}$-structures, we recognize the classical harmonic measures on the limit set.
Another definition of the harmonic measures is based on the so-called Furstenberg boundary map, which was designed in [@furstenberg], based on the discretization of Brownian motion in ${\mathbb{H}}$ (see also [@margulis Theorem 3] for a different approach). Furstenberg shows that if $\Gamma\subset \text{PSL}(2,\mathbb R)$ is a cofinite Fuchsian group and $\rho:\Gamma{\rightarrow}{\mathrm{PSL}(2,\mathbb C)}$ is a non-elementary representation, there exists a unique measurable equivariant mapping $\theta: \mathbb P^1(\mathbb R){\rightarrow}{{\mathbb{P}^1}}$ defined a.e. with respect to Lebesgue measure (here $\mathbb P^1(\mathbb R)$ is viewed as ${\partial}{\mathbb{H}}$) Choose a biholomorphism $\widetilde X\simeq {\mathbb{H}}$, thereby identifying $\pi_1(X)$ with a cofinite Fuchsian group. For $\tau\in {\mathbb{H}}$, recall that the classical harmonic measure $m_\tau$ is a probability measure with smooth density on $\mathbb P^1(\mathbb R)$, defined as the exit distribution of Brownian paths issued from $\tau$. The harmonic measure $\nu_x$ is then defined by $\nu_x = \theta_*m_\tau$, where $x\in \widetilde{X}$ corresponds to $\tau \in {\mathbb{H}}$, and $\theta$ is associated to ${\mathsf{dev}}$. From this perspective it is clear that, the measures $\nu_x$ are mutually absolutely continuous and depend harmonically on $x$.
The main results
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The main result in this paper is the following formula, relating the Lyapunov exponent and the degree of a ${{\mathbb{P}^1}}$-structure.
\[theo:formula\] Let $\sigma$ be a parabolic holomorphic $\mathbb{P}^1$ structure on a hyperbolic Riemann surface $X$ of finite type. Let as above $\chi(\sigma)$, $\delta(\sigma)$, and $\deg(\sigma)$ respectively denote the Lyapunov exponent, the unnormalized degree and the degree of $\sigma$. Then the following formula holds: $$\label{eq:formula}
\displaystyle \chi (\sigma) = \frac{1}{2} + 2\pi\delta(\sigma) = \frac12+ \frac{\deg(\sigma)}{{\left\vert\mathrm{eu(X)}\right\vert}}.$$
Surprisingly enough, the proof is based on the ergodic theory of holomorphic foliations. Indeed, to any representation $\rho: \pi_1(X){\rightarrow}{\mathrm{PSL}(2,\mathbb C)}$ one classically associates its [*suspension*]{}, a flat $\mathbb P^1$-bundle $M_\sigma \rightarrow X$ whose monodromy is $\rho$. In more concrete terms, it is the quotient of $\widetilde X\times {{\mathbb{P}^1}}$ under the diagonal action of $\pi_1(X)$. The horizontal foliation of $
\widetilde X\times {{\mathbb{P}^1}}$ descends to a holomorphic foliation on $M_\sigma$ transverse to the ${{\mathbb{P}^1}}$ fibers, with monodromy $\rho$. This “dictionary" between ${{\mathbb{P}^1}}$-structures and transverse sections of flat ${{\mathbb{P}^1}}$-bundles, was investigated e.g. in [@loray; @marin].
For $\rho = \mathsf{hol}_\sigma$, we analyze this foliation from the point of view of Garnett’s theory of foliated harmonic measures and currents. This interplay was already explored by Bonatti and Gomez-Mónt [@bgm] and Alvarez [@alvarez]. The key of the proof of the theorem is to interpret $\chi$ and $\delta$ as cohomological quantities on $M_{\sigma}$. The idea that foliated Lyapunov exponents can be computed in cohomology stems from the first author’s thesis (see [@deroin; @levi; @plate Appendice]).
When $X$ is not compact, to prove the result we compactify both $M$ and the foliation. The computations then become much more delicate because the compactified foliation is singular. The details are carried out in Sections \[s:degree\], \[sec:lyapunov\] and \[sec:proof\].
Theorem \[theo:formula\] is mostly interesting for the purpose of studying the space of projective structures on $X$. Let $P(X)$ be the space of parabolic projective structures on $X$ which are compatible with the complex structure. It is well known that $P(X)$ is naturally isomorphic to an affine space of quadratic differentials on $X$ of dimension $3g-3+n$ (see §\[subs:Teichmuller\] below for more details). The Bers simultaneous uniformization theorem implies that the Teichmüller space of marked conformal structures on $X$ embeds as a bounded open subset $B(X)\subset P(X)$ (the [*Bers slice*]{}), whose geometry has been extensively studied. The Sullivan dictionary is a very fruitful set of analogies between the dynamics of rational transformations on ${{\mathbb{P}^1}}$ and the theory of Kleinian (and more generally Möbius) groups. In [@mcm; @renormalization], McMullen draws a fundamental parallel between the Bers slice $B(X)$ and the Mandelbrot set. We take one step further here by relating $P(X)$ and the space of polynomials. In this respect, Theorem \[theo:formula\] should be understood as the analogue of the familiar Manning-Przytycki formula [@manning; @prz] for the Lyapunov exponent of the maximal entropy measure of a polynomial. This analogy should be used as a guide for the forthcoming results.
It was shown in [@Bers1] that $\sigma\mapsto \chi(\sigma)$ is a continuous (Hölder) plurisubharmonic (psh for short) function on $P(X)$, hence it follows from Theorem \[theo:formula\] that $\deg$ is continuous and psh, too. In addition we see that $\chi(\sigma)$ reaches its minimal value $\frac12$ exactly when $\deg(\sigma)=0$. As already observed, $\deg =0$ on $\overline{B(X)}$, so in particular $\chi = \frac12$ there.
A first result which parallels exactly the dynamics of polynomials concerns the Hausdorff dimension of the harmonic measures.
\[theo:dimension\] Let $X$ be a hyperbolic Riemann surface of finite type and $\sigma$ be a parabolic projective structure on $X$. Let as above $\chi$ be its Lyapunov exponent and $(\nu_x)_{x\in \widetilde X}$ be the associated family of harmonic measures. Then for every $x$, $$\dim_H(\nu_x) \leq \frac{1}{2\chi}\leq 1.$$ Furthermore $\dim_H(\nu_x) = 1$ if and only if $\sigma$ belongs to the closure of the Bers slice ${B(X)}$.
Notice that since the measures $\nu_x$ are mutually absolutely continuous, $\dim_H(\nu_x)$ is independent of $x$, so abusing notation, we often simply denote it as $\dim_H(\nu)$. The proof is an adaptation of Ledrappier [@ledrappier Thm 1].
So, as in the polynomial case, Theorem \[theo:dimension\] provides an alternate approach to the classical bound $\dim_H(\nu)\leq 1$ for the harmonic measure on boundary of discontinuity components of finitely generated Kleinian groups, which follows from the famous results of Makarov [@makarov] and Jones-Wolff [@jones; @wolff]. In addition, with this method we are also able to show that $\dim_H(\nu)< 1$ when the component is not simply connected. Indeed we have the more precise bound $\dim_H(\nu)\leq \frac{A}{2\chi}$, where $0\leq A\leq 1$ is an invariant of the flat foliation, and $A< 1$ when ${\mathsf{hol}}$ is not injective. We also see that the value of the dimension of the harmonic measures detects exotic quasifuchsian structures, that is, projective structures with quasifuchsian holonomy which do not belong to the Bers slice.
As a third application of Theorem \[theo:formula\], we recover a result due to Shiga [@shiga].
\[theo:convex\] Let $X$ be a hyperbolic Riemann surface of finite type (of genus $g$ with $n$ punctures). The closure of the Bers embedding $B(X)$ is a polynomially convex compact subset of the space $P(X)\simeq {\mathbb{C}}^{3g-3+n}$ of holomorphic projective structures on $X$. As a consequence, $B(X)$ is a polynomially convex (or Runge) domain.
Recall that a compact set $K$ in ${\mathbb{C}}^N$ is polynomially convex if $\widehat K = K$, where $$\widehat{K} = {\left\{z\in {\mathbb{C}}^N, \ {\left\vertP(z)\right\vert}\leq \sup_K {\left\vertP\right\vert} \text{ for every polynomial }P\right\}}.$$ An open set $U\subset {\mathbb{C}}^N$ is said to be polynomially convex (or Runge) if for every $K\Subset U$, $\widehat K\subset U$. The theorem may be reformulated by saying that $\overline{B(X)}$ is defined by countably many polynomial inequalities of the form ${\left\vertP\right\vert}\leq 1$. This is not an intrinsic property of Teichmüller space, but rather a property of its embedding into the space $P(X)$ of holomorphic projective structures on $X$ (as opposed to the Bers-Ehrenpreis theorem that Teichmüller spaces are holomorphically convex).
Shiga’s proof is based on the Grunsky inequality on univalent functions. Only the polynomial convexity of $B(X)$ was asserted in [@shiga], but the proof covers the case of $\overline{B(X)}$ as well. Our approach is based on the elementary fact that the minimum locus of a global psh function on ${\mathbb{C}}^N$ is polynomially convex.
In [@Bers1] we showed that ${T_{\mathrm{bif}}}:= dd^c\chi$ is a [*bifurcation current*]{}, in the sense that its support is precisely the set of projective structures whose holonomy representation is not locally structurally stable in $P(X)$. Equivalently, the complement $\operatorname{Supp}({T_{\mathrm{bif}}})^c$ is the interior of the set of projective structures with discrete holonomy $P_{D}(X)$. A theorem due to Shiga and Tanigawa [@shiga; @tanigawa] and Matsuzaki [@matsuzaki] asserts that $\mathrm{Int} (P_D(X)) = P_{QF}(X)$, the set of projective structures with quasifuchsian holonomy, so we conclude that $\operatorname{Supp}({T_{\mathrm{bif}}}) = (P_{QF}(X))^c$.
Analogous bifurcation currents have been defined for families of rational mappings on ${{\mathbb{P}^1}}$. It turns out that the exterior powers ${T_{\mathrm{bif}}}^k$ are interesting and rather well understood objects in that context (see [@survey] for an account). In particular, in the space of polynomials of degree $d$, the maximal exterior power ${T_{\mathrm{bif}}}^{d-1}$ is a positive measure supported on the boundary of the connectedness locus, which is the right analogue in higher degree of the harmonic measure of the Mandelbrot set [@preper].
For bifurcation currents associated to spaces of representations, nothing is known in general about the exterior powers ${T_{\mathrm{bif}}}^k$. In our situation, we are able to obtain some information.
\[theo:tbif\] Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $T_{\mathrm{bif}} = dd^c\chi$ be the natural bifurcation current on $P(X)$. Then ${\partial}B(X)$ is contained in $\operatorname{Supp}({T_{\mathrm{bif}}}^{3g-3})$.
Notice that $3g-3$ is the maximum possible exponent. It is likely that the support of ${T_{\mathrm{bif}}}^{3g-3}$ is much larger than ${\partial}B(X)$. The reason for the compactness assumption here is that the proof is based on results of Otal [@otal] and Hejhal [@hejhal; @schottky] that are known to hold only when $X$ is compact.
If $\gamma$ is a geodesic on $X$, we let $Z(\gamma)$ be the subvariety of $P(X)$ defined by the property that $\operatorname{tr}^2(\mathsf{hol}(\gamma)) = 4$ (i.e. $\mathsf{hol}(\gamma)$ is parabolic or the identity). As a consequence of Theorem \[theo:tbif\] and of the equidistribution theorems of [@Bers1] we obtain the following result, which contrasts with the description of ${\partial}B(X)$ “from the inside" in terms of maximal cusps and ending laminations ([@minsky; @bcm], see also [@lecuire] for a nice account).
\[coro:deterministic\] For every ${\varepsilon}>0$ there exist $3g-3$ closed geodesics $\gamma_1, \ldots , \gamma_{3g-3}$ on $X$ such that ${\partial}B(X)$ is contained in the ${\varepsilon}$-neighborhood of $Z(\gamma_1)\cap \cdots \cap Z(\gamma_{3g-3})$.
We observe that the value 4 for the squared trace is irrelevant here. As the proof will show, the result holds a.s. when $\gamma_1, \cdots, \gamma_k$ are independent random closed geodesics of length tending to infinity.
Notation
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${{\mathbb{P}^1}}= \mathbb P^1(\mathbb C)$ is the Riemann sphere. $z$ often denotes the variable in $\mathbb P^1$
$\mathbb H$ the upper half plane. $\tau$ a variable in $\mathbb H$
$X$ the finite type Riemann surface on which is defined a projective structure
$\sigma$ a parabolic projective structure on $X$.
$\mathsf{dev} = \mathsf{dev}_\sigma: \widetilde{X} \rightarrow \mathbb P^1$ a developing map ${\mathsf{hol}}= \mathsf{hol}_\sigma: \pi_1(X) \rightarrow \text{PSL} (2,\mathbb C)$ the holonomy representation
$\pi : M_\sigma \rightarrow X$ the flat $\mathbb P^1$-bundle over $X$ $\mathbb P^1 _x = \pi^{-1}(x)$ the fiber over $x\in X$.
$\varpi : \widetilde{X} \times \mathbb P^1 \rightarrow M_\sigma$ the quotient map.
$s : X\rightarrow M_\sigma$ a holomorphic section of $\pi$
$\mathcal F $ the holomorphic foliation defined by the flat connexion
$T$ the harmonic current on $M_\sigma$
$\overline M_\sigma$, $\overline{\mathcal F}$, $\overline{s}$, $\overline{T}$, etc. are the compactifications of the corresponding objects when $X$ is not compact.
Harmonic measures and harmonic currents
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In this section we introduce a number of geometric objects which that will be fundamental in our study: the suspension $M_\sigma$ (as well as its compactification $\overline M_\sigma$), and the foliation $\mathcal{F}$ (resp. $\overline{\mathcal{F}}$). We also study the ergodic theoretic properties of $\mathcal{F}$, by way of three closely related, though slightly different tools: a family of harmonic measures on the fibers of $M_\sigma$, a foliated harmonic current, and its associated foliated harmonic measure.
Generalities {#ss:generalities}
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We fix once for all a Riemann surface $X$ of finite type, that is, $X$ is biholomorphic to a compact Riemann surface $\overline{X}$ with finitely many points deleted. We assume that $X$ has negative Euler characteristic. If necessary, we endow $X$ with its hyperbolic metric, which is of finite volume.
A $\mathbb P^1$-bundle over a Riemann surface $X$ is a holomorphic fibration $M\rightarrow X$ with $\mathbb P^1$ fibers. It is always the projectivization of a rank $2$ holomorphic vector bundle over $X$. If the Riemann surface is compact, the compact complex surface $M$, being a $\mathbb P^1$-bundle over a curve, is algebraic (by the GAGA principle), thus in particular it is Kähler. We refer to [@bpvdv V.4].
A holomorphic $\mathbb P^1$-bundle always admits a smooth section $X\rightarrow M$ (and even a holomorphic one, see [@bpvdv V.4, p. 139]). When $X$ is compact, the parity of the self-intersection of such a smooth section depends only on the fibration; it is even iff $X$ is diffeomorphic to the trivial bundle $X\times \mathbb P^1$, and odd otherwise.
Let $s$ and $f$ respectively denote a smooth section and a fiber of $M\rightarrow X$, then we have that $$\label{eq:homology fiber bundle} H^2 (M, \mathbb C ) := \mathbb C [s] \oplus \mathbb C [f] ,$$ where $[s]$ and $[f]$ are the cohomology classes dual to $s$ and $f$ respectively. (Throughout this paper, we consistently identify the section $s$ and its graph in $M$).
In particular, since we can always choose $s$ to be holomorphic, and since $M$ is Kähler, we obtain an isomorphism between the Dolbeaut cohomology group $H^{1,1}_{\overline{\partial}} (M,\mathbb C)$, the Bott-Chern cohomology group $H^{1,1}_{{\partial}\overline{\partial}}(M,\mathbb C)$, and $H^2 (M,\mathbb C)$.
Parabolic flat $\mathbb P^1$-bundles {#ss:model}
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Given a parabolic projective structure $\sigma$ on $X$, we introduce the flat $\mathbb P^1$-bundle $\mathbb P^1 \rightarrow M_{\sigma}\stackrel{\pi}{\rightarrow} X$ with monodromy $\mathsf{hol}_\sigma$, namely the quotient of the flat bundle $\widetilde{X} \times \mathbb P^1$ under the action of $\pi_1(X)$ given by $$\label{eq:action of Gamma} \gamma (x,z) = (\gamma x, \mathsf{hol}_\sigma (\gamma)z).$$ We denote by $\varpi : \widetilde{X} \times \mathbb P^1 \rightarrow M_{\sigma}$ the natural projection. Also, we let $\mathcal F$ be the holomorphic foliation on $M_\sigma$ obtained by taking the quotient of the horizontal fibration $\widetilde{X}\times {\left\{z\right\}}$ of $\widetilde{X} \times \mathbb P^1$.
When $X$ is not compact, we can compactify the bundle $M_\sigma$ as a bundle over $\overline{X}$. The flat connexion $\nabla$ extends as a meromorphic connexion $\overline{\nabla}$ on $\overline M_\sigma $, and the foliation $\mathcal F$ extends as a singular holomorphic foliation $\overline{\mathcal F}$.
Here are the details. Consider the following model for a $\mathbb P^1$-bundle over the unit disk equipped with a meromorphic flat connexion having a pole over $0$, defined by the differential equation $$\label{eq:model} \frac{ dv}{du} = \frac{i}{2\pi u}$$ in coordinates $(u,v)\in \mathbb D\times \mathbb C$. We denote the induced foliation on $\mathbb D \times \mathbb P^1$ by $\mathcal F_m$. The monodromy of the connexion around $u=0$ is the parabolic map $v\mapsto v+1$. Since by assumption the holonomy representation is parabolic, we can glue this local model to each of the cusps of $ M_\sigma$ to obtain the desired $\mathbb P^1$-bundle $\overline{M}_{\sigma}$ over $\overline{X}$ equipped with a meromorphic flat connexion $\overline{\nabla}$ and singular holomorphic foliation $\overline{\mathcal F}$.
Parabolic $\mathbb P^1$-structures and holomorphic sections of flat ${{\mathbb{P}^1}}$-bundles
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Let now $\mathsf{dev}: \widetilde{X} \rightarrow \mathbb P^1$ be a developing map of the parabolic $\mathbb P^1$-structure $\sigma$. The map $\widetilde X\ni x\mapsto (x, \mathsf{dev}(x))\in \widetilde X\times {{\mathbb{P}^1}}$ is $\pi_1(X)$-equivariant, hence it descends to a section $s : X\rightarrow M_\sigma$ of the bundle $M_\sigma \rightarrow X$. This section will play an important role in what follows.
\[l:section\] The section $s$ extends to a section $\overline{s} : \overline{X} \rightarrow \overline M_\sigma$ which is transverse to the foliation $\overline{\mathcal F}$.
To make the compactification more explicit, consider a neighborhood $N$ of a puncture equipped with a coordinate $x$ in which the projective structure is defined in the punctured unit disk by its developing map $x\mapsto \log x$ (see §\[ss:parabolic type PS\]). Write $x= \exp (2i\pi \tau)$ with $\tau\in {\mathbb{H}}$. The $\mathbb P^1$-bundle $M_{\sigma }$ over the punctured disk $ N \simeq \mathbb D ^*$ is the quotient of $\mathbb H \times \mathbb P^1$ by the cyclic group generated by $(\tau, z)\mapsto (\tau + 1, z + 1)$. The map $$\label{eq:identification} (\tau, z) \longmapsto (u= \exp(2i\pi \tau) , v= z-\tau )$$ is invariant under this transformation, and it maps the horizontal foliation to the foliation defined by $dz= dv + d\tau = \big(dv + \frac{du}{2i\pi u}\big) = 0$. Hence provides the identification between the bundle $ M_\sigma$ over $ N \simeq \mathbb D^*$ and the model .
The section $s$ of $ M_\sigma$ is defined in the coordinates $(\tau, z)$ to be the diagonal $z \mapsto (z,z)$, so in the coordinates $(u,v)$ it is given by $u \mapsto (u,0)$. Hence the section $s$ extends as a section $\overline{s}$ of $\overline{ M_\sigma}$.
Fiberwise harmonic measures {#ss:fiberwise}
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Fix an biholomorphism between $\widetilde{X}$ and $\mathbb H$, thereby identifying $\pi_1(X)$ with a lattice $\Gamma$ in ${\mathrm{PSL}(2,\mathbb R)}$. Recall from [@furstenberg; @margulis] that if $\rho: \Gamma{\rightarrow}{\mathrm{PSL}(2,\mathbb C)}$ is a non elementary representation, there exists a unique (Lebesgue) measurable $\rho$-equivariant map $\Phi : \mathbb P^1(\mathbb R) \rightarrow \mathbb P^1$ defined almost everywhere. Likewise, if $\text{Prob}(\mathbb P^1)$ denotes the compact convex set of probability measures on $\mathbb P^1$ (endowed with the weak\* topology), then the map $a \in \mathbb P^1(\mathbb R) \mapsto
\delta_{\Phi(a)} \in \text{Prob} (\mathbb P^1)$ is the unique measurable $\rho$-equivariant map.
A measurable family of probability measures $(m_{\tau})_{\tau\in {\mathbb{H}}}$ on ${{\mathbb{P}^1}}$ is said to be [*harmonic*]{} if for every test function $\psi$, the function ${\mathbb{H}}\ni \tau\mapsto \int \psi m_\tau $ is harmonic.
\[p:harmonic measures\] Let $\Gamma$ be a lattice in ${\mathrm{PSL}(2,\mathbb R)}$ and $\rho: \Gamma{\rightarrow}{\mathrm{PSL}(2,\mathbb C)}$ be a non elementary representation. Then there exists a unique measurable family of probability measures $\{ \nu_\tau \} _{\tau\in \mathbb H} $ on ${{\mathbb{P}^1}}$ such that :
1. $\tau \mapsto \nu_\tau$ is harmonic;
2. $\nu_{\gamma \tau } = \rho(\gamma)_* \nu_\tau$ for every $\gamma \in \pi_1(X)$ and every $\tau\in \mathbb H$.
In particular, we have the formula $$\label{eq:harmonic current}
\text{ for every } \tau \in \mathbb H,\ \nu_\tau = \frac{1}{\pi} \Phi _* \big( \frac{\Im \tau \ d a}{ |\tau - a |^2} \big),$$ and these measures coincide with those defined in Definition \[defprop:harmonic measure\].
The family ${\left\{\nu_\tau, \tau\in {\mathbb{H}}\right\}}$ will be simply referred to as the [*family of harmonic measures*]{} associated to the ${{\mathbb{P}^1}}$-structure.
It will be convenient to view $\mathbb P^1({\mathbb{R}})$ as the boundary of the upper half plane. First, it follows from the Poisson formula and the equivariance of $\Phi$ that defines a family of harmonic measures $\nu_\tau$ satisfying (i) and (ii).
To establish the uniqueness statement, fix a family of probability measures $\{\nu'_\tau\}_{\tau \in {\mathbb{H}}}$ on $\mathbb P^1$ satisfying (i) and (ii). Then, Fatou’s theorem on boundary values on bounded harmonic functions implies that there is a measurable map $\widehat \nu' : \partial \mathbb H \rightarrow \text{Prob} (\mathbb P^1)$ with values in the set of probability measures on $\mathbb P^1$, such that the Poisson formula holds, namely for every $\tau \in \mathbb H$ $$\nu'_\tau = \frac{1}{\pi}\int_{\mathbb R} \frac{\Im \tau }{ |\tau - a |^2} \widehat \nu' (a) da .$$ The map $\widehat \nu'$ is $\rho$-equivariant since the family $\{\nu'_\tau\}_{\tau \in \mathbb H}$ is. Hence by the observations preceding the proposition, we get that a.s. $\widehat{\nu'}(a)$ is the Dirac mass at $\Phi (a)$, and we are done. The fact that these measures coincide with the ones from Definition-Proposition \[defprop:harmonic measure\] follows from this uniqueness. Indeed, the family of measures defined by Definition \[defprop:harmonic measure\] clearly satisfies the equivariance property (ii). To check (i) we adapt the classical argument for the harmonic dependence of the harmonic measures with respect to the starting point. Indeed let $\mathrm e (\omega)$ be the endpoint mapping defined in Definition \[defprop:harmonic measure\]. Let $B\subset{{\mathbb{P}^1}}$ be any Borel set. Let us prove that $u:x\mapsto {\mathbb{P}}_x(e(\omega)\in B)$ is harmonic. For this, identify $ \widetilde X$ with the unit disk and $D$ be a small disk centered at $x\in \widetilde X$. For $\omega\in \Omega_x$, let $T = \inf{\left\{t>0, \omega(t)\notin D\right\}} $. It follows from the strong Markov property of Brownian motion in $\widetilde X$, that $$u(x) = {\mathbb{E}}_x{\left({\mathbb{P}}_{{\omega(T)}}(\mathrm e(\omega(\cdot - T)) \in B\right)} = {\mathbb{E}}_x\big(u(\omega(T))\big) = \int_{{\partial}D} u.$$ Therefore $u$ satisfies the mean value property and the result follows.
\[r:measurable isomorphism\] The map $\Phi$ is in general not injective on any full measure subset of $\mathbb P^1(\mathbb R)$. However a theorem of Ledrappier shows that this is the case when $\rho$ is faithful and discrete, see [@ledrappier; @poisson; @boundary]. This result will be used in section \[s:dimension\].
Harmonic currents {#ss:harmonic currents}
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Given a foliated complex surface $(M,\mathcal F)$ (possibly with singularities), a [*directed*]{} (or [*foliated*]{}) [*harmonic current*]{} (often simply abbreviated as “harmonic current" in the sequel) is a positive current of bidegree $(1,1)$ which is $\partial \overline{\partial}$-closed, and such that $\langle T, \psi \rangle\geq 0$ if $\psi$ is a (1,1) form which is positive along the leaves.
Such currents have the following local structure outside the singular set $\text{sing} (\mathcal F)$: in a foliation box biholomorphic to the bidisk $\mathbb D \times \mathbb D$ in which the foliation is the horizontal fibration, there exists a finite positive measure $m$ on $\mathbb D$, and a non-negative bounded measurable function such that $$\label{eq:foliated}
T = \int \varphi \ [\mathbb D \times {\left\{w\right\}}] \ dm(w) ,$$ and moreover $\varphi (.,w)$ is harmonic for $m$-a.e. $w\in \mathbb D$. Here as usual $[\mathbb D \times {\left\{w\right\}}]$ stands for the current of integration on $\mathbb D \times {\left\{w\right\}} $. The product $\varphi m$ is a well-defined object, which can be thought of as a transverse measure for the foliation $\mathcal F$. In particular, if $C\subset M$ is a holomorphic curve disjoint from $\text{sing}(\mathcal{F})$, we can define the restriction $T{ \arrowvert_{C}}$ of $T$ to $C$, also referred to as the geometric intersection $T{\dot \wedge}[C]$ between $T$ and $C$). Observe that the same makes sense for any current of the form , whenever harmonic or not.
The existence of a harmonic current directed by the foliation is classical when $M$ is compact and $\mathcal{F}$ is non-singular (see e.g. [@Ghys]); the singular case was treated in [@BS].
Assume now that $\mathcal{F}$ is the foliation by flat sections of $M_\sigma$. There is a 1-1 correspondence between foliated harmonic currents and the fiberwise harmonic measures of §\[ss:fiberwise\]. Indeed, consider a foliated harmonic current $T$ on $M_\sigma$, normalized so that one (hence all) of its vertical slices is of unit mass. Lifting $T$ to the universal cover ${\mathbb{H}}\times {{\mathbb{P}^1}}$, we obtain a harmonic current $\widetilde{T}$ directed by the horizontal fibration, that is invariant with respect to the action of $\Gamma\simeq \pi_1(X)$ defined in . Restricting to the vertical fibers ${\left\{\tau\right\}}\times {{\mathbb{P}^1}}$ we get a family of measures $\nu_\tau$ which is easily seen to satisfy the assumptions of Proposition \[p:harmonic measures\].
Conversely, any family of measures $(\nu_\tau)_{\tau\in {\mathbb{H}}}$ on ${\left\{\tau\right\}}\times {{\mathbb{P}^1}}$ satisfying the assumptions of Proposition \[p:harmonic measures\] gives rise to a foliated harmonic current on $M_\sigma$. For this, working first on ${\mathbb{H}}\times {{\mathbb{P}^1}}$, we construct from (i) a harmonic current $\widetilde T$ directed by the horizontal fibration. Indeed, the Poisson formula asserts that $\nu_\tau$ is a convex combination of measures of the form $h(\tau, a)\delta_{\Phi(a)}$, where $\tau\mapsto h(\tau, a)$ is harmonic. Then we get $\widetilde T$ by taking the corresponding combination of currents of the form $h(\tau, a)[{\mathbb{H}}\times {\left\{\Phi(a)\right\}}]$. From the equivariance property (ii), $\widetilde{T}$ descends to a foliated harmonic current on $M_\sigma$ and we are done.
The following uniqueness statement will be of utmost importance to us. When $X$ is compact it was already established in [@dk].
\[p:extension\] Let $X$, $\sigma$ and $M_\sigma$ be as above. The singular foliation $\overline{\mathcal{F}}$ on the compactified suspension $\overline{M}_\sigma$ admits a unique normalized foliated harmonic current, carrying no mass on the fibers over the punctures.
In $M_\sigma$, the existence and uniqueness of a foliated harmonic current $T$ giving mass 1 to the vertical fibers follows from the above discussion, together with Proposition \[p:harmonic measures\]. Thus, the point is to show that $T$ admits an extension to a harmonic current $\overline{T}$ on $\overline{M}_\sigma$ with no mass on the fibers over the punctures, which is then necessarily unique.
Recall that the foliation $\mathcal{F}$ has a well defined rigid model $\mathcal{F}_m$ in a neighborhood of each puncture, which was defined in §\[ss:model\]. The key is the following lemma.
\[l:extension model\] Consider the model foliation $ \mathcal F_m$ on $\mathbb D\times \mathbb P^1$, as defined in §\[ss:model\]. Let $T$ be any foliated harmonic current in $\mathbb D^* \times \mathbb P^1$, normalized so that the restriction of $T$ to any fiber $u\times \mathbb P^1$, $u\neq 0$ is a probability measure. Then the restriction of $T$ to the curve $s^* = \{ 0< |u|\leq e^{-2\pi},\ v= 0\}$ has finite mass.
From this and Lemma \[l:section\] (see also Figure \[fig:foliation\]), we deduce that the harmonic current extends to $\mathbb D \times \mathbb P^1 \setminus \text{sing} (\mathcal F_m)$. It then follows from general extension results for harmonic currents (see e.g. [@dee Thm 5]) that it also compactifies at the singular points of $\mathcal F_m$. The proposition follows.
The harmonic current $T$ lifts as a harmonic current $\widetilde{T}$ on $\mathbb H\times \mathbb P^1$ which is defined in the $(\tau,z)$-coordinates by a family of measures $\{ \nu_\tau \}_{\tau \in \mathbb H}$ satisfying $$\label{eq:equivariance} \nu_{\tau +1} = (z+1)_* \nu_\tau ,$$ and depending harmonically on $\tau$. As in the proof of Proposition \[p:harmonic measures\], the Poisson formula implies that there exists a family of probability measures $\{ \nu_a \}_{a\in \mathbb R}$ defined for a.e. $a\in \mathbb R$ and depending measurably on $a$, such that for every $\tau\in \mathbb H$ $$\nu_\tau = \int _{\mathbb R} \frac{\Im \tau }{(\Re \tau - a)^2 + \Im \tau ^2} \nu_a da.$$ The equivariance relation implies that $$\nu_{a+1} = (z+1)_* \nu_a$$ almost everywhere. A canonical example of such a family of measures is given by $\nu^{\rm can} _a = \delta_a$ the Dirac mass at the point $a$. It defines a harmonic current $T^{can}$ (corresponding to the harmonic current on the suspension corresponding to the identity representation).
A fundamental domain for the pull-back of $s^*$ in $\mathbb H \times \mathbb P ^1$ is the subset $D\times D$ of the diagonal in $ {\mathbb{H}}\times {{\mathbb{P}^1}}$, where $$D = {\left\{ \frac{-1}{2}\leq \Re \tau \leq \frac{1}{2} , \ \Im \tau \geq 1 \right\}}\subset \mathbb H.$$ Therefore, we need to prove that the integral $$I = \int _{\mathbb R} da \cdot \int _D \frac{\Im \tau }{(\Re \tau - a) ^2 + (\Im \tau )^2} \nu_a(d\tau)$$ is finite. Performing the change of variable $a= b+n$ yields $$I =\int _0 ^1 db \cdot \sum _{n\in \mathbb Z} \int _D \frac{\Im \tau }{(\Re \tau - (b+n)) ^2 + (\Im \tau )^2} \nu_{b+n}(d\tau) .$$ The equivariance relation $(z+n)_* \nu_b = \nu_{b+n}$ gives $$\int _D \frac{\Im \tau }{(\Re \tau - (b+n)) ^2 + (\Im \tau )^2} (z+n)_* \nu_b(d\tau) = \int _{D-n} \frac{\Im \tau }{(\Re \tau - a) ^2 + (\Im \tau )^2} \nu_a(d\tau)$$ where $D-n = \{ \tau -n \ |\ \tau \in D\}$, and we conclude that $$I = \int_0 ^1 da \cdot \int_{\Im \tau \geq 1} \frac{\Im \tau }{(\Re \tau - a) ^2 + (\Im \tau )^2} \nu_a(d\tau) \leq 1$$ since $\nu_a$ is a probability measure on $\mathbb P^1$ and $\ \frac{\Im \tau }{(\Re \tau - a) ^2 + (\Im \tau )^2} \leq 1$ if $\Im \tau \geq 1$. The proof is complete.
\[r:measurable conjugacy\] Identify $\widetilde{X}$ with $\mathbb H$ via a biholomorphism. For any $\mathbb P^1$-structure $\sigma$, the map $\Phi$ can be used to construct a measurable map $M_{\sigma_{ \rm Fuchs}} \rightarrow M_\sigma$ mapping biholomorphically every leaf of $\mathcal F_{\sigma_{ \rm Fuchs}}$ to a leaf of $\mathcal F_\sigma$. At the level of the universal covers, this map is simply given by $(\tau, z) \mapsto (\tau , \Phi(z))$. Observe furthermore, that the normalized current $T_{\sigma_{ \rm Fuchs}}$ is mapped to $T_\sigma$ (indeed, this holds for the fiber harmonic measures). If in addition the holonomy is faithful with discrete image, Remark \[r:measurable isomorphism\] shows that the foliations $(M_{\sigma_{ \rm Fuchs}}, \mathcal F_{\sigma_{ \rm Fuchs}}, T_{\sigma_{ \rm Fuchs}})$ and $(M_\sigma, \mathcal F_\sigma, T_\sigma)$ are actually measurably conjugated.
Foliated harmonic measures: Garnett’s theory {#ss:garnett}
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In this paragraph we briefly review Garnett’s theory of foliated Brownian motion [@garnett] (see also [@candel]), and adapt it to our non compact situation. Let us define the normalized measure $$\label{eq:harmonic measure} \mu = \frac{1}{\text{vol} (X) } \text{vol}_P \wedge T .$$ This measure is a *harmonic measure in the sense of Garnett*, namely it satisfies the equation $\Delta_{\mathcal F} \mu = 0$ in the weak sense, here $\Delta_{\mathcal F}$ is the leafwise laplacian relative to the leafwise Poincaré metric. (This is immediate from the fact that $T$ itself is harmonic.) We will refer to such measure as *foliated harmonic measures*.
Let $\Pi = \{ \Pi_t \}_{t\geq 0}$ be the Markov semi-group of operators acting on $C^0 _c (M_\sigma)$, whose infinitesimal generator is $\Delta_{\mathcal F}$. It is convenient to consider it at the level of the universal cover $\widetilde{X} \times \mathbb P^1$. There it expresses as $$\label{eq:markov operator} \Pi_t f(x,z) = \int _{\widetilde{X}} p(x,y,t) f(y,z) \text{vol}(dy)$$ where $p(x,y,t)$ is the fundamental solution of the heat equation $\frac{\partial}{\partial t} = \Delta_{\rm Poin}$ on the hyperbolic plane. Then, since $\mu$ satisfies $\Delta_{\mathcal F} \mu = 0$, it is invariant under the semi-group $\Pi$.
The following is essentially a reformulation of Proposition \[p:extension\]. The proof will be left to the reader.
\[p:ergodic\] The measure $\mu$ is the only normalized foliated harmonic measure in the sense of Garnett for $\mathcal{F}$ on $M_\sigma$. In particular any measurable subset of $M_\sigma$ which is saturated by $\mathcal F$ has zero or full $\mu$-measure.
Consider the Markov process on $ M_\sigma$ induced by the leafwise Brownian motion, with respect to the Poincaré metric (recall that the Brownian motion is generated by the operator $ \Delta$). More precisely, we let $\Omega^\mathcal{F}$ be the set of semi-infinite continuous paths $\omega : [0,\infty ) \rightarrow M_\sigma$ which are contained in a leaf of $\mathcal F$, and $\sigma = \{ \sigma_t \}_{t\in [0,+\infty)}$ be the shift semi-group acting on $\Omega$ by $\sigma_t (\omega) (s) = \omega (t+s)$. Let $$W^{\mathcal F}_\mu := \int W^{\mathcal F}_x \ d\mu (x)$$ on $\Omega^\mathcal{F}$, where $W^{\mathcal F}_{x}$ is the Wiener measure on the subset $\Omega^{\mathcal F}_x$ of paths starting at $x$. We also sometimes use the Wiener space $(\Omega^X, W^X)$ of Brownian paths on $X$.
The following proposition is contained in [@candel §6]. For the sake of convenience we sketch the proof.
The measure $ W^{\mathcal F}_\mu$ is $\sigma$-invariant and the dynamical system $(\Omega^\mathcal{F}, \sigma, W^{\mathcal F}_\mu)$ is ergodic.
Let us first show that $W^{\mathcal F}_{\mu}$ is $\sigma$-invariant. Let $E \subset \Omega$ be a measurable subset. By the Markov property, for every $x\in M_\sigma$ and every $t\geq 0$ we have that $$\label{eq:invariance} W^{\mathcal F}_x (\sigma_{t}^{-1} E) = \int_{L_x} p(x,y,t) W^{\mathcal F}_y ( E) d y .$$ Consider the function $f:(t, x) \mapsto W^{\mathcal F}_x (\sigma_{t}^{-1} E)$. Equation shows that $f$ satisfies the heat equation, with initial condition $f(0, x) = W^{\mathcal F}_x (E)$, hence for every $t\geq 0$, $f(t,.) = \Pi_t f(0,.)$. Since $\mu$ is invariant under the heat semi-group, we deduce that $$W^{\mathcal F}_\mu (\sigma^{-t} (E))= \int _{X_\rho} f(t,x) d\mu(x) = \int _{X_\rho} f(0,x) d\mu(x) = W^{\mathcal F}_\mu (E),$$ hence proving the first part of the proposition.
We now prove that $(\Omega^\mathcal{F}, \sigma, W^{\mathcal F}_\mu)$ is ergodic. Let $E$ be any $\sigma$-invariant subset. The function $x\mapsto f(0,x) = W^{\mathcal F}_x(E)$ is then measurable, bounded, and harmonic along $\mu$-a.e. leaf. We claim that it is constant. Indeed, observe that for any $c\in \mathbb Q$, the function $g= \max (f,c)$ is leafwise subharmonic on a.e. leaf, so we get that for every $t\geq 0$, $\Pi_t g \geq g$ on a.e. leaf. On the other hand, $\int \Pi_t g \; d\mu = \int g \; d\mu$, so we infer that on a set of full measure $\Pi_t g = g$ holds for every rational $t\geq 0$. This proves that $g$ is harmonic on $\mu$-a.e. leaf. This being true for every $c$, it follows that $f$ is constant along a.e. leaf. Now, since $E$ is shift invariant, belonging to $E$ is a a tail property, so by applying the 0-1 law [@candel Prop. 6.5] we infer that $E$ has zero or full measure on a.e. leaf. Applying Proposition \[p:ergodic\] then concludes the proof.
The degree {#s:degree}
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In this section we introduce the concept of the degree of a ${{\mathbb{P}^1}}$-structure on $X$. We justify its existence in §\[ss:existence of degree\] by proving Proposition \[defprop:degree\]. Then in §\[ss:vanishing\], we characterize projective structures with vanishing degree, and in §\[ss:cohomological degree\] we show that it can be expressed in terms of cohomological data.
Existence of the degree and equidistribution of large leafwise discs {#ss:existence of degree}
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This subsection is devoted to the proof of Definition-Proposition \[defprop:degree\]. Recall that we are given a developing map $\mathsf{dev}:\widetilde X{\rightarrow}{{\mathbb{P}^1}}$ of a parabolic $\mathbb P^1$-structure with non-elementary holonomy, and wish to show that ${\frac{1}{\mathrm{vol} (B(x_n, R_n))}} \#{\left\{B(x_n, R_n)\cap \mathsf{dev}^{-1}(z_n)\right\}}$ converges to some limit $\delta$, independent of the choices. To ease notation we set $\mathrm{vol}(R_n) = \mathrm{vol} (B(x_n, R_n))$. Using the equivariance we may assume without loss of generality that $(x_n)$ is relatively compact in $\widetilde X$. Recall that the graph of the developing map in $\widetilde X\times {{\mathbb{P}^1}}$ descends to a section $s$ of $M_\sigma$ transverse to $\mathcal{F}$. Recast in geometric language, we need to show that ${\frac{1}{\mathrm{vol}(R_n)}}\#(B(x_n, R_n))\times{\left\{z_n\right\}} ) \cap \mathrm{Graph}(\mathsf{dev})$ converges to some value $\delta$. Pushing forward by $\varpi$, this amounts to proving that the geometric intersection number $$\int_{M_\sigma} \varpi_*{\left(\frac{1}{\mathrm{vol}(R_n)} [ B(x_n,R_n)\times{\left\{z_n\right\}} ]\right)}{\dot \wedge}[s]$$ converges to $\delta$. Put $\Delta(R_n) =\varpi_*{\left( \frac{1}{\mathrm{vol}(R_n)} [ B(x_n,R_n)\times{\left\{z_n\right\}} ]\right)}$, which is a current with boundary supported in a leaf of $\mathcal F$. It is perhaps useful to stress here that $\Delta(R_n)$, may be decomposed into pieces of varying multiplicities (according to the self- overlapping properties of $\varpi(B(x_n, R_n))$), and that these multiplicities are taken into account in the geometric wedge product ${\dot \wedge}$.
The key is the following equidistribution result for large leafwise discs in parabolic flat $\mathbb P^1$-bundles, which is due to Bonatti and Gómez-Mont [@bgm], given the positivity of the foliated Lyapunov exponent, a fact that was established in this generality in our previous work [@Bers1].
\[p:bgm\] Let $\rho : \pi_1 (X) \rightarrow \text{PSL} (2,\mathbb C)$ be a non elementary representation. Let $(x_n)_{n\geq 0}$ be a sequence in $\widetilde X$ such that $(c(x_n))_{n{\rightarrow}\infty}$ is relatively compact in $X$. Let $(R_n)$ be a sequence of positive real numbers tending to $+\infty$, and $(z_n)$ be any sequence of points on the Riemann sphere. Then the projection in $M_\sigma$ of the sequence of integration currents $\Delta(R_n) = \varpi_*{\left( \frac{1}{\mathrm{vol}(R_n)} \left[ B(x_n,R_n) \times{\left\{ z_n\right\}}\right]\right)}$ converges to ${\frac{1}{\operatorname{vol}(X)}}T$ when $n$ tends to infinity.
Let $\text{vol}$ denote the Poincaré volume form along the leaves of $\mathcal F$. Remark that since all currents are directed by the foliation, the convergence $ { \Delta(R_n)}\underset{n{\rightarrow}\infty}{\longrightarrow} \frac{1}{\mathrm{vol} (X)} T $ is equivalent to that of $\varpi_*{\left(\frac{1}{\mathrm{vol}(R_n)} \text{vol} { \arrowvert_{B(x_n,R_n)\times{\left\{z_n\right\}} }}\right)} $ towards the measure $\mu :=\frac{1}{\mathrm{vol} (X)} T\wedge \text{vol}$.
By [@bgm Thm 2], for this it is enough to show that the top Lyapunov exponent of the cocycle induced by $\rho$ over the geodesic flow on $T^1X$ is positive. The representation $\rho$ being non elementary, this positivity was shown in [@Bers1 Rmk 2.19]. The result follows.
We see that to prove the desired result, it is enough to show that $$\label{eq:counting}
\int_{M_\sigma} {\Delta(R_n)}{\dot \wedge}[s] \underset{n{\rightarrow}\infty}{\longrightarrow} \frac{1}{\mathrm{vol} (X)}
\int_{M_\sigma} T{\dot \wedge}[s].$$ We note that it follows from the previous proposition that if $\alpha$ is any smooth form along the leaves of $\mathcal{F}$, ${\left\langle \Delta(R_n), \alpha\right\rangle}$ converges to ${\left\langle {\frac{1}{\operatorname{vol}(X)}} T, \alpha\right\rangle}$. The proof of will be carried out in several steps. As it is common in such counting issues, special attention must be paid to boundary effects.
*Step 1.* Here we prove on compact subsets of $M_\sigma$. Since $s$ is a section of $M_\sigma {\rightarrow}X$, it is enough to test the convergence on test functions of the form $\pi^* \psi$, with $ \psi\in \mathcal{C}_c(X)$, which we simply denote by $ \psi$, that is, we need to show that ${\left\langle \Delta(R_n){\dot \wedge}[s], \psi\right\rangle} {\rightarrow}\frac{1}{\mathrm{vol} (X)} {\left\langle T{ \arrowvert_{s}}, \psi\right\rangle}$. Fix ${\varepsilon}>0$. To lighten notation, we put $B_n(R_n) = B(x_n, R_n)\times {\left\{z_n\right\}}$.
We first construct a regularization of $ \psi[s]$. Since $s$ is transverse to $\mathcal{F}$ we can extend $ \psi$ locally around $s$ to be constant along the leaves. Fix a non-negative smooth function $\theta_{\varepsilon} : [0,\infty) \rightarrow [0,\infty)$ with support contained in $[0,\varepsilon]$, and such that $\int_{\mathbb D} \theta_{\varepsilon} (d_{\rm Poin}(0,x)) \operatorname{vol}(dx) = 1$. Let now $\Delta$ be a foliated current, expressed as $\Delta = \int \varphi [{\mathbb{D}}\times {\left\{w\right\}}] dm(w)$ in a flow box around a point of $s$, in which $s$ corresponds to ${\left\{0\right\}}\times {\mathbb{D}}$. Define a form along the leaves by $$( \psi[s])_{\varepsilon}= \theta_{\varepsilon}(d_{\mathcal{F}}(\cdot, s))\psi \operatorname{vol}_\mathcal{F},$$ where $d_{\mathcal{F}}$ (resp. $\operatorname{vol}_{\mathcal{F}}$) is the leafwise Poincaré distance (resp. volume form). If $\varphi$ is continuous, then clearly $\Delta\wedge ( \psi[s])_{\varepsilon}$ is close to $\Delta{\dot \wedge}( \psi[S])$ (when $\Delta= \Delta(R_n)$, this will happen when ${\partial}\Delta(R_n)$ is far from $s$).
We then write $$\begin{aligned}
\label{eq:Delta}
& \int \Delta(R_n){\dot \wedge}\psi [s] - \frac{1}{\mathrm{vol} (X)} \int T{\dot \wedge}\psi[s] = {\left(
\int \Delta(R_n) \wedge ( \psi [s] )_{\varepsilon}- \frac{1}{\mathrm{vol} (X)} T\wedge ( \psi [s] )_{\varepsilon}\right)} + \\ \notag
&+ \int \Delta(R_n) {\dot \wedge}\psi [s] -\Delta(R_n) \wedge ( \psi [s] )_{\varepsilon}+\frac{1}{\mathrm{vol} (X)} \int T\wedge ( \psi [s] )_{\varepsilon}-T{\dot \wedge}\psi [s] \end{aligned}$$ as a sum of three terms $ I+II+III$. Since $( \psi [s] )_{\varepsilon}$ is smooth along the leaves, Proposition \[p:bgm\] implies that $I$ converges to zero as $n{\rightarrow}\infty$. Since in the representation the density $\varphi$ of $T$ along the leaves is harmonic, the mean value formula implies that the integral $III$ vanishes.
We will decompose the integral $II$ as a sum of two contributions. We declare that a point in $B_n(R_n)\cap \varpi^{-1}(s)$ is a good intersection if the ball $B_{Poin}(p,{\varepsilon})$ of radius ${\varepsilon}$ relative to the Poincaré metric is disjoint from ${\partial}(B_n( R_n))$. Therefore, $ B_n( R_n)\cap \varpi^{-1}\operatorname{Supp}( \psi[s])_{\varepsilon}$ is a union of good and bad components. Notice that bad components are contained in a leafwise $2{\varepsilon}$-neighborhood of $ {\partial}B_n( R_n)$. Pushing forward again by $\varpi$ we let $\Delta_n^{\rm bad}$ be the part of $\Delta(R_n)$ corresponding to bad components and $\Delta_n^{\rm good}$ be its complement (notice that $\Delta_n^{\rm good}$ is larger than the union of good components). Since $ \psi$ is constant along the leaves near $s$, by definition of $( \psi[s])_{\varepsilon}$ we get that $$\int \Delta_n^{\rm good}{\dot \wedge}( \psi[s]) = \int \Delta_n^{\rm good}\wedge ( \psi[s])_{\varepsilon}) .$$ To estimate the contribution of the bad part, observe that bad components of $\Delta(R_n)$ become good in $\varpi (B_n( R_n+2{\varepsilon}))$ as well as in the annulus $\varpi (B_n( R_n+2{\varepsilon})\setminus B_n( R_n-2{\varepsilon}))$. So we infer that $$\begin{aligned}
\int \Delta_n^{\rm bad} {\dot \wedge}\psi [s] &
\leq \int {\frac{1}{\mathrm{vol}(R_n)}}{\left(\varpi_* \left[B_n( R_n+2{\varepsilon})\setminus B_n( R_n-2{\varepsilon})\right]\right)}^{\rm good} {\dot \wedge}\psi[s] \\
&= \int {\frac{1}{\mathrm{vol}(R_n)}}{\left(\varpi_* \left[B_n( R_n+2{\varepsilon})\setminus B_n( R_n-2{\varepsilon})\right]\right)}^{\rm good} \wedge( \psi[s])_{\varepsilon}\\
&\leq \int {\frac{1}{\mathrm{vol}(R_n)}} {\varpi_* \left[B_n( R_n+2{\varepsilon})\setminus B_n( R_n-2{\varepsilon})\right]} \wedge ( \psi[s])_{\varepsilon}\\
&\underset{n{\rightarrow}\infty}{\longrightarrow} {\left(e^{2{\varepsilon}} - e^{-2{\varepsilon}}\right)} \int T \wedge ( \psi[s])_{\varepsilon}= {\left(e^{2{\varepsilon}} - e^{-2{\varepsilon}}\right)} \int T {\dot \wedge}\psi[s] =O({\varepsilon})
\end{aligned}$$ where the convergence in the last line follows from Proposition \[p:bgm\] and the fact that $ {\mathrm{vol}(R_n+2{\varepsilon})}\underset{n{\rightarrow}\infty}\sim e^{2{\varepsilon}}{\mathrm{vol}(R_n)}$. We thus conclude that the difference of integrals in is arbitrary small as $n{\rightarrow}\infty$, and Step 1 is complete.
*Step 2.* To show that the convergence holds throughout $M_\sigma$, we work in the compactification $\overline M_\sigma$. Let $\mathbb P^1_p$ be the fiber of $\overline M_\sigma{\rightarrow}\overline{X}$ over a puncture $p$. We know from Lemma \[l:extension model\] that the measure $\overline T{\dot \wedge}[\overline s]$ has finite mass. Since $\overline T$ carries no mass on $\mathbb{P}^1_p$, from the local picture of $\mathcal{F}$ and $s$ given in §\[ss:model\], we infer that the measure $\overline T{\dot \wedge}[\overline s]$ has no atom at $\overline s(p)$. Therefore, to prove the desired convergence it is enough to show that the mass of $\Delta(R_n){\dot \wedge}[s]$ near $\overline s(p)$ is uniformly small with $n$.
We use the local model for $\sigma$ near $p$. Fix a coordinate $z$ in which $N(p)$ is identified to ${\mathbb{D}}^*$ and the projective structure is given by $\log z$. Let $N_{\varepsilon}(p) = {\left\{0<{\left\vertz\right\vert}<{\varepsilon}\right\}}$. Then any connected component of $c^{-1}(N_{\varepsilon}(p))$ is the interior of a horocycle in ${\mathbb{H}}$. The crucial point is that the developing map is injective in any component of $c^{-1}(N_{\varepsilon}(p))$. In particular the cardinality of $\mathsf{dev}^{-1} (z_n) \cap B_n(R_n) \cap c^{-1}(N_{\varepsilon}(p))$ is bounded by the number of connected components of $c^{-1}(N_{\varepsilon}(p))$ intersecting $B_n(R_n)$. Now we observe that there is a universal constant $\alpha>0$ such that if $U$ is such a component, then the area of $B_n(R_n +1)\cap U$ is at least $\alpha$. From this we infer that $$\label{eq:component}
\#{\left\{\mathsf{dev}^{-1}(z_n) \cap B_n(R_n) \cap c^{-1}(N_{\varepsilon}(p))\right\}} \leq {\frac{1}{\alpha}}
\mathrm{vol}_{\mathbb{H}}{\left(B_n(R_n + 1) \cap c^{-1}(N_{\varepsilon}(p))\right)}.$$ It is well known that the image of $B_n(R_n + 1)$ under $c$ becomes asymptotically equidistributed in $X$ as $n{\rightarrow}\infty$. This may be obtained as a consequence of Proposition \[p:bgm\], but it already follows from Margulis [@margulis; @these]. From this and , we conclude that $${\frac{1}{\mathrm{vol}(R_n)}} \#{\left\{D^{-1}(z_n) \cap B_n(R_n) \cap c^{-1}(N_{\varepsilon}(p))\right\}}$$ is bounded by $C \operatorname{vol}_X(N_{\varepsilon}(p))$ and the result follows.
This proof shows that rather than a simple number, it is more precise to view the degree as a positive measure on $X$, defined by the formula $\deg(\sigma) = \pi_* (T{\dot \wedge}[s])$. In particular it makes sense to speak of the degree of ${\mathsf{dev}}$ restricted to some $\pi_1(X)$-invariant subset of $\widetilde X$. This measure is canonically associated to the $\mathbb P^1$-structure on $X$ (that is it does not depend on the chosen developing map).
The support of the degree measure can be described as follows. Let $({\mathsf{dev}}, {\mathsf{hol}})$ be a development-holonomy pair for the structure. Let $
\Lambda \subset \mathbb P^1$ be the limit set of ${\mathsf{hol}}$. The closed subset ${\mathsf{dev}}^{-1} (\Lambda)$ being invariant by $\pi_1(X)$, it defines a closed subset $\Lambda_\sigma \subset X$. This set is canonically associated to the projective structure and does not depend on the chosen development-holonomy pair. An instructive example is given by $\mathbb P^1$-structures with Fuchsian holonomy. In this case the set ${\Lambda}_\sigma$ is a union of the boundaries of disjoint annuli embedded in $X$. It was studied e.g. by Goldman to prove that such structures are obtained from $2\pi$-graftings, see [@Goldman; @grafting]. We claim that the support of the degree is the set $\Lambda_\sigma$. Indeed, at the level of the universal cover, the pull-back of the degree is the intersection of $\widetilde{T}$ with the graph of ${\mathsf{dev}}$. In particular ence by the Harnack inequality, it is absolutely continuous with respect to the pull-back by ${\mathsf{dev}}$ of *any* harmonic measure, with density bounded from above and below by positive constants. The claim then follows from the fact that the support of the harmonic measures is the limit set of ${\mathsf{hol}}$. This argument shows more: namely, that the degree has the same Hausdorff dimension of that of the harmonic measures. In particular, our Theorem \[theo:dimension\] shows that the Hausdorff dimension of the degree measure is always smaller than $1$ (since the equality case happens only when $\deg(\sigma)=0$).
Theorem \[theo:formula\] shows that the mass of the degree defines a psh function on the moduli space of $\mathbb P^1$-structures on $X$. It would be interesting to know if the degree is also psh considered as a measure.
Projective structures with vanishing degree {#ss:vanishing}
-------------------------------------------
Recall that $\sigma$ is a [*covering projective structure*]{} on $X$ if its developing map is a covering of some proper open subset ${\Omega}\subset{{\mathbb{P}^1}}$. In other words, if $\sigma$ is the quotient of the orbit of a component of the discontinuity set of a Kleinian group. Such projective structures were studied e.g. by Kra [@kra1; @kra2] who showed that a parabolic projective structure is of covering type if and only if its developing map is not surjective.
Projective structures of covering type may also be characterized in terms of their degree.
\[p:vanishing\] Let $X$ be a Riemann surface of finite type and $\sigma$ be a parabolic projective structure on $X$. Then $\mathrm{deg}(\sigma) = 0$ if and only if $\sigma$ is of covering type.
The proof of Definition-Proposition \[defprop:degree\] shows that the degree vanishes if and only if the support of the foliated harmonic current $T$ is disjoint from $s$. Equivalently, the image of the developing map is disjoint from the support of the harmonic measures. Hence the developing map is not surjective and the result follows from the remarks preceding the proposition.
Cohomological expression of the degree {#ss:cohomological degree}
--------------------------------------
Observe that the harmonic current $\overline{T}$ on $\overline M_\sigma$ naturally defines an element of the dual of the Bott-Chern cohomology group $H^{1,1}_{\partial \overline{\partial}} (\overline{M}_\sigma, \mathbb C)$. By the $\partial \overline{\partial}$-lemma, the natural map $ H_{{\partial}\overline {\partial}}^{1,1} (\overline{M}_\sigma,\mathbb C)
\rightarrow H_{\overline{\partial}} ^{1,1} (\overline{M}_\sigma, \mathbb C)$ with values in the Dolbeaut cohomology group is an isomorphism. Thus by duality, the current $\overline{T}$ defines a cohomology class $[\overline{T}]$ in $H^{1,1} (\overline{M}_\sigma,\mathbb C)$. In more concrete terms, if $\alpha_1$ and $\alpha_2$ are smooth (1,1) forms defining the same class $[\alpha]$ in $H^{1,1}
(\overline{M}_\sigma,\mathbb C)$, then $\alpha_1 - \alpha_2 = dd^c u$ for some smooth function $u$, and we get that ${\left\langle \overline T, \alpha_1\right\rangle} = {\left\langle \overline
T, \alpha_2\right\rangle}$. So it makes sense to speak about the pairing between $\overline T$ and $\alpha$ which we simply denote it by $\overline T\cdot \alpha$. Observe also that any curve $C\subset M_\sigma$ admits a class in $H^{1,1}(M_\sigma)$, which is the cohomology class dual to the cycle $C$ (or equivalently, that of the integration current $[\overline s]$).
Recall from §\[ss:existence of degree\] that $\deg(\sigma)$ is the mass of $\overline T{\dot \wedge}{\overline s}$. The next result –presumably part of the folklore– asserts that this geometric intersection number can be computed in cohomology.
\[p:cohomological expression of the degree\] Let $\sigma$ be a parabolic projective structure on a Riemann surface of finite type, and $\deg(\sigma)$ be be its degree, as defined in Definition \[defprop:degree\]. Then $$\deg(\sigma) = \overline{T} \cdot \overline{s}.$$
The difficulty is that we cannot simply regularize the integration current $[s]$ within smooth positive forms because, as we will see later, $\overline s^2 <0$. Pick a smooth closed $(1,1)$ form cohomologous to $[\overline s]$, and write $[\overline s]=\alpha + dd^cu$, where $u$ is a quasi-psh function, smooth outside $\overline s$, with logarithmic singularities along $s$. Then by definition, $\overline T \cdot \overline s = {\left\langle T, \alpha\right\rangle}$. Recall that $s$ stays far from the singularities of the foliation $\overline {\mathcal F}$ and is everywhere transverse to it. Consider a tubular neighborhood ${N}_{\varepsilon}$ of $\overline s$, such that if $p\in \overline s$ and $L_p$ is the leaf through $p$, then $L_p\cap N_{\varepsilon}$ is a small disk about $p$, contained in a flow box. We modify $u$ by replacing it inside $N_{\varepsilon}$ by any smooth function $u_{\varepsilon}$ such that $u = u_{\varepsilon}$ near ${\partial}N_{\varepsilon}$. We denote by $u_{\varepsilon}$ the resulting function on $\overline M_\sigma$. By construction, $[\overline s]_{\varepsilon}:= \alpha + dd^cu_{\varepsilon}$ is a smooth form cohomologous to $[\overline s]$, so $ \overline T\cdot [\overline s] ={\left\langle \overline T, [\overline s]_{\varepsilon}\right\rangle} $.
Now consider a flow box ${\mathbb{B}}$ endowed with local coordinates $(z,w)\in {\mathbb{D}}^2$ where $\overline{\mathcal{F}}$ becomes the horizontal foliation and $\overline s$ is a vertical graph. Then in this flow box, $[\overline s] = dd^c v $ for some psh function $v$ and $[\overline s]_{\varepsilon}= dd^cv_{\varepsilon}$ with $v=v_{\varepsilon}$ in a neighborhood of ${\partial}{\mathbb{D}}\times {\mathbb{D}}$. With notation as in , we see that the local contribution of ${\left\langle \overline T, [\overline s]_{\varepsilon}\right\rangle} $ is equal to $${\left\langle \overline T, [\overline s]_{\varepsilon}\right\rangle}{ \arrowvert_{{\mathbb{B}}}} = \int {\left( \int_{{\mathbb{D}}\times {\left\{w\right\}} } \varphi dd^c v_{\varepsilon}\right)} dm(w) =
\int {\left(\int_{{\mathbb{D}}\times {\left\{w\right\}} } \varphi dd^c v \right)} dm(w) = T{\dot \wedge}\overline s { \arrowvert_{{\mathbb{B}}}},$$ where the middle equality follows from the Green formula and the harmonicity of $\varphi$. The result follows.
The Lyapunov exponent {#sec:lyapunov}
=====================
In this section we relate the exponent $\chi$ defined in Definition \[def:lyap\] to a foliated Lyapunov exponent introduced by the first author in [@deroin; @levi; @plate Appendice]. This leads in §\[ss:cohomological exponent\] to a cohomological formula for $\chi$ analogous to that obtained for the degree.
The foliated Lyapunov exponent
------------------------------
Using [@Bers1 Proposition 2.2], we start by introducing a Lipschitz family of spherical metrics on $M_\sigma$, simpy denoted by ${\left\Vert\cdot\right\Vert}$. By this, we mean a smooth family of conformal metrics of curvature $+1$ on the fibers, with the property that there exists $C>0$ such that for every smooth path $\omega: [0, 1]{\rightarrow}X$, $\log {\left\VertD h_\rho(\omega)\right\Vert}_\infty \leq C \mathrm{length}(\omega)$, where $h_\rho(\omega)$ is the holonomy of $\omega$ and ${\left\VertD h_\rho(\omega)\right\Vert}_\infty$ is the supremum of the norm of the fiber derivative relative to the spherical metrics on $\pi^{-1}(\omega(0))$ and $\pi^{-1}(\omega(1))$. We will recall some details of the construction below in \[ss:cohomological exponent\]. More generally, the notation $\mathrm{length}(\omega) $ will stand for the [*homotopic length*]{} of $\omega$, that is, the minimal length of a smooth path homotopic to $\omega$ with fixed endpoints. In particular this notion makes perfect sense for a Brownian sample path.
Since $\mathcal{F}$ is transverse to the fibers, this induces a smooth metric on the normal bundle $N_{\mathcal F}$. Later on we will study the extension properties of ${\left\Vert\cdot\right\Vert}$ to a singular metric on the fibers of $\overline M_\sigma$.
Notice that in our situation, the data of a Brownian sample path along a leaf is equivalent to that of its projection on $X$, together with its starting point in the initial fiber. So if the starting point $x$ is given, the projection $\pi$ gives an identification between $W^{\mathcal{F}}_x$ and $W_{\pi(x)}^X$. In this way we can speak of the holonomy, or homotopic length of a leafwise Brownian path, by simply projecting it to $X$. We now consider the family of functions $$\Omega^\mathcal{F}\ni\omega\longmapsto K_t ({\omega}) = \log {\left\VertD _{\omega(0)} h (\omega{ \arrowvert_{[0,t] }}) \right\Vert}.$$ This is a cocycle, in the sense that $K_{t+s} (\omega) = K_s(\omega) + K_t (\sigma_s \omega)$ for every $t,s\geq 0$. As explained above, the estimate $${ K_t({\omega}) } \leq C \cdot \text{length} (\omega{ \arrowvert_{[0,t]}}) ,$$ holds, for some $C$ is independent of $\omega$. The superexponential decay of the heat kernel on the hyperbolic plane [@davies §5.7] then implies that $K_t$ is $W^{\mathcal F}_\mu$-integrable for every $t\geq 0$. The ergodic theorem shows that for $W^{\mathcal F}_\mu$-almost every path $ {\omega}$ the limit ${\lambda}= \lim_{t{\rightarrow}\infty}\frac{K_t ({\omega}) } {t}$ exists and does not depend on $\omega$. By definition ${\lambda}$ is the [*foliated Lyapunov exponent*]{}.
We can now compare ${\lambda}$ and $\chi$.
\[p:comparison exponents\] Let $\sigma$ be a parabolic projective structure on a Riemann surface of finite type. Let $\chi(\sigma) = \chi_{\rm Brown}(\mathsf{hol}_\sigma)$ be the Lyapunov exponent of $\sigma$, as defined in §\[ss:lyapunov\]. Then if ${\lambda}$ is as above we have $\lambda = -2 \chi (\sigma) $.
The proof relies on the following result:
\[l:technical step\] Assume that $\rho$ is non elementary. Then for every $x\in X$ and $W^X_x$-a.e. $\omega: [0,\infty) \rightarrow X$ starting at $x$, there exists $r( \omega) \in \mathbb P^1 _x$ such that the pointwise convergence $$\label{eq:norm} \lim_{t\rightarrow \infty} \frac{1}{t} \log {\left\Vert D_y h (\omega{ \arrowvert_{[0,t]}}) \right\Vert}
\rightarrow -2\chi_{\rm Brown}(\rho)$$ holds uniformly on compact subsets of $\mathbb P^1 _x \setminus \{r(\omega) \}$. Moreover, the distribution of the exceptional point $r(\omega)$ is the harmonic measure $\nu_x$ on $\mathbb P^1_x$.
In order to apply the Oseledets theorem, consider a measurable trivialization $M_\sigma \simeq X \times \mathbb P^1$ and set $\chi = \chi_{\rm Brown}(\rho)$. For every continuous $\omega$, and every $t> 0$, the map $h_t = h_{\omega{ \arrowvert_{[0,t]}}} : \mathbb P^1_{\omega(0)} \rightarrow \mathbb
P^1 _{\omega(t)}$ can be lifted to a matrix $\widetilde{h_t}$ in $\text{SL}(2,\mathbb C)$ which is well defined up to sign. The family $
\widetilde{h} = \{ \widetilde{h_t} \}_{t\geq 0}$ on $\Omega$ is a cocyle modulo signs, namely it satisfies $\widetilde{h_{t+s}} (\omega) = \pm \widetilde{h_t} (\sigma _s(\omega)) \widetilde{h_s} (\omega)$ for every $\omega \in \Omega$ and every $s,t\geq 0$. Moreover, from [@Bers1 Proposition 2.5], we have that for every $x\in X$ and $W^X_x$ a.e. $\omega$, $$\lim _{t\rightarrow \infty} \frac{1}{t} \log {\left\Vert\widetilde{h_t}\right\Vert} = \chi ,$$ where ${\left\Vert\cdot\right\Vert}$ is the matrix norm to the usual hermitian norm ${\left\Vert\cdot\right\Vert}_2$ on ${{{\mathbb{C}}^2}}$. Since $\rho$ is non elementary, by [@Bers1 Thm 2.7], $\chi>0$. Since in addition $h$ takes values in $\text{SL}(2,\mathbb C)$, the Lyapunov exponents of $h$ over $(\Omega_X , \sigma_X , W^X)$ are $\chi $ and $-\chi$. The Oseledets theorem tells us that for $W$-a.e. $\omega: [0,\infty) \rightarrow X$, there exists a complex line $E= E(\omega) \subset \mathbb C^2$ such that for every $Y\in \mathbb C^2$, $Y\neq 0$, $ \frac{1}{t} \log {\left\Vert\widetilde{h_t} (Y)\right\Vert}_2 $ converge to $-\chi$ as $t{\rightarrow}\infty$ when $Y\in E$, while this quantity converges uniformly to $\chi$ on compact subsets of $\mathbb C^2 \setminus E$. Finally, we observe that for the usual spherical derivative, we have that $$\label{eq:spherical derivative} {\left\VertD h_t (y) \right\Vert}_s = \frac{{\left\VertY\right\Vert}_2^2}{{\left\Vert\widetilde{h_t}(Y)\right\Vert}_2^2}, \text{ where $Y$ is a lift of $y$},$$ hence holds, with $r(\omega) = \mathbb P E(\omega) \in \mathbb P^1_{\omega(0)}$.
It remains to show that the distribution of $r$ when $\omega$ is conditioned to start at $x$ is the harmonic measure $\nu_x$. For this, we consider the mapping $\omega^\mathcal{F}\ni\omega \mapsto r(\omega)$, which is defined $W_\mu^\mathcal{F}$-a.e. The push-forward of $W_\mu^\mathcal{F}$ is a shift invariant measure on $M_\sigma$, so we conclude by the unique ergodicity of $\mathcal{F}$ (Proposition \[p:ergodic\]).
Let ${x}\in M_\sigma$. As observed before, we can identify $(\Omega^X _ {\pi(x)}, W^X_{\pi(x)})$ in $X$ and $(\Omega^{\mathcal F}_{ {x}}, W^{\mathcal F}_{ {x}})$ in $M_\sigma$ by lifting. Since the harmonic measure $\nu_{\pi(x)}$ on $\mathbb P^1_{\pi(x)}$ has no atoms, we infer that for $W_{{x}}^\mathcal{F}$ a.e. $\omega$, the point $r(\omega)$ defined in Lemma \[l:technical step\] is distinct from ${x}$. Hence $ \lim_{t\rightarrow \infty} \frac{1}{t} \log {\left\Vert D_{x} h (\omega{ \arrowvert_{[0,t] }}) \right\Vert} \rightarrow -2\chi$ for $W_{{x}}$-a.e. $\omega$, and the conclusion follows.
Cohomological expression of $\chi$ {#ss:cohomological exponent}
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Let $P$ be the set of punctures of $X$. To avoid confusion with the Lyapunov exponent, we denote by $\mathrm{eu}(X)$ the Euler characteristic of $X$, $\mathrm{eu}(X) = 2-2g -\# P$. Recall the Gauss-Bonnet formula $\operatorname{vol}(X) = 2\pi {\left\vert\mathrm{eu}(X)\right\vert}$.
In this section, we begin the proof of the following result, which will be complete only after proving Theorem \[theo:formula\]
\[p:computation of Lyapunov exponent\] Let $\sigma$ be a parabolic projective structure on a Riemann surface of finite type with puncture set $P$. Then $\chi (\sigma) = \displaystyle \frac{1}{2{\left\vert\mathrm{eu}(X)\right\vert} } ( N_{\overline{\mathcal F}} \cdot \overline{T} +\#P)$.
When $X$ is compact ($P = \emptyset$) this result follows from the cohomological formula derived in [@deroin; @levi; @plate Appendice A] for the foliated Lyapunov exponent, and from Proposition \[p:comparison exponents\]. The proof in the non compact case follows the same strategy but serious technical difficulties arise from the parabolic cusps.
Recall that if $X$ is a complex surface, $E\rightarrow X$ is a holomorphic line bundle, and ${\left\Vert\cdot\right\Vert}$ is a hermitian metric on $E$, its curvature form is defined by $\Theta({\left\Vert\cdot\right\Vert}) = \frac{1}{2 i \pi }\partial \overline{\partial} \log {\left\Verts \right\Vert}^2 $, where $s$ is any non vanishing local holomorphic section of $E$. In our situation we choose a Lipschitz family of spherical metrics on the fibers of $M_\sigma$, which, since $\mathcal{F}$ is transverse to the fibers, induces a hermitian metric on the normal bundle $N_{\mathcal F}$. Recall that the value of the Lyapunov exponent does not depend on this choice. We denote by $\Theta$ the curvature form of this metric.
The first result is obtained exactly as in the compact case [@deroin; @levi; @plate Appendice A] (see also [@candel §8]).
\[l:computation lyapunov exponent\] $\displaystyle \chi (\sigma) = \frac{\pi}{\operatorname{vol}(X)} \int \Theta \wedge T$.
We keep notation as in §\[ss:lyapunov\]. From the fact that $K$ is a cocycle, we deduce that the function $t\mapsto \int K_t(\omega) W^\mathcal{F}_\mu(d\omega)$ is linear. So its slope is equal to its derivative at 0, and we get that $$\label{eq: formula lyapunov exponent} \lambda= \frac{d}{dt} \Bigl\vert_{t=0}
\int K_t ( {\omega} ) W_\mu^\mathcal{F}(d\omega)
= \int _{M_\sigma} \frac{d}{dt}\Bigl\vert_{t=0} \mathbb E^{ {x}} \big( K_t ( {\omega} ) \big) d\mu ({x}) .$$ Let $x_0$ be a point of $X$. We use local coordinates $x=(\xi, \eta)$, to parametrize points in $M_\sigma$ via $s(\xi, \eta)$, that is, ${x}$ belongs to the fiber of $\xi$, $\eta$ belongs to a neighborhood of $\eta_0$ in $\mathbb P^1_\xi$. and $s(\xi, \eta) = h _{\xi, \eta} ({x})$ is the flat section passing through the point ${x}$, defined over a neighborhood of $\xi_0$ in $X$. Using the heat equation, the formula can be written in these coordinates $$\lambda = \int _{X_\rho} \Delta_{\xi} \log {\left\Vert\frac{{\partial}}{{\partial}\eta} h_{\xi,\eta} ({x})\right\Vert}\ \mu ( d{x})$$ Observe that the curvature form $\Theta$ of the Lipschitz metric on $N_{\mathcal F}$, restricted to the tangent bundle of $\mathcal F$, is given by the expression $$\Theta { \arrowvert_{T\mathcal F}} = \frac{1}{2 i \pi }\partial \overline{\partial}_\mathcal{F} \log {\left\Vert\frac{{\partial}}{{\partial}\eta} h_{\xi,\eta} ({x} )\right\Vert}^2 .$$ Because we have $\Delta_{\rm Poin} f \cdot \text{vol} _{\rm Poin} = 2 i \partial \overline{\partial} f$ for every function $f$ defined on the hyperbolic plane, we infer that $$\Delta_{\xi} \log {\left\Vert \frac{{\partial}}{{\partial}\eta} h_{\xi,\eta} ({x} )\right\Vert} \text{vol} _{\rm Poin} = - 2\pi\ \Theta{ \arrowvert_{T\mathcal F}} .$$ Using the fact that $T \wedge \text{vol}_{\rm Poin} = \text{vol(X)} \mu$, we finally obtain $$\lambda = -\frac{2\pi}{\text{vol} (X)} \int \Theta \wedge T,$$ which, together with Proposition \[p:comparison exponents\] finishes the proof of Lemma \[l:computation lyapunov exponent\].
When $X$ is compact, it immediately follows from Lemma \[l:computation lyapunov exponent\] that $$\label{eq:cohomological formula compact case} \chi =\frac{\pi} {\text{vol}(X)} T\cdot N_{\mathcal F} =
\frac{1}{2{\left\vert\mathrm{eu}(X)\right\vert}} N_{ {\mathcal F}} \cdot {T},$$ and the proof of Proposition \[p:computation of Lyapunov exponent\] is complete.
In the general case, however, this calculation is no longer valid, and in the remaining part of the argument we need to understand the contribution of the punctures to this formula. For the moment, we content ourselves with the following weakening of Proposition \[p:computation of Lyapunov exponent\].
\[p:weaker\] Under the assumptions of Proposition \[p:computation of Lyapunov exponent\], there exists a universal constant $I$ such that $\chi (\sigma) = \displaystyle \frac{1}{2{\left\vert\mathrm{eu}(X)\right\vert} } ( N_{\overline{\mathcal F}} \cdot \overline{T} + I \cdot \#P)$.
The proof occupies the remainder of this section. It will be carried out in several steps, mostly dealing with the local study of the model foliation $\mathcal{F}_m$ introduced in §\[ss:model\].
[*Step 1. A smooth metric.*]{}
Let $p\in P$ be a puncture of $X$, and let us work in a neighborhood $\pi^{-1}(U(p))$ of $\pi^{-1}(p)$ in $\overline M_\sigma$, in the coordinates $(u,v)$ introduced in §\[ss:model\]. We claim that the metric $$\label{eq:smooth metric}
{\left\Vert\cdot\right\Vert}_s = |u| \frac{|dv|}{1+|v|^2}$$ defines a smooth metric on $N_{\overline{\mathcal F}}$. To see this, observe that a non-vanishing holomorphic section of the normal bundle of $\overline{\mathcal F} = \mathcal{F}_m$ on $\mathbb D \times \mathbb C$ in the $(u,v)$-coordinates is defined by $n=\frac{1}{u} \frac{\partial}{\partial v}$. Indeed, $\omega =
du + 2i \pi u dv$ is a form defining $\overline{\mathcal F}$, and $\omega (n) = 2i\pi$. We see that ${\left\Vertn\right\Vert}_s = \frac{1}{1+|v|^2}$, so ${\left\Vert\cdot\right\Vert}_s$ extends smoothly along the line ${\left\{0\right\}}\times \mathbb C $. To analyse what happens close to the point $(0,\infty)$, we introduce the new coordinates $(u,V) = (u,\frac{1}{v})$. In these coordinates, the foliation is defined by the equation $2\pi u dV + i V^2 du = 0$. A non-vanishing section of the normal bundle is then given by $n = \frac{1}{u} \frac{\partial} {\partial V}$, and a straighforward computation yields ${\left\Vert\cdot\right\Vert}_s = |u| \frac{|dV|}{1+|V|^2}$. Hence the situation is symmetric and we conclude that ${\left\Vert\cdot\right\Vert}_s$ defines a smooth metric on $N_{\overline{\mathcal F}}$, as claimed.
[*Step 2. The Lipschitz metric.* ]{}
Here we give an explicit expression for a Lipschitz family of spherical metrics on $ M_\sigma$ close to $p$. Recall that a model for the bundle $\pi^{-1}(\mathcal U(p))\subset M_\sigma$ is the quotient of $\mathbb H \times \mathbb P^1$ by the identification $(\tau, z) \sim (\tau + 1 , z + 1)$. A Lipschitz family of spherical metrics on this model is defined by $$\label{eq:Lipschitz spherical metric} {\left\Vert\cdot\right\Vert}_{\tau} = \frac{\Im \tau \ |dz|}{|z-\Re \tau|^2 + \Im^2 \tau }$$ It is constructed by starting with the spherical metric ${\left\Vert\cdot\right\Vert}_i = \frac{|dz|}{1+|z|^2}$, which is already invariant by the stabilizer $\text{PSO}(2,\mathbb R)$ of the point $i$, and then by extending it by the formula $M^* {\left\Vert\cdot\right\Vert}_{M\tau} = {\left\Vert\cdot\right\Vert}_\tau$ for any $
\tau\in \mathbb H$ and $M\in \mathrm{PSL}(2,\mathbb R)$. The proof of [@Bers1 Prop. 2.2] shows that ${\left\Vert\cdot\right\Vert}_\tau$ is indeed Lipschitz. The family $\{ {\left\Vert\cdot\right\Vert}_\tau \}_{\tau\in \mathbb H}$ then induces a family of spherical metrics $\{ {\left\Vert\cdot\right\Vert}_u \}_{u\in \mathbb D^*}$ on the quotient bundle $\simeq \mathbb D ^* \times \mathbb P^1$ which is given by the formula $$\label{eq:Lipschitz spherical metric 2} {\left\Vert\cdot\right\Vert}_u = \frac{\frac{1}{2\pi}\log \big( \frac{1}{|u|}\big) |dv|}
{|v+ \frac{i}{2\pi}\log \big( \frac{1}{|u|}\big) |^2 + \frac{1}{4\pi^2}\log^2 \big( \frac{1}{|u|}\big)}
= \frac{2\pi} {\log \big( \frac{1}{|u|}\big)} \cdot \frac{|dV|}{|\frac{2\pi }{\log (\frac{1}{|u|})} + iV| ^2 + |V|^2}.$$
[*Step 3. The induced singular metric on $N_{\overline{\mathcal F}}$.*]{}
The family of spherical metrics constructed above on $ M_\sigma$ induces a metric ${\left\Vert\cdot\right\Vert}$ on the normal bundle of the foliation $\overline{\mathcal F}$ which possesses singularities along the fibers over the cusps of $X$, that we compute here. In the $(u,V)$-coordinates, we have that $$\label{eq:singularities} \frac{{\left\Vert\cdot\right\Vert}}{{\left\Vert\cdot\right\Vert}_s} = \frac{2\pi }{|u| \log (\frac{1}{|u|})} \cdot \Phi(u,V)
\text{ where } \Phi(u,V) = \frac{1+ |V|^2}{ | \frac{2\pi }{\log(\frac{1}{|u|})} + i V|^2 + |V|^2}.$$ The reader can check that $\Phi$ has a pole only at the point $(u,V)=(0,0)$, extends continuously and extends continuously elsewhere.
[*Step 4. Defining a foliation index.* ]{} For [*any*]{} harmonic current $T$ on $\mathbb D \times \mathbb P^1$ directed by $\mathcal F _m$, we define $$\label{eq:index} I(T):= \int_{\mathbb D^* \times \mathbb P^1} \frac{1}{i\pi} \partial \overline{\partial} \Psi \wedge T$$ where $\Psi : \mathbb D^* \times \mathbb P^1 $ is a smooth function supported in a domain $ D_r ^* \times \mathbb P^1$ for some $0<r<1$, and such that $\Psi = \log \frac{{\left\Vert\cdot\right\Vert}}{{\left\Vert\cdot\right\Vert}_s}$ in a neighborhood of $0\times \mathbb P^1$. Observe that this number does not depend on the chosen function $\Psi$, since the current $T$ is harmonic.
\[l:convergent\] The integral is convergent.
It suffices to proves the lemma for $\Psi = \log \frac{{\left\Vert\cdot\right\Vert}}{{\left\Vert\cdot\right\Vert}_s}$. In this case the integral $I(T)$ is nothing but the $T$-integral of the differences between the curvature of ${\left\Vert\cdot\right\Vert}$ and that of ${\left\Vert\cdot\right\Vert}_s$. Because ${\left\Vert\cdot\right\Vert}_s$ is smooth and hence $T$-integrable, it is enough to prove that the curvature of ${\left\Vert\cdot\right\Vert}$ is $T$-integrable. We claim that the restriction of the curvature of ${\left\Vert\cdot\right\Vert}$ along the leaves is bounded in modulus by the leafwise Poincaré metric. This is sufficient for our purposes since the Poincaré metric is $T$-integrable (due to the fact that $T$ projects on the integration current on $\mathbb D$) and that the $T$-integral of a $(1,1)$-form depends only on its restriction to $\mathcal F$.
To prove this claim, we work in the $(\tau,z)$-uniformizing coordinates, and use formula to get that the curvature of ${\left\Vert\cdot\right\Vert}$ along the leaf $\mathbb H^2 \times z$ is $$\frac{1}{i} \partial \overline{\partial}_\tau \log \big( \frac{\Im \tau }{|z-\Re \tau|^2 + \Im^2 \tau } \big) .$$ Then, writing $\tau = x+iy$, we compute $$\frac{1}{i} \partial \overline{\partial}_\tau \log \big( \frac{ y }{|z- x|^2 + y^2 } \big) = \big( \frac{-1}{y^2} + \frac{2 \Im ^2 z }{\big( (x- \Re z) ^2 + y^2 + \Im^2 z\big)^2 } \big) dx\wedge dy$$ and the result follows since $$0\leq \frac{2 \Im ^2 z }{\big( (x- \Re z) ^2 + y^2 + \Im^2 z\big)^2 } \leq \frac{2}{ y^2}.$$
The index is defined so as to have the following formula, which corrects formula . For every puncture $p$ of $X$, we define $I(T,p)$ to be the index of the canonical foliated harmonic current defined in subsection \[ss:harmonic currents\] at the puncture $p$.
\[l:lyapunov exponent 1\] $\displaystyle\chi(\sigma) = \frac{\pi} {\mathrm{vol}(X)} \left( N_{\overline{\mathcal F}} \cdot T + \sum _p I(T, p) \right)$.
Let $\Psi$ be the function on $M_\sigma$, defined in a neighborhood of the exceptional fibers, as just constructed. Introduce a smooth family of metrics on the fibers of $\overline M_\sigma$, which coincides with ${\left\Vert\cdot\right\Vert}_s$ near the punctures. Such a family is not Lipschitz, so we multiply it by a function of the form $e^\Psi$, to make it coincide with the local model discussed above, ${\left\Vert\cdot\right\Vert} = {\left\Vert\cdot\right\Vert}_s \cdot e^\Psi$. Then we infer that $$\int \Theta \wedge T = \int \Theta_{{\left\Vert\cdot\right\Vert}_s} \wedge T + \int \frac{1}{i\pi} \partial \overline{\partial} \Psi \wedge T= N_{\overline{\mathcal F}} \cdot T + \sum _p I(T, p)$$ and result follows from Lemma \[l:computation lyapunov exponent\].
[*Step 5. An invariance property for the index*]{}
\[p:invariance\] The index $I(T)$ takes the same value on all foliated harmonic currents $T$ on $\mathcal F_m$ that give mass $1$ to the fibers ${\left\{u\right\}}\times \mathbb P^1$.
Let us introduce two families of symmetries for the foliation $\mathcal F_m$. They are induced by the translations $(\tau,z) \mapsto (\tau +x, z)$ and $(\tau,z) \mapsto (\tau, z+c)$ for $x\in \mathbb R$ and $c\in \mathbb C$ at the level of the universal cover : $$\label{eq:symmetries} H_x(u,V) = (e^{2i\pi x} u, \frac{V}{1-xV})\ \ \ \text{and}\ \ \ V_c (u, V) = (u, \frac{V}{1+cV}).$$ The following result is the key of the argument:
\[l:invariance\] If $T$ is as in Proposition \[p:invariance\], then for all $x\in \mathbb R$ and $c\in \mathbb C$, $I(( H_x)_*T)=I((V_c)_* T)= I(T)$.
We treat the case of $H_x$, the proof being similar (and in fact easier) for $V_c$. We have that $$I(( H_x)_*T)- I(T) = \int_{\mathbb D^* \times \mathbb P^1} \frac{1}{2i\pi}
\partial \overline{\partial} (\Psi \circ H_x - \Psi) \wedge T .$$ Let us split this function as a sum $$\label{eq:splitting} \Psi \circ H_x - \Psi = \Gamma + \Gamma _s,$$ where $\Gamma$ and $\Gamma_s$ are smooth functions on $\mathbb D^* \times \mathbb P^1$ supported in $\mathbb D^* _r\times \mathbb P^1$ for some $0<r<1$ and such that in a neighborhood of the divisor ${\left\{u=0\right\}}$, $\Gamma = \log \frac{(H_x)_*{\left\Vert\cdot\right\Vert}}{{\left\Vert\cdot\right\Vert}}$ and $\Gamma_s =\log \frac{(H_s)_*{\left\Vert\cdot\right\Vert}_x}{{\left\Vert\cdot\right\Vert}_s}$. Observe that $$\label{eq:gamma}
\int \frac{1}{2i\pi} \partial \overline{\partial} \Gamma_s \wedge T =0$$ since ${\left\Vert\cdot\right\Vert}_s$ is a smooth metric, thus $\Gamma_s$ is a smooth function. Now $\Gamma$ is smooth if $u\neq0$, tends to $0$ uniformly when $u$ tends to $0$, and the derivative of $\Gamma$ along the leaves is bounded by the Poincaré metric (since ${\left\Vert\cdot\right\Vert}$ is a Lipschitz metric). Since the leafwise Poincaré metric is given in $u$-coordinates by $\frac{|du|}{|u|\log |u|}$, we get that $$\label{eq:bound} | d_{\mathcal F} \Gamma |\leq \frac{|du|}{|u|\log |u|} .$$ From and we are left to prove that $$\label{eq:computation} \int \frac{1}{2i\pi} \partial \overline{\partial} \Gamma \wedge T =0$$ (the fact that this integral makes sense follows from Lemma \[l:convergent\]). To do this, we introduce a family of smooth functions $\theta_r: \mathbb D^* \rightarrow [0,1]$ such that $\theta_r(u)=1$ if $|u|\geq r$, $\theta_r (u) = 0$ if $|u|\leq r/2$, ${\left\Vertd\theta_r \right\Vert}_{\infty} = O (\frac{1}{r})$, and ${\left\Vert\partial \overline {\partial} \theta_r\right\Vert}_{\infty} = O(\frac{1}{r^2})$. Since ${\partial}\overline {\partial}\Gamma\wedge T$ is of order 0, to get , it is enough to prove that $$\lim_{r\rightarrow 0} \int \theta_r \partial \overline{\partial} \Gamma \wedge T = 0 .$$ To compute this integral, we observe that since $T$ is harmonic $\int \partial \overline{\partial} (\theta_r \Gamma) \wedge T = 0$ , hence we get that $$\int \theta_r \partial \overline{\partial} \Gamma \wedge T = - \int \Gamma \partial \overline{\partial} \theta_r \wedge T -
2\Re \int \partial \theta_r \wedge \overline{\partial} \Gamma\wedge T =: - A_r - B_r.$$ To conclude the proof, we will show that both integrals $A_r$ and $B_r$ tend to $0$ with $r$. To estimate the former, we write $${\left\vertA_r\right\vert} \leq \delta(\Gamma,r) O (\frac{1}{r^2}) \int _{ \frac{r}{2} \leq |u|\leq r} idu\wedge d\overline{u} \wedge T \leq O (\delta(\Gamma, r) )$$ where $\delta(\Gamma,r) = \sup _{\frac{r}{2}\leq |u| \leq r, v\in \mathbb P^1} \Gamma(u,v) $, and the last inequality holds because $T$ projects on the current of integration on $\mathbb D$. As observed above, $\delta(\Gamma,r)= o(1)$ whence $ \lim_{r\rightarrow 0} A_r = 0 $. The same argument works for the second integral: indeed by using and the bound on ${\left\Vertd\theta_r \right\Vert}_{\infty}$, we get that $${\left\vertB_r\right\vert} \leq O(\frac{1}{r^2 \log (\frac{1}{r})}) \int _{ \frac{r}{2} \leq |u|\leq r} idu\wedge d\overline{u} \wedge T \leq O(\frac{1}{\log (\frac{1}{r})}) ,$$ which completes the proof.
Let us resume the proof of Proposition \[p:invariance\]. Recall from §\[ss:harmonic currents\] that a foliated harmonic current $T$ for $\mathcal{F}_m$ lifts as a harmonic current $\widetilde{T}$ on ${\mathbb{H}}\times {{\mathbb{P}^1}}$, which by the Poisson formula is induced by a family of probability measures ${\left\{\nu_a\right\}}_{a\in {\mathbb{R}}}$ on ${{\mathbb{P}^1}}$, depending measurably on $a$, and satisfying the relation $\nu_{a+1} = (z+1)_*\nu_a$. From this equivariance, the data of such a family of measures is in turn equivalent to that of a probability measure on $[0,1)\times {{\mathbb{P}^1}}$. Such a measure is a convex combination of Dirac masses on $[0,1)\times {{\mathbb{P}^1}}$. This shows that any family ${\left\{\nu_a\right\}}$ as above is a convex combination of families of the form $\nu_{{(a_0, z_0)}}$, $a\in [0,1)$, $z_0\in {{\mathbb{P}^1}}$, where $(\nu_{(a_0, z_0)})_a = 0$ if $a\neq a_0$ mod. ${\mathbb{Z}}$ and $(\nu_{(a_0, z_0)})_{a_0+k} = \delta_{z_0+k}$. (Notice that the point $\infty\in {{\mathbb{P}^1}}$, corresponding to the separatrix of the singularity of $\mathcal{F}_m$, plays a special role here. Nevertheless we do not need to take it in to account since our measures and currents are diffuse.) Going back to currents, $\nu_{(a_0, z_0)}$ corresponds to a certain harmonic current $T_{(a_0, z_0)}$ and all foliated harmonic currents for $\mathcal{F}_m$ are obtained from these by taking convex combinations.
Now it is clear that $(H_x)_*(T_{(a_0, z_0)}) = T_{({a_0+x, z_0})}$ and $(V_c)_*(T_{(a_0, z_0)}) = T_{({a_0, z_0+c})}$, hence we infer from Lemma \[l:invariance\] that the index $I$ takes the same value on all the extremal points $T_{(a_0, z_0)}$, and we are done.
Proof of Theorem \[theo:formula\] (and of Proposition \[p:computation of Lyapunov exponent\]) {#sec:proof}
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The proof is based on some basic cohomological computations in $H^2(\overline{M }_\rho, {\mathbb{C}})$. Recall from §\[ss:generalities\] that $H^2(\overline{M }_\rho, {\mathbb{C}}) = {\mathbb{C}}\overline [\overline{s}] \oplus {\mathbb{C}}[f]$. We will need the following fact: if $\mathcal G$ is a singular holomorphic foliation on a complex surface, and $C $ is a non singular compact holomorphic curve that is everywhere transverse to $\mathcal{G}$, then $$N_{\mathcal G} \cdot C = \mathrm{eu} (C).$$ Indeed, under these assumptions, $N_\mathcal{G}{ \arrowvert_{C}} \simeq T_C = -K_C$, and by the genus formula, $K_C\cdot C = -\mathrm{Eu(C)}$. Hence in our situation, working in $\overline M_\sigma$ we get that $$\label{eq:npoint}
N_{\overline{\mathcal{F} } }\cdot \overline s = \mathrm{eu} (\overline s) = \mathrm{eu} (\overline X) \text{ and }
N_{\overline{\mathcal{F}}}\cdot f = \mathrm{eu}({{\mathbb{P}^1}}) = 2.$$ The intersection form in $H^2(\overline{M }_\rho, {\mathbb{C}})$ is characterized by the identities $$\overline s \cdot f =1, \ f^2 = 0, \text{ and } \overline s^2 = \mathrm{eu} (\overline X).$$ The first two equalities are obvious, and the justification of the third one is as follows: since the section $\overline s$ is everywhere tangent to $\mathcal{F}$ and to the fibers, we get an isomorphism between the tangent bundle and normal bundle to $\overline s$. Therefore the adjunction formula yields $$\overline s^2 = \mathrm{deg}(N_{\overline s}{ \arrowvert_{\overline{s}}}) = - \mathrm{deg}(K_{\overline s}{ \arrowvert_{\overline{s}}}) =
\mathrm{eu} (\overline s) = \mathrm{eu} (\overline X).$$ From this and we easily deduce that $$\label{eq:normal bundle} [N_{\overline{\mathcal F}} ] = 2 [\overline{s}] - \mathrm{eu}(\overline{X} ) [f] .$$ Now Proposition \[p:weaker\] asserts that $$\chi (\sigma) = \displaystyle \frac{1}{2{\left\vert\mathrm{eu}(X)\right\vert} } ( N_{\overline{\mathcal F}} \cdot \overline{T} + I \cdot \#P).$$ Also, from Proposition \[p:cohomological expression of the degree\] we have that $\delta ={\frac{1}{\operatorname{vol}(X)}} \overline{T} \cdot \overline{s}$, and it is obvious that $\overline{T} \cdot f = 1$. Using the fact that $\mathrm{eu}(\overline X)
=\mathrm{eu}(X) + \#P$, altogether this yields $$\chi (\sigma) = \frac12 + 2\pi \delta+ (I-1) \frac{\#P}{2{\left\vert\mathrm{eu}(X)\right\vert}}.$$ Therefore, to finish the proof it is enough to show that $I=1$. This is done by considering the particular case of the canonical projective structure induced by the uniformization of $X$, since in this case we have that $\delta = 0$ and $\chi = \frac12$ (see the remarks following [@Bers1 Def. 2.1]). The proof is complete.
The dimension of harmonic measure: proof of Theorem \[theo:dimension\] {#s:dimension}
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In this part we provide the proof of Theorem \[theo:dimension\]. We start with a result originating in the work of S. Frankel (see [@frankel]).
\[p:frankel\] Let $\varphi$ be the density of the desintegration of $T$ (resp. $\mu$) along the leaves of $\mathcal F $ (see (\[eq:foliated\])). Then the integral $$\label{eq:alpha} A = - \int _{X_\rho} \Delta_{\mathcal F} \log \varphi \ d\mu$$ is convergent, and moreover $$\label{eq:bound on alpha} 0\leq A \leq 1.$$
Notice that $\varphi$ is defined only up to a multiplicative factor which is constant along the leaves, which shows that $\Delta_{\mathcal F} \log \varphi$ is well defined. The quantity $A$ is called the [*action*]{} of $T$.
The positivity and harmonicity of the density $\varphi$ implies that $\Delta_{\mathcal F} \log \varphi = - {\left\Vert\nabla \log \varphi \right\Vert}^2$. Thus by the Harnack inequality for positive harmonic functions we infer that $ {\left\Vert\nabla \log \varphi \right\Vert} $ is uniformly bounded, whence the convergence of the integral in . To get the bound , we observe that $\varphi$ lifts to the universal cover of $\mu$-a.e. leaf as a positive harmonic function, hence the half-plane version of the Harnack inequality (obtained by taking conjugate harmonic functions and applying the Schwarz-Pick lemma) yields $ {\left\Vert\nabla \log \varphi \right\Vert} \leq 1 $. This proves (see [@deroin; @levi; @plate] for more details).
The main step of the proof is the following probabilistic estimate of the measure of a ball inside a fiber. For every $ {x} \in M_\sigma$, and $\rho >0$, we denote by $B_{{\mathbb{P}^1}}(\tilde{x},\rho)$ the ball of radius $\rho$ centered at $\tilde{x}$ inside the fiber $\mathbb P^1_{\pi(x)}$. To ease notation, from now on if $x\in M_\sigma$ we denote its fiber by $\mathbb{P}^1_x$ (resp. the corresponding harmonic measure by $\nu_x$).
\[p:sup dimension\] Let $A$ be as in Proposition \[p:frankel\]. For every $\varepsilon >0$, there exists $r_{\varepsilon}>0$ such that for every $0< r < r_{\varepsilon}$, $$\mu \left( x\in M_\sigma ,\ \ \nu_x (B_{{\mathbb{P}^1}}({x} , r)) \geq r ^{\frac{A}{2\chi}+\varepsilon}) \right) \geq 1 -\varepsilon.$$
The proof follows an argument of Ledrappier’s [@ledrappier Thm 4.1, p. 372], with the difference that the discrete random walk is replaced by a cocycle over the ergodic system $(\Omega^{\mathcal{F}} , \sigma , W^{\mathcal{F}}_\mu)$.
In view of the next lemma it is useful to recall that the data of a foliated Brownian path is equivalent to that of a Brownian path in $X$ together with its starting point in the fiber.
\[l:measures of images\] Let $x\in X$ and $C_x$ be a measurable subset of $\mathbb P^1 _x$ such that $\nu_x (C_x) >0$. Then for $W^X_x$-a.e. $\omega: [0,\infty) \rightarrow X$, letting $h_t = h(\omega{ \arrowvert_{[0,t]}})$, we have that $$\limsup _{t\rightarrow \infty} -\frac{\log \nu_{\omega(t)} (h_t (C_x)) }{t} \leq A.$$
We first work with foliated Brownian motion. Let us introduce the family of functions $(L_t)_{t\geq 0}$, defined on ${\Omega}^{\mathcal{F}}$ by $$L_t (\omega ) = - \log \frac{ (h_t)^{-1} \nu_{\omega(t)} }{\nu_{\omega(0)}} (\omega (0) ) .$$ It is immediate that $L=( L_t )_{t\geq 0}$ is a cocycle, namely it satifies the relations $$L_{t+s} (\omega) = L_s (\omega) + L_t (\sigma _s (\omega)) , \text{ for every } s,t\geq 0 \text{ and }
\omega \in \Omega^{\mathcal{F}}.$$ Moreover, in terms of the densities $\varphi$, $L_t$ expresses as $ L_t (\omega ) = - \log \frac{\varphi (\omega(t))}{\varphi(\omega(0))}$, hence by applying the Harnack inequality ${\left\Vert\nabla_\mathcal{F} \log \varphi\right\Vert}\leq 1$ we obtain the estimate $$\label{eq:Harnack} |L_t(\omega) | \leq \text{length} (\omega{ \arrowvert_{[0,t]}}), \text{ for every } t\geq 0 \text{ and } \omega \in \Omega^{\mathcal{F}}$$ (recall that the length in question here is the homotopic length of $\pi(\omega)$). Thus the super-exponential decay of the heat kernel on the upper half plane [@davies §5.7] implies that $L_t$ is $W^{\mathcal F}_\mu$-integrable. The subadditive ergodic theorem applied to the cocycle $L$ and the ergodic system $(\Omega ^{\mathcal{F}}, \sigma , W^{\mathcal{F}}_\mu)$ shows that $\frac{L_t(\omega)} {t}$ converges a.s. to a limit independent of $\omega$. Arguing exactly as in Lemma \[l:computation lyapunov exponent\] shows that this limit equals $A$, that is, $$\label{eq:cocycle L} \frac{L_t(\omega)} {t} \underset{t\rightarrow \infty}\longrightarrow A,
\text{ for $W^{\mathcal{F}}_\mu$-a.e. } \omega \in \Omega^{\mathcal{F}}.$$ From this, we infer that for a.e. $x\in M_\sigma$, and $W_x^{\mathcal{F}}$ a.e. $\omega$, if $C_x \subset \mathbb P^1 _x$ is a measurable subset such that $\nu_x(C_x ) >0$, then $$\label{eq:inversion limit} \lim _{t\rightarrow \infty} \frac{1}{\nu_x(C_x )} \int _{C_x } -\frac{1}{t} \log \left( \frac{ (h_t)^{-1} \nu_{\omega(t)} }{\nu_{\omega(0)}} (\omega(0) ) \right) d\nu_{\omega(0)} = A .$$ This follows from , Fubini’s theorem and the dominated convergence theorem, since for a generic $\omega$, $\text{length} (\omega{ \arrowvert_{[0,t]}})= O(t)$ so by the argument of the integral in is bounded independently of $t$. We now use the convexity of the function $-\log$ which by Jensen’s inequality implies that $$\frac{1}{\nu_x(C_x )} \int _{C_x } -\frac{1}{t} \log \left(
\frac{ (h_t)^{-1} \nu_{\omega(t)} }{\nu_{\omega(0)}}(\omega(0) )\right) d\nu_{\omega(0)} \geq
- \frac{1}{t} \log \left( \frac{\nu_{\omega(t)} (h_t (C_x) ) }{\nu_{\omega(0)} (C_x) } \right) ,$$ and so we deduce that for a.e. $x\in X$, as soon as $\nu_x(C_x) >0$, we have that $$\limsup _{t\rightarrow \infty} - \frac{1}{t} \log \nu_{\omega(t)} (h_t (C_x) ) \leq A.$$
Notice that this property makes no reference to the starting point in the fiber, so it can be stated as well for a.e. $x\in X$ and $W^X_x$ a.e. $\omega\in {\Omega}^X_x$
To finish the proof it remains to see that this statement holds for *every* $x$. For this, we first observe that if $C_x$ has positive measure, then for $W^X_x$-a.e. $\omega$, $\nu_{\omega(1)} ( h_1 (C_x)) >0$. Furthermore, the distribution of $\omega(1)$ is absolutely continuous, so that the previous estimates hold when $x$ is replaced by $\omega(1)$. The assertion then follows from the Markov property of Brownian motion.
Let us resume the proof of Proposition \[p:sup dimension\]. Fix ${\varepsilon}>0$, and put $\eta = \frac{4\chi+A}{4\chi^2} {\varepsilon}$. By Lemma \[l:technical step\], for $W^{\mathcal F}_{\mu}$-a.e. $\omega$, the convergence $\frac{1}{t} \log {\left\VertDh_t (y)\right\Vert}\underset{t\rightarrow\infty}\longrightarrow \lambda$, holds uniformly on compact subsets of $\mathbb P_{\omega(0)} ^1 \setminus {\left\{r(\omega)\right\}}$. So if we let $R = \frac{1}{2}d_{{\mathbb{P}^1}}(\omega(0), r(\omega))$, there exists $t_1 = t_1 (\omega, \varepsilon)$ such that if $ t\geq t_1$, $$h_t ( B(\omega(0), R ))\subset B( \omega(t), e^{(\lambda + \eta)t} ) .$$ On the other hand by the previous lemma, for $W_\mu^{\mathcal{F}}$ a.e. $\omega$, there exists $t_2 = t_2 (\omega, \varepsilon)$ such that for $t\geq t_2$, $$\nu_{\omega(t)} ( h_t( B(\omega(0), \eta ) ) \geq e^{ -(A +\eta) t} .$$ So we infer that for $t\geq \max(t_1, t_2)$, $$\label{eq:measures of images} \nu_{\omega(t)} (B( \omega(t), e^{(\lambda + \eta)t} ) ) \geq e^{ -(A +\eta) t} .$$ For every $t>0$ let $\Omega_t$ be the set of paths $\omega\in \Omega$ such that $\max (t_1,t_2) \leq t$. Clearly $W^{\mathcal{F}}_\mu(\Omega_t)\geq 1-{\varepsilon}$ for $t\geq t({\varepsilon})$. Setting $r = e^{ (\lambda + \eta) t} $, if $\omega\in {\Omega}_t$ and $x=\omega(t)$, for $t\geq t({\varepsilon})$ we have that $$\nu_{ {x} } (B ( {x} , r ) ) \geq r ^{-\frac{A + \eta}{\lambda + \eta}} = r^{\frac{A + \eta}{2\chi - \eta}} .$$
This finishes the proof since the image of $W^{\mathcal{F}}_{\mu}$ under $\omega \mapsto \omega(t)$ is the measure $\mu$, and $\eta$ was chosen so that $\frac{A + \eta}{2\chi - \eta}<\frac{A}{2\chi}+{\varepsilon}$.
An estimate similar to that of Proposition \[p:sup dimension\] holds in every fiber.
\[c:volume balls\] Let $x\in X$ and $\varepsilon >0$. There exists $r_{\varepsilon}>0$ such that if $0<r <r_{\varepsilon}$ then $$\nu_x \left( \tilde x \in \mathbb P _x^1 \ :\ \nu_x ( B_{{\mathbb{P}^1}}(\tilde x , r ) ) \geq r ^{\frac{A}{2\chi}+2\varepsilon} \right) \geq 1-\varepsilon.$$
This is due to the fact that the holonomy map $h_\gamma$ corresponding to a path $\gamma: [0,1]\rightarrow X$ of length $\ell$ is bilipschitz with constant depending only on $\ell$, and moreover it sends the measure $\nu_x$ to a measure absolutely continuous with respect to $\nu_y$, whose density is bounded from above and below by positive constants depending only on $\ell$. Now if $\ell$ is fixed, the proportion of points in a given fiber lying at leafwise distance $\ell$ from a point satisfying the conclusion of Proposition \[p:sup dimension\] tends to 1 when ${\varepsilon}{\rightarrow}0$, so we are done.
We are now ready to finish the proof of Theorem \[theo:dimension\]. Fix a real number $s> \frac{A}{2\chi}$ and ${\varepsilon}$ such that $0<2 {\varepsilon}< s- \frac{A}{2\chi}$. With $r_{\varepsilon}$ as in Corollary \[c:volume balls\], for every $r<r_{\varepsilon}$, consider the set $$E_{r,{\varepsilon}} = {\left\{\tilde x\ \in \mathbb{P}^1_x , \ \nu_x ( B(\tilde{x},r) \geq r ^{\frac{A}{2\chi} + 2\varepsilon}\right\}}.$$ From Corollary \[c:volume balls\] we know that $\nu_x (E_{r/5 ,{\varepsilon}})> 1-{\varepsilon}$. Furthermore, a classical covering argument gives an estimate of the $s$-dimensional Hausdorff measure of $E_{r/5,{\varepsilon}}$. Indeed by the Vitali covering lemma there exists a covering of $E_{r/5, {\varepsilon}}$ by balls $B_{{\mathbb{P}^1}}(\tilde x_i, r)$ centered on $E_{r/5, {\varepsilon}}$ and of radius $r$ such that the corresponding balls of radius $\frac{r}{5}$ are disjoint. This disjointness together with the measure estimate imply that this set of balls has cardinality at most $$N \leq {\left(\frac{r}{5}\right)}^{-{\left(\frac{A}{2\chi}+2{\varepsilon}\right)}}.$$ Therefore, $$\label{eq:hs}
\mathcal{H}_s(E_{r/5 ,{\varepsilon}}) \leq \sum_i (2r)^s \leq 2^s 5^{\frac{A}{2\chi}+2{\varepsilon}} r^{s- \frac{A}{2\chi}-2{\varepsilon}}.$$
We now set ${\varepsilon}_n =2^{-n}$, $r_n = r({\varepsilon}_n)/5$ , and put $F_k = \bigcap_{n\geq k} E_{r_n,{\varepsilon}_n}$. Since for every $n \geq k$, $F_k \subset E_{r_n, {\varepsilon}_n}$, from we infer that $\mathcal{H}_s(F_k) = 0$. On the other hand $\nu_x(F_k)\geq 1-{\left({\frac{1}{2}}\right)}^{k-1}$, so if we let $F = \bigcup_{k\geq 1} F_k$ we have that $\nu_x(F)=1$ and $\mathcal{H}_s(F) =0$, hence $\mathrm{dim}_H(\nu_x)\leq s$. Since $s> \frac{A}{2\chi}$ was arbitrary, we conclude that $\mathrm{dim}_H(\nu_x)\leq \frac{A}{2\chi}\leq {\frac{1}{2\chi}}$, as asserted.
In particular, it follows from Theorem \[theo:formula\] that $\mathrm{dim}_H(\nu_x)\leq 1$, and if equality holds then $\delta=0$ and $A = 1$. Lemma \[lem:zero degree\] below shows that if $\mathrm{dim}_H(\nu_x) = 1$, then $\sigma\in \overline{B(X)}$. Conversely, if $\sigma\in \overline{B(X)}$, it follows from Makarov’s celebrated theorem [@makarov] that the harmonic measures are supported by a set of dimension 1 (the measures $\nu_x$ coincide with the classical harmonic measure in this case, see the next lemma). This completes the proof of the theorem.
\[lem:zero degree\] Let $\sigma$ be a parabolic projective structure with $\deg(\sigma) = 0$. Let $(m_y)_{y\in {\mathsf{dev}}(\widetilde X)}$ be the usual harmonic measure of the open set ${\mathsf{dev}}(\widetilde X)\subset {{\mathbb{P}^1}}$. Then for every $x\in \widetilde X$, $\nu_x = m_{{\mathsf{dev}}(x)}$.
In addition the action $A$ equals $1$ if and only if ${\mathsf{dev}}$ is injective, that is, $\sigma\in \overline{B(X)}$ (see the discussion on the density theorem in §\[subs:Teichmuller\]).
The first part is proved by using the conformal invariance of Brownian motion. Indeed, any Brownian path $\eta:[0,\infty[$ (relative to the spherical metric on $\mathbb P^1$, say) starting at $y$ hits a.s. the boundary of ${\mathsf{dev}}(\widetilde{X})$ at a first moment $S>0$. We denote by $p= \eta (S)$. The distribution of $p$ is by definition the harmonic measure $m_y$. The path $\eta{ \arrowvert_{[0,S)}}$ can be lifted to a continuous path $\widetilde{\eta} : [0,S) \rightarrow \widetilde{X}$ starting at $\widetilde{\eta}(0)= x$, and satisfying ${\mathsf{dev}}\circ
\widetilde{\eta} =\eta$. Let $\omega :[0,T)\rightarrow \widetilde{X}$ be the reparametrization of $\widetilde{\eta}$ defined by $ \omega (t) =
\widetilde{\eta}(s) $, where $$\label{eq:conformal invariance} t = \int _0 ^s {\left\VertD {\mathsf{dev}}^{-1}(\eta(u)) \right\Vert}^2 du.$$ Here ${\mathsf{dev}}^{-1}$ is understood as the analytic continuation along $\eta$ of the inverse of ${\mathsf{dev}}$ defined at the neighborhood of $y$ and such that ${\mathsf{dev}}^{-1} (y)=x$. The conformal invariance can be stated in the following form: $\omega$ is a model for a Brownian path starting at $x$ for the Poincaré metric (see e.g. [@carne Section 1]). Since $\omega$ tends to infinity in $\widetilde{X}$ when $t$ tends to $T$, we see that $T=+ \infty$ a.s. Moreover, a.s. $\lim_{t\rightarrow +\infty} {\mathsf{dev}}(\omega(t)) = p$, which implies using Definition- Proposition \[defprop:harmonic measure\] that $m_y= \nu_x$.
Let now address the second part of the lemma. We first prove that $A < 1$ if ${\mathsf{dev}}$ is not injective. We need the concept of an extremal positive harmonic function on the universal covering of $\widetilde{X}$. Such a function is (by definition) the composition of a biholomorphism from $\widetilde{X}$ to $\mathbb H$ with the imaginary part function $\Im : \mathbb H \rightarrow (0,\infty)$. It will be important to notice that the subgroup of $\text{Aut}(\widetilde{X})$ that preserves an extremal positive function is abelian (this is the group of translations in the coordinate where the function is the imaginary part). The following statement is a consequence of the case of equality in the Schwarz-Pick lemma:
*A function $\varphi : \widetilde{X} \rightarrow (0,\infty)$ is an extremal positive harmonic function if and only if at some (and hence all) point $x\in \widetilde{X}$ one has ${\left\Vert\nabla \log \varphi (x)\right\Vert} = 1$.*
Now assume that ${\mathsf{dev}}$ is not injective. In such a situation, the covering group $\text{ker} ({\mathsf{hol}})$ is a non trivial normal subgroup of $\pi_1(X)$. Recall (item (ii) of Proposition \[p:harmonic measures\]) that the family of harmonic measures $\{ \nu_x \}_{x\in \widetilde{X}} $ satisfies the equivariance relation $\nu_{\gamma x} = {\mathsf{hol}}(\gamma) _* \nu_x$ for every $x\in \widetilde{X} $ and every $\gamma \in \pi_1(X)$. Hence, the density of the disintegration of $\widetilde{T}$ along the leaves is a function $\varphi : \widetilde{X}\times \mathbb P^1 \rightarrow (0,\infty)$ which belongs to $L^1_{loc} (\text{vol} \otimes \nu)
$ and is invariant under the group $\text{ker} ({\mathsf{hol}})$. This subgroup being non trivial and normal, its limit set as a subgroup of isometries of $\text{Aut}(\widetilde{X})$ for the Poincaré metric is the whole $\partial \widetilde{X}$. In particular, it contains non abelian free subgroups [@beardon]. As a consequence, the density $\varphi(\cdot, z)$ of the disintegration of $T$ cannot be extremal. In particular, ${\left\Vert\nabla_{\mathcal F} \log \varphi \right\Vert}<1$ a.s. This proves that $A < 1$, as required.
It remains to prove that if ${\mathsf{dev}}$ is injective, then $A = 1$. In this case, the holonomy representation is injective with image a discrete subgroup of $\text{PSL} (2,\mathbb C)$. In particular, using Remark \[r:measurable conjugacy\] we see that the foliated bundle $(M_\sigma,\mathcal F_\sigma, T_\sigma)$ is measurably conjugate to the bundle $(M_{\sigma_{\rm Fuchs}},\mathcal F_{\sigma_{\rm Fuchs}}, T_{\sigma_{\rm Fuchs}})$ where $\sigma_{\rm Fuchs}$ is the uniformizing structure on $X$. Hence $A =A_{{\rm Fuchs}}$. But the densities of the disintegration of $T_{\rm Fuchs}$ along the leaves are given by the Poisson kernel in the uniformization coordinates, in particular these are extremal positive harmonic functions. We conclude that $A = A_{\rm Fuchs} = 1$, and the proof of the lemma is complete.
\[rmk:makarov\] Let ${\Omega}_0 \subset {{\mathbb{P}^1}}$ be any component of the discontinuity set of a finitely generated Kleinian group $\Gamma$. Using Theorem \[theo:dimension\] we can recover the classical Jones-Wolff theorem [@jones; @wolff] that the dimension of the harmonic measure of $\partial {\Omega}_0$ is bounded by $1$, with strict inequality unless ${\Omega}$ is simply connected (see [@puz] for another dynamical proof of this fact). Indeed, from Lemma \[lem:zero degree\], it is enough to show there exists a Riemann surface $X$ of finite type and a parabolic projective structure on $X$ with zero degree such that ${\Omega}_0 = {\mathsf{dev}}(\widetilde X)$. This simply follows from the Ahlfors finiteness theorem: first take a finite index torsion free subgroup $\Gamma' \subset \Gamma$, and define $X:= \Gamma_0' \backslash {\Omega}_0$ where $\Gamma_0'\subset \Gamma'$ is the stabilizer of ${\Omega}_0$.
Applications to Teichmüller theory {#sec:teich}
==================================
Preliminaries {#subs:Teichmuller}
-------------
All this is well-known, but not so easy to locate in the literature when $X$ is non-compact.
Recall that $X$ is assumed to be a Riemann surface of finite type, that is biholomorphic to $\overline{X} \setminus P$ where $\overline{X}$ is compact and $P$ is a finite set of punctures. Introduce a projective structure $\overline{\sigma}$ on $\overline{X}$, which can always can be done e.g. by uniformization. For every projective structure $\sigma$ on $X$, consider the holomorphic quadratic differential on $X$ defined by $$q = \{ w , x \} dx^2$$ where $x$ and $w$ are projective coordinates for the projective structures $\overline{\sigma}$ and $\sigma$ respectively, and as usual $\{ w, x \}$ is the Schwarzian derivative. By the cocycle property of the Schwarzian, we infer that $\{ w,x\}dx^2 = \{ z, x\} dx^2 + \{ w,z \} dz^2 $. Remembering that $\{ w,x\}$ vanishes if $w$ is a Moebius transformation, we see that the differential $q$ is well-defined, that is does not depend on the chosen coordinates $x$ and $w$. Moreover, a result due to Fuchs and Schwarz shows that a projective structure on $X$ is parabolic if and only if the Laurent series expansion of $q$ at the neighborhood of every point of $P$ takes the form $q(x) = \big(\frac{1}{2x^2} + \text{l.o.t.}\big)dx^2$, see [@unif Théorème 10.1.1, p. 291]. Hence, the space of parabolic projective structures on $X$ is an complex affine space directed by the set of meromorphic quadratic differentials having poles on $P$ of order at most $1$. This space is the set of sections of the line bundle $L = 2K + O(P)$, where $K$ is the canonical divisor of $\overline{X}$. By Riemann-Roch $$h^0 (\overline{X}, L) - h^0 (\overline{X}, K - L ) = \mathrm{deg} (L) + 1 - g .$$ Since $K-L= -K-O(P)$ has no non trivial sections, and that $\mathrm{deg} (L) = 4g - 4 + |P|$, we deduce $$h^0 (\overline{X}, L) = 3g-3 + |P| .$$ Thus the set of parabolic projective structures on $X$ is a complex affine space of dimension $3g- 3 + n$, where $n$ is the number of punctures. Observe that for the once punctured torus or the fourth punctured sphere, the dimension equals $1$.
We denote by $ T_{g,n}$ the Teichmüller space of equivalence classes of marked Riemann surfaces biholomorphic to a compact Riemann surface of genus $g$ punctured at $n$ distinct points. Here a marking of the Riemann surface $Y$ will refer to the data of a universal covering $\widetilde{Y} \rightarrow Y$ together with an identification of the covering group $\pi_1(Y)$ of this covering with $\pi_1(X)$. Two marked surfaces are considered as equivalent if there exists an equivariant holomorphic diffeomorphism between the universal covers.
Let $Y \in \mathcal T_{g,n}$. Denote by $c(Y)$ the complex conjugation of $Y$, keeping the marking fixed. The Bers simultaneous uniformization theorem [@Bers Theorem 1] asserts that there exists a faithful discrete representation $\rho _{X,Y} : \pi_1 (X) \rightarrow \text{PSL} (2,\mathbb C)$, uniquely defined up to conjugation, such that the Riemann sphere admits a $\rho$-invariant partition of the form $$\label{eq:quasi-Fuchsian} \mathbb P^1 = D_X\cup \Lambda \cup D_Y,$$ where $D_X$ and $D_Y$ are two simply connected domains, $\Lambda$ is a topological circle, and such that the marked Riemann surfaces $\rho (\pi_1(X)) \backslash D_X$ and $\rho (\pi_1(X)) \backslash D_Y$ are respectively equivalent to $X$ and $c(Y)$. A representation with an invariant decomposition such as is called quasi-Fuchsian. Thus an element $Y\in \mathcal T_{g,n}$ produces a parabolic $\mathbb P^1$-structure $b(Y)\in P(X)$ on $X$, defined as the $\rho$-equivariant identification between $\widetilde{X} $ and $D_X$.
It turns out that the map $Y\in T_{g,n} \mapsto b(Y) \in P(X)$ is a holomorphic embedding onto a bounded open subset $B(X)\subset P(X)$, known as the [*Bers embedding*]{} (or [*Bers slice*]{}) of $T_{g,n}$. The holomorphicity of $b$ follows from the holomorphic dependence of the solution of the Beltrami equation with respect to parameters. The boundedness of $B(X)$ follows from Nehari’s estimate for the Schwarzian of univalent meromorphic functions defined on the hyperbolic disc, and its openness from the so-called Ahlfors-Weill extension lemma (see e.g. [@gardiner; @lakic]).
Due to deep recent advances in Kleinian group and 3-manifold theory, there is a now good understanding of the structure of $\overline{B(X)}$. To be precise, the [*density theorem*]{} (formerly known as the Bers density conjecture), specialized to our context, asserts that $\sigma\in \overline B(X)$ if and only if ${\mathsf{dev}}_\sigma$ is injective. This means in particular that the image of ${\mathsf{dev}}$ is a simply connected component of the discontinuity set of ${\mathsf{hol}}_\sigma(\pi_1(X))$, which uniformizes $X$. An equivalent formulation is that $\sigma\in \overline{B(X)}$ if and only if $\deg(\sigma)=0$ and ${\mathsf{hol}}_\sigma $ is faithful. When $X$ is compact, this was explicitly proved by Bromberg [@bromberg; @annals]. In the general case, this statement is generally accepted by the experts as being a consequence of the ending lamination theorem of Minsky [@minsky] and Brock, Canary and Minsky [@bcm] (see Ohshika [@ohshika] and Namazi-Souto [@namazi; @souto] for the derivation of the density theorem from the ending lamination theorem in the whole character variety). It seems, however, that no detailed proof of this fact has appeared yet.
Holomorphic convexity of Bers slices
------------------------------------
Here we prove Theorem \[theo:convex\]. It is known that every component of the interior of a polynomially convex set is polynomially convex (i.e. a Runge domain) (see e.g. [@fs2 Prop. 2.7]), but the converse is false (this fails e.g. for $U = D(0, 2)\setminus D(1,1) \subset {\mathbb{C}}$). In particular the second statement of the theorem (the polynomial convexity of $B(X)$) follows from the first (the polynomial convexity of $\overline{B(X)}$). Actually, one may derive the polynomial convexity of $B(X)$ from a simple, direct argument. Indeed, consider in $X$ the set ${\left\{\deg = 0\right\}}$ of projective structures with vanishing degree. Equivalently, by Proposition \[p:vanishing\] such a structure is of quotient type. It was shown in [@kra; @maskit] that ${\left\{\deg = 0\right\}}$ is a compact subset of $P(X)$. Since in ${\mathbb{C}}^n$ convexity with respect to polynomials and psh functions coincide [@stout Thm 1.3.11], we infer that ${\left\{\deg=0\right\}}$ is polynomially convex. Now it is a result due to Shiga and Tanigawa [@shiga; @tanigawa] and Matsuzaki [@matsuzaki] that the interior in $P(X)$ of the set of projective structures with discrete holonomy is the set of projective structures with quasifuchsian holonomy. Thus $\operatorname{Int}{\left\{\deg = 0\right\}} = B(X)$, and the polynomial convexity of $B(X)$ follows.
We now turn to the polynomial convexity of $\overline{B(X)}$. A connected component of a polynomially convex set is polynomially convex [@stout Cor. 1.5.5], so it is enough to show that $\overline{B(X)}$ is a connected component of ${\left\{\deg = 0\right\}}$. This will be based on the following amusing lemma.
\[lem:connex\] Let $(\rho_{{\lambda}})_{{\lambda}\in {\Lambda}}$ be a holomorphic family of representations of a finitely generated group $G$ into ${\mathrm{PSL}(2,\mathbb C)}$, parameterized by some complex manifold ${\Lambda}$. If $K\subset {\Lambda}$ is a compact connected set of discrete and non-elementary representations then $\ker (\rho_{\lambda})$ is constant over $K$. In particular if $K$ contains a faithful representation, then all representations in $K$ are faithful.
Admitting this result for the moment, let us finish the proof. Let $K$ be the connected component of ${\left\{\deg = 0\right\}}$ containing $\overline{B(X)}$. By Proposition \[p:vanishing\] for every $\sigma\in K$, $\mathsf{hol}_\sigma$ is discrete. Applying Lemma \[lem:connex\], we infer that all representations in $K$ are faithful. By the density theorem (see the end of §\[subs:Teichmuller\]), the inclusion $K\subset \overline{B(X)}$ holds, hence $K = \overline{B(X)}$, and the proof is complete.
Without loss of generality we may assume that ${\Lambda}$ is connected. Consider the set of subgroups $\ker (\rho_{\lambda})$ as ${\lambda}$ ranges in ${\Lambda}$. Our first claim is that this set is at most countable. The argument is based on basic finiteness (Noetherian) properties in analytic geometry. Let ${{\lambda_0}}\in {\Lambda}$ and put $K_0 = \ker (\rho_{{\lambda_0}})$. Define $$Z'(K_0) = {\left\{{\lambda}\in {\Lambda},\ \forall g\in K_0, \rho_{\lambda}(g) = \mathrm{id}\right\}},$$ and let $Z(K_0)$ be the component of $Z'(K_0)$ containing ${{\lambda_0}}$. If ${\lambda}\in Z(K_0)$, $\ker(\rho_{\lambda}) \supset K_0$ nevertheless equality needn’t hold. On the other hand we observe that if ${\lambda}$ is a generic point in $Z(K_0)$, that is, chosen outside a countable family of proper analytic subvarieties, then $\ker(\rho_{\lambda}) = K_0$. Indeed for every $g\in G\setminus K_0$, the set ${\left\{{\lambda}\in Z(K_0), \ \rho_{\lambda}(g) = \mathrm{id}\right\}}$ is a proper subvariety of $Z(K_0)$, since it does not contain ${{\lambda_0}}$.
Conversely, a similar argument shows that if $V\subset {\Lambda}$ is any irreducible variety, the subgroup $$K(V) = {\left\{g\in G, \forall {\lambda}\in V, \rho_{\lambda}(g) =\mathrm{id}\right\}}$$ is the kernel of generic representations in $V$.
So if $K_0$ is as above, $Z'(K_0)$ has at most countably many irreducible components, each of which associated with a generic kernel (for $Z(K_0)$, this is precisely $K_0$). Now we observe that locally $Z'(K_0)$ is defined by finitely many equations, that is there exists a finite number of elements $g_i\in G$, $i=1\ldots N$, such that for ${\Lambda}'\Subset {\Lambda}$, $$Z'(K_0) \cap {\Lambda}' = {\left\{{\lambda}\in {\Lambda}',\ \forall i =1\ldots N, \rho_{\lambda}(g_i) = \mathrm{id}\right\}}.$$ This leaves only countably many possibilities for the generic kernels, and our claim is proved.
Under the assumptions of the lemma, label all kernels of representations in $K$ as $(H_i)_{i\in {\mathbb{N}}}$ and write accordingly $K$ as a disjoint union $K = \bigcup K_i$, where $K = {\left\{{\lambda},\ \ker (\rho_{\lambda})= H_i\right\}}$. The next claim is that for every $i$, $K_i$ is closed. For this we use the precise version of the Chuckrow (Margulis-Zassenhaus-Jorgensen) theorem stated in [@kapovich Thm 8.4 p.170]: if $\rho_p$ is a sequence of discrete faithful representations of some non-radical group $\Gamma$ into ${\mathrm{PSL}(2,\mathbb C)}$, algebraically converging to some representation $\rho$ of $\Gamma$, then $\rho$ is also discrete and faithful. Recall that a group $\Gamma$ is said non-radical if it does not admit infinite normal nilpotent subgroups. A non-elementary subgroup of ${\mathrm{PSL}(2,\mathbb C)}$ contains rank $2$ non abelian free subgroups and in particular is non-radical.
Let now $({\lambda}_p)\in K_i^{\mathbb{N}}$ be sequence converging to some ${\lambda}\in K$. For every $p$, $\rho_{{\lambda}_p}$ is a discrete faithful non-elementary representation of $G/H_i$, which is therefore non-radical. It is clear that $H_i\subset \ker(\rho_{\lambda})$ so $\rho_{\lambda}$ can be viewed as a representation of $G/H_i$. Hence by Chuckrow’s theorem $\ker(\rho_{\lambda}) = H_i$, and we conclude that $K_i$ is closed.
We have thus written $K$ as an at most countable union of disjoint closed sets $K_i$. A (not so well-known!) theorem of Sierpiński [@sierpinski] asserts that such a decomposition must be trivial. The result is proved.
If $\dim(P(X)) =1$, it is not necessary to use the density theorem. Indeed in dimension 1, polynomial convexity simply means that $K^c$ has no bounded component (there is no such simple topological characterization in higher dimension). What our argument says is that $\overline{B(X)}$ is contained in a polynomially convex set $K$ with $\mathring{K} = B(X)$. This directly implies that $\overline{B(X)}^c$ has no bounded components (this is left as an exercise to the reader).
Exterior powers of the bifurcation current
------------------------------------------
The argument is based on the fact that an isolated minimum of the continuous psh function $\delta$ must belong to $\operatorname{Supp}(dd^c\delta)^{3g-3}$. This is a consequence of the so-called [*comparison principle*]{} for psh functions [@bedford; @taylor Thm A]. A similar idea appears in the work of Bassanelli and Berteloot (see [@basber Prop. 6.3]).
It is a theorem due to Hejhal [@hejhal; @schottky Thm 6] that covering projective structures with Schottky holonomy are isolated points of ${\left\{\delta=0\right\}}$. Therefore, for such a $\sigma_0$, we infer that $\sigma_0 \in \operatorname{Supp}({T_{\mathrm{bif}}}^{3g-3})$. Finally, we use a result due to Otal [@otal] (see also Ito [@ito]): when $X$ is compact, ${\partial}B(X)$ is contained in the accumulation set of projective structures with degree 0 and Schottky holonomy. We conclude that ${\partial}B(X)\subset \operatorname{Supp}({T_{\mathrm{bif}}}^{3g-3})$.
Corollary \[coro:deterministic\] is an immediate consequence of the following equidistribution result in the spirit of [@Bers1 Thm C]. If $\gamma$ is a closed geodesic on $X$ we let $$Z(\gamma, t) = {\left\{\sigma\in P(X), \ \operatorname{tr}^2(\mathsf{hol}_\sigma) = t\right\}}$$ (notice that since $\mathsf{hol}_\sigma$ is well-defined up to conjugacy, so it makes sense to speak of its trace). Fix a sequence $(r_n)_{n\geq 1}$ such that for every $c>0$, the series $\sum e^{-cr_n}$ converges. The notion of a random sequence of geodesics of length at most $r_n$ was discussed at length in [@Bers1]. We say that a holomorphic family of representations $(\rho_{\lambda})_{{\lambda}\in {\Lambda}}$ is [*reduced*]{} if the associated mapping ${\Lambda}{\rightarrow}\mathcal X(\pi_1(X), {\mathrm{PSL}(2,\mathbb C)})$ to the character variety has discrete fibers. Notice that since $\mathsf{hol}:P(X){\rightarrow}\mathcal X(\pi_1(X), {\mathrm{PSL}(2,\mathbb C)})$ is injective, this property is satisfied in the context of Corollary \[coro:deterministic\].
Let $X$ be a hyperbolic Riemann surface of finite type and $(\rho_{{\lambda}})_{{\lambda}\in {\Lambda}}$ be a reduced holomorphic family of non-elementary representations of $\pi_1(X)$, with $\dim ({\Lambda})\geq k$. Let $(r_n)$ be as above, and fix $t\in {\mathbb{C}}$. For $i= 1,\ldots , k$ fix a sequence $(\gamma_n^i)$ of independent random closed geodesics of length at most $r_n$. Then almost surely, $$\label{eq:tbifk}
\lim_{n_1{\rightarrow}\infty} \cdots \lim_{n_k{\rightarrow}\infty}
{\frac{1}{4^k\prod_{i=1}^k \mathrm{length}(\gamma_{n_i}^i)}} \big[Z(\gamma_{n_1}^1, t)\cap \cdots \cap Z(\gamma_{n_k}^k, t)\big]
= T_{\rm bif}^k,$$
Note that in the intersections are counted with multiplicity (i.e. in the sense of holomorphic chains, see [@chirka Chap. 12]).
The proof is similar to that of [@preper Thm 6.16]. We argue by induction on $k$. For $k=1$ this is [@Bers1 Thm C]. Now assume that the result has been proved for $k$. Since $T_{\rm bif}$ has continuous potential, it follows from that $${\frac{1}{4^k\prod_{i=1}^k \mathrm{length}(\gamma_{n_i}^i)}} \big[Z(\gamma_{n_1}^1, t)\cap \cdots \cap Z(\gamma_{n_k}^k, t)\big] \wedge
{T_{\mathrm{bif}}}$$ converges to ${T_{\mathrm{bif}}}^{k+1}$ as $n_k, \ldots , n_1{\rightarrow}\infty$ successively. Now, when $n_1, \ldots , n_k$ are large and fixed, we apply [@Bers1 Thm C] to the family $(\rho_{{\lambda}})_{{\lambda}\in Z(\gamma_{n_1}^1, t)\cap \cdots \cap Z(\gamma_{n_k}^k, t)}$ (the reducedness assumption is used here to ensure that this family is non constant) to get that $$\begin{aligned}
\lim_{n_{k+1}{\rightarrow}\infty}
{\frac{1}{4^{k+1}\prod_{i=1}^{k+1} \mathrm{length}(\gamma_{n_i}^i)}} & \big[Z(\gamma_{n_1}^1, t)\cap \cdots \cap Z(\gamma_{n_{k+1}}^{k+1}, t)\big] \\ &= {\frac{1}{4^k\prod_{i=1}^k \mathrm{length}(\gamma_{n_i}^i)}} \big[Z(\gamma_{n_1}^1, t)\cap \cdots \cap Z(\gamma_{n_k}^k, t)\big] \wedge {T_{\mathrm{bif}}}\end{aligned}$$ and we are done.
Branched projective stuctures {#sec:appendix}
=============================
A branched $\mathbb P^1$-structure on $X$ is by definition an equivalence class of development-holonomy pairs $({\mathsf{dev}}, {\mathsf{hol}})$, where ${\mathsf{hol}}$ is a representation of $\pi_1(X) $ with values in $\text{PSL}(2,\mathbb C)$ and ${\mathsf{dev}}: \widetilde{X} \rightarrow \mathbb P^1$ is a ${\mathsf{hol}}$-equivariant (non constant) meromorphic map. Two development-holonomy pairs are considered as equivalent if they are of the form $(\mathsf{dev},\mathsf{hol})$ and $(A\circ
\mathsf{dev},A \circ \mathsf{hol}\circ A^{-1})$ for some $A\in {\mathrm{PSL}(2,\mathbb C)}$. Thus the only difference with the classical (unbranched) case is that the developing maps are allowed to have critical points. These points are organized as a finite number of orbits under $\pi_1(X)$, and their projections in $X$ are called the branched points. If $X$ is a punctured Riemann surface, we can define a notion of branched parabolic $\mathbb P^1$-structure exactly as before, by specifying it to be induced by $\log z$ near the cusps.
Examples of branched $\mathbb P^1$-structures come from conformal metrics with constant curvature $-1,0,1$ on $X$ and conical angles multiple of $2\pi$. In particular, a non constant meromorphic map from $X$ to $\mathbb P^1$ defines a branched $\mathbb P^1$-structure with trivial holonomy. Quadratic differentials are other kind of examples associated with a flat metric. We refer more generally to [@veech]. Those examples of non negative curvature have elementary holonomies. Nevertheless, the holonomy of a branched $\mathbb P^1$-structure induced by a conformal metric of curvature $-1$ and conical angle multiple of $2\pi$ is always non elementary. We refer to [@troyanov] for the construction of many examples.
When $X$ is compact, Gallo, Kapovich and Marden [@gkm] showed that if $\rho: \pi_1(X)\rightarrow \text{PSL}(2,\mathbb C)$ is a non elementary representation which does not lift to $\text{SL}(2,\mathbb C)$, then $\rho$ is the holonomy of a branched $\mathbb P^1$- structure with exactly one branch point of angle $4\pi$ (for a certain Riemann surface structure on $X$ which depends on $\rho$). On the other hand $\rho$ is not the holonomy of a unbranched $\mathbb P^1$-structure.
Some of our results extend [*mutatis mutandis*]{} to branched projective structures with non elementary holonomy. For instance, the degree of a branched $\mathbb P^1$-structure is defined exactly as in Definition-Proposition \[defprop:degree\], the proof being identical to the unbranched case. The Lyapunov exponent depends only on the Riemann surface structure and on the holonomy representation, hence it has already been defined in our previous work [@Bers1]. Finally, Hussenot’s definition of the harmonic measures was actually introduced in the context of branched $\mathbb P^1$-structures with non elementary holonomy.
In this appendix we indicate how our formula relating the Lyapunov exponent to the degree needs to be modified in the branched case.
\[theo:formula branched\] Let $\sigma$ be a parabolic branched $\mathbb{P}^1$ structure on a hyperbolic Riemann surface $X$ of finite type. Let $k$ denote the number of branched points, counted with multiplicity. Then, with notation as in Theorem \[theo:formula\], the following formula holds: $$ \displaystyle \chi (\sigma) = \frac{1}{2} + 2\pi\delta(\sigma) - \frac{k}{{\left\vert\mathrm{eu(X)}\right\vert}} = \frac12+ \frac{\deg(\sigma) - k }{{\left\vert\mathrm{eu(X)}\right\vert}}$$
\[Sketch of proof\] Introduce as in the unbranched case the flat bundle $(\overline{M_\sigma} , \overline{\mathcal F_\sigma})$, the holomorphic section $\overline{s}$ (the compactification of the graph of the developing map at the level of the universal cover) and the normalized harmonic current $\overline{T}$ giving mass $1$ to the generic fibers. Proposition \[p:weaker\] holds without modification, as well as the computation of the index $I$ made in §\[sec:proof\] (which is local near the punctures), so we infer that $$\chi (\sigma) = \displaystyle \frac{1}{2{\left\vert\mathrm{eu}(X)\right\vert} } ( N_{\overline{\mathcal F}} \cdot \overline{T} + \#P).$$
Now if $\mathcal G$ is a singular holomorphic foliation on a complex surface, and $C $ is a non singular compact holomorphic curve not everywhere tangent to $\mathcal G$, and not intersecting the singular set of $\mathcal G$, we have $$N_{\mathcal G} \cdot C = \mathrm{eu} (C) + | \mathrm{tang} (\mathcal G, C) |,$$ where the tangency points are counted with multiplicities (see [@brunella] for details). Hence formula has to be replaced by $$\label{eq:npoint2}
N_{\overline{\mathcal{F} } }\cdot \overline s = \mathrm{eu} (\overline s) + |\mathrm{tang} (\overline{\mathcal F}, \overline{s} ) |= \mathrm{eu} (\overline X) + k \text{ and }
N_{\overline{\mathcal{F}}}\cdot f = \mathrm{eu}({{\mathbb{P}^1}}) = 2.$$ We also have $$\overline{s}^2 = \mathrm{eu} (\overline{s}) + |\mathrm{tang} (\overline{\mathcal F}, \overline{s} ) | = \mathrm{eu} (\overline{s}) + k.$$ So we infer that $$[ N_{\overline{\mathcal F}} ] = 2 [\overline{s}] - (\mathrm{eu}(\overline{X} ) + k) [ f ] ,$$ and we conclude as in the proof of Theorem \[theo:formula\].
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abstract: 'Magnetised turbulence is ubiquitous in astrophysical systems, where it notoriously spans a broad range of spatial scales. Phenomenological theories of MHD turbulence describe the self-similar dynamics of turbulent fluctuations in the inertial range of scales. Numerical simulations serve to guide and test these theories. However, the computational power that is currently available restricts the simulations to Reynolds numbers that are significantly smaller than those in astrophysical settings. In order to increase computational efficiency and, therefore, probe a larger range of scales, one often takes into account the fundamental anisotropy of field-guided MHD turbulence, with gradients being much slower in the field-parallel direction. The simulations are then optimised by employing the reduced MHD equations and relaxing the field-parallel numerical resolution. In this work we explore a different possibility. We propose that there exist certain quantities that are remarkably stable with respect to the Reynolds number. As an illustration, we study the alignment angle between the magnetic and velocity fluctuations in MHD turbulence, measured as the ratio of two specially constructed structure functions. We find that the scaling of this ratio can be extended surprisingly well into the regime of relatively low Reynolds number. However, the extended scaling becomes easily spoiled when the dissipation range in the simulations is under-resolved. Thus, taking the numerical optimisation methods too far can lead to spurious numerical effects and erroneous representation of the physics of MHD turbulence, which in turn can affect our ability to correctly identify the physical mechanisms that are operating astrophysical systems.'
author:
- 'Joanne Mason$^1$, Jean Carlos Perez$^{2,3}$, Fausto Cattaneo$^1$, Stanislav Boldyrev$^2$'
title: Extended Scaling Laws in Numerical Simulations of MHD Turbulence
---
Introduction
============
Magnetised turbulence pervades the universe. It is likely to play an important role in the transport of energy, momentum and charged particles in a diverse range of astrophysical plasmas. It is studied with regards to its influence on the generation of magnetic fields in stellar and planetary interiors, small-scale structure and heating of stellar winds, the transport of angular momentum in accretion discs, gravitational collapse and star formation in molecular clouds, the propagation and acceleration of cosmic rays, and interstellar scintillation [e.g., @biskamp_03; @kulsrud_04; @mckee_o07; @goldstein_rm95; @brandenburg_n11; @schekochihin_c07]. The effects of magnetised turbulence need to be taken into account when analysing astrophysical observations and also when modelling astrophysical processes.
The simplest theoretical framework that describes magnetised plasma turbulence is that of incompressible magnetohydrodynamics (MHD), $$\label{eq:mhd-elsasser}
{\left(}\frac{\partial}{\partial t}\mp{\mathbf{V}}_A\cdot\nabla{\right)}{\mathbf{z}}^\pm+\left({\mathbf{z}}^\mp\cdot\nabla\right){\mathbf{z}}^\pm = -\nabla P + \nu \nabla^2 {\mathbf{z}}^{\pm}+{\mathbf{f}}^\pm,$$ $$\label{eq:div}
\nabla \cdot {\bf z}^{\pm}=0,$$ where the Elsässer variables are defined as ${\mathbf{z}}^\pm={\mathbf{v}}\pm{\mathbf{b}}$, ${\mathbf{v}}$ is the fluctuating plasma velocity, ${\mathbf{b}}$ is the fluctuating magnetic field normalized by $\sqrt{4 \pi \rho_0}$, ${\bf
V}_A={\bf B}_0/\sqrt{4\pi \rho_0}$ is the Alfvén velocity based upon the uniform background magnetic field ${{\mathbf{B}}_0}$, $P=(p/\rho_0+b^2/2)$, $p$ is the plasma pressure, $\rho_0$ is the background plasma density, ${\mathbf{f}}^\pm$ represents forces that drive the turbulence at large scales and for simplicity we have taken the case in which the fluid viscosity $\nu$ is equal to the magnetic resistivity. Energy is transferred to smaller scales by the nonlinear interactions of oppositely propagating Alfvén wavepackets [@kraichnan_65]. This can be inferred directly from equation (\[eq:mhd-elsasser\]) by noting that in the absence of forcing and dissipation, if ${\bf z}^\mp({\bf x},t)\equiv 0$ then any function ${\bf z}^\pm({\bf x},t)=F^\pm({\bf
x}\pm {\bf V}_A t)$ is an exact nonlinear solution that propagates parallel and anti-parallel to ${\mathbf{B_0}}$ with the Alfvén speed. The efficiency of the nonlinear interactions splits MHD turbulence into two regimes. The regime in which the linear terms dominate over the nonlinear terms is known as ‘weak’ MHD turbulence, otherwise the turbulence is ‘strong’. In fact, it has been demonstrated both analytically and numerically that the MHD energy cascade occurs predominantly in the plane perpendicular to the guiding magnetic field. This ensures that even if the turbulence is weak at large scales it encounters the strong regime as the cascade proceeds to smaller scales. MHD turbulence in astrophysical systems is therefore typically strong.
For strong MHD turbulence, @goldreich_s95 argued that the linear and nonlinear terms in equations (\[eq:mhd-elsasser\]) should be approximately balanced at all scales, known as the critical balance condition. Consequently, [@goldreich_s95] postulated that the wave packets get progressively elongated in the direction of the guide field as their scale decreases (with the field-parallel lengthscale $l$ and field-perpendicular scale $\lambda$ related by $l\propto \lambda^{2/3}$) and that the field-perpendicular energy spectrum takes the Kolmogorov form $E_{GS}(k_\perp)\propto k_\perp^{-5/3}$.
Recent high resolution direct numerical simulations with a strong guide field ($v_A\geq
5v_{rms}$) do indeed verify the strong anisotropy of the turbulent fluctuations, however, the field-perpendicular energy spectrum appears to be closer to $E(k_\perp)\propto k_\perp^{-3/2}$ [e.g., @maron_g01; @muller_g05; @mason_cb08; @perez_b09; @perez_b10]. A resolution to this contradiction was proposed in [@boldyrev_06]. Therein it was suggested that in addition to the elongation of the eddies in the direction of the guiding field, the fluctuating velocity and magnetic fields at a scale $\lambda\sim 1/k_{\perp}$ are aligned within a small scale-dependent angle in the field perpendicular plane, $\theta_\lambda
\propto \lambda ^{1/4}$. In this model the wavepackets are three-dimensionally anisotropic. Scale-dependent dynamic alignment reduces the strength of the nonlinear interactions and leads to the field-perpendicular energy spectrum $E(k_{\perp}) \propto k_{\perp}^{-3/2}$.
Although the two spectral exponents $-5/3$ and $-3/2$ are close together in numerical value, the physics of the energy cascade in each model is different. The difference between the two exponents is especially important for inferring the behaviour of processes in astrophysical systems with extended inertial intervals. For example, the two exponents can lead to noticeably different predictions for the rate of turbulent heating in coronal holes and the solar wind [e.g., @chandran_etal10; @chandran10; @podesta_b10]. Thus, there is much interest in accurately determining the spectral slope from numerical simulations. Unfortunately, the Reynolds numbers that are currently accessible by most direct numerical simulations do not exceed a few thousand, which complicates the precise identification of scaling exponents. Techniques for careful optimisation of the numerical setup and alternative ways of differentiating between the competing theories are therefore much sought after.
Maximising the extent of the inertial range is often achieved by implementing physically motivated simplifying assumptions. For example, since the turbulent cascade proceeds predominantly in the field-perpendicular plane it is thought that the shear-Alfvén waves control the dynamics while the pseudo-Alfvén waves play a passive role (see, e.g., [@maron_g01]). If one neglects the pseudo-Alfvén waves (i.e. removes the fluctuations parallel to the strong guide field) one obtains a system that is equivalent to the reduced MHD system (RMHD) that was originally derived in the context of fusion devices by [@kadomtsev_p74] and [@strauss_76] (see also [@biskamp_03; @oughton_dm04]). Incompressibility then enables the system to be further reduced to a set of two scalar equations for the Elsässer potentials, resulting in a saving of approximately a factor of two in computational costs. Further computational savings can be made by making use of the fact that the wavepackets are elongated. Hence variations in the field-parallel direction are slower than in the field-perpendicular plane and a reduction in the field-parallel resolution would seem possible. Indeed, this is widely used as an optimisation tool in numerical simulations of the inertial range of field-guided MHD turbulence [e.g., @maron_g01; @mason_cb08; @grappin_m10; @perez_b10]. The accumulated computational savings can then be re-invested in reaching larger Reynolds numbers for the field-perpendicular dynamics.
Additionally, it is advantageous to seek other ways of probing the universal scaling of MHD turbulence. In this work we point out a rather powerful method, which is based on the fact that there may exist certain quantities in MHD turbulence that exhibit very good scaling laws even for turbulence with relatively low Reynolds numbers. The situation here is reminiscent of the well known phenomenon of extended self-similarity in hydrodynamic turbulence [@benzi_etal93]. We propose that one such “stable” object is the alignment angle between the velocity and magnetic fluctuations, which we measure as the ratio of two specially constructed structure functions. This ratio has been recently measured in numerical simulations in an attempt to differentiate among various theoretical predictions ([@beresnyak_l06; @mason_cb06]). Also, it has recently been shown by [@podesta_etal09] that the same measurement is accessible through direct observations of solar wind turbulence. Scale-dependent alignment therefore has practical value: its measurement may provide an additional way of extracting information about the physics of the turbulent cascade from astrophysical observations. In the present work we conduct a series of numerical simulations with varying resolutions and Reynolds numbers. We find that as long as the simulations are well resolved, the alignment angle exhibits a universal scaling behavior that is virtually independent of the Reynolds number of the turbulence. Moreover, we find that the [*length*]{} of scaling range for this quantity extends to the smallest resolved scale, independently of the Reynolds number. This means that although the dissipation spoils the power-law scaling behaviour of each of the structure functions, the dissipation effects cancel when the ratio of the two functions is computed and the universal inertial-range scaling extends deep in the dissipation region.
The described method allows the inference of valuable scaling laws from numerical simulations, experiments, or observations of MHD turbulence with limited Reynolds number. However, one can ask how well the extended-scaling method can be combined with the previously mentioned optimisation methods relying on the reduced MHD equations and a decreased parallel resolution. We check that reduced MHD does not alter the result. However, when the dissipation region becomes under-resolved (as can happen, for example, when the field-parallel resolution is decreased), the extended scaling of the alignment angle deteriorates significantly. Thus the optimisation technique that works well for viewing the inertial range of the energy spectra should not be used in conjunction with the extended-scaling measurements that probe deep into the dissipation region. The remainder of this paper will report the findings of a series of numerical measurements of the alignment angle in simulations with different Reynolds numbers and different field-parallel resolutions in both the MHD and RMHD regimes. The aim is to address the need to find an optimal numerical setting for studying strong MHD turbulence and to raise caution with regards to the effects that implementing simplifying assumptions in the numerics can have on the solution and its physical interpretation.
Numerical results
=================
We simulate driven incompressible magnetohydrodynamic turbulence in the presence of a strong uniform background magnetic field, ${\mathbf{B_0}}=B_0{\mathbf{\hat e_z}}$. The MHD code solves equations (\[eq:mhd-elsasser\],\[eq:div\]) on a periodic, rectangular domain with aspect ratio $L_{\perp}^2 \times L_\|$, where the subscripts denote the directions perpendicular and parallel to ${\mathbf{B_0}}$ and we take $L_{\perp}=2\pi$. A fully dealiased 3D pseudospectral algorithm is used to perform the spatial discretisation on a grid with a resolution of $N_{\perp}^2\times N_\|$ mesh points. The RMHD code solves the reduced MHD counterpart to equations (\[eq:mhd-elsasser\],\[eq:div\]) in which ${\mathbf{z}}^\pm=(z_x^\pm, z_y^{\pm},0)$ (see [@perez_b08]).
The domain is elongated in the direction of the guide field in order to accommodate the elongated wavepackets and to enable us to drive the turbulence in the strong regime while maintaining an inertial range that is as extended as possible (see [@perez_b10]). The random forces are applied in Fourier space at wavenumbers $2\pi/L_{\perp} \leq k_{\perp} \leq 2 (2\pi/L_{\perp})$, $(2\pi/L_\|) \leq
k_\|\leq (2\pi/L_\|)n_\|$, where we shall take $n_\|=1$ or $n_\|=2$. The forces have no component along $z$ and are solenoidal in the $xy$-plane. All of the Fourier coefficients outside the above range of wavenumbers are zero and inside that range are Gaussian random numbers with amplitudes chosen so that $v_{rms}\sim 1$. The individual random values are refreshed independently on average every $\tau=n_{\tau} L_{\perp}/2\pi v_{rms}$, i.e. the force is updated approximately $1/n_{\tau}$ times per turnover of the large-scale eddies. The variances $\sigma_{\pm}^2=\langle |{\mathbf{f}}^{\pm} |^2\rangle$ control the average rates of energy injection into the $z^+$ and $z^-$ fields. The results reported in this paper are for the balanced case $\sigma_+\approx \sigma_-$. In all of the simulations performed in this work we will set the background magnetic field $B_0=5$ in velocity $rms$ units. Time is normalised to the large scale eddy turnover time $\tau_0=L_\perp/2\pi v_{rms}$. The field-perpendicular Reynolds number is defined as $Re_{\perp}=v_{rms}(L_\perp/2\pi)/\nu$.
The system is evolved until a stationary state is reached, which is confirmed by observing the time evolution of the total energy of the fluctuations, and the data are then sampled in intervals of the order of the eddy turnover time. All results presented correspond to averages over approximately 30 samples. We conduct a number of MHD and RMHD simulations with different resolutions, Reynolds numbers and field-parallel box sizes. The parameters for each of the simulations are shown in Table \[tab:params\].
[ccccccc]{} M1 & MHD & 256 & 256 & 5 & 800 & 1\
M2 & MHD & 512 & 512 & 5 & 2200 & 1\
M3 & MHD & 512 & 512 & 5 & 2200 & 0.1\
M4 & MHD & 512 &512 & 10 & 2200 & 0.1\
M5 & MHD & 512 & 256 & 10 & 2200 &0.1\
R1 & RMHD & 512 & 512 & 6 & 960 & 0.1\
R2 & RMHD & 512 & 512 & 6 & 1800 & 0.1\
R3 & RMHD & 256 & 256 & 6 & 960 &0.1\
\[tab:params\]
For each simulation we calculate the scale-dependent alignment angle between the shear-Alfvén velocity and magnetic field fluctuations. We therefore define velocity and magnetic differences as $\delta {\bf v}_{r}={\bf v}({\bf x}+{\bf r})-{\bf v}({\bf x})$ and $\delta {\bf b}_{r}={\bf b}({\bf x}+{\bf r})-{\bf b}({\bf x})$, where ${\bf r}$ is a point-separation vector in the plane perpendicular to ${\bf B}_0$. In the MHD case the pseudo-Alfvén fluctuations are removed by subtracting the component that is parallel to the local guide field, i.e. we construct $\delta {\tilde {\bf v}}_{r}=\delta {\bf v}_{r}-(\delta
{\bf v}_{r}\cdot {\bf n}){\bf n}$ (and similarly for $\delta {\tilde {\bf b}}_{r}$) where ${\bf n}={\bf B}({\bf x})/\vert {\bf B}({\bf x})\vert$. In the RMHD case fluctuations parallel to ${\mathbf{B_0}}$ are not permitted and hence the projection is not necessary. We then measure the ratio of the second order structure functions $$\label{eq:angle1}
\frac{S^{v \times b}_r}{S^{vb}_r}=\frac{\langle | \mathbf{\delta \tilde v_r} \times \mathbf{\delta \tilde b_r} | \rangle }{\langle | \mathbf{\delta \tilde v_r}| |\mathbf{\delta \tilde b_r}| \rangle}$$ where the average is taken over different positions of the point ${\mathbf{x}}$ in a given field-perpendicular plane, over all such planes in the data cube, and then over all data cubes. By definition of the cross product $$\label{eq:angle2}
\frac{S^{v \times b}_r}{S^{vb}_r} \approx \sin(\theta_r)\approx \theta_r,$$ where $\theta_r$ is the angle between $\delta \tilde {\mathbf{v}}_r$ and $\delta \tilde {\mathbf{b}}_r$ and the last approximation is valid for small angles. We recall that the theoretical prediction is $\theta_r \propto r^{1/4}$ [@boldyrev_06].
Figure \[fig:angle\_n\_re\] illustrates the ratio (\[eq:angle2\]) as a function of the separation $r=|{\mathbf{r}}|$ for two MHD simulations (M1 and M2) corresponding to a doubling of the resolution from $256^3$ to $512^3$ mesh points with the Reynolds number increased from $Re_{\perp}=800$ to $2200$. Excellent agreement with the theoretical prediction $\theta \propto r^{1/4}$ is seen in both cases. As the resolution and Reynolds number increase, the scale-dependence of the alignment angle persists to smaller scales. Indeed, we believe that the point at which the alignment saturates can be identified as the dealiasing scale, $k_d=N/3=85,170$ corresponding in configuration space to $r_d \approx 1/2k_d \approx 0.006, 0.003$ for the $256^3,512^3$ simulations, respectively. This is verified in Figure \[fig:angle\_re\] that shows that alignment is largely insensitive to the Reynolds number (provided that the system is turbulent) and Figure \[fig:angle\_n\] that shows that the saturation point decreases by a factor of approximately 2 as the resolution doubles at fixed Reynolds number. Thus as computational power increases, allowing higher resolution simulations to be conducted, we expect to find that scale-dependent alignment persists to smaller and smaller scales.
The fact that even in the lower Reynolds number cases scale-dependent alignment is clearly seen over quite a wide range of scales is particularly interesting, as in those cases only a very short inertial range can be identified in the field-perpendicular energy spectrum, making the identification of spectral exponents difficult (see Figure 1 in [@mason_cb06]). In the larger $Re$ cases, we can estimate the inertial range of scales in configuration space to be the range of $r \sim 1/2k$ over which the energy spectrum displays a power law dependence. The field-perpendicular energy spectrum for the Case M2 is shown in Figure 1 in [@mason_cb08], with the inertial range corresponding to approximately $4 \lesssim k \lesssim 20$, i.e. $0.025 \lesssim r \lesssim 0.125$. Comparison with Figure \[fig:angle\_n\_re\] shows that a significant fraction of the region over which the scaling $\theta_r \propto r^{1/4}$ is observed corresponds to the dissipative region, i.e. that ratios of structure functions appear to probe deeper than the inertial range that is suggested by the energy spectra.
We now consider the effect on the alignment ratio of decreasing the field-parallel resolution. Figure \[fig:angle\_nz\] shows the results from three MHD simulations (M3, M4 & M5) for which the field-parallel resolution decreases by a factor of two, twice. As the resolution decreases the extent of the self-similar region diminishes and the scale-dependence of the alignment angle becomes shallower. If one were to calculate the slope for the lowest field-parallel resolution case (M5) one would find a scale-dependence that is shallower than the predicted power law exponent of $1/4$. This may lead one to conclude (incorrectly) that scale-dependent alignment is not a universal phenomenon in MHD turbulence. However, the effect is obviously a result of the poor resolution rather than being an attribute of the alignment mechanism itself.
Finally, we mention that for the three cases illustrated in Figure \[fig:angle\_nz\], the field-perpendicular energy spectra (not shown) display no appreciable difference. Since the Reynolds number is moderate the inertial range in $k$-space is quite short. However, when the spectra are compensated with $k^{3/2}$ and $k^{5/3}$ the former results in a better fit in all cases. This happens for two reasons. First, the stronger deviation from the alignment scaling $\theta \propto r^{1/4}$ occurs deeper in the dissipation region, that is, further from the inertial interval where the energy spectrum is measured. Second, according to the relationship between the scaling of the alignment angle and the energy spectrum, a noticeable change in the scaling of the alignment angle leads to a relatively small change in the scaling of the field-perpendicular energy spectrum.
\[tbp\]
\[tbp\]
Conclusion
==========
There are two main conclusions that can be drawn from our results. The first is that the measurement of the alignment angle, which is composed of the ratio of two structure functions, appears to display a self-similar region of significant extent, even in the moderate Reynolds number case which requires only a moderate resolution. We have checked that plotting the numerator and the denominator of the alignment ratio separately as functions of the increment $r$ displays only a very limited self-similar region, from which scaling laws cannot be determined. A clear scaling behaviour is also not found when one plots the numerator versus the denominator as is the case in extended self-similarity [@benzi_etal93]. The result is interesting in its own right. It also has important practical value as it allows us to differentiate effectively between competing phenomenological theories through numerical simulations conducted in much less extreme parameter regimes than would otherwise be necessary. The result could be especially useful if it extends to ratios of structure functions for which an exact relation, such as the @politano_p98b relations, is known for one part, as it would then allow the inference of the scaling of the other structure function. Reaching a consensus on the theoretical description of magnetised fluctuations in the idealised incompressible MHD system represents the first step towards the ultimate goal of building a theoretical foundation for astrophysical turbulence.
The second main result that can be drawn from our work is that the measurement of the alignment angle appears to probe deep into the dissipation region and hence it is necessary to adequately resolve the small scale physics. As the field-parallel resolution is decreased, numerical errors contaminate the physics of the dissipative range and affect measurement of the alignment angle. As the decrease in resolution is taken to the extreme, the errors propagate to larger scales and may ultimately spoil an inertial range of limited extent. We propose that similar contamination effects should also arise through any mechanism that has detrimental effects on the dissipative physics. Mechanisms could include pushing the Reynolds number to the extreme or using hyperdiffusive effects. For example, our results may provide an explanation for the numerical findings by [@beresnyak11] who noticed a flattening of the alignment angle in simulations of MHD turbulence with a reduced parallel resolution and strong hyperdiffusivity.
We also point out that the result recalls the phenomenon of extended self-similarity in isotropic hydrodynamic turbulence [@benzi_etal93], which refers to the extended self-similar region that is found when one plots one structure function versus another, rather than as a function of the increment. Our finding is fundamentally different however, in the sense that the self-similar region only becomes apparent when one plots ratios of structure functions versus the increment, rather than structure functions versus other structure functions. Our result appears to be due to a non-universal features in the amplitudes of the functions, rather than their arguments, cancelling when the ratios are plotted. Whether such a property holds for other structure functions in MHD turbulence is an open and intriguing question. This is a subject for our future work.
We would like to thank Leonid Malyshkin for many helpful discussions. This work was supported by the NSF Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas at the University of Chicago and the University of Wisconsin - Madison, the US DoE awards DE-FG02-07ER54932, DE-SC0003888, DE-SC0001794, and the NSF grants PHY-0903872 and AGS-1003451. This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357.
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abstract: 'We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.'
author:
- |
Carl Mueller$^1$\
Department of Mathematics\
University of Rochester\
Rochester, NY 15627 USA\
email: cmlr@math.rochester.edu
- |
David Nualart$^2$\
Department of Mathematics\
University of Kansas\
Lawrence, Kansas, 66045 USA\
email: nualart@math.ku.edu
title: Regularity of the density for the stochastic heat equation
---
Introduction
============
Consider the one-dimensional stochastic heat equation on $[0,1]$ with Dirichlet boundary conditions, driven by a two-parameter white noise, and with initial condition $u_{0}$: $$\frac{\partial u}{\partial t}=\frac{\partial ^{2}u}{\partial x^{2}}+b(t,x,u(t,x))+\sigma (t,x,u(t,x))\frac{\partial ^{2}W}{\partial t\partial x}. \label{e1}$$Assume that the coefficients $b(t,x,u),\sigma(t,x,u)$ have linear growth in $t,x$ and are Lipschitz functions of $u$, uniformly in $(t,x)$.
In [@PZ] Pardoux and Zhang proved that $u(t,x)$ has an absolutely continuous distribution for all $(t,x)$ $\ $such that $t>0$ and $x\in (0,1)$, if $\sigma (0,y_0,u_{0}(y_0))\not=0$ for some $y_0\in (0,1)$. Bally and Pardoux have studied the regularity of the law of the solution of Equation (\[e1\]) with Neumann boundary conditions on $[0,1]$, assuming that the coefficients $b(u)$ and $\sigma(u) $ are infinitely differentiable functions, which are bounded together with their derivatives.
Let $u(t,x)$ be the solution of Equation (\[e1\]) with Dirichlet boundary conditions on $[0,1]$ and assume that the coefficients $b$ and $\sigma $ are infinitely differentiable functions of the variable $u$ with bounded derivatives. The aim of this paper is to show that if $\sigma (0,y_0,u_{0}(y_0))\not=0$ for some $y_0\in
(0,1)$, then $u(t,x)$ has a smooth density for all $(t,x)$ $\ $such that $t>0$ and $x\in (0,1)$. Notice that this is exactly the same nondegeneracy condition imposed in [@PZ] to establish the absolute continuity. In order to show this result we make use of a general theorem on the existence of negative moments for the solution of Equation (\[e1\]) in the case $b(t,x,u)=B(t,x)u$ and $\sigma (t,x,u)=H(t,x)u$, where $B$ and $H$ are some bounded and adapted random fields.
Preliminaries
=============
First we define white noise $W$. Let $$W=\{W(A),A \mbox{ a Borel subset of } \mathbb{R}^{2},|A|<\infty \}$$ be a Gaussian family of random variables with zero mean and covariance$$E\big[W(A)W(B)\big]=|A\cap B|,$$where $|A|$ denotes the Lebesgue measure of a Borel subset of $\mathbb{R}^{2} $, defined on a complete probability space $(\Omega ,\mathcal{F},P)$. Then $W(t,x)=W([0,t]\times \lbrack 0,x])$ defines a two-parameter Wiener process on $[0,\infty )^{2}$.
We are interested in the following one-dimensional heat equation on $[0,\infty )\times \lbrack 0,1]$$$\frac{\partial u}{\partial t}=\frac{\partial ^{2}u}{\partial x^{2}}+b(t,x,u(t,x))+\sigma (t,x,u(t,x))\frac{\partial ^{2}W}{\partial t\partial x}, \label{a1}$$with initial condition $u(0,x)=u_{0}(x)$, and Dirichlet boundary conditions $u(t,0)=u(t,1)=0$. We will assume that $u_{0}$ is a continuous function which satisfies the boundary conditions $u_{0}(0)=u_{0}(1)=0$. This equation is formal because the partial derivative $\frac{\partial ^{2}W}{\partial t\partial x}\,\ $ does not exist, and (\[a1\]) is usually replaced by the evolution equation $$\begin{aligned}
u(t,x)
&=&\int_{0}^{1}G_{t}(x,y)u_{0}(y)dy+\int_{0}^{t}\int_{0}^{1}G_{t-s}(x,y)b(s,y,u(s,y))u(s,y)dyds \notag \\
&&+\int_{0}^{t}\int_{0}^{1}G_{t-s}(x,y)\sigma (s,t,u(s,y))u(s,y)W(dy,ds),
\label{a2}\end{aligned}$$where $G_{t}(x,y)$ is the fundamental solution of the heat equation on $[0,1]$ with Dirichlet boundary conditions. Equation (\[a2\]) is called the mild form of the equation.
If the coefficients $b$ and $\sigma \,\ $are have linear growth and are Lipschitz functions of $u$, uniformly in $(t,x)$, there exists a unique solution of Equation (\[a2\]) (see Walsh [@Wa]).
The Malliavin calculus is an infinite dimensional calculus on a Gaussian space, which is mainly applied to establish the regularity of the law of nonlinear functionals of the underlying Gaussian process. We will briefly describe the basic criteria for existence and smoothness of densities, and we refer to Nualart [@Nu] for a more complete presentation of this subject.
Let $\mathcal{S}$ denote the class of smooth random variables of the the form $$F=f(W(A_{1}),\dots ,W(A_{n})), \label{a5}$$where $f$ belongs to $C_{p}^{\infty }(\mathbb{R}^{n})$ ($f$ and all its partial derivatives have polynomial growth order), and $A_{1},\dots ,A_{n}$ are Borel subsets of $\mathbb{R}_{+}^{2}$ with finite Lebesgue measure. The derivative of $F$ is the two-parameter stochastic process defined by$$D_{t,x}F=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}(W(A_{1}),\dots
,W(A_{n}))\mathbf{1}_{A_{i}}(t,x).$$In a similar way we define the iterated derivative $D^{(k)}F$. The derivative operator $D$ (resp. its iteration $D^{(k)}$)$\ $ is a closed operator from $L^{p}(\Omega )$ into $L^{p}(\Omega ;L^{2}(\mathbb{R}^{2}))$ (resp. $L^{p}(\Omega ;L^{2}(\mathbb{R}^{2k}))$) for any $p>1$. For any $p>1$ and for any positive integer $k$ we denote by $\mathbb{D}^{p,k}$ the completion of $\mathcal{S}$ with respect to the norm$$\left\| F\right\| _{k,p}=\left\{ E(|F|^{p})+\sum_{j=1}^{k}E\left[ \left(
\int_{\mathbb{R}^{2j}}\left( D_{z_{1}}\cdots D_{z_{j}}F\right)
^{2}dz_{1}\cdots dz_{j}\right) ^{\frac{p}{2}}\right] \right\} ^{\frac{1}{p}}.$$Set $\mathbb{D}^{\infty }=\cap _{k,p}\mathbb{D}^{k,p}$.
Suppose that $F=(F^{1},\ldots ,F^{d})$ is a $d$-dimensional random vector whose components are in $\mathbb{D}^{1,2}$. Then, we define the Malliavin matrix of $F$ as the random symmetric nonnegative definite matrix$$\sigma _{F}=\left( \left\langle DF^{i},DF^{j}\right\rangle _{L^{2}(\mathbb{R}^{2})}\right) _{1\leq i,j\leq d}.$$The basic criteria for the existence and regularity of the density are the following:
\[t1\] Suppose that $F=(F^{1},\ldots ,F^{d})$ is a $d$-dimensional random vector whose components are in $\mathbb{D}^{1,2}$. Then,
1. If $\det \sigma _{F}>0$ almost surely, the law of $F$ is absolutely continuous.
2. If $F^{i}\in \mathbb{D}^{\infty }$ for each $i=1,\ldots ,d$ and $E\left[ (\det \sigma _{F})^{-p}\right] <\infty $ for all $p\geq 1$, then the $F$ has an infinitely differentiable density.
Negative moments
================
\[t2\] Let $u(t,x)$ be the solution to the stochastic heat equation$$\begin{aligned}
\frac{\partial u}{\partial t}&=&\frac{\partial ^{2}u}{\partial x^{2}}+Bu+Hu\frac{\partial ^{2}W}{\partial t\partial x}, \label{e3} \\
u(0,x) &=& u_0(x) \notag\end{aligned}$$on $x\in[0,1]$ with Dirichlet boundary conditions. Assume that $B=B(t,x)$ and $H=H(t,x)$ are bounded and adapted processes. Suppose that $u_{0}(x)\ $ is a nonnegative continuous function not identically zero. Then, $$E[u(t,x)^{-p}]<\infty$$ for all $p\geq 2$, $t>0$ and $0<x<1$.
For the proof of this theorem we will make use of the following large deviations lemma, which follows from Proposition A.2, page 530, of Sowers [@So].
\[lem1\] Let $w(t,x)$ be an adapted stochastic process, bounded in absolute value by a constant $M$. Let $\epsilon>0$. Then, there exist constants $C_0$, $C_1>0$ such that for all $\lambda>0$ and all $T>0$ $$P\left( \sup_{0\le t\le T} \sup _{0\le x\le 1} \left| \int_0^t \int_0^1 G_{t-s} (x,y) w(s,y) W(ds,dy) \right| >\lambda \right) \le C_0 \exp \left( -\frac {C_1 \lambda^2} {T^{\frac 12-\epsilon} }\right).$$
We also need a comparison theorem such as Corollary 2.4 of [@Sh]; see also Theorem 3.1 of Mueller [@Mu] or Theorem 2.1 of Donati-Martin and Pardoux [@DP]. Shiga’s result is for $x\in\mathbb{R}$, but it can easily be extended to the following lemma, which deals with $x\in[0,1]$ and Dirichlet boundary conditions.
\[lem-compare\] Let $u_i(t,x): i=1,2$ be two solutions of $$\begin{aligned}
\frac{\partial u_i}{\partial t}&=&\frac{\partial ^{2}u_i}{\partial x^{2}}+B_iu_i+Hu_i\frac{\partial ^{2}W}{\partial t\partial x}, \label{e3a} \\
u_i(0,x) &=& u_0^{(i)}(x) \notag\end{aligned}$$where $B_i(t,x), H(t,x), u^{(i)}_0(x)$ satisfy the same conditions as in Theorem \[t2\]. Also assume that with probability one for all $t\geq0,x\in[0,1]$ $$\begin{aligned}
B_1(t,x) &\leq& B_2(t,x) \\
u_0^{(1)}(x) &\leq& u_0^{(2)}(x).\end{aligned}$$ Then with probability 1, for all $t\geq0, x\in[0,1]$. $$u_1(t,x)\leq u_2(t,x).$$
We shall repeatedly use the comparison lemma, Lemma \[lem-compare\], along with the following argument. Observe that if $0<w(t,x)\leq u(t,x)$ with probability one, and $p>0$, then $$E\left[u(t,x)^{-p}\right] \leq E\left[w(t,x)^{-p}\right].$$ Thus, to bound $E[u(t,x)^{-p}]$, it suffices to find a nonnegative function $w(t,x)\leq u(t,x)$ and to prove a bound for $E[w(t,x)^{-p}]$. Such a function $w(t,x)$ might be found using the comparison lemma, Lemma \[lem-compare\].
Suppose that $|B(t,x)|\leq K$ almost surely for some constant $K>0$. By the comparison lemma, Lemma \[lem-compare\], it suffices to consider the solution to the equation$$\begin{aligned}
\frac{\partial w}{\partial t}&=&\frac{\partial ^{2}w}{\partial x^{2}}-Kw+Hw\frac{\partial ^{2}W}{\partial t\partial x} \label{e4} \\
w(0,x) &=& u_0(x) \notag\end{aligned}$$on $x\in[0,1]$ with Dirichlet boundary conditions. Indeed, the comparison lemma implies that a solution $w(t,x)$ of (\[e4\]) will be less than or equal to a solution $u(t,x)$ of (\[e3\]). Then we can use the argument outlined in the previous paragraph to conclude that the boundedness of $E\left[w(t,x)^{-p}\right]$ implies the boundedness of $E\left[u(t,x)^{-p}\right]$.
Set $u(t,x)=e^{-Kt}w(t,x)$, where $u(t,x)$ is not the same as earlier in the paper. Simple calculus shows that $u(t,x)$ satisfies$$\begin{aligned}
\frac{\partial u}{\partial t}&=&\frac{\partial ^{2}u}{\partial x^{2}}+Hu\frac{\partial ^{2}W}{\partial t\partial x}. \label{e4b} \\
u(0,x) &=& u_0(x) \notag\end{aligned}$$and we have $$E\left[w(t,x)^{-p}\right] = e^{Ktp}E\left[u(t,x)^{-p}\right].$$ So, we can assume that $K=0$, that is that $u(t,x)$ satisfies (\[e4b\]). The mild formulation of Equation (\[e4b\]) is$$u(t,x)=\int_{0}^{1}G_{t}(x,y)u_{0}(y)dy+\int_{0}^{t}\int_{0}^{1}G_{t-s}(x,y)H(s,y)u(s,y)W(ds,dy).$$ Suppose that $u_{0}(x)\geq \delta >0$ for all $x\in \lbrack a,b]\subset (0,1)
$. Since (\[e4b\]) is linear, we may divide this equation by $\delta $, and assume $\delta =1$, and also $u_{0}(x)=\mathbf{1}_{[a,b]}(x)$. Fix $T>0$, and consider a larger interval $[a,b]\subset [c,d]$ of the form $d=b+\gamma T$ and $c=a-\gamma T$, where $\gamma>0$. We are going to show that $E((u(T,x)^{-p})<\infty$ for $x\in [c,d]$ and for any $p\ge 1$. Define $$c= \inf_{0\le t+s \le T,} \inf_{a-\gamma (t+s)\le x\le b+\gamma (t+s)} \int_{a-\gamma s}^{b+\gamma s} G_t(x,y)dy$$ and note that $0<c<1$ for each $\gamma>0$ and $(a,b)\in(0,1)$. Next we inductively define a sequence $\left\{ \tau _{n},n\geq 0\right\} $ of stopping times and a sequence of processes $v_{n}(t,x)$ as follows. Let $v_{0}(t,x)$ be the solution of ([e4b]{}) with initial condition $u_{0}=\mathbf{1}_{[a,b]}$ and let $$\tau_0= \inf\left\{t>0: \inf_{a-\gamma t \le x\le b+\gamma t}
v_0(t,x)=\frac{c}{2} \, \mbox{ or }\, \sup_{0\le x\le 1} v_0(t,x)= \frac{2}{c}\right\}.$$ Next, assume that we have defined $\tau_{n-1}$ and $v_{n-1}(t,x)$ for $\tau_{n-2}\le t\le\tau_{n-1}$. Then, $\{v_n(t,x), \tau_{n-1}\le t\}$ is defined by (\[e4b\]) with initial condition $v_n(\tau_{n-1},x)=(\frac c2)^{n} \mathbf{1}_{[a-\gamma \tau_{n-1}, b+\gamma \tau_{n-1}]}(x)$. Also, let $$\begin{aligned}
\lefteqn{
\tau_n= \inf\Bigg\{t>\tau_{n-1}: \inf_{a-\gamma t \le x\le b+\gamma t}
v_n(t,x)=\left(\frac c2\right)^{n+1} }\\
&& \hspace{1.5in} \mbox{ or } \sup_{0\le x \le 1}
v_n(t,x)=\left(\frac 2c\right)^{-n+1}\Bigg\}.\end{aligned}$$ It is not hard to see that $\tau_n <\infty$ almost surely. Notice that $$\inf_{a-\gamma \tau_n \le x \le b+\gamma \tau_n} v_n(\tau_n ,x) \ge \left(\frac c2\right)^{n+1}.$$ By the comparison lemma, we have that $$u(t,x)\ge v_n(t,x) \label{e5}$$ for all $(t,x)$ and all $n\ge 0$. For all $p\ge 1$ we have $$\begin{aligned}
E\left[u(T,x)^{-p}\right]
&\leq& P\Big(u(T,x)\geq 1\Big) \notag \\
&& + \sum_{n=0}^{\infty}\left( \frac 2c\right)^{np}P\left(u(T,x)\in\Big[ \left(\frac c2\right)^{n+1},\left(\frac c2\right)^n\Big)\right) \notag\\
&\leq& 1 + \sum_{n=0}^{\infty}\left( \frac 2c\right)^{np} P\left(u(T,x) < \left(\frac c2\right)^n \right).
\label{e44}\end{aligned}$$ Taking into account (\[e5\]), the event $\{ u(T,x) <(\frac c2)^n\}$ is included in $\mathcal{A}_n=\{\tau_n <T\}$. Set $\sigma_n=\tau_n- \tau_{n-1}$, for all $n\ge 0$, with the convention $\tau_{-1}=0$.
We have $$\begin{aligned}
P\left(\sigma_i <\frac 2n \bigg| \mathcal{F}_{\tau_{i-1}}\right) &\le&
P\left( \sup_{\tau_{i-1} <t <\tau_{i-1} +\frac 2n,} \sup_{0\le x\le 1} v_i(t,x) >\left(\frac 2c\right)^{-i+1}\right) \\
&& + P\left( \inf_{\tau_{i-1} <t <\tau_{i-1} +\frac 2n,} \inf_{a-\gamma t \le x\le b+\gamma t} v_i(t,x) >\left(\frac c2\right)^{i+1} \right) \\\end{aligned}$$ Notice that, for $\tau_{i-1} <t< \tau_i$ we have $$\begin{aligned}
\lefteqn{
\left(\frac 2c\right)^iv_i(t,x) = \int_{a-\gamma \tau_{i-1}}^{b+\gamma \tau_{i-1}} G_{t-\tau_{i-1}}(x,y) dy }\\
&&+\int_{\tau_{i-1}} ^t \int_0^1 G_{t-s}(x,y) H(s,y)
\left(\left[\left(\frac 2c\right)^i v_i(s,y)\right]\wedge
\frac2c \right)W(ds,dy).\end{aligned}$$ As a consequence, by Lemma \[lem1\] $$\begin{aligned}
P\left( \sigma_i <\frac 2n \bigg| \mathcal{F}_{\tau_{i-1}}\right) & \le & P\left( \sup_{\tau_{i-1} \le t\le \tau_{i-1}+\frac 2n,} \notag
\sup_{0\le x\le 1} \left| N_i(t,x)\right| > 1 \right) \\
&\le& C_0 \exp \left( - C_1 n^{\frac 12 -\epsilon} \right). \label{e11}\end{aligned}$$
Next we set up some notation. Let $\mathcal{B}_n$ be the event that at least half of the variables $\sigma_i: i=0,\ldots,n$ satisfy $$\tau_i < \frac{2T}{n}$$ Note that $$\mathcal{A}_n \subset \mathcal{B}_n$$ since if more than half of the $\sigma_i: i=1,\ldots,n$ are larger than or equal to $2T/n$ then $\tau_n>T$.
For convenience we assume that $n=2k$ is even, and leave the odd case to the reader. Let $\Xi_n$ be all the subsets of $\{1,\ldots,n\}$ of cardinality $k=n/2$. Using Stirling’s formula, the reader can verify that as $n\to\infty$ $${n \choose n/2} = O(2^n) \label{e2}$$ Then, $$\begin{aligned}
P(\mathcal{B}_n)
&\leq& P\left(\bigcup_{\{i_1,\ldots,i_k\}\in\Xi_n}\bigcap_{j=1}^{k}
\left\{\sigma_{i_j}<\frac{2T}{n}\right\}\right) \\
&\leq& \sum_{\{i_1,\ldots,i_k\}\in\Xi_n} P\left(\bigcap_{j=1}^{k} \sigma_{i_j}<\frac{2T}{n}\right) \end{aligned}$$ Using the estimate (\[e11\]) and (\[e2\]) yields $$\begin{aligned}
P(\mathcal{B}_n)&\leq& C_0 2^{n} \exp\left(-C_1n^{1/2-\varepsilon}\right)^n \\
&\leq& C_0 \exp\left(-C_1 n^{3/2-\varepsilon}+C_2n\right) \\
&\leq& C_0 \exp\left(-C_1 n^{3/2-\varepsilon}\right) \end{aligned}$$ where the constants $C_0,C_1$ may have changed from line to line. Hence, $$P\left( u(T,x) <\left(\frac c2\right)^n\right) \le C_0 \exp\left(-C_1 n^{3/2-\varepsilon} \right)
\label{e33}$$ Finally, substituting (\[e3\]) into (\[e4\]) yields $E\left[u(T,x)^{-p}\right]<\infty$.
Smoothness of the density
=========================
Let $u(t,x)$ be the solution to Equation (\[a1\]). Assume that the coefficients $b$ and $\sigma $ are continuously differentiable with bounded derivatives. Then $u(t,x)$ belongs to the Soboev space $\mathbb{D}^{1,p}$ for all $p>1$, and the derivative $D_{\theta ,\xi }u(t,x)$ satisfies the following evolution equation$$\begin{aligned}
D_{\theta ,\xi }u(t,x) &=&\int_{\theta
}^{t}\int_{0}^{1}G_{t-s}(x,y)\frac{\partial b}{\partial u}(s,y,u(s,y))D_{\theta ,\xi
}u(s,y)dyds \notag \\
&&+\int_{\theta }^{t}\int_{0}^{1}G_{t-s}(x,y)\frac{\partial\sigma}{\partial u}
(s,y,u(s,y))D_{\theta ,\xi }u(s,y)W(dy,ds) \notag \\
&&+\sigma (u(\theta ,\xi ))G_{t-\theta }(x,\xi ), \label{e22}\end{aligned}$$if $\theta <t$ and $D_{\theta ,\xi }u(t,x)=0$ if $\theta >t$. That is, $D_{\theta ,\xi }u(t,x)$ is the solution of the stochastic partial differential equation$$\frac{\partial D_{\theta ,\xi }u}{\partial t}=\ \frac{\partial ^{2}D_{\theta
,\xi }u}{\partial x^{2}}+ \frac{\partial b}{\partial u}(t,x,u(t,x)) D_{\theta ,\xi }u
+ \frac{\partial\sigma}{\partial u}
(t,x,u(t,x))D_{\theta ,\xi }u\frac{\partial ^{2}W}{\partial t\partial x}$$on $[\theta ,\infty )\times \lbrack 0,1]$, with Dirichlet boundary conditions and initial condition $\sigma (u(\theta ,\xi ))\delta _{0}(x-\xi )
$.
Let $u(t,x)$ be the solution of Equation (\[a1\]) with initial condition $u(0,x)=u_{0}(x)$, and Dirichlet boundary conditions $u(t,0)=u(t,1)=0$. We will assume that $u_{0}$ is an $\alpha$-Hölder continuous function for some $\alpha>0$, which satisfies the boundary conditions $u_{0}(0)=u_{0}(1)=0$. Assume that the coefficients $b$ and $\sigma $ are infinitely differentiable functions with bounded derivatives. Then, if $\sigma (0,y_{0},u_{0}(y_{0}))\not=0$ for some $y_{0}\in (0,1)$, $u(t,x)$ has a smooth density for all $(t,x)$ $\ $such that $t>0$ and $x\in (0,1)$.
From the results proved by Bally and Pardoux in [@BP] we know that $u(t,x)$ belongs to the space $\mathbb{D}^{\infty }$ for all $(t,x)$. Set $\ $$$C_{t,x}=\int_{0}^{t}\int_{0}^{1}\left( D_{\theta ,\xi }u(t,x)\right)
^{2}d\xi d\theta .$$Then, by Theorem \[t1\] it suffices to show that $E(C_{t,x}^{-p})<\infty$ for all $p\geq 2$.
Suppose that $\sigma (0,y_{0},u_{0}(y_{0}))>0$. By continuity we have that $\sigma (0,y,u(0,y))\geq \delta >0$ for all $y\in \lbrack a,b]\subset (0,1)$. Then $$\begin{aligned}
C_{t,x} \ge \int_{0}^{t}\int_{a}^{b}\left( D_{\theta ,\xi }u(t,x)\right)
^{2}d\xi d\theta
\ge
\int_{0}^{t} \left(\int_{a}^{b} D_{\theta ,\xi }u(t,x) d\xi\right)
^{2} d\theta.\end{aligned}$$ Set $Y^{\theta}_{t,x}=\int_{a}^{b}D_{\theta,\xi }u(t,x)d\xi $. Fix $r<1$ and $\varepsilon >0$ such that $\varepsilon
^{r}<t$. From$$\ \varepsilon ^{r} \left( Y^0_{t,x}\right)^2 \leq \
\int_{0}^{\varepsilon ^{r}} \left| \left( Y^0_{t,x}\right)^2-\left( Y^\theta_{t,x}\right)^2
\right| d\theta +C_{t,x}$$we get$$\begin{aligned}
P(C_{t,x} <\varepsilon )&\leq& P\left( \int_{0}^{\varepsilon
^{r}}\left| \left( Y^0_{t,x}\right)^2-\left( Y^\theta_{t,x}\right)^2
\right| d\theta >\varepsilon \right) \\
&&+P\left( Y^0_{t,x} <
\sqrt{2} \varepsilon ^{\frac{1-r}2} \right) \\
&=&P(A_{1})+P(A_{2}).\end{aligned}$$Integrating equation (\[e22\]) in the variable $\xi $ yields the following equation for the process $\{Y^\theta_{t,x}, t\ge \theta, x\in [0,1]\}$ $$\begin{aligned}
Y^\theta_{t,x} &=&\int_{\theta}^{t}\int_{0}^{1}G_{t-s}(x,y)\frac{\partial b}{\partial u}
(s,y,u(s,y))Y^\theta_{s,y}dyds \notag \\
&&+\int_{\theta}^{t}\int_{0}^{1}G_{t-s}(x,y)\frac{\partial\sigma}{\partial u}
(s,y,u(s,y))Y^\theta_{s,y}W(dy,ds) \notag \\
&&+\int_{a}^{b}\sigma ( u(\theta,\xi ))G_{t-\theta}(x,\xi )d\xi . \label{eq3}\end{aligned}$$In particular, for $\theta=0$, the initial condition is $Y^0_{0,\xi}=\sigma(0,\xi,u(0,\xi)) \mathbf{1}_{[a,b]}(\xi)$, and by Theorem \[t2\] the random variable $Y^0_{t,x}$ has negative moments of all orders. Hence, for all $p\ge 1$, $$P(A_2) \le \varepsilon^p$$ if $\varepsilon \le \varepsilon_0$. In order to handle the probability $P(A_1)$ we write $$P(A_{1})\leq \varepsilon ^{(r-1)q}\sup_{ 0\leq \theta \leq
\varepsilon ^{r} } \left(E\left[ \left| Y^\theta_{t,x} -Y^0_{t,x} \right|^{2q}
\right] E\left[ \left| Y^\theta_{t,x} + Y^0_{t,x} \right|^{2q}
\right] \right)^{1/2}.$$We claim that $$\label{c1}
\sup_{0\le \theta \le t} E\left[ \left| Y^{\theta}_{x,t} \right|^{2q} \right] <\infty,$$ and $$\label{c2}
\sup_{0\le \theta \le \varepsilon^r } E\left[ \left| Y^\theta_{t,x} -Y^0_{t,x} \right|^{2q} \right] <\varepsilon^{2sq},$$ for some $s>0$. Property (\[c1\]) follows easily from Equation (\[eq3\]). On the other hand, the difference $Y^\theta_{t,x} -Y^0_{t,x} $ satisfies $$\begin{aligned}
Y^\theta_{t,x} -Y^0_{t,x} & =& \int_{\theta}^{t}\int_{0}^{1}G_{t-s}(x,y)
\frac{\partial b}{\partial u}
(s,y,u(s,y))(Y^\theta_{s,y} -Y^0_{x,t}) dyds \notag \\
&&+\int_{\theta}^{t}\int_{0}^{1}G_{t-s}(x,y) \frac{\partial\sigma}{\partial u}
(s,y,u(s,y))(Y^\theta_{s,y}- Y^0_{x,t}) W(dy,ds) \notag \\
&& +\int_0^{\theta}\int_{0}^{1}G_{t-s}(x,y) \frac{\partial b}{\partial u}
(s,y,u(s,y))Y^0_{s,y}dyds \notag \\
&& +\int_0^{\theta}\int_{0}^{1}G_{t-s}(x,y) \frac{\partial\sigma}{\partial u}
(s,y,u(s,y))Y^0_{s,y}W(dy,ds) \notag \\
&& +\int_{a}^{b} (\sigma ( u(\theta,\xi )) G_{t-\theta}(x,\xi ) -\sigma(u_0(\xi)) G_{t }(x,\xi ))d\xi \notag\\
&=& \sum_{i=1} ^5 \Psi_i(\theta). \notag\end{aligned}$$Applying Gronwall’s lemma and standard estimates, to show (\[c2\]) it suffices to prove that $$\label{c3}
\sup_{0\le \theta \le \varepsilon^r } E\left( \left| \Psi_i(\theta) \right|^{2q} \right) <\varepsilon^{2sq},$$ for $i=3,4,5$ and for some $s>0$. The estimate (\[c3\]) is clear for $i=3,4$ and for $i=5$ we use the properties of the heat kernel and the Hölder continuity of the initial condition $u_0$. Finally, it suffices to choose $r>1-s$ and we get the desired estimate for $P(A_1)$. The proof is now complete.
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abstract: 'We study the ground state and the phase transitions of the bilayered spin-$S$ antiferromagnetic Heisenberg model using the Schwinger boson mean field theory. The interplane coupling initially stabilizes but eventually destroys the long-range antiferromagnetic order. The transition to the disordered state is continuous for small $S$, and first order for large $S$. The latter is consistent with an argument based on the spin wave theory. The phase diagram and phase transitions in corresponding model in fractional dimensions are also discussed.'
address: |
$^{1}$Department of Physics, University of Cincinnati, Cincinnati, OH 45221\
$^{2}$Department of Physics, Hong Kong University of Science and Technology\
Clear Water Bay, Kowloon, Hong Kong\
$^{3}$Department of Physics, Chinese University of Hong Kong\
Satin, New Territories, Hong Kong
author:
- 'Kwai-Kong Ng$^{1}$, Fu Chun Zhang$^{1,2}$, and Michael Ma$^{1,3}$'
title: 'Phase Transitions of the Bilayered Spin-$\bf {S}$ Heisenberg Model and Its Extension to Fractional Dimensions'
---
———- X-Sun-Data-Type: default X-Sun-Data-Name: bilayer.tex X-Sun-Content-Lines: 600
Introduction
============
Recently there has been considerable interest in quantum spin liquids, which are magnetic systems without LRO at low temperature. While in general, the ground state of quantum spin systems lack true LRO in 1D, the ground state of the 2D Heisenberg antiferromagnet exhibits Neel ordering even for $S = 1/2$, albeit with an sublattice magnetization that is considerably decreased from its classical value. Since spin is quantized, the spin value cannot be decreased beyond $1/2$, hence the model does not have a spin liquid ground state. On the other hand, when two planes of antiferromagnetic spins are coupled together [@Millis; @Hida1; @Hida2; @Hida3; @Dagotto; @Sandvik1; @Sandvik2; @Yoshioka; @Maekawa], and if the interplane coupling is strong enough, the ground state is easily seen to be one of valence bond solid of interplane singlets (IVBS). Thus, there should be a transition from the LRO Neel state to a spin liquid state as the interplane coupling is increased. It has been suggested that the unusual magnetic properties of YBCO, with its basic unit of a pair of coupled CuO planes, may be due to its lying close to this quantum transition [@Millis].
It is of interest to study the nature of this quantum transition. Within a non-linear sigma-model (NLSM) description, Haldane [@Haldane] has pointed out that for a single plane of spins, topological Berry phase terms exist which differ between half-integer, odd integer, and even integer spins. One way to understand this is to consider the degeneracy of the valence bond solid states which maximize the number of resonating plaquettes in each case (4-fold, 2-fold, and non-degenerate respectively). On the other hand, the mapping of the two-plane system to the NLSM does not yield a topological term, which is consistent with the valence bond solid state for two planes with large interplane vs. intraplane coupling being zero-dimensional like and non-degenerate. Since the 2+1 D NLSM has only one phase transition which is second order, this suggests the same for the 2D quantum Heisenberg antiferromagnet at T = 0. However, the NLSM mapping assumes slow variation on the scale of lattice spacing, and so additional disordered phases and/or first order transition cannot be ruled out conclusively.
In this paper we investigate the ground state of the 2D bilayered Heisenberg AF for general $S$ using the Schwinger boson mean field theory [@Arovas; @Sarker] with no additional approximation . Our calculation complements previous calculations for $S = 1/2$ only and/or using additional approximations, as well as a calculation using the related Takahasi bosons approach [@Hida1; @Hida2; @Hida3]. Our results show that a first order transition is favored by decreasing quantum fluctuations, i.e.. increasing $S$. In particular, in agreement with those previous works, the transition for $S = 1/2$ is first order. The critical $S$ separating first from second order transition equals $0.35$. While increasing interplane coupling $J_ {\perp}$ eventually destroys LRO, ordering is stabilized by small $J_ {\perp}$, and the critical $S$ for no LRO is shifted from the single plane value of $0.2$ to $0.13$. A simple argument using spin wave theory helps to explain why first order transition occurs for large $S$. Since quantum fluctuations increases both with decreasing $S$ and decreasing $d$, we also study the dimensionality dependence of the “bilayer” hypercubic system. For $S = 1/2$, we find that for $d < 1.86$ , the first order transition is replaced by second order one. En route, we also calculate the $S$ vs. $d$ phase diagram for $J_ {\perp} =
0$, i.e. the hypercubic Heisenberg AF. In addition to contradicting the NLSM description by having the possibility of a first order phase transition, whenever the transtion is continuous, the Schwinger boson MFT gives an additional phase transition between two disordered phases for all $S$, corresponding to a jump in the ratio of short-ranged intraplane to interplane correlations. However, since the Schwinger boson order parameter is not related directly to any physical symmetry breaking, it is likely this does not constitute a real phase transition but a sharp cross-over in behavior.
Bilayered Antiferromagnet in 2D
===============================
We begin with a quick review of Schwinger boson mean field theory [@Sarker] as applied to the translationally invariant nearest neighbor Heisenberg antiferromagnet on a bipartite lattice. The Hamiltonian is
$$\begin{aligned}
H=\sum_{\langle ij \rangle} J_{ij}{\bf{S}}_i\cdot{\bf{S}}_j; ~~~~~
J_{ij}>0,~~ \langle
ij \rangle =n.n.
\nonumber\end{aligned}$$
In the Schwinger boson representation, spin operators in each lattice site are replaced by spin 1/2 bosons as follows:
$$\begin{aligned}
S^{\dag}_{i} &=& b_{i\uparrow}^{\dag} b_{i\downarrow},~~~~ S^{-}_{i} =
b_{i\downarrow}^{\dag} b_{i\uparrow} \nonumber\\
S^{z}_{i} &=& \frac{1}{2}( b_{i\uparrow}^{\dag} b_{i\uparrow} -
b_{i\downarrow}^{\dag} b_{i\downarrow}), \nonumber\end{aligned}$$
The number of bosons at each lattice site is subject to the constraint:
$$\begin{aligned}
\sum_{\sigma}b^{\dag}_{i\sigma}b_{i\sigma} = 2 S, \nonumber\end{aligned}$$
which can be implemented by introducing a Lagrange multiplier on each site. The Hamiltoninan can now be written as
$$\begin{aligned}
H = -2 \sum_{\langle ij \rangle} J_{ij}\tilde{A}^{\dag}_{ij} \tilde{A}_{ij}
+ \frac{1}{2} NzJS^{2} + \sum_{i}
\lambda_i(\tilde{b}^{\dag}_{i\sigma}\tilde{b}_{i\sigma} -
2S), \nonumber\end{aligned}$$
where $\tilde{A}^{\dag}_{ij}=\frac{1}{2}\sum_{\sigma}
\tilde{b}^{\dag}_{i\sigma}
\tilde{b}^{\dag}_{j\sigma}$ and $\tilde{b}_{i\uparrow}=b_{i\downarrow},~\tilde{b}_{i\downarrow}
=-b_{i\uparrow}$ for sites on one sublattice and $\tilde{b}_{i\sigma} =
b_{i\sigma}$ for sites on the other sublattice. Physically, the product $\tilde{A}^{\dag}_{ij} \tilde{A}_{ij}$ acts as the valence bond (singlet) number operator for sites (i,j). In the mean field approximation, this product is decoupled by the Hartree-Fock decomposition. In addition, the exact local constraint is relaxed to one for the average:
$$\begin{aligned}
\langle\sum_{\sigma}b^{\dag}_{i\sigma}b_{i\sigma}\rangle = 2 S \nonumber,\end{aligned}$$
leading to the mean field Hamiltonian
$$\begin{aligned}
H_{MF} =
E_{0}+\lambda\sum_{i\sigma}\tilde{b}^{\dag}_{i\sigma}\tilde{b}_{i\sigma}
- 2\sum_{\langle ij \rangle} J_{ij} A_{ij}
(\tilde{A}^{\dag}_{ij}+\tilde{A}_{ij}),
\nonumber\end{aligned}$$
where we have taken $A_{ij}=\langle \tilde{A}_{ij}\rangle$ to be real.
First consider the case that all the bonds are identical by symmetry, and assuming no spontaneous dimerization, then all $A_{ij}$ must be the same $A_{ij}=A$, . In this case $E_{0}=\frac{1}{2}NzJS^{2} - 2\lambda NS +
JA^{2}Nz$, where $z$ is the coordiantion number. $H_{MF}$ can be diagonalized by going to momentum space and perfroming the Bogoliubov transformation:
$$\begin{aligned}
H_{MF}=E_{0}-\lambda N
+\sum_{\bf{k}}\omega_{\bf{k}}(\alpha^{\dag}_{\bf{k}}\alpha_{\bf{k}} +
\beta^{\dag}_{\bf{k}}\beta_{\bf{k}} + 1), \nonumber\end{aligned}$$
where $\omega_{\bf{k}} = [\lambda^{2} -
(J\tilde{A}z\gamma_{\bf{k}})^{2}]^{\frac{1}{2}}$, $\gamma_{\bf{k}} =
\frac{1}{z} \sum_ \delta e^{i\bf{k} \cdot \delta} = \sum_{i=1}^{d} \cos
k_{i}/d$. At $T=0$, the energy should be minimized with respect to $\lambda$ and $A$, yielding the set of self-consistent equations:
$$\begin{aligned}
S + \frac{1}{2} &=& \frac{1}{2N}\sum_{\bf{k}}\frac{\mu} {(\mu^{2}-
\gamma_{\bf{k}}^{2})^{\frac{1}{2}}}, \nonumber\\
\tilde{A} &=& \frac{1}{2N}\sum_{\bf{k}}\frac{\gamma^{2}_{\bf{k}}} {(\mu^{2}-
\gamma_{\bf{k}}^{2})^{\frac{1}{2}}}, \nonumber\end{aligned}$$
where we define $\mu\equiv\lambda/(J\tilde{A}z)$. An essential point of the theory is that a non-zero mean field amplitude $A$, which gives rise to boson hopping, indicates short-ranged antiferromagnetic order. Long-ranged order is achieved if the hopping amplitude is sufficiently large to give Bose condensation. This occurs when these eqs. cannot be satisfied by having $\mu>1$, in which case $\mu=1$, and the $k=0$ term gives a finite contribution when converting the momentum sums into integrals:
$$\begin{aligned}
S+\frac{1}{2} &=& m_{s} + \frac{1}{2} \int^{\pi}_{-\pi} \frac{d^{d}{\bf{k}}}
{(2\pi)^{d}} \frac{1} {(1-\gamma_{\bf{k}}^{2})^{\frac{1}{2}}}, \nonumber\\
A &=& m_{s}+\frac{1}{2} \int^{\pi}_{-\pi} \frac{d^{d}{\bf{k}}} {(2\pi)^{d}}
\frac{\gamma_{\bf{k}}^{2}} {(1-\gamma_{\bf{k}}^{2})^{\frac{1}{2}}}, \label{ms2}\end{aligned}$$
It has been shown that the condensate density $m_{s}$ is also the sublattice magnetization. For the Heisenberg antiferromagnet on a square lattice, it was found that Bose condensation occurs for all $S>S_{c}$, where $S_{c}=0.2$, with a gapless linear excitation spectrum characteristic of spin waves. For $S<S_{c}$, $\mu>1$, and there is an energy gap for excitations. Thus, for all physical values of $S$, there is AFLRO.
On the other hand, if two such planes are coupled together antiferromagnetically, and the interplane coupling is very large compared to intraplane coupling, the ground state is obviously a valence bond solid of interplane singlets, and the intraplane correlation length is zero. Thus, there must be at least one phase transition as the interplane coupling is increased. We now analyze this for general $S$ using Schwinger boson MFT.
The Hamiltonian in this case is
$$\begin{aligned}
H=J\sum_{\langle ij \rangle} {\bf{S}}_{i} {\bf{S}}_{j} +
J_{\perp}\sum_{\langle ij
\rangle_{z}} {\bf{S}}_{i} {\bf{S}}_{j}, \nonumber\end{aligned}$$
where $\sum_{\langle ij \rangle}$ sums over n.n. on the same plane and $\sum_{\langle ij \rangle _{z}}$ sums over n.n. on different planes. Since there is still translational invariance, the mean field Lagrange multiplier will be the same on all sites. However, the lack of symmetry between intraplane and interplane bonds means two mean field ampitudes must be introduced for the bond decoupling. Letting these be $A$ and $B$ respectively, and taking them to be both real, the mean field Hamiltonian is now
$$\begin{aligned}
H_{MF} = &&E_{0} + \lambda\sum_{i\sigma} (\tilde{b}^{\dag}_{i\sigma}
\tilde{b}_{i\sigma}) - 2JA\sum_{\langle ij \rangle}
(\tilde{A}^{\dag}_{ij}+\tilde{A}_{ij})
\nonumber\\
&-& 2J_{\perp}B\sum_{\langle ij \rangle _{z}}
(\tilde{A}^{\dag}_{ij}+\tilde{A}_{ij})\nonumber,\end{aligned}$$
where $E_{0} = 2NJS^{2}-2\lambda NS + 4JA^{2}N + NJ_{\perp}S^{2}/2 +
J_{\perp}B^{2}N$. As before, we diagonalize $H_{MF}$ through the Bogoliubov transformation, giving
$$\begin{aligned}
H_{MF}=E_{0}-\lambda N
+\sum_{\bf{k}\sigma}\omega_{\bf{k}\sigma}(\alpha^{\dag}_{\bf{k}\sigma}\alpha
_{\bf{k}\sigma} + \beta^{\dag}_{\bf{k}\sigma}\beta_{\bf{k}\sigma} + 1),
\nonumber\end{aligned}$$
The excitation energies are given by
$$\begin{aligned}
\omega_{{\bf{k}},\sigma} = [\lambda^{2} - (2JA\sum_{i=1}^{d} \cos k_{i} +
J_{\perp}B\sigma)^{2}] ^{\frac{1}{2}}. \nonumber\end{aligned}$$
where $\sigma= \pm 1$. Minimizing $H_{MF}$ with respect to $\lambda$, $A$, and $B$ gives the self-consistent equations:
$$\begin{aligned}
S+\frac{1}{2} &=& \frac{1}{2N} \sum_{\bf{k},\alpha} \frac{\mu} {(\mu^{2} -
\Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}} \nonumber\\
A &=& \frac{1}{2N} \sum_{\bf{k},\alpha} \frac{\Gamma_{\bf{k},\alpha}}
{(\mu^{2} - \Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}}
(\frac{\sum_{i=1}^{d} \cos k_{i}} {d})\nonumber\\
B &=& \frac{1}{2N} \sum_{\bf{k},\alpha} \frac{\Gamma_{\bf{k},\alpha}
\alpha} {(\mu^{2} - \Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}},
\label{consist1}\end{aligned}$$
where $\Gamma_{\bf{k},\alpha} = (\sum_{i=1}^{d} \cos k_{i} + Q \alpha)/d$ and $Q=J_{\perp}B/(2JA)$. Note that $\mu$, the excitation gap, must be greater than or equal $1+Q/d$. In particular, in the case of bose condensation, the value of $\mu$ is fixed to $1+Q/d$ and hence the summations in Eq. (\[consist1\]) turn out to be a function of the parameter $Q$ only. The magnetization $m_{s}$ is calculated by solving the self-consistent equations with the summations converted into intergrals:
$$\begin{aligned}
S+\frac{1}{2} &=& m_{s} + \frac{1}{4} \int \frac{d^{d} {\bf{k}}} {(2\pi)^{d}}
\sum_{\alpha} \frac{\mu_{0}} {(\mu_{0}^{2} -
\Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}} \nonumber\\
\tilde{A} &=& m_{s}+\frac{1}{4} \int \frac{d^{d} {\bf{k}}} {(2\pi)^{d}}
\sum_{\alpha} \frac{\Gamma_{\bf{k},\alpha}} {(\mu_{0}^{2} -
\Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}} (\frac{\sum_{i=1}^{d} \cos
k_{i}} {d} )\nonumber\\
\tilde{B} &=& m_{s}+\frac{1}{4} \int \frac{d^{d} {\bf{k}}} {(2\pi)^{d}}
\sum_{\alpha} \frac{\Gamma_{\bf{k},\alpha} \alpha} {(\mu_{0}^{2} -
\Gamma_{\bf{k},\alpha}^{2})^{\frac{1}{2}}}, \label{consist2}\end{aligned}$$
where $\mu_{0} = 1+Q/d$. These equations [@Millis] hold so long as they give $m_{s}>0$, ie., Bose condensation, otherwise Eqs. (\[consist2\]) should be used with $\mu$ also as an unknown parameter. In principle, we can solve for $Q$ and $m_{s}$ or $\mu$. In practice, the form of these equations allow us to avoid this by plugging in an arbitrary values of $Q$ into the equations to find out $m_{s}$ or $\mu$, and then $A$ and $B$. The self-consistency is then reduced to using the values of $A$, $B$, and $Q$ to determine the value of $\beta \equiv J_{\perp}/J$.
The behavior of $Q$ as a function of $\beta$ for $S=1/2$ is shown in Fig. \[qvsj\], and is representative of all $S$. Discounting the trivial solutions $Q=0$ and $Q=\infty$, corresponding to independent planes and IVBS respectively, there are $Q\neq 0$ solutions indicating both intraplane and interplane correlations. For small $\beta$, there is only one solution, with Q increasing from $0$ with $\beta$. For $\beta > 4(S+1/2)^{2}$, a second branch of solution, beginning at infinity appears. The two solutions merge at some larger value of $\beta$, and beyond that, only the trivial solutions remain. The significance of these solutions can be understood if we consider the energy $E(Q)$ obtained by minimizing the energy with respect to all other parameters except $Q$. Then, the non-trivial solutions are extrema of $E(Q)$. Thus, for $\beta <
\beta_{0}=4(S+1/2)^{2}$, the solution of $Q$ corresponds to a global minimum in $E(Q)$, and describes the ground state. For $\beta >
\beta_{0}$, the upper branch corresponds to a local maximum while the lower branch remains a local minimum. The local maximum begins at $Q=\infty$ at $\beta_{0}$, and moves towards the local minimum with increasing $\beta$. Eventually, the two extrema merge into a saddle point at $\beta_{2}$. Beyond that, $E(Q)$ is strictly decreasing with $Q$. By continuity, this means somewhere between $\beta_{0}$ and $\beta_{2}$, $E(\infty)$ must cross from being greater than $E(Q_{1})$ to less than it, where $Q_{1}$ is the lower branch solution. Thus, at this value $\beta_{1}$, the ground state jumps from the 2D correlated state described by $Q_{1}$ to the interplane VBS state.
We can solve for the value of $m_{s}$ at the non-trivial solutions. Initially, $m_{s}$ increases with increasing $Q$, but eventually will decrease, vanishing at some $Q_{c}$. For sufficiently large $S$, $Q_{c}$ will belong to the upper branch (maximum energy) solution. More importantly, in the case where $Q_{c}$ lies in the lower branch, its $\beta$ value changes from less than to greater than $\beta_{1}$ with increasing $S$. Thus, the transition from the LRO’ed state to disordered state is second order for small $S$, but becomes first order for larger $S$. In the former case, there is a subsequent transition from a disordered state with finite $Q$, hence with both interplane and intraplane short-ranged correlations, to the $Q=\infty$ state with only interplane correlations. Along with the jump in $Q$ is a discontinous jump in the gap. It is tempting to associate this jump as a transition from some disordered state associate with a single plane to the non-degenerate IVBS. More likely this transition is probably an artifact of the Schwinger boson MFT, and indicates a relative sharp drop in the intraplane correlation length and a sharp rise in the gap. This is similar to the finite temperature MFT solution for a single plane, where $A$, hence short-ranged correlation, drops to zero above some finite temperatures [@Sarker]. In the latter case of first order transition in sublattice magnetization, the ground state jumps from one with LRO to the IVBS state. Since this latter state should be the correct ground state only in the $\beta$ goes to infinity limit, we interpret this as the MFT way of showing a transition into a disordered state with a very short intraplane correlation length. The behavior of $m_{s}$ and the gap $\Delta$ as a function of $\beta$ is shown in Fig. \[fig2\] for representative values of $S$. Fig. 2c shows an example of reentrance, where LRO first develops with increasing $\beta$, but is subsequently destroyed when $\beta$ gets too large. This occurs for $S$ smaller than approximately 0.2, the MFT value of $S$ below which the ground state has no LRO, but greater than approximately 0.13, the minimum value of $S$ for LRO at some $\beta$.
The phase diagram of $S$ vs. $\beta$ is shown in Fig. \[phase\]. For $S <
0.13$, the ground state is always disordered. For $0.13<S<0.2$, the system undergoes first a disorder-order and then a order-disorder continuous transition with increasing $\beta$. For $S>0.2$, there is LRO for $\beta =
0$, and only the order-disorder transition remains. This transition is continuous until it terminates at a tricritical point at $S \approx 0.35,
\beta \approx 2.92$, beyond which the continuous transition is preempted by a first order transition. Thus, for $S>0.35$, there are values of $\beta$ where the LRO’ed state is not the ground state, but is nevertheless metastable. The continuous transition phase boundary remains metastable until $S \approx
0.4$, beyond which the $m_{s}=0_{+}$ state moves into the upper branch and becomes unstable. In all cases of $S$ where a disordered ground state with finite $Q$ exists, a subsequent “first-order transition” occurs, with a discontinuous jump in $Q$ and the gap $\Delta$. As mentioned above, we interpret the jump as unphysical, and represents in reality a relatively sharp drop in the 2D correlation length.
We can understand why large $S$ favors a first order transition quite simply in terms of spin wave theory [@Hida1; @Hida2]. The Neel state energy is $E_{N} =
S^{2}(2Jz+J_{\perp})$ while the energy of the IVBS state is $E_{V}=J_{\perp}S(S+1)$. Equating the two implies an estimate for the first order transition at $\beta_{1}$ of the order of $S$ for large $S$. Within spin wave theory, the sublattice magnetization is given by $m_{s}$ in Eq. (\[consist2\]) with $B/A=1$. For large $\beta$, the integral on the LHS scales as $\sqrt{\beta}$. If we set $m_{s}=0$ as an estimate for the critical value $\beta_{c}$ for continuous transition, then $\beta_{c}$ is of order $S^{2}$. Thus, for large $S$, $\beta_{1}$ is much less than $\beta_{c}$.
Within MFT, the tricritical point and even the metastable continuous transition boundary occurs below the minimum physical value of $S=1/2$. Thus, a first order transition is predicted for all physical systems described by the model. In fact, the sublattice magnetization jump at transition for $S=1/2$ is about 30% of that at $\beta=0$, clearly contradicting the results of numerical work on the model for $S=1/2$, which supports a continuous transition in the same universality class as the finite temperature transition of the $3D$ classical Heisenberg model. On the other hand, there is no reason to expect the Schwinger boson MFT to give the exact answer, so the true position of the tricritical point might very well be above $S=1/2$. This is particularly so since by relaxing the local constraint to a global one, unphysical states are included in the mean field solution, and the MF energy is not even variational. Thus, using these MF energies to find the position of first order transition is necessarily suspect. Nevertheless, we believe the prediction of larger $S$ favoring a first order transition to be correct, and the nature of phase transition in the bilayer system is non-universal. For example, the transition for $S=1/2$ may become first order if there is a sufficiently large next nearest neighbor ferromagnetic interaction. Conversely, first order transition may become continuous if frustration is introduced. In other words, the value of $S$ at the tricritical point can be changed by enlarging the parameter space. The seeming contradiction to the fact that the 2+1 $D$ NLSM has only continuous transition is resolved by noting that the mapping of the Heisenberg model into the NLSM is legitimate only if the correlation length is long, which does not have to be the case of the disordered state close to a first order transition. Also of interest is that with sufficient frustration, the single-layered system can be disordered for $S=1/2$ or other physical values, and the reentrance behavior for small $S$ discussed above in the bilayered system can be physically observed.
Within our MFT and according to general arguments, first order transition implies the existence of metastable states with finite sublattice magnetization. This may lead to observeable dynamics characteristic of macroscopic quantum tunneling. It would also be of significance with respect to Monte Carlo type numerical calculations [@Hida3; @Dagotto; @Sandvik1; @Sandvik2] due to problems of being “stuck” in the metastable minimum. For example, the first order transition may be missed if the metastability persists till the would-be continuous transition.
Extension to the Fractional Dimensions
======================================
We have seen that for the 2D bilayered square lattice antiferromagnet, Schwinger boson MFT shows the physically interesting case of $S=1/2$ as undergoing a first order transition. Since the continuous transition is favored by small $S$, hence increasing quantum fluctuations, one way of getting a continuous transition in MFT for $S=1/2$ is to go to a dimension below $2$. In this section, we show that indeed this is the case. En route, we present the phase diagram of $S$ vs. $d$ (Fig. \[critdim\]) for a single layer. These non-integer dimension results may be relevant to the physics of the Heisenberg antiferromagnet on percolating clusters, which have fractal dimensionality.
We first perform the Schwinger boson MFT for a single hypercubic lattice in $d$ dimension (square lattice for $d=2$). The self-consistent equations (Eq. (\[ms2\])) depend on $d$ only through the momentum sum, which must be analytically continued to non-integer dimensions. This can be done by using the gaussian identity:
$$\begin{aligned}
\frac{1} {(1 \pm \gamma_{\bf{k}})^{\frac{1}{2}}} = 2 \sqrt{\frac{d}{\pi}}
\int^{\infty}_{0} dx ~e^{-x^{2}(d \pm \sum^{d}_{i=1} \cos k_{i})},
\label{gauss}\end{aligned}$$
to rewrite Eq. (\[ms2\]) as
$$\begin{aligned}
\frac{2d}{\pi} \int^{\infty}_{0} dxdy~ e^{-(x^{2}+y^{2})d} (\int^{\pi}_{-\pi}
\frac{dk}{2\pi} e^{-\cos k (x^{2}-y^{2})} ) ^{d}. \label{bessel}\end{aligned}$$
In this form, the analytic continuation to arbitrary $d$ is obvious (see Appendix). In Fig. \[msgap\], the result for $S=1/2$ is shown. As $d$ is decreased below $2$, $m_{s}$ decreases and vanishes at some critical dimension $d_{c}=1.46$. Below $d_{c}$, the excitation spectrum has a gap, which rises with decreasing $d$. These behaviors are representative for all $S$. However, it is known that for the simple Heisenberg Hamiltonian, $1/2$ integer spin chains and integer spin chains are intrinsically different in that the former should be gapless, which can be understood as due to the presence of a topological term the appropriate NLSM. Thus MFT must break down for $1/2$ integer spins even qualitatively as $d$ gets sufficiently close to $1$, and $\Delta$ must decrease again.
Next we generalize our bilayer calculation to two coupled hypercubes in $d$ dimension. The analytic continuation of Eqs (\[consist2\]) can again be done using the gaussian identity (\[gauss\]). As expected, a continuous transition can now be observed within MFT if $d$ is reduced sufficiently from $2$. For $d <1.46$, the critical dimension for LRO for a single hypercube, the reentrance seen with increasing $\beta$ for $S<0.2$ is seen for $S=1/2$. So far, we have concentrated on lowering the dimension from $2$. Of course, raising it would have the opposite effect. For example, for two $3D$ hypercubes, the $S=1/2$ transition would be strongly first order within MFT, and so even taking into account inaccuracy of MFT, strongly implies a first order transition.
Finally we discuss the critical phenomena of the continuous transition of this model. Analyzing Eqs. (2) and (3) close to the transition, we find the staggered magnetization vanishes linearly, while the gap vanishes as $(\beta -
\beta_{c})^{s}$ with $s=1/(d-1)$ for $d<3$, and $s=1/2$ for $d>3$ (there are logarithm corrections at $d=3$). These MF exponents are the same as those for the finite temperature transition of a single hypercube with the substitution $d\rightarrow d+1$, reflecting the quantum nature of the present transition.
Appendix
========
Notice that in Eq. (\[bessel\]), the integral inside the bracket is a modified Bessel function of the first kind $I_{0}(x^{2}-y^{2})$. Therefore the first equation of Eq. (\[consist2\]) can be written as:
$$\begin{aligned}
S+\frac{1}{2} = m_{s} + \frac{2d}{\pi} \int^{\infty}_{0} dxdy
(e^{-(x^{2}+y^{2})} I_{0}(x^{2}-y^{2}))^{d}. \nonumber\end{aligned}$$
Take the transformation:
$$\begin{aligned}
u=x^{2}-y^{2}, ~~~~~~v=x^{2}+y^{2} \nonumber\end{aligned}$$
with $v\geq |u|$, for the integration variable will give us:
$$\begin{aligned}
S+\frac{1}{2} &=& m_{s} + \frac{2d}{\pi} \int^{\infty}_{-\infty} du
(I_{0}(u))^{d} \int^{\infty}_{|u|} dv \frac{e^{-vd}}{\sqrt{v^{2}-u^{2}}}
\nonumber \\
&=& m_{s} \frac{2d}{\pi} \int^{\infty}_{0} du (I_{0}(u))^{d}
K_{0}(ud).\nonumber\end{aligned}$$
It reduces the final formula into a single integral of $I_{0}$ and $K_{0}$, modified Bessel function of the second kind. In this form, the integral indeed converges much faster than the original form in Eq. (\[consist2\]) and consequently save much of the computation time. The same trick is also applied to both interplane and intraplane mean field equations.
D. P. Arovas and A. Auerbach, Phys. Rev. B [**38**]{}, 316 (1988). S. Sarker, C. Jayaprakash, H. R. Krishnamurthy and M. Ma, [**40**]{}, 5028 (1989). A. J. Millis and H. Monien, Phys. Rev. B [**50**]{}, 16606 (1994). T. Matsuda and K. Hida, J. Phys. Soc. Jpn. [**59**]{}, 2223 (1990). K. Hida, J. Phys. Soc. Jpn. [**59**]{}, 2230 (1990). K. Hida, J. Phys. Soc. Jpn. [**61**]{}, 1013 (1992). E. Dagotto, J. Riera, D. Scalapino, Phys. Rev. B [**45**]{}, 5744 (1992). A. W. Sandvik and D. J. Scalapino, Phys. Rev. Lett. [**72**]{}, 2777 (1994). A. W. Sandvik and M. Veki$\acute{c}$, J. Low Temp. Phys. [**99**]{}, 367 (1995). D. Yoshioka et. al., to be published. S. Maekawa et. al., to be published. F. D. M. Haldane, Phys. Rev. Lett. [**61**]{}, 1029 (1988).
|
---
author:
- |
JLQCD Collaboration: $^{a}$[^1], S. Hashimoto$^{a,b}$, T. Kaneko$^{a,b}$, J. Koponen$^{a}$\
[$^{a}$ High Energy Accelerator Research Organization (KEK), Ibaraki 305-0801, Japan]{}\
$^{b}$ School of High Energy Accelerator Science, SOKENDAI (The Graduate University for Advanced Studies), Ibaraki, 305-0801, Japan\
title: 'Study of intermediate states in the inclusive semileptonic $B \rightarrow X_c l \nu$ decay structure function'
---
Introduction
============
The semileptonic decays of $B$ meson to excited states of $D$ meson have not been well understood. In this work, we focus on the decays $B\to D^{**}\ell\nu$, where $D^{**}$ stands for one of orbitally excited states $D_0^*$, $D_1^\prime$, $D_1$ and $D_2^*$. Among them, the former two have large width of order $200-400$ MeV, while the other two are narrower, $30 - 50$ MeV [@Tanabashi:2018oca]. There are some theoretical estimates on these semileptonic decay rates, which suggest that the narrower states have much larger rates than the broader ones [@Uraltsev:2000ce]. The experimental data do not support this expectation and the problem remains for more than a decade. In this study, we use lattice QCD calculation to obtain some insight into this problem.
In general, excited states are more difficult to treat in lattice calculations, because of larger statistical noise. Preparing an interpolating operator that efficiently creates the desired state is also challenging since the states have non-trivial structures. Here, we use the forward scattering matrix element of $B$ meson, which was developed for a calculation of inclusive decay structure function [@Hashimoto:2017wqo]. It does not require an explicit identification of the individual states. Rather, the states are created through a flavor changing vector or axial-vector current just as in the physical process of $B\to D^{**}\ell\nu$. From a relevant four-point function, we are able to identify the corresponding contributions.
The rest of the paper is organized as follows. Section \[section2\] and \[section2.1\] introduce the form factors for the P-wave states. Section \[section3\] presents our lattice computation strategy. Section \[section4\] contains our results and conclusions for the zero and non-zero recoil cases. Finally, Section \[section6\] presents our conclusions.
P-wave states $D^{**}$ and their form factors {#section2}
=============================================
In the static limit $(m_b, m_c \to \infty)$, the heavy quark symmetry emerges and the meson spectrum can be constructed by combining the spin $1/2$ of the heavy quark with the total angular momentum and parity $j^{P}$ of the light degrees of freedom (light quarks and gluons). For instance the S-wave states of total spin-parity $0^{-}$ and $1^{-}$ become degenerate. For the P-wave states, the light degrees of freedom may have $j^{P} = (1/2)^{+}$ or $(3/2)^{+}$, which combined with the heavy quark spin produce the states of $J^{P} = (0^{+}, 1^{+})$ as well as $(1^{+}, 2^{+})$, respectively. When $m_b$ and $m_c$ are finite the states are classified according to their parity $P$ and total angular momentum $J$. They are named $D^{*}_0$, $D_1^{'}$, $D_1$ and $D_2^{*}$, respectively.
In the heavy quark limit the relevant matrix elements for $B \to D^{**} \ell\nu$ decays can be parametrized by two form factors, the Isgur-Wise functions $\tau_{1/2}$ and $\tau_{3/2}$ [@Isgur:1991wr]:
\[isgur1\] & \_[1/2]{}(w)(v-v’)\_,\
& \_[3/2]{}(w)((w+1) \^[\*]{}\_v\^ - \^[\*]{}\_ v\^v\^ v\^\_ ),
where $v$ and $v'$ are the velocities associated with the $B$ and $D^{**}$ mesons respectively, $w = (v' \cdot v)$ and $\epsilon$ is the polarization tensor of the $D^{**}$ mesons. Previous theoretical estimates were obtained through sum rules. The most relevant one in this context was derived by Uraltsev [@Uraltsev:2000ce], $$\begin{aligned}
\label{ulratsev}
\sum_{n} \left( |\tau_{3/2}^{(n)}(1)|^2 - |\tau_{1/2}^{(n)}(1)|^2 \right) = \frac{1}{4},\end{aligned}$$ where $\tau^{(0)}_{1/2} \equiv {\tau}_{1/2}$, $\tau^{(0)}_{3/2} \equiv {\tau}_{3/2}$ and the sum over $n$ is done for all $j =$ $1/2$ and $3/2$ states. One may expect saturation from the ground states, leading to $ |\tau_{3/2}^{(0)}(1)|^2 - |\tau_{1/2}^{(0)}(1)|^2 \approx \frac{1}{4} $ and consequently to $|\tau_{1/2}(1)| < |\tau_{3/2}(1)|$. The sum rule concerns the zero-recoil limit $(w = 1)$, where the $B$ and the $D^{**}$ mesons have the same velocity. To obtain the decay rates, however, one has to integrate over $w$, and one assumes that the inequality remains for $w \neq 1$.
BaBar [@Aubert:2008ea] and BELLE [@Liventsev:2007rb] have measured the composition of the semileptonic $B \to X_c l \nu$ decay. It turned out that $X_c$ is $70\%$ composed by $D$ and $D^{*}$ mesons (S-wave states) and $15\%$ by $D_{1}$ and $D_2$ ($j = 3/2$ states). A natural candidate for the remaining $15\%$ would be the $D_{0}$ and $D^{\prime}_1$ ($j = 1/2$ states) mesons. However, this proposal seems to be in conflict with the theoretical prediction discussed above, which has been called the “$1/2$ versus $3/2$ puzzle”.
Form factors of $B \to D^{**}l\nu$ {#section2.1}
==================================
The Isgur-Wise form factors (\[isgur1\]) are not used directly in our work. We use the conventional definition of the $B \to D^{**}$ form factors given by
- P-wave $j^{+} = \frac{3}{2}^{+}$ states:
\[P\_1\] &\^[-1]{} = f\_[V\_1]{}\^[\*]{}\_+(f\_[V\_2]{}v\_+f\_[V\_3]{}v’\_)(v),\
&\^[-1]{} = - if\_A\_\^[\*]{}\_v\_v’\_,
- P-wave $j^{+} = \frac{1}{2}^{+}$ states:
\[P\_2\] & = 0,\
&\^[-1]{} = g\_[+]{}(v\_ +v’\_) + g\_[-]{}(v\_ - v’\_),\
&\^[-1]{} = g\_[V\_1]{}\^[\*]{}\_ + (g\_[V\_2]{}v\_ + g\_[V\_3]{}v’\_)(\^[\*]{}v),\
&\^[-1]{} = ig\_A \_\^[\*]{}\_v\_v’\_.
Here, $f_{V_1}$, $f_{V_2}$, $f_{V_3}$ and $f_A$ are form factors for $j^{+} = {3}/{2}^{+}$ and $g_+$, $g_-$, $g_{V_1}$, $g_{V_2}$, $g_{V_3}$ and $g_A$ represent the ones for $j^{+} = {1}/{2}^{+}$. They are functions of $w = v \cdot v'$.
In the heavy quark expansion, these form factors can be written in terms of the Isgur-Wise functions $\tau_{3/2}(w)$, $\tau_{1/2}(w)$ plus the terms to represent the $1/m_c$ and $1/m_b$ corrections. Such calculation was performed for the P-wave decay modes [@Leibovich:1997em], which we use in the following.
Lattice computation strategy {#section3}
============================
We utilize the forward-scattering matrix element, which represents the inclusive decay of the $B$ meson. It contains contributions from all possible final states with a certain weight factor. We compute a four-point function corresponding to the matrix element: $$\begin{aligned}
\label{4point}
C_{\mu\nu}^{JJ}(t;\vec{q}) = \int d^3\vec{x}e^{i\vec{q}\cdot {\vec{x}}}\frac{1}{2M_B}\braket{B(\vec{0})|J_{\mu}^{\dagger}(\vec{x},t)J_{\nu}(0)|B(\vec{0})},\end{aligned}$$
![\[results\_1\] Four-point correlator of double insertions of vector and axial currents.](result.png){width="23pc"}
which can be extracted by taking a ratio of the four-point function to two-point functions as $$\begin{aligned}
\label{ratios}
\frac{C_{\mu\nu}^{SJJS} (t_{snk}, t_1, t_2, t_{src})}{C^{SL}(t_{snk}, t_2)C^{LS}(t_1, t_{src})} \to \frac{\frac{1}{2M_B}\braket{B(\vec{0})|J_{\mu}(\vec{q}, t_1)^{\dagger}J_{\nu}(\vec{q}, t_2) |B(\vec{0})}}{ \frac{1}{2M_B}|\braket{0|P^L|B(\vec{0})}|^2 } .\end{aligned}$$ For details about other uses of (\[4point\]) we refer to [@Hashimoto:2017wqo]. An example of the correlator ratio (\[ratios\]) is shown in Fig. \[results\_1\]. It is plotted as a function of $t_2 - t_1$, the separation between two current insertions. One can see the exponential fall-off due to charm quark propagation.
In the zero-recoil limit, $\vec{q} = \vec{0}$, there are two distinct channels. One is those for the temporal vector current $V^0$ and the spatial axial-vector current $A^k$, which are the upper two lines in Fig. \[results\_1\] and correspond to the S-wave final states $D$ and $D^{*}$, while for $A^0$ and $V^k$ the final states have an opposite parity and they correspond to the P-wave states shown by the lower two lines. Therefore, for sufficiently large separations $t_2 - t_1$, the final states are dominated by the $D^{**}$ states and the exponential fall-off may be written in terms of the corresponding decay form factors:
\[fithere\] &C\^[A\_0A\_0]{}(t) = |g\_[+]{}(1)|\^2e\^[-m\_[D\^[\*]{}\_[0]{}]{}t]{},\
&C\^[V\^kV\^k]{}(t) = e\^[-m\_[D\^[\*]{}\_1]{}t]{} + e\^[-m\_[D\_1]{}t]{}.
Then, we are able to extract the form factor $|g_{+}(1)|$ as well as a combination of $|g_{V_1}(1)|$ and $|{f_{V_1}}(1)|$ by fitting the lattice data at large time separations $t = t_2 - t_1$. The contribution from $|g_{V_1}(1)|$ is small for a reason that we describe later.
Results {#section4}
=======
We have performed a set of lattice QCD simulations with $2+1$ flavors of dynamical quarks using the tree-level improved Symanzik gauge action and the Möbius domain-wall fermions. Our computation preserves chiral symmetry and has a large lattice cutoff $a^ {-1} \simeq 2.5 - 4.5$ GeV. The strange quark mass $m_s$ is simulated close to its physical value, whereas the degenerate up and down quark mass $m_{ud}$ corresponds to pion masses as low as $M_{\pi} \sim 230$ MeV. In this work, we use a $48^3 \times 96$ lattice of $a = 0.055$ fm. The spatial lattice size $L$ satisfies the condition $M_{\pi} L \gtrsim 4 $ to control finite volume effects. The charm quark mass $m_c$ is set to its physical value, whereas we take bottom quark mass $m_b = 1.25^{4}m_c$, which is smaller than the physical value. The spectator quark in this calculation is strange, so that the relevant decays are actually those of $B_s \to D^{**}_{s} l \nu$. More information about our lattice data can be found in [@Nakayama:2016atf]. The statistics is $100$ independent gauge configurations with $4$ source locations on each configuration.
The form factors for P-wave final states introduced in (\[P\_1\]) and (\[P\_2\]) can be expanded in $1/m_c$ and $1/m_b$ as [@Leibovich:1997em]
\[uma\] &f\_[V\_1]{}(w) = - \[w\^2 - 1 + 8\_c (|’ - |)\] (w) + ...,\
&g\_[+]{}(w) = - (\_c + \_b)(|\^[\*]{} - |)(w) + ...,\
\[duas\] &g\_[V\_1]{}(w) = \[w-1+(\_c - 3\_b)(|\^[\*]{} - |)\](w)+...,
where $\zeta(w) = 2\tau_{1/2}(w)$ and $\tau(w) = \sqrt{3}\tau_{3/2}(w)$ are the Isgur-Wise functions, $(\bar{\Lambda}^\prime - \bar{\Lambda})$ and $(\bar{\Lambda}^* - \bar{\Lambda})$ stand for the mass difference between S-wave and P-wave states and $\epsilon_c = 1/2m_c$, $\epsilon_b = 1/2m_b$. As anticipated, the form factors for the P-wave states vanish in the zero-recoil limit when $\epsilon_b = \epsilon_c = 0$, because of the parity conservation. Away from the heavy quark limit, small contribution arises at the order of $\epsilon_c$ and $\epsilon_b$, which explain the small amplitudes of two lower lines found in Fig. \[results\_1\].
Using $(\bar{\Lambda}^* - \bar{\Lambda}) = 0.36$ MeV and $(\bar{\Lambda}^\prime - \bar{\Lambda}) = 0.40$ MeV following [@Bernlochner:2016bci], we extract $\tau(1)$ and $\zeta(1)$ from the lattice data of (\[fithere\]). The contribution of $|g_{V_1}(1)|$ is neglected because $\epsilon_c - 3\epsilon_b$ in (\[duas\]) is numerically small. Our results are $\tau_{3/2}(1) = 0.45(7)$ and $\tau_{1/2}(1) = 0.39(6)$. This suggests that $\tau_{3/2}(1) \sim \tau_{1/2}(1)$, which is in agreement with the experimental results $\Gamma(B \to D^{**}_{1/2}l \nu) \approx \Gamma (B \to D^{**}_{3/2} l \nu)$. Our result is also consistent with the phenomenological analysis of the experimental data [@Bernlochner:2016bci]. A previous lattice calculation was done in the heavy quark limit [@Blossier:2009vy]. Its results $\tau_{3/2}(1)~=~0.53(2)$ and $\tau_{1/2}(1) = 0.30(3)$ favor the sum rule expectation $\tau_{3/2}(1) > \tau_{1/2}(1)$. Our result is not inconsistent with theirs within a large error. However, statistical error has to be reduced before drawing any firm conclusions.
Inserting finite momentum in the final state, both S-wave and P-wave states contribute to (\[4point\]). Since the P-wave amplitude is relatively small, we need to carefully subtract the S-wave states to extract the P-wave contributions. In Fig. \[V1V1\_001\_wsub\] (upper plot) the curve entitled “$V_1V_1$ from $B \to D$” represents the expected S-wave states, which is a reconstructed from the lattice calculation dedicated for the $B \to D^{(*)} l \nu$ decay [@Kaneko:2018mcr]. The curve named “P-wave Contributions” is obtained after subtracting the S-wave contribution. One can clearly find that the lower energy state corresponding to the S-wave contribution is removed and the higher energy state is left (see also the effective mass plot Fig. \[V1V1\_001\_wsub\] (bottom) before (square) and after the subtraction (star)).
![\[V1V1\_001\_wsub\] On the left-side we have the four-point functions ratios for the different lattice data. On the right-side we present the effective energy result for the P-wave states.](V1V1_001_wsub "fig:"){width="23pc"}![\[V1V1\_001\_wsub\] On the left-side we have the four-point functions ratios for the different lattice data. On the right-side we present the effective energy result for the P-wave states.](V1V1_lattice2019_eff "fig:"){width="23pc"}
With a momentum insertion in the Z direction, i.e. momentum $p' = \frac{2\pi}{L} (0,0,1)$, we extract the form factors $f_{V_1}$ and $f_{V_3}$ from perpendicular $V_1V_1$ and longitudinal $V_3V_3$ vector channels, res-pectively. Following the “Approximation $A$” of [@Leibovich:1997em; @Bernlochner:2016bci], which means ${O}(w-1) \sim {O}(\epsilon_{c,b})$ and their higher orders are truncated, we use
\[approxA\] &f\_[V\_1]{}(w) = \[(1-w\^2)(w) - 4\_c(w+1)(w|’ - |)(w) \]
and
\[approxA1\] &wf\_[V\_1]{}(w) + (w\^2 - 1)f\_[V\_3]{}(w) = \[(1-w\^2)(w) - 4\_c(w+1)(w|’ - |)\] + (w-2)(w)
to extract the Isgur-Wise form factor $\tau(w)$. In the same way, we also obtain $\zeta(w)$ from the form factor $g_{+}$ through the $A_0A_0$ channel. Our results are $\tau(w)=0.539(33)$ for $V_1V_1$, $\tau(w) = 0.455(27)$ for $V_3V_3$ and $\zeta(w) = 1.21(14)$, at $w = 1.027$. The inconsistency between $V_1V_1$ and $V_3V_3$ may be due to the approximation involved in the analysis.
The slopes of the Isgur-Wise functions $\tau'(w)$ and $\zeta'(w)$ defined through $$\begin{aligned}
\tau(w) = \tau(1)[1+\tau'(w-1)],\\
\zeta(w) = \zeta(1)[1+\zeta'(w-1)],\end{aligned}$$ are obtained combining with the zero-recoil results. We obtain $\tau'(1) = -7.8(6.3)$ for $V_1V_1$, $\tau'(1) = -12.2(5.3)$ for $V_3V_3$ and $\zeta'(1) = 21(12)$. Even with the large error, our final results are in agreement with the phenomenological results [@Bernlochner:2016bci].
Discussions {#section6}
===========
The results shown in this write-up are a by-product of a calculation of the inclusive decay structure functions [@Hashimoto:2017wqo]. By inspecting the energy and amplitude of the final states contributing to the forward-scattering matrix elements, we are able to identify those states as the P-wave D mesons, which is natural since they are contributing to the physical processes as experimentally observed. The method to extract the excited state contribution is not particularly superior compared to dedicated calculations because the statistical noise is larger for four-point functions. It may be useful however, when a proper interpolating operator is not known for the excited states like those for the $j = 1/2$ states. Also, this work provides a good consistency test of the strategy to obtain the inclusive decay structure functions.
Acknowledgment {#acknowledgment .unnumbered}
--------------
Numerical computations are performed on Oakforest-PACS at JCAHPC. This work was supported in part by JSPS KAKENHI Grant Number JP18H03710 and by MEXT as “Priority Issue on post-K computer”.
[9]{} M. Tanabashi [*et al.*]{} \[Particle Data Group\], Phys. Rev. D [**98**]{} (2018) no.3, 030001. doi:10.1103/PhysRevD.98.030001 N. Uraltsev, Phys. Lett. B [**501**]{} (2001) 86 \[arXiv:hep-ph/0011124\]. S. Hashimoto, PTEP [**2017**]{} (2017) no.5, 053B03 \[arXiv:hep-lat/1703.01881\]. N. Isgur, M. B. Wise and M. Youssefmir, Phys. Lett. B [**254**]{} (1991) 215. B. Aubert [*et al.*]{} \[BaBar Collaboration\], Phys. Rev. Lett. [**101**]{} (2008) 261802 \[arXiv:hep-ex/0808.0528\]. D. Liventsev [*et al.*]{} \[Belle Collaboration\], Phys. Rev. D [**77**]{} (2008) 091503 \[arXiv:hep-ex/0711.3252\]. A. K. Leibovich, Z. Ligeti, I. W. Stewart and M. B. Wise, Phys. Rev. D [**57**]{} (1998) 308 \[arXiv:hep-ph/9705467\]. K. Nakayama, B. Fahy and S. Hashimoto, Phys. Rev. D [**94**]{} (2016) no.5, 054507 \[arXiv:hep-lat/1606.01002\]. F. U. Bernlochner and Z. Ligeti, Phys. Rev. D [**95**]{} (2017) \[arXiv:hep-ph/1606.09300\]. B. Blossier [*et al.*]{} \[European Twisted Mass Collaboration\], JHEP [**0906**]{} (2009) 022 \[arXiv:hep-lat/0903.2298\]. T. Kaneko [*et al.*]{} \[JLQCD Collaboration\], PoS LATTICE [**[2019]{}**]{} (2019) 139
[^1]: E-mail: gabriela@kek.jp
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[**Unruh detectors and quantum chaos in JT gravity** ]{}
\
**Abstract**
We identify the spectral properties of Hawking-Unruh radiation in the eternal black hole at ultra low energies as a probe for the chaotic level statistics of quantum black holes. Level repulsion implies that there are barely Hawking particles with an energy smaller than the level separation. This effect is experimentally accessible by probing the Unruh heat bath with a linear detector. We provide evidence for this effect via explicit and exact calculations in JT gravity building on a radar definition of bulk observables in the model. Similar results are observed for the bath energy density. This universal feature of eternal Hawking radiation should resonate into the evaporating setup.
Introduction {#sect:1}
============
One of the main features of finite volume AdS/CFT is that the CFT in question is a unitary and discrete system. The latter follows from the fact that the boundary on which the CFT lives is compact. This implies that quantum gravity in AdS must be a unitary and discrete quantum system. There is however at least one further constraint on the spectrum of quantum gravity in AdS on which we want to focus in this work. It is a consequence of the fact that black holes are chaotic quantum systems [@bhrm].
Roughly speaking there are two hallmarks of chaotic quantum systems. The first is exponential dependence on changes in initial conditions. In a quantum theory this translates into the exponential growth of operators with time [@operatorgrowth]. One popular way to probe this effect is by computing out-of-time-ordered correlators [@SS; @SSmultiple; @Shenker:2014cwa; @bound]. The gravitational translation of this fast scrambling is the exponential redshift close to gravitational horizons, resulting in gravitational shockwaves [@SS; @SSmultiple; @Shenker:2014cwa].
The second hallmark of quantum chaotic systems concerns their level statistics. In particular quantum chaotic systems are very well characterized by the property of level repulsion. Roughly speaking two subsequent energy levels of a chaotic system are rarely close together. A more precise version of this is that the local spectral statistics of any quantum chaotic system can be described using random matrix statistics.[^1]
Black holes are fast scramblers [@sekinosusskind] and shockwave interactions are a universal feature of quantum gravity. Therefore quantum black holes are expected to be chaotic quantum systems [@bhrm]. But if quantum gravity is quantum chaotic, then what is the bulk gravitational interpretation of level repulsion and of random matrix level statistics? Recently this question has been addressed within Jackiw-Teitelboim (JT) gravity [@sss2].[^2] The answer in this theory is that Euclidean wormhole contributions to the gravitational path integral are responsible for level repulsion and random matrix level statistics in the spectrum of the gravitational theory. We expect this conclusion to be true universally.\
\
We conclude that any acceptable theory of quantum gravity in AdS is a discrete and unitary quantum system with random matrix level statistics. One prime consequence of this spectral behavior is the specific late time behavior of boundary correlators. Consider for example the two-point function of a discrete quantum chaotic system with an $L$ dimensional Hilbert space with levels $\lambda_1\dots \lambda_L$: $$\begin{aligned}
{\left\langle {\mathcal{O}}(0){\mathcal{O}}(t) \right\rangle}_\beta
&= \int_{\mathcal{C}}d E_1\,e^{-\beta E_1}\,\int_{\mathcal{C}}d E_2\,e^{it(E_1-E_2)}\,\rho(E_1,E_2)\,\rvert{\mathcal{O}}_{E_1 E_2}\rvert^2\,,\label{erraticbdy}\end{aligned}$$ where $$\rho(E_1,E_2)=\sum_{i=1}^L \delta(E_1-\lambda_i)\sum_{j=1}^L \delta(E_2-\lambda_j)\,.\label{spikes}$$ For a quantum chaotic system the eigenvalue thermalization hypothesis [@eth1; @eth2; @bhrm; @phil] states that $\rvert{\mathcal{O}}_{E_1 E_2}\rvert^2$ is a smooth function of $E_1$ and $E_2$. At early times the Fourier transform in cannot distinguish the delta functions from a coarse-grained version of this spectrum. The result is that at early times this correlator decays exponentially with time. This is the quasinormal mode decay known from quantum fields on a black hole background.
At exponentially late times however the correlator essentially oscillates erratically around an in general nonzero average [@maldainfo]. The nonzero averaged value is explained by random matrix theory [@bhrm]. The erratic oscillations are testimony to the fundamental discreteness of the theory. In fact this erratically oscillating behavior can itself be viewed as distinguishing a chaotic from a regular quantum system [@haake].
In gravity, the bulk explanation for the late time behavior of this correlator involves rather exotic gravitational effects. In particular in order to explain the nonzero average one needs to sum over an infinite number of Euclidean wormhole amplitudes in the Euclidean path integral [@sss; @sss2; @Blommaert:2019hjr; @phil]. To explain the erratic oscillations in JT gravity from the Euclidean path integral one furthermore needs to realize that a single discrete quantum chaotic system can be described by a version of JT gravity which includes a specific set of branes in the gravitational path integral [@paper6; @maxfieldmarolf; @wophilbert].\
\
In this work we would like to introduce a structurally different probe of the chaotic level statistics of quantum black holes. We propose to investigate the low-energy spectral properties of the eternal Hawking-Unruh radiation as detected by a linear Unruh-DeWitt detector. There are two main reasons why we believe this to be an interesting observable.
1. It has recently been advocated [@rw1; @rw2] that the same Euclidean wormholes which play a crucial role in explaining the late time behavior of correlators [@sss; @sss2; @Blommaert:2019hjr; @phil] are key to understanding unitary black hole evaporation from the bulk point of view. Ultimately we want to understand black hole evaporation by tracking what happens to all Hawking quanta emitted from the horizon during the evaporation process, which is more fine-grained bulk information than the early-late entanglement entropy considered in [@rw1; @rw2]. We expect to already see traces of the mechanism in the eternal Hawking-Unruh radiation.
2. Ultimately we want to have a pure bulk gravitational intuition of quantum gravity. In this sense our observable stands out as compared to for example late time boundary correlators. It is inherently a bulk observable. One further incentive to investigate bulk probes in quantum gravity is that in de Sitter or in flat space it is not so natural to formulate questions in the dual theory and so to make progress there we will need to strengthen our understanding of bulk observables in quantum gravity.[^3]
This work is organized as follows.
In **section \[sect:1\]** we argue on general grounds for a modification of the semiclassical formula for the Unruh-Hawking emission probability which accounts for the fact that black holes have chaotic level statistics. We then introduce the setup of JT gravity which we will use to gather concrete evidence for these ideas.
In **section \[s:udw\]** we explicitly verify these expectations. In particular, we couple a massless scalar field to JT gravity and compute the response rate of an Unruh-DeWitt detector which couples linearly to this scalar field. We analyze the detector response in three layers of approximation. Firstly we use the semiclassical approximation. Secondly we discuss the effects of coupling to the Schwarzian reparameterization mode. Finally we include the effects of Euclidean wormholes which give rise to random matrix level statistics. We observe a depletion of the detector response rate at extremely low energies as testimony to the chaotic level statistics of quantum black holes. We generalize to bulk fermionic matter and to other detector couplings.
In **section \[s:hb\]** we compute the spectral energy density in the Unruh heat bath. For the Schwarzian system (disk topology) this was investigated in [@Mertens:2019bvy]. Perhaps surprisingly this spectral energy density is not precisely identical to the detector response. We explain that this is due to ordering ambiguities that arise when promoting classical expressions to gravitational operators. Despite these subtle differences with the detector setup, we find a similar depletion at ultra low energies due to level repulsion.
In **section \[s:concl\]** we comment on gravitational dressings, higher genus contributions to bulk correlators and evaporation. Certain more technical aspects of the story have been relegated to appendices.
Unruh detectors and level repulsion {#s:exp}
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The main conceptual point in this work is that the semiclassical Planckian black body law for Hawking radiation does not take into account the chaotic level statistics of the quantum black holes which emit these quanta.\
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We first briefly review the expected level statistics of black holes. For an arbitrary quantum mechanical system we denote the probability to find an energy level between $E_1$ and $E_1+ d E_1$ and a second energy level between $E_2$ and $E_2+d E_2$ as $$\rho(E_1,E_2)\, d E_1\,d E_2.$$ For a system of which the level spacings are Poisson distributed, the probability to find a level between $E_1$ and $E_1+d E_1$ is independent of the probability to find a second level between $E_2$ and $E_2+d E_2$ $$\rho(E_1)\rho(E_2)\, d E_1\, d E_2.$$ In other words different levels are uncorrelated. An example of a system with implicit Poisson level statistics is a particle in a very large box, typically used to derive the Planckian black body law.
For chaotic quantum systems, the story is quite different. The local level statistics of a quantum chaotic system are described by random matrix theory [@haake; @mehta]. Level statistics in random matrix theory is universal: the multi-density correlators of essentially any chaotic quantum systems (without time-reversal symmetry) are those of the Gaussian unitary ensemble (GUE).[^4] The GUE two-level correlator is[^5] $$\rho(E_1,E_2) = \rho(E_1)\rho(E_2)-\rho(E_1)\rho(E_2)\,{\text{sinc}}^2\, \pi \rho(E_1)(E_1-E_2).\label{re1e2}$$ This leads to the normalized two-density correlator $$\frac{\rho(E_1,E_2)}{\rho(E_1)\rho(E_2)}=\quad \raisebox{-15mm}{\includegraphics[width=50mm]{rep.pdf}}\quad.\label{8}$$ This depletion around $E_1=E_2$ is the characteristic level repulsion of quantum chaotic systems. In particular for the GUE we have quadratic level repulsion meaning there is a quadratic zero in level correlators when any two eigenvalues approach.[^6]\
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Now let’s think about a process where a system with such chaotic level statistics emits an energy quantum. In particular let’s consider the probability for a black hole with energy $E$ to emit a massless scalar particle with energy $\omega$ which we then detect in our detector. In $d$ spacetime dimensions this probability is proportional to the usual Planckian black body law[^7] $$R_{\text{BE}}(\omega) \sim \frac{\omega^{d-1}}{e^{\beta \omega}-1}.\label{be}$$ The Hawking modes in the Unruh heat bath originate from a decay process between two black hole energy levels. So every energy level $\omega$ of the Hawking radiation corresponds to the difference of two energy levels of the quantum chaotic black hole system $$\omega= E_1-E_2.\label{difference}$$ Therefore the level density for the Unruh modes in the heat bath is by definition sensitive and proportional to the two-level spectral density $\rho(E_1,E_2)$ of the underlying quantum black hole.
For this reason we expect that the intrinsically chaotic level statistics of quantum black holes modifies the detection formula for the probability to detect a massless scalar Hawking particle with energy $\omega$ that has been emitted by a black hole with energy $E$ as $$\boxed{
R(\omega) \, \sim\, R_{\text{BE}}(\omega)(1-{\text{sinc}}^2 \pi \rho(E)\,\omega\,) \, \sim\, R_{\text{BE}}(\omega) \frac{\rho(E,E-\omega)}{\rho(E)\rho(E-\omega)}} \,.
\label{expectationunruh}$$ This appropriately takes into account level repulsion.[^8] Of course this level repulsion is only visible at ultra low energies $\omega$. At such low energies the semiclassical Planckian black body low goes like $\sim \omega^{d-2}$. Level repulsion modifies this behavior at ultra low energies to $\sim \omega^d$. The effect is most clearly visible in 2d where the total detection probability looks like at very low energies $\omega$.
Before proceeding, let us note that in quantum gravity there will be further modifications to the detector response for highly energetic Hawking modes $\omega\sim E$ for which we may no longer approximate the setup as a light probe particle travelling on a heavy black hole background $E\gg 1$. Related quantum gravitational effects will kick in for Planck size black holes $E\sim 1$. These effects are not governed by random matrix theory.
The model {#s:setup}
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In the remainder of this work we will gather concrete evidence in favor of via exact calculations in JT gravity. The action of JT gravity is [@Jackiw:1984je; @Teitelboim:1983ux][^9] $$\label{JTaction}
S_{\text{JT}}[g,\Phi] = S_0\,\chi + \frac{1}{16\pi G}\int d x\, \sqrt{g}\,\Phi \left(R + 2\right) + \frac{1}{8\pi G}\int_\partial d\tau\, \sqrt{h}\,\Phi_{\partial} \left(K- 1 \right)\,.$$ Here the Euler character $\chi$ comes from the usual Einstein-Hilbert action in 2d. Its only effect is a weighting of different topologies. The extremal entropy is $S_0 = \Phi_0/4 G$.[^10] Integrating over the bulk values of the Lagrange multiplier dilaton field $\Phi$ localizes the gravitational path integral on hyperbolic Riemann surfaces $R+2=0$. Any hyperbolic Riemann surface can be built be gluing together different subregions of the [Poincaré ]{}upper half plane. In Lorentzian signature $$ds^2=\frac{d Z^2-d F^2}{Z^2} = -\frac{4d U d V}{(U-V)^2}\quad, \quad Z>0,\label{poinc}$$ where we introduced the Poincaré lightcone coordinates $U=F+Z$ and $V=F-Z$. There is an asymptotic boundary at $Z=0$ where we will have to impose interesting boundary conditions.
Focusing on one such asymptotic boundary, there are several ways of discussing the reduction of the dynamics of this model into a purely boundary degree of freedom [@Almheiri:2014cka; @jensen; @malstanyang; @ads2]. Here we briefly review an intrinsically real-time approach [@ads2]. In order to treat this model within the holographic paradigm, we envision a dynamical boundary curve $(F(t),Z(t))$ as UV-cutoff as $Z\to 0$. This curve is specified by off-shell boundary conditions on the metric and dilaton field.
The geometry is taken in Fefferman-Graham gauge, which entails partially fixing the diff-group near the boundary such that the leading part of the geometry is the Poincaré metric and the interesting dynamics is in the subleading pieces as $Z \to 0$. Physically this simply means one can only compare different spacetimes if they share the same asymptotics. This leads to the constraint $Z(t) = \varepsilon\, F'(t)$, determining the wiggly boundary in terms of a single reparameterization $F(t)$ mapping the [Poincaré ]{}time $F$ to the proper time $t$ of an observer following the boundary trajectory. This single function $F(t)$ generates the 1d conformal group (as the residual diff’s that preserve Fefferman-Graham gauge) and this is the usual endpoint in AdS/CFT.
However, it can be shown that this system is special in the sense that also the dilaton field $\Phi$ blows up near the boundary $Z=0$. This means it has to be treated on the same footing as the metric, since once again one cannot compare spacetimes with different (dilaton) asymptotics. So, as a second constraint, we choose the boundary curve to satisfy $\Phi_\partial = a/\varepsilon$ in terms of a dimensionful quantity $a$ that determines the theory of interest.[^11]
Combining the ingredients, one finds that the Lorentzian version of the action reduces to a Schwarzian derivative action [@Almheiri:2014cka; @jensen; @malstanyang; @ads2] $$\label{SSchL}
S_L[F] = C\int dt \, \text{Sch}(F,t)\quad, \quad \text{Sch}(F,t) =\frac{F'''}{F'} - \frac{3}{2}\left(\frac{F''}{F'}\right)^2.$$ The coupling constant $C=a/8\pi G$ has units of length and controls the gravitational fluctuations of the wiggly boundary of the disk.[^12] Since path-integral computations are always performed in Euclidean signature, we Wick-rotate $t \to -i\tau$ and $F \to -i F$ to get the Euclidean Schwarzian action $$\label{SSch}
S[F] =- C\int d\tau \, \text{Sch}(F,\tau)$$ With multiple boundaries one has to glue several such Schwarzian systems together [@sss2; @Blommaert:2018iqz]. Higher genus contributions can be included as explained in [@sss2]. In the lowest genus (disk) case it is more convenient to describe the model using the time reparametrization $f(\tau)$ defined as $$F(\tau) = \tan \frac{\pi}{\beta} f(\tau)\quad, \quad f(\tau+\beta) = f(\tau) + \beta\quad, \quad \dot{f}(\tau) \geq 0 \, .$$ The last two equations characterize $f(\tau)$ as reparametrizing a circle that is the boundary of the 2d Euclidean disk. The quantity $\beta$ is the boundary length and is interpretable as the inverse temperature. We will adhere to this notation of $F(\tau)$ and $f(\tau)$ in the remainder of this work.\
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Using a myriad of techniques, JT gravity has been exactly solved for an entire class of boundary correlation functions. Gravitational contributions to correlation functions in JT gravity come in several flavors in terms of $G$.
1. There are perturbative $G$ corrections. These can be viewed as boundary graviton interactions and can be obtained via Schwarzian perturbation theory [@malstanyang; @Stanford:2017thb; @Qi:2019gny].
2. There are nonperturbative $G$ corrections associated with an exact solution of Schwarzian correlation functions [@altland; @altland2; @schwarzian; @Mertens:2018fds; @paper3; @Blommaert:2018iqz; @kitaevsuh; @zhenbin; @Iliesiu:2019xuh; @suh].
3. There are furthermore nonperturbative $G$ corrections associated with Euclidean wormhole contributions to the Euclidean path integral [@sss; @sss2; @Blommaert:2018iqz; @phil; @wophilbert]. These represent higher genus Riemann surfaces ending on the wiggly boundary. Indeed via we see that such contributions are perturbative in $e^{S_0}$ and hence nonperturbative in $G$.
4. Finally there are nonperturbative $e^{S_0}$ contributions which are hence doubly nonperturbative in $G$. In the gravitational language these are due to brane effects. JT gravity has a dual formulation as a double-scaled random matrix model [@sss2]. The doubly nonperturbative effects can be considered as hallmarks of this dual matrix integral description.
One of the properties which makes JT gravity so interesting is that we have analytic control over all these types of corrections. The centerpiece formulas and take all such corrections into account.
Detectors in the Unruh heat bath {#s:udw}
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In this sector we will consider a massless scalar field minimally coupled to JT gravity:[^13] $$\label{fieldact}
S= \frac{1}{2} \int d x\,\sqrt{-g}\, g^{\mu\nu}\,\partial_\mu \phi\, \partial_\nu \phi.$$ In particular, we aim to probe the emission spectrum of massless scalar Hawking-Unruh particles by a 2d quantum black hole. To do so, we imagine an experiment where we probe the heat bath using a linear Unruh-DeWitt detector. We will isolate the effects of different types of gravitational interactions by working in three improving layers of approximation.
Semiclassical analysis
----------------------
Let us first consider physics on the gravitational saddle, which is a black hole with inverse Hawking temperature $\beta$ $$ds^2 = \frac{\pi^2}{\beta^2}\frac{ dz^2-dt^2}{\sinh^2 \frac{2\pi}{\beta} z}\quad, \quad z>0\,.\label{thermal}$$ The semiclassical relation between the ADM mass $M$ of the black hole and the Hawking temperature is $\sqrt{M}=2\pi C/\beta$. We will henceforth set $C=1/2$. The asymptotic boundary is at $z=0$ and the semiclassical horizon is at $z=\infty$. This is just a conformal rescaling of flat space. The massless field $\phi$ is insensitive to this conformal rescaling of the metric and hence the solutions to the equations of motion of are left-and right-moving plane waves. Introducing lightcone coordinates $u=t+z$ and $v=t-z$ we have the mode expansion $$\label{modex}
\phi(u,v) = \int_0^\infty \frac{d\omega}{\sqrt{4\pi \omega}}\,\left( a_\omega e^{-i\omega u } + a^\dagger_\omega e^{i\omega u } - a_\omega e^{-i\omega v } - a^\dagger_\omega e^{i\omega v }\right),$$ implementing Dirichlet boundary conditions on the boundary $z=0$ of the half plane . The modes are orthogonal with respect to the usual Klein-Gordon inner product, and the modes satisfy $$[a_\omega, a^{\dagger}_{\omega'}] = \delta(\omega-\omega').\label{24}$$ The Wightman two-point function in the thermal state of the CFT is [@spradlin] $${\left\langle \phi(u_1,v_1)\phi(u_2,v_2) \right\rangle}_{{\scriptscriptstyle \text{CFT}}} = - \frac{1}{4\pi} \ln \abs{\frac{\sinh \frac{\pi}{\beta}(u_1-u_2)\sinh \frac{\pi}{\beta}(v_1-v_2)}{\sinh \frac{\pi}{\beta}(u_1-v_2)\sinh \frac{\pi}{\beta}(v_1-u_2)}}.\label{25}$$ All matter correlators will be denoted by ${\left\langle \dots \right\rangle}_{{\scriptscriptstyle \text{CFT}}}$. This expression can be equivalently read as evaluated in the Poincaré vacuum state, defined by taking positive-frequency modes with respect to the [Poincaré ]{}time $F$. As is well known the correlator looks thermal in $t$ coordinates because of the thermal coordinate transformation $F = \tanh \frac{\pi}{\beta} t$ relating the two [@spradlin]. We will be interested in understanding the frequency content of this correlator, and in particular on what it has to say about the underlying black hole.\
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The Unruh-DeWitt detector is a simple quantum mechanical detector model [@Unruh:1976db; @DeWitt:1980hx]. It linearly couples a quantum mechanical system with degree of freedom $\mu(t)$ to the scalar quantum field $\phi(u(t),v(t))$ via the local interaction Hamiltonian: $$H_\text{int}(t) = g \, \mu(t)\, \phi(u(t),v(t)).\label{26}$$ Typically one models the detector system to be a two-level system described by its “monopole moment” $\mu$. Here $g \ll 1$ is a tiny coupling and $(u(t),v(t))$ is the worldline of the detector. We assume the detector is initially in its ground state $\ket{0}$.
We want to compute the probability $P(\omega)$ for the final state of the detector to be the energy eigenstate $\ket{\omega}$ in first order perturbation theory in the detector coupling $g$. Within perturbation theory, the Hilbert space factorizes as $\mathcal{H}_{\text{det}} \otimes \mathcal{H}_{\text{matter}}$. Since the detector is ignorant of the final state $\ket{\psi}$ of the matter sector, one finds: $$\label{27}
P(\omega) =\sum_{\psi} \left|\bra{0,M} -i\int_{-\infty}^{+\infty} d t H_\text{int}(t)\ket{\omega,\psi}\right|^2.$$ Here $\ket{M}$ denotes the thermal state of the matter sector. Summing over $\psi$, one finds the response rate $R(\omega)$ which is defined as the probability per unit time to see the detector transition:[^14] $$\begin{aligned}
\label{29} R(\omega)=& \, g^2\,\abs{\bra{\omega}\mu(0)\ket{0}}^2\\&\lim_{{{\small \text{T}}} \to +\infty}\frac{1}{{\small \text{T}}} \int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_1\int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_2\, e^{-i\omega (t_1-t_2)}{\left\langle \phi(u(t_1),v(t_1))\phi(u(t_2),v(t_2)) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}.\nonumber\end{aligned}$$ This quantity represents the probability rate for the detector to get excited to energy level $\omega$. By energy conservation the matter state gets depleted by a similar energy $\omega$.
In the vacuum associated to the time coordinate $t$ the result would be zero but this is not so in the Poincaré vacuum. This is the essence of the Unruh effect. The analogous situation in flat space is by interpreting the black hole time coordinate $t$ as Rindler (or Schwarzschild) time, and the Poincaré time $T$ as Minkowski time. For a thorough recent review see [@Crispino:2007eb]. In the [Poincaré ]{}vacuum one finds the semi-classical thermal Unruh population by computing the integral on the second line of using and assuming a stationary detector worldline. The answer is $$\label{210}
R(\omega) = g^2 \abs{\bra{\omega}\mu(0)\ket{0}}^2 \, 2\, \frac{\sin^2 \omega z}{\omega^2}\, \frac{\omega}{e^{\beta \omega} -1}\,.$$ One indeed recognizes the Planckian black body law in two dimensions .
Coupling to Schwarzian reparameterizations
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Our goal for this subsection and the following is to compute the Fourier transform of the bulk two-point function on the second line of in two different levels of approximation. In this subsection we will ignore Euclidean wormhole (or higher genus) contributions to the Euclidean JT gravity path integral. This means we incorporate the gravitational corrections of only items 1 and 2 of the list in section \[s:setup\].\
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Within quantum gravity, physical bulk locations and bulk observables must be defined in a diff-invariant manner. To do so in a holographic context we are led to define a point in the bulk by anchoring the bulk point to the asymptotic boundary [@Donnelly:2015hta; @gid3; @Almheiri:2017fbd; @ref3; @Lewkowycz:2016ukf; @ref4; @ref5; @ref6; @Chen:2017dnl; @Chen:2018qzm; @Engelhardt:2016wgb]. One particularly natural way to do this in this context is by geodesic localizing using lightrays [@Engelhardt:2016wgb; @Blommaert:2019hjr; @Mertens:2019bvy].
In JT gravity the boundary is one-dimensional. Therefore the physical coordinates used to define bulk points are boundary time coordinates. In particular we need two such time coordinates $t_1$ and $t_2$. We associate these to the lightcone coordinates $u$ and $v$ of a point in the bulk. Here $v$ is the physical boundary time at which an observer sends a signal to a given bulk point and $u$ is the physical boundary time at which the observer receives back the signal after reflecting off some fictitious mirror.
The boundary curve $(F(t),Z(t))$ to which we anchor a bulk point is described by a single function $F(t)$, where $Z$ is determined in terms of $F$ by the boundary conditions. The actual wiggling of the boundary as explained above , is negligible for $\varepsilon\ll 1$. The field $F(t)$ represents the map between [Poincaré ]{}coordinates and the physical boundary time coordinate $t$ [@Almheiri:2014cka; @jensen; @malstanyang; @ads2]. Consequently in terms of the two boundary times $u$ and $v$, the location of the bulk point in [Poincaré ]{}coordinates is defined as $U= F(u)$ and $V =F(v)$. For a more detailed explanation of this construction, see [@Blommaert:2019hjr; @Mertens:2019bvy; @wopjordan]. The bulk metric constructed in this fashion is $$ds^2=-\frac{F'(u)F'(v)}{(F(u)-F(v))^2}\,du\,dv.\label{189}$$ Following this same logic, we are led to define massless scalar bulk observables $\Phi$ which implicitly depend on the Schwarzian reparameterization $f$ as: $$\label{dress}
\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u,v] = \phi(f(u),f(v)).$$ This is just a regular massless scalar field but the location of the insertion of this operator in [Poincaré ]{}coordinates $(Z,F)$ depends on the details of the wiggly boundary $F(t)$ $$\Phi[f_1{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u,v]= \raisebox{-17mm}{\includegraphics[width=21mm]{points1.pdf}}\quad,\quad \Phi[f_2{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u,v]= \raisebox{-17mm}{\includegraphics[width=22.7mm]{points2.pdf}}\quad.$$ This implicit dependence on $F(t)$ of bulk operators couples bulk matter to the Schwarzian reparameterization mode. We note that the definition of bulk operators in quantum gravity corresponds in the language of [@Donnelly:2015hta; @gid1; @gid2; @gid3; @gid4; @gid5; @gid6; @gid7] to specifying a particular gravitational dressing of a bare bulk matter operator such that the dressed operator is diff-invariant.\
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In every fixed metric , the two-point function of two operators of the type is then by definition of just the reparameterization of $$\label{twoqft}
{\left\langle \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1] \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}} = - \frac{1}{4\pi}\left|\ln \frac{\sinh \frac{\pi}{\beta}(f(u_1)-f(u_2))\sinh \frac{\pi}{\beta}(f(v_1)-f(v_2))}{\sinh \frac{\pi}{\beta}(f(u_1)-f(v_2))\sinh \frac{\pi}{\beta}(f(v_1)-f(u_2))}\right|.$$ In this same way we define the entire trajectory of the Unruh-DeWitt detector:[^15] $$\includegraphics[width=0.2\textwidth]{bulkframeUDW.pdf}$$ This shows the constant $z$ worldlines of the Unruh-DeWitt detector for two different clock ticking patterns $F(t)$ (blue and red) at the same radial coordinate $z$. Close to the horizon and semi-classically, these become constant accelerated worldlines. Close to the boundary at $z=\epsilon$, this becomes the wiggly boundary curve itself.
The interaction Hamiltonian now features the diff-invariant dressed field $\Phi$ $$H_\text{int}(t) = g\, \mu(t)\,\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u(t),v(t)].\label{216}$$ Including gravitational degrees of freedom and perturbatively in $g$ the Hilbert space may be considered to factorize as $\mathcal{H}_\text{det}\otimes \mathcal{H}_\text{matter+gravity}$. The second factor is the whole coupled system of matter and gravity. Computing the probability for the detector to evolve from an initial state $\ket{0}$ to a final state $\ket{\omega}$ one finds an identical formula but with the interaction replaced by and with $\psi$ summed over $\mathcal{H}_\text{matter+gravity}$. Doing the sum over the final states $\psi$ results in $$\begin{aligned}
\bra{\mM}\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1]\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2]\ket{\mM} =\int_{\text{micro M}} [{\mathcal{D}}f]\, {\left\langle \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1] \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}\,e^{-S[f]}.\label{215}\end{aligned}$$ This implements a matter-coupled quantum gravity path integral in the microcanonical ensemble of fixed energy $M$, defined by inverse Laplace transforming the canonical ensemble path integral. Here the Schwarzian action is .\
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We can now proceed with the actual computation, for which we use a technical trick [@Blommaert:2019hjr]: $${\left\langle \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1] \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}=\int_{v_1}^{u_1} d t_1\int_{v_2}^{u_2}dt_2\,{\left\langle {\mathcal{O}}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_1]{\mathcal{O}}[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}.\label{hkll}$$ Here the matrix element on the right hand side is a thermal boundary two-point function of a massless scalar: $${\left\langle {\mathcal{O}}[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_1]{\mathcal{O}}[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}=-\frac{1}{4\pi}\,\frac{ f'(t_1)f'(t_2)}{\frac{\beta^2}{\pi^2}\sinh^2 \frac{\pi}{\beta}(f(t_1)-f(t_2))}.\label{218}$$ We notice that is essentially implementing the HKLL bulk reconstruction in each of the metrics [@hkll1; @hkll2; @kll; @kl; @Lowe:2008ra]. This reverse-engineered version of bulk reconstruction is pivotal since the Schwarzian path integral in over the right hand side of can be easily computed [@Blommaert:2019hjr], as the disk boundary-to-boundary propagator in JT gravity: $$\int [{\mathcal{D}}f]\, {\left\langle {\mathcal{O}}[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_1]{\mathcal{O}}[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}t_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}\, e^{-S[f]}\,=\quad\raisebox{-10mm}{\includegraphics[width=45mm]{1lbiloc.pdf}}\quad.\label{40}$$ The answer is [@altland; @altland2; @schwarzian; @Mertens:2018fds; @paper3; @Blommaert:2018iqz; @kitaevsuh; @zhenbin; @Iliesiu:2019xuh]:[^16] $$\frac{1}{Z(\beta)}\int_0^\infty d M\,e^{-\beta M}\int_{0}^{+\infty} dE \, e^{-i(t_1-t_2)(E-M)}\, \rho_0(M)\rho_0(E)\,\rvert {\mathcal{O}}_{M E}^1\rvert^2. \label{41}$$ Here we introduced the notation:[^17] $$\label{221}
\rho_0(E)=\frac{e^{S_0}}{2\pi^2}\sinh 2\pi \sqrt{E}\quad,\quad \rvert{\mathcal{O}}_{E_1E_2}^\ell\rvert^2 = e^{-S_0}\frac{\Gamma(\ell\pm i\sqrt{E_1}\pm i\sqrt{E_2})}{\Gamma(2\ell)}\, .$$ Within the microcanonical ensemble of fixed energy $M$, the boundary-to-boundary correlator becomes: $$\int_{0}^{+\infty} d E \, e^{-i(t_1-t_2)(E-M)}\, \rho_0(E)\,\rvert {\mathcal{O}}_{M E}^1\rvert^2 \, .$$ The response rate for a detector at $z_1=z_2=z$ is now readily found by performing the elementary integrations in and . One obtains: $$\begin{aligned}
\label{response}
R(\omega) &= g^2\,\abs{\bra{\omega}\mu(0)\ket{0}}^2\, 2 \, \frac{\sin^2 \omega z}{\omega^2}\, \rho_0(M-\omega)\, \rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2 \,.\end{aligned}$$ Notice that by it is independent of the ground-state degeneracy $e^{S_0}$. The “greybody” interference factor $2 \sin^2 \omega z/\omega^2$ is determined in part by choosing Dirichlet boundary conditions for the massless scalar field in .[^18] For other boundary conditions on the scalar field, the interference pattern changes, see appendix \[app:obc\].
In the semiclassical regime $M \gg 1$ and $\omega\ll M$, reproduces the classical answer . For more generic values of $\omega$ however there are gravitational backreaction effects. The most prominent such backreaction effect is that the response function abruptly stops at $\omega = M$ with a square root edge via .\
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In order to better understand from a physical point of view why vanishes at $\omega=M$, it is convenient to go back to the mode expansions . We can use it to define raising and lowering operators in the matter Hilbert space: $$\begin{aligned}
\label{creaan}
a_\omega &= \frac{i}{\sqrt{\pi \omega}} \int_{-\infty}^{+\infty} du \,\partial_u \phi(u) \, e^{i\omega u}, \qquad a^{\dagger}_\omega = -\frac{i}{\sqrt{\pi \omega}} \int_{-\infty}^{+\infty} du \, \partial_u \phi(u) \, e^{-i\omega u} \, .\end{aligned}$$ Quite analogously, we could define raising and lowering operators of gravitationally dressed matter fields in the Hilbert space of the coupled system of matter and gravity $$\begin{aligned}
\label{creabn}
A_\omega[f] &= \frac{i}{\sqrt{\pi \omega}} \int_{-\infty}^{+\infty} du \,\partial_u \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u] \, e^{i\omega u}, \qquad A^{\dagger}_\omega[f] = -\frac{i}{\sqrt{\pi \omega}} \int_{-\infty}^{+\infty} du \, \partial_u \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u] \, e^{-i\omega u}\, .\end{aligned}$$ Working with these dressed (or diff-invariant) raising and lowering operators, we find that the response rate is proportional to a number operator expectation value[^19] $$\begin{aligned}
\label{num1}
\bra{\mM}A^\dagger_\omega[f] A_\omega[f] \ket{\mM} &= \frac{V}{2\pi\omega} \,\rho_0(M-\omega)\, \rvert\mathcal{O}_{M M-\omega}^1\rvert^2\,.\end{aligned}$$ This formula explains the step function in a quite natural manner: the dressed operator $A_\omega$ extracts an energy $\omega$ from the gravity system. Of course it is impossible to extract more energy from this system than the finite energy $M$ which it had to begin with and so we have $$A_\omega[f] \ket{\mM}= 0\quad,\quad \omega>M.\label{229}$$ Therefore and should be expected to vanish for $\omega >M$. Using similar techniques, one could compute more involved matrix elements. For example $$\bra{\mM}A^\dagger_{\omega_1}[f]\dots A^\dagger_{\omega_n}[f]A_{\omega_n}[f]\dots A_{\omega_1}[f]\ket{\mM}.$$ One finds, in accordance with the fact that operators such as $A_\omega$ deplete the system of an energy $\omega$, that this amplitude vanishes if $\omega_1+\dots \omega_n>M$.
Level repulsion {#s:levrep}
---------------
We would now like to include Euclidean wormhole contributions to the JT gravity path integral which computes the massless bulk two point function. This includes all 4 items of the list of gravitational corrections in section \[s:setup\].\
\
It is not a priori obvious how to dress the massless scalar bulk two-point function, whose radar construction was inherently Lorentzian, with higher genus contributions to the Euclidean path integral. We here propose a very natural way of doing so.
We start with the disk (genus zero) contribution first. One may then use the bulk reconstruction formula to write the bulk matter two-point function in terms of a Euclidean disk JT gravity path integral with a boundary-to-boundary matter propagator $$\bra{\mM}\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1]\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2]\ket{\mM}\,=\int_{v_1}^{u_1}dt_1\int_{v_2}^{u_2} dt_2 \quad\raisebox{-10mm}{\includegraphics[width=45mm]{1lbiloc.pdf}}\quad.\label{231}$$ Including Euclidean wormhole connections for the boundary two-point function on the right hand side has recently been understood [@phil; @Blommaert:2019hjr; @wophilbert]. One just sums over all higher genus Riemann surfaces which end on the union of the boundary circle and the boundary-to-boundary bilocal line.
We now define bulk correlators by applying the bulk reconstruction formula to these higher-topology boundary correlators. So our definition of bulk operators is really a “bulk reconstruction first” approach. This includes for example a contribution of the type $$\bra{\mM}\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1]\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2]\ket{\mM}\,\supset\int_{v_1}^{u_1}dt_1\int_{v_1}^{u_2} dt_2 \quad\raisebox{-10mm}{\includegraphics[width=45mm]{1lbilochandle.pdf}}\quad.\label{232}$$ This definition ends up reproducing our generic intuition which we view as an argument that this definition makes sense. Similarly in [@wophilbert] we will find that this definition ends up reproducing other intuitive expectations about bulk observables.
As it turns out, it is quite feasible to sum over all such amplitudes via the definition of JT gravity as a double-scaled matrix integral [@sss2]. In the end, the effect of including such contributions is quite elegant. In terms of the formulas, one ends up effectively replacing $\rho_0(M)\rho_0(E)$ in the boundary two-point function by $\rho(M,E)$ which is the universal answer from random matrix theory[^20] $$\frac{1}{Z(\beta)}\int_{\mathcal{C}}d M\,e^{-\beta M}\int_{\mathcal{C}}dE \, e^{-i(t_1-t_2)(E-M)}\, \rho(M,E)\,\rvert {\mathcal{O}}_{M E}^1\rvert^2. \label{233}$$ Here we have $$\rho(M,E) = \rho(M)\rho(E)-\rho(M)\rho(E)\,{\text{sinc}}^2\, \pi \rho(M)(M-E) + \rho(M)\delta(M-E). \label{49}$$ This formula holds if both $M$ and $E$ are far enough from the spectral edge $E\gg e^{-2S_0/3}$. Closer to the spectral edge, we can resort to similarly universal formulas for the two-level spectral density in the Airy model [@sss2; @paper6]. The first term corresponds geometrically to the disconnected contribution of adding higher topology to each of the two sides of the bilocal line in . The second term is due to Riemann surfaces connecting both sides of the bilocal such as the annulus in . Each of these terms furthermore includes nonperturbative contributions in $e^{-S_0}$ due to brane effects [@sss2; @phil; @paper6]. These are oscillatory and hence not necessarily small. The final term represents a contact term that may or may not have a geometric interpretation.[^21]
We can now immediately write down the analogue of which takes into account these Euclidean wormholes via the substitution $$\boxed{R(\omega) = g^2\,\abs{\bra{\omega}\mu(0)\ket{0}}^2\,2\,\frac{\sin ^2\omega z}{\omega^2}\, \frac{\rho(M,M-\omega)}{\rho(M)}\, \rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2.}\label{51}$$ It is straightforward to similarly write down the modifications to the expectation values of the dressed number operator . Notice that now there is no step function. This is because in the matrix integral description there is a tiny but nonzero probability for eigenvalues to be in the forbidden region $E<0$ of the energy contour ${\mathcal{C}}$ [@sss2].
We note that that the prefactor $g^2\,\abs{\bra{\omega}\mu(0)\ket{0}}^2$ is intrinsic to the detector, which we (the observer) can determine within the free detector theory. Dividing out this known prefactor, and computing in the probe approximation $M\gg 1$ and $\omega\ll M$ we find $$\frac{R(\omega)}{2 \, \sin^2 \omega z/\omega^2} = \frac{\omega}{e^{\beta\omega}-1} (1-{\text{sinc}}^2 \pi \rho(M)\,\omega\,) \, . \label{237}$$ This explicitly confirms our general expectation about the effects of level repulsion on the detection rate, via a bulk JT gravity calculation. Schematically the detector finds the following result $$R(\omega) = 2 \, \frac{\sin^2 \omega z}{\omega^2}\quad \raisebox{-15mm}{\includegraphics[width=65mm]{depl.pdf}}\quad\label{234}$$ The red curve denotes the semiclassical Planckian black body law with initial linear decay. At extremely low energies with $\omega$ of order $e^{-S_0}$ we explicitly see the depletion of the Hawking-Unruh spectrum due to level repulsion in quantum gravity.\
\
To get a better grip on the physics we are probing we can briefly consider the bulk gravitational dual to one single quantum chaotic system with Hamiltonian $H$. In JT gravity, we can effectively achieve this by considering a new definition of JT gravity for which the path integral itself includes a bunch of additional branes [@sss2; @paper6; @maxfieldmarolf; @wophilbert]. The data of these branes are one-to-one with the spectral data of the dual quantum mechanical system. For this setup in particular, we want to consider a version of JT gravity that includes branes which fix the eigenvalues $\lambda_1\dots \lambda_L$ of the Hamiltonians [@paper6]. Essentially we replace in and $$\rho(E,M)=\sum_{i=1}^L \delta(E-\lambda_i)\sum_{j=1}^L \delta(M-\lambda_j).\label{239}$$ We imagine probing in a state of the coupled matter and gravity system with energy $M$ identical to one of the eigenvalues $\lambda$. We find via and $$R(\omega)= 2\,\frac{\sin^2 \omega z}{\omega^2} \, \sum_{i=1}^L \delta(\omega-\lambda+\lambda_i)\, \rvert {\mathcal{O}}_{\lambda \lambda_i}^1\rvert^2.$$ In the probe approximation $\lambda\gg 1$ and $\omega\ll \lambda$ this approximates to[^22] $$R(\omega)_{\lambda} = 2\,\frac{\sin^2 \omega z}{\omega^2}\, \frac{\omega}{e^{\beta \omega}-1}\sum_{i=1}^L \frac{1}{\rho(\lambda_i)}\delta(\omega-\lambda+\lambda_i) .\label{55}$$ This makes explicit formula which states that the energy levels $\omega$ of the Unruh-Hawking modes must match to energy differences of the quantum chaotic black hole system: the gravitationally dressed raising and lowering operators generate level transitions within the quantum gravitational system $$\raisebox{-15mm}{\includegraphics[width=48mm]{trans.pdf}}\quad\label{242}$$ We remove (orange) an energy difference $\omega_{ij}$ from an initial state (red) with energy $\lambda_i$ and are left with a final state (blue) with energy $\lambda_j$.\
\
The result carries an interesting interpretation. It is commonly assumed that the detailed information about the microstate of a gravitational system is somehow located behind the horizon. In however we have an explicit experiment which can be performed by an observer hovering outside of the semiclassical horizon. From the resulting set of delta spikes as observed by his detector, he is in principle able to determine the energy spectrum of all the gravitational microstates. Here we take the details of one gravitational microstate to mean knowledge of all the levels $\lambda_1\dots \lambda_L$.
Fermionic matter
----------------
The result is expected to hold quite universally. We may provide evidence for this by testing it in more general situations. Therefore in this section we consider an Unruh-DeWitt detector coupled to massless fermionic bulk matter in JT gravity. In the next section \[s:proper\] we consider other detector couplings.
We note that massless bulk fermions coupled to JT gravity are expected to play quite an important role in the elusive 2d gravity theory dual to the SYK model. Furthermore a large number of such massless bulk fermionic fields drastically enhance the evaporation rate for 4d magnetically charged black holes (which have a near horizon description as JT gravity) [@Maldacena:2018gjk; @Maldacena:2020skw].\
\
We will consider as detector interaction the simplest possible coupling of a bosonic detector $\tilde{\mu}(t)$ to a bulk Dirac fermion $\psi(u,v)$ [@Takagi:1986tf] $$\begin{aligned}
H_\text{int} &= g \, \tilde{\mu}(t) \, (\bar{\psi} \psi)(u(t),v(t)).\label{ferco}\end{aligned}$$ See [@Hummer:2015xaa] for a comprehensive study of different Unruh-DeWitt detector types and their effects and [@Gray:2018ifq; @Louko:2016ptn] for recent studies. The normal ordering defined by $(\bar{\psi} \psi) = \bar{\psi}\psi - \left\langle \bar{\psi}\psi\right\rangle_{{\scriptscriptstyle \text{CFT}}}$ is required in order to sensibly define this detector coupling. The detector variable is chosen as $$\label{rel}
\tilde{\mu}(t) = \frac{L_{\text{det}}}{\Omega(u(t),v(t))} \,\mu(t) \, .$$ We recognize the same detector variable $\mu(t)$ from the bosonic detector . Furthermore we introduced the Weyl factor $\Omega(u(t),v(t))$ of the metric $$ds^2=-\frac{1}{\Omega(u,v)^{2}}\,du\, dv = -\frac{F'(u)F'(v)}{(F(u)-F(v))^2} du\, dv \, .$$ The relation can be motivated by dimensional analysis. The coupling $\mu(t)$ in the bosonic detector has units of inverse length whereas $\tilde{\mu}(t)$ is dimensionless. The Weyl factor $\Omega(u(t),v(t))$ transforms between the detector length scale $L_\text{det}$ and the local length scale. We will suppress the detector length scale $L_{\text{det}}$. It is however not difficult to consider a detector with coupling that does not include this Weyl factor. We comment on this in section \[s:proper\].
Notice that because of the coupling to a composite operator in , we we are not probing for a fermionic emission spectrum, but instead are measuring the emission of a fermion-antifermion pair. This jives with the interpretation that we are probing with the Unruh-DeWitt detector the probability of level transitions within an underlying bosonic black hole system, for which level transitions are only possible upon emission of bosonic quanta.\
\
Let us develop this example in more detail. The 2d massless curved spacetime Dirac equation has the following Weyl rescaling property: if $\psi$ is a solution in the metric $g_{\mu\nu}$, then $\Omega^{1/2}\psi$ is a solution in the metric $\Omega^{-2}g_{\mu\nu}$. We can use this to write the mode expansion of the Dirac field in AdS$_2$ in [Poincaré ]{}coordinates ($U,V$) as:[^23] $$\begin{aligned}
\label{modefermi}
\frac{\psi(U,V)}{(U-V)^\frac{1}{2}}&=\psi(U)+\psi(V)\\&= \,\,\frac{1}{\sqrt{4\pi}}\,\left(\begin{array}{c} 1 \\ i \end{array}\right)\,\sum_{\omega>0} \left(e^{- i \omega U} a_{\omega} +e^{ i \omega U} b_{\omega}^{\dagger}\right) +\,\frac{1}{\sqrt{4\pi}}\,\left(\begin{array}{c} i \\ 1 \end{array}\right)\,\sum_{\omega>0} \left(e^{- i \omega V} a_{\omega} +e^{ i \omega V} b_{\omega}^{\dagger}\right) \, .\nonumber\end{aligned}$$ We have implicitly adopted Dirichlet boundary conditions here.[^24] Using this mode expansion, it is easy to obtain the Wightman bulk two point function in the [Poincaré ]{}vacuum: $$\frac{{\left\langle \psi_\alpha(U_1,V_1)\bar{\psi}_\beta(U_2,V_2) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}}{ (U_1-V_1)^\frac{1}{2}\,(U_2-V_2)^\frac{1}{2}} = S_{\alpha\beta}(U_1,V_1,U_2,V_2) \, ,$$ where $$S(U_1,V_1,U_2,V_2)=\frac{1}{2}\left[ \begin{array}{cc}
\frac{1}{U_1-U_2} - \frac{1}{U_1-V_2} - \frac{1}{V_1-V_2} + \frac{1}{V_1-U_2} & \frac{i}{U_1-U_2} + \frac{i}{U_1-V_2} + \frac{i}{V_1-V_2} + \frac{i}{V_1-U_2} \\
\frac{i}{U_1-U_2} - \frac{i}{U_1-V_2} + \frac{i}{V_1-V_2} - \frac{i}{V_1-U_2} & -\frac{1}{U_1-U_2} - \frac{1}{U_1-V_2} + \frac{1}{V_1-V_2} + \frac{1}{V_1-U_2}
\end{array}\right] \nonumber \, .$$ In terms of the detector we are led to compute the bulk two-point function of fermion-antifermion pairs. By taking Wick contractions, one finds $$\begin{aligned}
\label{contr}
\frac{\left\langle (\bar{\psi} \psi) (U_1,V_1) (\bar{\psi} \psi) (U_2,V_2)\right\rangle_{{\scriptscriptstyle \text{CFT}}}}{(U_1-V_1)\,(U_2-V_2)}&= \sum_{\alpha\,\beta}\contraction{}{\overline{\psi}_\alpha}{\psi_\alpha \overline{\psi}_\beta}{\psi_\beta}
\contraction{\overline{\psi}_\alpha}{\psi_\alpha}{}{\overline{\psi}_\beta} \overline{\psi}_\alpha \psi_\alpha \overline{\psi}_\beta \psi_\beta = - \Tr S^2(U_1,V_1,U_2,V_2)\\&= -\frac{1}{(U_1-V_2)^2} - \frac{1}{(V_1-U_2)^2} + \frac{2}{(U_1-U_2)(V_1-V_2)}\nonumber \,.\end{aligned}$$ Notice that by definition of normal-ordering, there are no contractions to be considered within each composite operator $(\bar{\psi}\psi)(U,V)$.\
\
As in for the bosonic case, we can couple this observable to the Schwarzian by applying a coordinate reparameterization and then computing the path integral. Before doing so, let us note that the Weyl factor in the detector couplings cancels with the Weyl factors on the left hand side of in the detector transition rate . So from hereon let us drop all such factors.
In reparameterized bulk metrics and ignoring the Weyl factors, the fermion pair two-point function becomes: $$\begin{aligned}
\label{contr2}
\nonumber &{\left\langle (\bar{\Psi} \Psi) [f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1,v_1] (\bar{\Psi} \Psi) [f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2,v_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}} \\&= -\frac{F'(u_1)F'(v_2)}{(F(u_1)-F(v_2))^2} - \frac{F'(v_1)F'(u_2)}{(F(v_1)-F(u_2))^2} + 2\, \frac{F'(u_1)F'(u_2)}{(F(u_1)-F(u_2)}\,\frac{F'(v_1)F'(v_2)}{(F(v_1)-F(v_2)}\,,\end{aligned}$$ in terms of the gravitationally dressed field $\Psi$. The conformal scaling factors $F'$ are explained because the holomorphic and antiholomorphic components $\psi(u)$ and $\psi(v)$ in are $\ell=1/2$ conformal primaries. Following and normalizing by the operator intrinsic prefactor on the first line of , one now computes the response rate as: $$\begin{aligned}
R(\omega)=&\lim_{{\small \text{T}}\to \infty}\frac{1}{{\small \text{T}}}\int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_1\int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_2\,e^{-i\omega(t_1-t_2)}\nonumber\\&\qquad\qquad\qquad\qquad\bra{\mM}(\bar{\Psi} \Psi) [f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_1(t_1),v_1(t_2)] (\bar{\Psi} \Psi) [f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u_2(t_1),v_2(t_2)]\ket{\mM}\, .\label{254}\end{aligned}$$ As in the bosonic case we will consider a detector following a trajectory at fixed $z$. In implementing the Schwarzian path integral of there are a few subtleties related with operator ordering ambiguities which we must first address.
1. All three terms in correspond to the product of two Schwarzian $\ell=1/2$ bilocals. In our case the quantum mechanical operator ordering is fixed by the nested Wick contractions in , leading to three nested Schwarzian four-point functions.
2. The first two terms of correspond to two Schwarzian bilocals with the same start and end points. The Schwarzian correlator can be simplified using the following property: $$\quad\raisebox{-10mm}{\includegraphics[width=45mm]{l1.pdf}}\quad=\quad\raisebox{-10mm}{\includegraphics[width=45mm]{l1plus.pdf}}\quad.\label{256}$$ Both these JT gravity amplitudes can be immediately evaluated using the result of [@schwarzian; @Mertens:2018fds; @paper3; @Blommaert:2018iqz]. Labeling the energies as $E_1$, $E$ and $E_2$ from left to right of the diagram , we have the result:[^25] $$\begin{aligned}
\int_0^\infty d E\, \rho_0(E)\, \rvert{\mathcal{O}}^{\ell_1}_{E_1 E}\rvert^2\,\rvert{\mathcal{O}}^{\ell_2}_{E E_1}\rvert^2 = \rvert{\mathcal{O}}^{\ell_1+\ell_2}_{E_1 E_2}\rvert^2\, .\label{257}
\end{aligned}$$ This means we can use the classical identity $$\left(\frac{F'_1 F'_2}{(F_1-F_2)^2} \right)^{\ell_1 } \left(\frac{F'_1 F'_2}{(F_1-F_2)^2} \right)^{\ell_2 } = \left(\frac{F'_1 F'_2}{(F_1-F_2)^2} \right)^{\ell_1+\ell_2 } \, ,$$ also at the quantum level. We also have the more general identity: $$\quad\raisebox{-10mm}{\includegraphics[width=45mm]{l1it.pdf}}\quad=\quad\raisebox{-10mm}{\includegraphics[width=45mm]{l1plus.pdf}}\quad.\label{259}$$ This property also reduces the Schwarzian path integral of the last term in (for which $c=2i z$) to a single bilocal computation.
Proceeding by the calculation of we find that this boils down to computing three Schwarzian diagrams of the same type. Their prefactors combine as an interference factor as $ - \frac{1}{2}e^{i\omega 2z} - \frac{1}{2}e^{-i\omega 2z} + 1 = 2 \sin^2 \omega z$, leading to the result $$R(\omega) = 2\,\sin^2 \omega z\, \rho_0(M-\omega)\, \rvert{\mathcal{O}}_{M M-\omega}^1 \rvert^2\,.\label{260}$$ Finally, we should include Euclidean wormholes to this computation. The calculation is essentially identical to the one which led to . We note that does not technically hold when including such Euclidean wormholes because we could have Euclidean wormholes connecting the middle region with the two other regions. Such corrections are important in the intermediate $E$ integral whenever $E \approx M$ and or $E\approx M-\omega$ where the contribution to the $E$ integral is pushed to zero by quadratic level repulsion. This region has energy width $\sim e^{-S_0}$ and order $1$ height. Doing the $E$ integral results in an order $e^{-S_0}$ correction to the response rate. This is negligible. In other words we may safely still use even when including Euclidean wormhole contributions to both diagrams.
The final result is hence essentially identical to the bosonic answer and provides a second example of our generic expectation in the probe approximation $M\gg 1$ and $\omega \ll M$. The difference sits only in the interference factor which differs by an overall $\omega^2$. This comes purely from dimensional reasons since there is also a length scale $L_{\text{det}}$ in the coupling . The relative prefactor between and is the dimensionless combination $L_{\text{det}}^2 \, \omega^2$.
More general detector couplings {#s:proper}
-------------------------------
The detector couplings $\mu(t)$ and $\tilde{\mu}(t)$ we have defined previously, transform as scalar densities under coordinate transformations, and correspond to time measurements on the boundary clock. It is straightforward to consider coupling to the proper bulk time of the worldline, by changing the interaction terms into $$\begin{aligned}
\label{propcou}
S_\text{int} &= g\int d\tau_p \, \mu(\tau_p)\, \phi(u(\tau_p),v(\tau_p)) = g\int dt \, \Omega(u(t),v(t))^{-1} \, \mu(t)\, \phi(u(t),v(t)) \, \\
S_\text{int} &= g\int d\tau_p \, \tilde{\mu}(\tau_p) \, (\bar{\psi} \psi)(u(\tau_p),v(\tau_p)) = g \int dt \, \Omega(u(t),v(t))^{-1} \, \tilde{\mu}(t) \, (\bar{\psi} \psi)(u(t),v(t)) \, . \nonumber \end{aligned}$$ Here $dt \, \Omega(u(t),v(t))^{-1} = d\tau_p$ is the proper time along the bulk worldline. At the quantum gravity level, the above coupling is replaced by a Hermitian coupling including the dressed fields $$\begin{aligned}
S_\text{int} = \frac{g}{2}\int dt \, \mu(t) \Bigl( \Omega(u(t),v(t))^{-1} \,\Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u(t),v(t))] + \Phi[f{\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u(t),v(t))]\Omega(u(t),v(t))^{-1} \Bigr) \nonumber \, .\end{aligned}$$ Notably also $\Omega^{-1}$ depends on the gravity variable $f$. An analogous formula holds for the fermion detector. The explicit Weyl factors $\Omega^{-1}$ in this expression yield additional Schwarzian bilocal lines with length $2z$ between their endpoints. Diagrammatically we have: $$\label{weyl}
\frac{1}{2} \quad\raisebox{-10mm}{\includegraphics[width=45mm]{ProperFig.pdf}}\quad+ \frac{1}{2}\quad\raisebox{-10mm}{\includegraphics[width=45mm]{ProperFig2.pdf}}\quad.$$ The bilocal lines coming from the explicit Weyl factors are represented by the red lines. The important point is that since this additional structure only depends on $z$, it does not participate in the Fourier transform over $t$, and factorizes from the amplitudes. Furthermore Euclidean wormholes connecting to these new regions in only give subdominant contributions because we are integrating over the corresponding energy labels of these regions. Hence the only effect of including these Weyl factors in is that and receive an additional overall prefactor which depends on $z$ but which is crucially independent of $\omega$. By consequence one still finds the same physics as in for example . The fermionic coupling requires exactly the same treatment with Weyl factors on either side of the $\ell=1/2$ pair of lines in .
We note to conclude that one could consider more generic detector couplings, of higher order in the fields, such as: $$H_\text{int} = g\,\mu(t)\,f(\phi(u(t),v(t))\quad,\quad f(\phi(u(t),v(t)) = \sum_{n} c_n (\phi^n)(u(t),v(t)) \, .$$ The precise Schwarzian and Euclidean wormhole computations are more involved. The resulting detector response would be obtained as the same Taylor series expansion: $$R_g(\omega)=\sum_n c_n R_n(\omega) \, .$$ Intuitively, we expect to see level repulsion in any such response rate given its universal role in random matrix correlation functions.
Energy density in the Unruh heat bath {#s:hb}
=====================================
In this section we investigate the spectral energy density in the Unruh heat bath. Semi-classically this is given by $\bra{\mM}\omega\,a_\omega^\dagger a_\omega \ket{\mM}$, so we might expect the gravitational corrections to be accounted for by considering instead $\bra{\mM}\omega\,A_\omega^\dagger[f] A_\omega[f] \ket{\mM}$. However, this is not true. This traces back to operator ordering ambiguities when promoting quantum field observables in a fixed metric to operators in a theory of matter-coupled quantum gravity.\
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Let us first focus on coupling to the Schwarzian reparametrization. The local energy density in the Unruh heat bath can be computed independently using the coincident limit of the bulk two-point function (regularized via point splitting) [@spradlin]. The result is: $$\label{Unruhflux}
\left\langle :T_{uu}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u]:\right\rangle_{{\scriptscriptstyle \text{CFT}}} = -\frac{1}{24\pi}\text{Sch}\left(\tanh\frac{\pi}{\beta}f,u \right), \quad \left\langle :T_{vv}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}v]:\right\rangle_{{\scriptscriptstyle \text{CFT}}} = -\frac{1}{24\pi}\text{Sch}\left(\tanh\frac{\pi}{\beta}f,v\right) \, .$$ Doing the gravitational path integral one finds [@Mertens:2019bvy] $$\begin{aligned}
\label{Unruhflux2}
\bra{\mM}:T_{uu}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u]:\ket{\mM} &= \bra{\mM}:T_{vv}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}v]:\ket{\mM} =\frac{M}{12 \pi}\, .\end{aligned}$$ The total energy in the heat bath is then[^26] $$E_{\text{bath}} = \int_{-\infty}^{+\infty} du\, \bra{\mM}:T_{uu}[f {\,\rule[-1.2pt]{.2pt}{1.7ex}\,}u]:\ket{\mM}=\frac{V M}{12\pi}\, .\label{33}$$ The bath energy $E_{\text{bath}}$ is defined operationally by summing the bath energy densities $T_{uu}$ across an entire spatial slice. However, this is not the total matter energy that backreacts on the geometry: including the vacuum energy gives a net zero result. In flat spacetime this famously leads to the statement that the energy in the Unruh heat bath does not deform the flat geometry one started with.\
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Our goal here is to find a spectral quantity that reproduces this bath energy $E_{\text{bath}}$, and the first candidate is $\bra{\mM}\omega\,A_\omega[f]^\dagger A_\omega[f] \ket{\mM}$. However, one finds very explicitly from that: $$\int_{0}^{+\infty}d\omega \bra{\mM}\omega\,A_\omega^\dagger[f] A_\omega[f] \ket{\mM}\, \neq \, E_{\text{bath}}\, .\label{34}$$ The potential confusion is closely related to the fact that the operators $A_\omega$ constructed in are not quite the same as raising and lowering operators acting on the matter Hilbert space (nor where they constructed to be). For example, we may compute: $$\begin{aligned}
\bra{\mM}[A_{\omega_1},A^\dagger_{\omega_2}]\ket{\mM} = \frac{\delta(\omega_1-\omega_2)}{\omega_1}\,&\rho_0(M+\omega_1)\,\rvert{\mathcal{O}}_{M \, M+\omega_1}^1\rvert^2\nonumber \\
&-\frac{\delta(\omega_1-\omega_2)}{\omega_1}\,\rho_0(M-\omega_1)\,\rvert {\mathcal{O}}_{M\, M-\omega_1}^1\rvert^2\, .\label{35}\end{aligned}$$ In the probe approximation $M\gg 1$ and $\omega\ll M$ this does reduce to $\delta(\omega_1-\omega_2)$. However for more generic values of $M$ and $\omega$, it does not.
We will below define the spectral energy density $\omega N_\omega[f]$ that does satisfy this relation: $$\int_{0}^{+\infty}d\omega \bra{\mM}\omega N_\omega[f]\ket{\mM} = E_{\text{bath}} \, ,$$ and find that it corresponds to a symmetrized version of $\omega A^{\dagger}_\omega A_\omega$.
Symmetric number operator
-------------------------
The goal here is to introduce a symmetric number operator $N_\omega[f]$, which does compute the bath spectral energy density. Consider first semi-classical matter. Following one defines a number operator $${\left\langle a^\dagger_\omega a_\omega \right\rangle}_{{\scriptscriptstyle \text{CFT}}} = \frac{1}{\pi\omega}\int_{-\infty}^{+\infty} du_1\int_{-\infty}^{+\infty}du_2\, e^{-i\omega(u_1-u_2)}\, {\left\langle \partial\phi(u_1) \partial\phi(u_2) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}\, .\label{38}$$ This features the Wightman bulk two-point function. Using the canonical commutation relation $$a^\dagger_\omega a_\omega =\frac{1}{2}a^\dagger_\omega a_\omega+\frac{1}{2} a_\omega a^\dagger_\omega -\frac{1}{2}\delta(0)\, ,\label{39}$$ we can write this in a more symmetric way as: $$\begin{aligned}
\label{symoper}
{\left\langle a^\dagger_\omega a_\omega \right\rangle}_{{\scriptscriptstyle \text{CFT}}} &= \frac{1}{\pi\omega}\int_{-\infty}^{+\infty} du_1\int_{-\infty}^{+\infty}du_2\, e^{-i\omega(u_1-u_2)}\, {\left\langle \partial\phi(u_1) \partial\phi(u_2) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}\nonumber \\&\qquad+ \frac{1}{\pi\omega}\int_{-\infty}^{+\infty} du_1\int_{-\infty}^{+\infty}du_2\, e^{-i\omega(u_1-u_2)}\, {\left\langle \partial\phi(u_2) \partial\phi(u_1) \right\rangle}_{{\scriptscriptstyle \text{CFT}}} - \frac{1}{2} \delta(0)\,.\end{aligned}$$ Semiclassically, this manipulation is completely harmless. The point is that and turn out not to be equivalent when promoting the Wightman two-point functions to operators in quantum gravity. This has to do with operator orderings.
The two Wightman bulk two-point functions are not precisely identical. Including an appropriate $i\varepsilon$ regulator we have $$\begin{aligned}
\label{wight}
{\left\langle \partial\phi(u_1) \partial\phi(u_2) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}&=-\frac{1}{4\pi}\,\frac{1}{\frac{\beta^2}{\pi^2}\sinh^2 \frac{\pi}{\beta}(u_1-u_2+i\epsilon)}\nonumber\\ {\left\langle \partial\phi(u_2) \partial\phi(u_1) \right\rangle}_{{\scriptscriptstyle \text{CFT}}}&=-\frac{1}{4\pi}\,\frac{1}{\frac{\beta^2}{\pi^2}\sinh^2\frac{\pi}{\beta}(u_1-u_2-i\epsilon)}\,.\end{aligned}$$ In Schwarzian quantum gravity, we can then likewise define the symmetrized version of the dressed operators $A^{\dagger}_\omega A_\omega$ and $A_\omega A^{\dagger}_\omega$ as: $$\begin{aligned}
B^\dagger_\omega[f] B_\omega[f] &= \frac{1}{2}A^\dagger_\omega[f] A_\omega[f] + \frac{1}{2}A_\omega[f] A^\dagger_\omega[f] -\frac{1}{2}\delta(0)\, \\
B_\omega[f] B^\dagger_\omega[f] &= \frac{1}{2}A^\dagger_\omega[f] A_\omega[f] + \frac{1}{2}A_\omega[f] A^\dagger_\omega[f] + \frac{1}{2}\delta(0)\, .\end{aligned}$$ By construction these modes $B_\omega$ automatically satisfy the commutator relation: $$[B_{\omega_1}[f],B_{\omega_2}^\dagger[f]]=\delta(\omega_1-\omega_2)\, .\label{313}$$
Dressing the bulk Wightman two-point functions in to include for Schwarzian gravitational interactions as in , we write for $N_\omega[f] = B^\dagger_\omega B_\omega$ [@Mertens:2019bvy]: $$\begin{aligned}
\label{planckp}
&{\left\langle N_\omega[f] \right\rangle}_{{\scriptscriptstyle \text{CFT}}} \\\nonumber &= -\frac{1}{8\pi^2\omega}\int_{-\infty}^{+\infty} d u_1 \int_{-\infty}^{+\infty} du _2\, e^{-i\omega (u_1-u_2)}\,\frac{f'(u_1)f'(u_2)}{\frac{\beta^2}{\pi^2}\sinh^2 \frac{\pi}{\beta}(f(u_1)-f(u_2)+i\varepsilon)} - \frac{1}{(u_1-u_2+i\varepsilon)^2}\\\nonumber &\quad -\frac{1}{8\pi^2\omega}\int_{-\infty}^{+\infty} d u_1 \int_{-\infty}^{+\infty} du _2\, e^{-i\omega (u_1-u_2)}\,\frac{f'(u_1)f'(u_2)}{\frac{\beta^2}{\pi^2}\sinh^2 \frac{\pi}{\beta}(f(u_1)-f(u_2)-i\varepsilon)} - \frac{1}{(u_1-u_2-i\varepsilon)^2}\,.\end{aligned}$$ Here we have rewritten the delta-function in as a term which explicitly subtracts the poles in the Wightman two-point functions.[^27] With this operator, we first compute the energy: $$\int_0^\infty d\omega\,{\left\langle \omega N_\omega[f] \right\rangle}_{{\scriptscriptstyle \text{CFT}}} \, .$$ By swapping the integration variables $u_1$ and $u_2$ in the integrals of , we see that the terms on the middle and last line are mapped into one another if we change the sign of $\omega$. Therefore, including a $1/2$ factor to enlarge the integration over $\omega$ along the entire real axis, we obtain a factor $$\int_{-\infty}^{+\infty} d\omega\,e^{-i\omega(u_1-u_2)}=2\pi \delta(u_1-u_2) \, .$$ One then finds directly from Taylor expanding [^28] $$\begin{aligned}
- \frac{1}{8\pi} \int_{-\infty}^{+\infty} du_1\int_{-\infty}^{+\infty} d u_2 \,\delta(u_1-u_2)\,&\frac{f'(u_1)f'(u_2)}{\frac{\beta^2}{\pi^2}\sinh^2 \frac{\pi}{\beta}(f_1-f_2-i\epsilon)} - \frac{1}{(u_{1}-u_{2}-i\epsilon)^2} + (\epsilon \to - \epsilon) \nonumber \\&\qquad\qquad= - \frac{1}{24\pi} \int d y\, \text{Sch}\left(\tanh \frac{\pi}{\beta}f,y\right) \, .\end{aligned}$$ As classical functions this is true in any case. However, in the Schwarzian path integral this is only true when we work with the symmetrically dressed operators $B_\omega$. The reason is that the two Wightman two-point functions are not equal to one another after the Schwarzian path integral.[^29] The above trick $\omega \to - \omega$ maps one into the other, but since we started with a symmetric combination this has no impact. This confirms that we may view the Schwarzian path integral of as the spectral energy density $\bra{\mM}\omega N_\omega[f] \ket{\mM}$ in the Unruh heat bath. This agrees with the formulas presented in [@Mertens:2019bvy].
Analytical analysis
-------------------
Doing the Schwarzian path integrals in , we find $$\begin{aligned}
\bra{\mM}\omega N_\omega[f]\ket{\mM} = \frac{V}{4\pi}\,\rho_0(M+\omega)\,\rvert {\mathcal{O}}_{M M+\omega}^1\rvert^2+\frac{V}{4\pi}\,\rho_0(M-\omega)\,\rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2-\frac{\omega}{2}\,\delta(0) \, .\label{319}\end{aligned}$$ To deal with the delta-function, we will choose to calibrate our measurement to the zero energy [Poincaré ]{}state[^30] $$\bra{0}\omega N_\omega[f]\ket{0}=\frac{V}{4\pi}\,\rho_0(\omega)\,\rvert {\mathcal{O}}_{0\, 0+\omega}^1\rvert^2+\frac{V}{4\pi}\,\rho_0(-\omega)\,\rvert {\mathcal{O}}_{0\,0-\omega}^1\rvert^2-\frac{\omega}{2}\,\delta(0)\, .\label{320}$$ We will henceforth only discuss this relative spectral energy density $$\begin{aligned}
\bra{\mM}\omega N_\omega[f]\ket{\mM}=&\frac{V}{4\pi}\,\rho_0(M+\omega)\,\rvert {\mathcal{O}}_{M M+\omega}^1\rvert^2-\frac{V}{4\pi}\,\rho_0(\omega)\,\rvert {\mathcal{O}}_{0\, 0+\omega}^1\rvert^2\nonumber\\&+\frac{V}{4\pi}\,\rho_0(M-\omega)\,\rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2-\frac{V}{4\pi}\,\rho_0(-\omega)\,\rvert {\mathcal{O}}_{0\,0-\omega}^1\rvert^2 \, .\label{321}\end{aligned}$$ In the semi-classical probe approximation $M\gg 1$ and $\omega\ll M$ one recovers the classical answer for the spectral energy density $$\bra{\mM}\omega N_\omega[f]\ket{\mM}=\frac{V}{2\pi}\frac{\omega}{e^{\beta\omega}-1}\, .\label{322}$$ The Schwarzian answer which combines and is accurate for any $M\gg e^{-2S_0/3}$ and for any $\omega\gg e^{-S_0}$. This includes Planck sized black holes where $M\sim 1$. For such tiny black holes and with $\omega < M$ one finds slightly lower spectral energy density in as compared to . See figure \[Urandomz2\]. On the other hand for $\omega>M$ the Schwarzian result gives a slightly higher occupation as compared to the classical answer .
Including Euclidean wormhole corrections to the Schwarzian correlations is done analogously as in , and leads to: $$\begin{aligned}
\nonumber \bra{\mM}\omega N_\omega[f]\ket{\mM}=&\frac{V}{4\pi}\,\frac{\rho(M,M+\omega)}{\rho(M)}\,\rvert {\mathcal{O}}_{M M+\omega}^1\rvert^2-\frac{V}{4\pi}\,\frac{\rho(0,0+\omega)}{\rho(0)}\,\rvert {\mathcal{O}}_{0\,0+\omega}^1\rvert^2\nonumber\\&+\frac{V}{4\pi}\,\frac{\rho(M,M-\omega)}{\rho(M)}\,\rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2-\frac{V}{4\pi}\,\frac{\rho(0,0-\omega)}{\rho(0)}\,\rvert {\mathcal{O}}_{0\,0-\omega}^1\rvert^2\, .\label{323}\end{aligned}$$ The behavior of this function is in fact quite similar to . In particular we see quadratic level repulsion for $\omega \sim e^{-S_0}$. One notable difference with is the effect of the zero energy subtractions. As we review in appendix \[s:zeroref\] these come which much slower wiggles as compared to the oscillations in . The last term in is in the forbidden region and is suppressed as $\sim e^{-S_0}$ in any case, making it negligible in practice. Supersymmetric JT gravity and the resulting random matrix completion from the Altland-Zirnbauer ensembles have a hard spectral edge at $\omega=0$ [@Stanford:2019vob] removing the forbidden region alltogether.
We have checked numerically for that this indeed computes the spectral energy density , by satisfying $$\label{toten}
\int_0^{+\infty} d\omega \bra{\mM}\omega N_\omega[f]\ket{\mM} = \frac{V M}{12\pi} \,.$$ Upon including higher topology in , there are several tiny corrections. This is perfectly fine since neither nor is precise when including Euclidean wormhole corrections. We note that via one finds that such corrections are at least suppressed by a factor $e^{-S_0}$. Therefore they can be neglected.[^31]
For completeness, we note that the contact term contribution in implies a further contribution to of the form: $$\begin{aligned}
\bra{\mM}\omega N_\omega[f]\ket{\mM} \, \supset \, \delta(\omega)\,\frac{V}{2\pi}\, \bigl(\rvert {\mathcal{O}}_{M M}^1\rvert^2-\,\rvert{\mathcal{O}}_{0\,0}^1\rvert^2 \, \bigr) \,.\end{aligned}$$ It is impossible to measure precisely zero energy so this contribution seems less interesting. On the other hand, there are several known examples of zero energy modes being important in a black hole context [@Donnelly:2014fua; @Donnelly:2015hxa; @Blommaert:2018rsf; @Blommaert:2018oue].
Numerical analysis
------------------
It is clarifying to explicitly plot the spectral energy density as computed in the three levels of improving approximation in , and . We note that the zero-energy kernels in are different from the universal answer , which only holds far enough from the spectral edge. For the zero-energy kernels we may utilize instead results from the exactly solvable Airy model.[^32] We present some details in appendix \[s:zeroref\].\
\
The effects which we aim to see in the plots are
1. The main effect of the Schwarzian corrections is that the Schwarzian answer (red) lies below the semiclassical answer (green) for small $\omega$ and above the semiclassical answer for large $\omega$.
2. Secondly and most importantly, we can clearly see the signs of level repulsion. There is a depletion in the spectral density of the Unruh heat bath for $\omega\ll e^{-S_0}$ and furthermore we see high-frequency wiggles in the regime where $\omega$ is order $e^{-S_0}$.
3. Finally, there are similar wiggles associated to the zero-energy subtraction in . These will play a role when $\omega$ is order $e^{-2S_0/3}$. For $M\gg 1$ these Airy wiggles effectively become invisible as all contributions to grow exponentially with $M$.
We will consider a parametric regime where all these effects are clearly visible. Therefore we take $M=2$ and $S_0=10$ in both figures \[Urandomz2\] and \[Urandomfull\].
![(left) Zoom-in on the region with $\omega$ or order $e^{-S_0}$ where we see level repulsion and high-frequency wiggles in the exact (blue) result. (right) Zoom-in on the region $\omega$ of order $e^{-2S_0/3}$ where we see the slower Airy wiggles from the zero energy reference . Furthermore one sees that the Schwarzian answer (red) is lower than the semi-classical answer (green).[]{data-label="Urandomz2"}](Urandomzv2.pdf){width="95.00000%"}
![The Schwarzian effects (right) and level repulsion effects (left) are simultaneously visible in this log plot of $\bra{\mM}\omega N_\omega[f]\ket{\mM}$. One may compare the exact answer (blue) with the Schwarzian answer (red) and the semiclassical answer (green).[]{data-label="Urandomfull"}](Urandomfull.pdf){width="95.00000%"}
Concluding remarks {#s:concl}
==================
The main goal of this work was to advocate and provide evidence that Hawking-Unruh radiation is highly sensitive (at ultra low energies) to level repulsion in the chaotic spectrum of the underlying quantum black hole.
We made this explicit by probing a massless scalar field coupled to JT gravity using an Unruh-DeWitt detector. The calculation involves including Euclidean wormhole corrections to a massless scalar bulk two-point function in JT gravity. Due to random matrix universality for quantum black holes, we expect our conclusion to be quite universal and to qualitatively hold in any number of dimensions. One immediate way to test the universality of these ideas is to consider an Unruh-DeWitt detector in super JT gravity or to consider charged versions of JT gravity. Results in this direction are forthcoming.\
\
We end this work by emphasizing three features of our setup that deserve more work.\
\
***Gravitational dressings***\
\
One important aspect of this work is navigating through different types of operator dressings and operator ordering ambiguities. In particular, for the gravitationally dressed matter modes $A_\omega[f]$ in , we obtained the expectation value of their correlator . Schematically, we way write this in a canonical language as $$[A_{\omega_1}^\dagger[f], A_{\omega_2}[f]]=\delta(\omega_1-\omega_2)+ \text{gravitational corrections}.$$ The first contribution is due to the canonical oscillator algebra of the undressed matter modes $a_\omega$. The second contribution is due to canonical commutators of the gravitational variables. Indeed, the matter modes $A_\omega[f]$ include a dressing with gravitational variables. See for example [@Donnelly:2015hta; @gid3] and section 5 of [@ads2]. This should be contrasted with the behavior of the dressed matter modes $B_\omega[f]$ in for which we write by construction $$[B_{\omega_1}^\dagger[f], B_{\omega_2}[f]]=\delta(\omega_1-\omega_2).$$ In defining $B_\omega[f]^\dagger B_\omega[f]$ via we have specified a symmetric gravitational dressing for this combination of modes, aimed in a precise way such that the canonical algebra of the modes would receive no gravitational corrections. This means a single mode $B_\omega[f]$ is by itself not a diff-invariant observable but the gravitational dressed composite operator $N_\omega[f] = B_\omega[f]^\dagger B_\omega[f]$ is. On the other hand, the modes $A_\omega[f]$ are well defined diff-invariant operators. The definition of the Unruh-DeWitt detector requires we use such individually well-defined diff-invariant modes.\
\
More broadly speaking there are numerous operator ordering ambiguities whenever we are promoting a correlator or a field in semiclassical physics to an operator in quantum gravity. One might refer to all of these as choices associated with gravitational dressings [@Donnelly:2015hta; @gid3]. However in the real world we are all gravitationally dressed composite objects with an implicit choice of dressing. This begs the questions: which dressings are actually natural in quantum gravity and which are just mathematical curiosities? Given a particular physical context, what is the appropriate dressing?
In our particular case of Hawking-Unruh radiation, we were able to pinpoint a natural type of dressing for two distinct experiments. For the Unruh-DeWitt detector experiment we were led to work with the dressed matter modes $A_\omega[f]$ whereas in an experiment which measures the spectral energy density in the heat bath the modes $B_\omega[f]$ turned out to be relevant. Nevertheless, these dressing ambiguities remain largely elusive.\
\
***Higher genus bulk correlators***\
\
Another important aspect of our story is a prescription for how to include Euclidean wormhole corrections to inherently Lorentzian bulk observables in JT gravity. Let us summarize the general idea by an algorithm
1. Find an expression for the bulk matter correlator in the reparameterized metrics .
2. Attempt to rewrite that expression as a combination of Schwarzian bilocal operators. An algorithmic way of doing so is to reverse engineer bulk reconstruction in each of the metrics . Write each term as a Euclidean JT gravity path integral on the disk including corresponding boundary to boundary matter propagators.
3. Include Euclidean wormhole contributions to each such individual JT gravity boundary correlator as done in e.g. [@phil]. The result is a correlation function in a double-scaled matrix integral. This is identical to summing over all Riemann surfaces which end on the union of the bilocal lines and the boundary.[^33] Effectively this leads to replacing $\rho_0(E_1)\dots \rho_0(E_n)$ with $\rho(E_1\dots E_n)$ in the Schwarzian correlators. The correlators $\rho(E_1\dots E_n)$ are multi-level correlators of a double-scaled matrix integral and may be obtained via universal random matrix cluster functions [@mehta; @paper6].
4. Sum over all these boundary correlators in order to obtain the bulk correlator.
One pragmatic way to argue for this prescription is that the resulting bulk correlators end up showcasing certain generic physical principles. One example is level repulsion in Hawking-Unruh radiation as discussed in this work. A second example is the behavior of bulk matter correlators at large distances in a finite entropy system as will be discussed in [@wophilbert].\
\
***Implications for evaporation?***\
\
Our discussion on the Unruh-DeWitt detector experiment is an idealization of a more realistic experiment where the measurement takes place over an infinite amount of time. In a more realistic experiment, we would measure for a finite time ${\small \text{T}}$. This can be implemented in the formulas by introducing in the coupling an additional switching function $\chi(t)$ which has a width of order ${\small \text{T}}$. This introduces a factor $\chi(t_1)\chi(t_2)$ in the integrand on the second line of . If we denote the Fourier transform of $\chi(t)$ by $\hat{\chi}(\omega)$ then is replaced by a convolution of the previous answer with the frequency content of the switching function $$R(\omega) \sim \frac{1}{{\small \text{T}}} \int_{\mathcal{C}} d\tilde{\omega} \, \hat{\chi}^2(\tilde{\omega}-\omega) \, \frac{\rho(M,M - \tilde{\omega})}{\rho(M)} \left|\mathcal{O}^1_{M \, M - \tilde{\omega}}\right|^2\, \frac{\sin^2 z \tilde{\omega}}{\tilde{\omega}^2} \, .$$ This convolution replaces but also for example with a version that is smooth on frequency scale of order $1/{{\small \text{T}}}$.
This implies we must measure for a time ${{\small \text{T}}}\gg 1/\beta$ in order to resolve the semiclassical Planckian black body law. The level repulsion in and the delta spikes in can only be resolved if we measure for a time ${{\small \text{T}}} \gg e^{S_0}$. In a non-evaporating setup this remains a sensible experiment. However for an evaporating black hole on these time scales we are in the regime where the Page curve is decreasing [@rw1] and any eternal approximation no longer applies. Nevertheless it is quite natural that the information about the microstates in our eternal model of quantum gravity can only be accessed from a measurement that takes longer than what would be the Page time in an evaporating setup.
At the very least this work emphasizes that we should expect extraordinary nonperturbative effects in gravity to highly affect Hawking radiation at long time scales ${{\small \text{T}}}\gg e^{S_0}$. These effects can be probed directly in the quantum gravity bulk using an Unruh-DeWitt detector.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Jordan Cotler, Julius Engelsoy and Joaquin Turiaci for useful discussions. AB and TM gratefully acknowledge financial support from FWO Vlaanderen.
Detectors with other boundary conditions {#app:obc}
========================================
In section \[s:udw\] we studied an Unruh-DeWitt detector that couples to a massless scalar field with Dirichlet boundary conditions . One might be interested in bosonic bulk matter with other boundary conditions. The result generalizes as follows. For example, imposing Neumann boundary conditions one finds $$\begin{aligned}
&{\left\langle \Phi[f\rvert u_1,v_1]\Phi[f\rvert u_2,v_2] \right\rangle}_{{\scriptscriptstyle \text{CFT}}}\nonumber\\&\quad=-\frac{1}{4\pi}\ln (F(u_1)-F(u_2))(F(v_1)-F(v_2))(F(v_1)-F(u_2))(F(u_2)-F(v_2))\, .\label{a1}\end{aligned}$$ As a Schwarzian insertion, notice that this operator is not SL$(2,\mathbb{R})$ invariant. For the Fourier transform, we can integrate by parts twice: $$\begin{aligned}
\nonumber &\lim_{{{\small \text{T}}} \to +\infty}\frac{1}{{\small \text{T}}} \int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_1\int_{-{\small \text{T}}}^{+{\small \text{T}}} dt_2\, e^{-i\omega (t_1-t_2)}\,\ln (F(u_1)-F(u_2))\\&\qquad =\frac{1}{\omega^2}\lim_{{{\small \text{T}}} \to +\infty}\frac{1}{{\small \text{T}}} \int_{-{\small \text{T}}}^{+{\small \text{T}}}dt_1\int_{-{\small \text{T}}}^{+{\small \text{T}}} dt_2\, e^{-i\omega (t_1-t_2)}\,\frac{F'(u_1)F'(u_2)}{(F(u_1)-F(u_2))^2} \, .\end{aligned}$$ Notice that hence by integrating the above operator , the result *is* SL$(2,\mathbb{R})$ invariant. The full answer is a sum over four such terms, each with a generically different phase factor. Summing those phase factors given a greybody factor $\cos^2 z\omega / \omega^2$ replacing the $\sin^2 z\omega / \omega^2$ in the Dirichlet case. Including Euclidean wormhole corrections, one obtains $$R(\omega) = 2\frac{\cos^2 z\omega}{\omega^2}\, \frac{\rho(M,M-\omega)}{\rho(M)}\, \rvert {\mathcal{O}}_{M M-\omega}^1\rvert^2\, .\label{a4}$$ In terms of an experimental response the effect in this case is quite different to . The semiclassical answer now blows up for $\omega\ll 1$ due to the double pole in the interference factor. Level repulsion regulates this pole and results in a finite answer for $\omega\ll e^{-S_0}$. For generic conformally invariant boundary conditions, the greybody factor is $$\frac{1-a \cos 2\omega z}{\omega^2}\,.$$ Hera $a=1$ is Dirichlet, $a=-1$ is Neumann, and $a=0$ corresponds to transparant boundary conditions, where one only takes the first two factors in the logarithm of . So we see that the behavior of for $\omega\ll e^{-S_0}$ is an exception to the general rule where the semiclassical answer has a double pole for $\omega\ll 1$ which is regulated by level repulsion.
Airy model and zero energy reference term {#s:zeroref}
=========================================
Let us briefly address the zero energy [Poincaré ]{}reference contribution to the spectral energy density $$\bra{0}\omega N[f|\omega]\ket{0}=\frac{V}{4\pi}\,\frac{\rho(0,\omega)}{\rho(0)}\Gamma(1 \pm i\sqrt{\omega})^2+\frac{V}{4\pi}\,\frac{\rho(0,-\omega)}{\rho(0)}\Gamma(1 \pm i\sqrt{\omega})^2 \, .\label{b1}$$ We note that the contribution from the second term in is essentially negligible for every $\omega$ as it is evaluated in the forbidden region where there is barely any density.\
Consider first $\rho(0,\omega)$ for $\omega\ll 1$. Both energies are close to the spectral edge and therefore we must use the two-level spectral density of the Airy model instead of : $$\frac{\rho(E_1,E_2)}{\rho(E_2)}=\rho(E_1)+\delta(E_1-E_2)-\frac{K(E_1,E_2)^2}{\rho(E_2)} \, .$$ Here the spectral density is $$\rho(E) = e^{\frac{2S_0}{3}}\,{\text{Ai}}'(- e^{\frac{2S_0}{3}} E)^2-e^{\frac{2S_0}{3}}\xi {\text{Ai}}(- e^{\frac{2S_0}{3}} E)^2\,.\label{b2}$$ Furthermore the Airy kernel is[^34] $$\label{airykernel}
K(E_1,E_2) = \frac{{\text{Ai}}'(-e^{\frac{2S_0}{3}}E_1){\text{Ai}}(-e^{\frac{2S_0}{3}}E_2) - {\text{Ai}}'(-e^{\frac{2S_0}{3}}E_2){\text{Ai}}(-e^{\frac{2S_0}{3}}E_1)}{E_1-E_2}\, .$$ This kernel is only relevant for $E_1-E_2$ of order $e^{-S_0}$. Otherwise one finds: $$\frac{\rho(0,\omega)}{\rho(0)}=\rho(\omega)\quad,\quad \omega \gg e^{-S_0}\, .$$ Using this one finds: $$\bra{0}N_\omega[f]\ket{0}=\frac{V}{4\pi}\quad,\quad \omega\gg 1 \, .\label{b6}$$ This zero-energy subtraction ensures that the relative spectral energy density $\bra{M}\omega N_\omega[f]\ket{M}$ in goes to zero smoothly for $\omega \gg M$.
We furthermore note that is positive definite for any $M>0$. This is explicit in figure \[Urandomz2\]. For $M\gg 1$ is is also quite easy to check this explicitly. This whole zero energy contribution is strongly suppressed by exponentials of $M$ at any $\omega$ scale.\
\
In order to construct a plot of we use the Airy result for $\omega\ll 1$, and the smooth JT gravity spectral density $\rho(\omega)$ as determined in [@sss2] $$\rho(\omega)=\rho_0(\omega)-\frac{e^{-S_0}}{4\pi\omega}\,\cos(2\pi e^{S_0}\int_0^\omega d E\,\rho_0(E)) \, .\label{b7}$$ This can be trusted for $\omega\gg e^{-2S_0/3}$. Since $S_0 \gg 1$, these regions overlap so we can construct the whole plot. Both $\rho(\omega)$ and $\bra{0}N_\omega[f]\ket{0}$ are plotted for any positive $\omega$ in figure \[Urandomref\] which uses $S_0=10$. Notice the wiggles in the latter which shine through in figure \[Urandomz2\] (right).
![ (left) Spectral density $\rho(\omega)$ in JT gravity for any positive energy $\omega$. The Airy answer is accurate for $\omega$ much smaller then the right dotted line. The JT gravity answer of [@sss2] is accurate for $\omega$ much greater then the first dotted line. (right) $\bra{0}N_\omega[f]\ket{0}$ for any positive $\omega$. The asymptotic limit is shown (green). []{data-label="Urandomref"}](airyLRv2.pdf){width="95.00000%"}
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[^1]: For a system with internal symmetries level repulsion or random matrix statistics emerges within subsets of energy levels where each subset has fixed values of quantum numbers related to these internal symmetries. See for example [@mehta; @haake; @Kapec:2019ecr].
[^2]: Gravitational shockwaves in this model have been studied in [@malstanyang; @shockwaves].
[^3]: For example in order to probe scrambling in de Sitter one is led to investigate shockwaves and out-of-time-ordered correlators in the bulk [@aalsmashiu; @wopjordan]. In this setup we do not have access to intuition from a unitary dual theory [@cotleremergent].
[^4]: We will focus on such systems here. The discussion is immediately modified to other ensembles [@Stanford:2019vob].
[^5]: There is furthermore a contact term contribution $\rho(E_1)\delta(E_1-E_2)$ which is largely irrelevant to our story here. We define $${\text{sinc}}\, x =\frac{\sin x}{x}.$$
[^6]: The Gaussian orthogonal ensemble (GOE) and Gaussian symplectic ensemble (GSE) have linear respectively quartic level repulsion.
[^7]: One recognizes the Bose-Einstein distribution. The factor $\omega^{d-1}$ is proportional to the level density for a particle in a $d-1$ dimensional box. This formula should furthermore be augmented with suitable greybody factors associated with the centrifugal barrier and with interference terms associated with reflections off of asymptotic boundaries. We leave them implicit here.
[^8]: The middle formula is the unique rate which probes the two level spectral density for transitions of energy $\omega$ close to a heavy black hole of mass $E$ and which matches with the semiclassical answer for $\omega\gg 1/\rho(E)$.
[^9]: We work in units where the AdS length $L=1$.
[^10]: For example if we want to interpret JT gravity as the low-energy limit of a 4d near-extremal black hole, we have $S_0 \sim \frac{Q}{G}$ where $Q$ is the charge of the extremal black hole.
[^11]: On-shell, the equations of motion link the dilaton field to the matter content in the theory, and these constraints are then sufficient to find the Schwarzian equation of motion [@ads2].
[^12]: It is conventional to choose units such that $C=1/2$.
[^13]: For earlier discussion in this context see for example [@Blommaert:2019hjr; @Mertens:2019bvy].
[^14]: The Fourier transform appears due to $$\bra{\omega}\mu(t)\ket{0}=e^{i\omega t}\bra{\omega}\mu(0)\ket{0}.$$
[^15]: Infinitesimally separated bulk points along such a worldline are separated by $$ds^2 = -\frac{F'(t+z)F'(t-z)}{(F(t+z)-F(t-z))^2}dt^2$$ and since $F' \geq 0$ are hence for any off-shell $F$ time-like separated, proving that the resulting trajectory is always timelike. It is possible for the trajectory to become lightlike at points where time stops flowing $F'=0$.
[^16]: There is an implicit $i\varepsilon$ in the exponential $t_1-t_2-i\varepsilon$ as a Euclidean damping factor [@schwarzian]. This is related to the particular ordering of both operators in the Wightman two-point function. We will be more explicit about this in section \[s:hb\] where both operator orderings are relevant.
[^17]: One should multiply all signs.
[^18]: Note that for $z\ll 1$ this quantity goes to zero like $z^2$ as demanded by the extrapolate dictionary for a massless scalar field.
[^19]: Here the factor $V$ is the total volume outside the semiclassical black hole horizon and is an artifact of the matter theory.
[^20]: The integration contour is chosen as in [@sss2]. It follows the positive real axis. However in the classically forbidden region $E<0$ one needs to choose an appropriate contour for convergence. The contributions to observables from the forbidden part of the integration contour is highly suppressed in all cases by powers of $e^{-S_0}$. Therefor we can essentially neglect these contributions.
[^21]: We note [@phil] that is conform the idea that this expression represents the ensemble average over different Hamiltonians $H$ of the two point function in a discrete quantum chaotic system. The idea is to take such a set of discrete quantum chaotic systems and to first ensemble average over unitaries $U$ which diagonalize $H$. One then invokes a version of the eigenvalue thermalization hypothesis $$\sum_{a,b}{\mathcal{O}}_{a}{\mathcal{O}}_{b}\int d U\,U_{i a}\,U_{j a}^*\,U_{j b}\,U_{i b}^* = \rvert{\mathcal{O}}_{E_i E_j}\rvert^2.$$ The assumption is that one point functions in the averaged theory vanish (which they do in JT gravity). Furthermore $\rvert{\mathcal{O}}_{E_i E_j}\rvert^2$ are smooth functions on energy scales of order the typical level spacing $e^{-S_0}$. The result is . Furthermore ensemble averaging over the eigenvalues of the Hamiltonians (with a well chosen potential $V(H)$ for Hamiltonians $H$ in the ensemble average) one then indeed finds .
[^22]: Here $\rho(\lambda_i)$ is to be understood as a function that varies only on scales much larger than $e^{-S_0}$.
[^23]: We use the Dirac algebra convention where $\gamma_0= i \sigma_1$ and $\gamma_1 = \sigma_3$, in terms of the Pauli $\sigma$-matrices.
[^24]: Variation of the massless Dirac action gives a boundary condition: $$\left. \bar{\psi} \gamma^1 \psi \right\rvert_\partial = 0 \, .$$ This equation holds also in curved spacetime as one checks that all Weyl scaling factors end up cancelling. In terms of spinor components $\psi_\alpha, \, \alpha=1,2$, this becomes: $$\psi_1^*\, \psi_2\rvert_\partial = \psi_1 \,\psi_2^*\rvert_\partial \, .$$ The Dirichlet boundary discussed above corresponds to: $$\psi_1\rvert_\partial =\psi_2\rvert_\partial \, .$$ This results eventually in the $4\sin^2\,\omega z$ greybody interference factor in . Other possibilities include for example setting $\psi_1=0$ or $\psi_2=0$. This results instead in a $4 \cos^2\, \omega z$ interference factor. See also appendix \[app:obc\].
[^25]: This follows from a generalization of the Barnes identities and is identical to the orthonormality relation of the zeroth Wilson polynomial [@groenevelt]. In particular the identity is: $$\begin{aligned}
\frac{1}{4\pi i}\int_{-\infty}^{+\infty}ds \frac{\Gamma(a \pm s) \Gamma(b \pm s) \Gamma( c \pm s) \Gamma (d\pm s)}{\Gamma(\pm 2s)} = \frac{\Gamma(a+b)\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)\Gamma(c+d)}{\Gamma(a+b+c+d)}.
\end{aligned}$$ This identity holds when $\Re a,b,c,d >0$. We checked it numerically as well.
[^26]: We have included the contribution from the $v$-lightcone component by using the doubling trick.
[^27]: This is directly found by using the integral representation of the step function. See also [@Fabbri:2004yy] for the semi-classical versions of these equations.
[^28]: The validity of such a series expansion within Schwarzian correlators is not entirely straightforward. We refer to [@schwarzian; @Mertens:2019tcm] for some comments and to [@toth] for a thorough analysis.
[^29]: Unlike the semi-classical answer , the Schwarzian answers and its complex conjugate differ by more than an infinitesimal term.
[^30]: Other options exist, and will be explored elsewhere.
[^31]: Actually, for all intents and purposes this suppression is exact. The sine kernel and contact term contributions in cancel perfectly upon integration (by construction). Furthermore there are only quick wiggles but their integral gives a suppressed effect as well.
[^32]: The kernel is accurate as long as both $E, M\gg e^{-2S_0/3}$.
[^33]: This is true in a precise sense for the two-point function considered in this work. However it is a slight oversimplification for higher point functions. For example in an out-of-time-ordered disk four-point functions, two bilocals cross. This crossing can be avoided on higher genus surfaces if one bilocal traverses a handle whilst the other travels underneath it. In the end however such contributions can also be written in terms of multi-level correlators $\rho(E_1\dots E_n)$ [@wophilbert].
[^34]: For a recent derivation of this kernel from a brane correlator computation, see appendix A.4 of [@paper6].
|
---
abstract: 'Network analysis is rapidly becoming a standard tool for studying functional magnetic resonance imaging (fMRI) data. In this framework, different brain areas are mapped to the nodes of a network, whose links depict functional dependencies between the areas. The sizes of the areas that the nodes portray vary between studies. Recently, it has been recommended that the original volume elements, voxels, of the imaging experiment should be used as the network nodes to avoid artefacts and biases. However, this results in a large numbers of nodes and links, and the sheer amount of detail may obscure important network features that are manifested on larger scales. One fruitful approach to detecting such features is to partition networks into modules, *i.e.* groups of nodes that are densely connected internally but have few connections between them. However, attempting to understand how functional networks differ by simply comparing their individual modular structures can be a daunting task, and results may be hard to interpret. We show that instead of comparing different partitions, it is beneficial to analyze differences in the connectivity between and within the very same modules in networks obtained under different conditions. We develop a network coarse-graining methodology that provides easily interpretable results and allows assessing the statistical significance of observed differences. The feasibility of the method is demonstrated by analyzing fMRI data recorded from 13 healthy subjects during rest and movie viewing. While independent partitioning of the networks corresponding to the the two conditions yields few insights on their differences, network coarse-graining allows us to pinpoint *e.g.* the increased number of intra-module links within the visual cortex during movie viewing. Given the computational and visualization challenges due to increasing resolution and accuracy of brain imaging data, we expect that the importance of methods such as network coarse-graining will become increasingly important in helping to interpret the data.'
address:
- 'Department of Computer Science, School of Science, Aalto University, Helsinki, Finland'
- 'Department of Neuroscience and Biomedical Engineering, School of Science, Aalto University, Helsinki, Finland'
author:
- Rainer Kujala
- Enrico Glerean
- Raj Kumar Pan
- 'Iiro P. Jääskeläinen'
- Mikko Sams
- Jari Saramäki
bibliography:
- 'references.bib'
title: 'Graph coarse-graining reveals differences in the module-level structure of functional brain networks'
---
functional magnetic resonance imaging, functional brain networks, modules, coarse-graining
**This is the pre-peer reviewed version of the following article: Kujala, R., Glerean, E., Pan, R. K., Jääskeläinen, I. P., Sams, M., Saramäki, J. (2016), Graph coarse-graining reveals differences in the module-level structure of functional brain networks. European Journal of Neuroscience, 44: 2673–2684, which has been published in final form at <http://dx.doi.org/10.1111/ejn.13392>. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.**
Introduction
============
Methods of network science are increasingly used for analyzing functional magnetic resonance imaging (fMRI) data [@Newman2010; @Sporns2009; @Power2011Neuron; @papo2014functional]. In this framework, fMRI time series are mapped to a network, where the nodes correspond to different brain areas and the links between nodes indicate dependencies between the blood-oxygen-level dependent (BOLD) time series of these areas. Whenever the dependency between the BOLD signals corresponding to two brain areas is strong enough, they are thought to be functionally related.
Network analysis has revealed many insights on the functional structure of the brain. For instance, on the scale of network nodes, the use of various network centrality measures has allowed consistent detection of certain hub regions [@vandenHeuvel2013]. At the network level, the entire structure of functional brain networks has been found to be of the ‘small-world’ type [@Watts1998; @Eguiluz2005; @Salvador2005], meaning that the average number of steps required to reach any other node in a network is low while the network is locally clustered. In addition to revealing such general properties, the network framework has also been used for investigating how brain dynamics depend on different stimuli [@lahnakoski2012naturalistic], mental health [@Achard2012a; @AlexanderBloch2012; @Glerean2015], and age [@Meunier2009Age].
Despite being widely applied, the network approach to brain functional networks can still be considered as somewhat immature, and methodological variations persist in the literature [@Stanley2013; @Garrison2015]. In particular, there is no standard set of brain areas to be used as network nodes; rather, different studies use different definitions of what constitutes a node [@Stanley2013]. Approaches for defining network nodes range from the use of anatomical atlas-based Region-Of-Interest (ROI) parcellations to the use of the original fMRI imaging voxels of a few mm$^3$ in size [@Stanley2013]. Typically, the number of nodes in atlas-based definition schemes is of the order of $10^2$, while the number of nodes in voxel-based node definition schemes is of the order of $10^3-10^4$. These differences between node definitions make comparison of results across studies challenging.
The use of small number of nodes is understandable as it makes network analysis easier, and a low number of nodes even allows meaningfully visualizing the networks. However, such low resolution also means that the nodes of a network may cover multiple functionally specific areas. As the representative BOLD signal for each node is typically computed as the average over the node’s spatial extent, this can lead to significant loss of information and node-level signals that are not representative of true function. The recommended remedy is to instead use individual voxels as the network nodes [@Stanley2013]. Given that with high-field (*e.g.* 7 Tesla) fMRI it has been possible to reach a spatial resolution smaller than 1 mm$^3$ [@yacoub2008high], the volumes that voxel-based nodes represent can be expected to decrease in the future, while their number increases.
Constructing and analyzing networks that consist of voxel-level nodes is challenging because of the large number of nodes and links. While computing node-wise centrality measures or global network characteristics may be possible for voxel-level networks, understanding the *the overall organization of a network’s links* becomes difficult. Because the number of potential links is of the order of millions, mere visualization of such networks is extremely challenging, and *e.g.* comparing two networks on a link-to-link basis becomes meaningless.
One approach for going around this problem is to look at network structure on an intermediate level between the level of nodes and the level of the entire network. This intermediate level is often approached by splitting the network into *modules*, *i.e.* groups of nodes that are densely connected internally but have few connections between them [@Fortunato2010]. Modules are typically discovered using stochastic algorithms that partition the network into non-overlapping node groups. For examples of applications of module detection to functional brain network analysis, see [@Meunier2009; @Meunier2010; @Uehara2012; @Power2011Neuron; @betzel2016dynamic; @sporns2016modular].
Any observed modular structure of functional brain networks may arise from experimental conditions (different stimuli), or reflect more persistent, underlying features of the subject brain. Some recent studies have investigated how network modules differ between healthy controls and patients suffering from schizophrenia [@AlexanderBloch2012], patients with autism spectrum disorders [@Glerean2015], or patients who are comatose [@Achard2012a]. However, due to various intricacies in the detection of network modules and the difficulty of comparing different module partitions, results have been difficult to interpret and their statistical significance has remained elusive. Although one can statistically approach the stability of modules [@Moussa2012; @Glerean2015] or the similarity of partitions between groups of networks [@AlexanderBloch2012], verifying the significance of specific differences in modules remains an open problem. Therefore, there is a need for appropriate methods for analysing and comparing functional network structure at the intermediate level of modules. We argue that instead of focusing on how the network modules themselves differ between groups of networks, it is more fruitful and methodologically sound to assess the differences in the numbers of connections between and within a fixed set of modules that is used as a frame of reference for all groups. The boundaries of these modules are defined using a specific network or group of networks, and applied as such to the rest of available networks. This coarse-graining approach allows transparent, statistically verifiable investigation of module-level differences between networks.
Below, we first demonstrate the difficulties of comparing network modules with the help of a toy example, and proceed to show how the coarse-graining approach overcomes these problems. Then, to show the applicability of our approach in practice, we apply it to networks constructed from fMRI data recorded for 13 subjects during rest and movie viewing.
Comparing functional networks: coarse-graining and alternative approaches {#sec:coarse-grain-toy}
=========================================================================
Perhaps the simplest approach for studying differences in the link structure of functional brain networks is to investigate how the existence or weight of individual links differs between groups of networks. In this case, assessing the statistical significance of the observed differences is straightforward. However, with voxel-based functional brain networks that have thousands of nodes and millions of links this approach is impractical for several reasons. First, the number of statistical tests that need to be performed reaches millions, requiring a large amount of computational resources. Second, the spatial locations of functionally specific parts of the brain differ across people even after imaging data have been transformed into standard coordinates. Consequently, there is no one-to-one correspondence between specific nodes and links across subjects. Third, visualizing all differences between groups of networks becomes impossible. Because of the above, some studies have focused on quantifying differences between networks at the level of modules that have been discovered using stochastic network partitioning algorithms [@AlexanderBloch2012; @Glerean2015]. It is worth noting that there is no universally agreed definition for network modules [@Fortunato2010] – rather, each algorithm introduces its own definition of a network module that dictates how it partitions a network and how modules should look like. Subsequently, different algorithms give different results. Therefore, proper interpretation of module partitions requires a profound understanding of the underlying mathematical ideas and of the actual implementation of the algorithm.
If interpreting what module partitions signify can be difficult, it is even far more difficult to interpret differences between partitions. Often, small differences in a network’s link structure give rise to large differences in the partitioning of the network into modules. Yet, at times, major structural differences result in little or no differences in the discovered modules. We illustrate this with Fig. \[fig:intro-example\]. In this toy example, we consider two networks, A and B, that have a clearly modular structure. Both networks share the same set of nodes and their link structures are almost the same, with 4 links being different wired differently in B. We then apply the Louvain algorithm [@Blondel2008], one of the most popular methods for detecting network modules, to the two networks separately. While the removal of three links between the yellow and brown modules of Network A results in no differences in the modules detected in network B, the addition of only one extra link between the two smaller green and violet modules in Network A forces them to merge into one large blue module in Network B. This simple example illustrates that it is not straightforward to infer the underlying differences in the overall organization of links by comparing only the network partitions: the cause and the apparent effect size can be disproportionate.
When dealing with experimental, noisy fMRI data the challenges are even greater. The randomness in module partitions that arises from the stochasticity of the algorithms is further amplified by the noise in the data, which obscures real structural differences between networks. In addition, there are no statistical frameworks for directly assessing the statistical significance of differences between partitions – *e.g.*, is the merging of the two network A modules into the the larger blue network B module statistically significant or just due to chance? Thus, even when the structure of network modules may shed some light on the functioning of the brain under different conditions, drawing definite conclusions from differences between network partitions is very difficult.
![**Network modules, difficulties in comparing them, and the strengths of network coarse-graining.** **Top row** shows two networks A (left) and B (middle) that differ from each other only by a relocation of four links (right); blue links are present in A but missing in B, whereas red links are there in B but not in A. **Second row** shows the network modules corresponding to networks A and B as identified by the Louvain algorithm that optimizes modularity [@Newman2004; @Blondel2008]. On right, the differences in the discovered modular structure are visualized as an alluvial diagram [@Rosvall2010]: the left side of the diagram represents the modules of network A and the right side represents the modules of network B. Ribbons connecting the left and right side show how the modules of A and B match each other in terms of their node composition: it is seen that the only difference between A and B is that the green and violet modules of A correspond to only one module in network B (blue). This difference arises from the addition of a single link between the green and violet modules. To the contrary, the three links between the yellow and brown modules in network A that are relocated in B do not give rise to differences in the modules; the rest of changes (see top row, right) do not produce any changes in modules either. These examples highlight the difficulty in inferring differences in the link structure of networks based on the modular structure alone. **Third row** shows coarse-grained versions of networks A and B and their difference, the coarse-graining being based on the modules detected for A. Here, the width of the link between two modules corresponds to the number of links between their constituent nodes in the original network. Similarly, the width of the arc around each module represents the number of links within the module. The differences between the coarse-grained networks are shown on the right. For the blue link, there are more between-module links in A than B, and for the red links, B has more between-module links. Note,how the coarse-grained difference network is able to compactly summarize the differences between networks A and B. **Bottom row** shows the same information as the third row, but in the matrix form. Each (square) element of the matrix corresponds to a module, and the area of the square is proportional to the number of links between modules (off-diagonal) or within modules (diagonal). The row and column colors correspond to the modules of network A.[]{data-label="fig:intro-example"}]({fig1}.pdf){width="0.95\linewidth"}
We have seen that directly comparing module partitions is difficult, as is making comparisons at the fine-grained level of individual links. However, as we will show below, combining both points of view yields fruitful results. In particular, we argue that first producing a fixed set of modules, whose boundaries will be the same in all networks, and then comparing inter- and intra-module connections across networks reveals their differences more clearly. Conceptually, this corresponds to network coarse-graining, where each module corresponds to a node of the coarse-grained network, and the number of links between two modules in the original network corresponds to the weight of the link in the coarse-grained network. The number of internal links within each module is taken into account as the weight of the self-link (connection from the node to itself) of each module-node of the coarse-grained network. The usefulness of the coarse-graining approach is demonstrated in Fig. \[fig:intro-example\] with the same toy networks as earlier. Notably, the comparison of coarse-grained networks reveals differences in a more transparent way than attempting to compare modules. In addition, when groups of networks are to be compared, the coarse-graining approach allows statistical testing of the differences in the mean number of links within a module or between two modules.
Materials and Methods
=====================
Participants {#participants .unnumbered}
------------
The participants were 13 healthy native Finnish speakers (ages 22-43 years, 2 females, 2 left-handed, no neurological or psychiatric history, no hearing impairments, normal vision). The ethical committee of the Hospital district of Helsinki and Uusimaa granted permission for this study which was conducted in accordance with the guidelines of the declaration of Helsinki. Each subject gave written informed consent prior to participation.
Stimulus paradigm {#stimulus-paradigm .unnumbered}
-----------------
The stimulus used in this paper has also been used previously [@Lahnakoski2012; @Salmi2014] and consisted of an edited version of the Finnish movie ‘The Match Factory Girl’ (Aki Kaurismäki 1990). The film was projected on a semi-transparent screen behind the subject’s head and the audio track was delivered via plastic tubes through porous earplugs. Each subject went through three sessions with the following order: resting state (15 minutes, 450 volumes), free viewing of the film (22min 58s, 689 volumes), resting state (15 minutes, 450 volumes). In this study we analyze data recorded during rest before viewing the movie, and during movie viewing. After preprocessing, the movie session was truncated to match the length of the rest session to avoid any biases from different scan durations.
Data acquisition {#data-acquisition .unnumbered}
----------------
MR imaging was conducted on a 3.0T GE Signa Excite MRI scanner, with a quadrature 8-channel head coil. A total of 29 functional gradient-echo planar axial slices (thickness 4 mm, 1 mm gap between slices, in-plane resolution 3.4 mm $\times$ 3.4 mm, imaging matrix 64 $\times$ 64, TE 32 ms, TR 2000 ms, flip angle 90$^\circ$,). T1-weighted images were also acquired (TE 1.9 ms, TR 9 ms, flip angle 15$^\circ$, SPGR pulse sequence) with in-plane resolution of 1 mm $\times$ 1 mm, matrix size 256 $\times$ 256 and slice thickness 1 mm with no gap.
Preprocessing {#preprocessing .unnumbered}
-------------
Preprocessing of the fMRI data was carried out with FSL (release 4.1.6 www.fmrib.ox.ac.uk/fsl). The first 10 volumes of each session were discarded from the analysis. Motion correction was performed with McFlirt and the data were spatially smoothed using a Gaussian kernel with 6mm full-width half maximum, and high-pass filtered with a 100s cutoff. Functional data were co-registered with FLIRT to the anatomical image allowing 7 degrees of freedom. Furthermore, the data were registered from the anatomical space to the MNI152 2mm standard template (Montreal Neurological Institute), allowing 12 degrees of freedom. The signal was bandpass filtered with a passband of 0.01–0.08Hz in accordance with standard functional connectivity procedures. To control for motion artefacts, motion parameters were regressed out from the data with linear regression (36 Volterra expansion based signals, see Ref. [@Power2014]). As it is known that head motion affects connectivity results, we controlled for motion with framewise displacement [@Power2012]: all subjects had at least 95% of time points under the suggested displacement threshold of 0.5mm. For this reason, we decided not to use the scrubbing technique and utilize all time points. When looking at the individual mean frame-wise displacement, there was no significant difference between conditions ($p$=0.2732). Finally, to further control for artefacts, voxels at the edge between brain and skull where the signal power was less than 2% of the individual subject€’™s mean signal power were excluded from the analysis. This resulted in 5562 6-mm isotropic voxels of brain grey matter covering the whole cerebral cortex, subcortex and cerebellum. After removing the first and last 15 data points due to bandpass filtering artifacts [@Power2014], we obtained for each 6-mm voxel a BOLD time-series with 410 time points corresponding to a duration of 13min 40s.
Network construction {#network-construction .unnumbered}
--------------------
The functional dependency of two voxels $i$ and $j$ can be measured in many ways, given their BOLD time series $s_i(t)$ and $s_j(t)$ [@Smith2011]. As there is no consensus on the best measure, we opt for simplicity and use the Pearson correlation coefficient, which has been shown to capture a major proportion of pairwise dependencies in fMRI data [@Hlinka2011]. For each subject and condition, we then compute a correlation matrix $\textbf{R}$, whose elements $r_{ij}$ are the estimated Pearson correlation coefficients between voxels $i$ and $j$. Given that we have 2 different conditions and 13 subjects, this yields 26 correlation matrices in total.
There are several ways of constructing networks from such matrices by thresholding them so that only chosen elements remain. One common approach is to use a constant threshold value, so that only nodes pairs whose correlation coefficients (link weights) exceed this value are connected by a link. Another typical approach is to include a fixed fraction of the strongest links in the network. Here, we adopt the latter approach as it has been shown to provide more stable estimates of various network measures [@Garrison2015].
We construct networks from correlation matrices as follows: for each matrix, we begin with an empty network, where each node corresponds to a voxel. We then first compute the maximal spanning tree (MST) of the correlation matrix, and insert the corresponding links to the network. As the MST connects all nodes, this guarantees that no nodes or groups of nodes remain isolated in the network; isolated modules would cause technical difficulties in the later stages of our pipeline. Next, we sort all correlation coefficients, and insert links corresponding to the strongest positive coefficients until the network contains $\frac{1}{2}N(N-1)\rho$ links in total, where $\rho \in [0,1]$ is a pre-defined network density. As the end result, we obtain 26 undirected, unweighted networks that all share the same set of nodes, and have the same number of links.
Selection of network density {#selection-of-network-density .unnumbered}
----------------------------
Choosing the fixed density for the thresholded networks is not a straightforward problem. There are no commonly-accepted criteria for choosing an optimal density. If the density is very low, too much information is discarded and features of interest may remain hidden. On the other hand, the presence of too many links may obscure relevant structures. Sometimes this problem can be overcome by investigating network structure across different network densities [@AlexanderBloch2012; @Lord2012]. However, carrying out detailed structural analysis of a large number networks of different densities quickly becomes overwhelming. Then, selecting a reasonable specific network density may be a better option. This is also the case in this study, where we compare the module-level structure of groups of networks in detail.
To guide our choice of network density, we investigate the similarity of pairs of networks as a function of the network density. If two networks share the same set of nodes and have the same number of links, the most straightforward approach to measure their similarity between is to count the number of shared links. This is the approach we adopt. Given two networks $G$ and $G'$ that both have $\frac{1}{2}N(N-1)\rho$ links, we monitor how the fraction of links $f(G,G')$ common to the networks changes with the network density $\rho$. The fraction of common links $f(G,G')$ is defined as the number of shared links divided by the total number of links in one network: $$f(G, G') = \frac{\text{number of common links in $G$ and $G'$}}{\frac{1}{2}N(N-1)\rho} \in [0,1].$$ In Fig. \[fig:frac-com-links\] we show $f$ as a function of the network density when averaged over pairs of networks, so that each pair represents (i) the same subject in different conditions, (ii) different subjects in the rest condition, (iii) different subjects in the movie condition, and (iv) different subjects in different conditions. For all cases, the fraction of common links first increases until $\rho \approx 0.1\%$; one might envision that at this density, a common “backbone” shared by networks is well captured. Then $f$ decreases until $\rho \approx$ 2%, after which it begins to monotonously increase as the networks become denser and more and more links are necessarily shared.
Based on the observed behaviour of $\rho$, we pick the value $\rho=2\%$ to be used in all subsequent analyses, as it appears to maximize non-trivial variation between network pairs (of course, there is more variation for excessively low $\rho$). This value was also used in Ref. [@AlexanderBloch2012] for networks with smaller numbers of nodes. In our data, the 2% network density translates to 309303 links in each thresholded network; for different networks, this on average corresponds to a correlation coefficient threshold of 0.56$\pm$ 0.04 (std) (no statistically significant difference between conditions).
Fig. \[fig:frac-com-links\] also reveals some further insights. First, the network similarity $f(G,G')$ is remarkably higher for networks corresponding to the same subject in different conditions than for different subjects in the same condition. This indicates that individual variation dominates over differences caused by different stimuli. Subsequently, it is essential to take the paired nature of the data into account when validating any results statistically. Second, when the similarity of networks of different subjects is assessed, network pairs corresponding to the movie condition are seen to be more similar than network pairs corresponding to rest condition or different conditions. This is expected, as the viewing of a well-directed movie stimulus has been found to synchronize the subjects’ brains [@hasson2010reliability], which results in increased functional connectivity compared to the similarities of resting state networks arising from shared functional anatomy and connectivity.
![Fraction of common links as a function of the network density, averaged over pairs of networks representing same subject in different conditions $(\circ)$, different subjects viewing the movie($\triangledown$), different subjects at rest ($\triangle$), and different subjects under different conditions ($\square$). Shaded areas denote 95% bootstrap confidence intervals of the mean. The dashed vertical line denotes the network density that corresponds to the inclusion of the minimal spanning tree (MST) (= $\frac{1}{N_\text{nodes}-1}$), and the solid vertical line denotes the 2% network density which we use in the actual module-level analyses. The plot shows that the data is strongly paired: the average fraction of common links for networks representing the same subject is significantly higher than the average fraction of common links for network pairs corresponding to different subjects. However, if we consider pairs of networks corresponding to different subjects, networks measured for movie-viewing tend to be more similar than other network pairs. []{data-label="fig:frac-com-links"}](fraction_of_common_links_vs_density_with_mst.pdf)
Computation of condition-wise consensus partitions {#computation-of-condition-wise-consensus-partitions .unnumbered}
--------------------------------------------------
Given the networks for all subjects S1-S13 and conditions, our next target is to compute representative network partitions for each condition. As outlined in Fig. \[fig:module-pipeline\], this is realized in two steps: First, we partition the networks using a popular partitioning algorithm, the Louvain algorithm [@Blondel2008], which is based on modularity optimization [@Newman2004]. In more detail, we run the stochastic Louvain algorithm 100 times for each network and select the network partition with the highest value of modularity. Then, for each condition, we summarize all 13 selected partitions with the help of the MCLA meta-clustering algorithm [@Strehl2003] which yields one representative network partition as an output. In addition to the network partitions, MCLA requires the user to input a parameter determining the upper bound for the number of modules in the consensus partition. In this study, the value of this parameter was set to the median number of modules in the 13 partitions.
To summarize, our pipeline transforms the subject-specific networks corresponding to one condition to a single representative *consensus partition* $\mathcal{P}$ consisting of $m$ modules $C_1, ..., C_m$ which in turn are sets of network nodes such that each node belongs to exactly one module.
![Pipeline for computing consensus modules for a networks corresponding to one experimental condition. First, for each subject S1-S13 and their functional network, we run the Louvain algorithm 100 times, and preserve the partition with the best modularity. Then, all 13 best partitions are summarized as a single consensus partition using the MCLA algorithm.[]{data-label="fig:module-pipeline"}](consensus_schematic_best_of_100.pdf){width="\linewidth"}
Comparing groups of coarse-grained networks {#sec:methods-coarse-graining .unnumbered}
-------------------------------------------
The network coarse-graining process briefly introduced in Fig. \[fig:intro-example\] is defined as follows: Given a network $G$ and a partition $\mathcal{P}$ consisting of a set of modules $\{C_1, C_2, ..., C_m\}$, the coarse-graining process yields a matrix $\mathbf{W}$ that has the following properties: The non-diagonal matrix element $\mathbf{W}_{i,j}$ represents the total number of links between the nodes of $C_i$ and $C_j$. Similarly, each diagonal element $\mathbf{W}_{i,i}$ represents the number of links within module $C_i$.
To evaluate differences between experimental conditions, we first coarse-grain each subject’s network into its matrix representation $\mathbf{W}^{\text{condition,i}}$, where $i$ stands for the index of the subject. Then, we average the coarse-grained matrix representations of all 13 subjects over each condition: $$\langle \mathbf{W} \rangle ^{\text{rest}} = \frac{1}{13}\sum_{i=1}^{13} \mathbf{W}^{\text{rest},i},$$ and $$\langle \mathbf{W} \rangle ^{\text{movie}} = \frac{1}{13}\sum_{i=1}^{13} \mathbf{W}^{\text{movie},i}.$$ Then, by investigating the elements of the mean difference matrix $\Delta \mathbf{W} = \langle\mathbf{W} \rangle^{\text{movie}} - \langle \mathbf{W} \rangle ^{\text{rest}}$, we can quantify the level of differences in the numbers of connections between and within modules. Thus, the value of $\Delta \mathbf{W}_{1,2}$ indicates how many more connections there are on average between modules 1 and 2 in the movie condition than in the rest condition.
To test the statistical significance of our findings, we perform paired permutation tests separately on each of the matrix elements. As we have a limited number of samples, we use the full permutation distribution yielding in total $2^{13} = 8192$ different permutations. All $p$-values we report are two-sided, and we correct for multiple comparisons using the original Benjamini-Hochberg (BH) FDR correction [@Benjamini1995].
Code {#code .unnumbered}
----
The Python code used in our analysis is freely available at <http:github.com/rmkujala/brainnets>.
Results and Discussion
======================
Consensus modules are similar to previously reported resting-state modules {#consensus-modules-are-similar-to-previously-reported-resting-state-modules .unnumbered}
--------------------------------------------------------------------------
The consensus modules computed for the rest and movie-viewing conditions are shown in Fig. \[fig:results\]A with different colors on the cortical surface. A browsable display of each module is available at NeuroVault <http://neurovault.org/collections/1080/> [@gorgolewski2015neurovault], where the modules are weighted by their consistency as measured by scaled inclusivity [@Steen2011] (see Supporting Information for more details). In the center of Fig. \[fig:results\]A, the matching of the rest and movie consensus partitions is also visualized as an alluvial diagram [@Rosvall2010].
For the rest condition, we identified 11 consensus modules (Fig. \[fig:results\], left hand side of alluvial diagram, from bottom to top): 1) Limbic (LIM)€“ subcortical midbrain structures; 2) Cerebellum/ventro-temporal (CRBL/VT); 3) Default mode (DM); 4) Precuneus (PCUN); 5) Visual (VIS); 6) Auditory (AUD); 7) Salience (SAL); 8) Fronto-Parietal (FP); 9) Visual-extrastriate (VISx); 10) Sensorimotor (SM); 11) Language (LAN). For the movie condition, we obtained 10 modules: 1) Ventro-temporal limbic (VTL); 2) Cerebellum (CRBL); 3) Default mode (DM); 4) Cuneus (CUN); 5) Visual (VIS); 6) Auditory (AUD); 7) Salience (SAL); 8) Fronto-parietal (FP); 9) Dorsal attention (DA); 10) Sensorimotor (SM). For details on how the modules were assigned a label, please see Supplementary Information.
There is a good agreement in the literature on the module-level structure of resting networks, which have been identified using various methods such as multidimensional clustering [@Yeo2011], Infomap graph clustering [@Power2011Neuron], and independent component analysis [@Smith2009]. Overall, the resting state consensus modules we obtained are in line with the previously reported resting state modules (see Table S1).
However, there is no general agreement on the module structure during movie viewing, or more generally, during a task. While one study has found that task and rest are highly similar [@cole2014intrinsic], another study has found remarkable differences in subcortical, limbic regions as well as primary sensory and motor cortices [@mennes2013extrinsic]. In our case, overall, the movie consensus modules are similar to the resting state modules, and *e.g.* the dorsal attention module that has been previously reported in resting state studies [@Yeo2011; @cole2014intrinsic] is even better identified amongst in our movie consensus partition than in the rest consensus partition (see Tables S1 and S2). Interestingly, we also identified a VTL subnetwork present only during movie viewing in agreement with [@Glerean2015], possibly suggesting stronger functional couplings between brain areas involved in the processing of social and emotional events in the movie.
Differences in condition-wise consensus modules are difficult to directly interpret {#differences-in-condition-wise-consensus-modules-are-difficult-to-directly-interpret .unnumbered}
-----------------------------------------------------------------------------------
As shown in Fig. \[fig:results\]A, the consensus network partitions obtained for the movie and rest conditions are broadly speaking similar: for most rest modules, there is a clear counterpart among the movie modules. At the same time, almost all rest modules overlap with multiple movie modules (and vice versa) – there are no simple relationships such as one module splitting into two, and the varying amount of overlap between modules results in a diagram that is not straightforward to interpret beyond the clear matches.
As there are no statistical frameworks that can be used for measuring the significance of the relationships between multiple modules, the differences in the alluvial diagram lack statistical validation. Some insights into the significance of the transitions in the alluvial diagram could be obtained by investigating the consistency of the modules [@Moussa2012; @Glerean2015]. However, these methods do not directly assess the significance of individual splits and joinings of modules between partitions. Thus, it remains challenging to draw conclusions about network differences based on the alluvial diagram alone.
Network coarse-graining allows simple, statistically verifiable interpretations {#network-coarse-graining-allows-simple-statistically-verifiable-interpretations .unnumbered}
-------------------------------------------------------------------------------
In Fig \[fig:results\]B, we show the matrix representations of the average coarse-grained networks corresponding to both conditions, as well as the coarse-grained difference network, where the resting-state consensus modules have been used as the basis for coarse-graining. As expected, the matrices representing the coarse-grained networks have high values on their diagonals, indicating that the density of links within consensus modules is higher than between them. Off-diagonal element provide an overview of how strongly the modules are connected.
In the coarse-grained difference matrix, we observe multiple elements that survive the 0.05 Benjamini-Hochberg FDR correction. These are also listed in Table \[tab:rest-fdr\]. All surviving elements are positive, indicating that there are more connections between the modules in the movie networks. In particular, the VIS module displays significantly more external as well as internal connections in the movie condition. The number of connections between the AUD and DM modules is also increased in the movie condition. The coarse-graining method thus succeeds in highlighting task-driven changes at visual areas as well as inferior temporal structures. For similar coarse-graining results where the movie modules are used as the frame of reference, please see Supporting Information.
Module pair $i,j$ $p$-value $\Delta W_{i,j}$ relative increase
------------------- ----------- ------------------ -------------------
VIS,VIS 0.00122 15184.7 46%
VIS,PCUN 0.00024 2624.6 60%
VIS,VISx 0.00171 7865.6 47%
VIS,CRBL/VT 0.00146 3312.1 170%
AUD,DM 0.00293 3646.3 100%
: Differences in link numbers within and between rest consensus modules that survive the 0.05 Benjamini-Hochberg FDR correction.[]{data-label="tab:rest-fdr"}
There are of course some similarities between the coarse-grained difference network and the alluvial diagram presented in Fig. \[fig:results\]. As an example, in both it is seen that the resting-condition VIS module and part of the resting-condition VISx module merge to form the larger VIS module in the movie condition. A simple explanation for this would be that there are more connections between the VIS and VISx rest modules in the movie condition. However, as motivated in Sec. \[sec:coarse-grain-toy\], this is not the only possible reason, and further, the statistical significance of the observation cannot be deduced using the module-matching/alluvial-diagram approach. However, the coarse-grained difference matrix clearly indicates that there is a statistically significant increase (47%) in the number of links between the VIS and VISx modules in the movie condition. Thus, while the alluvial diagram can be used for formulating hypotheses on changes in network structure, the coarse-graining process allows verifying that the observed differences are not due to random chance.
In addition to assessing differences in the connectivity between modules, the coarse-graining approach also allows investigating the internal connectivity of modules. This cannot be directly done by comparison of matched modules; further, if the nodes are from the beginning defined as larger entities (*e.g.* using anatomical atlases), this information is lost.
{width="95.00000%"}
Methodological considerations {#methodological-considerations .unnumbered}
-----------------------------
In this work, we have applied the introduced coarse-graining approach only to the analysis of undirected and unweighted networks. However, the approach itself is more general and could be easily extended to the analysis of weighted networks that also take into account the strength of correlation between voxels, or directed networks that depict causal relationships between brain areas. The coarse-graining approach could also be applied in clinical settings by comparing the module-level differences between different groups of individuals, extending the previous attempts of comparing brain modules [@AlexanderBloch2012; @Glerean2015].
There are many choices to be made in deciding on the particulars of the module detection pipeline, especially the choice of the module detection algorithm (and its parameters, if any). Naturally, these choices will also affect the exact outcome. However, we expect that regardless of the exact way of defining modules, the benefits of using coarse-graining over comparing partitions remain. This is because the problems in interpreting the differences between partitions are universal, *e.g.* for any module detection method, there is nearly always a borderline case where the existence or absence of one single link affects boundaries between modules. Naturally, the choice of a module detection method also affects the resulting coarse-grained networks; however, when comparing across conditions and using the modules of one condition as the frame of reference, the differences should still be straightforward to interpret. Also, which of the conditions is used as the basis for coarse-graining should not be a critical choice: e.g. the observed differences in the network structure remain relatively similar for rest and movie modules as basis (see Supporting Information).
It is also essential to point out that, most likely, there is no perfect partition of the brain into distinct areas. Thus, even though different module detection algorithms yield different network partitions, these may still capture at least some meaningful aspect of the network’s organization. Instead of considering this multiplicity of possible partitions as a problem, it is rather an opportunity, as coarse-graining networks using different partitions can actually turn out to be very useful. In the same way as one can deduce the shape of a 3D object from its 2D projections, a network’s structure can be better understood by investigating its different coarse-grained representations. Therefore, coarse-graining could be useful even using an anatomical brain atlas as the basis, as the atlas provides a frame of reference that is well known to researchers in the field.
Conclusions
===========
We developed a coarse-graining method to analyze differences in the modular structure of functional brain networks during rest and task. The coarse-graining approach focuses on the differences in the connectivity between and within larger brain areas without sacrificing the spatial accuracy of fMRI data already at the network construction stage. The method yields results that summarizes differences in connectivity on the module level, using a set of modules as a frame of reference across groups or conditions. In contrary to some alternative approaches studying differences in the module-level connectivity of the human brain, the results produced by our method are both easy to interpret and verify statistically – they allow to “see the forest for the trees”. Because data on the structural and functional human connectome are becoming more and more detailed, we believe such methods will play an increasingly important role in understanding the module-level structure of functional networks.
Acknowledgments
===============
Computing resources provided by the Aalto Science IT project are acknowledged. JS acknowledges financial support from the Academy of Finland, project n:o 260427. EG acknowledges aivoAALTO Project Grant from the Aalto University, doctoral program “Brain & Mind". We thank Onerva Korhonen for comments on the manuscript.
References {#references .unnumbered}
==========
Supporting information {#supporting-information .unnumbered}
======================
Text S1: Labeling of the consensus modules {#text-s1-labeling-of-the-consensus-modules .unnumbered}
------------------------------------------
For labeling the movie and rest consensus modules, we first computed their spatial overlap with known major modules reported in the literature [@Power2011Neuron] and [@Yeo2011]. The final module labels were chosen manually and, where possible, matched with the spatial overlap results presented in Tables \[table:labels-rest\] and \[table:labels-movie\].
In detail, the computation of the spatial overlaps was done as follows: For each network node, we computed a ‘stability’ measure to describe how well they on average belonged to the consensus module in one condition. The node-wise stability we used was *scaled inclusivity* [@Steen2011] that has also been previously applied to brain network analyses [@Moussa2012]. For the movie and rest conditions, we then separately computed the average scaled inclusivity values for each node defined as $$SI_i = \frac{1}{13} \sum_{J \in \{1,..., 13\}} \frac{|C_i^{\text{REF}} \cap C_i^{\text{J}}| }{ |C_i^{\text{REF}}|} \frac{|C_i^{\text{REF}} \cap C_i^{\text{J}}| }{ |C_i^{\text{J}}| } \in [0,1],$$ where $|C_i^{\text{REF}} \cap C_i^{\text{J}}| $ corresponds to the number of nodes that belong both to node $i$’s consensus cluster and node $i$’s cluster in the best partition found for subject $J$. ($|C|$ denotes the number of nodes in cluster $C$)
Our spatial maps of the consensus modules were weighted by the scaled inclusivity values that were separately computed for each condition. Then we computed the spatial overlap defined as the Pearson correlation coefficient between the spatial maps as done in Refs. [@Smith2009; @Glerean2015]. The code used for computing spatial correlations between brain modules is available at <https://github.com/eglerean/hfASDmodules/compare_modules>.
Module ID Yeo et al 2011 corr. Power et al 2011 corr. Name given Abbreviation
----------- ------------------- ------- -------------------------------- ------- ------------------------------ --------------
rest 1 Limbic 0.066 Subcortical 0.84 Limbic LIM
rest 2 Limbic 0.21 Default mode 0.1 Cerebellum / ventro temporal CRBL/VT
rest 3 Default mode 0.54 Default mode 0.66 Default mode DM
rest 4 Visual 0.064 Memory retrieval 0.29 Precuneus PCUN
rest 5 Visual 0.5 Visual 0.6 Visual VIS
rest 6 Somatomotor 0.28 Auditory 0.53 Auditory AUD
rest 7 Ventral Attention 0.27 Cingulo-opercular task control 0.46 Salience SAL
rest 8 Frontoparietal 0.41 Fronto-parietal task control 0.68 Fronto-parietal FP
rest 9 Visual 0.31 Visual 0.53 Visual-extrastriate VISx
rest 10 Somatomotor 0.39 Sensory/ somatomotor hand 0.78 Sensorimotor SM
rest 11 Default mode 0.079 Fronto-parietal task control 0.13 Language LAN
: Spatial correlation of the rest consensus modules with reported modules in the literature.[]{data-label="table:labels-rest"}
\[table:labels-movie\]
Module ID Yeo et al 2011 corr. Power et al 2011 corr. Name given Abbreviation
----------- ------------------ ------- ------------------------------ ------- ------------------------ --------------
movie 1 Limbic 0.24 Subcortical 0.83 Ventro-temporal limbic VTL
movie 2 - - Default mode 0.02 Cerebellum CRBL
movie 3 Default 0.39 Default mode 0.46 Default mode DM
movie 4 Visual 0.14 Visual 0.21 Cuneus CUN
movie 5 Visual 0.62 Visual 0.80 Visual VIS
movie 6 Somatomotor 0.19 Auditory 0.38 Auditory AUD
movie 7 Frontoparietal 0.16 Salience 0.43 Salience SAL
movie 8 Frontoparietal 0.39 Fronto-parietal task control 0.59 Fronto-parietal FP
movie 9 Dorsal attention 0.4 Dorsal attention 0.42 Dorsal attention DA
movie 10 Somatomotor 0.42 Sensory/ somatomotor hand 0.71 Sensorimotor SM
: Spatial correlation of the movie consensus modules with reported modules in the literature.[]{data-label="tab:title"}
Text S2: Node labels {#text-s2-node-labels .unnumbered}
--------------------
In the alluvial diagram (Fig.4A in the main text), most ribbons connecting the two partitions have labels attached to them. These labels correspond to the labels of individual nodes, and a label is shown when there are at least 15 nodes with the same label in a ribbon. The labels for individual nodes were labelled automatically by matching each node with its corresponding automatic atlas labeling (AAL) or Harvard Oxford (HO) label. AAL was used for cerebral cortical areas and HO for subcortical and cerebellar areas. Furthermore, if both L(eft) and R(ight) parts are present, then no L/R is shown.
--------------------------------------- --------------------------------------------------------------- -------------------- ----------------------------------------------------
\[tab:abbreviations\]
Anatomical name AAL (cerebral cortex) & HO (sub-cortex and cerebellum) labels Major region label Acronym (as per doi:10.1371/ journal.pone.0005226)
Precentral gyrus (L) Precentral\_L Frontal PreCG(L)
Precentral gyrus (R) Precentral\_R Frontal PreCG(R)
Superior frontal gyrus (L) Frontal\_Sup\_L Frontal SFGdor(L)
Superior frontal gyrus (R) Frontal\_Sup\_R Frontal SFGdor(R)
Orbital superior frontal gyrus (L) Frontal\_Sup\_Orb\_L Frontal ORBsup(L)
Orbital superior frontal gyrus (R) Frontal\_Sup\_Orb\_R Frontal ORBsup(R)
Middle frontal gyrus (L) Frontal\_Mid\_L Frontal MFG(L)
Middle frontal gyrus (R) Frontal\_Mid\_R Frontal MFG(R)
Orbital middle frontal gyrus (L) Frontal\_Mid\_Orb\_L Frontal ORBmid(L)
Orbital middle frontal gyrus (R) Frontal\_Mid\_Orb\_R Frontal ORBmid(R)
Opercular inferior frontal gyrus (L) Frontal\_Inf\_Oper\_L Frontal IFGoperc(L)
Opercular inferior frontal gyrus (R) Frontal\_Inf\_Oper\_R Frontal IFGoperc(R)
Triangular inferior frontal gyrus (L) Frontal\_Inf\_Tri\_L Frontal IFGtriang(L)
Triangular inferior frontal gyrus (R) Frontal\_Inf\_Tri\_R Frontal IFGtriang(R)
Orbital inferior frontal gyrus (L) Frontal\_Inf\_Orb\_L Frontal ORBinf(L)
Orbital inferior frontal gyrus (R) Frontal\_Inf\_Orb\_R Frontal ORBinf(R)
Rolandic operculum (L) Rolandic\_Oper\_L Frontal ROL(L)
Rolandic operculum (R) Rolandic\_Oper\_R Frontal ROL(R)
Supplementary motor area (L) Supp\_Motor\_Area\_L Frontal SMA(L)
Supplementary motor area (R) Supp\_Motor\_Area\_R Frontal SMA(R)
Olfactory cortex (L) Olfactory\_L Frontal OLF(L)
Olfactory cortex (R) Olfactory\_R Frontal OLF(R)
Medial superior frontal gyrus (L) Frontal\_Sup\_Medial\_L Frontal SFGmed(L)
Medial superior frontal gyrus (R) Frontal\_Sup\_Medial\_R Frontal SFGmed(R)
Orbital medial frontal gyrus (L) Frontal\_Mid\_Orb\_L Frontal ORBsupmed(L)
Orbital medial frontal gyrus (R) Frontal\_Mid\_Orb\_R Frontal ORBsupmed(R)
Gyrus rectus (L) Rectus\_L Frontal REC(L)
Gyrus rectus (R) Rectus\_R Frontal REC(R)
Insula (L) Insula\_L Insula INS(L)
Insula (R) Insula\_R Insula INS(R)
Anterior cingulum (L) Cingulum\_Ant\_L Cingulate ACG(L)
Anterior cingulum (R) Cingulum\_Ant\_R Cingulate ACG(R)
Middle cingulum (L) Cingulum\_Mid\_L Cingulate DCG(L)
Middle cingulum (R) Cingulum\_Mid\_R Cingulate DCG(R)
Posterior cingulum (L) Cingulum\_Post\_L Cingulate PCG(L)
Posterior cingulum (R) Cingulum\_Post\_R Cingulate PCG(R)
Parahippocampal gyrus (L) ParaHippocampal\_L Occipital PHG(L)
Parahippocampal gyrus (R) ParaHippocampal\_R Occipital PHG(R)
Calcarine gyrus (L) Calcarine\_L Occipital CAL(L)
Calcarine gyrus (R) Calcarine\_R Occipital CAL(R)
Cuneus (L) Cuneus\_L Occipital CUN(L)
Cuneus (R) Cuneus\_R Occipital CUN(R)
Lingual gyrus (L) Lingual\_L Occipital LING(L)
Lingual gyrus (R) Lingual\_R Occipital LING(R)
Superior occipital gyrus (L) Occipital\_Sup\_L Occipital SOG(L)
Superior occipital gyrus (R) Occipital\_Sup\_R Occipital SOG(R)
Middle occipital gyrus (L) Occipital\_Mid\_L Occipital MOG(L)
Middle occipital gyrus (R) Occipital\_Mid\_R Occipital MOG(R)
Inferior occipital gyrus (L) Occipital\_Inf\_L Occipital IOG(L)
Inferior occipital gyrus (R) Occipital\_Inf\_R Occipital IOG(R)
Fusiform gyrus (L) Fusiform\_L Occipital FFG(L)
Fusiform gyrus (R) Fusiform\_R Occipital FFG(R)
Postcentral gyrus (L) Postcentral\_L Parietal PoCG(L)
Postcentral gyrus (R) Postcentral\_R Parietal PoCG(R)
Superior parietal lobule (L) Parietal\_Sup\_L Parietal SPG(L)
Superior parietal lobule (R) Parietal\_Sup\_R Parietal SPG(R)
Inferior parietal lobule (L) Parietal\_Inf\_L Parietal IPL(L)
Inferior parietal lobule (R) Parietal\_Inf\_R Parietal IPL(R)
Supramarginal gyrus (L) SupraMarginal\_L Parietal SMG(L)
Supramarginal gyrus (R) SupraMarginal\_R Parietal SMG(R)
Angular gyrus (L) Angular\_L Parietal ANG(L)
Angular gyrus (R) Angular\_R Parietal ANG(R)
Precuneus (L) Precuneus\_L Parietal PCUN(L)
Precuneus (R) Precuneus\_R Parietal PCUN(R)
Paracentral lobule (L) Paracentral\_Lobule\_L Parietal PCL(L)
Paracentra lobule (R) Paracentral\_Lobule\_R Parietal PCL(R)
Heschl gyrus (L) Heschl\_L Temporal HES(L)
Heschl gyrus (R) Heschl\_R Temporal HES(R)
Superior temporal gyrus (L) Temporal\_Sup\_L Temporal STG(L)
Superior temporal gyrus (R) Temporal\_Sup\_R Temporal STG(R)
Temporal pole (superior) (L) Temporal\_Pole\_Sup\_L Temporal TPOsup(L)
Temporal pole (superior) (R) Temporal\_Pole\_Sup\_R Temporal TPOsup(R)
Middle temporal gyrus (L) Temporal\_Mid\_L Temporal MTG(L)
Middle temporal gyrus (R) Temporal\_Mid\_R Temporal MTG(R)
Temporal pole (middle) (L) Temporal\_Pole\_Mid\_L Temporal TPOmid(L)
Temporal pole (middle) (R) Temporal\_Pole\_Mid\_R Temporal TPOmid(R)
Inferior temporal gyrus (L) Temporal\_Inf\_L Temporal ITG(L)
Inferior temporal gyrus (R) Temporal\_Inf\_R Temporal ITG(R)
Thalamus (L) Left\_Thalamus Subcortex THA(L)
Caudate (L) Left\_Caudate Subcortex CAU(L)
Putamen (L) Left\_Putamen Subcortex PUT(L)
Pallidum (L) Left\_Pallidum Subcortex PAL(L)
Brainstem Brain-Stem Subcortex BST
Hippocampus (L) Left\_Hippocampus Subcortex HIP(L)
Amygdala (L) Left\_Amygdala Subcortex AMYG(L)
Nucleus accumbens (L) Left\_Accumbens Subcortex NAcc(L)
Thalamus (R) Right\_Thalamus Subcortex THA(R)
Caudate (R) Right\_Caudate Subcortex CAU(R)
Putamen (R) Right\_Putamen Subcortex PUT(R)
Pallidum (R) Right\_Pallidum Subcortex PAL(R)
Hippocampus (R) Right\_Hippocampus Subcortex HIP(R)
Amygdala (R) Right\_Amygdala Subcortex AMYG(R)
Nucleus accumbens (R) Right\_Accumbens Subcortex NAcc(R)
Cerebellar lobule I-IV (L) Left\_I-IV Cerebellum I-IV(L)
Cerebellar lobule I-IV (R) Right\_I-IV Cerebellum I-IV(R)
Cerebellar lobule V (L) Left\_V Cerebellum V(L)
Cerebellar lobule V (R) Right\_V Cerebellum V(R)
Cerebellar lobule VI (L) Left\_VI Cerebellum VI(L)
Cerebellar vermis VI Vermis\_VI Cerebellum VI-vermis
Cerebellar lobule VI (R) Right\_VI Cerebellum VI(R)
Cerebellar crus I (L) Left\_Crus\_I Cerebellum XI(L)
Cerebellar crus I (R) Right\_Crus\_I Cerebellum XI(R)
Cerebellar crus II (L) Left\_Crus\_II Cerebellum XII(L)
Cerebellar vermis crus II Vermis\_Crus\_II Cerebellum XII-vermis
Cerebellar crus II (R) Right\_Crus\_II Cerebellum XII(R)
Cerebellar lobule VIIb (L) Left\_VIIb Cerebellum VIIb(L)
Cerebellar lobule VIIb (R) Right\_VIIb Cerebellum VIIb(R)
Cerebellar lobule VIIIa (L) Left\_VIIIa Cerebellum VIIIa(L)
Cerebellar vermis VIIIa Vermis\_VIIIa Cerebellum VIIIa-vermis
Cerebellar lobule VIIIa (R) Right\_VIIIa Cerebellum VIIIa(R)
Cerebellar lobule VIIIb (L) Left\_VIIIb Cerebellum VIIIb(L)
Cerebellar vermis VIIIb Vermis\_VIIIb Cerebellum VIIIb-vermis
Cerebellar lobule VIIIb (R) Right\_VIIIb Cerebellum VIIIb(R)
Cerebellar lobule IX (L) Left\_IX Cerebellum IX(L)
Cerebellar vermis IX Vermis\_IX Cerebellum IX-vermis
Cerebellar lobule IX (R) Right\_IX Cerebellum IX(R)
Cerebellar lobule X (L) Left\_X Cerebellum X(L)
Cerebellar vermis X Vermis\_X Cerebellum X-vermis
Cerebellar lobule X (R) Right\_X Cerebellum X(R)
--------------------------------------- --------------------------------------------------------------- -------------------- ----------------------------------------------------
: Abbreviations of node labels
Text S3: Average coarse-grained movie networks {#text-s3-average-coarse-grained-movie-networks .unnumbered}
----------------------------------------------
Fig. \[fig:movie-coarse-grained\] shows the results of the coarse-graining process, when the movie consensus modules have been used as the reference modules for the coarse-graining process. The differences that survive the 0.05 Benjamini-Hochberg FDR correction are additionally listed in Table \[tab:movie-fdr\].
Module pair $i,j$ $p$-value $\Delta W_{i,j}$ relative increase (+) / decrease (-)
------------------- ----------- ------------------ --------------------------------------
VIS,VIS 0.0010 26204.4 +56%
VIS,CUN 0.0029 2228.9 +59%
DA,SAL 0.0007 -1120.0 -54%
DA,FP 0.0017 -1981.3 -35%
: Differences in link numbers within and between movie consensus modules that survive the 0.05 Benjamini-Hochberg FDR correction.[]{data-label="tab:movie-fdr"}
{width="100.00000%"}
|
---
abstract: 'This paper is devoted to the interpolation principle between spaces of weak type. We characterize interpolation spaces between two Marcinkiewicz spaces in terms of Hardy type operators involving suprema. We study general properties of such operators and their behavior on Lorentz gamma spaces.'
address:
- '^1^Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic'
- '^2^Institute of Applied Mathematics and Information Technologies, Faculty of Science, Charles University, Albertov 6, 128 43 Praha 2, Czech Republic'
author:
- Vít Musil^1^
- 'Rastislav Ohava^1,2^'
title: Marcinkiewicz theorem for Lorentz gamma spaces
---
[^1]
introduction
============
Let $\dom=(\dom,\mu)$ be non-atomic $\sigma$-finite measure space with $\mu(\dom)=R$, where $0<R\le\infty$. Let $\mathcal{M}(\dom,\mu)$ denote the collection of all extended real-valued $\mu$-measurable and a.e. finite functions on $\dom$.
This paper deals with Marcinkiewicz interpolation theorem between spaces of weak type where the norm is defined by $$\|f\|_{M_\varphi(\dom)} = \sup_{0<s<R} \varphi(s) f^{**}(s) .$$ Here $\varphi$ is so-called quasiconcave function (for the definition see Section \[sec:back\]), the double stars stand for the maximal function defined as a Hardy average of $f^*$, $$f^{**}(t) = \frac{1}{t} \int_{0}^{t} f^{*}(s)\,\d s,$$ in which $f^*$ represents the non-increasing rearrangement of $f$, given by $$f^*(t)=\inf
\bigl\{
\lambda > 0;\, \mu
(\{
x\in \dom;\, |f(x)|>\lambda
\})
\leq t
\bigr\},
\quad t\in[0,R).$$ The collection of all functions $f\in\mathcal{M}(\dom,\mu)$ with $\|f\|_{M_\varphi(\dom)}$ finite is called Marcinkiewicz space $M_\varphi(\dom,\mu)$.
In our main result we prove that the boundedness of a certain operator is ensured by that of the supremum operators or, more precisely, Hardy-type operators involving suprema $S_\fI$ and $T_\fII$ defined by $$S_{\fI} f(t) = \frac {1} {\fI(t)} \sup_{0<s<t} \fI(s) f^*(s),
\quad t\in(0,R),$$ $$T_{\fII} f(t) = \frac {1} {\fII(t)} \sup_{t<s<R} \fII(s) f^*(s),
\quad t\in(0,R),$$ where $\fI$ and $\fII$ are quasiconcave functions. Such a result was first proved by Dmitriev and Kreĭn in [@DK]; however, the supremum operators appeared only implicitly. Later, Kerman and Pick in [@KP:Forum] and [@KP:Studia] showed the equivalence of the boundedness of the operators of such kind and certain Sobolev-type embeddings and they also used their result in the search of optimal pairs of r.i. spaces for which these embeddings hold. Consequently, Kerman, Phipps and Pick in [@KPP] found simple criteria for the boundedness of the supremum operators on Orlicz spaces and Lorentz Gamma spaces and they obtained corresponding Marcinkiewicz interpolation theorems. However, all the above-mentioned results concern only power functions in place of $\varphi$. In this work, we want to fill this gap.
The principal innovation of this paper consists not only in a significant extension of the known results but also in the new and more elegant comprehensive approach that enables us to establish proofs which are more enlightening and illustrative and less technical than those applied in earlier works.
We will work in the general setting of rearrangement-invariant (r.i. for short) Banach function spaces $X(\dom, \mu)$ as collections of all $\mu$-measurable functions finite a.e. on $\dom$ such that $\|f\|_{X(\dom,\mu)}$ is finite.
One can define an r.i. space $X(\dom,\mu)$ on a general measure space $(\dom,\mu)$ using rearrangement invariance of the given r.i. space $X(0,R)$, $$\|f\|_{X(\dom,\mu)} =\|f^*\|_{X(0,R)},
\quad f\in\mathcal{M}(\dom,\mu).$$ On the other hand, there is also a representation of each norm of a given r.i. space $X(\dom,\mu)$ by r.i. norm on interval due to the Luxemburg representation theorem. For further information regarding r.i. norms see [@BS Chapter 1 and 2]. At the places where no confusion is likely to happen, we shall use a shorter form $X(\dom)$ instead of $X(\dom,\mu)$ and $X$ in the case when $(\dom,\mu)$ is the interval $[0,R)$ equipped with Lebesgue measure.
We also exhibit the general properties of the supremum operators $S_\varphi$ and $T_\fII$ like the endpoint embeddings in the r.i. class (Section \[sec:endpoints\]) or the relation to the maximal function (Section \[sec:stars\]). It turns out that a certain averaging condition on the quasiconcave function plays a key role here. It reads as $$\frac{1}{t} \int_0^t \frac{\d s}{\varphi(s)}
\lesssim \frac{1}{\varphi(t)},
\quad t\in(0,R).$$ We shall refer to this relation as a $B$-condition and write $\varphi\in B$. More details about quasiconcave functions and $B$-condition can be found in Section \[sec:back\].
Our principal result now reads as follows.
Let $\dom_1=(\dom_1,\mu_1)$ and $\dom_2 = (\dom_2,\mu_2)$ be non-atomic $\sigma$-finite measure spaces for which $\mu_1(\dom_1) = \mu_2(\dom_2)=R$. Suppose that a quasilinear operator $T$ satisfies $$T\colon M_{\fI}(\dom_1) \to M_{\fI}(\dom_2)
\quad\text{and}\quad
T\colon M_{\fII}(\dom_1) \to M_{\fII}(\dom_2)$$ for quasiconcave functions $\fI,\fII$ defined on $[0,R)$, both satisfying the B-condition and let $X_i(\dom_i)$, $i=1,2$, be r.i. spaces satisfying $$M_{\fI}(\dom_i)\cap M_{\fII}(\dom_i)
\subset X_i(\dom_i) \subset
M_{\fI}(\dom_i) + M_{\fII}(\dom_i),
\quad i=1,2.$$ Then $$T\colon X_1 (\dom_1) \to X_2 (\dom_2)$$ whenever $$S_{\fI}\colon X_1(0,R) \to X_2(0,R)
\quad\text{and}\quad
T_{\fII}\colon X_1(0,R) \to X_2(0,R).
\label{eq:DKP}$$ \[thm:DKP\]
Our next result concerns the criteria to guarantee in specific class of r.i. spaces, namely in the classical Lorentz gamma spaces $\Gamma^p_\phi(\dom)$ where the norm is given as $$\|f\|_{\Gamma^p_\phi(\dom)} = \biggl( \int_{0}^{R} [ f^{**}(t) ]^p \phi(t)\,\d t \biggr)^\frac{1}{p}.$$ Here $1\le p<\infty$ and $\phi$ is some positive and locally integrable function, so-called weight. We require $\int_{1}^{\infty} s^{-p}\phi(s)\,\d s <\infty$ when $R=\infty$ and $\int_{0}^{R} s^{-p}\phi(s)\,\d s = \infty$ when $R<\infty$ otherwise $\Gamma^p_\phi = \{0\}$ in the first case and $\Gamma^p_\phi = L^1$ in the second one. Such requirements are called nontriviality conditions.
If we deal with the operator $S_\varphi$ acting between Lorentz gamma spaces with discontinuous $\varphi$, we moreover admit additional nontriviality conditions, i.e, we assume $$\int_0^R \varphi^{-p}(s)\phi(s) \,\d s < \infty
\label{nontriv}$$ and $$\lim_{t\to 0^+} t^p \int_{t}^{R} s^{-p} \phi(s)\,\d s > 0.
\label{Emb-g-ln}$$ As we explain in Remark \[rem:discontinuity\], such requirements are necessary and cause no loss of generality.
Let $\dom_1 = (\dom_1,\mu_1)$ and $\dom_2 = (\dom_2,\mu_2)$ be non-atomic $\sigma$-finite measure spaces with $\mu_1(\dom_1)=\mu_2(\dom_2)=R$, $\fI$ and $\fII$ be quasiconcave functions defined on $[0,R)$ satisfying the B-condition, $\phi_1$ and $\phi_2$ be nontrivial weights on $(0,R)$. In the case $\varphi$ is not continuous, let, in addition, $\phi_1$ and $\phi_2$ satisfy and respectively. Let $p$ be an index, $1\leq p<\infty$, such that $$M_{\fI}(\dom_i)\cap M_{\fII}(\dom_i) \subset \Gamma^p_{\phi_i}(\dom_i) \subset M_{\fI}(\dom_i)+ M_{\fII}(\dom_i),
\quad i=1,2.$$ Suppose $T$ is a quasilinear operator that satisfies $$T \colon M_{\fI}(\dom_1) \to M_{\fI}(\dom_2)
\quad \text{and} \quad
T \colon M_{\fII}(\dom_1) \to M_{\fII}(\dom_2);$$ then, a sufficient condition for the embedding $$T \colon \Gamma^p_{\phi_1}(\dom_1) \to \Gamma^p_{\phi_2}(\dom_2)$$ is as follows $$\sup_{0<t<R}\frac{
\fII^p(t) \int_{0}^{t} \fII^{-p}(s)\phi_2(s)\,\d s
+ \fI^p(t) \int_{t}^{R} \fI^{-p}(s)\phi_2(s)\,\d s
}{
\int_0^t \phi_1(s)\,\d s
+
t^p \int_t^R s^{-p}\phi_1(s)\,\d s
} < \infty.
\label{STggpp}$$ \[thm:A\]
The proof of this result follows from a characterization of the boundedness of the supremum operators $S_\varphi$ and $T_\varphi$ between two Lorentz gamma spaces, of independent interest, formulated in the following two theorems.
Let $1\le p <\infty$, let $\varphi$ be a quasiconcave function on $[0,R)$ satisfying the $B$-condition and let $\phi_1$, $\phi_2$ be nontrivial weights on $(0,R)$. In the case $\varphi$ is not continuous, let, in addition, $\phi_1$ and $\phi_2$ satisfy and respectively. Then $$S_\varphi\colon \Gamma^p_{\phi_1}(0,R) \to \Gamma^p_{\phi_2}(0,R)
\label{Sgg}$$ holds if and only if $$\sup_{0<t<R}
\frac{\int_0^t \phi_2(s)\,\d s
+ \varphi^p(t) \int_t^R \varphi^{-p}(s) \phi_2(s) \,\d s}
{\int_0^t \phi_1(s)\,\d s
+
t^p \int_t^R s^{-p}\phi_1(s)\,\d s}
< \infty.
\label{Sggpp}$$ \[thm:Sgg\]
Let $1\le p <\infty$, let $\fII$ be a quasiconcave function on $[0,R)$ satisfying the $B$-condition and let $\phi_1$, $\phi_2$ be nontrivial weights on $(0,R)$. Then $$T_{\fII}\colon \Gamma^p_{\phi_1}(0,R) \to \Gamma^p_{\phi_2}(0,R)
\label{Tgg}$$ holds if and only if $$\sup_{0<t<R}
\frac{\fII^p(t) \int_0^t {\fII}^{-p}(s) \phi_2(s) \,\d s
+ t^p \int_t^R s^{-p}\phi_2(s)\,\d s}
{\int_0^t \phi_1(s)\,\d s
+ t^p \int_t^R s^{-p}\phi_1(s)\,\d s}
< \infty.
\label{Tggpp}$$ \[thm:Tgg\]
The proofs of these results appear in Section \[sec:proofs\]. One may also notice that we sometimes avoid stating the results in the full generality. For instance, one may try to extend Theorems \[thm:Sgg\] and \[thm:Tgg\] to various exponents on the left and the right hand sides or avoid the $B$-condition. The reason is similar here; the general situation can be treated by discretization methods while we want to keep the approach as simple as possible.
quasiconcave functions {#sec:back}
======================
Let us recall that if a non-negative function defined on $[0,R)$, $\varphi$, satisfies
1. $\varphi(t)=0$ if and only if $t=0$;
2. $\varphi$ is non-decreasing;
3. $\varphi(t)/t$ is non-increasing on $(0,R)$,
then $\varphi$ is said to be *quasiconcave*. If we denote $\tilde{\varphi}(t) = t/\varphi(t)$ for $t\in(0,R)$ and $\tilde{\varphi}(0)=0$ then $\tilde{\varphi}$ is also a quasiconcave function. We say that $\tilde{\varphi}$ is complementary function to $\varphi$.
A quasiconcave function $\varphi$ is continuous in every positive argument from its domain. Any jump at such a point would lead to the contradiction with the monotonicity of complementary function $\tilde{\varphi}$ or $\varphi$ itself. Only possible point of discontinuity of quasiconcave functions is zero.
In the next theorem we will give a equivalent form of the $B$-condition. The idea is based on [@Stromberg:Indiana Lemma 2.3]. Here and in the sequel we will use the notation $A\lesssim B$ if $A\le CB$ where $C$ is a constant independent of all quantities obtained in $A$ and $B$. In the case $A\lesssim B$ and $B\lesssim A$ we will use $A\simeq B$.
Let $\varphi$ be a quasiconcave function on $[0,R)$. Then the following conditions are equivalent.
1. $\varphi$ satisfies $B$-condition;
2. It holds $$\int_0^t \tilde{\varphi}(s) \frac{\d s}{s}
\lesssim \tilde{\varphi}(t),
\quad t\in (0,R);$$
3. There exists a constant $c\in(0,1)$ such that $$\inf_{0<t<R} \frac{\widetilde{\varphi}(t)}{{\widetilde\varphi}(ct)} > 1.$$
\(i) is equivalent to (ii) simply by the definition of $\tilde\varphi$. Next, suppose that (iii) holds. There exists a constant $r>1$ such that $${\widetilde{\varphi}(t)}
\ge r{\widetilde{\varphi}(ct)},
\quad t\in (0,R).$$ Using this inequality iteratively for $t, ct, c^2t, \dots$, we get $${\widetilde{\varphi}(t)}
\ge r{\widetilde{\varphi}(ct)}
\ge r^k{\widetilde{\varphi}(c^{k}t)},
\quad t\in (0,R),
\quad k\in\N.$$ Now, we slice the integration domain of the integral in the first condition and since $\widetilde{\varphi}$ is increasing, we get $$\begin{aligned}
\int_0^t \widetilde{\varphi}(s) \frac{\d s}{s}
&= \sum_{k=0}^\infty \int_{c^{k+1}t}^{c^{k}t} \widetilde{\varphi}(s) \frac{\d s}{s}
\le \sum_{k=0}^\infty \widetilde{\varphi}(c^{k}t) \int_{c^{k+1}t}^{c^{k}t} \frac{\d s}{s}
\\
& = \log(1/c) \sum_{k=0}^\infty \widetilde{\varphi}(c^{k}t)
\le \widetilde{\varphi}(t) \log(1/c) \sum_{k=0}^\infty r^{-k}
\\
&=\frac{r\log(1/c)}{{r-1}} \widetilde{\varphi}(t),
\quad t\in (0,R),
\end{aligned}$$ so (ii) holds.
In the opposite direction let us assume that (iii) is not satisfied and (ii) is; in other words, for some positive constant $K$, $$\int_0^t \widetilde{\varphi}(s) \frac{\d s}{s}
\le K\widetilde{\varphi}(t),
\quad t\in (0,R).$$ Now, fix an arbitrary $r>1$. Then, for every constant $c\in(0,1)$ there exists $t\in (0,R)$ such that $${\widetilde{\varphi}(t)}
< r{\widetilde{\varphi}(ct)}.$$ Using all this, we obtain $$\begin{aligned}
K\widetilde{\varphi}(t)
&\ge \int_0^{t} \widetilde{\varphi}(s) \frac{\d s}{s}
\ge \int_{ct}^{t} \widetilde{\varphi}(s) \frac{\d s}{s} \\
&\ge \widetilde{\varphi}(ct) \log(1/c)
> \frac{1}{r}\widetilde{\varphi}(t) \log(1/c).
\end{aligned}$$
Thus, we have a contradiction since $Kr\ge \log(1/c)$ cannot hold for every $c\in(0,1)$ and the proof is complete.
\[lemm:marc\_one\_star\] Let $\varphi$ be a quasiconcave function on $[0,R)$. Then $$\label{eq:mos}
\sup_{0<t<R} \varphi(t) f^{**}(t)
\simeq \sup_{0<t<R} \varphi(t) f^{*}(t)$$ for every measurable $f$ if and only if $\varphi\in B$.
Necessity follows immediately by setting $f=f^*=1/\varphi$.
Now suppose that $\varphi\in B$. Since $f^{*}\le f^{**}$ the left hand side of dominates the right hand side of . For the opposite inequality denote the right hand side of by $M$. We then have $$f^*(t) \le M \frac{1}{\varphi(t)},
\quad t\in (0,R).$$ Integrating this inequality over $(0,s)$ and dividing by $s$ we get $$f^{**}(s)
\le \frac{M}{s} \int_0^s \frac{\d t}{\varphi(t)}
\lesssim M \frac{1}{\varphi(s)},
\quad s\in (0,R),$$ hence $$\sup_{0<s<R} \varphi(s)f^{**}(s) \lesssim M$$ as we wished to show.
Note that for a given measurable function $f$, both $T_\fII f$ and $S_\varphi f$ are non-increasing functions. Indeed, $$\begin{aligned}
S_\varphi f (t)
& = \frac{ 1 }{\varphi(t) } \sup_{0<s<t} \varphi(s) \sup_{s<y<R} f^*(y)
\\
& = \frac{ 1 }{\varphi(t) } \sup_{0<y<R} f^{*}(y) \sup_{0<s<\min \{ t,y \} } \varphi(s)
\\
& = \sup_{0<y<R} f^{*}(y) \min\biggl\{ 1, \frac{\varphi(y)}{\varphi(t)}\biggr\}\end{aligned}$$ which is clearly non-increasing. The case concerning $T_\fII f$ is obvious.
endpoint estimates {#sec:endpoints}
==================
\[lemm:endpoint\_T\] Let $\varphi$ be a quasiconcave function on $[0,R)$. Then
1. $$T_\varphi \colon L^1 \to L^1
\quad \text{if and only if}\quad
\varphi \in B;$$
2. $$T_\varphi \colon M_\varphi \to M_\varphi
\quad \text{if and only if}\quad
\varphi \in B.$$
For the necessity of the $B$-condition we just put $f=\chi_{(0,t)}$. The calculations are straightforward. For the sufficiency in (i) we split the integration in two parts, namely $$\begin{aligned}
\bigl\|T_\varphi f\bigr\|_{L^1}
& = \int_0^R \frac{1}{\varphi(t)} \sup_{t<s<R} \varphi(s) f^{*}(s)\,\d t
\\
& \le \int_0^R \frac{1}{\varphi(t)} \sup_{t<s<R} \bigl( \varphi(s) - \varphi(t) \bigr) f^{*}(s)\,\d t
+ \int_0^R \sup_{t<s<R} f^{*}(s)\,\d t
\\
& = \text{I} + \text{II}.
\end{aligned}$$ The second part equals to the $L^1$ norm of $f$, while the first part needs some estimates. We have $$\begin{aligned}
\text{I}
& = \int_0^R \frac{1}{\varphi(t)}\sup_{t<s<R} \Bigl( \int_t^s \varphi'(y)\,\d y \Bigr) f^*(s)\,\d t
\\
& \le \int_0^R \frac{1}{\varphi(t)}\sup_{t<s<R} \Bigl( \int_t^s \varphi'(y) f^*(y)\,\d y \Bigr) \d t
\\
& = \int_0^R \frac{1}{\varphi(t)} \int_t^R \varphi'(y)f^{*}(y)\,\d y\,\d t
\\
& = \int_0^R \varphi'(y)f^{*}(y) \int_0^y \frac{\d t}{\varphi(t)} \,\d y
\tag{by the Fubini theorem} \\
& \lesssim \int_0^R \frac{y}{\varphi(y)}\varphi'(y)f^{*}(y)\,\d y
\tag{since $\varphi\in B$} \\
& \le \int_0^R f^{*}(y)\,\d y.
\tag{by quasiconcavity} \\
\end{aligned}$$ This completes the proof of the part (i). For the sufficiency in the part (ii), recall that $T_\varphi f$ is non-increasing and hence we have $$\begin{aligned}
\|T_\varphi f \|_{M_\varphi}
& = \sup_{0<t<R} \varphi(t) \bigl(T_\varphi f \bigr)^{**}(t)
\\
& = \sup_{0<t<R} \frac{\varphi(t)}{t} \int_0^t \frac{1}{\varphi(s)} \sup_{s<y<R}\varphi (y)f^*(y)\,\d s
\\
& \le \sup_{0<t<R} \frac{\varphi(t)}{t} \int_0^t \frac{1}{\varphi(s)} \sup_{0<y<R}\varphi (y)f^{**}(y)\,\d s
\\
& = \| f \|_{M_\varphi} \sup_{0<t<R} \frac{\varphi(t)}{t}\int_0^t \frac{\d s}{\varphi(s)}
\end{aligned}$$ and the last supremum is finite because of the $B$-condition for $\varphi$.
\[lemm:endpoint\_S\] Let $\varphi$ be a quasiconcave function on $[0,R)$. Then
1. $$S_\varphi \colon M_\varphi \to M_\varphi
\quad \text{if and only if}\quad
\varphi \in B;$$
2. $$S_\varphi \colon L^\infty \to L^\infty
\quad \text{for every quasiconcave $\varphi$}.$$
Let us consider part (i). For the necessity we set $f=\chi_{(0,a)}$. We obtain $$S_\varphi \chi_{(0,a)} (t)
= \min\biggl\{ 1, \frac{\varphi(a)}{\varphi(t)} \biggr\},
\quad t\in (0,R),
\quad a\in (0,R),$$ and thus for every $a\in(0,R)$ we have $$\begin{aligned}
\| S_\varphi \chi_{(0,a)} \|_{M_\varphi}
& = \sup_{0<t<R} \frac{\varphi(t)}{t} \int_0^t \min\biggl\{ 1, \frac{\varphi(a)}{\varphi(s)} \biggr\}\d s
\\
& = \sup_{0<t<R} \varphi(t) \chi_{(0,a)}(t) +
\frac{\varphi(t)}{t}\biggl( a + \varphi(a) \int_a^t \frac{\d s}{\varphi(s)} \biggr) \chi_{(a,R)}(t)
\\
& \ge \varphi(a)\sup_{a<t<R} \frac{\varphi(t)}{t} \int_a^t \frac{\d s}{\varphi(s)}.
\end{aligned}$$ Clearly $\|\chi_{(0,a)}\|_{M_\varphi} = \varphi(a)$ and since $S_\varphi$ is bounded on $M_\varphi$ we get $$\varphi(a)\sup_{a<t<R} \frac{\varphi(t)}{t} \int_a^t \frac{\d s}{\varphi(s)}
\le \varphi(a),
\quad a\in(0,R).$$ The term $\varphi(a)$ cancels and by taking the limit $a\to 0^+$ we get the $B$-condition.
Now suppose that $\varphi\in B$. Taking Lemma \[lemm:marc\_one\_star\] and the monotonicity of $S_\varphi f$ into account, we have $$\begin{aligned}
\| S_\varphi f \|_{M_\varphi}
& = \sup_{0<t<R} \varphi(t) \bigl( S_\varphi f \bigr)^{**}(t)
\simeq \sup_{0<t<R} \varphi(t) \bigl( S_\varphi f \bigr)^{*}(t)
\\
& = \sup_{0<t<R} \varphi(t) \bigl( S_\varphi f \bigr)(t)
= \sup_{0<t<R} \sup_{0<s<t} \varphi(s) f^*(s)
\\
& \simeq \sup_{0<s<R} \varphi(s) f^{**}(s)
= \|f\|_{M_\varphi}.
\end{aligned}$$
Part (ii) is trivial.
starfalls {#sec:stars}
=========
Let $\varphi$ be a quasiconcave function on $[0,R)$. Then
1. $$\bigl( T_\varphi f \bigr)^{**}
\lesssim T_\varphi f + f^{**}
\quad \text{if and only if}\quad
\varphi \in B;$$
2. $$T_\varphi f^{**}
\simeq T_\varphi f + f^{**}
\quad \text{if and only if}\quad
\varphi \in B.$$
\[lemm:prop\_T\]
To prove the necessity, we test the inequalities by characteristic function $f=\chi_{(0,a)}$. We compute $$T_\varphi \chi_{(0,a)}(t)
= \chi_{(0,a)}(t) \frac{\varphi(a)}{\varphi (t)}$$ and $$T_\varphi \chi^{**}_{(0,a)}(t)
= \chi_{(0,a)}(t) \frac{\varphi(a)}{\varphi (t)}
+ \chi_{(a,R)}(t) \frac{a}{t}$$ and also $$\bigl( T_\varphi \chi_{(0,a)}\bigr)^{**} (t)
= \chi_{(0,a)}(t) \frac{\varphi(a)}{t}\int_0^t \frac{\d s}{\varphi(s)}
+ \chi_{(a,R)}(t) \frac{\varphi(a)}{t}\int_0^a \frac{\d s}{\varphi(s)}$$ for every pair $a$ and $t$ in $(0,R)$. The necessity of the $B$-condition then follows by comparing appropriate quantities for arbitrary $t<a$.
To prove the sufficiency in (i), we divide the outer integral into three parts. $$\begin{aligned}
\frac1t \int_0^t \frac{1}{\varphi(y)} \sup_{y<s<R} \varphi(s) f^*(s)\,\d y
& \le \frac1t \int_0^t \frac{1}{\varphi(y)} \sup_{y<s<t} \bigl( \varphi(s) - \varphi(y) \bigr) f^*(s)\,\d y
\\
& \qquad + \frac1t \int_0^t \sup_{y<s<t} f^*(s)\,\d y
\\
& \qquad + \frac1t \int_{0}^{t} \frac{1}{\varphi(y)}\sup_{t<s<R} \varphi(s) f^*(s)\,\d y
\\
& \quad = \text{I} + \text{II} + \text{III}.
\end{aligned}$$ The first term can be treated in the same way as in the proof of Lemma \[lemm:endpoint\_T\], part (i). We get $\text{I}\lesssim f^{**}(t)$ The term II clearly equals $f^{**}(t)$. Finally, since $\varphi\in B$, $$\text{III}
\lesssim \frac{1}{\varphi(t)} \sup_{t<s<R} \varphi(s)f^*(s)
= T_\varphi f(t),
\quad t\in(0,R).$$ Adding all these estimates together we have $$\bigl( T_\varphi f \bigr)^{**}(t)
\lesssim f^{**}(t) + T_\varphi f (t),
\quad t\in (0,R).$$
Let us show the equivalence (ii) assuming $\varphi\in B$. One inequality is obvious since $f^{**} \le T_\varphi f^{**}$. The reversed inequality can be observed by the splitting argument similar to that in part (i). For $t\in(0,R)$, we have $$\begin{aligned}
T_{\varphi} f^{**} (t)
& = \frac{1}{\varphi (t) }\sup_{t<s<R} \frac{\varphi(s)}{ s }\int_0^s f^*(y)\,\d y
\\
& \le \frac{1}{\varphi (t) }\sup_{t<s<R} \frac{\varphi(s)}{s }\int_t^s f^*(y)\,\d y
+ \frac{1}{\varphi (t) }\sup_{t<s<R} \frac{\varphi(s)}{s }\int_0^t f^*(y)\,\d y
\\
& = \text{I} + \text{II}.
\end{aligned}$$ Now, we can continue by $$\begin{aligned}
\text{I}
& = \frac{1}{\varphi (t) } \sup_{t<s<R} \frac{\varphi(s)}{s }\int_t^s \varphi(y)f^*(y)\frac{\d y}{\varphi(y)}
\\
& \le \frac{1}{\varphi (t) } \sup_{t<y<R} \varphi(y)f^*(y) \sup_{t<s<R} \frac{\varphi(s)}{ s }\int_t^s \frac{\d y}{\varphi(y)}
\tag{by taking the supremum out}
\\
& \le T_{\varphi} f (t) \sup_{0<s<R} \frac{\varphi(s)}{s }\int_0^s \frac{\d y}{\varphi(y)}
\tag{by taking $t=0$}
\\
& \lesssim T_{\varphi} f (t)
\tag{since $\varphi \in B$}
\end{aligned}$$ and surely $\text{II} = f^{**}(t)$.
\[lemm:prop\_S\] Let $\varphi$ be a quasiconcave function on $[0,R)$. Then
1. $$\bigl( S_\varphi f \bigr)^{**}
\lesssim S_\varphi f^{**}
\quad \text{if and only if}\quad
\varphi \in B;$$
2. $$S_\varphi f^{**}
\lesssim S_\varphi f
\quad \text{if and only if}\quad
\varphi \in B.$$
Part (i). The necessity follows by plugging $f=\chi_{(0,a)}$ into the inequality. We have $$S_\varphi \chi_{(0,a)} (t)
= S_\varphi \chi^{**}_{(0,a)} (t)
= \min\biggl\{ 1, \frac{\varphi(a)}{\varphi(t)} \biggr\},
\quad t\in (0,R),
\quad a\in(0,R).$$ Similarly as in the proof of Lemma \[lemm:endpoint\_S\] we calculate $$\bigl( S_\varphi \chi_{(0,a)} \bigr)^{**}(t)
= \chi_{(0,a)}(t) + \frac{1}{t}\biggl( a + \varphi(a) \int_a^t \frac{\d s}{\varphi(s)} \biggr) \chi_{(a,R)}(t),
\quad a\in(0,R),
\quad t\in(0,R),$$ hence for $t>a$ we have $$\bigl( S_\varphi \chi_{(0,a)} \bigr)^{**}(t)
\ge \frac{\varphi(a)}{t}\int_a^t \frac{\d s}{\varphi(s)},$$ therefore for those $t$ and $a$ we have $$\frac{\varphi(a)}{t}\int_a^t \frac{\d s}{\varphi(s)}
\lesssim \frac{\varphi(a)}{\varphi(t)}.$$ The term $\varphi(a)$ cancels and by taking the limit $a\to 0^+$ we obtain the $B$-condition.
On the other side, we have $$\begin{gathered}
\varphi(t)\bigl( S_\varphi f\bigr)^{**}(t)
\le \sup_{0<s<t} \varphi(s)\bigl( S_\varphi f\bigr)^{**}(s)
\\
= \bigl\| S_\varphi f\bigr\|_{M_\varphi(0,t)}
\lesssim \|f\|_{M_\varphi(0,t)}
= \sup_{0<s<t} \varphi(s) f^{**}(s)
\end{gathered}$$ thanks to Lemma \[lemm:endpoint\_S\]. Dividing by $\varphi(t)$ we get the result.
Part (ii) follows immediately with the help of Lemma \[lemm:marc\_one\_star\] by $$\varphi(t)\, S_\varphi f (t)
= \sup_{0<s<t} \varphi(s) f^{*}(s)
\simeq \sup_{0<s<t} \varphi(s) f^{**}(s)
= \varphi(t)\, S_\varphi f^{**} (t),
\quad t\in(0,R).$$
\[lemm:stars\_out\] Let $0<R\le\infty$ and let $\fI$ and $\fII$ be quasiconcave functions on $(0,R)$. Then $$S_{\fI} f^{**} + T_{\fII} f^{**}
\simeq
S_{\fI} f + T_{\fII} f$$ for every measurable $f$ if and only if both $\fI\in B$ and $\fII\in B$ hold.
The claim is a corollary of Lemma \[lemm:prop\_T\] since $$T_{\fII} f^{**}
\le T_{\fII} f + f^{**}
\le T_{\fII} f + S_{\fI} f^{**}$$ and Lemma \[lemm:prop\_S\] which ensures that $$S_{\fI} f^{**} \lesssim S_{\fI} f.$$ The opposite inequality and the necessity are obvious.
proof of the main results {#sec:proofs}
=========================
Let us fix $f\in \mathcal{M}(\dom_1)$ and $t\in(0,R)$. We decompose $f=f^t+f_t$ by $$\begin{aligned}
f^t(x) &= \max\bigl\{ |f(x)| - f^*(t),0\bigr\} \sgn f(x),
\\
f_t(x) &= \min\bigl\{ |f(x)|, f^*(t)\bigr\} \sgn f(x).\end{aligned}$$ We then have $$\begin{aligned}
\bigl(f^t\bigr)^{**}(s)
& \le \frac{t}{s} f^{**} (t),
\quad s\in (0,R),
\label{hh} \\
\bigl(f_t\bigr)^{**}(s)
& \le f^{**} (t),
\quad s\in (0,R).
\label{dh}\end{aligned}$$ Thus $$\begin{aligned}
\bigl( Tf \bigr)^{**} (t)
& \lesssim \bigl( Tf^t + Tf_t \bigr)^{**} (t)
\tag{by quasilinearity}
\\
& \le \bigl( Tf^t \bigr)^{**} (t) + \bigl( Tf_t \bigr)^{**} (t)
\tag{by subaditivity of ${}^{**}$}
\\
& \le \frac{1}{\fI(t)} \sup_{0<s<R} \fI(s) \bigl( Tf^t \bigr)^{**} (s)
+ \frac{1}{\fII(t)} \sup_{0<s<R} \fII(s) \bigl( Tf_t \bigr)^{**} (s)
\\
& = \frac{1}{\fI(t)} \| Tf^t \|_{M_{\fI}}
+ \frac{1}{\fII(t)} \| Tf_t \|_{M_{\fII}}
\\
& \lesssim \frac{1}{\fI(t)} \| f^t \|_{M_{\fI}}
+ \frac{1}{\fII(t)} \| f_t \|_{M_{\fII}}
\tag{by the boundedness of $T$ on $M_{\fI}$ and $M_\fII$}
\\
& = \frac{1}{\fI(t)} \sup_{0<s<R} \fI(s) ( f^t )^{**} (s)
+ \frac{1}{\fII(t)} \sup_{0<s<R} \fII(s) ( f_t )^{**} (s)
\\
& = \text{I} + \text{II}.\end{aligned}$$ Next, $$\begin{aligned}
\text{I}
& \le \frac{1}{\fI(t)} \sup_{0<s<t} \fI(s) f^{**} (s)
+ \frac{1}{\fI(t)} \sup_{0<s<t} \fI(s) ( f_t )^{**} (s)
+ \frac{1}{\fI(t)} \sup_{t<s<R} \fI(s) ( f^t )^{**} (s)
\\
& \le S_{\fI} f^{**} (t)
+ f^{**}(t) \frac{1}{\fI(t)} \sup_{0<s<t} \fI(s)
+ f^{**}(t) \frac{t}{\fI(t)} \sup_{t<s<R} \frac{\fI(s)}{s}
\tag{by \eqref{dh} and \eqref{hh}}
\\
& \lesssim S_{\fI} f^{**} (t)
+ f^{**}(t)
\\
& \lesssim S_{\fI} f^{**} (t).\end{aligned}$$ Similarly, $$\begin{aligned}
\text{II}
& \le \frac{1}{\fII(t)} \sup_{t<s<R} \fII(s) f^{**} (s)
+ \frac{1}{\fII(t)} \sup_{t<s<R} \fII(s) ( f^t )^{**} (s)
+ \frac{1}{\fII(t)} \sup_{0<s<t} \fII(s) ( f_t )^{**} (s)
\\
& \le T_{\fII} f^{**} (t)
+ f^{**}(t) \frac{t}{\fII(t)} \sup_{t<s<R} \frac{\fII(s)}{s}
+ f^{**}(t) \frac{1}{\fII(t)} \sup_{0<s<t} \fII(s)
\\
& \lesssim T_{\fII} f^{**} (t).\end{aligned}$$ Adding both parts together, we obtain $$( Tf )^{**} (t)
\lesssim S_{\fI} f^{**} (t)
+ T_{\fII} f^{**} (t).$$ Now thanks to Lemma \[lemm:stars\_out\] we can put the double stars away and continue by $$( Tf )^{**} (t)
\lesssim S_{\fI} f (t)
+ T_{\fII} f (t)
\lesssim ( S_{\fI} f + T_{\fII} f )^{**} (t).$$ Now, the claim of the theorem follows by Hardy’s lemma [@BS Chapter 2, Corollary 4.7].
\[rem:discontinuity\] Before we get to the proof of Theorem \[thm:Sgg\] let us first say a few words about additional assumptions and in the case of discontinuous quasiconcave function $\fI$.
Denote $\varphi(0{\scriptstyle +})=\lim_{t\to 0^+} \fI(t)>0$. Since $$S_{\fI} f(t) = \frac {1} {\fI(t)} \sup_{0<s<t} \fI(s) f^*(s)\geq \frac {\varphi(0{\scriptstyle +})} {\fI(t)}\|f\|_{L^\infty},
\quad t\in(0,R),$$ we get $S_{\fI} f(t)=\infty$ on whole $(0,R)$ for every unbounded $f$. Thus, in the sake of nontriviality, we are only interested in the situation when $\Gamma^p_{\phi_1}\embed L^\infty$. The embeddings of this type were studied in many papers. By methods of [@CPSS Remark 2.3], this embedding is equivalent to $\sup_{0<t<R}1/\varphi_{\Gamma^p_{\phi_1}}<\infty$, which rewrites as $$\label{sic1}
\inf_{0<t<R}\biggl(\int_0^t \phi_1(s)\,\d s
+ t^p \int_{t}^{R} s^{-p} \phi_1(s)\,\d s\biggr)>0.$$ However, since $\phi_1$ is assumed to be positive, is equivalent to .
Nontriviality also depends on the interplay between quasiconcave function $\fI$ and weight $\phi_2$. Indeed, $$\begin{aligned}
\| S_\varphi f \|^p_{\Gamma^p_{\phi_2}}
& = \int_0^R \bigl[\bigl( S_\varphi f \bigr)^{**}(s)\bigr]^p \phi_2(s)\,\d s
\\
& \simeq \int_0^R \bigl[\bigl( S_\varphi f \bigr)(s)\bigr]^p \phi_2(s)\,\d s
\tag{by Lemma \ref{lemm:prop_S}}\\
& = \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} [f^{*}(y)]^p \varphi^p(y) \,\d s
\\
&\geq \varphi^p(0{\scriptstyle +})\|f\|_{L^\infty}^p\int_0^R \varphi^{-p}(s)\phi_2(s) \,\d s\end{aligned}$$ and as we can see the is a necessary assumption in order to avoid the situation when $S_\varphi f \not\in {\Gamma^p_{\phi_2}}$ for any nontrivial $f$.
The necessity follows by plugging the characteristic function into . As for the sufficiency let us first deal with the case of continuous $\fI$. Take an arbitrary function $f\in \mathcal{M}(0,R)$ and estimate $$\begin{aligned}
\| S_\varphi f \|^p_{\Gamma^p_{\phi_2}}
& = \int_0^R \bigl[\bigl( S_\varphi f \bigr)^{**}(s)\bigr]^p \phi_2(s)\,\d s
\\
& \lesssim \int_0^R \bigl[\bigl( S_\varphi f \bigr)(s)\bigr]^p \phi_2(s)\,\d s
\tag{by Lemma \ref{lemm:prop_S}}\\
& = \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} [f^{*}(y)]^p \varphi^p(y) \,\d s
\\
& \simeq \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} [f^{*}(y)]^p \int_0^y \varphi^{p-1}(t)\varphi'(t)\,\d t \,\d s
\\
& \le \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} \int_0^y [f^{*}(t)]^p \varphi^{p-1}(t)\varphi'(t)\,\d t \,\d s
\\
& = \int_0^R \varphi^{-p}(s)\phi_2(s) \int_0^s [f^{*}(t)]^p \varphi^{p-1}(t)\varphi'(t)\,\d t \,\d s
\\
& = \int_0^R [f^{*}(t)]^p \varphi^{p-1}(t)\varphi'(t) \int_t^R \varphi^{-p}(s)\phi_2(s)\,\d s \,\d t.\end{aligned}$$ Thus, we only need that $$\label{sic2}
\int_{0}^{R} [f^*(t)]^p w(t)\,\d t
\lesssim \int_{0}^{R} [f^{**}(t)]^p \phi_1(t)\,\d t$$ where $$\label{eq:wdef}
w(t)=p\varphi^{p-1}(t)\varphi'(t) \int_t^R \varphi^{-p}(s)\phi_2(s)\,\d s,
\quad t\in(0,R).$$ By [@Neugebauer Theorem 3.2], the inequality holds if and only if $$\label{eq:Neucond}
\int_0^t w(s)\,\d s
\lesssim
\int_0^t \phi_1(s)\,\d s
+ t^p \int_{t}^{R} s^{-p} \phi_1(s)\,\d s,
\quad t\in(0,R),$$ which is equivalent to by integration by parts.
For the sufficiency in the case $\fI$ is discontinuous, we start similarly $$\begin{aligned}
\| S_\varphi f \|^p_{\Gamma^p_{\phi_2}}
& \lesssim \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} [f^{*}(y)]^p \left(p\int_0^y \varphi^{p-1}(t)\varphi'(t)\,\d t+\varphi^p(0{\scriptstyle +})\right) \d s
\\
& \simeq \int_0^R \varphi^{-p}(s)\phi_2(s) \sup_{0<y<s} [f^{*}(y)]^p \int_0^y \varphi^{p-1}(t)\varphi'(t)\,\d t \,\d s
\\
& \quad + \|f\|_{L^\infty}^p\int_0^R \varphi^{-p}(s)\phi_2(s) \,\d s.\end{aligned}$$ The second term is estimated by a constant multiple of $\|f\|^p_{\Gamma_{\phi_1}^p}$, thanks to the assumptions. As for the first one, we proceed in the same way as above and again, due to [@Neugebauer Theorem 3.2], we obtain the sufficiency of where $w$ is defined as in . Now, by integration by parts of the left side, we get $$\begin{aligned}
\int_0^t w(s)\,\d s
& =
\int_0^t \phi_2(s)\,\d s
+ \varphi^p(t) \int_t^R \varphi^{-p}(s) \phi_2(s) \,\d s
\\
& \quad - \varphi^p(0{\scriptstyle +})\int_0^R \varphi^{-p}(s)\phi_2(s) \,\d s,
\quad t\in (0,R),\end{aligned}$$ and clearly is also sufficient for in this case.
Assume that holds. We have $$\begin{aligned}
\| T_\psi f \|^p_{\Gamma^p_{\phi_2}}
& = \int_0^R \bigl[\bigl( T_\psi f \bigr)^{**}(s)\bigr]^p \phi_2(s)\,\d s
\\
& \lesssim \int_0^R \bigl[ T_\psi f (s)\bigr]^p \phi_2(s)\,\d s
+ \int_0^R [ f^{**}(s) ]^p \phi_2(s)\,\d s
\tag{by Lemma \ref{lemm:prop_T}}
\\
& = \text{I} + \text{II}.\end{aligned}$$ Next, $$\begin{aligned}
\text{I}
& \lesssim \int_0^R \psi^{-p}(s) \phi_2(s) \sup_{s<y<R}
\bigl( \psi^p(y) - \psi^p(s) \bigr) [f^*(y)]^p\,\d s
\\
& \quad + \int_0^R \psi^{-p}(s) \phi_2(s) \sup_{s<y<R} \psi^p(s) [f^*(y)]^p\,\d s
\\
& = p \int_0^R \psi^{-p}(s) \phi_2(s) \sup_{s<y<R} [f^*(y)]^p
\int_{s}^{y} \psi^{p-1}(t)\psi'(t) \,\d t\, \d s
\\
& \quad + \int_0^R \phi_2(s) \sup_{s<y<R} [f^*(y)]^p\,\d s
\\
& \le p \int_0^R \psi^{-p}(s) \phi_2(s)
\int_{s}^{R} [f^*(t)]^p \psi^{p-1}(t)\psi'(t) \,\d t\, \d s
\\
& \quad + \int_0^R [f^*(s)]^p \phi_2(s) \,\d s
\\
& \le p\int_{0}^{R} [f^*(t)]^p \psi^{p-1}(t)\psi'(t)
\int_0^t \psi^{-p}(s) \phi_2(s)\,\d s\, \d t
\\
& \quad + \int_0^R [f^*(s)]^p \phi_2(s) \,\d s
\\
& = \int_{0}^{R} [f^*(t)]^p w(t)\,\d t\end{aligned}$$ where we set $$w(t) = \phi_2(t) + p\psi^{p-1}(t)\psi'(t)
\int_0^t \psi^{-p}(s) \phi_2(s)\,\d s,
\quad t\in(0,R).$$ Now, it suffices to show that implies $$\label{sic3}
\int_{0}^{R} [f^{**}(t)]^p \phi_2(t)\,\d t
\lesssim \int_{0}^{R} [f^{**}(t)]^p \phi_1\,\d t$$ and also $$\label{sic4}
\int_{0}^{R} [f^*(t)]^p w(t)\,\d t
\lesssim \int_{0}^{R} [f^{**}(t)]^p \phi_1\,\d t.$$ The embedding holds if and only if $$\int_{0}^{t} \phi_2(s)\,\d s + t^p\int_{t}^R s^{-p}\phi_2(s)\,\d s
\lesssim \int_0^t \phi_1(s)\,\d s
+ t^p \int_t^R s^{-p}\phi_1(s)\,\d s,
\quad t\in(0,R),
\label{GG}$$ due to [@GHS Theorem 3.2], while is by [@Neugebauer Theorem 3.2] equivalent to $$\int_{0}^{t} w(s)\,\d s
\lesssim \int_0^t \phi_1(s)\,\d s
+ t^p \int_t^R s^{-p}\phi_1(s)\,\d s,
\quad t\in(0,R),$$ which is the same as $$\psi^p(t) \int_{0}^{t} \psi^{-p}(s) \phi_2(s)\,\d s
\lesssim \int_0^t \phi_1(s)\,\d s
+ t^p \int_t^R s^{-p}\phi_1(s)\,\d s,
\quad t\in(0,R),
\label{GL}$$ by integration by parts. Finally, since $$\int_{0}^{t} \phi_2(s)\,\d s
\le \psi^p(t) \int_{0}^{t} \psi^{-p}(s) \phi_2(s)\,\d s,
\quad t\in (0,R),$$ due to the fact that $\psi$ is increasing, ensures both and .
The necessity follows again by evaluating both sides of on characteristics functions.
Let us first show that the validity of both conditions for the boundedness of $S_\fI$ and $T_\fII$ on Lorentz gamma spaces and is equivalent to the condition . Indeed, since $\fII(s)$ and $s/\fI(s)$ are both increasing we have $$\int_0^t \phi_2(s)\,\d s
\leq
\fII^p(t) \int_0^t \fII^{-p}(s) \phi_2(s) \,\d s$$ and $$t^p \int_t^R s^{-p}\phi_2(s)\,\d s
\leq
\fI^p(t) \int_t^R \fI^{-p}(s) \phi_2(s) \,\d s.$$ Our result then follows from Theorem \[thm:Tgg\] and Theorem \[thm:Sgg\] used together with Theorem \[thm:DKP\].
acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to express their thanks to Luboš Pick and Martin Křepela for valuable comments and suggestions.
[1]{}
C. Bennett and R. Sharpley. , volume 129 of [*Pure and Applied Mathematics*]{}. Academic Press, Inc., Boston, MA, 1988.
M. Carro, L. Pick, J. Soria, and V. D. Stepanov. On embeddings between classical [L]{}orentz spaces. , 4(3):397–428, 2001.
V. I. Dmitriev and S. G. Kre[ĭ]{}n. Interpolation of operators of weak type. , 4(2):83–99, 1978.
M. L. Goldman, H. P. Heinig, and V. D. Stepanov. On the principle of duality in [L]{}orentz spaces. , 48(5):959–979, 1996.
R. Kerman, C. Phipps, and L. Pick. Marcinkiewicz interpolation theorems for [O]{}rlicz and [L]{}orentz gamma spaces. , 58(1):3–30, 2014.
R. Kerman and L. Pick. Optimal [S]{}obolev imbeddings. , 18(4):535–570, 2006.
R. Kerman and L. Pick. Optimal [S]{}obolev imbedding spaces. , 192(3):195–217, 2009.
C. J. Neugebauer. Weighted norm inequalities for averaging operators of monotone functions. , 35(2):429–447, 1991.
J. O. Str[ö]{}mberg. Bounded mean oscillation with [O]{}rlicz norms and duality of [H]{}ardy spaces. , 28(3):511–544, 1979.
[^1]: This research was partly supported by the grant P201-13-14743S of the Grant Agency of the Czech Republic and by the grant SVV-2016-260335 of the Charles University.
|
---
abstract: 'Recent modelling of solar energetic particles (SEPs) propagation through the heliospheric turbulence, also discussed in this workshop, has investigated the role of the pitch-angle scattering and the perpendicular transport in spreading particles in heliolongitude, as shown by multi-spacecraft measurements (STEREO A/B, ACE, SOHO, etc.) at 1 AU in various energy ranges. In some events the first-order pitch-angle anisotropy of the particles distribution is not-negligible. We calculate the average perpendicular displacement due to the gradient/curvature drift in an inhomogeneous turbulence accounting for pitch-angle dependence for two MHD turbulence models: (a) 3-D isotropic, (b) anisotropic as conjectured by Goldreich-Sridhar. We find in both cases that the drift scales as $(1-\mu^2)^2$ with the cosine of pitch-angle $\mu$, in contrast with previous models for transport of SEPs. This result can impact the models of propagation of SEPs through the heliosphere.'
author:
- 'F. Fraschetti[^1]'
title: 'Cross-field transport and pitch-angle anisotropy of solar energetic particles in MHD turbulence'
---
Introduction
============
In recent years, the multi-spacecraft monitoring by STEREO A/B, ACE, SOHO or *Ulysses*, has shown that SEP evciteents spread in longitude more than $180^{\circ}$ at $1$ AU [@r14] and also in latitude as far as $70^{\circ}$ [@m09]. The interpretation of these measurements invokes two main agents: (1) extended or moving source in the interplanetary medium (CME-driven shocks); (2) strong perpendicular transport across the spiral magnetic field. Measurements of first-order anisotropy suggest that the pitch-angle dependence of the particle distribution is relevant to the interpretation of the longitudinal spread [@d14]. [In situ]{} measurements of longitudinal spread are particularly valuable in that they might probe whether the transport perpendicular to the average magnetic field is dominated by the meandering of the field lines or the transport away from the field lines; this issue has been recently investigated in [@fj11].
Test-particle simulations [@fg12] show that even in a weak three-dimensional isotropic turbulence charged particles decorrelate from the unperturbed magnetic field on a time scale comparable to the gyroperiod. In addition to repeated scattering in the pitch-angle by which particles move back and forth along a given field line, a prompt decorrelation reduces the initial anisotropy of the particles streaming from the source and enhances the angular spread. These results pinpoint to a propagation regime wherein early-time perpendicular transport cannot be neglected.
Recent phenomenological models of charged particle transport in the heliosphere include a dependence of the perpendicular diffusion coefficient on the pitch-angle of the particle velocity with respect to the global average magnetic field ($\mu$) given by $\sqrt{1-\mu^2}$, suggested by a proportionality of the perpendicular diffusion coefficient to the particle gyroradius [@d14]. Different pitch-angle dependencies were considered in [@sf15].
In this short note we outline the calculation of the perpendicular transport coefficient due to gradient/curvature drift originating from the inhomogeneity of the magnetic turbulence for an anisotropic pitch-angle distribution of the particles. The results for the 3-D isotropic turbulence and the MHD anisotropic turbulence conjectured by [@gs95], hereafter GS95, are compared. In [@fj11] we have calculated the average over an isotropic pitch-angle distribution; in this short note we relax such an assumption. We do not calculate here the contribution to the perpendicular transport due to field line meandering.
Gradient/curvature drift transport {#sect_drift}
==================================
Three-dimensional isotropic
---------------------------
We consider a spatially homogeneous time-independent magnetic field with superimposed fluctuations. The amplitude of the fluctuation (${\delta B}$) is assumed to be much smaller than the average field magnitude ($B_0$). We represent such a magnetic field as ${\mathbf{B}}(\mathbf{x}) =\mathbf{B}_0 + \delta {\mathbf{B}(\mathbf{x})}$, with an average component $\mathbf{B}_0 = B_0
\mathbf{e}_z$; we assume that the average over field realisations vanishes, $\langle \delta \mathbf{B}(\mathbf{x}) \rangle = 0$, and $\delta B(\mathbf{x})/B_0 \ll 1$. We make use of the gyroperiod averaged guiding-center velocity transverse to the field $\mathbf{B} (\mathbf{x})$, i.e., $\mathbf{V}_{\perp}^G (t)$, to the first order in $\delta B(\mathbf{x})/B_0$, given for a particle of speed $v$, momentum $p$ and charge $Ze$ by $$\mathbf{V}_{\perp}^G (t) = \frac{vpc}{Ze B} \left[ \frac{1-\mu^2}{2}
\frac{\mathbf{B}\times \nabla B}{B^2} + \mu^2 \frac{\mathbf{B} \times (\mathbf{B} \cdot \nabla ) \mathbf{B})}{B^3} \right]
\label{Vperp}$$ where $\mu$ is the cosine of the pitch-angle with respect to the average field $B_0$ [@ro70; @fj11]. The average square transverse displacement of the particle guiding center from the direction of local $B$ due to drift along the axis $x_i$, $d_{D_{ii}} (t)$, at time $t$ is written as $$d_{D_{ii}} (t) = \int _0 ^t d\xi \langle {\bf V}_{\perp, i}^G (t') {\bf V}_{\perp, i}^G (t' + \xi) \rangle .
\label{dXX1}$$
In the Eq. (\[Vperp\]) we make use of the Fourier representation for $\delta
\mathbf{B}(\mathbf{x})$ and assume an inertial range magnetic turbulence power spectrum which is uncorrelated at different wavenumber vectors: $\langle
\delta B_r({\bf k}) \delta B_q ^*(\mathbf{k}') \rangle = \delta (\mathbf{k} - \mathbf{k}')
P_{rq}(\mathbf{k})$. As in [@fj11] we use $$P_{rq} ({\bf k}) = \frac{G(k)}{8\pi k^2} \left[ \delta_{lm} - \frac{k_l k_m}{k^2}\right], \\$$ with $$G(k)= \left\{
\begin{array}{cc}
G_0 k ^{-q} & \rm{if} k_{\it min} < k < k_{\it max} \, \\
G_0 k_{\it min} ^{-q} & \rm{if} k_0< k < k_{\it min}\, ,
% \label{spectrumk}
\end{array}
\right.$$ where $k^{\it max}$ corresponds to the scale where the dissipation rates of the turbulence overcomes the energy cascade rate, the coherence length is given by $L = 2\pi/k^{\it min}$, and the physical scale of the system by $2\pi/k ^0$.
With these assumptions, the cross-field diffusion coefficient (limit for large times of the average square displacement) associated to gradient/curvature drift of the guiding center due to the turbulence inhomogeneity (in other words the diffusive motion of the guiding center away from the field lines) reads $$D^i _D (t) \rightarrow { \pi\over 16} \left( \frac{\delta B}{B_0} \right)^2 \left
| \frac{q-1}{q-2} \right | \frac{2\pi v^3}{L\Omega^2} \left ( \frac{1 - \mu^2}{2}\right)^2
\label{dDiso_asy}$$
We note that $1/2 \int_{-1}^{1} d\mu D^i _D (t) = d^i _D (t)$, where $d^i _D
(t)$ is the time-dependent average square displacement in Eq. (45) of [@fj11]. The condition that the perpendicular transport time-scale is much shorter than the parallel one is translated into the restriction of Eq. (\[dDiso\_asy\]) to times $t < 1/k_{\it min} v_\parallel \simeq L/ 2 \pi
v_\parallel$; thus, for higher energy particles the physical time this regime applies is shorter because the perpendicular scale grows with the particle gyroradius $r_\mathrm{g}$ and the perpendicular transport becomes relevant. For a $25$ MeV proton (within the energy range of STEREOs and SOHO instruments) at $1$ AU, $(\delta B/B_0)^2 = 0.1$, $r_\mathrm{g}/L \simeq 0.1$ ($L = 0.01$ AU) and $q =
11/3$, we have $D^i_{D} (t) \simeq 4.5 \times 10^{17} (1 - \mu^2)^2 $cm$^2$s$^{-1}$ for $t < 1/k_{\it min} v_\parallel \simeq 3.5$s.
MHD anisotropic turbulence GS95
-------------------------------
Also in this case we consider a spatially homogeneous, fluctuating, time-independent global magnetic field $\mathbf{B}_0$ and decompose the field as ${\mathbf{B}}(\mathbf{x}) =\mathbf{B}_0 + \delta {\mathbf{B}(\mathbf{x})}$, with a large-scale average component $\mathbf{B}_0 = B_0 \mathbf{e}_z$ and a fluctuation $\langle \delta \mathbf{B}(\mathbf{x}) \rangle = 0$. We assume that the inertial range extends from $L$ down to some scale wherein injected energy ultimately must dissipate. We introduce a scale $\hat L$ such that $r_\mathrm{g} \ll
\hat L \ll L$. The fluctuations are large compared to the total magnetic field at scale $\sim L$; however, at scales $< \hat L$ the fluctuations are small compared to the local average field, and the first-order orbit theory applies to eddies up to scale $\hat L$. By using similar assumptions, [@c00] made use of the quasi-linear theory to calculate parallel transport coefficient in GS95.
According to the GS95 conjecture, the pseudo-Alfvén modes are carried passively by the shear-Alfvén modes with no contribution to the turbulence cascade to small scales which is seeded by collisions of shear modes only. [*[@f15]*]{} shows that the transport perpendicular to the local average field due to gradient/curvature drift is dominated by the power of the pseudo-Alfvén modes along the local average field given by $$P_{33} (\hat k_\parallel, \hat k_\perp) = \frac{\hat k_\perp ^2}{\hat k^2}
\Pi(\hat k_\parallel, \hat k_\perp), \\$$ with $${ \Pi
(\hat k_\parallel, \hat k_\perp) = \frac{\varepsilon {\cal N} B_0^2}{ \ell^{1/3}}
\hat k_\perp^{-10/3} {\exp}\left( -\frac{\ell^{1/3} \hat k_\parallel}{\hat k_\perp^{2/3}}\right)} .
\label{pseudo}$$ where $\hat k_\parallel, \hat k_\perp$ indicate wave numbers parallel and perpendicular, respectively, to the local average field, $\ell$ is the outer scale of the pseudo-modes, $\varepsilon$ is the power in the pseudo- relative to shear-modes ($0 < \varepsilon < 1$) and ${\cal N}$ is a normalisation constant accounting for both polarisation modes given by ${\cal N} \simeq {1\over 3\pi} (\delta B/B_0)^2
(1 + \varepsilon (L/\ell))^{-1}$, that includes also scales between $\hat L$ and $L$, as it can be easily seen that the contribution to the turbulent power from perpendicular scales $\hat L < k_\perp ^{-1} < L$ is exponentially suppressed.
Following the calculation in the Appendix of [@f15], we find that the average square displacement due to gradient/curvature drift is given by $$D^a_{D} (t) \simeq \frac{1}{8} \left(\frac{\delta B}{B_0} \right)^2 \frac{\varepsilon}{1+\varepsilon L/\ell}
\left(\frac{L}{\ell} \right)^{1/3} \frac{v^4}{\Omega ^3 L^2} {\Omega t} (k_\perp ^M L)^{2.2} \left ( \frac{1 - \mu^2}{2}\right)^2 .
\label{series_tot_drift}$$
Due to the lack of space, we omit here the corresponding result for the shear- modes. For a 25MeV proton at 1 AU, by taking $L = l$, $\varepsilon =
0.5$, $(\delta B/B_0)^2 = 0.1$ and $(k_\perp ^M)^{-1} \simeq 10^{-5}$ AU, we have $D^a_{D} (t) \simeq 2.5 \times 10^{21} (1 - \mu^2)^2 t [{\it s}] $ cm$^2$s$^{-1}$ for $t < 1/k_{\parallel}^{\it min} v_\parallel \simeq 3.5$s. We emphasise that such a drift cumulates for hours, that is the typical time-scale of the longitudinal spread of SEP, resulting in a possibly significant contribution.
Conclusions
===========
We have calculated the time-dependent average square displacement DD(t) due to gradient/curvature drift for two distinct MHD turbulence models: (a) 3-D-isotropic and (b) anisotropic as conjectured by Goldreich and Sridhar (1995); the assumption of the pitch-angle isotropy of the particle distribution function has been relaxed to account for recently measured first order anisotropy. In both cases, we find $D_D \propto (1 - \mu^2)^2$; such a dependence arises from the gradient drift that is proportional to the particle kinetic energy normal to the field ($p_\perp ^2 / 2m$ for a particle of mass $m$). We conclude that spacecraft data compatible with the scaling $(1 - \mu^2)^2$ would support at once the Goldreich-Sridhar conjecture and our model for perpendicular transport. Although drifts cannot be neglected in the interpretation of multi-spacecraft SEP data across the Parker spiral [@m13], the meandering of the field lines is expected to be larger than drifts in perpendicular transport at scales close to the outer scale. This effect will be assessed in a separate work.
Acknowledgements
================
The author acknowledges useful discussions with W. Dröge and R. D. Strauss and constructive feedback of the referees. This work was supported, in part, by NASA under grant NNX13AG10G. This work benefited from discussions at the team meetings “First principles physics for charged particle transport in strong space and astrophysical magnetic turbulence” at ISSI in Bern, Switzerland.
Chandran, B. D. G.: Scattering of Energetic Particles by Anisotropic Magnetohydrodynamic Turbulence with a Goldreich-Sridhar Power Spectrum, Phys. Rev. Lett., 85, 4656–4659, 2000.
Dröge, W., Kartavykh, Y. Y., Dresing, N., Heber, B., and Klassen, A.: Wide longitudinal distribution of interplanetary electrons following the 7 February 2010 solar event: Observations and transport modeling, J. Geophys. Res., 119, 6074–6094, 2014.
Fraschetti, F.: Cross-field transport in Goldreich-Sridhar MHD turbulence 2015, pre-print: arXiv/1512.05352.
Fraschetti, F. and Giacalone, J.: Early-time velocity auto-correlation for charged particles diffusion and drift in static magnetic turbulence, Astrophys. J., 755, 114, 9 pp., 2012.
Fraschetti, F. and Jokipii, J. R.: Time-dependent perpendicular transport of fast charged particles in a turbulent magnetic field, Astrophys. J., 734, 83, 8 pp., 2011.
Goldreich, P. and Sridhar, S.: Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence, Astrophys. J., 438, 763–775, 1995.
Malandraki, O. E., Marsden, R. G., Lario, D., Tranquille, C., Heber, B., Mewaldt, R. A., Cohen, C. M. S., Lanzerotti, L. J., Forsyth, R. J., Elliott, H. A., Vogiatzis, I. I., and Geranios, A.: Energetic Particle Observations and Propagation in the Three-dimensional Heliosphere During the 2006 December Events, Astrophys. J., 704, 469–476, 2009.
Marsh, M. S., Dalla, S., Kelly, J., and Laitinen, T.: Drift-induced Perpendicular Transport of Solar Energetic Particles, Astrophys. J., 774, 4, 9 pp., 2013.
Richardson, I. G., von Rosenvinge, T. T., Cane, H. V., Christian, E. R., Cohen, C. M. S., Labrador, A. W., Leske, R. A., Mewaldt, R. A., Wiedenbeck, M. E., and Stone, E. C.: $>$25MeV Proton Events Observed by the High Energy Telescopes on the STEREO A and B Spacecraft and/or at Earth During the First Seven Years of the STEREO Mission, Sol. Phys., 289, 3059–3107, 2014.
Rossi, B. and Olbert, S.: Introduction to the Physics of Space, McGraw-Hill, New York, 1970.
Strauss, R. D. and Fichtner, H.: On aspects pertaining to the perpendicular diffusion of solar energetic particles, Astrophys. J., 801, 29, 2015.
[^1]: Departments of Planetary Sciences and Astronomy, University of Arizona, USA
|
---
abstract: 'There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to obtain. In this research project we do not only determine existence, but study the more general problem of counting them. We formulate the enumeration problem as a problem in real algebraic geometry and then use state of the art computational methods in semidefinite programming and representation theory to derive lower bounds for the number of monochromatic arithmetic progressions in any finite group.'
author:
- Erik Sjöland
title: 'Enumeration of monochromatic three term arithmetic progressions in two-colorings of any finite group'
---
Introduction
============
In Ramsey theory colors or density are commonly used to force structures. It follows from Szemerédi’s theorem that there exist monochromatic arithmetic progressions of any length if we color the integers with a finite number of colors. Similarly there are arithmetic progressions of any length in any subset of the integers of positive density.
We can explore finite versions of these statements if we replace the integers by $[n]$, or $\mathbb{Z}_n$. In this article we will focus on the cyclic group, as it is easier to analyze because of the symmetries. In most of the literature the posed question has been about how large $n$ has to be for us to find a monochromatic progression of a desired length when $\mathbb{Z}_n$ is colored by a fixed number of colors. A more difficult problem is to determine how many monochromatic progressions there are of desired length when $\mathbb{Z}_n$ is colored by a fixed number of colors given $n$.
Many results on monochromatic progressions are first obtained for cyclic groups and then extended to results for other groups. Many results can be extended to abelian groups using the same machinery as for the cyclic group, whereas results about non-abelian groups are usually very difficult and require a completely different set of tools.
In this paper we reformulate the problem of counting monochromatic arithmetic progressions to a problem in real algebraic geometry that can be attacked by state of the art optimization theory. The methods allows us to find a lower bound to the amount of monochromatic progressions in any finite group, including non-abelian groups. One could find optimal lower bounds for small groups by explicitly counting them for all different colorings, but as this can be done only for very small groups it has not been considered in this article.
In the next section we present our results. In later sections we present our methods and proofs.
Results
=======
The main theorem of this paper holds for any finite group $G$, including non-abelian groups for which very little is known about arithmetic progressions. The only information that is needed to get a lower bound for a specific group $G$ is the number of elements of the different orders of $G$. The lower bound is sharp for some groups, for example $\mathbb{Z}_p$ with $p$ prime, but is not optimal for most groups. We have further included Table \[tab:examples\] where we have calculated the lower bound for some small symmetric groups.
\[thm:groupcase\] Let $G$ be any finite group and let $R(3,G,2)$ denote the minimal number of monochromatic $3$-term arithmetic progressions in any two-coloring of $G$. Let $G_k$ denote the set of elements of $G$ of order $k$, $N=|G|$ and $N_k=|G_k|$. Denote the Euler phi function $\phi(k)=|\{ t \in \{1,\dots,k\}: t \textrm{ and } k \textrm{ are coprime}\}|$. Let $K=\{k \in \{5,\dots,n\} : \phi(k) \geq \frac{3k}{4}\}$. For any $G$ there are $\sum_{k=4}^n \frac{N\cdot N_k}{2} + \frac{N \cdot N_3}{24}$ arithmetic progressions of length 3. At least $$\begin{array}{rl}
R(3,G,2) \geq &\displaystyle \sum_{k \in K} \frac{N\cdot N_k}{8}(1- 3\frac{k-\phi(k)}{\phi(k)})
\end{array}$$ of them are monochromatic in a 2-coloring of $G$.
Group $G$ Number of 3-APs Lower bound for $R(3,G,2)$
----------- ----------------- ----------------------------
$S_5$ $4540$ $90$
$S_6$ $205440$ $3240$
$S_7$ $11307660$ $306180$
$S_8$ $774278400$ $16208640$
: Calculation of $\sum_{k \in K} \frac{N\cdot N_k}{8}(1- 3\frac{k-\phi(k)}{\phi(k)})$ for small symmetric groups[]{data-label="tab:examples"}
Polynomial optimization {#sec:poly}
=======================
To obtain the main theorem we use methods from real algebraic geometry. It is important to note that the proof of the main theorem can be understood without this section, even though the methods played a vital role when finding the sum of squares based certificate in the proof. In this article we only give the elementary definitions relating to polynomial optimization that are needed to prove the results. For a more extensive review of the topic we refer to [@Laurent2009], and for implementation aspects we refer to [@Sjoland_methods].
Given polynomials $f(x),g_1(x),\dots,g_m(x)$ we define a polynomial optimization problem to be a problem on the form $$\begin{array}{rll}
\rho_* = \inf & \displaystyle f(x) \\
\textnormal{subject to} & \displaystyle g_1(x) \geq 0, \dots, g_m(x) \geq 0, \\
& \displaystyle x \in \mathbb{R}^n,
\end{array}$$ The problem can be reformulated as follows $$\begin{array}{rll}
\rho^* = \sup & \lambda \\
\textnormal{subject to} & \displaystyle f(x) - \lambda \geq 0, g_1(x) \geq 0, \dots, g_m(x) \geq 0 \\
& \displaystyle \lambda \in \mathbb{R}, x \in \mathbb{R}^n
\end{array}$$ where $f(x), g_1(x),\dots,g_m(x)$ are the same polynomials as before. We refer to the book by Lasserre [@Lasserre2010] for an extensive discussion on the relationship between these problems. It is for example easy to see that $\rho_* = \rho^*$.
One of the most challenging problems in real algebraic geometry is to find the most useful relationships between nonnegative polynomials and sums of squares. These relationships are known as Positivstellensätze. Let $X$ be a formal indeterminate and let us introduce the following optimization problem: $$\begin{array}{rll}
\sigma^* = \sup & \lambda \\
\textnormal{subject to} & \displaystyle f(X)-\lambda= \sigma_0 + \sum_{i=1}^m \sigma_ig_i \\
& \displaystyle \sigma_i \textrm{ is a sum of squares.}
\end{array}$$ One can easily see that $\sigma^* \leq \rho^*$ holds, and under some technical conditions (Archimedean) it was proven by Putinar that $\sigma^* = \rho^*$. The equality holds because of Putinar’s Positivstellensatz, which is discussed further in [@Sjoland_methods]. The optimization problem can be relaxed by bounding the degrees of all involved monomials to $d$, lets call the solution $\sigma^*_d$. It holds that $\sigma_{d_1}^* \leq \sigma_{d_2}^*$ for $d_1 < d_2$, and it was proven by Lasserre [@Lasserre2001] that if the Archimedean condition hold, then $$\lim_{d \rightarrow \infty} \sigma^*_d = \sigma^* = \rho^* = \rho_*.$$ The positivity condition is rewritten as a sum of squares condition because the latter is equivalent to a semidefinite condition: $f(X)$ is a sum of squares of degree $2d$ if and only if $f(X) = v_d^T Q v_d$ for some positive semidefinite matrix $Q$, where $v_d$ is the vector of all monomials up to degree $d$. This makes it possible to use semidefinite programming to find $\sigma^*_d$ as well as a sum of squares based certificate for the lower bound of our original polynomial, $f(X) = \sigma^*_d + \sigma_0 + \sum_{i=1}^m \sigma_i g_i \geq \sigma^*_d$.
Exploiting symmetries in semidefinite programming {#sec:Sym}
=================================================
This section can be skipped by the reader who is just interested in the final proof of this article. The methods in this section were used and implemented to find parts of the numerical results that lead to the final proof, and is thus intended for the reader who wants to understand the full process from problem formulation to the end certificate, or solve similar problems with symmetries.
Let $C$ and $A_1, \dots, A_m$ be real symmetric matrices and let $b_1,\dots,b_m$ be real numbers. To reduce the order of the matrices in the semidefinite programming problem $$\max \{\mathrm{tr}(CX) ~ | ~X \textrm{ positive semidefinite}, \mathrm{tr}(A_i X)=b_i \textrm{ for } i = 1, \dots, m\}$$ when it is invariant under all the actions of a group is the goal of this section.
As in [@KPSchrijver2007] and [@Klerk2011], we use a $\ast$–representation to reduce the dimension of the problem. For the reader interested in $\ast$–algebras we recommend the book by Takesaki [@Takesaki2002]. A collection of several new methods, including the one we use, to solve invariant semidefinite programs can be found in [@Bachoc2012]. Other important recent contributions in this area include [@Kanno2001; @Gatermann2004; @Vallentin2009; @Maehara2010; @Murota2010; @Riener2013].
A *matrix $\ast$-algebra* is a collection of matrices that is closed under addition, scalar multiplication, matrix multiplication and transposition.
Let $G$ be a finite group that acts on a finite set $Z$. Define a homomorphism $h : G \rightarrow S_{Z}$, where $S_{Z}$ is the group of all permutations of $Z$. For every $g \in G$ there is a permutation $h_g=h(g)$ of the elements of $Z$ with the properties $h_{g g'}=h_{g}h_{g'}$ and $h_{g^{-1}}=h_{g}^{-1}$. For every permutation $h_{g}$, there is a corresponding permutation matrix $M_{g} \in \{0,1\}^{Z \times Z}$, element-wise defined by $$(M_{g})_{i,j}=
\left\{
\begin{array}{rl}
1 & \textrm{ if } h_{g}(i )= j, \\
0 & \textrm{otherwise}
\end{array}
\right.$$ for all $i,j \in Z$. Let the span of these permutation matrices define the matrix $\ast$-algebra $$\mathcal{A} = \left\{ \sum_{g \in G} \lambda_g M_{g} ~ | ~ \lambda_g \in \mathbb{R} \right\}.$$ The matrices $X$ satisfying $XM_{g}=M_{g}X$ for all $g \in G$ are the *matrices invariant under the action of $G$*. The $\ast$-algebra consisting of the collection of all such matrices, $$\mathcal{A'} = \{X \in \mathbb{R}^{n \times n} | XM=MX \textrm{ for all } M \in \mathcal{A} \},$$ is known as the *commutant* of $\mathcal{A}$. We let $d=\dim\mathcal{A'}$ denote the dimension of the commutant.
The commutant has a basis of $\{0,1\}$-matrices, which we denote $E_1,\dots,E_d$, with the property that $\sum_{i=1}^d E_i = J$, where $J$ is the all-one $Z \times Z$-matrix.
We form a new basis for the commutant by normalizing the $E_i$s $$B_i = \frac{1}{\sqrt{tr(E_i^TE_i)}}E_i.$$ The new basis has the property that $\mathrm{tr}(B_i^TB_j) = \delta_{i,j}$, where $\delta_{i,j}$ is the Kronecker delta.
From the new basis we introduce *multiplication parameters* $\lambda_{i,j}^k$ by $$B_iB_j = \sum_{k=1}^d \lambda_{i,j}^kB_k$$ for $i,j,k = 1,\dots,d$.
We introduce a new set of matrices $L_1,\dots,L_d$ by $$(L_k)_{i,j} = \lambda_{k,j}^i$$ for $k,i,j =1,\dots,d$. The matrices $L_1,\dots,L_d$ are $d \times d$ matrices that span the linear space $$\mathcal{L}=\{\sum_{i=1}^d x_iL_i : x_1,\dots,x_d \in \mathbb{R} \}.$$
The linear function $\phi : \mathcal{A'} \rightarrow \mathbb{R}^{d \times d}$ defined by $\phi(B_i) = L_i$ for $i=1,\dots,d$ is a bijection, which additionally satisfies $\phi(XY)=\phi(X)\phi(Y)$ and $\phi(X^T)=\phi(X)^T$ for all $X,Y \in \mathcal{A}'$.
$\sum_{i=1}^d x_iB_i$ is positive semidefinite if and only if $\sum_{i=1}^d x_iL_i$ is positive semidefinite. \[cor:schrijver\]
If it is possible to find a solution $X \in \mathcal{A}'$, then Corollary \[cor:schrijver\] can be used to reduce the size of the matrix in the linear matrix inequality. Let us show that this is possible:
\[lem:groupaverage\] There is a solution $X \in \mathcal{A}'$ to a $G$-invariant semidefinite program $$\max \{\mathrm{tr}(CX) ~ | ~X \textrm{ positive semidefinite}, \mathrm{tr}(A_i X)=b_i \textrm{ for } i = 1, \dots, m\}.$$
Let $C,A_1,\dots,A_m$ be $Z \times Z$ matrices commuting with $M_g$ for all $g \in G$. If $X$ is an optimal solution to the optimization problem then the group average, $X'=\frac{1}{|G|} \sum_{g \in G}M_g X M_g^T$, is also an optimal solution: It is feasible since $$\begin{array}{rl}
\displaystyle \mathrm{tr}(A_jX') &= \displaystyle \mathrm{tr}(A_j\frac{1}{|G|} \sum_{g \in G}M_g X M_g^T) \\
&= \displaystyle \mathrm{tr}(\frac{1}{|G|} \sum_{g \in G}M_g A_jX M_g^T) \\
&= \displaystyle \mathrm{tr}(\frac{1}{|G|} \sum_{g \in G}A_jX ) \\
&= \displaystyle \mathrm{tr}(A_jX),
\end{array}$$ where we have used that the well-known fact that the trace is invariant under change of basis. By the same argument $\mathrm{tr}(CX') =\mathrm{tr}(CX)$, which implies that $X'$ is optimal. It is easy to see that $X' \in \mathcal{A}'$.
The following theorem follows, which gives a tremendous computational advantage when $d$ is significantly smaller than $|Z|$:
\[thm:Schrijver\] The $G$-invariant semidefinite program $$\max \{\mathrm{tr}(CX) ~ | ~X \succeq 0, \mathrm{tr}(A_i X)=b_i \textrm{ for } i = 1, \dots, m\}$$ has a solution $X = \sum_{i=1}^d x_iB_i$ that can be obtained by $$\max \{\mathrm{tr}(CX) ~ | ~ \sum_{i=1}^d x_iL_i \succeq 0, \mathrm{tr}(A_i \sum_{j=1}^d B_jx_j)=b_i \textrm{ for } i = 1, \dots, m\}.$$
Problem formulated as a semidefinite program {#sec:methodscolor}
============================================
We follow the convention that an arithmetic progression in $G$ of length $k$ is a set of $k$ distinct element $\{a,b \cdot a,\dots,b^{k-1} \cdot a \}$ where $a \in G$, $b \in \mathbb{Z}^+$. Also, as $\{1,2,3\}$, $\{1,3,2\}$, $\{2,1,3\}$, $\{2,3,1\}$, $\{3,1,2\}$ and $\{3,2,1\}$ denote the same set they are considered as the same arithmetic progression, thus when summing over all arithmetic progressions the set is only considered once.
Let $\chi : G \rightarrow \{-1,1\}$ be a $2$-coloring of the finite group $G$, and for simplicity let $x_g = \chi(g)$ for all $g \in G$. Furthermore, let $x_{_G}$ denote the vector of all variables $x_g$. Let us introduce the polynomial $$\begin{array}{rl}
p(x_a,x_b,x_c) &= \displaystyle \frac{(x_a+1)(x_b+1)(x_c+1)-(x_a-1)(x_b-1)(x_c-1)}{8} \\
&= \displaystyle \frac{x_ax_b+x_ax_c+x_bx_c+1}{4},
\end{array}$$ where $a,b,c \in G$, which has the property that $$p(x_a,x_b,x_c) = \left\{
\begin{array}{ll}
1 & \text{if } x_a=x_b=x_c\\
0 & \text{otherwise.}
\end{array} \right.$$ The polynomial $p$ is one when $a,b,c$ are the same color and zero otherwise. It follows that $$R(3,G,2)= \min_{x_{_G} \in \{-1,1\}^{|G|}} \displaystyle\sum_{\{a,b,c\} \textrm{ is an A.P. in } G}p(x_a,x_b,x_c).$$
The integer problem is relaxed to a problem on the hypercube to obtain a lower bound for $R(3,G,2)$ . Since any solution of the integer program is also a solution to the hypercube problem we have $$\begin{array}{rl}
R(3,G,2) &\displaystyle \geq \min_{ x_{_G} \in [-1,1]^{|G|}} \sum_{\{a,b,c\} \textrm{ is an A.P. in } G}p(x_a,x_b,x_c) \\
& = \displaystyle \min_{ x_{_G} \in [-1,1]^{|G|}} \sum_{\{a,b,c\} \textrm{ is an A.P. in } G} \frac{x_ax_b+x_ax_c+x_bx_c+1}{4} \\
& = \displaystyle \min_{ x_{_G} \in [-1,1]^{|G|}} \frac{p_{_G}}{4}+\sum_{\{a,b,c\} \textrm{ is an A.P. in } G} \frac{1}{4}
\end{array}$$ where $$p_{_G} = \sum_{\{a,b,c\} \textrm{ is an A.P. in } G}x_ax_b+x_ax_c+x_bx_c.$$
We immediately get a lower bound for $R(3,G,2)$ by finding a lower bound for the homogeneous degree 2 polynomial $p_{_G}$. The coefficient of $x_ax_b$ in $p_{_G}$ equals the number of times the pair $(a,b)$ occurs in a 3-arithmetic progression, which depends on the group $G$. After the coefficients are found the state-of-the-art methods surveyed in Sections \[sec:poly\] and \[sec:Sym\] are used to find a lower bound for $$\min_{ x_{_G} \in [-1,1]^{|G|}} p_{_G}.$$
Let us use the degree 3 relaxation of Putinar’s Positivstellensatz, and let $\lambda^*$ denote the maximal lower bound using this relaxation. Denote the elements of $G$ by $g_1,\dots,g_{|G|}$ and let $v$ be the vector of all monomials of degree less or equal than one in the formal indeterminates $X_{g_1},\dots,X_{g_{|G|}}$; $v=[1, X_{g_1}, \dots, X_{g_{|G|}}]^T$. We get $$\begin{array}{rl}
\lambda^* = \max & \lambda \\
\textnormal{subject to:} & \displaystyle p_{_G} - \lambda = v^T Q_0 v + \sum_{g \in G} v^T Q_g^+ v(1+X_g) + \sum_{g \in G} v^T Q_g^- v (1-X_g) \\
& \lambda \in \mathbb{R} \\
& \displaystyle Q_0,Q_g^+,Q_g^- \succeq 0 \textnormal{ for all } g\in G.
\end{array}$$
For simplicity, let the $Q$-matrices entries be indexed by the set $\{1$, $g_1$, $\dots$, $g_{|G|}\}$. Let $A \rtimes B$ denote the semidirect product of $A$ and $B$, defined such that $(a,b) \in A \rtimes B$ takes $i$ to $a+bi$ (here $+$ denotes the action of the group $G$, and $bi$ denotes the repeated action of $i$ on itself $b$ times: $bi=\sum_{j=1}^b i$). Arithmetic progressions are invariant under affine transformations: if $(a,b) \in G \rtimes \mathbb{Z}^+$, and $\{i,j,k\}$ is an arithmetic progression, then $(a,b) \cdot \{i,j,k\}=\{a+bi,a+bj,a+bk\}$ is also an arithmetic progressions. It follows that the semidefinite program is invariant under affine transformations, and thus that we can find an invariant solution by Lemma \[lem:groupaverage\]. This implies that the degree 3 relaxation has a solution for which $Q_0(g_i,g_j)=Q_0(a + b g_i, a + b g_j)$ and $Q_0(1,g_i) = Q_0(g_i,1) =Q_0(1,a+b g_i)$ for all $(a,b) \in G \rtimes \mathbb{Z}^+$ and $g_i,g_j \in G$. Also $Q_{g_i}^+(g_j,g_k) = Q_{a + b g_i}^+(a + b g_j, a + b g_k)$, $Q_{g_i}^+(1,g_k) = Q_{g_i}^+(g_k,1) = Q_{a + b g_i}^+(1, a + b g_k)$ and $Q_{g_i}^+(1,1) = Q_{a + b g_i}^+(1, 1)$. By the same argument we get similar equalities for the indices of the matrices $Q_{g_i}^-$. From the equalities It is easy to see that the matrices $Q_{g_i}^+$ and $Q_{g_i}^-$ are obtained by simultaneously permuting the rows and columns of $Q_{g_j}^+$ and $Q_{g_j}^-$ respectively, and hence it is enough to require that $Q_{g_1}^+$ and $Q_{g_1}^-$ are positive semidefinite where $g_1 \in G$ is the identity element.
In conclusion we see that the number of variables can be reduced significantly, and that only three $|G|+1 \times |G|+1$-matrices are required to be positive semidefinite instead of the $2|G|+1$ matrices required in the original formulation. The dimension of the commutant is small for some groups, and in those cases the size of these matrices can be reduced further using Theorem \[thm:Schrijver\].
The certificates for the lower bounds we get from these methods are numerical, and to make them algebraic additional analysis of the numerical data, and possibly further restrictions, must be done. There is no general way to find algebraic certificates, and for many problem it might not even be possible. In this paper it is a vital step to find as the numerical certificates only gives certificates for one group at the time whereas we need a certificate for all groups.
It is in general very difficult to go from numerical patterns to algebraic certificates. For a specific group all information required to find a lower bound can be found in an eigenvalue decomposition of the involved matrices, but there is no general way of finding the optimal algebraic lower bound when the different eigenvalues and eigenvectors have decimal expansions that cannot trivially be translated into algebraic numbers. If one is interested in a rational approximation to the lower bound one can use methods by Parrilo and Peyrl [@Parrilo2008]. These methods gives a good certificate for a specific group but does not help when one want to find certificates for an infinite family of groups.
To find a certificate for all groups, one of the tricks we used was to restrict the SDP above by requiring that some further entires equal one another, in order to at least get a lower bound for $\lambda^*$. There are also many other ways to restrict the SDP further, including setting elements to zero. Restricting the SDP either decreases the optimal value or reduces the number of solutions to the original problem. In the ideal scenario one can keep restricting the SDP until there is only one solution, without any change in the optimal value.
Proof of Theorem \[thm:groupcase\]
==================================
To make the computations in the proofs that follow readable we introduce additional notation: $$\sigma(a;b_0,b_1,\ldots,b_{n-1})=a+ \sum_{i,j\in \mathbb{Z}_n} b_{j-i} X_iX_j.$$
By elementary calculations we have the following equalities, which are needed in the proofs:
$$I_1=\displaystyle \sum_{i\in \mathbb{Z}_{n}} (1-X_i^2) = \sigma(n; -1, 0, \dots , 0),$$ $$I_2=\displaystyle \Big( \sum_{i\in \mathbb{Z}_{n}} X_i \Big)^2 = \sigma\Big( 0; 1, 2, \dots, 2 \Big),$$ $$I_3^j=\displaystyle \frac{1}{2}\sum_{i \in \mathbb{Z}_{n}}(X_i-X_{i+j})^2 = \sigma\Big( 0; 1 , 0 , \dots, 0 ,-1 , 0, \dots, 0,-1,0,\dots,0 \Big),$$
The first one is non-negative since we for the problem require that $-1 \leq x_i \leq 1$ and the other polynomials are non-negative since they are sums of squares.
Let us count all arithmetic progressions $\{a,b \cdot a,b \cdot b \cdot a\}$, with $(a,b) \in G \times G_k$. There is a cyclic subgroup of $G$ with elements $\{1,b,b^2,\dots,b^{k-1}\}$. Let us write $k=p_1^{e_1}p_2^{e_2} \cdots p_s^{e_s}$ for $1<p_1<\dots<p_s$ distinct primes and $e_1,\dots,e_s$ positive integers. The elements $\{b^{p_i},b^{2p_i},\dots,b^{(\frac{k}{p_i}-1)p_i}\}$ are of order less than $k$ for all $i$ whereas the elements $U=\{b^t : t$ and $k$ are coprime$\}$ are of order $k$. Note that $\phi(k)=|U|$. If $k<3$ there are no arithmetic progressions, so in all the following calculations we will always assume that $k \geq 3$. If $3 \nmid k$, then all triples $\{(1,b^i,b^{2i}) : b^i \in U \}$ will be distinct arithmetic progressions. Since there are $\frac{N_k}{\phi(k)}$ different cyclic subgroups of $G$ of this type there are $\phi(k)\frac{N_k}{\phi(k)}=N_k$ arithmetic progressions of the form $\{1,b^i,b^{2i}\}$, where $b^i$ is an element of order $k$. Since $\{a, b\cdot a, b\cdot b\cdot a\}$ is an arithmetic progression if and only if $\{1,b,b\cdot b\}$ is an arithmetic progression it follows that there are $\frac{N\cdot N_k}{2}$ arithmetic progressions with $(a,b) \in G \times G_k $ if $3 \nmid k$. If $3 | k$ we have to be careful so that we do not count arithmetic progressions of the form $\{a,b^{\frac{k}{3}}\cdot a,b^{\frac{2k}{3}}\cdot a\}$ three times. Since $b^{\frac{k}{3}}$ will be of lower order than $k$ if $k>3$ it follows that this kind of arithmetic progressions only occur for $(a,b) \in G \times G_3$. We conclude that there are $\frac{N\cdot N_k}{2}$ arithmetic progressions with $(a,b) \in G \times G_k $ if $k>3$, and $\frac{N\cdot N_3}{6}$ if $(a,b) \in G \times G_3$.
The number of monochromatic arithmetic progressions is given by $$\begin{array}{rl}
R(3,G,2) &= \displaystyle\sum_{\{a,b,c\} \textrm{ is an A.P. in } G}p(x_a,x_b,x_c) \\
&= \displaystyle\sum_{\{a,b,c\} \textrm{ is an A.P. in } G}\frac{x_ax_b+x_ax_c+x_bx_c+1}{4}.
\end{array}$$ Let us rewrite this as $$R(3,G,2)=\frac{\sum_{k=4}^n \frac{N\cdot N_k}{2} +p_k }{4} + \frac{N \cdot N_3 }{24} + \frac{p_3}{4},$$ where $p_k$ is the sum of all polynomials $x_sx_{t \cdot s}+x_sx_{t \cdot t \cdot s}+x_{t \cdot b}x_{t \cdot t \cdot s}$ where $t$ is of order $k$. Let us also define the further reduced polynomial $p_k^{(a,b)}$ for a fixed pair $(a,b) \in G \times G_k$ by $$p_k^{(a,b)}=\sum_{0 \leq i < j \leq k-1}c_{b^i \cdot a,b^j \cdot a}x_{b^i \cdot a}x_{b^j \cdot a}$$ where $c_{b^i \cdot a,b^j\cdot a}$ denotes how many times the pair $(b^i\cdot a,b^j\cdot a)$ is in an arithmetic progression $\{s,t \cdot s,t \cdot t \cdot s\}$ with $t$ of order $k$. When $a=1$ we will have a set $\{b_{i_1},\dots,b_{i_{N_k/\phi(k)}} \}$ of elements of order $k$ such that $b_{i_j}^{c_1} \neq b_{i_k}^{c_2}$ for all $j,k \in \{1,\dots,N_k/\phi(k) \}$ and all $c_1,c_2 \in \{1,\dots,k-1\}$, and can write $$p_k= \frac{1}{k} \sum_{a\in G} \sum_{j=1}^{N_k/\phi(k)} p_k^{(a,b_{i_j})}.$$ It follows that if $p_k^{(1,b)} \geq c$ for a $b$ of order $k$, then $p_k \geq \frac{N}{k}\frac{N_k}{\phi(k)}c $.
[**Let k mod 2 = 1, k mod 3 $\neq$ 0:**]{} It is fairly easy to see that if $b^i$ is of order $k$, then so is $b^{-i}$ and $b^{2i}$. Also, if $2 | i$ then $b^{i/2}$ is of order $k$, if $2 \nmid i$ then $b^{(n+i)/2}$ is of order $k$. Hence any $b^i$ is in both arithmetic progressions $b^{-i},1,b^i$ and $1,b^i,b^{2i}$, and also in either $1,b^{i/2},b^i$ or $1,b^{(n+i)/2},b^i$. These are all arithmetic progressions containing both $1$ and $b^i$. Note that the elements that are not of order $k$ (apart from the identity element) are not in any of these arithmetic progressions, hence if $k=p_1^{e_1}\dots p_s^{e_s}$, then $$p_k^{(1,b)}(X)=\sum_{i,j\in \mathbb{Z}_k} b_{j-i} X_iX_j=\sigma(0;b_0,b_1,\ldots,b_{k-1})$$ where $b_{tp_i}=0$ for $t\in \{0,\dots,k-1\}$ and $i\in \{1,\dots,s\}$, and $b_t=3$ for all other $t$. In particular we get $$p_k^{(1,b)}(X) = \frac{3}{2}I_2+\frac{3}{2}\sum_{t,i}I_3^{tp_i} +\frac{3}{2}(1+k-\phi(k)-1)I_1 - \frac{3}{2}(k-\phi(k))k \geq - \frac{3}{2}(k-\phi(k))k,$$ and it follows that $$p_k \geq - \frac{N}{k}\frac{N_k}{\phi(k)} \frac{3}{2}(k-\phi(k))k=- \frac{3}{2} \frac{N \cdot N_k}{\phi(k)} (k-\phi(k)),$$ and furthermore that $$\frac{N\cdot N_k}{2} +p_k \geq \frac{N\cdot N_k}{2}(1- 3\frac{k-\phi(k)}{\phi(k)}).$$
In cases when $1- 3\frac{k-\phi(k)}{\phi(k)} < 0$, that is when $\phi(k) <\frac{3k}{4}$, we will instead use the trivial bound $$\frac{N\cdot N_k}{2} +p_k \geq 0$$
[**k mod 2 = 0:** ]{} $b \in G_k$, and let us color all elements $\{1,b^2,b^4,\dots,b^{k-2}\}$ blue and $\{b,b^3,b^5,\dots,b^{k-1}\}$ red. In an arithmetic progression $\{a,c\cdot a,c\cdot c\cdot a \}$ with $(a,c) \in \{0,b,b^2,\dots,b^{k-1} \} \times \{0,b,b^2,\dots,b^{k-1} \} \cap G_k$ it holds that $a$ and $c\cdot a$ are of different colors. The coloring can be extended too all pairs $(a,c) \in G \times G_k$, and thus there is a coloring without monochromatic arithmetic progressions. Hence we cannot hope to do better than the trivial bound using these methods: $$\frac{N\cdot N_k}{2} +p_k \geq 0.$$
[**k mod 3 = 0:** ]{} Let us color the elements $\{0,b,b^3,b^4,\dots,b^{k-3},b^{k-2}\}$ blue and $\{b^2,b^5,b^8,\dots,b^{k-1}\}$ red. As when k mod 2 = 0 we consider arithmetic progressions $\{a,c\cdot a,c\cdot c\cdot a\}$ with $(a,c) \in \{0,b,b^2,\dots,b^{k-1} \} \times \{0,b,b^2,\dots,b^{k-1} \} \cap G_k$. It is easy to see that either $a$ and $c\cdot a$ or $a$ and $c \cdot c \cdot a$ are of different colors. Again there is a coloring without monochromatic arithmetic progressions and so we cannot do better than the trivial bound: $$\frac{N\cdot N_k}{2} +p_k \geq 0$$ for $k>3$ and $$\frac{N\cdot N_3}{6} +p_3 \geq 0$$ for $k=3$.
Let $K=\{k \in \{5,\dots,n\} : \phi(k) \geq \frac{3k}{4}\}$. If $2$ or $3$ divide $k$ then $\phi(k) < \frac{3k}{4}$, and hence none of those numbers are included in $K$. Summing up all cases we get $$R(3,G,2) \geq \frac{\sum_{k \in K} \frac{N\cdot N_k}{2}(1- 3\frac{k-\phi(k)}{\phi(k)})}{4}.$$
Methods for longer arithmetic progressions
==========================================
Let $\chi : G \rightarrow \{-1,1\}$ be a $2$-coloring of the group $G$, and let $x_g = \chi(g)$ for all $g \in G$. Let also $x_{_G}$ denote the vector of all variables $x_g$. For $a,b,c \in G$, let us introduce the polynomial $$p(x_{a_1},\dots,x_{a_k}) = \displaystyle \frac{(1+x_{a_1})\cdots(1+x_{a_k})+(1-x_{a_1})\cdots(1-x_{a_k})}{2^k},$$ which has the property that $$p(x_{a_1},\dots,x_{a_k}) = \left\{
\begin{array}{ll}
1 & \text{if } x_{a_1}=\dots=x_{a_k}\\
0 & \text{otherwise.}
\end{array} \right.$$ In other words, the polynomial $p$ is one when $\{a_1,\dots,a_k\}$ is a monochromatic arithmetic progression and zero otherwise. It follows that $$R(k,G,2)= \min_{x_{_G} \in \{-1,1\}^{|G|}} \displaystyle\sum_{\{a_1,\dots,a_k\} \textrm{ is an A.P. in } G}p(x_{a_1},\dots,x_{a_k}).$$
To find a lower bound for $R(k,G,2)$ we relax the integer quadratic optimization problem to an optimization problem on the hypercube. Since it is a relaxation, i.e. any solution of the integer program is also a solution to the hypercube problem, we have $$R(k,G,2) \geq \min_{ x_{_G} \in [-1,1]^{|G|}} \displaystyle\sum_{\{a_1,\dots,a_k\} \textrm{ is an A.P. in } G}p(x_{a_1},\dots,x_{a_k}),$$ an optimization problem that we can find lower bounds for using Putinar’s Positivstellensatz and the Lasserre Hierarchy.
Related problems
================
The methods developed in this article may also be applied in a wide variety of similar problems.
Let $R(3,[n],2)$ denote the minimal number of monochromatic arithmetic progressions of length $3$ in a $2$-coloring of $[n]$. Asymptotic bounds for $R(3,[n],2)$ have been found for large $n$ using other methods [@Parrilo2008_2]: $$\frac{1675}{32768}n^2(1+o(1)) \leq R(3,[n],2) \leq \frac{117}{2192}n^2(1+o(1)).$$ The author has in collaboration with Oscar Kivinen found numerical results suggesting that the lower bound can be improved to $0.052341 \dots$ using a degree $3$-relaxation ($\frac{1675}{32768} = 0.05111 \dots$ and $\frac{117}{2192} =0.05337 \dots$) . The major challenge is to use the numerical results to obtain an algebraic certificate because of the lack of symmetries in the problem. For large $n$ the numerical information seems to converge towards a fractal, and to obtain the improved lower bound based on the proposed methods one would need to fully understand the behavior of the fractal. Even if it is not possible to understand the fractal, one can probably do approximations that would improve on the lower bound. This is work under progress.
Let $R(3,\mathbb{Z}_n,2)$ denote the minimal number of monochromatic arithmetic progressions of length $3$ in a $2$-coloring of the cyclic group $\mathbb{Z}_n$. Optimal, or a constant from optimal, lower bounds for $R(3,\mathbb{Z}_n,2)$ have been found for all $n$ [@Sjoland_cyclic]: $$n^2/8-c_1n+c_2 \leq R(3,\mathbb{Z}_n,2) \leq n^2/8-c_1n+c_3,$$ where the constants depends on the modular arithmetic and are tabulated in the following table. $$\begin{array}{c|c|c|c}
n \mod 24 & c_1 & c_2 & c_3 \\
\hline
1,5,7,11,13,17,19,23 & 1/2 & 3/8 & 3/8 \\
8,16 & 1 & 0 & 0 \\
2,10 & 1 & 3/2 & 3/2 \\
4,20 & 1 & 0 & 2 \\
14,22 & 1 & 3/2 & 3/2 \\
3,9,15,21 & 7/6 & 3/8 & 27/8 \\
0 & 5/3 & 0 & 0 \\
12 & 5/3 & 0 & 18 \\
6,18 & 5/3 &1/2 & 27/2 \\
\end{array}$$ A corollary is that we can find an optimal, or a constant from optimal, lower bound for the number of monochromatic arithmetic progressions for the dihedral group $D_{2n}$ for any $n$: $$R(3,D_{2n},2)=2R(3,\mathbb{Z}_n,2).$$ In particular $$n^2/4-2c_1n+2c_2 \leq R(D_{2n};3) \leq n^2/4-2c_1n+2c_3$$ where the constants can be found in the table above.
Let $R(4,\mathbb{Z}_n,2)$ denote the minimal number of monochromatic arithmetic progressions of length $4$ in a $2$-coloring of the cyclic group $\mathbb{Z}_n$. Asymptotic bounds for $R(4,\mathbb{Z}_n,2)$ have been found for large $n$ in [@Wolf2010], and the bounds have since then been improved in [@Lu2012 Theorem 1.1, 1.2, 1.3] to: $$\frac{7}{192}p^2(1+o(1)) \leq R(4,\mathbb{Z}_p,2) \leq \frac{17}{300}p^2(1+o(1))$$ when $p$ is prime, and for other $n$ $$c_1n^2(1+o(1)) \leq R(4,\mathbb{Z}_n,2) \leq c_2n^2(1+o(1))$$ where the constants depends on the modular arithmetic on $n$ in accordance with the following table $$\begin{array}{c|c|c}
n \mod 4 & c_1 & c_2 \\
\hline
1,3 & 7/192 & 17/300\\
0 & 2/66 & 8543/1452000 \\
2 & 7/192 & 8543/1452000 \\
\end{array}$$ Furthermore [@Lu2012 Theorem 1.5] $$\underline{\lim}_{n \rightarrow \infty} R(4,\mathbb{Z}_n,2) \leq \frac{1}{24},$$ and it is conjectured that [@Lu2012 Conjecture 1.1]: $$\inf_n\{R(4,\mathbb{Z}_n,2) \} = \frac{1}{24}.$$ The author has tried to improve the bounds using a degree 4-relaxations with further restriction to simplify the problem, but as the author obtained algebraic bounds that are far from the current best bounds it seems like one needs to omit the restrictions and possibly use a higher degree relaxation to get relevant bounds. One of the main challenges is that the solution depends heavily on the modular arithmetic of $n$, and so one needs to numerically find solutions for fairly high values of $n$, which is not feasible for a high degree relaxation. Doing a full degree 4-relaxation is possible but tedious, and since one can only get numerical results for fairly small $n$ it is difficult to find general patterns. One cannot exclude the possibility that this would be enough to improve the bounds, but what is more likely is that it would require additional tricks or techniques after one has found numerical certificates for small $n$.
Let $R(4,[n],2)$ denote the minimal number of monochromatic arithmetic progressions of length $4$ in a 2-coloring of $[n]$. Let furthermore $R(5,\mathbb{Z}_n,2)$ and $R(5,[n],2)$ denote the minimal number of monochromatic arithmetic progressions of length $5$ in $\mathbb{Z}_n$ and $[n]$ respectively. Upper bounds have been found for $R(4,[n],2)$, $R(5,\mathbb{Z}_n,2)$ and $R(5,[n],2)$. For $n$ large enough [@Lu2012 Equation (12)]: $$R(4,[n],2) \leq \frac{1}{72}n^2(1+o(1)).$$ When $n$ is odd and large enough we have [@Lu2012 Theorem 1.4]: $$R(5,\mathbb{Z}_n,2) \leq \frac{3629}{131424}n^2(1+o(1)).$$ When $n$ is even and large enough we have [@Lu2012 Theorem 1.4]: $$R(5,\mathbb{Z}_n,2) \leq \frac{3647}{131424}n^2(1+o(1)).$$ Furthermore [@Lu2012 Theorem 1.5]: $$\underline{\lim}_{n \rightarrow \infty} R(5,\mathbb{Z}_n,2) \leq \frac{1}{72}.$$ Finally when $n$ is large enough we have [@Lu2012 Equation (13)]: $$R(5,[n],2) \leq \frac{1}{304}n^2(1+o(1)).$$ These are all upper bounds that have been obtained through good colorings. Lower bounds to all these problems can possibly be found using the methods developed in this article using a relaxation of high enough order. As it is numerically very difficult to find solutions to relaxations of high orders, one should be aware that this is not guaranteed to work in practice.
Another type of problems that one can use the methods developed in this article to solve are enumeration problems in fixed density sets. As in this article it is of interest to count arithmetic progressions. Let $W(k,S,\delta)$ denote the minimal number of arithmetic progressions of length $k$ in any subset of $S$ of cardinality $|S|\delta$. One can for example let $S$ be a group or $[n]$. Note that if one could find strict lower bounds for $W(k,[n],\delta)$ for all $n$, $k$ and $\delta$ this would imply optimal quantitative bounds for Szemerédi’s theorem. Although it might be too ambitious to try to find strict lower bounds it might still be possible to find bounds good enough to generalize Szemerédi’s theorem. This kind of results can be obtained by methods very similar to the ones used in this article as discussed in [@Sjoland_density].
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Alexander Engström for our long discussions on this topic and for all his valuable feedback. I would also want to thank Markus Schweighofer and Cynthia Vinzant for their comments, which have been incorporated in the article.
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|
---
abstract: |
We report the discovery of a massive (${\ensuremath{M_{p}}}= 9.04 \pm
0.50\,{\ensuremath{M_{\rm Jup}}}$) planet transiting the bright ($V=8.7$) F8 star , with an orbital period of $5.63341\pm0.00013$ days and an eccentricity of $e=0.520\pm0.010$. From the transit light curve we determine that the radius of the planet is ${\ensuremath{R_{p}}}= 0.982\pm^{0.038}_{0.105}\,{\ensuremath{R_{\rm Jup}}}$. (also coined HAT-P-2b) has a mass about 9 times the average mass of previously-known transiting exoplanets, and a density of ${\ensuremath{\rho_{p}}}=
11.9\,{\ensuremath{\rm g\,cm^{-3}}}$, greater than that of rocky planets like the Earth. Its mass and radius are marginally consistent with theories of structure of massive giant planets composed of pure H and He, and may require a large ($\gtrsim 100{\ensuremath{M_\earth}}$) core to account for. The high eccentricity causes a 9-fold variation of insolation of the planet between peri- and apastron. Using follow-up photometry, we find that the center of transit is $T_{mid}=2,\!454,\!212.8559 \pm 0.0007$ (HJD), and the transit duration is $0.177\pm0.002$d.
author:
- 'G. Á. Bakos, G. Kovács, G. Torres, D. A. Fischer, D. W. Latham, R. W. Noyes, D. D. Sasselov, T. Mazeh, A. Shporer, R. P. Butler, R. P. Stefanik, J. M. Fernández, A. Sozzetti, A. Pál, J. Johnson, G. W. Marcy, J. Winn, B. Sipőcz, J. Lázár, I. Papp & P. Sári'
---
Introduction {#sec:intro}
============
To date 18 extrasolar planets have been found which transit their parent stars and thus yield values for their mass and radius[^1]. Masses range from 0.3[$M_{\rm J}$]{} to about 1.9[$M_{\rm J}$]{}, and radii from 0.7[$R_{\rm J}$]{}to about 1.4[$R_{\rm J}$]{}. The majority fit approximately what one expects from theory for irradiated gas giant planets [e.g. @fortney06 and references therein], although there are exceptions: has a small radius for its mass [@sato05], implying that it has a large heavy core [$\sim$70[$M_\earth$]{}; @laughlin05], and several (, HAT-P-1b, WASP-1) have unexpectedly large radii for their masses, perhaps suggesting some presently unknown source of extra internal heating [@guillot02; @bodenheimer03]. The longest period and lowest density transiting exoplanet (TEP) detected so far is HAT-P-1b with $P=4.46$d [@bakos07]. All TEPs have orbits consistent with circular Keplerian motion.
From existing radial velocity (RV) data, it might be expected that there are some close-in (semi-major axis $\lesssim0.07$AU, or $P\lesssim10$d) giant planets with masses considerably larger than any of the 18 transiting planets now known. A well-known example, considering only objects below the Deuterium burning threshold [$\sim 13\,{\ensuremath{M_{\rm J}}}$, e.g. @burrows97], is $\tau$ Boo b, which was detected from RV variations, and has a minimum mass of [${\ensuremath{M_{p}}}\sin i$]{}=3.9[$M_{\rm J}$]{} and orbits only 0.046 AU from its star [@butler97]. Another example is HIP14810 b [@wright07] with similar mass, and orbital period of 6.7d and semi-major axis of 0.069AU. At this orbital distance the [*a priori*]{} probability of such a planet transiting its star is about 10%. Thus “super-massive” planets should sometimes be found transiting their parent stars. We report here the detection of the first such TEP, and our determination of its mass and radius. This is also the longest period TEP, and the first one to exhibit highly eccentric orbit.
Observations and Analysis {#sec:anal}
=========================
Detection of the transit in the HATNet data {#sec:dete}
-------------------------------------------
is an F8 star with visual magnitude 8.7 and Hipparcos parallax $7.39\pm0.88$mas [@perryman97]. It was initially identified as a transit candidate in our internally labeled field G193 in the data obtained by HATNet’s[^2] [@bakos02; @bakos04] HAT-6 telescope at the Fred Lawrence Whipple Observatory (FLWO) of the Smithsonian Astrophysical Observatory (SAO). The detection of a $\sim$5mmag transit with a 5.63d period in the [light curve]{} consisting of $\sim$7000 data-points (with a 5.5min cadence) was marginal. Fortunately the star was in the overlapping corner with another field (G192) that has been jointly observed by HATNet’s HAT-9 telescope at the Submillimeter Array (SMA) site atop Mauna Kea, Hawaii, and by the Wise HAT telescope [WHAT, Wise-Observatory, Israel; @shporer05], for an extended period that yielded $\sim$6700 and $\sim$3900 additional data-points, respectively. The transit was independently detected and confirmed with these data-sets. By chance the candidate is in yet another joint field (G149) of HATNet (HAT-7 at FLWO) and WHAT, contributing $\sim$6200 and $\sim$2200 additional data points, respectively. Altogether this resulted in a [light curve]{} with exceptional time-coverage (570 days), an unprecedented number of data-points (26461 measurements at 5.5min cadence), and an rms of 5mmag. It is noteworthy that the network coverage by WHAT (longitude 35 E), HATNet at FLWO (111 W) and HATNet at Hawaii (155 W) played an important role in detecting such a long period and shallow transit. Data were reduced using astrometry from @pal06, and with a highly fine-tuned aperture photometry. We applied our external parameter decorrelation (EPD) technique on the [light curves]{}, whereby deviations from the median were cross-correlated with a number of “external parameters”, such as the $X$ and $Y$ sub-pixel position, hour-angle, and zenith distance. We have also applied the Trend Filtering Algorithm [TFA; @kovacs05 hereafter ] along with the Box Least Squares [BLS; @kovacs02] transit-search algorithm in our analysis. TFA and BLS were combined in signal-reconstruction mode, assuming general signal shape, as described in . The detection of this relatively shallow transit is a good demonstration of the strengths of TFA. The upper panel of [Fig. \[fig:lc\]]{} shows the unbinned [light curve]{} with all 26400 data points, whereas the middle panel displays the transit binned to 1/2000 of the period (4 minutes). We note that due to the large amount of data, the binned [light curve]{} is of similar precision as a single-transit observation by a 1m-class telescope.
After several failed attempts (due to bad weather and instrumental failure) to carry out high-precision photometric follow-up observations from FLWO, Wise Observatory, Konkoly Observatory, and the Clay Center (Boston), we finally succeeded in observing a full transit using the KeplerCam detector on the telescope [see @holman07] on UT 2007 April 22. The Sloan $z$-band light curve is shown in the lower panel of [Fig. \[fig:lc\]]{}. From the combined HATNet and KeplerCam photometry, spanning a baseline of 839 days, we derive a period of $5.63341\pm0.00013$d and an epoch of mid-transit of $T_{mid}=2,\!454,\!212.8559 \pm 0.0007$d (HJD). From the data alone (and the analytic [light curve]{} fit as described later), the length of transit is $0.177\pm0.002$d (4 hours, 15 minutes), the length of ingress is $0.012\pm0.002$d (17.5 minutes), and the depth (at the middle of the transit) is 0.0052mag.
Early spectroscopy follow-up
----------------------------
Initial follow-up observations were made with the CfA Digital Speedometer [DS; @latham92] in order to characterize the host star and to reject obvious astrophysical false-positive scenarios that mimic planetary transits. These observations yielded values of ${\ensuremath{T_{\rm eff}}}=6250\,K$, ${\ensuremath{\log{g}}}=4.0$ and ${\ensuremath{v \sin{i}}}=22\,{\ensuremath{\rm km\,s^{-1}}}$, corresponding to a moderately-rotating main sequence F star. The radial velocity (RV) measurements showed an rms residual of $\sim$0.82[$\rm km\,s^{-1}$]{}, slightly larger than the nominal DS precision for a star with this rotation, and suggested that they may be variable. With a few dozen additional DS observations, it was found that the RV appeared periodic with $P\approx5.63$d, semi-amplitude $\sim$1[$\rm km\,s^{-1}$]{}, and phasing in agreement with predictions from the HATNet+WHAT light curve. This gave strong evidence that there really was an RV signal resulting from Keplerian motion, although the precision was insufficient to establish the orbit with confidence. Altogether we collected 53 individual spectra spanning a time-base of more than a year ().
[lrrr]{} $53981.7775$ & $-556.0$ & $8.4$ & Keck\
$53982.8717$ & $-864.1$ & $8.5$ & Keck\
$53983.8148$ & $-62.9$ & $8.8$ & Keck\
$53984.8950$ & $280.6$ & $8.6$ & Keck\
$54023.6915$ & $157.8$ & $9.9$ & Keck\
$54186.9982$ & $120.2$ & $5.5$ & Keck\
$54187.1041$ & $104.6$ & $5.7$ & Keck\
$54187.1599$ & $130.1$ & $5.3$ & Keck\
$54188.0169$ & $168.5$ & $5.3$ & Keck\
$54188.1596$ & $198.2$ & $5.5$ & Keck\
$54189.0104$ & $68.9$ & $5.7$ & Keck\
$54189.0889$ & $69.7$ & $6.2$ & Keck\
$54189.1577$ & $25.2$ & $6.1$ & Keck\
$54168.9679$ & $-152.7$ & $42.1$ & Lick\
$54169.9519$ & $542.4$ & $41.3$ & Lick\
$54170.8619$ & $556.8$ & $42.6$ & Lick\
$54171.0365$ & $719.1$ & $49.6$ & Lick\
$54218.8081$ & $-1165.2$ & $88.3$ & Lick\
$54218.9856$ & $-1492.6$ & $90.8$ & Lick\
$54219.9373$ & $-28.2$ & $43.9$ & Lick\
$54219.9600$ & $-14.8$ & $43.9$ & Lick\
$54220.9641$ & $451.6$ & $38.4$ & Lick\
$54220.9934$ & $590.7$ & $37.1$ & Lick\
High-precision spectroscopy follow-up
-------------------------------------
In order to confirm or refute the planetary nature of the transiting object, we pursued follow-up observations with the HIRES instrument [@vogt94] on the W. M. Keck telescope and with the Hamilton Echelle spectrograph at the Lick Observatory [@vogt87]. The spectrometer slit used at Keck is $0\farcs 86$, yielding a resolving power of about $55,\!000$ with a spectral coverage between about 3200 and 8800Å. The Hamilton Echelle spectrograph at Lick has a similar resolution of about $50,\!000$. These spectra were used to i) more fully characterize the stellar properties of the system, ii) to obtain a radial velocity orbit, and to iii) check for spectral line bisector variations that may be indicative of a blend. We gathered 13 spectra at Keck (plus an iodine-free template) spanning 207 days, and 10 spectra at Lick (plus template) spanning 50 days. The radial velocities measured from these spectra are shown in , along with those from the CfA DS.
Stellar parameters {#sec:stelpar}
==================
A spectral synthesis modeling of the iodine-free Keck template spectrum was carried out using the SME software [@valenti96], with the wavelength ranges and atomic line data described by @valenti05. Results are shown in . The values obtained for the effective temperature ([$T_{\rm eff}$]{}), surface gravity ([$\log{g}$]{}), and projected rotational velocity (${\ensuremath{v \sin{i}}}$) are consistent with those found from the CfA DS spectra. As a check on [$T_{\rm eff}$]{}, we collected all available photometry for in the Johnson, Cousins, 2MASS, and Tycho systems, and applied a number of color-temperature calibrations [@ramirez05; @masana06; @casagrande06] using 7 different color indices. These resulted in an average temperature of $\sim$6400$ \pm
100$K, somewhat higher than the spectroscopic value but consistent within the errors.
Based on the Hipparcos parallax ($\pi = 7.39\pm0.88$mas), the apparent magnitude $V = 8.71 \pm 0.01$ [@droege06], the SME temperature, and a bolometric correction of $BC_V = -0.011 \pm 0.011$mag [@flower96], application of the Stefan-Boltzmann law yields a stellar radius of ${\ensuremath{R_\star}}= 1.84\pm 0.24\,{\ensuremath{R_\sun}}$.
[lll]{}
A more sophisticated approach to determine the stellar parameters uses stellar evolution models along with the observational constraints from spectroscopy. For this we used the Y$^2$ models by @yi01 and @demarque04, and explored a wide range of ages to find all models consistent with $T_{\rm eff}$, $M_V$, and \[Fe/H\] within the observational errors. Here $M_V = 3.05 \pm 0.26$ is the absolute visual magnitude, as calculated from $V$ and the Hipparcos parallax. This resulted in a mass and radius for the star of ${\ensuremath{M_\star}}= 1.42\pm^{0.10}_{0.12}\,{\ensuremath{M_\sun}}$ and ${\ensuremath{R_\star}}= 1.85\pm^{0.31}_{0.28}\,{\ensuremath{R_\sun}}$, and a best-fit age of $2.7^{+1.4}_{-0.6}$Gyr. Other methods that rely on the Hipparcos parallax, such as the Padova[^3] stellar model grids [@girardi02], consistently yielded a stellar mass of $\sim1.4\,{\ensuremath{M_\sun}}$ and stellar radius $\sim1.8\,{\ensuremath{R_\sun}}$.
If we do [*not*]{} rely on the Hipparcos parallax, and use [$\log{g}$]{} as a proxy for luminosity (instead of $M_V$), then the Y$^2$ stellar evolution models yield a smaller stellar mass of ${\ensuremath{M_\star}}= 1.29\pm^{0.17}_{0.12}\,{\ensuremath{M_\sun}}$, radius of ${\ensuremath{R_\star}}= 1.46\pm^{0.36}_{0.27}\,{\ensuremath{R_\sun}}$, and best fit age of $2.6^{+0.8}_{-2.5}$Gyr. The surface gravity is a sensitive measure of the degree of evolution of the star, as is luminosity, and therefore has a very strong influence on the radius. However, $\log g$ is a notoriously difficult quantity to measure spectroscopically and is often strongly correlated with other spectroscopic parameters.
It has been pointed out by @sozzetti07 that the normalized separation $a/{\ensuremath{R_\star}}$ can provide a much better constraint for stellar parameter determination than [$\log{g}$]{}. The $a/{\ensuremath{R_\star}}$ quantity can be determined directly from the photometric observations, without additional assumptions, and it is related to the density of the central star. As discussed later in , an analytic fit to the [light curve]{}, taking into account an eccentric orbit, yielded $a/{\ensuremath{R_\star}}= 9.77^{+1.10}_{-0.02}$. Using this as a constraint, along with [$T_{\rm eff}$]{} and [\[Fe/H\]]{}, we obtained ${\ensuremath{M_\star}}= 1.30\pm^{0.06}_{0.10}\,{\ensuremath{M_\sun}}$, ${\ensuremath{R_\star}}= 1.47\pm^{0.04}_{0.17}\,{\ensuremath{R_\sun}}$ and age of $2.6^{+0.8}_{-1.4}$Gyr. The ${\ensuremath{\log{g}}}= 4.214\pm^{0.085}_{0.015}$ derived this way is consistent with former value from SME.
As seen from the above discussion, there is an inconsistency between stellar parameters depending on whether the Hipparcos parallax is employed or not. Methods relying on the parallax (Stefan-Boltzmann law, stellar evolution models with $M_V$ constraint, etc) prefer a larger mass and radius ($\sim1.4\,{\ensuremath{M_\sun}}$, $\sim1.8\,{\ensuremath{R_\sun}}$, respectively), whereas methods that do not rely on the parallax (stellar evolution models with [$\log{g}$]{} or $a/{\ensuremath{R_\star}}$ constraint) point to smaller mass and radius ($\sim1.3\,{\ensuremath{M_\sun}}$, $\sim1.46\,{\ensuremath{R_\sun}}$, respectively). We have chosen to rely on the $a/{\ensuremath{R_\star}}$ method, which yields considerably smaller uncertainties and a calculated transit duration that matches the observations. Additionally, it implies an angular diameter for the star ($\phi = 0.127^{+0.021}_{-0.014}$ mas) that is in agreement with the more direct estimate of $\phi = 0.117 \pm 0.001$ mas from the near-infrared surface-brightness relation by @kervella04. The later estimate depends only on the measured $V-K_s$ color and apparent $K_s$ magnitude (ignoring extinction) from 2MASS [@skrutskie06], properly converted to the homogenized Bessell & Brett system for this application [following @carpenter01]. We note that our results from the $a/{\ensuremath{R_\star}}$ method imply a somewhat smaller distance to than the one based on the Hipparcos parallax. The final adopted stellar parameters are listed in .
Stellar jitter
--------------
Stars with significant rotation are known to exhibit excess scatter (“jitter”) in their radial velocities [e.g., @wright05 and references therein], due to enhanced chromospheric activity and the associated surface inhomogeneities (spottedness). This jitter is in addition to the internal errors in the measured velocities, and could potentially be significant in our case. We note that after pre-whitening the [light curve]{} with the transit component, we found no significant sinusoidal signal above 0.3mmag amplitude. From this we conclude that there is no very significant spot activity on the star (in the observed 500 day window). In order to estimate the level of chromospheric activity in the star, we have derived an activity index from the H and K lines in our Keck spectra of ${\ensuremath{\log{\ensuremath{R^{\prime}_{HK}}}}}=
-4.72\pm0.05$. For this value the calibration by [@wright05] predicts velocity jitter ranging from 8 to 16[$\rm m\,s^{-1}$]{}. An earlier calibration by @saar98, parametrized in terms of the projected rotational velocity, predicts a jitter level of up to 50[$\rm m\,s^{-1}$]{} for our measured ${\ensuremath{v \sin{i}}}$ of $20\,{\ensuremath{\rm km\,s^{-1}}}$. A different calibration by the same authors in terms of [$R^{\prime}_{HK}$]{} gives 20[$\rm m\,s^{-1}$]{}. An additional way to estimate the jitter is to compare to stars of the Lick Planet Search program [@cumming99] that have similar properties ($0.4<B-V<0.5$, ${\ensuremath{v \sin{i}}}> 15\,{\ensuremath{\rm km\,s^{-1}}}$). There are four such stars (J. Johnson, private communication), and their average jitter is $45\,{\ensuremath{\rm m\,s^{-1}}}$. A more direct measure for the particular case of may be obtained from the multiple exposures we collected during a 3-night Keck run in 2007 March. Ignoring the small velocity variations due to orbital motion during any given night, the overall scatter of these 8 exposures relative to the nightly means is $\sim$20[$\rm m\,s^{-1}$]{}. This may be taken as an estimate of the jitter on short timescales, although it could be somewhat larger over the entire span of our observations. Altogether, it is reasonable to expect the jitter to be at least 10[$\rm m\,s^{-1}$]{}, and possibly around 30–50[$\rm m\,s^{-1}$]{} for this star.
Spectroscopic orbital solution
==============================
We have three velocity data sets available for analysis: 13 relative radial velocity measurements from Keck, 10 from Lick, and 53 measurements from the CfA DS, which are nominally on an absolute scale (). Given the potential effect of stellar jitter, we performed weighted Keplerian orbital solutions for a range of jitter values from 10 to 80[$\rm m\,s^{-1}$]{} with 10[$\rm m\,s^{-1}$]{} steps. These jitter values were added in quadrature to all individual internal errors. We performed separate fits for the star orbited by a single planet, both with and without the CfA DS measurements, since these have errors ($\sim$600[$\rm m\,s^{-1}$]{}) significantly larger than Keck (5–9[$\rm m\,s^{-1}$]{}) or Lick (40–90[$\rm m\,s^{-1}$]{}). In all of these solutions we held the period and transit epoch fixed at the photometric values given earlier. The parameters adjusted are the velocity semi-amplitude $K$, the eccentricity $e$, the longitude of periastron $\omega$, the center-of-mass velocity for the Keck relative velocities $\gamma$, and offsets $\Delta v_{KL}$ between Keck and Lick and $\Delta v_{KC}$ between Keck and CfA DS. The fitted parameters were found to be fairly insensitive to the level of jitter assumed. However, only for a jitter of $\sim$60[$\rm m\,s^{-1}$]{} (or $\sim$70[$\rm m\,s^{-1}$]{}when the CfA DS data are included) did the $\chi^2$ approach values expected from the number of degrees of freedom. There are thus two possible conclusions: if we accept that has stellar jitter at the 60[$\rm m\,s^{-1}$]{} level, then a single-planet solution such as ours adequately describes our observations. If, on the other hand, the true jitter is much smaller ($\lesssim 20\,{\ensuremath{\rm m\,s^{-1}}}$), then the extra scatter requires further explanation (see below). Our adopted orbital parameters for the simplest single-planet Keplerian solution are based only on the more precise Keck and Lick data, and assume the jitter is 60[$\rm m\,s^{-1}$]{} (). The orbital fit is shown graphically in the upper panel of [Fig. \[fig:rv\]]{}. In this figure, the zero-point of phase is chosen to occur at the epoch of mid-transit, $T_{mid}=2,\!454,\!212.8559$ (HJD). The most significant results are the large eccentricity ($e =
0.520 \pm 0.010$), and the large velocity semi-amplitude ($K=1011\pm38\,{\ensuremath{\rm m\,s^{-1}}}$, indicating a very massive companion). As we show in the next section (), the companion is a planet, i.e. , which we hereafter refer to as HAT-P-2b.
As a consistency check we also fitted the orbits by fixing only the period, and leaving the transit epoch as a free parameter. We found that for all values of the stellar jitter the predicted time of transit as derived from the RV fit was consistent with the photometric ephemeris within the uncertainties. We also found that in these fits the orbital parameters were insensitive to the level of jitter and to whether or not the CfA DS data were included. The eccentricity values ranged from 0.51 to 0.53.
Solutions involving two planets
-------------------------------
If we assume that the true stellar jitter is small, then the excess scatter in the RV fit could be explained by a third body in the system, i.e., a hypothetical HAT-P-2c. In addition, such a body could provide a natural dynamical explanation for the large eccentricity of HAT-P-2b at this relatively short period orbit. Preliminary two-planet orbital fits using all the data yielded solutions only significant at the 2-sigma level, not compelling enough to consider as evidence for such a configuration. Additional RV measurements are needed to firmly establish or refute the existence of HAT-P-2c.
We also exploited the fact that the HATNet [light curve]{} has a unique time coverage and precision, and searched for signs of a second transit that might be due to another orbiting body around the host star. Successive box-prewhitening based on the BLS spectrum and assuming trapezoidal-shape transits revealed no secondary transit deeper than the 0.1% level and period $\lesssim 10$ days.
Excluding blend scenarios {#sec:blend}
=========================
As an initial test to explore the possibility that the photometric signal we detect is a false positive (blend) due to contamination from an unresolved eclipsing binary, we modeled the light curve assuming there are three coeval stars in the system, as described by @torres04. We were indeed able to reproduce the observed light curve with a configuration in which the brighter object is accompanied by a slightly smaller F star which is in turn being eclipsed by a late-type M dwarf. However, the predicted relative brightness of the two brighter objects at optical wavelengths would be $\sim0.58$, and this would have been easily detected in our spectra. This configuration can thus be ruled out.
The reality of the velocity variations was tested by carefully examining the spectral line bisectors of the star in our more numerous Keck spectra. If the velocity changes measured are due only to distortions in the line profiles arising from contamination of the spectrum by the presence of a binary with a period of 5.63 days, we would expect the bisector spans (which measure line asymmetry) to vary with this period and with an amplitude similar to the velocities [see, e.g., @queloz01; @torres05]. The bisector spans were computed from the cross-correlation function averaged over 15 spectral orders blueward of 5000Å and unaffected by the iodine lines, which is representative of the average spectral line profile of the star. The cross correlations were performed against a synthetic spectrum matching the effective temperature, surface gravity, and rotational broadening of the star as determined from the SME analysis. As shown in [Fig. \[fig:rv\]]{}, while the measured velocities exhibit significant variation as a function of phase (upper panel), the bisector spans are essentially constant within the errors (lower panel). Therefore, this analysis rules out a blend scenario, and confirms that the orbiting body is indeed a planet.
Planetary parameters {#sec:planpar}
====================
For a precise determination of the physical properties of HAT-P-2b we have modeled the Sloan $z$-band photometric data shown in [Fig. \[fig:lc\]]{}. The model is an eccentric Keplerian orbit of a star and planet, thus accounting for the nonuniform speed of the planet and the reflex motion of the star. Outside of transits, the model flux is unity. During transits, the model flux is computed using the formalism of @mandel02, which provides an analytic approximation of the flux of a limb-darkened star that is being eclipsed. The free parameters were the mid-transit time $T_{mid}$, the radius ratio $R_p/R_\star$, the orbital inclination $i$, and the scale parameter $a/{\ensuremath{R_\star}}$, where $a$ is the semimajor axis of the relative orbit. The latter parameter is determined by the time scales of the transit (the total duration and the partial-transit duration), and is related to the mean density of the star (see ). The orbital period, eccentricity, and argument of pericenter were fixed at the values determined previously by fitting the radial velocity data. The limb darkening law was assumed to be quadratic, with coefficients taken from @claret04.
To solve for the parameters and their uncertainties, we used a Markov Chain Monte Carlo algorithm that has been used extensively for modeling other transits [see, e.g. @winn07; @holman07]. This algorithm determines the [*a posteriori*]{} probability distribution for each parameter, assuming independent (“white”) Gaussian noise in the photometric data. However, we found that there are indeed correlated errors. Following Gillon et al. (2006), we estimated the red noise $\sigma_r$ via the equation $$\sigma_r^2 = \frac{\sigma_N^2 - \sigma_1^2/N}{1-1/N},$$ where $\sigma_1$ is the standard deviation of the out-of-transit flux of the original (unbinned) light curve, $\sigma_N$ is the standard deviation of the light curve after binning into groups of $N$ data points, and $N=40$ corresponds to a binning duration of 20 minutes, which is the ingress/egress time scale that is critical for parameter estimation. With white noise only, $\sigma_N=\sigma_1/\sqrt{N}$ and $\sigma_r=0$. We added $\sigma_r$ in quadrature to the error bar of each point, effectively inflating the error bars by a factor of 1.25.
The result for the radius ratio is ${\ensuremath{R_{p}}}/{\ensuremath{R_\star}}=0.0684\pm0.0009$, and for the scale parameter $a/{\ensuremath{R_\star}}= 9.77^{+1.10}_{-0.02}$. The [*a posteriori*]{} distribution for $a/{\ensuremath{R_\star}}$ is very asymmetric because the transit is consistent with being equatorial: $i>84\fdg 6$ with 95% confidence. We confirmed that these uncertainties are dominated by the photometric errors, rather than by the covariances with the orbital parameters $e$, $\omega$, and $P$, and hence we were justified in fixing those orbital parameters at constant values. Based on the inclination, the mass of the star () and the orbital parameters (), the planet mass is then $9.04 \pm 0.50\,{\ensuremath{M_{\rm J}}}$. Based on the radius of the star () and the above ${\ensuremath{R_{p}}}/{\ensuremath{R_\star}}$ determination, the radius of the planet is ${\ensuremath{R_{p}}}=
0.982\pm^{0.038}_{0.105}\,{\ensuremath{R_{\rm J}}}$. These properties are summarized in .
[lr]{} Period (d) & $5.63341\pm0.00013$\
$T_{mid}$ (HJD) & $2,\!454,\!212.8559\pm0.0007$\
Transit duration (day) & $0.177\pm 0.002$\
Ingress duration (day) & $0.012\pm 0.002$\
Stellar jitter ([$\rm m\,s^{-1}$]{}) & 60\
$\gamma$ ([$\rm m\,s^{-1}$]{}) & $-278\pm20$\
$K$ ([$\rm m\,s^{-1}$]{}) & $1011\pm38$\
$\omega$ (deg) & $179.3\pm3.6$\
$e$ & $0.520\pm0.010$\
$T_{peri}$ (HJD) & $2,\!454,\!213.369\pm0.041$\
$\Delta v_{KL}$ ([$\rm m\,s^{-1}$]{}) & $-380\pm35$\
$\Delta v_{KC}$ ([$\rm km\,s^{-1}$]{}) & $19.827\pm0.087$\
$f(M)$ ([$M_\sun$]{}) & $(376 \pm 42) \times 10^{-9}$\
${\ensuremath{M_{p}}}\sin i$ ([$M_{\rm J}$]{}) & $7.56 \pm 0.28 ([{\ensuremath{M_\star}}+ {\ensuremath{M_{p}}}]/M_{\sun}])^{2/3}$\
${\ensuremath{a_\star}}\sin i$ (km) & $(0.0669 \pm 0.0025) \times 10^6$\
$a_{\rm rel}$ (AU) & $0.0677\pm0.0014$\
[$i_{p}$]{}(deg) & $>84.6\arcdeg$ (95% confidence)\
[$M_{p}$]{}([$M_{\rm J}$]{}) & $9.04\pm0.50$\
[$R_{p}$]{}([$R_{\rm J}$]{}) & $0.982\pm^{0.038}_{0.105}$\
[$\rho_{p}$]{}([$\rm g\,cm^{-3}$]{}) & $11.9\pm^{4.8}_{1.6}$\
[$g_{p}$]{}($m\,s^{-2}$) & $227\pm^{44}_{16}$\
Discussion {#sec:disc}
==========
In comparison with the other 18 previously known transiting exoplanets, HAT-P-2b is quite remarkable ([Fig. \[fig:exomr\]]{}, [Fig. \[fig:Pg\]]{}). Its mass of $9.04\pm0.50\,{\ensuremath{M_{\rm J}}}$ is $\sim$5 times greater than any of these 18 other exoplanets. Its mean density $\rho = 11.9\pm^{4.8}_{1.6}\,{\ensuremath{\rm g\,cm^{-3}}}$ is $\sim$9 times that of the densest known exoplanet (OGLE-TR-113b, $\rho
= 1.35\,{\ensuremath{\rm g\,cm^{-3}}}$) and indeed greater than that of the rocky planets of the Solar System ($\rho$ = 5.5[$\rm g\,cm^{-3}$]{}). Its surface gravity of $227\pm^{44}_{16}\,{\ensuremath{\rm m\,s^{-2}}}$ is 7 times that of any of the previously known TEPs, and 30 times that of HAT-P-1b ([Fig. \[fig:Pg\]]{}).
We may compare the mass and radius for HAT-P-2b with evolutionary models, including irradiation, as recently presented by @fortney06 (hereafter ). Given the inferred stellar luminosity (), and the time-integral of the insolation over an entire period (taking into account the orbital parameters, notably $e$ and $a_{rel}$), the equivalent semi-major axis $a_{eq}$ for the same amount of irradiation if the central star were solar is 0.036 AU. At that separation, find for a pure hydrogen/helium planet of mass 9[$M_{\rm J}$]{} and age of 4.5Gyr a planetary radius about 1.097[$R_{\rm J}$]{}. A 100[$M_\earth$]{} core has a negligible effect on the radius (yielding 1.068[$R_{\rm J}$]{}), which is not surprising, since the mass of such a core is only a few percent of the total mass. For younger ages of 1Gyr and 0.3Gyr the radii are larger: 1.159[$R_{\rm J}$]{} and 1.22[$R_{\rm J}$]{} for coreless models, respectively. Our observed radius of $0.982\,{\ensuremath{R_{\rm J}}}$ is smaller than any of the above values (4.5, 1, 0.3Gyr, with or without 100[$M_\earth$]{} core). Since the $1\sigma$ positive error-bar on our radius determination is 0.038[$R_{\rm J}$]{}, the inconsistency is only marginal. Nevertheless, the observed radius prefers either larger age or bigger core-size, or both. Given the age of the host star (2.6Gyr, ) the larger age is an unlikely explanation. The required core-size for this mass and radius according to would be 300[$M_\earth$]{}, which amount of icy and rocky material may be hard to account for.
[Fig. \[fig:exomr\]]{} also shows a theoretical mass-radius relation for objects ranging from gas giant planets to stars [@baraffe98; @baraffe03]. Note that HAT-P-2b falls on the relation connecting giant planets to brown dwarfs to stars. It thus appears to be intermediate in its properties between Jupiter-like planets and more massive objects like brown dwarfs or even low mass stars. According to theories, stars with mass $\gtrsim 0.2\,{\ensuremath{M_\sun}}$ have a core, where internal pressure is dominated by classical gas (ions and electrons), and the $R \propto M$ radius–mass relation holds in hydrostatic equilibrium [for a review and details on the following relations see e.g. @chabrier00]. Below $\sim 0.075\,{\ensuremath{M_\sun}}\ (80\,{\ensuremath{M_{\rm J}}})$ mass, however, the equation of state in the core becomes dominated by degenerate electron gas ($R \propto
M^{-1/3}$ for full degeneracy), yielding an expected minimum in the mass–radius relationship (around 73[$M_{\rm J}$]{}). Below this mass, the partial degeneracy of the object and the classical ($R\propto M^{1/3}$) Coulomb pressure together yield an almost constant radius ($R\propto
M^{-1/8}$). HAT-P-2b is a demonstration of this well known phenomenon. (The approximate relation breaks below $M\sim4\,{\ensuremath{M_{\rm J}}}$, where the degeneracy saturates, and a classical mass–radius behaviour is recovered).
Compared to the other 18 known transiting planets, HAT-P-2b is also unique in having an orbit with remarkably high eccentricity. The primary question is how such an eccentricity was created in the first place. One possible explanation could be that the planet was scattered inward from a larger orbit, acquiring a high eccentricity in the process [@ford07; @chatterjee07]. If so, then the scattering event might have caused its new orbital plane to be inclined relative to the plane of the original disk, and hence out of the equatorial plane of the parent star [e.g. @fabrycky07]. This angle between these two planes should be readily measurable from the Rossiter-McLaughlin effect [@winn05]. Indeed, the star is an ideal subject for studying this effect, because its rapid rotation should lead to a relatively large Rossiter-McLaughlin signal.
There are a number of other interesting issues related to the high eccentricity of HAT-P-2b. During its 5.63 day orbit, the insolation reaching the planet’s surface varies by a factor of 9. Assuming an albedo of 0.1 [@rowe06] and complete redistribution of insolation energy over the surface of the planet, the equilibrium temperature varies from about 2150K at periastron to 1240K at apastron. This would have a major influence on atmospheric dynamics and photochemistry.
It is interesting to compare the properties of the HAT-P-2 system with the $\tau$ Boo system, which – as already noted – harbors a close-in planet with minimum mass ${\ensuremath{{\ensuremath{M_{p}}}\sin i}}= 3.9\,{\ensuremath{M_{\rm J}}}$. Similarities of the two parent stars include the nearly identical masses, effective temperature, and the rapid rotation, although $\tau$ Boo, with ${[Fe/H]}= +0.28$, is somewhat more metal rich than , with ${[Fe/H]}=
+0.12$. A striking difference is that, while the orbital eccentricity of HAT-P-2b is 0.5, the eccentricity of $\tau$ Boo b is not measurably different from zero. However, $\tau$ Boo b’s orbital period, 3.3 days, is almost half that of HAT-P-2b. A large fraction of close-in planets with $5<P<10$ days have significant eccentricities ($0.1<e<0.3$) although not as large as HAT-P-2b. For discussion on the eccentricity distribution see @juric07. As circularization timescales are thought to be very steep functions of the orbital semi-major axis [@terquem98], one could then argue that HAT-P-2b’s large value of $e$ is due to either the fact that the planet’s orbit is not yet circularized (while $\tau$ Boo b’s instead is), or to the presence of a second planet in the HAT-P-2 system, or to rather different formation/migration scenarios altogether.
, with visual magnitude 8.71, is the fourth brightest among the known stars harboring transiting planets. Therefore it has special interest because of the possibilities for followup with large space or ground-based telescopes.
Operation of the HATNet project is funded in part by NASA grant NNG04GN74G. Work by GÁB was supported by NASA through Hubble Fellowship Grant HST-HF-01170.01-A. GK wishes to thank support from Hungarian Scientific Research Foundation (OTKA) grant K-60750. We acknowledge partial support from the Kepler Mission under NASA Cooperative Agreement NCC2-1390 (DWL, PI). GT acknowledges partial support from NASA Origins grant NNG04LG89G. TM thanks the Israel Science Foundation for a support through grant no. 03/233. The Keck Observatory was made possible by the generous financial support of the W. M. Keck Foundation. DAF is a Cottrell Science Scholar of Research Corporation, and acknowledges support from NASA grant NNG05G164G. We would like to thank Joel Hartman (CfA), Gil Esquerdo (CfA), Ron Dantowitz and Marek Kozubal (Clay Center) for their efforts to observe HAT-P-2b in transit, and Howard Isaacson (SFSU) for obtaining spectra at Lick Observatory. We wish to thank Amit Moran for his help in the observations with the Wise HAT telescope. We owe special thanks to the directors and staff of FLWO, SMA and Wise Observatory for supporting the operation of HATNet and WHAT. We would also like to thank the anonymous referee for the useful suggestions that improved this paper.
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[^1]: Extrasolar Planets Encyclopedia: http://exoplanet.eu
[^2]: http://www.hatnet.hu
[^3]: http://pleiadi.pd.astro.it/
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