# A Hall of Statistical Mirrors

Gabriel Khan  and  Jun Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The primary focus of information geometry are _statistical manifolds_ , which
are parametrized families of probability measures, induced with the Fisher-Rao
metric and a pair of torsion-free conjugate connections. In recent work
[ZK20], the authors considered parametrized probability distributions as
_partially-flat statistical manifolds admitting torsion_ and showed that there
is a complex to symplectic duality on the tangent bundles of such manifolds,
based on the dualistic geometry of the underlying manifold.

In this paper, we explore this correspondence further in the context of
Hessian manifolds, in which case the conjugate connections are both curvature-
and torsion-free, and the associated dual pair of spaces are Kähler manifolds.
We focus on several key examples and their geometric features. In particular,
we show that the moduli space of univariate normal distributions gives rise to
a correspondence between the Siegel half-space and the Siegel-Jacobi space,
which are spaces that appear in the context of automorphic forms.

##  1\. Introduction

In complex and symplectic geometry, mirror symmetry is a duality between
Calabi-Yau manifolds, in which two distinct manifolds have closely related
geometry. This correspondence has played an important role in string theory as
well enumerative algebraic geometry (see [CK99, Mor97] for more details).
Mirror symmetry provides a correspondence between the geometry of two
different Calabi-Yau manifolds $\mathbb{M}$ and $\mathbb{W}$. Such manifolds
are generally topologically and geometrically distinct, so this correspondence
is not induced by a mapping between the manifolds. Instead, it is more
appropriate to think that the spaces appear in pairs where it is possible to
understand some aspects of the complex geometry of the primal space in terms
of the symplectic geometry of the dual manifold.

At present, mirror symmetry is a somewhat mysterious phenomena; given a
Calabi-Yau manifold it is not clear how to construct its mirror. However, it
seems to be deeply linked with T-duality,111The notion of T-duality appears
both in mathematics as well as physics, where two seemingly different physical
systems turn out to be equivalent. In this paper, we adopt the mathematical
definition of T-duality given in [Leu00, FP19b]. which is a duality between
semi-flat Kähler manifolds induced by inverting the length-scale of the
fibers. In particular, the SYZ conjecture states that away from a singular
locus, the mirror manifold can be constructed using T-duality [SSTZ96].

In this paper, we study a related duality in the context of the tangent
bundles of statistical manifolds.222Strictly speaking, non-Kähler statistical
mirror symmetry is defined for partially-flat statistical manifolds admitting
torsion, rather than traditional statistical manifolds, which are torsion-free
(see Section 3.2 for details). Due to the similarities to the semi-flat case
of Calabi-Yau manifolds, we call this correspondence statistical mirror
symmetry [ZK20]. From a topological and symplectic perspective, this
correspondence is often very simple. However, from the perspective of complex
geometry, rich phenomena arise.

The goal of this paper is to explore the geometric properties of statistical
mirror symmetry in more detail. We will focus primarily on examples whose
underlying Riemannian manifold is hyperbolic and where the mirror pair are
Kähler manifolds, and use these examples to motivate the more general results.

###  1.1. Overview of the paper

In Section 2, we discuss the moduli space of normal distributions (i.e.,
Gaussian distributions) and use this example to introduce statistical mirror
symmetry. This induces a duality between Siegel half-space with its metric of
constant holomorphic sectional curvature and the Siegel-Jacobi space with a
$Sp(2,\mathbb{R})$-invariant metric. We study the geometry of these two spaces
in depth, and use them to highlight several important geometric features that
differ from Calabi-Yau manifolds. Furthermore, we show that both of these
spaces are Kähler-Ricci solitons which remain coupled for all time under
Kähler-Ricci flow. Using this example as a guide, we find two curvature
properties which are preserved by conjugate flow (i.e, the dual to Kähler-
Ricci flow). Finally, we make some speculative remarks the relationship
between this duality and the automorphic forms on these two spaces. In
particular, we propose that this correspondence may give rise to the so-called
Saito-Kurokawa lift, which takes elliptic modular forms to Siegel modular
forms of degree 2.

In Section 3, we provide a more general definition for statistical mirror
symmetry. In this section, we describe how this construction is related to the
duality of semi-flat Calabi-Yau manifolds. We also discuss a non-Kähler
generalization of this concept, which is defined on the tangent bundle of an
affine Riemannian manifold.

In Section 4, we study other examples of statistical mirror pairs whose
underlying statistical manifold has constant negative sectional curvature.
These examples show several important principles about statistical mirror
symmetry. For instance, by explicitly constructing a second example of a
mirror pair (induced from the moduli of negative trinomial distributions)
where one of the metrics has constant holomorphic sectional curvature, we show
that two tube domains can be locally holomorphically isometric, but have non-
isometric dual metrics. These examples also show that the dual of a space of
constant holomorphic sectional curvature need not even have constant scalar
curvature. We will also use this geometric perspective to provide new
statistical results. In particular, by studying the moduli of Inverse Gaussian
distributions, we compute the isometry group of the parameter space.

In Section 5, we discuss statistical mirror pairs whose underlying Hessian
manifolds are Riemannian flat, which are better known as Frobenius manifolds.
These spaces can be used to construct pairs of solutions to the Witten-
Dijkgraaf-Verlinde-Verlinde (WDVV) equations [Kit99], so have applications in
theoretic physics. We use our geometric approach to establish several basic
properties. For instance, by considering the scalar curvature, we show that it
is possible to simplify the WDVV equations to a single equation in dimension
two. Using the Ricci curvature, we show that the full system of equations
simplifies to six equations in dimension three. We also provide several
examples of such metrics.

Finally, in Section 6, we discuss the question of whether a given Riemannian
metric admits a Hessian structure, from both a global and local perspective.
We also discuss the fact that such structures fail to be unique; metrics which
admit one Hessian structure generally admit many others as well.

##  2\. The univariate normal and its corresponding mirrors

To introduce statistical mirror symmetry, it is helpful to do so in the
context of a specific example. In this section, we show that by considering
the family of univariate normal distributions, we obtain a correspondence
between the Siegel-Jacobi space $\mathbb{C}\times\mathbb{H}$ with an
$Sp(2,\mathbb{R})$-invariant metric and the Siegel half-space with the Bergman
metric. We will use this as an archetypal case which motivates the more
general definition in Section 3. In the interest of clarity, we will not
include long derivations of curvature tensors in this paper. Instead, we have
written a Mathematica notebook which can be used to calculate all of the
relevant quantities [Kha20].

###  2.1. The geometry of the normal family

A univariate Gaussian distribution is a probability distribution $\rho$ of the
form

$\rho(\zeta|\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}}\exp\left(-\frac{(\zeta-\mu)^{2}}{2\sigma^{2}}\right).$

In this expression, $\zeta$ represents the random variable, which takes values
in the sample space $\mathcal{S}=\mathbb{R}$. There are two parameters, $\mu$
and $\sigma$ which correspond to the mean and variance of the distribution,
respectively. We can consider the space of all univariate normal distributions
over the reals as a parametrized family, which is a family of probability
distributions specified by some number of parameters (in this case, $\mu$ and
$\sigma$). Furthermore, we can consider the space of all univariate normal
distributions as a statistical manifold, where the parameters serve as a
global coordinate chart. This gives this space the structure of a smooth
manifold.333In fact, it is a domain in Euclidean space, but it is beneficial
to consider it as a manifold.

For any parametrized family of probability distributions, it is possible to
define an associated Riemannian metric, which is known as the Fisher metric
(or Fisher-Rao metric. Denoting the parameters of the family of probability
density functions as $\\{x\\}_{i=1}^{n}$, the Fisher metric is given by the
expression444Here, the reference measure $d\zeta$ is the usual Lebesgue
measure.

$g_{jk}(x)=\int_{\mathcal{S}}\frac{\partial\log p(\zeta|x)}{\partial
x^{j}}\frac{\partial\log p(\zeta|x)}{\partial x^{k}}p(\zeta|x)\,d\zeta.$

This expression originates in statistics, where it is also known as the Fisher
information. It can be interpreted as the infinitesimal form of the relative
entropy, which provides a non-symmetric notion of the distance (also known as
a divergence) between two mutually absolutely continuous probability measures.
The Fisher metric has found many uses in physics and other areas of science,
but a full discussion of its properties would take us too far from our main
focus, so we refer the interested reader to [AN00] [CSS15].

For the space of normal distributions (which we denote $\mathcal{M}_{\tt
Normal}$), it is possible to compute the Fisher metric explicitly in the
$(\mu,\sigma)$-coordinates. Doing so, we find that

$g=\frac{1}{\sigma^{2}}(d\mu^{2}+2d\sigma^{2}).$

This computation shows that the moduli of normal distributions is a hyperbolic
space.

######  Proposition 1 (Amari [Ama80]).

The statistical manifold of normal distributions $\mathcal{M}_{\tt Normal}$ is
a half-plane with constant negative curvature.

###  2.2. Normal distributions as an exponential family

In order to construct a pair of mirror Kähler manifolds from the space of
normal distributions, we first must reparametrize $\mathcal{M}_{\tt Normal}$
as an exponential family.

######  Definition 1 (Exponential family).

Given a sample space $\mathcal{S}$ with random variable $\zeta$, an
exponential family is a parametrized family of probability distributions whose
probability density functions are of the form

(1)
$\rho(\zeta\,|\,x)=h(\zeta)\exp\left(\sum_{i=1}^{n}x^{i}F_{i}(\zeta)-\Phi(x)\right).$

Here $h:\mathcal{S}\to\mathbb{R}$ is a known function which serves to fix a
base measure on $S$. The parameters are denoted by the
$x=(x^{1},\cdots,x^{n})$ and take values in some domain
$\Omega\subset\mathbb{R}^{n}$. When an exponential family is parametrized in
this way, the $x^{i}$’s are known as the _natural parameters_. The functions
$F=(F_{1},\cdots,F_{n}):\mathcal{S}\to\mathbb{R}^{n}$ are known as the
sufficient statistics. Finally, the function $\Phi:\Omega\to\mathbb{R}$ is
known as the _log-partition function_ , which serves to renormalize the
distribution so that the total mass is one.

In the context of normal distributions, we want to find functions $x^{1}$ and
$x^{2}$ which are functions of $\mu$ and $\sigma$ so that the probability
density function takes the form 1. To this end, we set $F=(F_{1},F_{2})$ to be

(2) $F_{1}(\zeta)=\zeta,\,\,F_{2}(\zeta)=\zeta^{2}$

and

(3) $x^{1}=\frac{\mu}{\sigma^{2}}\,\,\textrm{ and
}\,\,x^{2}=-\frac{1}{2\sigma^{2}}.$

These functions are defined on the domain
$\Omega=\\{(x^{1},x^{2})\in\mathbb{R}^{2}~{}|~{}x^{2}<0\\}$.

Finally, the function $\Phi(x)$ is the following:

$\Phi=-\frac{x^{1}\cdot
x^{1}}{4x^{2}}-\frac{1}{2}\log\left(-\frac{x^{2}}{\pi}\right).$

Exponential families play an essential role throughout this paper, so we
mention some general properties about them now.

######  Observation (Important facts about exponential families, part I).

For any exponential family, the natural parameters have the following
properties.

  1. (1)

The domain $\Omega$ of the natural parameters is a convex subset of
$\mathbb{R}^{n}$.

  2. (2)

When parametrized in terms of the natural parameters $x$, the Fisher metric of
an exponential family is given by the Hessian of the log-partition function
$\Phi$. That is to say, we have

$g_{ij}=\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}\Phi.$

Specializing to $\mathcal{M}_{\tt Normal}$, this means that when we use the
$x$-coordinates, the Fisher metric is given by

$\displaystyle g$ $\displaystyle=$ $\displaystyle\frac{\partial^{2}}{\partial
x^{i}\partial x^{j}}\left(\frac{-x^{1}\cdot
x^{1}}{4x^{2}}-\frac{1}{2}\log\left(-\frac{x^{2}}{\pi}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2x^{2}}\left[\begin{array}[]{cc}-1&\frac{x^{1}}{x^{2}}\\\
\frac{x^{1}}{x^{2}}&\frac{-x^{1}\cdot x^{1}+x^{2}}{x^{2}\cdot x^{2}}\\\
\end{array}\right].$

In some sense, the natural parameters are “preferred coordinates” for an
exponential family,555For geometers, the notion of preferred coordinates might
be somewhat anathema. Conceptually, one can rephrase this definition in terms
of a flat affine connection, in which case these coordinates are those for
which the Christoffel symbols vanish. and give the manifold $\mathcal{M}_{\tt
Normal}$ the structure of a Hessian manifold.666We will later give another
definition for Hessian manifolds that does not use coordinates (Definition
Definition).

######  Definition 2.

A Riemannian manifold $(\mathcal{M},g)$ is a Hessian manifold if

  1. (1)

$\Omega$ admits an atlas of coordinate charts, such that

  2. (2)

the transition maps between charts in this atlas are affine functions, and

  3. (3)

in each coordinate chart, there is a potential function
$\Phi:\Omega\to\mathbb{R}$, such that

$g=\frac{\partial^{2}}{\partial x^{i}\partial x^{j}}\Phi.$

For reasons that will later become apparent, we will denote Hessian manifolds
using the notation $(\mathcal{M},g,D)$, where $D$ indicates the affine
structure.777More precisely, $D$ is a flat affine connection which satisfies a
compatibility condition with $g$ known as _Codazzi coupling_.

####  2.2.1. The dual parametrization

For an exponential family, the natural parameters provide an important
coordinate system. There is also a dual set of coordinates, called the
expectation parameters, which are induced by the sufficient statistics.

######  Observation (Important facts about exponential families, part II).

  1.   2. (1)

Given an exponential family $\rho(\zeta|x)$, the expected value of the
sufficient statistics

$u_{i}=\int_{\mathcal{S}}F_{i}(\zeta)\rho(\zeta|x)\,d\zeta$

also form a coordinate chart for the exponential family.

  3. (2)

The domain $\Omega^{\ast}$ of the expectation parameters $u$ is a convex
subset of $\mathbb{R}^{n}$.

  4. (3)

If we use these expected values as coordinates, the Fisher metric is given by
the Hessian of the Legendre dual $\Phi^{\ast}$ of the log-partition function
$\Phi$. In other words, we have

$g\left(\frac{\partial}{\partial u_{i}},\frac{\partial}{\partial
u_{j}}\right)=\frac{\partial^{2}}{\partial u^{i}\partial u^{j}}\Phi^{\ast},$

where

$\Phi^{\ast}=\sup_{x\in\Omega}\langle x,u\rangle-\Phi(x).$

  5. (4)

The coordinate systems $x$ and $u$ are bi-orthogonal, which means they satisfy
the identity

$g\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
u_{j}}\right)=\delta^{i}_{j}.$

To construct the expectation parameters for normal distributions, we start by
considering the sufficient statistics $F(\zeta)=(\zeta,\zeta^{2})$. To
translate these quantities into coordinates $u=(u_{1},u_{2})$ for the
statistical manifold $\mathcal{M}_{\tt Normal}$, we compute the expected value
of $F$ for a given normal distribution $\rho(\zeta\,|\,x)$:

$u_{i}=\int\rho(\zeta\,|\,x)F_{i}(\zeta)d\zeta.$

Computing these explicitly in terms of the mean and variance, we find that

$u_{1}=\mu\qquad u_{2}=\mu^{2}+\sigma^{2}.$

As such, each normal distribution corresponds to a unique point in $u$-space,
which means that we can use the expectation parameters as coordinates for the
statistical manifold. The $u$ coordinates are defined on the set
$\Omega^{\ast}=\\{(u_{1},u_{2})\in\mathbb{R}^{2}~{}|~{}u_{1}\cdot
u_{1}-u_{2}<0\\}$. The Legendre dual of the log-partition function $\Phi$ is

$\Phi^{\ast}=-\frac{1}{2}-\frac{1}{2}\log(u_{2}-u_{1}\cdot u_{1}).$

As such, in the $u$-coordinates, the Fisher metric is given by

$g=\frac{1}{(u^{1}\cdot u^{1}-u^{2})^{2}}\left[\begin{array}[]{cc}u^{1}\cdot
u^{1}+u^{2}&-u^{1}\\\ -u^{1}&\frac{1}{2}\\\ \end{array}\right].$

####  2.2.2. The symmetries of the normal manifold

Before constructing a statistical mirror pair from the moduli space of normal
distributions, let us discuss the isometry group of the $\mathcal{M}_{\tt
Normal}$ and its statistical meaning. It is a classic fact in hyperbolic
geometry that the symmetry group of the half-plane is the projective special
linear group $PSL(2,\mathbb{R})$, which acts by Möbius transformations.

(5) $z\mapsto\frac{az+b}{cz+d},$

where $a,b,c,d\in\mathbb{R}$ with $ad-bc=1$.

In the context of normal distributions, there is a distinguished subgroup
induced by affine transformations of the underlying sample space $\mathbb{R}$.
More precisely, if we consider an affine map of the random variable $\zeta$
defined on $\mathbb{R}$ (i.e. $\zeta\mapsto a\zeta+b$), then under push-
forward of measures, this induces a self-map of $\mathcal{M}_{\tt Normal}$. In
fact, we can write out the embedding of the affine group into
$PSL(2,\mathbb{R})$ explicitly as

$\displaystyle Aff^{+}\to PSL(2,\mathbb{R})$ $\displaystyle
ax+b\mapsto\left[\begin{array}[]{cc}\sqrt{a}&\frac{b}{\sqrt{a}}\\\
0&\frac{1}{\sqrt{a}}\\\ \end{array}\right]$

The phenomena that diffeomorphisms of the sample space induce isometries of
the moduli space is more general, and holds for any parametrized family of
probability distributions.

######  Theorem 2 (Chentsov [Cen00]).

The Fisher metric is invariant under the diffeomorphism group of the sample
space $\mathcal{S}$.

This result was originally proven by Chentsov888It is worth noting this
theorem is actually a corollary of Chentsov’s main result, which states that
the Fisher metric is invariant under _sufficient statistics_ , which are
statistics $T(Z)$ (for data $Z$) where the data processing inequality becomes
an equality:  $I\bigl{(}x;T(Z)\bigr{)}=I(x;Z).$ when the sample space is
finite, but it holds in much greater generality as well (see, e.g., Corollary
3.6 of [AJVLS15]). In view of this result, we say that an isometry of the
Fisher metric is a Fisher-Chentsov isometry if it is induced by map of the
underlying random variable. It is worth noting that $\mathcal{M}_{\tt Normal}$
also has non-Fisher-Chentsov isometries, which correspond to Mobius
transformations where $c$ in Equation (5) is non-zero.

###  2.3. Constructing the statistical mirror pair

With these preliminary calculations finished, we can now construct the
statistical mirror pair. We have seen that the moduli space of univariate
normal distributions is a hyperbolic manifold with dual Hessian structures,
one induced by the natural parameters and the other induced by the sufficient
statistics. Using these dual Hessian manifolds, we can construct two Kähler
manifolds $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$, which are
said to be a statistical mirror pair.

The primal manifold (denoted $\mathbb{M}_{\tt Normal}$), is constructed on the
tube domain999Observant readers will notice that this notation is reminiscent
of tangent bundles. As we will see in Section 3, this is not coincidental.
$T\Omega\subset\mathbb{C}^{2}$, which is defined as

$T\Omega=\\{x^{j}+\sqrt{-1}y^{j}~{}|~{}y^{j}\in\mathbb{R}^{n},(x^{1},x^{2})\in\Omega\\}\subset\mathbb{C}^{2}.$

Recall that $\Omega$ is a half space in $\mathbb{R}^{2}$, so $T\Omega$ is a
half space in $\mathbb{C}^{2}$. We use the log-partition function $\Phi$ to
define a Kähler metric on $\mathbb{M}_{\tt Normal}$. To do so, we define the
lift of $\Phi$, denoted $\Phi^{h}$, as

$\Phi^{h}(x+\sqrt{-1}y)=\Phi(x)$

and consider the Kähler metric

$\omega_{\mathbb{M}_{\tt
Normal}}=\frac{\sqrt{-1}}{2}\frac{\partial^{2}\Phi^{h}}{\partial x^{i}\partial
x^{j}}dz^{i}\wedge d\overline{z}^{j}.$

In other words, the Kähler potential is simply $\Phi$, extended so that it is
constant with respect to the imaginary directions $y$. Since $\Phi$ is
strongly convex, $\Phi^{h}$ is strongly pluri-subharmonic, so induces a Kähler
metric on $\mathbb{M}_{\tt Normal}.$

To build the mirror manifold $\mathbb{W}_{\tt Normal}$, we consider the tube
domain over the parabola $\Omega^{\ast}$, i.e.
$T\Omega^{\ast}\subset\mathbb{C}^{2}$ with holomorphic coordinates
$\\{w^{j}=u^{j}+\sqrt{-1}v^{j}\\}_{j=1}^{n}$ for
$(u^{1},u^{2})\in\Omega^{\ast}$, the domain of the dual variable. Its Kähler
metric is induced by lifting the conjugate potential $\Phi^{\ast}$ to
$T\Omega^{\ast}$ and defining

$\omega_{\mathbb{W}_{\tt
Normal}}=\frac{\sqrt{-1}}{2}\frac{\partial^{2}(\Phi^{\ast})^{h}}{\partial
u_{i}\partial u_{j}}dw^{i}\wedge d\overline{w}^{j}.$

We will provide a general definition for statistical mirror symmetry in
Section 3 (Definition 3). For now, we simply define the Kähler manifolds
$\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ to be a statistical
mirror pair.

###  2.4. The geometry of $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt
Normal}$

The spaces $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ have been
studied extensively in the literature, and a summary of their geometric
properties is provided in Table 1. The space $\mathbb{W}_{\tt Normal}$ is more
commonly known as the _Siegel half-space_ and has been studied in the context
of automorphic forms and Abelian varieties. $\mathbb{M}_{\tt Normal}$ is known
as the _Siegel-Jacobi space_ , and has also been studied in the context of
automorphic forms.101010The associated Kähler metric is occasionally referred
to as the Kähler-Brandt metric (see, e.g., [Mol14]), although we will not use
this terminology. The first observation that we make is that $\mathbb{M}_{\tt
Normal}$ and $\mathbb{W}_{\tt Normal}$ are distinct from the viewpoint of
complex geometry.

| $\mathbb{M}_{\tt Normal}$ | $\mathbb{W}_{\tt Normal}$
---|---|---
also known as | the Siegel-Jacobi space | the Siegel half-space (n=2)
base coordinates | $(x^{1},x^{2})=\left(\frac{\mu}{\sigma^{2}},\,-\frac{1}{2\sigma^{2}}\right)$ | $(u_{1},u_{2})=(\mu,\,\mu^{2}+\sigma^{2})$
base domain | $\Omega=\\{(x^{1},x^{2})\in\mathbb{R}^{2}:x^{2}\geq 0\\}$ | $\Omega^{\ast}=\\{(u_{1},u_{2})\in\mathbb{R}^{2}:u_{1}\cdot u_{1}<u_{2}\\}$
holomorphic coordinates | $\\{z^{j}=x^{j}+\sqrt{-1}\,y^{j}\\}_{j=1,2}\in T\Omega$ | $\\{w_{j}=u_{j}+\sqrt{-1}\,v_{j}\\}_{j=1,2}\in T\Omega^{\ast}$
biholomorphic to | the half-space in $\mathbb{C}^{2}$ | the unit ball in $\mathbb{C}^{2}$
convex potential | $\Phi(x)=\frac{x^{1}\cdot x^{1}}{4(-x^{2})}-\frac{1}{2}\log\left(\frac{-x^{2}}{\pi}\right)$ | $\Phi^{\ast}(u)=-\frac{1}{2}\log(u_{2}-u_{1}\cdot u_{1})$
Kähler potential | $\Phi^{h}(x+\sqrt{-1}\,y)=\Phi(x)$ | $(\Phi^{\ast})^{h}(u+\sqrt{-1}\,v)=\Phi^{\ast}(u)$
Kähler metric | $\frac{\sqrt{-1}}{2}\frac{\partial^{2}\Phi^{h}}{\partial x^{i}\partial x^{j}}dz^{i}\wedge d\overline{z}^{j}$ | $\frac{\sqrt{-1}}{2}\frac{\partial^{2}(\Phi^{\ast})^{h}}{\partial u_{i}\partial u_{j}}dw^{i}\wedge d\overline{w}^{j}$
metric known as | the Kähler-Brandt metric | Bergman metric
symmetry group | $SL(2,\mathbb{R})\ltimes\mathbb{R}^{2}$ | $PU(2,1)$
geometric properties | cscK expanding Kähler-Ricci soliton | complex hyperbolic space form
encountered in | automorphic forms | automorphic forms/Abelian varieties
Table 1. Summary of the mirror pair $\mathbb{M}_{normal}$ and
$\mathbb{W}_{normal}$

######  Observation.

$\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ are not biholomorphic.

###### Proof.

It is a classic fact in complex analysis that the Siegel half-space (i.e.,
$\mathbb{W}_{\tt Normal}$) is biholomorphic to the unit ball
$\mathbb{B}^{2}\subset\mathbb{C}^{2}$, by the map:

$F(w_{1},w_{2})=\left(\frac{2w_{1}}{\sqrt{-1}+w_{2}},\frac{1+\sqrt{-1}w_{2}}{\sqrt{-1}+w_{2}}\right).$

For the sake of contradiction, suppose there were a biholomorphism
$\phi:\mathbb{M}_{\tt Normal}\to\mathbb{B}^{2}$. Then the function
$z^{2}\to\phi(\sqrt{-1},z^{2})$ would be entire on $\mathbb{C}$ and bounded in
each of its coordinates, which is impossible.∎

Furthermore, $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ have
different automorphism groups, which shows that the duality between them does
not preserve the automorphism group. Among complex domains, the ball has the
largest possible automorphism group [Isa04], so the automorphism group of
$\mathbb{M}_{\tt Normal}$ is smaller. These observations should not be
particularly surprising; the Legendre transformation can have a complicated
effect on the domain of a convex function, which corresponds to deforming the
complex structure of its tube domain111111In coordinate-invariant language,
Legendre duality does not preserve the affine structure of an affine manifold.
(see [Shi00] for a details on the automorphism group of tube domains).

With Calabi-Yau manifolds, mirror symmetry can change the topology (it acts to
rotate the Hodge diamond [Kon95]), so mirror pairs are generally not
diffeomorphic. However, $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt
Normal}$ are both tube domains over a convex subset of $\mathbb{R}^{2}$, and
thus are diffeomorphic.121212More conceptually, statistical mirror pairs are
defined on the tangent bundle $T\mathcal{M}$ of an affine manifold
$\mathcal{M}$, so they are always diffeomorphic (see Definition 3).

Finally, we remark that $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt
Normal}$ are both domains of holomorphy, as they are tube domains whose base
is convex [Yan82]. This fact holds for any statistical mirror pair induced by
an exponential family, since the domain of the natural parameters and
expectation parameters are always convex.

######  Observation.

For any exponential family ${\tt S}$, the Kähler mirror pair $\mathbb{M}_{\tt
S}$ and $\mathbb{W}_{\tt S}$ are domains of holomorphy.

####  2.4.1. Symmetries of $\mathbb{M}_{\tt Normal}$

We now study the geometry of $\mathbb{M}_{\tt Normal}$ in more detail. For a
more complete reference on this space, we refer the reader to the following
papers [Yan07, YY16, YYH+13]. The first two of these study the geometry of
this space from the perspective of automorphic forms and spectral analysis.
The third computes the curvature tensor explicitly. We also recommend the work
of Molitor [Mol14], which studied this space from the perspective of
mathematical physics.

We start by considering the symmetries of $\mathbb{M}_{\tt Normal}$. The group
of holomorphic isometries is the affine symplectic group
$SL(2,\mathbb{R})\ltimes\mathbb{R}^{2}$ [Mol14]. As a result, the group of
holomorphic symmetries is (real) $5$-dimensional, and acts transitively.

Intuitively, it is reasonable that the symmetry group of this space is real
five-dimensional. In particular, any two-dimensional tube domain has a natural
$\mathbb{R}^{2}$-action induced by translation in the fibers and the symmetry
group of the half-plane is three-dimensional. As we shall see shortly, it is
not always the case that the holomorphic symmetry groups of tube domains are
the semi-direct product of the affine symmetries group on the base and the
$\mathbb{R}^{n}$-symmetry induced by translation in the fibers. However, this
is indeed the case for $\mathbb{M}_{\tt Normal}$.

It is worth noting that $\mathbb{M}_{\tt Normal}$ is often studied in the
context of the Jacobi group $G^{J}$, which is defined as

$G^{J}:=SL(2,\mathbb{R})\ltimes\operatorname{Heis}(\mathbb{R}),$

where $\operatorname{Heis}(\mathbb{R})$ is the Heisenberg group (see [Yan07]
for details). There is a natural action of this group on $\mathbb{M}_{\tt
Normal}$, under which the metric is invariant. However, this action is not
faithful, which is why the isometry group is smaller.131313The affine
symplectic group is not a subgroup of the Jacobi group. The latter is a
central extension of the former. Nonetheless, the Jacobi group plays an
important role in the study of modular forms on this space, which is why we
introduce it now.

From the fact that $\mathbb{M}_{\tt Normal}$ is homogeneous, it immediately
follows that $\mathbb{M}_{\tt Normal}$ is complete. However, there is a more
general result, which was proven by Molitor.

######  Proposition (Proposition 2.15 [Mol14]).

A Kähler manifold defined on the tangent bundle $T\mathcal{M}$ of a Hessian
manifold $\mathcal{M}$ is complete if and only if the underlying Hessian
manifold $(\mathcal{M},g,D)$ is complete.

The horizontal submanifolds

$\mathcal{M}_{a,b}=\\{(z^{1},z^{2})~{}|~{}y^{1}=a,y^{2}=b\\}$

induce a totally geodesic foliation of $\mathbb{M}_{\tt Normal}$. Since each
of these submanifolds is hyperbolic (and the transition maps to the standard
half-plane model are given by Equation (3)), we can calculate horizontal
geodesics and distances explicitly. The vertical submanifolds (where the
$x$-coordinates are fixed) are Lagrange submanifolds, but are not totally
geodesic and so we are not aware of a similar characterization for geodesics
whose vertical displacement is non-zero.

####  2.4.2. The Ricci and scalar curvature of $\mathbb{M}_{\tt Normal}$

We now study some curvature properties of $\mathbb{M}_{\tt Normal}$. On a
Kähler manifold, the Ricci curvature can be expressed as a $(1,1)$-form, which
can be computed as

$\displaystyle Ric_{i\bar{j}}$ $\displaystyle=$
$\displaystyle-\sqrt{-1}\frac{\partial}{\partial
z^{i}}\frac{\partial}{\partial\bar{z}^{j}}\log(\det[g]).$

For $\mathbb{M}_{\tt Normal}$, the Kähler form is given by
$\omega=\partial\bar{\partial}\Phi^{h}$. Computing the Ricci curvature in the
$(x_{1},y_{1},x_{2},y_{2})$ coordinates, we find that it is given by

$Ric=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&-\frac{3}{2x_{2}^{2}}&0&0\\\
0&0&0&0\\\ 0&0&0&-\frac{3}{2x_{2}^{2}}\end{array}\right].$

For $\mathbb{M}_{\tt Normal}$, the Kähler potential $\Phi^{h}$ is a function
of the base (i.e. the $x$-coordinates) alone. As such, we can calculate the
Ricci tensor by computing

$\frac{\partial^{2}}{\partial x^{i}\partial
x^{j}}\log\left(\det\left[\frac{\partial^{2}}{\partial x^{a}\partial
x^{b}}\Phi\right]\right).$

This reduces the original $4\times 4$ matrix of Ricci components to a $2\times
2$ matrix, which greatly simplifies the analysis.

From this computation, we can see that the Ricci curvature is non-positive.
However, this metric is not Kähler-Einstein. It is worth emphasizing that the
Ricci tensor of $\mathbb{M}_{\tt Normal}$ is not the Ricci tensor of
$\mathcal{M}_{\tt Normal}$ (which is a hyperbolic Riemann surface). Taking the
trace of the Ricci curvature with the inverse metric, we also find that
$\mathbb{M}_{\tt Normal}$ has constant scalar curvature $-6$.

####  2.4.3. The bisectional and anti-bisectional curvatures

In Riemannian geometry, it is often of interest to consider metrics whose
sectional curvature is either positive or negative. Oftentimes, this sort of
assumption is necessary to obtain analytic results.

In Kähler geometry, it is often too restrictive to assume that the sectional
curvature has a sign. However, due to the additional structure and symmetries
introduced by the complex structure (see Chapter 4 of [Bal06] for an
introduction), it is possible to define other curvature conditions which play
an important role in the analysis of Kähler manifolds. Below, we note some of
the curvature properties of $\mathbb{M}_{\tt Normal}$. This list is not
intended to be exhaustive, and it is likely that there are other interesting
curvature properties of this metric, which may play an important role in its
analysis.

  1. (1)

The holomorphic sectional curvature does not have a sign [Mol14]. At every
point there are directions with both negative and positive holomorphic
sectional curvature. Similarly, the orthogonal bisectional curvature does not
have a sign.

  2. (2)

The _orthogonal anti-bisectional curvature_ is non-negative definite. The
orthogonal anti-bisectional curvature plays an important role in the
regularity theory of optimal transport. In particular, if a convex potential
$\Phi$ induces a Kähler metric with non-negative anti-bisectional curvature,
the associated cost function $c(x,y)=\Phi(x-y)$ is _weakly regular_. In other
words, there are no local obstructions to establishing continuity for
solutions to the Monge problem with respect to this cost.141414More precisely,
the orthogonal anti-bisectional curvature is proportional to the MTW tensor,
and so non-negative orthogonal anti-bisectional curvature of the Kähler metric
is equivalent to the cost function satisfying the MTW(0) condition. For
details on the orthogonal anti-bisectional curvature and its relationship with
optimal transport, we refer the reader to [KZ20a]. We discuss this particular
metric in Example 9.

####  2.4.4. The geometry of $\mathbb{W}_{\tt Normal}$

We now turn our attention to the mirror Kähler manifold $\mathbb{W}_{\tt
Normal}$. As we mentioned previously, this is the Siegel half-space, which has
been studied extensively in the context of automorphic forms. Let us note some
important properties of this space.

  1. (1)

The Kähler metric is the Bergman metric. As a result, it is invariant under
biholomorphisms and so its isometry group is the same as its automorphism
group.

  2. (2)

The space is homogeneous and, as such, complete.

The first property has the following important consequence, which is a
classical fact in several complex variables (see, e.g., [Sha92]).

######  Observation.

The isometry group of $\mathbb{W}_{\tt Normal}$ is the projective unitary
group $PU(2,1)$.

Since $PU(2,1)$ is real six-dimensional, the symmetry group is larger than the
semi-direct product of translations on $\mathbb{R}^{2}$ and the symmetries of
$\mathcal{M}_{\tt Normal}$. This occurs because there are automorphisms of
$\mathbb{W}_{\tt Normal}$ which do not respect the tube domain structure.

######  Observation.

Statistical mirror symmetry need not preserve the isometry group of a Kähler
manifold.

It is worth noting that $\mathbb{W}_{\tt Normal}$ is the only strictly pseudo-
convex tube domain in complex dimension two which is
homogeneous151515$\mathbb{M}_{\tt Normal}$ is homogeneous and pseudo-convex,
but not strictly so. (and its higher dimensional analogues are the only such
domains in $\mathbb{C}^{n}$ [EMS18]).

As before, the horizontal submanifolds

$\mathbb{W}_{a,b}=\\{(w_{1},w_{2})~{}|~{}v_{1}=a,v_{2}=b\\}$

induce a totally geodesic foliation of the $\mathbb{W}_{\tt Normal}$.
Furthermore, since $\mathbb{W}_{\tt Normal}$ is a complex space form, it is
possible to calculate distances and geodesics explicitly (see [San96] for
details).

####  2.4.5. Curvature properties of $\mathbb{W}_{\tt Normal}$

Similarly to $\mathbb{M}_{\tt Normal}$, we can understand the geometry of
$\mathbb{W}_{\tt Normal}$ by investigating its curvature.

  1. (1)

The metric has constant negative holomorphic sectional curvature.161616There
is a general theorem due to Shima which shows that if the a Kähler manifold
constructed in this fashion has constant holomorphic sectional curvature, the
underlying Riemannian manifold $\mathcal{M}$ must have constant sectional
curvature [Shi95]. This gives an alternate proof that the space
$\mathcal{M}_{\tt Normal}$ is hyperbolic.

  2. (2)

Since $\mathbb{W}_{\tt Normal}$ has constant holomorphic sectional curvature,
it is necessarily Kähler-Einstein and has constant negative scalar curvature.
This space is distinguished in that it is the unique strictly pseudo-convex
domain for which the Bergman metric is Kähler-Einstein [FW97, HX16].

  3. (3)

The orthogonal anti-bisectional curvature vanishes and the space has negative
cost-curvature (see [KZ20b] for a definition of cost-curvature).

By contrasting the geometry of $\mathbb{W}_{\tt Normal}$ with that of
$\mathbb{M}_{\tt Normal}$, we find the following fact.

######  Observation.

The dual of a of Kähler metric with constant holomorphic sectional curvature
need not be Kähler-Einstein.

As we will see in Subsection 4.1, the statistical mirror pair of a Kähler
manifold with constant holomorphic sectional curvature need not even have
constant scalar curvature. Recently, Maeta171717In his paper, the results are
phrased in terms of Hesse-Einstein and Hesse solitons, but this statement is
equivalent to this claim. showed that the statistical mirror of a Kähler-
Einstein space is a Kähler-Ricci soliton (see Corollary 5.3 of [Mae21]).

This phenomena contrasts with mirror symmetry for compact Calabi-Yau
manifolds, where the mirror pairs both have vanishing Ricci curvature. From a
more conceptual perspective, the Legendre transform inverts the Hessian matrix
of a potential. For a semi-flat Kähler metric, when the determinant of the
Hessian is constant (which corresponds to Ricci flatness in the compact case),
the determinant of its dual is also constant. However, for Kähler-Einstein
metrics with non-zero scalar curvature, the determinant of the Hessian matrix
will not be constant, and will instead satisfy a different Monge-Ampère
equation, which is not preserved by Legendre duality.181818However, there are
some simple examples where the dual of a Kähler-Einstein metric is again
Kähler-Einstein. For instance, consider
$\mathbb{H}=\\{z=x+\sqrt{-1}y~{}|~{}y>0\\}$ with the Kähler potential
$f(z)=-\log(y)$.

###  2.5. Kähler-Ricci flow, conjugate flow, and statistical mirror symmetry

Thus far, the geometric properties of $\mathbb{M}_{\tt Normal}$ and
$\mathbb{W}_{\tt Normal}$ we have discussed were previously known in the
literature (although the duality between these spaces was not studied in
depth). To motivate the study of statistical mirror symmetry, it is natural to
ask for new insights into the geometry of these spaces. In this subsection, we
do so by considering the relationship between this duality and Kähler-Ricci
flow. The main result of this subsection is the following.

######  Proposition 3.

The spaces $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ are Kähler-
Ricci solitons191919$\mathbb{W}_{\tt Normal}$ is a complex space form, so the
content of this proposition is that $\mathbb{M}_{\tt Normal}$ is a soliton and
that the coupling persists under the flow. which are immortally coupled under
Kähler-Ricci flow.

Before we discuss this result and some of its consequences, we first provide a
brief background on Kähler-Ricci flow. The Ricci flow was first introduced by
Hamilton [Ham82] and evolves a Riemannian metric by its Ricci curvature:

$\frac{\partial}{\partial t}g_{ij}=-2Ric_{ij}.$

This flow plays a central role in modern geometric analysis, largely due to
its role in the proof of the the Poincaré and Geometrization conjectures
[Per02, Per03]. If one starts with a Kähler metric, the evolving metrics will
remain Kähler (with the same complex structure), and the resulting flow is
called the Kähler-Ricci flow [Cao85]. This flow has also been used to prove
many results in complex geometry (see, e.g., [Shi97, Mok88]).

Tube domains form a natural class of metrics where the Kähler-Ricci flow is
particularly simple to study. In particular, the Kähler-Ricci flow induces a
flow of the underlying Hessian metric, whereby the Hessian potential $\Phi$
evolves via the parabolic Monge-Ampère equation202020The flow on the
underlying Hessian manifold is known as Hesse-Koszul flow, and was studied by
Mirghafouri and Malek [MM17].

(7) $\frac{\partial}{\partial
t}\Phi=\log\left(\det\left[D^{2}\Phi\right]\right).$

####  2.5.1. Kähler-Ricci flow on $\mathbb{M}_{\tt Normal}$ and
$\mathbb{W}_{\tt Normal}$

It is possible to compute the Kähler-Ricci flow on $\mathbb{M}_{\tt Normal}$
explicitly. To do so, we consider the one-parameter family of potentials
(indexed by $t$):

$\Phi_{t}=-\frac{x_{1}^{2}}{x_{2}}-t\log(-x_{2})$

defined on the half space $\mathbb{H}=\\{(x_{1},x_{2})~{}|~{}x_{2}<0\\}$. We
then define a family of Kähler metrics212121The Kähler metric corresponding to
univariate normal distributions (in their natural parameters) corresponds to
$\mathbb{M}_{1/2}$. Here, we can ignore the constant term
$\frac{1}{2}\log(\pi)$ since it does not affect the metric. on the tube domain
$\mathbb{H}\times\sqrt{-1}\mathbb{R}^{2}$ by lifting the potentials
$\Phi_{t}$:

$\omega_{t}=\sqrt{-1}\partial\bar{\partial}\Phi^{h}_{t}.$

It is straightforward to verify that the potentials $\Phi_{t}$ solve Equation
(7), and so induce solutions to the Kähler-Ricci flow. In fact, the space
$\mathbb{M}_{\tt Normal}$ is an expanding Kähler-Ricci soliton. Upon
rescaling, the flow acts by dilating the $z_{1}$ coordinate, but does not
alter the geometry.222222 This result gives a simple explanation to a fact
about these spaces which was originally observed by Yang et. al. [YYH+13].
They studied a two-parameter family of invariant metrics on this space (which
they denote $\mathbb{H}_{1,1},ds^{2}_{1,1,A,B}$) and found that the scalar
curvature does not depend on the $B$ parameter. In our language, changing the
$B$ parameter acts to rescale the $z_{1}$-coordinate, so the invariance of the
scalar curvature follows from the fact that $\mathbb{M}_{\tt Normal}$ is a
Kähler-Ricci soliton with constant scalar curvature.

####  2.5.2. The Conjugate flow

Given a solution to the Kähler-Ricci flow on a tube domain (or more generally
the tangent bundle of a Hessian manifold), we can use statistical mirror
symmetry to induce a conjugate flow on $\mathbb{W}$.

To construct this flow, we consider Kahler-Ricci flow on $\mathbb{M}$ and its
associated flow on the Hessian manifold $\mathcal{M}$. By taking the Legendre
transformation at each time $t$, we obtain a dual Hessian manifold defined on
a (time-dependent) domain $\Omega^{\ast}_{t}$. We can then consider a tube
domain $T\Omega^{\ast}_{t}$ and construct a Kähler metric by lifting the
potential $\Phi^{\ast}_{t}$.

This flow was previously studied by Fei and Picard [FP19b] as a model to
understand the relationship between T-duality and Hermitian curvature
flows.232323The main focus of their work is actually the “anomaly flow,” which
deforms conformally balanced Hermitian metrics. However, there are some deep
connections between this flow and Kähler-Ricci flow [FP19a]. In general, the
domain $\Omega^{\ast}_{t}$ will evolve with $t$. From the viewpoint of
statistical mirror symmetry, this shows that the complex structure of
$(T\Omega^{\ast}_{t},\Phi^{\ast}_{t})$ can change along the conjugate flow. On
the other hand, the _symplectic structure_ remains constant under conjugate
flow (which is one manifestation of the duality exchanging complex with
symplectic geometry). As such, conjugate flow changes both the metric and the
complex structure of $\mathbb{W}$, in such a way as to leave the symplectic
structure fixed.

In the case of the statistical manifold of normal distributions, we can write
out this flow explicitly by considering the one-parameter family of potentials
$\Phi_{t}$ and computing the Legendre transform at each time. Doing so, we
find that

(8) $\Phi_{t}^{\ast}=-t-t\log\left(\frac{u_{2}-u_{1}^{2}}{2t}\right)$

which is defined on the domain

$\Omega^{\ast}_{t}=\\{(u_{1},u_{2})~{}|~{}u_{2}-u_{1}^{2}\geq 0\\},$

(which in this case is independent of the time). As such, the conjugate flow
on $\mathbb{W}$ is the family of Kähler metrics on $T\Omega^{\ast}_{t}$ where
the metric is given by
$\omega^{\ast}=\sqrt{-1}\partial\bar{\partial}\Phi_{t}^{\ast}$.

This flow is defined for positive time, and for all time the associated Kähler
manifold is complex hyperbolic, so this flow expands the metric uniformly.
Furthermore, since $\mathbb{W}$ is a complex hyperbolic space form, we find
the following.

######  Proposition 4.

Kähler-Ricci flow on $\mathbb{M}_{\tt Normal}$ is identical to conjugate flow
on $\mathbb{W}_{\tt Normal}$.

To explain why this is noteworthy, let us define conjugate flow more
precisely. We say that a Hessian manifold (or its tube domain) evolves via
conjugate flow if its Hessian metric evolves via the equation

(9) $\frac{\partial}{\partial t}g_{jk}=\frac{1}{2}\frac{\partial^{2}}{\partial
x_{j}\partial
x_{k}}\log\operatorname{det}\left(g\right)-\frac{1}{2}\sum_{q}\frac{\partial}{\partial
x_{q}}\log\operatorname{det}\left(g\right)\frac{\partial}{\partial
x_{j}}g_{qk}$

Under this flow, the first term is the same as for Kähler-Ricci flow and the
second is a “correction term” which deforms the affine structure of the
Hessian manifold (or equivalently the complex structure of its tube domain).
From this, we can see that in general conjugate flow and Kähler-Ricci flow are
not the same.

####  2.5.3. Preserved curvature inequalities

One of the main motivations in finding Kähler-Ricci solitons is that they can
be used to derive strong inequalities about the flow (see, e.g. [Ham86,
Cao92]) and find preserved curvature conditions. Since $\mathbb{M}_{\tt
Normal}$ is a soliton with non-negative orthogonal anti-bisectional curvature
and $\mathbb{W}_{\tt Normal}$ is a soliton with negative cost curvature, it is
natural to ask whether either of these properties are preserved by Kähler-
Ricci flow. In recent work of the first named author and F. Zheng, we studied
these questions and showed that non-positive anti-bisectional curvature (as
with $\mathbb{W}_{\tt Normal}$) is preserved.

######  Theorem 5 (K-Zheng ‘20).

Let $(T\Omega,\omega_{t})$ be a family of complete Kähler metrics with bounded
curvature which solve the Kähler-Ricci flow. Suppose that the initial metric
$\omega_{0}$ has non-positive anti-bisectional curvature. Then for all
positive time, anti-bisectional curvature remains non-positive.

We furthermore show that for complex surfaces, non-negative orthogonal anti-
bisectional curvature (as with $\mathbb{M}_{\tt Normal}$) is preserved.

######  Theorem 6 (K-Zheng ‘20).

Suppose that $\Omega\subset\mathbb{R}^{2}$ is a convex domain and
$\Phi:\Omega\to\mathbb{R}$ is a strongly convex function so that the
associated Kähler manifold $(T\Omega,\omega_{0})$

  1. (1)

is complete,

  2. (2)

has bounded curvature, and

  3. (3)

has non-negative orthogonal anti-bisectional curvature.

Then the orthogonal anti-bisectional curvature remains non-negative along
Kähler-Ricci flow.

Using Legendre duality, we can translate these results into the dual setting
to find conditions which are preserved under conjugate flow.

######  Theorem 7.

Let $\Phi(x,t)$ be a convex potential on a (time-dependent) tube domain that
evolves via conjugate flow. Suppose that for all covectors
$u=\sum_{i}u_{i}dx^{i}$ and $v=\sum_{i}v_{i}dx^{i}$, the potential $\Phi(x,0)$
satisfies the inequality

(10) $\sum_{i,j,k,\ell}\mathfrak{W}^{ijk\ell}u_{i}u_{j}v_{k}v_{\ell}\leq 0,$

where $\mathfrak{W}^{ijk\ell}$ is defined to be the quantity242424In the
formula, we have written $\mathfrak{W}$ in this way so that the first two
terms have corresponding terms in the anti-bisectional curvature and the final
term is a “correction” term.

$\displaystyle\mathfrak{W}^{ijk\ell}={{{\sum}}}_{\alpha,\beta,\gamma,\delta,\epsilon,\eta,\zeta}\left(\begin{aligned}
&-\Phi^{i\alpha}\Phi^{j\beta}\Phi^{\gamma
k}\Phi^{\delta\ell}\Phi_{\alpha\beta\gamma\delta}\\\
&+\Phi^{i\alpha}\Phi^{j\beta}\Phi_{\alpha\beta\gamma}\Phi^{k\epsilon}\Phi^{\ell\zeta}\Phi^{\gamma\eta}\Phi_{\epsilon\zeta\eta}\\\
&-\frac{\partial}{\partial
x^{\delta}}\left(\Phi^{i\alpha}\Phi^{j\beta}\Phi^{\gamma
k}\right)\Phi^{\delta\ell}\Phi_{\alpha\beta\gamma}.\end{aligned}\right).$

Then this inequality persists for all time along the flow.

A calculation shows that $\mathfrak{W}$ is the anti-bisectional curvature of
the dual manifold $\mathbb{M}$ and thus Equation 10 states that the anti-
bisectional curvature of $\mathbb{M}$ is negative, so this theorem is a
restatement of Theorem 5 in terms of conjugate flow. Ideally, one would want
to use conjugate flow to prove new results about Kähler-Ricci flow, instead of
the other way around. However, we will leave this question for future work and
so leave Theorem 7 as a proof of concept for this approach.

###  2.6. Conjectural remarks

While both $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ are well
known spaces, their duality has not drawn as much attention. In this section,
we make some conjectural remarks to motivate further exploration on this
topic.

One of the important mathematical successes of mirror symmetry has been to
relate difficult questions in enumerative algebraic geometry to problems on
their mirror side that can calculated more readily [Mor97]. We hope that it
might be possible to use the duality between $\mathbb{M}_{\tt Normal}$ and
$\mathbb{W}_{\tt Normal}$ in a similar way.

####  2.6.1. Analytic number theory and the Saito-Kurokawa lift

Both the Siegel-Jacobi space and the Siegel half-space have been studied in
the context of number theory, and it is seems likely that their duality might
give additional insight.

An automorphic form is a function from a topological group $G$ to the complex
numbers $\mathbb{C}$ which is invariant with respect to some discrete subgroup
$\Gamma\subset G$. In this case, the relevant group on $\mathbb{M}_{\tt
Normal}$ is the Jacobi group (see [Yan15] for details) and the associated
group for $\mathbb{W}_{\tt Normal}$ is $Sp(2,\mathbb{R})$.

Since $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ are not
biholomorphic (and these groups are not the same), the relationship between
automorphic forms on these spaces will be somewhat complicated. However, the
existence of the Saito-Kurokawa lift strongly suggests there is some
connection.

The Saito–Kurokawa lift takes modular forms on the hyperbolic half plane
$\mathbb{H}$ to a distinguished class (a _Spezialschar_) of Siegel modular
forms on the Siegel half space. Its existence was conjectured independently by
Saito and Kurokawa and proven by Andrianov [And79], Maass [Maa79] and Zagier
[Zag79]. Even more striking, this lift can be understood as a composition of
three mappings, with the last map being a map from certain Jacobi forms (i.e.,
automorphic forms on $\mathbb{M}_{\tt Normal}$) to certain automorphic forms
on $\mathbb{W}_{\tt Normal}$ (i.e., the Spezialschar). Keeping in mind that
$\mathbb{W}_{\tt Normal}$ and $\mathbb{M}_{\tt Normal}$ are both models for
the tangent bundle of the hyperbolic half plane, this picture suggests that
there might be a natural way to understand the Saito-Kurokawa lift in terms of
statistical mirror symmetry.

${\left\\{\begin{aligned} \textrm{Jacobi forms on $\mathbb{M}_{\tt Normal}$
}\\\ \textrm{of weight $k$ and index
1}\end{aligned}\right\\}}$${\left\\{\begin{aligned} \textrm{ Spezialschar of
Siegel }\\\ \textrm{ modular forms of degree
2}\end{aligned}\right\\}}$${\left\\{\begin{aligned} \textrm{ Kohnen plus-space
of modular}\\\ \textrm{\\! \\! forms of level 4 and weight
$k-\frac{1}{2}$}\end{aligned}\right\\}}$${\left\\{\begin{aligned}
\textrm{Modular forms on }\mathbb{H}\\\ \textrm{ of weight
}2k-2\end{aligned}\right\\}}$MaassEichler and ZagierShimura
correspondenceSaito-Kurokawa lift Figure 1. The Saito-Kurokawa lift

######  Question 1.

Is there a way to understand the Saito-Kurokawa lift using the duality between
these two spaces? Are there any number-theoretic applications for this?

If this is indeed the case, it raises the following question.

######  Question 2.

Can this type of duality be used more generally in the study of automorphic
forms?

For the latter question, the natural spaces to consider are homogeneous tube
domains with an invariant metric (such as a Kähler-Einstein metric) and to
compute the Legendre transform of the underlying potential on the base to find
a dual space. After considering the Siegel half-space and the Siegel-Jacobi
space, the natural next class of examples are tube domains over homogeneous
convex cones [Rot66], in which case the dual space will be a tube domain over
the dual cone.

####  2.6.2. The Moduli of Abelian Varieties

There is another interpretation of the Siegel half space $\mathbb{W}_{\tt
Normal}$ as the moduli space of principally polarized Abelian varieties (see
Chapter 8 of [BL13]). It is of interest whether there is some moduli space
interpretation for the Siegel-Jacobi space $\mathbb{M}_{\tt Normal}$ as well.
Naively, one may hope that $\mathbb{M}$ is the moduli of _dual_ Abelian
varieties, and that this duality provides the right notion of correspondence
between these spaces.

######  Question 3.

Does $\mathbb{M}_{\tt Normal}$ have an interpretation as a moduli space of
dual Abelian varieties?

Furthermore, $\mathbb{M}_{\tt Normal}$ and $\mathbb{W}_{\tt Normal}$ are not
merely complex varieties, but also Kähler manifolds, which means that they
also have a natural notion of distance and curvature. It is also possible to
write down Darboux coordinates for these spaces (see Section 3.1.1), so it is
straightforward to compute volumes in these spaces.252525Neither of these
spaces have finite volume, so this does not give a natural way to discuss
_random_ Abelian varieties. As a result, it seems natural to ask whether these
metrics are intrinsically meaningful for the moduli of Abelian varieties.

######  Question 4.

For genus $g$ curves, the Teichmüller metric provides a natural metric on the
moduli space, which provides insight into the geometry of such surfaces. Does
the Kähler metric on $\mathbb{W}_{\tt Normal}$ provide an analogy of a
“Teichmüller metric” for principally polarized Abelian varieties? In other
words, can we interpret it as a canonical Kähler metric in order to induce the
“distance” between Abelian varieties? If so, is there a corresponding
interpretation for the Kähler metric on $\mathbb{M}_{\tt Normal}$ in terms of
dual Abelian varieties?

###  2.7. Generalization to the multivariate normal distribution

It is also possible to study the moduli space of multivariate normal
distributions (MulNor). However, if one considers multivariate normal
distributions with arbitrary covariance matrices, the resulting statistical
manifold is an exponential family, but the Fisher metric is not hyperbolic
[Sko84]. Instead, a more natural generalization is to consider the moduli
space of isotropic multivariate normal distributions, which are multivariate
normal distributions whose covariance matrix is $\sigma\,Id_{n}$ (where
$\sigma$ is a real parameter), which we denote by IsoMulNor,n. The associated
statistical manifold is isometric to hyperbolic $n$-space $\mathbb{H}^{n}$.
Furthermore, it is possible to parametrize isotropic multivariate normal
distributions as an exponential family to obtain a mirror pair of Kähler
manifolds. Doing so, one establishes a duality between $\mathbb{W}_{\tt
IsoMulNor,n}$, which is the Siegel upper half-space of degree $n$ and
$\mathbb{M}_{\tt IsoMulNor,n}$, which is the Siegel-Jacobi space
$\left(\mathbb{H}_{1,n},ds^{2}_{1,n,1,1}\right)$ (using the notation of Yang
[Yan07]). As with the univariate case, the dual space $\mathbb{W}_{\tt
IsoMulNor,n}$ is a complex space form. The geometry $\mathbb{M}_{\tt
IsoMulNor,n}$ is more complicated, but this perspective allows us to
immediately understand some of its properties.

To do so, we first compute the associated log-partition function and its
Legendre dual. Doing so, we find that (ignoring some additive constants)

$\Phi_{\tt
IsoMulNor,n}(x^{1},\ldots,x^{n})=\frac{1}{2}\left(\frac{1}{x^{1}}\sum_{i=2}^{n}x^{i}\cdot
x^{i}-\log\left(-x^{1}\right)\right)$

and its dual potential is

$\Phi_{\tt
IsoMulNor,n}^{\ast}(u_{1},\ldots,u_{n})=-\frac{1}{2}-\frac{1}{2}\log\left(2u_{1}-\sum_{i=2}^{n}u_{i}\cdot
u_{i}\right).$

From this calculation, we are able to compute the Ricci and scalar curvatures
of these metrics.

######  Proposition 8.

The Ricci potential $\rho_{\tt
IsoMulNor,n}=-\log\left(\det\left[D^{2}\Phi_{\tt IsoMulNor,n}\right]\right)$
of $\mathbb{M}_{\tt IsoMulNor,n}$ satisfies

$\rho=-(n+1)\log(-2\cdot x^{1}).$

As a result, the scalar curvature of $\mathbb{M}_{\tt IsoMulNor,n}$ is
$-(n+1).$

From this, we can see that the Ricci curvature is non-positive and the metrics
have constant scalar curvature. Using Yang’s notation, this answers Question 3
of [Yan19] for the spaces262626Note that we have used $n$ in the second index
to match the dimension of the multivariate normal distribution. However, Yang
normally denotes this index by $m$.
$\left(\mathbb{H}_{1,n},ds^{2}_{1,n,A,B}\right)$. It is also possible to
compute the orthogonal anti-bisectional curvature of these spaces. Using
computer algebra to simplify the expression, we find the following.

######  Proposition 9.

For all $n$, the space $\mathbb{M}_{\tt IsoMulNor,n}$ has non-negative
orthogonal anti-bisectional curvature.

As a final note, we pose as an open question the problem of finding
exponential families whose tube domains correspond to the other Siegel-Jacobi
spaces. Doing so would give a way to compute the Kähler potentials of these
spaces explicitly and determine other properties of their geometry in a
straightforward way.

##  3\. What is Statistical Mirror Symmetry?

While the correspondence between the Siegel-Jacobi space and the Siegel half-
plane is the archetypal example of statistical mirror symmetry, this duality
can be defined for more general Hessian manifolds. In this section, we provide
a precise definition for the concept and contrast it with mirror symmetry of
Calabi-Yau manifolds. We will focus on the case when the mirrors are Kähler.
It is possible to extend this idea to the non-Kähler case, and we will make
some comments about this more construction at the end of this section.

###  3.1. Kähler statistical mirror symmetry

From a high level perspective, Kähler statistical mirror symmetry can be
described quite simply. For any Hessian manifold $(\mathcal{M},g,D)$, there is
a dual Hessian manifold $(\mathcal{M},g,D^{\ast})$, where the affine structure
is induced by dual coordinates and the metric is given by the Legendre dual of
the original potential (in each affine chart). Furthermore, for any Hessian
manifold $(\mathcal{M},g,D)$, it is possible to define a Kähler metric on its
tangent bundle, by lifting the potential as we did in the previous section.
This map from a Hessian manifold to its tangent bundle is known in the
mathematical physics literature as the ‘r-map’ [AC09].

######  Definition 3 (Statistical mirror symmetry-Kähler case).

Two Kähler manifolds $\mathbb{M}$ and $\mathbb{W}$ are a statistical mirror
pair if they are constructed from the $r$-maps of dual Hessian structures
$(\mathcal{M},g,D)$ and $(\mathcal{M},g,D^{\ast})$.

This definition of statistical mirror symmetry is concise, but requires a fair
amount of background to be meaningful. It also does not provide any motivation
for studying this notion, or give any insight into the geometry of the mirror
pairs. As such, in the rest of this section we will construct such the two
spaces more concretely. Before doing so, however, let us note on the
connection between Definition 3 and mirror symmetry for Calabi-Yau manifolds.

Recall that Calabi-Yau manifolds are Kähler manifolds whose first Chern class
vanishes. By a celebrated result of Yau, such manifolds admit a metric of
vanishing Ricci curvature. In the context of Hessian manifolds and their
tangent bundles, the spaces $\mathbb{M}$ and $\mathbb{W}$ will be Ricci-flat
whenever the Hessian potential satisfies the Monge-Ampère equation

(11) $\det D^{2}\Phi=C.$

In this case, if we quotient the fibers $\mathbb{M}$ by a co-compact lattice
and quotient the fibers of $\mathbb{W}$ by its dual lattice, we obtain Leung’s
construction of mirror Calabi-Yau manifolds in the semi-flat case [Leu00]. As
such, statistical mirror symmetry is related to, but distinct from, mirror
symmetry for semi-flat Calabi-Yau manifolds.

The differences between the two constructions are two-fold. First, we do not
assume that our metrics are Ricci flat. In fact, the only compact Calabi-Yau
manifolds (with their Ricci-flat metrics) which arise via pulling back
solutions to Equation 11 over a compact affine base are complex tori [CY77].
As such, the non-trivial examples of statistical mirror symmetry will not be
Calabi-Yau.

Second, we do not compactify the fibers by quotienting by a co-compact lattice
and its dual. There are several reasons why it preferable to consider the full
tangent bundle in this context. First, for Hessian manifolds whose affine
structures are non-trivial, it is generally not possible to quotient by a co-
compact lattice in a consistent way. Second, even in the cases where the
affine structure is trivial, quotienting the fibers obscures aspects of the
correspondence between the primal and dual space. For instance, for the Siegel
half-space and the Siegel-Jacobi space, quotienting the fibers seems to
destroy any potential correspondence of the modular forms.

####  3.1.1. Constructing Kähler statistical mirror pairs

To construct statistical mirror pairs more concretely, it is helpful to
provide a second definition for Hessian manifolds which is equivalent to
Definition 2 and which makes the duality of Hessian manifolds much more
explicit (at the expense of suppressing the role the “Hessian” plays).

######  Definition (Hessian manifold – Information geometric).

A Riemannian manifold $(\mathcal{M},g)$ is said to be Hessian if it admits
dually flat connections. That is to say, it admits two flat (torsion- and
curvature-free) connections $D$ and $D^{*}$ satisfying

(12) $X(g(Y,Z))=g(D_{X}Y,Z)+g(Y,D^{*}_{X}Z)$

for all vector fields $X,~{}Y$, and $Z$. Because of these dual flat
connections, a Hessian manifold is often said to be dually flat.

The duality between the connections $D$ and $D^{\ast}$ is the crux of
statistical mirror symmetry. However, in order to view this as a
correspondence between complex and symplectic manifolds, we consider this
duality in terms of the tangent bundle. For any Riemannian manifold
$(\mathcal{M},g)$ together with an affine connection $D$ (not necessarily
curvature-free nor torsion-free), it is possible to construct an almost-
Hermitian structure on the tangent bundle $T\mathcal{M}$ known as the Sasaki
metric (for details on this construction, we refer the reader to [Dom62,
Sat07]).

For an arbitrary connection and metric, these structures are somewhat
complicated to write out explicitly (see Section 3.2 for the exact formulas).
However, for a flat connection $D$, this construction simplifies greatly if we
work in terms of the associated affine coordinates (in which the Christoffel
symbols vanish). Namely, if we let $x=\\{x^{1},\cdots,x^{n}\\}$, be such a
coordinate chart on $\mathcal{M}$, we consider the associated coordinates
$\phi:T\mathcal{M}\to\mathbb{R}^{2n}$ given by

$\phi\left(\left.y^{i}\frac{\partial}{\partial
x^{i}}\right|_{p}\right)=\left(x^{1}(p),\ldots,x^{n}(p),y^{1},\ldots,y^{n}\right).$

We then consider the complex valued coordinates
$\\{z^{i}=x^{i}+\sqrt{-1}y^{i}\\}_{i=1}^{n}$, which form holomorphic
coordinates on $T\mathcal{M}$. As a result, the tangent bundle is in fact a
complex manifold, not simply almost complex.272727The fact that the Sasaki
metric is complex is a consequence of the connection $D$ being flat (see
Proposition 10). We define the Hermitian metric $h$ as

$h(\partial_{x^{i}},\partial_{x^{j}})=g(\partial_{x^{i}},\partial_{x^{i}}),\,\,\,\,h(\partial_{y^{i}},\partial_{y^{j}})=g(\partial_{x^{i}},\partial_{x^{i}}),$

and

$h(\partial_{x^{i}},\partial_{y^{j}})=0.$

When $(\mathcal{M},g,D)$ is a Hessian manifold, it is possible to simplify
this expression by lifting the Hessian potential $\Phi$ to $T\mathcal{M}$. As
we did in the context of normal distributions, we define the horizontal lift

$\Phi^{h}(z)=\Phi(Re(z))$

and define the Kähler form

$\omega_{\Phi}=\frac{\partial^{2}}{\partial
z^{i}\partial\bar{z}^{j}}\Phi^{h}.$

With this form, $T\mathcal{M}$ is a Kähler manifold and it is straightforward
to show that the associated Hermitian metric is $h$. Furthermore, we can also
write out Darboux coordinates for this space explicitly. To do so, we note
that since $\mathcal{M}$ is a Hessian manifold, the dual connection $D^{\ast}$
is flat. As such, we can use it to induce coordinates $u$ on $\mathcal{M}$
which are bi-orthogonal to the $x$-coordinates (i.e. satisfy
$g(x^{i},u_{j})=\delta^{i}_{j}$). In fact, we can write out these coordinates
explicitly using the potential function as

$u_{i}=D_{x^{i}}\Phi.$

######  Observation.

The paired coordinates $\\{(u^{i},y^{i})\\}_{i=1}^{n}$ are Darboux coordinates
on $T\mathcal{M}$.

To construct the dual Kähler manifold $\mathbb{W}$, we repeat this
construction, but use the dual connection $D^{\ast}$ instead of the primal
one. Associated with the preferred coordinate of the dual connection is a
convex function $\Phi^{\ast}$, which is the convex dual of $\Phi$.
Furthermore, the primal $x$-coordinates can be obtained by differentiating
$\Phi^{\ast}$:

$x^{i}=D^{\ast}_{u_{i}}\Phi^{\ast}.$

####  3.1.2. Tube domains

The previous discussion provides an explicit description of the Kähler
geometry of the tangent bundle of a Hessian manifold. In this paper we will
primarily focus on a special case, which is when the associated Hessian
manifold is a domain in Euclidean space. If we consider a domain $\Omega$ in
Euclidean space, we can produce a flat connection simply by differentiating in
coordinates. In this case, the tangent bundle is simply the tube domain
$T\Omega$,

$T\Omega=\\{z=x+\sqrt{-1}y\in\mathbb{C}^{n}~{}|~{}x\in\Omega,y\in\mathbb{R}^{n}\\}.$

From a topological perspective, these spaces are simple (they are trivial when
$\Omega$ is convex). Nonetheless, statistical mirror symmetry produces
surprisingly rich structure from the perspective of complex geometry.

When $\mathcal{M}$ is a Hessian manifold which is defined on a convex domain
in Euclidean space, we can describe the geometry of $\mathbb{M}$ explicitly.
Since $\mathcal{M}$ is a convex domain, the Riemannian metric is given by

$g=D^{2}\Phi$

for some globally defined strongly-convex function
$\Phi:\mathcal{M}\to\mathbb{R}$. The Kähler potential on
$\mathbb{M}=T\mathcal{M}$ is then given by the horizontal lift of the convex
potential $\Phi$.

######  Definition 4 (Horizontal Lift).

A function $\Phi^{h}:\mathbb{M}\to\mathbb{R}$ is the horizontal lift of a
function $\Phi:\mathcal{M}\to\mathbb{R}$ if

(13) $\Phi^{h}(x,y)=\Phi(x).$

In other words, the horizontal lift ignores the “vertical” part of the
holomorphic coordinates. Since $\Phi$ is strongly convex on $\mathcal{M}$,
$\Phi^{h}$ is strongly pluri-subharmonic and so defines a Kähler metric.

We can also give an explicit characterization of mirror space $\mathbb{W}.$ We
first compute the Legendre transform of $\Phi$:

$\Phi^{\ast}(u)=\sup_{x\in\Omega}\,\langle u,x\rangle-\Phi(x),$

which is defined on
$\Omega^{\ast}=\\{u\in\mathbb{R}^{n}:\Phi^{\ast}(u)<\infty\\}$. Then we
construct the tube domain $T\Omega^{\ast}$, which will become $\mathbb{W}$.
The Kähler potential on $\mathbb{W}$ is just the horizontal lift
$(\Phi^{\ast})^{h}$.

###  3.2. The case of non-Kähler statistical mirror

It is possible to construct a non-Kähler version of statistical mirror
symmetry as well. This case was studied in depth in our previous work [ZK20],
and we will discuss it briefly here. Readers who are primarily interested in
the Kähler theory should feel welcome to skip this subsection.

At a fundamental level, statistical mirror symmetry can be understood in terms
of duality for affine connections. We have already seen dual connections in
the context of Hessian manifolds, but this duality can be defined for
arbitrary affine connections.282828The notion of conjugate connections is a
central notion in information geometry and has been studied in depth (see,
e.g., [FZ17, CMZ09, Mat10]).

######  Definition 5.

Given a Riemannian manifold with an affine connection $D$, the dual connection
$D^{\ast}$ is the unique connection which satisfies

$Z(g(X,Y))=g(D_{Z}X,Y)+g(X,D^{\ast}_{Z}Y),$

for vector fields $X,Y,$ and $Z$.

Let us recall some basic properties of conjugate connections.

  1. (1)

The dual connection $D^{\ast}$ is curvature-free if and only if the primal
connection $D$ is curvature-free.

  2. (2)

$D$ and $D^{\ast}$ have the same torsion tensor if and only if the quantity

$C^{D}:=Dg$

is a totally symmetric three-tensor. That is to say, if

$C^{D}(X,Y,Z)\equiv(D_{Z}g)(X,Y)$

for any three vector fields on $\mathcal{M}$. In this case, we say that the
pair $(g,D)$ is _Codazzi-coupled_.

  3. (3)

A straightforward computations shows that $C^{D^{\ast}}=-C^{D},$ so $(g,D)$ is
Codazzi-coupled if and only if $(g,D^{\ast})$ is.

####  3.2.1. General construction

To define non-Kähler version statistical mirror symmetry, we consider an
affine Riemannian manifold $(\mathcal{M},g,D)$ with a flat connection $D$, but
do not assume that $g$ is a Hessian metric (that is, $g$ cannot be written as
$D^{2}\Phi$ for some potential function $\Phi$). In this case, the connection
$D^{\ast}$ will be curvature-free, but will have torsion.292929Note that if
$g$ is not Hessian, $D$ and $g$ cannot be Codazzi coupled, because this would
imply that the dual connection $D^{\ast}$ is flat. It is worth noting that
such manifolds are not statistical manifolds, since statistical manifolds are
defined to be Riemannian manifold $(\mathcal{M},g)$ with an affine connection
$D$ such that both $D$ and $D^{\ast}$ are torsion-free [Lau87]. Instead, these
are examples of _statistical manifold admitting torsion_ [Kur07].

A manifold $(\mathcal{M},g,D,D^{\ast})$ where (i) $D$ is flat (both curvature-
and torsion-free) and (ii) $D^{\ast}$ is curvature-free but has non-zero
torsion is said to be _partially flat_ [HM19].303030In this paper, the authors
used the convention that $D^{\ast}$ is the flat connection, not $D$. However,
to be more consistent with the broader literature on mirror symmetry, we chose
to flip this definition in our previous work.

For any partially flat manifold, there are a pair of almost-Hermitian
structures on the tangent space $T\mathcal{M}$, with the first being a
Hermitian manifold $\mathbb{M}$ constructed from $D$ and the second an almost
Kähler manifold $\mathbb{W}$ constructed from $D^{\ast}$. These spaces are
said to be in mirror correspondence, and provide a non-Kähler generalization
of statistical mirror symmetry. To construct these spaces, we again consider
coordinates $x=\\{x^{1},\cdots,x^{n}\\}$ and the associated bundle coordinates

$\phi\left(\left.y^{i}\frac{\partial}{\partial
x^{i}}\right|_{p}\right)=\left(x^{1}(p),\ldots,x^{n}(p),y^{1},\ldots,y^{n}\right).$

Given an affine connection $D$ on $\mathcal{M}$ and a point $(x,y)\in
T\mathcal{M}$, we construct two lifts $X^{H}$ and $X^{V}$ of any vector $X\in
T_{x}\mathcal{M}$, which are vectors in $T_{(x,y)}(T\mathcal{M})$. These are
called the the horizontal and vertical lifts, from $T_{x}\mathcal{M}$ to
$T_{(x,y)}(T\mathcal{M})$, and are defined via the formulas

(14) $X^{V}=X^{i}\frac{\partial}{\partial y^{i}}\textrm{ \qquad and
\qquad}X^{H}=X^{i}\frac{\partial}{\partial
x^{i}}-X^{i}y^{j}\Gamma^{k}_{ij}\frac{\partial}{\partial y^{k}}.$

In this formula, the $\Gamma^{k}_{ij}$’s are the Christoffel symbols of $D$
with respect to the $x$-coordinates. In other words, if we consider the
vectors $e_{i}=\frac{\partial}{\partial x^{i}}$, the $\Gamma^{k}_{ij}$ terms
satisfy the identity

$D_{e_{i}}e_{j}=\Gamma^{k}_{ij}e_{k}.$

These lifts define a splitting of the tangent space $T_{(x,y)}(T\mathcal{M})$
as

$T_{(x,y)}(T\mathcal{M})=\mathcal{H}_{(x,y)}(T\mathcal{M})\oplus\mathcal{V}_{(x,y)}(T\mathcal{M}),$

where the spaces $\mathcal{H}$ and $\mathcal{V}$ are the images of the
horizontal and vertical lift, respectively. Note that the vertical lift (and
thus the vertical space313131More conceptually, the vertical space is the
kernel of the pushforward of the projection map
$\pi:T\mathcal{M}\to\mathcal{M}$.) is independent of the choice of connection
but the horizontal lift depends on the connection. We can then use this
splitting to define an almost complex structure $J_{D}$ and almost-Hermitian
metric $G_{D}$.

  1. (1)

The almost complex structure $J_{D}$ is defined by

$J_{D}(X^{H})=X^{V},\,\,\,\,\,\,\,\,\,J_{D}(X^{V})=-X^{H}.$

  2. (2)

The metric on $T\mathcal{M}$ is lifted from the metric $g$ on $\mathcal{M}$
via the formula

$\displaystyle G_{D}\left(X^{V},Y^{V}\right)=G_{D}\left(X^{H},Y^{H}\right)$
$\displaystyle=$ $\displaystyle g(X,Y),$ $\displaystyle
G_{D}\left(X^{V},Y^{H}\right)$ $\displaystyle=$ $\displaystyle 0,$

It follows immediately that the Riemannian metric $G_{D}$ and the almost
complex structure $J_{D}$ are orthogonal, which implies that the space
$(T\mathcal{M},J_{D},G_{D})$ is an almost Hermitian manifold.

######  Definition 6.

(Non-Kähler statistical mirror symmetry) Given a partially flat manifold
$(\mathcal{M},g,D,D^{\ast})$, the almost-Hermitian manifolds

  1. (i)

$\mathbb{M}=(T\mathcal{M},J_{D},G_{D})$, and

  2. (ii)

$\mathbb{W}=(T\mathcal{M},J_{D^{\ast}},G_{D^{\ast}})$

are said to be a statistical mirror pair.

To explain why this can be understood as a complex-to-symplectic duality, we
first observe that $\mathbb{M}$ is a Hermitian manifold (i.e., $J_{D}$ is
integrable).

######  Proposition 10 (Dombrowski [Dom62]).

The almost complex structure $J_{D}$ on $T\mathcal{M}$ is integrable if and
only $D$ is flat.

In fact, we can construct holomorphic coordinates in the same way we did
previously; by considering affine coordinates $x$ associated to $D$ and
considering the coordinates

$z=\left\\{z^{1},\ldots,z^{n}~{}|~{}z^{i}=x^{i}+\sqrt{-1}y^{i}\right\\}.$

On the mirror side, the following proposition shows that $\mathbb{W}$ is
symplectic.

######  Proposition 11 (Satoh [Sat07]).

The fundamental form $\omega^{\ast}=G_{D^{\ast}}(J_{D^{\ast}}\cdot,\cdot)$ is
closed (i.e., satisfies $d\omega^{\ast}=0$) if and only if $D$ is torsion-
free.

Furthermore, the symplectic form on $\mathbb{W}$ is the pull-back of the
canonical symplectic form on the cotangent bundle $T^{\ast}\mathcal{M}$.
However, since $D^{\ast}$ has non-vanishing torsion, these results show that
$\mathbb{M}$ is non-Kähler (since $d\omega\neq 0$) and that the complex
structure on $\mathbb{W}$ is not integrable.

####  3.2.2. Non-Kähler statistical mirrors for parametrized families

For any exponential family, it is possible to use the natural and expectation
parameters to construct a Kähler statistical mirror pair. However, exponential
families are a special class of statistical manifolds; for more general
parametrized families, it is generally not possible to find a Kähler mirror
pair (see Section 6 for details). As such, non-Kähler statistical mirror
symmetry is useful to understand statistical manifolds which are not
exponential families.

In most cases of interest, the parameter space of a family of probability
distributions is an open domain in Euclidean space, so most statistical
manifold can be induced with a partially-flat structure. For each global set
of coordinates on the domain, there is an associated flat connection which can
be used to construct a mirror pair (which will generally be non-Kähler). As
such, in order to find a canonical geometry, it seems necessary to impose
additional conditions on the geometry of $\mathbb{M}$ and $\mathbb{W}$. In
[ZK20], we proposed one possibility, which is to find a flat connection on
$\mathcal{M}$ so that $\mathbb{M}$ is balanced (i.e., satisfies
$d\omega^{n-1}=0$). The appeal of this geometric condition is that associated
system of PDEs is determined (i.e., has the same number of unknowns as
equations) and that (conformal) balancing arises in the Strominger system,
which has been studied in theoretical physics [GF16]. However, it is quite
possible that statistical considerations will motivate a different geometric
condition entirely.

######  Question 5.

Given a parametrized family of distributions which is not an exponential
family (and thus has no natural parameters), is there a natural condition to
impose on the parametrization so that the associated complex manifold
$\mathbb{M}$ has some desirable statistical properties?

##  4\. More examples of hyperbolic statistical mirrors

In Section 2, we saw that the moduli space of univariate normal distributions
induces a mirror correspondence between two Kähler manifolds, one of which was
a complex space form and the other of which was a Kähler-Ricci soliton. It
might initially be tempting to think of this as being the “canonical”
hyperbolic statistical manifold. However, it turns out that this example is
not unique, in that there are many other examples of statistical mirrors whose
underlying statistical manifold has constant negative sectional curvature. It
is even possible to find other examples where one of the Kähler metrics is a
space of constant negative holomorphic sectional curvature. In this section,
we will provide another such example, which is induced from the moduli space
of negative trinomial distributions. We will also provide one further example
of a statistical manifold $\mathcal{M}$ where the underlying Hessian metric is
hyperbolic, which is derived from the family of inverse Gaussian
distributions.

###  4.1. The geometry of the negative trinomial distribution (NegTri)

One of the common distributions one encounters in statistics is the binomial
distribution, which is the distribution of heads when one performs $k$
independent flips of a coin which is heads with probability $p$ and tails with
probability $1-p$. Conversely, one might flip such a coin repeatedly until
encountering the first tail. The number of heads that appear before the first
tail will be distributed according to the negative binomial distribution. The
negative multinomial distribution generalizes this distribution by having
multiple success events rather than a single one. For example, if one roles a
die repeatedly until a one appears, the distribution of the numbers two
through five will be given by a negative multinomial (in this case, a negative
sextinomial). More precisely, we consider repeated independent draws of a
fixed multinomial distribution where one event is considered a failure event.
The negative multinomial is the distribution of the number of draws of each of
the other events (i.e. the successes) before the first failure
occurs.323232This can be generalized to allow for a specified number of
failures $r$, but for simplicity, we will only consider the case where $r=1$.
This distribution is supported on the discrete space $\left(\mathbb{Z}_{\geq
0}\right)^{n-1}$ and the probability mass function is

$f(k_{1},\ldots,k_{n-1}~{}|~{}p_{1},\ldots,p_{n-1})=\Gamma\\!\left(\sum_{i=0}^{n-1}{k_{i}}\right)\prod_{i=1}^{n-1}{\frac{p_{i}^{k_{i}}}{k_{i}!}}.$

Here, $\Gamma$ is the gamma function and the $p_{i}$ are the probability of
each of the success events. The zero-th event is considered the failure event
and has probability $p_{0}.$

For the sake of concreteness, we will restrict our attention to the negative
trinomial (i.e., where there are two success events and one failure event).
However, all of the calculations in this section can be extended to the
negative multinomial. As we did with normal distributions, we consider the
moduli space of negative trinomial distributions as a statistical manifold,
and compute its Fisher-Rao metric.

$g=\frac{1}{\left(1-p_{1}-p_{2}\right)^{2}}\left[\begin{array}[]{cc}p_{1}(1-p_{2})&p_{1}p_{2}\\\
p_{1}p_{2}&p_{2}(1-p_{1})\\\ \end{array}\right].$

A straightforward calculation shows that this metric has constant curvature
$-1/2$, and so we see that $\mathcal{M}_{\tt NegTri}$ is hyperbolic. However,
unlike the Gaussian family, this metric is not complete. As such, the space of
negative trinomials is a proper subset of hyperbolic space. To determine the
global geometry, it is helpful to reparametrize the family in terms of the
parameters $s_{1}=\sqrt{p_{1}}$ and $s_{2}=\sqrt{p_{2}}$. In these new
coordinates, the Fisher metric for the negative trinomial is

$g_{ij}=\frac{\delta_{ij}}{2(1-s_{1}^{2}-s_{2}^{2})}+\frac{s_{i}s_{j}}{2\left(1-s_{1}^{2}-s_{2}^{2}\right)^{2}},$

which is precisely the Klein disk model for hyperbolic space restricted to the
first quadrant. This gives a picture of the global geometry of
$\mathcal{M}_{\tt NegTri}$, at least from the the perspective of Riemannian
geometry. From this, we see that the metric is unbounded, and that the
boundaries of the manifold occur where the probability of the failure event
approaches one.

The symmetry group of $\mathcal{M}_{\tt NegTri}$ is $\mathbb{Z}_{2}$, and is
generated by exchanging the probabilities of the first and second success
event.333333For the negative multinomial, the symmetry group is $S_{n-1}$ and
is induced by permuting success events. This is a Fisher-Chentsov isometry, as
it is induced by a map of the underlying random variable. To show that there
are no other isometries of $\mathcal{M}_{\tt NegTri}$, note that these are the
only isometries of the Klein disk (with its hyperbolic metric) which preserve
the first quadrant.

####  4.1.1. Negative trinomials as an exponential family

The family of negative trinomial distributions is an exponential family. The
natural parameters of this family are

(15) $\theta^{1}=\log(p_{1}),\,\,\,\,\,\,\,\,\theta^{2}=\log(p_{2}),$

which is defined on the domain

(16) $\Omega_{\tt
NegTri}=\left\\{(\theta^{1},\theta^{2})\in\mathbb{R}^{2}~{}|~{}\exp(\theta^{1})+\exp(\theta^{2})<1\right\\}$

The potential (log-partition) function is

(17) $\Phi_{\tt
NegTri}(\theta)=-\log\left(1-e^{\theta^{1}}-e^{\theta^{2}}\right).$

The corresponding sufficient statistics are

(18)
$\eta_{1}=\frac{p_{1}}{1-p_{1}-p_{2}},\,\,\,\,\,\,\,\,\eta_{1}=\frac{p_{2}}{1-p_{1}-p_{2}},$

which is defined on the set

(19) $\Omega^{\ast}_{\tt
NegTri}=\\{(\eta_{1},\eta_{2})~{}|~{}\eta_{1},\eta_{1}>0\\}.$

In order to transition from the $\theta$ to $\eta$ coordinates, we use the
formula $\eta=D_{\theta}\Phi$:

(20)
$\eta_{1}=\frac{e^{\theta^{1}}}{1-e^{\theta^{1}}-e^{\theta^{2}}},\,\,\,\,\,\,\,\,\eta_{2}=\frac{e^{\theta^{2}}}{1-e^{\theta^{1}}-e^{\theta^{2}}}.$

####  4.1.2. The Kähler geometry of $\mathbb{M}_{\tt NegTri}$

As with the normal family, we can use the natural parameters and sufficient
statistics of the negative trinomial distribution to construct a pair of
statistical mirrors. Let us first consider the primal manifold
$\mathbb{M}_{\tt NegTri}$.

This space is constructed on the tube domain $T\Omega_{\tt NegTri}$, which was
the domain of the natural parameters (15). To obtain a Kähler metric on
$T\Omega_{\tt NegTri}$, we compute the horizontal lift of the log-partition
function $\Phi_{\tt NegTri}(\theta)$ given by (17). We enumerate some of the
important properties of $\mathbb{M}_{\tt NegTri}$ below.

  1. (1)

It is incomplete (since $\mathcal{M}_{\tt NegTri}$ is incomplete).

  2. (2)

It has constant negative holomorphic sectional curvature. This implies that it
is Kähler-Einstein, has constant negative scalar curvature, negative
orthogonal bisectional curvature and vanishing orthogonal anti-bisectional
curvature.

  3. (3)

As a space of constant holomorphic sectional curvature, it is locally (but not
globally) holomorphically isometric to $\mathbb{W}_{\tt Normal}$.

  4. (4)

It is biholomorphic to a tube domain whose base is
$\\{(\theta^{1},\theta^{2})~{}|~{}\exp(\theta^{1})+\exp(\theta^{2})<1\\}$.

######  Question 6.

Is there a global isometric embedding from $\mathbb{M}_{\tt NegTri}$ to the
Siegel half-plane?

In other words, is it possible to consider $\mathbb{M}_{\tt NegTri}$ as a
subset of the Siegel half-space? We suspect that the answer to this question
is negative, but do not have a proof.343434The moduli space of multinomial
distributions induces an incomplete space of constant positive holomorphic
sectional curvature (see Subsection 5.3). This space has a natural immersion
into $\mathbb{CP}^{n}$, but no embedding. It is likely that the same phenomena
occurs with negative multinomials as well.

####  4.1.3. Properties of $\mathbb{W}_{\tt NegTri}$

The dual manifold $\mathbb{W}_{\tt NegTri}$ is induced by the sufficient
statistics (18) and is defined on the tube domain $T\Omega^{\ast}_{\tt
NegTri}$. Its Kähler potential is given by the horizontal lift of the Legendre
dual of the log-partition function, which is

(21) $\Phi_{\tt
NegTri}^{*}(\eta)=\eta_{1}\log(\eta_{1})+\eta_{2}\log(\eta_{2})-(1+\eta_{1}+\eta_{2})\log(1+\eta_{1}+\eta_{2}).$

We note some important properties of $\mathbb{W}_{\tt NegTri}$.

  1. (1)

It is incomplete.

  2. (2)

The Ricci curvature is negative. Showing this requires a bit of computation.
To show this, we first consider the matrix

$\operatorname{Ric}_{ij}=-\frac{\partial^{2}\log[\det[D^{2}\Phi_{\tt
NegTri}^{*}(\eta)]}{\partial\eta_{i}\partial\eta_{j}}.$

We then lift one of the indices and consider the matrix

$\operatorname{Ric}_{i}^{j}=\operatorname{Ric}_{ik}\,g^{jk}.$

The determinant of this matrix is

$\det[\operatorname{Ric}_{i}^{j}]=\frac{1+2\left(\eta_{1}+\eta_{1}\cdot\eta_{1}+\eta_{2}+\eta_{2}\cdot\eta_{2}+\eta_{1}\cdot\eta_{2}\right)}{\eta_{1}\eta_{2}(1+\eta_{1}+\eta_{2})},$

which is positive. On the other hand, the trace of
$\operatorname{Ric}_{i}^{j}$ (i.e., half the scalar curvature), is negative.

$\operatorname{Tr}[\operatorname{Ric}_{i}^{j}]=-3-\frac{1}{\eta_{1}}-\frac{1}{\eta_{2}}+\frac{1}{1+\eta_{1}+\eta_{2}}.$

From this it follows that the eigenvalues of the Ricci curvature are negative,
and that the scalar curvature is negative (but not constant).

  3. (3)

Neither the orthogonal bisectional curvature nor the holomorphic sectional
curvature have a sign.

  4. (4)

It is biholomorphic to a tube domain whose base is the first quadrant, which
is biholomorphic to the bidisk in $\mathbb{C}^{2}$. Although the domain admits
a transitive group of automorphisms, most of the automorphisms are not
isometries and the metric is not homogeneous .

  5. (5)

The spaces $\mathbb{M}_{\tt NegTri}$ and $\mathbb{W}_{\tt NegTri}$ are not
biholomorphic. To see this, note that

$\Phi(\theta)^{h}=\left(e^{\theta^{1}}+e^{\theta^{2}}-1\right)^{h}$

is a pluri-subharmonic exhaustion function on $T\Omega_{\tt NegTri}$ which is
strictly pseudo-convex on the boundary $\partial T\Omega$. However, the bidisc
is not biholomorphic to any strictly pseudo-convex domain.

####  4.1.4. Statistical mirror symmetry and biholomorphic isometries

This example of statistical mirror symmetry, in conjunction with the one in
Section 2, highlights a property of T-duality that is initially surprising.
Upon first encountering the concept, one might imagine that if two semi-flat
Kähler manifolds are locally biholomorphically isometric, that their dual
spaces are also locally isometric. This is the case for compact semi-flat
Calabi-Yau manifolds with their Ricci flat metrics, since such spaces are
simply complex tori. Such an isometry need not extend globally, and the
failure for a local isometry to extend can be seen at the length scale of the
co-compact lattice.

However, if we drop the assumption that the metric is Ricci-flat (as done in
[FP19b]), then the situation becomes more complicated. In particular, it is
possible to construct semi-flat metrics which have neighborhoods that are
locally biholomorphically isometric, but whose dual spaces are not locally
isometric. More precisely, if we quotient the fibers of $\mathbb{M}_{\tt
NegTri}$ and $\mathbb{W}_{\tt Normal}$ by a pair of co-compact lattices, the
resulting spaces will be locally biholomorphically isometric, since they are
spaces with constant holomorphic sectional curvature.353535There are other
examples of Hessian metrics whose $r$-map has constant holomorphic sectional
curvature. For instance, if we consider the domain
$\Omega=\\{(x^{1},x^{2})\in\mathbb{R}^{2}~{}|~{}x^{1}-\exp(x^{2})>0\\}$, the
convex potential $\Phi(x)=-\log(x^{1}-\exp(x^{2}))$ gives another. 363636As
with the case of flat tori, the failure of a global isometry will be apparent
at the length-scale of the co-compact lattices, since the dually-flat
structures are very different.  On the other hand, $\mathbb{W}_{\tt NegTri}$
and $\mathbb{M}_{\tt Normal}$ are distinct as Kähler manifolds and remain so
when quotiented by dual co-compact lattices. For instance, $\mathbb{M}_{\tt
Normal}$ has constant scalar curvature whereas $\mathbb{W}_{\tt NegTri}$ does
not. These spaces are non-compact (and cannot be quotiented to obtain compact
examples), but it is possible to locally patch these potentials in a compact
Hessian manifold (at the expense of deforming the potential elsewhere) to
obtain semi-flat tori $\mathbb{M}_{1}$ and $\mathbb{M}_{1}$ which have
biholomorphically isometric open subsets but whose T-duals $\mathbb{W}_{1}$
and $\mathbb{W}_{1}$ have no isometric neighborhoods.

###  4.2. Inverse Gaussian distributions

We now consider another exponential family whose underlying statistical
manifold is hyperbolic, which is the family of inverse Gaussian distributions.
This is the family of distributions whose probability density is given by

$f_{\tt
InvGau}(\zeta|~{}\lambda,\mu)=\sqrt{\frac{\lambda}{2\pi\zeta^{3}}}\exp\left[-\frac{\lambda(\zeta-\mu)^{2}}{2\mu^{2}\zeta}\right].$

where $\lambda$ and $\mu$ are the shape parameter and mean, respectively. As
the name suggests, there is a relationship between this distribution and the
Gaussian distribution. However, this connection requires some explanation
since it involves stochastic processes.

We consider the stochastic process $X(t)$ given by

$\displaystyle X(0)=0,\qquad\qquad X(t)=\nu t+W(t),$

where $W(t)$ is standard Brownian motion and $\nu$ is a fixed positive
constant which denotes the speed of the drift. We then consider the arrival
times $T_{\alpha}$ at a fixed level $\alpha>0$, which are given by

$T_{\alpha}=\inf\\{t>0~{}|~{}X(t)=\alpha\\}$

and are distributed as

$T_{\alpha}\sim{\tt
InvGau}\left(\frac{\alpha}{\nu},\left(\frac{\alpha}{\nu}\right)^{2}\right).$

For any pair of times, $W(t)-W(s)$ will be normally distributed, which gives
the connection between inverse Gaussian distributions and Gaussian
distributions.

As before, we consider the moduli of inverse Gaussian distributions as a
statistical manifold, and calculate its Fisher metric. In the $\mu,\lambda$
coordinates, this is given by

$g=\left[\begin{array}[]{cc}\frac{1}{2\lambda^{2}}&0\\\
0&\frac{\lambda}{2\mu^{3}}\\\ \end{array}\right]$

When we compute the curvature, we see that $\mathcal{M}_{\tt InvGau}$ has
constant negative curvature, so is hyperbolic. Furthermore, the metric is
unbounded but incomplete. To understand the global geometry of this space, it
is helpful to change of coordinates to

$x^{1}=\mu^{-1/2}\,\,\,\mbox{and}\,\,\,x^{2}=\lambda^{-2}.$

In the $(x^{1},x^{2})$ coordinates, $\mathcal{M}_{\tt InvGau}$ is the standard
half-plane model restricted to the first quadrant.

####  4.2.1. Symmetries of the moduli of inverse Gaussian distributions

By studying the geometry of the moduli of inverse Gaussians, it is possible to
prove new statistical results. For instance, the previous calculations allow
us to understand the symmetries of the inverse Gaussian family.

######  Theorem 12.

The isometry group of the inverse Gaussian family is the infinite dihedral
group $\mathbb{R}\ltimes\mathbb{Z}_{2}$ generated by the mappings

$(\mu^{-1/2},\lambda^{-2})\mapsto(\alpha\mu^{-1/2},\alpha\lambda^{-2})$

for $\alpha>0$ and inversions in the $(x^{1},x^{2})$-coordinates around a
circle centered at the origin.

###### Proof.

To see this, it suffices to compute the symmetries of the hyperbolic half-
plane which fix the line $x^{1}=0$. This is an infinite dihedral group,
generated by scaling and inversions around a circle centered at the origin. ∎

The fact that scaling generates a symmetry has a natural interpretation as it
corresponds to scaling the drift $\nu$ and the level $\alpha$ for the
stochastic process $X(t)$. This symmetry is not Fisher-Chentsov, but its
physical meaning is clear. However, we are not aware of a physical
interpretation for the isometry given by inversions.

######  Question 7.

Is there a natural geometric interpretation for the symmetry of the moduli of
inverse-Gaussians induced by inversions?

####  4.2.2. Properties of $\mathbb{M}_{\tt InvGau}$

Inverse Gaussian distributions are another example of an exponential family.
As such, we can use their moduli space to construct a Kähler statistical
mirror pair. To do so, we consider the natural parameters

$\displaystyle\theta^{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2\mu^{2}},$ $\displaystyle\theta^{2}$
$\displaystyle=$ $\displaystyle-\frac{\lambda}{2}$

and use the connection associated with these coordinates. In this case, the
log-partition function is given by

$\Phi(\theta)=-\sqrt{\theta^{1}\theta^{2}}-\frac{1}{2}\log(-\theta^{2}),$

which is supported on $\Omega=\\{\theta^{1},\theta^{2}<0\\}$. The Kähler
manifold $\mathbb{M}_{\tt InvGau}$ has the following properties.

  1. (1)

It has negative Ricci curvature (and hence negative scalar curvature).

  2. (2)

The holomorphic sectional curvature, antibisectional curvature and orthogonal
bisectional curvatures do not have a sign.

  3. (3)

$\mathbb{M}_{\tt InvGau}$ a tube domain whose base is the third quadrant. As
such, it is biholomorphic to the bidisk in $\mathbb{C}^{2}$.

####  4.2.3. Properties of $\mathbb{W}_{\tt InvGau}$

To construct the dual Kähler manifold, we consider the dual connection whose
associated affine coordinates are

$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial\Phi}{\partial\theta^{1}}=-\frac{1}{2}\sqrt{\frac{\theta^{2}}{\theta^{1}}}$
$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle\frac{\partial\Phi}{\partial\theta^{2}}=-\frac{1}{2}\sqrt{\frac{\theta^{1}}{\theta^{1}}}-\frac{1}{\theta^{2}}.$

The Hessian potential in these coordinates is is

$\Phi^{\ast}(\eta)=\frac{1}{2}\left[-1+\log\left(\frac{2\eta_{1}}{-1+4\eta_{1}\eta_{2}}\right)\right]$

which is defined on the hyperbola
$\Omega^{\ast}=\\{(\eta_{1},\eta_{2})\in\mathbb{R}^{2}~{}|~{}-1+4\eta_{1}\cdot\eta_{2}>0,\eta_{1}>0\\}$.
The Kähler manifold $\mathbb{W}_{\tt InvGau}$ is defined on the tube domain
$T\Omega^{\ast}$ and has the following properties.

  1. (1)

It has negative Ricci curvature (and so negative scalar curvature).

  2. (2)

It has negative orthogonal anti-bisectional curvature.

  3. (3)

Neither the orthogonal bisectional curvature nor the holomorphic sectional
curvature have a sign.

##  5\. Frobenius manifolds and their statistical mirrors

Thus far, we have considered examples where the underlying statistical
manifold has negative curvature. In this section, we turn our attention to
statistical manifolds which are Riemannian-flat and consider their
sttatistical mirrors. The primary motivation for studying such spaces is that
they induce solutions to the Witten–Dijkgraaf–Verlinde–Verlinde equations
(WDVV equations), which play an important role in mathematical physics.

###  5.1. WDVV equations

The WDVV equations are the system of equations

(22)
$\sum_{p,q}\Phi^{pq}\Phi_{jlp}\Phi_{ikq}=\sum_{p,q}\Phi_{ilp}\Phi_{jkq}\Phi^{pq},$

which are indexed in the variables $i,j,k$ and $\ell$. Here, we use the
notation $\Psi_{J}$ to denote

$\frac{\partial^{|J|}\Psi}{\partial u^{J}}$

for a multi-index $J$ and use $\Psi^{ij}$ to denote the $i,j$-th component of
the matrix inverse of the Hessian $\Psi_{\alpha\beta}$.

A solution of these equations give a flat Riemannian manifold
$(\mathcal{M},g)$ the structure of a Frobenius manifold. In other words, a
solution to these equations induces a commutative, associative product on the
tangent bundle. For a myriad of applications in topological field theory and
physics, see [Dub96] [Mag15] [Man99]. Furthermore, the WDVV equations are of
interest from the perspective of information geometry due to the following
correspondence.

######  Observation ([Kit99]).

The potential of a Riemannian-flat Hessian metric solves the WDVV equations.

To see this, we can calculate the curvature of a Hessian metric in the
associated affine coordinates (see [Tot04] for details). Doing so, we find the
following:

(23)
$R_{ijkl}=-\frac{1}{4}\sum_{p,q}\Phi^{pq}\left(\Phi_{jlp}\Phi_{ikq}-\Phi_{ilp}\Phi_{jkq}\right).$

As a word of caution, this quantity is the curvature of the underlying Hessian
manifold, and not the curvature of the Kähler metrics on its tangent bundle
(which involve fourth derivatives of the potential as well). From this, we see
that a convex function $\Phi$ solves the WDVV equations if and only if its
associated Hessian metric is Riemannian-flat. This correspondence gives a
geometric way to understand the WDVV-equations and to derive some of its
properties.

For instance, it provides a particularly simple proof that the Legendre
transformation of one solution to the WDVV-equations yields a second solution.

######  Observation.

Suppose that a convex potential $\Phi:\Omega\to\mathbb{R}$ for
$\Omega\subset\mathbb{R}^{n}$ satisfies Equations 22. Then the Legendre dual
$\Phi^{*}:\Omega^{\ast}\to\mathbb{R}$ also satisfies Equations 22.

To see this, note that the Hessian manifolds $(\Omega,\Phi)$ and
$(\Omega^{\ast},\Phi^{\ast})$ are isometric as Riemannian manifolds (in fact,
$\Omega$ and $\Omega^{\ast}$ can be interpreted as being two sets of
coordinates on the same underlying manifold). Furthermore, we can use this
perspective to simplify the WDVV equations in dimensions two and three.

######  Proposition 13.

  1. (1)

A convex potential $\Phi:\Omega\to\mathbb{R}$ for
$\Omega\subset\mathbb{R}^{2}$ satisfies Equations 22 if and only if it
satisfies the (single) equation

(24)
$\sum_{i,j,k,l,p,q}\Phi^{jl}\Phi^{ik}\Phi^{pq}\Phi_{jlp}\Phi_{ikq}=\sum_{i,j,k,l,p,q}\Phi_{ilp}\Phi_{jkq}\Phi^{pq}\Phi^{jl}\Phi^{ik}.$

  2. (2)

A convex potential $\Phi:\Omega\to\mathbb{R}$ for
$\Omega\subset\mathbb{R}^{3}$ satisfies Equations 22 if and only if it
satisfies the six equations (indexed symmetrically in $i$ and $k$)

(25)
$\sum_{j,l,p,q}\Phi^{jl}\Phi^{pq}\Phi_{jlp}\Phi_{ikq}=\sum_{j,l,p,q}\Phi_{ilp}\Phi_{jkq}\Phi^{pq}\Phi^{jl}.$

###### Proof.

Equation 24 is satisfied if and only if the scalar curvature of a Hessian
metric vanishes. For surfaces, the Riemannian curvature is completely
determined by the scalar curvature, so the metric is flat whenever the scalar
curvature vanishes.

Similarly, Equations 25 are satisfied whenever the Ricci curvature of the
associated Hessian metric vanishes. For three-folds, the Ricci curvature
completely determines the Riemannian curvature, so the metric is flat whenever
its Ricci curvature vanishes. ∎

###  5.2. Examples

As we will see in the next section, Frobenius structures exist in abundance.
Here, we have included a partial list of such potentials. For each of these
examples, the Legendre dual will induce a separate Frobenius structure as
well.

  1. (1)

The simplest example of a Frobenius manifold is $\mathbb{R}^{n}$ with the
quadratic potential

$\Phi=\sum_{i=1}^{n}\frac{x^{i}\cdot x^{i}}{2}.$

This is the unique structure which is self-dual.

  2. (2)

Another example of a Frobenius structure is $\mathbb{R}^{2}$ with the Hessian
potential

$\Phi(x^{1},x^{2})=\log(\cosh(x^{1})+\cosh(x^{2})).$

  3. (3)

Closely related to the previous example, we can replace the hyperbolic cosines
for normal cosines and consider the potential

$\Phi(x^{1},x^{2})=-\log(\cos(x^{1})+\cos(x^{2})),$

which is defined on a diamond in $\mathbb{R}^{2}$. The curvature of the
associated Kähler manifold is opposite that of the previous example.

  4. (4)

One well-known example is the domain

$\Omega=\\{x\in\mathbb{R}^{2}~{}|~{}x^{2}<|x^{1}|\\}$

with the potential

$\Phi(x^{1},x^{2})=-\log(x^{1}\cdot x^{1}-x^{2}\cdot x^{2}).$

This example is of interest because it is possible to quotient this cone by
co-compact automorphisms to construct a compact Hessian manifold. This example
can be generalized to arbitrary dimensions by reparametrizing the coordinates
as $\theta^{1}=x^{1}-x^{2}$ and $\theta^{2}=x^{1}+x^{2}$ and considering the
more general potential

$\Phi=-\sum_{i=1}^{n}\log(\theta^{i}),$

which is defined in the positive cone

$\Omega=\\{\theta=(\theta^{1},\cdots,\theta^{n})\in\mathbb{R}^{n}~{}|~{}\theta^{i}>0\\}.$

The $x$-parametrization indicates that there may be a connection between this
metric and the Siegel half space. More precisely, if we consider the
potentials

$\Phi_{\epsilon}=\log\left(\left(\epsilon
x^{1}+\epsilon^{-1}\right)^{2}-\frac{1}{\epsilon^{2}}-x^{2}\cdot
x^{2}\right),$

it might be possible to obtain the Siegel half-space as a suitably
renormalized limit.

######  Question 8.

Is there a precise sense in which the Siegel half-space is the limit of cones?
If so, are there any applications for this fact?

###  5.3. Positive curved statistical mirrors

We have now studied statistical mirror symmetry in both the flat case and the
hyperbolic case, so it is natural to consider positively curved Hessian
manifolds as well. However, when one tries to do so, one encounters the
immediate problem that any complete space of strongly positive sectional
curvature is not affine.373737To see this, note that Myer’s theorem implies
that the fundamental group of a manifold with strongly positive curvature is
finite, whereas the fundamental group of a compact affine manifold must be
infinite [AT02]. Therefore, it is only possible to construct Hessian metrics
on an open subset of spaces of strongly positive sectional curvature. In
[KZ20a], the authors studied one such example, which was given by the Hessian
potential

$\Phi(x)=\log\left(1+\sum_{j=1}^{n-1}e^{x^{j}}\right),$

which is defined on all of $\mathbb{R}^{n-1}$. This is the dually flat
geometry of the probability simplex

$\Delta^{n}=\left\\{(p_{1},\cdots,p_{n})\in\mathbb{R}^{n}~{}|~{}p_{i}\geq
0,\,\sum_{i=1}^{n}p_{i}=1\right\\}.$

This example is motivated by statistical considerations, since it induced by
the exponential family of multinomial distributions. It also has applications
in optimal transport and mathematical finance (see [PW16, PW18] for details).
Interestingly, the Kähler metric associated to this potential has constant
positive holomorphic sectional curvature and is defined on $\mathbb{C}^{n}$.
Since it is incomplete, this does not contradict the fact that complete Kähler
manifolds with positive constant holomorphic sectional curvature are
biholomorphically isometric to $\mathbb{CP}^{n}$.

If one drops the assumption of strongly positive curvature, it may be possible
to find examples of complete metrics which have positive curvature in some
weaker sense. In [KZZ20], the authors and Zheng studied this question and
constructed $O(n)$-symmetric metrics with non-negative orthogonal anti-
bisectional curvature. However, we showed that there are no non-trivial
$O(n)$-symmetric Kähler Sasaki metrics with positive bisectional curvature (or
orthogonal bisectional curvature for $n\geq 3$).

##  6\. The existence theory for Hessian metrics

Thus far, we have not discussed the existence theory for Hessian metrics in
any detail. It is a natural question of whether, given a Riemannian manifold
$(\mathcal{M},g)$ with $\dim(\mathcal{M})=n$, there exists a flat connection
$D$ (either globally or locally) and a potential $\Phi$ so that

(26) $g=D^{2}\Phi.$

From a global perspective, we can immediately see that in order for such a
structure to exist, the manifold $\mathcal{M}$ must be affine. That is, it
must admit a flat connection. This is a fairly restrictive condition from a
topological perspective. For instance, the only compact orientiable surface
which admits an affine structure is the torus.383838In fact, the torus admits
several affine structures which are inequivalent (see [Yag81] for details).
However, merely admitting an affine structure is not sufficient for a manifold
to admit a Hessian metric. For instance, the Hopf manifolds $S^{n}\times
S^{1}$ (for $n>1$) do not admit such a structure [Yag81].

It is also possible to study this question _locally_. For this, one considers
a Riemannian manifold $(\mathcal{M},g)$ and a point $x\in\mathcal{M}$. The
local question studies whether it is possible to find a connection $D$ which
is flat in a neighborhood of $x$ and which satisfies $D^{2}\Phi=g$?

When the dimension is greater than two, generic Riemannian metrics do not
admit a Hessian metric, even locally. Intuitively, this can be seen by
counting equations; a Hessian potential and flat connection can be specified
by $n+1$ local functions ($n$ functions for the associated affine coordinates
and a single function for the convex potential) whereas a generic Riemannian
metric can be understood as a map from each point (in a local chart) to the
space of $n\times n$ symmetric positive definite matrices, which has dimension
$\frac{(n+1)n}{2}$. In fact, for $n\geq 4$, there are point-wise curvature
obstructions to the existence of a Hessian metric (see [AA14] for details).
However, in dimension two, Amari and Armstrong proved that given a real
analytic metric, it is always possible to find a Hessian structure
locally.393939Another proof of this fact was given by Bryant in a MathOverflow
answer. [Bry]

The complement to the existence question is uniqueness. In other words, given
a Hessian manifold $(\mathcal{M},g,D)$, is it possible to find another
connection $D^{\prime}$ so that $(\mathcal{M},g,D^{\prime})$ is a Hessian
manifold? By considering the connection $D^{\prime}=D^{\ast}$, we can
immediately see that Hessian structures are not unique. In fact, our examples
show that there are potentially many different Hessian structures which induce
the same Riemannian geometry.

The moduli space of Hessian metrics (for some particular Riemannian metrics)
was studied by Kito [Kit99]. In particular, he considered the Euclidean plane
and showed that the space of associated Hessian metrics has the freedom of
three local functions of $\mathbb{R}$. He also showed that space of Hessian
metrics for flat Euclidean $\mathbb{R}^{n}$ has at least the freedom of $n$
functions. In practice, this means that flat Hessian metrics (and thus
solutions to the WDVV equations) exists in abundance. Furthermore, he
considered hyperbolic $n$-space and showed that the space of Hessian metrics
has at least the freedom of $n-1$ functions on $\mathbb{R}$. This observation
naturally raises the following question.

######  Question 9.

Are there other dually flat structures (other than the one from normal
distributions) defined on the entire hyperbolic plane which are _highly
symmetric_?

There are infinitely many dually flat structure on the hyperbolic plane,
though there are no other complete examples whose tube domains have constant
holomorphic sectional curvature. Still, it would be of interest to find other
examples with large automorphism group or which give rise to a Kähler-Einstein
metric.

##  7\. Acknowledgements

This projected is supported by DARPA/ARO Grant W911NF-16-1-0383 (“Information
Geometry: Geometrization of Science of Information”) and AFOSR Grant
FA9550-19-1-0213 (“Brain-Inspired Networks for Multifunctional Intelligent
Systems and Aerial Vehicles”), which also supported the first author when he
was at the University of Michigan. The first author is currently partially
supported by a Simons Collaboration Grant 849022 (“Kähler-Ricci flow and
optimal transport”). We thank Fangyang Zheng for some helpful conversations on
the curvature of Kähler manifolds. We also thank Teng Fei for providing
comments related to T-duality.

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