# Matrix Models, Integral Polyhedra and Toric Geometry

Aleksey Andreev<EMAIL_ADDRESS>
MIPT/TH-20/22

ITEP/TH-23/22

IITP/TH-22/22

Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia
NRC, “Kurchatov Institute”, Moscow 123182, Russia111former Institute for
Theoretical and Experimental Physics, Moscow 117218, Russia

ABSTRACT

We propose to take a look at a new approach to the study of integral
polyhedra. The main idea is to give an integral representation, or matrix
model representation, for the key combinatorial characteristics of integral
polytopes. Based on the well-known geometric interpretations of matrix model
digram techniques, we construct a new model that enumerates triangulations,
subdivisions, and numbers of integral points of integral polygons. This
approach allows us to look at their combinatorics from a new perspective,
motivated by knowledge about matrix models and their integrability. We show
how analogs of Virasoro constraints appear in the resulting model. Moreover,
we make a natural generalization of this matrix model to the case of polytopes
of an arbitrary dimension, considering already a tensor model. We also obtain
an analogue of Virasoro constraints for it and discuss their role in the
solvability of these models. The deep connection between the geometry of
convex polyhedra and toric geometry is the main reference point in the
construction of these models. We present considerations on specific ways of
applying this approach to the description of Batyrev’s mirror pairs. All this
allows us to formulate many interesting directions in the study of the
connection between matrix/tensor models and the geometry of toric varieties.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Convex integral polyhedra
    2. 2.2 Matrix models
  3. 3 Matrix model of triangulations
    1. 3.1 Constructing matrix model for polygons
    2. 3.2 Higher degree generalization
    3. 3.3 Ward identities/Virasoro algebra
    4. 3.4 Higher dimensional generalization and tensor model
  4. 4 Toric geometry applications
    1. 4.1 Toric geometry
    2. 4.2 Moduli space
  5. 5 Discussion
    1. 5.1 Main results
    2. 5.2 Open questions and further research
  6. 6 Acknowledgements
  7. 7 Appendix A. Proof of the Theorem 1
  8. 8 Appendix B. Virasoro-like algebra of trees

##  1 Introduction

In this paper, we propose to look at the combinatorics of convex polyhedra and
toric geometry from a new perspective. Namely, we propose to apply the method
of matrix models to the study of combinatorial problems about integral
polyhedra, which in turn are closely related to problems of toric geometry. It
seems to us that such an approach can be very useful in the future both from
the point of view of toric geometry and from the point of view of matrix
models.

Matrix models are an effective method for solving various problems in
combinatorics and enumerative geometry. One of the most famous examples is
Kontsevich matrix model [1], which enumerates integrals of $\psi$-classes of
moduli spaces of complex curves. The distinctive feature of matrix models is
the reach variety of analysis methods since they are always finite-dimensional
well-defined integrals [2]. It is often occur that matrix models turn out to
be solutions of some integrable hierarchy, which makes it possible to “solve
it completely”, which means obtaining exact answers for all (invariant)
correlators in this model [3, 4, 5]. For example, the Kontsevich model
satisfies the equations of the KdV hierarchy, and AMM/EO topological recursion
[6, 7] allows one to write recursive relations on correlators, which are
exactly integrals of $\psi$-classes over the moduli space. This example is far
from the only one, there are many other problems solved by matrix models (or
in which topological recursion can be found) [2, 8, 3, 9, 10, 11]. Inspired by
many of these examples, we try to apply this approach to describe the
combinatorics of convex integral polyhedra.

The combinatorics of convex integral polyhedra is amusing in itself, but we
are interested in it because of the well-known connection between the
combinatorics of polyhedra and the geometry of toric varieties [12, 13, 14].
In particular, we are especially interested in further applying the technique
of matrix models to problems of geometry of Batyrev’s mirror pairs [15], which
correspond to reflexive convex integral polyhedra.

In section 2 we give the basic information about integral polyhedra that we
need and a brief description of some techniques of matrix models using the
Hermitian one as an example.

In this paper, however, we do not discuss applications to Batyrev’s theory
yet, but we concentrate on the first natural problem in this direction, which
is to construct examples of matrix models that describe in some sense the
combinatorics of convex integral polyhedra. More specifically, we would like
to build matrix models that:

  * •

compute the number of integral points in an integral polyhedron,

  * •

enumerate triangulations (subdivisions into polytopes) of integral polyhedra

The first one, although simple, is quite interesting, because the Hodge
numbers of Calabi-Yau hypersurfaces in Batyrev’s construction are expressed in
terms of the numbers of integral points inside the (sub)polyhedron. The second
is directly related to the combinatorics of secondary polytope introduced by
Gelfand-Kapranov-Zelevinsky .

In order to build a matrix model, we use a simple observation about known
models with a cubic potential: it’s diagram technique enumerate various
triangulations of surfaces with certain weights. The simplest example of such
a matrix model is the Hermitian matrix model with a cubic potential whose
partition function has the form:

$Z=\int\mathcal{D}H\exp\left\\{-\frac{1}{2}Tr(H^{2})+t_{3}Tr(H^{3})\right\\}$
(1)

where the integration is over all Hermitian matrices and the expansion in a
series in $t_{3}$ under the integral is implied. Due to Wick’s rule, the terms
of the series in $t_{3}$ enumerate various triangulations of Riemann surfaces
of various genera with a fixed number of vertices. There are also more
complicated examples of such matrix models, where the correlators can be given
the appropriate geometric meaning: the Kontsevich model, the Kazakov model,
Hurwitz numbers model and so on. Based on similar considerations, we construct
a matrix model for triangulating two-dimensional integral (not necessarily
convex) polytopes in the subsection 3.1. To do this, we “deform” the Hermitian
matrix model. Unfortunately the deformation violate the unitary invariance,
but we can still apply some methods motivated by matrix models. From this
point of view our models, strictly speaking, are not matrix (or tensor). But
for the convenience, we will use this name in relation to the constructed
models.

In the context of matrix models, it is natural to add all higher powers to the
potential [16]. For example, considering Hermitian matrix model, we usually
replace cubic potential in the following way:

$-\frac{1}{2}Tr(H^{2})+t_{3}Tr(H^{3})\rightarrow-\frac{\mu}{2}Tr(H^{2})+\sum\limits_{n=0}^{\infty}t_{n}Tr(H^{n})$
(2)

We do the same for our matrix model of two-dimensional polytopes in subsection
3.2. It turns out that this is also natural from the point of view of integral
polyhedra, since it enumerate subdivisions into polytopes. In addition, in
subsection 3.3 this allows us to write the Virasoro operator algebra of Ward
identities in a similar way as for Hermitian matrix model (we briefly discuss
the HMM example in subsection 2.2):

$[L_{n},L_{m}]=(n-m)L_{n+m}$ (3)

The presence of such an algebra of operators indicates the presence of
integrability in such a model [17, 18, 19].

Finally, in subsection 3.4 we generalize this approach to the higher
dimensional integral polyhedra, but in terms of “tensor” models [20, 21, 22,
23, 24].

In section 4, we present our considerations regarding the applications of the
models we have obtained to the description of Batyrev’s mirror pairs [15, 25].
First, we give some basic definitions and statements from this area of
science, after which we discuss what facts allow us to assume the
applicability of our models to the enumeration of the Hodge numbers of Calabi-
Yau hypersurfaces in toric varieties and the description of the moduli spaces
of complex structures on them.

At the last section 5 we list our main results and discuss possible directions
for further research on this topic and most interesting questions in our
opinion.

##  2 Preliminaries

In this section, we give some background information with references. Since
our goal is to connect two subjects: matrix models and combinatorics of
polyhedra, there are two subsections here, one for each side. We do not give
any definitions and statements from toric geometry in this section, since in
fact, the main discussion below is devoted to matrix models describing
combinatorics of integral polyhedra. In section 4 we will discuss toric
geometry and our thoughts on its connections with our matrix model.

###  2.1 Convex integral polyhedra

From the side of combinatorics, the object we are studying – an integral
convex polyhedron – has a simple definition. Nevertheless, its combinatorics
isn’t trivial and quite deep. As mentioned in introduction, in future we would
like to connect the theory of matrix models with the theory of generic
hypersurfaces in toric varieties, so we interested in numbers $\ell(\Delta)$
and secondary polytope. We give both definitions here.

Let $M\simeq\mathbb{Z}^{n}$ be an $n$-dimensional lattice (free abelian
group). The corresponding $\mathbb{R}$-vector space defined as
$M_{\mathbb{R}}:=M\otimes\mathbb{R}\simeq\mathbb{R}^{n}$.

A convex polyhedron in $M_{\mathbb{R}}$ is the convex hull of finitely many
points in $M_{\mathbb{R}}$. A convex integral polyhedron in $M_{\mathbb{R}}$
is the convex hull of finitely many points from $M$. The dimension of a
polytope is the dimension of the minimal affine subspace containing this
polytope.

An affine hyperplane intersecting a polyhedron is called supporting if the
entire polyhedron lies in one of the two closed half-spaces defined by it. The
intersection of such a plane with a polyhedron is called a face (which, of
course, is a convex polyhedron). If the dimension of a face is equal to $n-1$,
then it is called facet.

Let us define the numbers $\ell(\Delta)$ and $\ell^{*}(\Delta)$ (the notation
is taken from Batyrev’s papers on dual polyhedra and Calabi-Yau hypersurfaces)
for a convex integral polytope $\Delta$. $\ell(\Delta)$ is just the number of
integral points (i.e., $M$ lattice points) in the polyhedron:

$\ell(\Delta):=\\#\\{x\in M|x\in\Delta\\}$ (4)

$\ell^{*}(\Delta)$ is the number of interior points of a polyhedron that do
not belong to its facets.

As mentioned above, another combinatorial object that interests us is the
secondary polytope [26].

The easiest way to construct a secondary polytope is to use the characteristic
functions of all possible $\Delta$ triangulations.

A triangulation $\tau$ of the integral polytope $\Delta$ is a decomposition of
it into a finite number of simplices (of dimension $n$), such that the
vertices of the simplices are integral points, and the intersection of any two
simplices is their common (possibly empty) face.

For each triangulation $\tau$ there is the associated characteristic function
$\phi_{\tau}$ on the set of vertices of this triangulation:

$\phi_{\tau}(a)=\sum\limits_{\\{\sigma|a\in\sigma\\}}\textrm{Vol}(\sigma),$
(5)

where the summation is over all simplices for which $a$ is a vertex.

Now one has to extend $\phi_{T}$ to all points in $\Delta\cap N$, taking
$\phi_{\tau}(a)=0$ if $a\in\Delta\cap N$ is not a vertex of the triangulation
$\tau$.

The secondary polytope $\mathrm{Sec}(\Delta)$ of the polytope $\Delta$ can be
defined as the convex hull of all vertices $(\phi_{\tau})$ in the vector space
$\mathbb{R}^{\Delta\cap N}$:

$\mathrm{Sec}(\Delta)=\textrm{Conv}(\\{(\phi_{\tau})\in\mathbb{R}^{\Delta\cap
N}|\tau\in\text{triangulations of }\Delta\\}).$ (6)

Note here that the point $\phi_{\tau}$ is a vertex of the secondary polytope
if and only if the triangulation is regular (or coherent, see [26]). The faces
of the secondary polyhedron can be described in a similar way. They correspond
to regular subdivisions of a polytope into convex polygons. Below we construct
the matrix model that enumerates subdivisions into polygons, which, however,
may not be convex.

###  2.2 Matrix models

In this subsection we discuss some of the main concepts of matrix models, we
are interested in, for the purposes of our paper. We demonstrate all of them
by example of Hermitian matrix model (HMM) since our model constructed in
subsection 3.1 is a “deformation” of this model.

The partition function of HMM is the following:

$Z_{N}(t)=\int_{formal}\mathcal{D}H\exp\left[N\left(-\mu/2Tr(H^{2})+\sum\limits_{q=0}^{\infty}t_{q}Tr(H^{k})\right)\right]$
(7)

where integration is over hermitian $N\times N$ matrices with the measure:

$\mathcal{D}H:=\prod\limits_{i}dH_{i}^{i}\prod\limits_{i<j}dRe(H_{i}^{j})dIm(H_{i}^{j})$
(8)

The notation $\int_{formal}$ means that the exponent under the integral must
first be expanded into a series in the variables $t=(t_{0},t_{1},\dots)$, and
then each coefficient separately for each monomial of $t$ should be
integrated.

We will be interested in expressions of the form (called correlators):

$\langle\dots\rangle_{0}:=\frac{\int\mathcal{D}H\dots\exp[-N/2Tr(H^{2})]}{\int\mathcal{D}H\exp[-N/2Tr(H^{2})]}$
(9)

where $\dots$ is an arbitrary polynomial expression in matrix elements of $H$.

Since the measure in (9) is Gaussian, there is Wick’s rule.

Wick’s rule. The correlator of an arbitrary product of matrix elements
$H_{i}^{j}$ is split into the sum:

$\langle H_{i_{1}}^{j_{1}}\dots
H_{i_{n}}^{j_{n}}\rangle_{0}=\sum\limits_{\text{pairings}}\prod\limits_{\text{pairs}}\langle
H_{i}^{j}H_{i^{\prime}}^{j^{\prime}}\rangle_{0}$ (10)

where the summation is carried out over all ways to split the factors into
pairs, and the product is taken over all pairs. Furthermore:

$\langle H_{i}^{j}H_{k}^{l}\rangle_{0}=\frac{\delta_{i}^{l}\delta_{j}^{k}}{N}$
(11)

Example. Suppose we want to calculate the correlator $\langle
Tr(H^{4})\rangle_{0}$. Wick’s rule allows one to do this in the following way:

$\langle NTr(H^{4})\rangle_{0}=N(\langle
H_{i}^{j}H_{j}^{k}H_{k}^{l}H_{l}^{i}\rangle_{0}=\langle
H_{i}^{j}H_{j}^{k}\rangle_{0}\langle H_{k}^{l}H_{l}^{i}\rangle_{0}+\langle
H_{i}^{j}H_{k}^{l}\rangle_{0}\langle H_{j}^{k}H_{l}^{i}\rangle_{0}+\langle
H_{i}^{j}H_{l}^{i}\rangle_{0}\langle H_{j}^{k}H_{k}^{l}\rangle_{0})=$
$=\frac{1}{N}(\delta^{j}_{j}\delta_{i}^{k}\delta^{l}_{l}\delta_{k}^{i}+\delta^{j}_{k}\delta^{l}_{i}\delta^{k}_{l}\delta^{i}_{j}+\delta^{j}_{l}\delta^{i}_{i}\delta^{l}_{j}\delta^{k}_{k})=2N^{2}+1$

All this makes it possible to give a geometric meaning to correlators by
considering an analog of Feynman diagrams. In the case of Hermitian matrix
model, one of the interpretations of the diagram technique has the following
form. Omitting details, let us consider a correlator of the form

$\langle NTr(H^{k_{1}})\dots NTr(H^{k_{n}})\rangle_{0}$ (12)

Each factor $NTr(H^{k_{i}})$ is associated with a $k_{i}$-gon, and then all
possible gluings of polygons along their edges into a closed orientable
surface are considered. It turns out that each gluing contributes

$\sim N^{\chi(G)}$ (13)

where $\chi(G)$ is the Euler characteristic of the resulting surface.

Example. In the previous example, we assotiate a square to the expression
under the correlator. The three terms below correspond to three ways of gluing
the edges of the square in pairs. Two of them correspond to the sphere $S^{2}$
and contribute $2N^{\chi(S^{2})}=2N^{2}$, while the last one corresponds to
the torus $T$ and contributes $N^{\chi(T)}=1$.

Thus, the correlators of the form $\langle NTr(H^{2n})\rangle_{0}$ in the
Hermitian matrix model enumerate the number of ways to glue a $2n$-gon along
the edges into oriented surfaces of different genera.

Similar diagram techniques also occur in other matrix models. For example, in
Kontsevich matrix model, summation is carried out over trivalent weighted
graphs describing the cellular decomposition of the decorated moduli space of
complex curves [27, 1].

The matrix model we are constructing uses a similar idea to enumerate
triangulations of integral polyhedra.

Ward identities. One can write equations on correlators in matrix models that
follow from the reparametrization invariance of the matrix integral. These
equations are called Ward identities and often allow one to “completely solve”
the matrix model, which means writing recursive relations on all correlators
(invariant under the conjugation $H\rightarrow UHU^{+}$ by unitary matrix),
allowing one to recover any correlator from a small a set of initial data.

Let’s look at Hermitian matrix model. Any reparametrization of the form
$H\rightarrow H+\delta H^{n}$ does not change the value of the integral, but
transforms the integrand, which leads to the Ward identities. We will,
however, use a different approach to derive them. It consists in the
following. Substituting a differential operator of the form
$\frac{\partial}{\partial H_{ij}}$ under the integral, we obtain 0, because
the Gaussian exponent vanishes the integrand at infinity:

$0=\int\mathcal{D}H\frac{\partial}{\partial
H_{i}^{j}}\dots\exp[-N/2Tr(H^{2})]$ (14)

where $\dots$ implies some polynomial expression of matrix elements of $H$.

Inserting the operator

$Tr\left(\frac{\partial}{\partial H}H^{n+1}\right)$ (15)

into the partition function (7) gives the Ward identities, which can be
written as:

$L_{n}Z=0,\>\geq-1,$ (16)

where

$L_{n}=-\mu\frac{\partial}{\partial
t_{n+2}}+\sum\limits_{k=0}^{\infty}kt_{k}\frac{\partial}{\partial
t_{k+n}}+\sum\limits_{k=0}^{n}\frac{\partial^{2}}{\partial t_{k}\partial
t_{n-k}}+2N\frac{\partial}{\partial t_{n}}+N^{2}\delta_{n,0}$ (17)

Surprizing fact here is that such operators form the Virasoro algebra:

$[L_{n},L_{m}]=(n-m)L_{n+m}$ (18)

The existence of such relations indicates the existence of a sufficient number
of identities to recover arbitrary invariant correlators [17].

The similar operators and relations will appear in our model, but this will
only allow us to recover the correlators, we are interested in, from the
“simple” ones. A detailed discussion of this issue is provided in the section
5.

##  3 Matrix model of triangulations

In this section, we build a matrix model, which we call the matrix model of
triangulations. As the name suggests, the main idea is to consider all
possible triangulations with vertices at integral points of integral polytope
$\Delta$. We first consider two-dimensional polytopes and then generalize the
model to higher dimensions. We think that it is quite natural to consider for
our purposes all possible triangulations for the following reasons:

  * •

Matrix models with a potential of the 3rd degree are naturally enumerate
triangulations of two-dimensional surfaces in their diagram technique [28].
Therefore, we only have to modify such a matrix model so that the vertices of
the triangles correspond to integral points and the two-dimensional surface is
simply a disk with a boundary coinciding with the boundary of the polytope.

  * •

The number of points in a maximal triangulation of the polytope is expressed
in terms of the number of triangles and doesn’t depend on particular choice of
such triangulation. Therefore, by introducing the parameter at the cubic
potential in the matrix model, we can calculate the number of triangles, and,
hence, the number of integral points $\ell(\Delta)$

  * •

The secondary polytope (6), which describes moduli spaces of toric
hypersurfaces in some sense [26], is the convex hull of the characteristic
functions of all possible triangulations. Therefore, enumeration of all
triangulations can be useful for studying moduli spaces

So we construct the simplex model in case of two dimensional polytopes in the
subsection 3.1. Then we generalise it to higher degree potential in subsection
3.2 which corresponds to subdivisions instead of triangulations. This
generalization allows us to obtain Ward identities and operator (17) analogue
in the subsection 3.3. Finally, we generalize our model to higher dimensional
case and do the same steps for it in the subsection 3.4.

###  3.1 Constructing matrix model for polygons

In this subsection, the dimension of the lattice $M$ is equal two. It is well
known that the matrix model with a potential of the third degree enumerates
triangulations of two-dimensional (in the sense of real dimension) oriented
surfaces [28]. We mentioned about it considering example of the Hermitian
matrix model (subsection 2.2). In our case, the following restrictions must be
imposed on triangulations:

  * •

firstly, the triangulation vertices must belong to the two-dimensional lattice
$M=\mathbb{Z}^{2}$

  * •

secondly, the triangles must not overlap each other in the lattice
$M=\mathbb{Z}^{2}$

  * •

thirdly, the surface glued from triangles has the boundary $S^{1}$, which is
the boundary of the polygon, and, furthermore, the surface has the topology of
the disk, which is the interior of the polygon

Let us give some considerations on how we are going to construct the
corresponding matrix model.

In order for the triangulation vertices to belong to the lattice, we define an
“index set” $I^{2}_{N}:=\\{m\in\mathbb{Z}^{2}:\>|m_{1}|,|m_{2}|\leq N\\}$.
This means that we will consider Hermitian matrices $(H_{ij})$ of size
$(2N+1)^{2}\times(2N+1)^{2}$, whose indices $i$ and $j$ take values from the
set $I^{2}_{N}$. Hence the triangulation vertices will correspond to points
from $I^{2}_{N}$. Taking the limit $N\rightarrow\infty$, we obtain the entire
lattice $\mathbb{Z}^{2}$.

In order for triangulations to have boundaries we will consider correlators of
special form:

$\langle\Delta_{i_{1},\dots,i_{n}}\rangle:=\left\langle
H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots H_{i_{n}i_{1}}\right\rangle$ (19)

where the sequence $i_{1},\dots,i_{n}\in I^{2}_{N}$ is formed by vertices of
fixed polygon $\Delta_{i_{1},\dots,i_{n}}$ in clockwise order (there is no
summation over indices $i_{1},\dots,i_{n}\in I^{2}_{N}$ here). Note that we
can set not only a boundary of a polygon in this way and not only a boundary
of a convex polygon. For now we will consider only boundaries of polygones,
possibly not convex one. We leave the discussion of other correlators to
further research.

Finally, in order for the triangles not to overlap, we require them to be
oriented counterclockwise with respect to the lattice and to be non-
degenerate. Then it is easy to check that if the boundary of the triangulation
is given by the boundary of an integral polyhedron, then there will be no
overlapping of triangles. Moreover, no triangles can arise outside the
interior of a polygon. In order to take into account only properly oriented
and non-degenerate triangles, we will have to significantly deform the cubic
potential, writing it in the form

$W(H):=\sum\limits_{i,j,k\in I^{2}_{N}}\omega_{ijk}H_{ij}H_{jk}H_{ki}$ (20)

where

$\omega^{ijk}\sim\begin{cases}1,\text{ for
}\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})>0\\\ 0,\text{ for
}\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})\leq 0\end{cases}.$ (21)

Recall that indices are takes values in the set $I^{2}_{N}$ which is a part of
two-dimensional lattice, so they have two components $i=(i_{1},i_{2})$.

Thus, we will consider the following matrix model:

$Z=\int_{formal}\mathcal{D}H\exp\left[-\frac{1}{2}Tr(H^{2})+\beta
W(H)\right].$ (22)

Here, $\int_{formal}$, as above, means that we consider the matrix model as a
formal series in $\beta$, that is, the integral is taken for each degree of
$\beta$ in the expansion of the exponent separately. Integration is over
Hermitian matrices, in the sense that $H_{ij}=\overline{H}_{ji}$ and the
measure $\mathcal{D}M$ has the same form as in (8).

We intentionally write both indices of $H$ as subscripts, because the matrix
model (22) is not unitary invariant. In this sense, our model is not formally
a matrix model, we only use the motivation of matrix models to write the
finite-dimensional integral in this form. On the other hand, it is possible to
preserve the matrix form (and, in further subsections, the tensor structure)
by writing similar expressions in a slightly more complex form, but this is
not required for our purposes.

Note that $Z=0$ due to the orientation of all triangles and the absence of
boundary in the sense of (19).

In order to clarify now the connection between correlators (19) and integral
polygons, we formulate the following theorem.

Theorem 1. Let the potential $W(H)$ be defined by $\omega_{ijk}$ of the form

$\omega_{ijk}=\begin{cases}e^{\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})(x^{i}+x^{j}+x^{k})},\text{
for }\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})>0\\\ 0,\text{ for
}\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})\leq 0\end{cases},$ (23)

where $x^{i}$ are formal variables ($i\in I^{2}_{N}$). Then the correlator

$\langle\Delta_{i_{1},\dots,i_{n}}\rangle:=\left\langle
H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots
H_{i_{n}i_{1}}\right\rangle=\sum\limits_{k=0}^{\infty}\frac{\beta^{k}}{k!}\left\langle
H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots H_{i_{n}i_{1}}(W(H))^{k}\right\rangle_{0},$
(24)

where the sequence of points $i_{1},\dots,i_{n}\in I^{2}_{N}$ forms the
boundary of the polygon $\Delta_{i_{1},\dots,i_{n}}$, for sufficiently large
$N$ is expressed in terms of the sum over triangulations $\Delta_{i_{1},\
dots,i_{n}}$ of the following form:

$\langle\Delta_{i_{1},\dots,i_{n}}\rangle=\sum\limits_{\tau\in\textrm{triangulations
of }\Delta}\beta^{|\tau|}e^{2\langle\phi_{tau},L(\Delta)\rangle}$ (25)

where $|\tau|$ is the number of triangles in the triangulation $\tau$,
$\langle\phi_{tau},L(\Delta)\rangle:=\sum\limits_{a\in
V(\tau)}\phi_{a}(\tau)x^{a}$ is a polynomial whose Newton polyhedron is
$\Delta$, and whose coefficients are the values of the triangulation $\tau$
characteristic function at the corresponding triangulation vertex $a\in
V(\tau)$.

Let us immediately note an important corollary which shows how the numbers
$\ell(\Delta)$ arise in this model.

Corollary. The highest degree of $\beta$ in the correlator
$\langle\Delta_{i_{1},\dots,i_{n}}\rangle$ is $2\ell^{*}(\Delta)+n-2$.

###  3.2 Higher degree generalization

There is some motivation to generalize the potential of the matrix model to
higher degrees (see [16]). Adding higher degree terms to potential can be
interpreted as “dressing process” in terms of renormalization group theory.
Such a procedure naturally leads to a Virasoro-like algebra of operators
corresponding to the Ward identities. Here we present a way to generalize the
above model in a similar way: we introduce higher powers into the matrix model
(22). We also discuss how the Theorem 1 changes.

Let us consider the following matrix model (inspired by analogous replacement
(2) for HMM):

$Z=\int\mathcal{D}H\exp\left\\{-\frac{\mu}{2}Tr(H^{2})+\sum\limits_{n=1}^{\infty}t_{n}W_{n}(H)\right\\}$
(26)

where

$W_{n}(H)=\sum\limits_{i_{1},\dots,i_{n+2}\in I^{2}_{N}}\omega^{n}_{i_{1}\dots
i_{n+2}}H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots H_{i_{n+2}i_{1}}$ (27)

We define $\omega^{n}_{i_{1}\dots i_{n+2}}$ by recursion relations having
sense of gluing triangles corresponding to $\omega^{1}_{ijk}$ into polygons:

  * •

$\omega^{1}_{ijk}$ is just $\omega_{ijk}$ defined in previous subsection. It
correspond to the triangle with vertices $i,j,k\in I^{2}_{N}$

  * •

Cyclic permutation of indices defines the same $n+2$-gon

$\omega^{n-2}_{i_{1}\dots i_{n-1}i_{n}}=\omega^{n-2}_{i_{n}i_{1}\dots
i_{n-1}}$ (28)

  * •

Gluing two polygons along a common edge gives a new polygon

$\omega^{m+n-4}_{i_{1}\dots i_{n}j_{3}\dots
j_{m}}=\begin{cases}\omega^{n-2}_{i_{1}\dots
i_{n}}\omega^{m-2}_{i_{1}i_{n}j_{3}\dots j_{m}},\text{ if }\Delta_{i_{1}\dots
i_{n}j_{3}\dots j_{m}}\text{ is convex}\\\ 0,\>\text{ otherwise}\end{cases}$
(29)

We can generalize Theorem 1 for this matrix model as follows:

Theorem 1’. Let the potential $W_{1}(H)$ be of the form (20), and the
remaining $W_{n}(H)$ satisfy the recursion relations described above. Then the
correlator

$\langle\Delta_{i_{1},\dots,i_{n}}\rangle:=\left\langle
H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots H_{i_{n}i_{1}}\right\rangle,$ (30)

where the sequence of points $i_{1},\dots,i_{n}\in I^{2}_{N}$ forms the
boundary of the polygon $\Delta_{i_{1},\dots,i_{n}}$, for sufficiently large
$N$ is expressed in terms of the sum over subdivisions into convex polygons of
the polygon $\Delta_{i_{1},\dots,i_{n}}$ of the following form:

$\langle\Delta_{i_{1},\dots,i_{n}}\rangle=\sum\limits_{\textrm{subdivisions of
}\Delta}\\#_{\text{subdivision}}1/\mu^{\\#\text{ of subdivision
edges}}\prod\limits_{q=1}^{\infty}t_{q}^{\\#\text{ of q-gons in
subdivision}}\prod\limits_{\square\in\text{ q-gons}}W_{q}(\square)$ (31)

where $W_{q}(\square)$ is the value of the potential $W_{q}(H)$ on a
particular polygon $\square$. $\\#_{\text{subdivision}}$ is some coefficient
depending on subdivision.

###  3.3 Ward identities/Virasoro algebra

Here we show how to obtain the Virasoro constraints for the matrix model (26).

We deform operators (15) as follows. From the Feynman diagram point of view
this operator corresponds to insertion of an $n$-gon:

$\hat{L}_{n}=\sum\limits_{i_{1}\dots i_{n+2}}\omega^{n}_{i_{1}i_{2}\dots
i_{n+2}}\frac{\partial}{\partial
M_{i_{2}i_{1}}}M_{i_{2}i_{3}}M_{i_{3}i_{4}}\dots M_{i_{n+2}i_{1}}$ (32)

As it was for Hermitian matrix model we obtain Ward identities:

$0=\hat{L}_{n-1}\langle\Delta_{j_{1}\dots j_{m}}\rangle$ (33)

We act by operators $\hat{L}_{n-1}$ on a correlator of the form (19), because
it can not be recovered from the partition function by the action of any
differential operator in variables $t_{i}$.

Theorem 2. The action of the operator (32) can be written in the following
form:

$\hat{L}_{n}\langle\Delta_{j_{1}\dots
j_{m}}\rangle=\left(-\mu\frac{\partial}{\partial
t_{n+2}}+\sum\limits_{k=1}^{\infty}kt_{k}\frac{\partial}{\partial
t_{k+n+2}}+\hat{I}_{n}\right)\langle\Delta_{j_{1}\dots j_{m}}\rangle$ (34)

where

$\hat{I}_{n}\langle\Delta_{j_{1}\dots
j_{m}}\rangle=\sum\limits_{l}\sum\limits_{i_{1},\dots,i_{n}}\omega^{n}_{j_{l}i_{1}\dots
i_{n}j_{l+1}}\langle M_{j_{1}j_{2}}\dots\hat{M}_{j_{l}j_{l+1}}\dots
M_{j_{m}j_{1}}(M_{j_{l}i_{1}}M_{i_{1}i_{2}}\dots M_{i_{n}j_{l+1}})\rangle=$
(35)
$=\sum\limits_{l}\sum\limits_{i_{1},\dots,i_{n}}\omega^{n}_{j_{l}i_{1}\dots
i_{n}j_{l+1}}\langle\Delta_{j_{1}\dots j_{l}i_{1}\dots i_{n}j_{l+1}\dots
j_{m}}\rangle$

And operators $\hat{L}_{n}$ form the Virasoro algebra:

$[\hat{L}_{n},\hat{L}_{m}]=(n-m)\hat{L}_{n+m}$ (36)

Note here that correlators obtained by the action of operator $\hat{I}_{n}$
does not necessarily correspond to boundaries of polygons. So we have to
consider correlators of the form (19) for an arbitrary sequence of points
$i_{1},\dots,i_{n}\in I^{2}_{N}$ in order to obtain recursion relations from
Virasoro constraints (33).

###  3.4 Higher dimensional generalization and tensor model

In this subsection, we give some considerations about higher dimensional
generalizations of the matrix model (22). We are going to construct a “matrix
model” that enumerates subdivisions of $d$-dimensional polytopes. But in order
to describe the gluing of simplices of higher dimensions, one has to write a
tensor model instead of a matrix one. Such models are poorly understood, and
in general it is not known whether their Ward identities are solvable.
However, in our case, this is not very significant, since as in the model
described above, the tensor structure would not be important to us. Indeed,
many methods of working with matrix models are based on the possibility to
diagonalize the matrix and integrate it over eigenvalues or, for example, use
the Itzykson-Zuber formula. Both of these methods are based on the unitary
invariance of the action, which is missing in our model and is not used in the
previous reasoning. Therefore, we expect that all previous statements can be
generalized to the case of a tensor model.

Note that there are some multidimensional generalizations of matrix models
which describe higher dimensional triangulations of pseudo manifolds [29].
Such generalizations are called group field theores (GFT) [20, 21, 24, 30].
One can consider our tensor model as a deformation of GFT in the same sense
that the matrix model of triangulations is the deformation of Hermitian one.
The Virasoro-like constraints (45) we obtain are quite similar to the
Virasoro-like constraints which arise in the large $N$ limit for GFT. They are
also can be naturally described in terms of trees [31]. Nonetheless, there is
a difference in how we glue trees and how they are glued in GFT. It would be
interesting to understand how these two Virasoro-like constraints are
connected, but we leave that for further research.

Now we fix the set in which the indices of tensors take values, in the
following obvious way
$I_{N}^{d}=\\{(m_{1},\dots,m_{d})\in\mathbb{Z}^{d}:|m_{i}|\leq N\>\forall
i\\}$.

We associate the $d-1$-simplex with vertices $i_{1}\dots i_{d}$ with the
tensor $T_{i_{1}\dots i_{d}}$ as it was for our matrix model: $H_{ij}$
corresponds to the edge of the triangle with vertices $ij$. In addition, we
impose the following condition on the tensor $T_{i_{1}\dots i_{d}}$:

$T_{i_{1}\dots i_{l}i_{l+1}\dots i_{d}}=\overline{T}_{i_{1}\dots
i_{l+1}i_{l}\dots i_{d}},\>\forall l\in\\{1,\dots,d-1\\}$ (37)

Then the analogue of the quadratic term will be:

$-1/2\sum\limits_{i_{1}\dots i_{n}}T_{i_{1}\dots
i_{d-1}i_{d}}T_{i_{2}i_{1}i_{3}\dots i_{d}}$ (38)

In order to write an analogue of the cubic action, we have to “glue” the
$d$-dimensional simplex from its boundaries. For example, the cubic potential
associated with 3-dimensional simplex can be written in the following way:

$\sum\limits_{i_{1}i_{2}i_{3}i_{4}}\Omega^{4}_{i_{1}i_{2}i_{3}i_{4}}T_{i_{2}i_{3}i_{4}}T_{i_{1}i_{3}i_{4}}T_{i_{1}i_{2}i_{4}}T_{i_{1}i_{2}i_{3}}$
(39)

In general, this potential can be written as:

$\sum\limits_{i_{0}\dots i_{d}}\Omega^{d+1}_{i_{0}\dots
i_{d}}\prod\limits_{k=0}^{d}T_{i_{0}\dots\hat{i}_{k}\dots i_{d}}$ (40)

If we want to obtain an analogue of the Ward identities (33) and the Virasoro
algebra (36), then we have to add higher powers in $T$ to the potential.
However, things are more complicated for them then it was for the matrix
model. We argue that they must correspond to “tree” simplicial complexes
instead polygons arising in the matrix model. We mean the following.

  * •

This simplicial complex is a simplicial manifold of dimension $d$ with
boundary, i.e. it is glued from $d$-simplices so that for each $d-1$-simplex,
at most 2 $d$-simplices are glued (in other words it is a triangulation of
$d$-dimensional manifold with triangulated boundary). The boundary of such a
simplicial complex is a simplicial sphere.

  * •

Therefore, the dual simplicial complex can be associated with it. Moreover,
this dual simplicial complex has to be a tree with ordered edges at each
vertex.

Therefore, for each degree of the potential, we have to write the sum over all
possible trees of this form with a fixed number of vertices. In the case of
the matrix model, this problem was greatly simplified, since all such trees
give equivalent partitions of the sphere $S^{1}$ into simplices. It was
expressed in the possibility of a cyclic permutation of
$\omega^{n}_{i_{1}\dots i_{n+2}}$ indices.

Let $\mathcal{T}$ be the set of trees described above. For each tree
$\tau\in\mathcal{T}$ there is the associated polytope $\Delta(\tau)$ and
associated term of potential $W_{\tau}(T)$. The partition function of our
tensor model has the following form:

$Z(t)=\int\mathcal{D}T\exp\left\\{-1/2\sum\limits_{i_{1}\dots
i_{n}}T_{i_{1}\dots i_{d-1}i_{d}}T_{i_{2}i_{1}i_{3}\dots
i_{d}}+\sum\limits_{\tau\in\mathcal{T}}t_{\tau}W_{\tau}(T)\right\\}$ (41)

with the set of variables $\\{t_{\tau}|\tau\in\mathcal{T}\\}$. We give a more
detailed description of the tensor model potential, trees and associated
$\hat{L}_{(\tau,v)}$ operators with them in Appendix B. Here we only present
the final results.

Proposition 1. (Generalization of the Theorem 1’). Let $\Delta$ be integral
simplicial polytope of dimension $d$ with the set of facets $F(\Delta)$. And
let $V(\sigma)\subset I^{d}_{N},\>\sigma\in F(\Delta)$ denote the set of
vertices of the simplex $\sigma$. We associate the correlator

$\langle\Delta\rangle=\left\langle\prod\limits_{\sigma\in
F(\Delta)}T_{V(\sigma)}\right\rangle$ (42)

to the polytope $\Delta$, where $T_{V(\sigma)}:=V_{i_{1}\dots
i_{d}},\>V(\sigma)=\\{i_{1},\dots,i_{d}\\}$. Then (for sufficiently large $N$)
this correlator can be expressed in terms of sudivisions of $\Delta$ in the
following way:

$\langle\Delta\rangle=\sum\limits_{\text{subdivisions' of
}\Delta}\\#_{\text{subdivision}}1/\mu^{\\#\text{ of
}d-1\text{-simplices}}\prod\limits_{\tau\in\mathcal{T}}t_{\tau}^{\\#\Delta(\tau)}W_{\tau}(\Delta(\tau))$
(43)

where summation is over all subdivisions into convex polytopes associated with
trees $\mathcal{T}$. $\\#\Delta(\tau)$ is the number of polytopes
$\Delta(\tau)$ in corresponding subdivision. $W_{\tau}(\Delta(\tau))$ is the
value of $W_{\tau}(T)$ on the polytope $\Delta(\tau)$.
$\\#_{\text{subdivision}}$ is some coefficient depending on subdivision.

Unfortunately, we obtain here generating function only for subdivisions of
special form. There exist polytopes of dimension $\geq 3$ that can not be
represented as $\Delta(\tau)$ for some $\tau\in\mathcal{T}$. The simplest
example is octahedron.

Proposition 2. (Generalization of the Theorem 2) The $\hat{L}_{(S,\sigma)}$
operators generate the Ward identities:

$\hat{L}_{(S,\sigma)}\langle\Delta\rangle=0$ (44)

and form a Virasoro-like algebra

$[\hat{L}_{(\tau_{1},v_{1})},\hat{L}_{(\tau_{2},v_{2})}]=\sum\limits_{\tau\in(\tau_{1}\\#(\tau_{2},v_{2})|v_{1})}\hat{L}_{(\tau,v_{1})}-\sum\limits_{\tau\in(\tau_{2}\\#(\tau_{1},v_{1})|v_{2})}\hat{L}_{(\tau,v_{2})}$
(45)

where $(\tau_{1}\\#(\tau_{2},v_{2})|v_{1})$ is the set of all trees obtained
by gluing the tree $\tau_{2}$ along the edge $v_{2}$ to an arbitrary edge of
$\tau_{1}$ except $v_{1}$; the marked edge in resulting tree is $v_{1}$

##  4 Toric geometry applications

In this section, we give some basic information about the connection between
combinatorics of integral polyhedra and toric geometry. We hope this will
clarify our motivation to study particular combinatorial objects described
above.

###  4.1 Toric geometry

Toric varieties correspond one-to-one with the so-called fans [12]. That is
why a lot of properties of toric varieties can be reformulated in
combinatorial terms of fans. Let’s give some examples.

  * •

A variety is complete (analytically compact) if and only if the union of all
cones in the fan coincides with the ambient lattice.

  * •

Toric variety is non-singular if and only if each cone of the corresponding
fan is generated by a part of the basis (that is, any integral point of the
cone can be obtained by a linear combination of generators that are part of
the basis of the lattice with positive integer coefficients).

  * •

Each cone of dimension $l$ corresponds to an orbit of a torus of dimension
$n-l$ and an invariant subvariety of dimension $n-l$, which are the closures
of these orbits.

  * •

One-dimensional cones correspond to $T$-invariant divisors.

  * •

Piecewise polynomial functions on the fan $\Sigma$ form a ring isomorphic to
the equivariant Chow ring $A^{*}_{T}(X_{\Sigma})$ of the corresponding toric
variety [32].

We are interested in complete projective toric varieties. It is more
convenient to describe such varieties in terms of convex integral polyhedra.
For each polytope of this kind, one can construct a fan and a strictly convex
piecewise linear function on it, which will define, respectively, a toric
variety and an ample divisor on it (or, which is the same, an embedding in the
projective space $\mathbb{C}P^{k}$ of sufficiently large dimension). In fact,
a toric variety is projective if and only if the fan comes from a polytope
(generally it is not true, the first counterexample occurs in dimension 4).
That is why the main combinatorial object studied in this paper is an integral
polyhedron.

Let us give here one of the ways to construct a projective toric variety from
the polytope $\Delta$. Consider the ring $\mathbb{C}[x_{0},x_{1}^{\pm
1},\dots,x_{n}^{\pm 1}]$, and the subring $S$ generated by monomials of the
form $x_{0}^{m_{0}}x_{1}^{m_{1}}\dots x_{n}^{m_{n}}$, where
$(\frac{m_{1}}{m_{0}},\dots,\frac{m_{n}}{m_{0}})\in\Delta$. Then the
corresponding toric variety

$X_{\Delta}=\textrm{Proj}(S).$ (46)

An interesting application of the theory of toric varieties is Batyrev’s
construction of Calabi-Yau mirror varieties. We give here some considerations
from his work.

Let $\Delta\subset M_{\mathbb{R}}$ be a convex polytope, then the dual one
defined as:

$\Delta^{\vee}=\left\\{x\in M_{\mathbb{R}}|\langle x,y\rangle\geq-1,\>\forall
y\in\Delta\right\\}.$ (47)

Obviously, $((\Delta)^{\vee})^{\vee}=\Delta$.

The combinatorics of the dual polytope $\Delta^{\vee}$ is simply related to
the combinatorics of the initial one $\Delta$. Each face $\Theta$ of dimension
$k$ in $\Delta$ corresponds to a dual face $\hat{\Theta}$ in $\Delta^{\vee}$
of dimension $n-1-k$. And the face embedding relation is inverted: if
$\Theta\subset\Gamma$ for $\Delta$, then $\hat{\Gamma}\subset\hat{\Theta}$ for
$\Delta^{\vee}$.

In the context of constructing Calabi-Yau varieties, reflexive polyhedra are
of particular interest. An integral convex polytope $\Delta$ is called
reflexive if $\Delta^{\vee}$ is integral.

Calabi-Yau varieties arise as hypersurfaces in toric projective varieties
$X_{\Delta}$, where $\Delta$ is reflexive. Strictly speaking, these
hypersurfaces are given as divisors of global sections of the anticanonical
sheaf in some desingularization of the toric variety associated with the
reflexive polytope.

However, there is a more explicit way to describe these hypersurfaces.
Associate each node $m$ of lattice $M$ with coordinates $(m_{1},\dots,m_{n})$
with the Laurent monomial
$x_{1}^{m_{1}}x_{2}^{m_{2}}\dots x_{n}^{m_{n}}$ and denote it by $x^{m}$. The
support of the Laurent polynomial $L=\sum\limits_{m\in M}a_{m}x^{m}$ with
coefficients $a_{m}\in\mathbb{C}$ is a subset of the lattice $M$ of the form:

$\mathrm{Supp}(L)=\\{m\in M|a_{m}\not=0\\}.$ (48)

The Newton polytope of the polynomial $L$ is the convex hull of the support.
The Newton polytope is obviously an integral polytope in $M_{\mathbb{R}}$.

Denote by $\mathbf{L}(\Delta)$ the space of polynomials with a fixed Newton
polyhedron $\Delta$. Any polynomial $L$ from $\mathbf{L}(\Delta)$ is
associated with the hypersurface $Z_{L,\Delta}=\\{X\in T|L(X)=0\\}$ in the
torus and its closure $\overline{Z}_{f,\Delta}$ in a projective toric variety
$X_{\Delta}$. The closure $\overline{Z}_{f,\Delta}$ in the maximal
desingularization of the variety $X_{\Delta}$ is exactly the Calabi-Yau
hypersurface.

One of the consequences of mirror symmetry [33] is that for a mirror pair $X$
and $X^{\circ}$ of Calabi-Yau varieties, the following Hodge number equality
holds:

$h^{1,1}(V)=h^{dim(V^{\circ})-1,1}(V^{\circ}),\>h^{1,1}(V^{\circ})=h^{dim(V)-1,1}(V)$
(49)

Batyrev proved in his papers that pairs of Calabi-Yau hypersurfaces
constructed from reflexive polyhedra are candidates for a mirror pair in the
sense that the equalities (49) hold. It follows from explicit combinatorial
formulas for the Hodge numbers of hypersurfaces:

$h^{1,1}(V)=\ell(\Delta^{\vee})-n-1-\sum\limits_{\Gamma^{\vee}}\ell^{*}(\Gamma^{\vee})+\sum\limits_{\Theta^{\vee}}\ell^{*}(\Theta^{\vee})\ell^{*}(\hat{\Theta}^{\vee}),$
(50)
$h^{n-2,1}(V)=\ell(\Delta)-n-1-\sum\limits_{\Gamma}\ell^{*}(\Gamma)+\sum\limits_{\Theta}\ell^{*}(\Theta)\ell^{*}(\hat{\Theta}),$
(51)

where $\ell(\Delta)$ is the number of integral points in $\Delta$ mentioned
above; $\Gamma$ and $\Gamma^{\vee}$ are faces of codimension one of the
polytope $\Delta$ and its dual $\Delta^{\vee}$, respectively; $\Theta$ and
$\Theta^{\vee}$ are faces of codimension two in $\Delta$ and $\Delta^{\vee}$,
respectively; $\hat{\Theta}$ is a face of dimension one in $\Delta^{\vee}$
corresponding to the face $\Theta$ of codimension two in $\Delta$.

Note that some formulations of mirror symmetry expect the Hodge diamond of one
variety to be the reflected Hodge diamond of another. As far as we know, the
calculation of all Hodge numbers of Calabi-Yau hypersurfaces in Batyrev’s
construction is still an open question.

###  4.2 Moduli space

A more general view on mirror symmetry is that there must be an isomorphism of
moduli spaces of complex and Kähler structures of Calabi-Yau mirror varieties
(with fixed Kähler and complex structures, respectively). Therefore, it would
be interesting for us to touch upon the combinatorics associated with these
moduli spaces. Here we give some known facts about the structure of moduli
spaces of complex structures on Calabi-Yau hypersurfaces in toric varieties.
In fact, we will consider a simplified moduli space of hypersurfaces in a
toric variety $\overline{\mathcal{M}}_{simp}$. It is naturally described in
terms of a secondary polyhedron [26], on the one hand, and partially describes
the real moduli space of complex structure on the Calabi-Yau hypersurface, on
the other. Nevertheless, the moduli space of hypersurfaces is interesting in
itself.

Let us give a brief description of connection between the moduli space of
complex structures and $\overline{\mathcal{M}}_{simp}$, moving from the first
to the second. First, to begin with, it is simpler to consider only the moduli
space of complex structures arising from the embedding of a hypersurface in a
toric variety denoted by $\mathcal{M}_{poly}$. In other words, it is the
closure of the space of hypersurfaces in general position in a toric variety
up to isomorphisms. It is natural to try to obtain this space from the space
of Laurent polynomials $\mathbf{L}(\Delta)$. Recall that each (general
position) polynomial in $\mathbf{L}(\Delta)$ corresponds to the hypersurface
$Z_{f,\Delta}$ in the torus $T$ and its closure $\overline{Z}_{f,\Delta}$ in
the toric variety $X_{\Delta}$.

One can describe the whole group of automorphisms of the toric variety
$\mathrm{Aut}(X_{\Delta})$, whose action extends to the projective space
$\mathbb{P}(\mathbf{L}(\Delta))$. This allows to define the following
space[25]:

$\mathcal{M}_{poly}:=\mathbb{P}(\mathbf{L}(\Delta))/\mathrm{Aut}(X_{\Delta}).$
(52)

It can be proved that the spaces $\mathcal{M}$ and $\mathcal{M}_{poly}$
coincide if the faces of $\Delta$ of codimension 2 do not contain integral
points in their interior. Otherwise $\mathcal{M}_{poly}\subset\mathcal{M}$.
This is consistent with the (51) formula in the following sense:

  * •

$\textrm{dim}(\mathcal{M}_{poly})=\ell(\Delta)-n-1-\sum\limits_{\Gamma}\ell^{*}(\Gamma)$,

  * •

$\textrm{dim}(\mathcal{M})=h^{n-2,1}=h^{n-2,1}(V)=\ell(\Delta)-n-1-\sum\limits_{\Gamma}\ell^{*}(\Gamma)+\sum\limits_{\Theta}\ell^{*}(\Theta)\ell^{*}(\hat{\Theta})$,

That is, the dimensions of the moduli spaces $\mathcal{M}$ and
$\mathcal{M}_{poly}$ differ by a term that vanishes exactly when the set of
interior integral points of codimension 2 faces is empty.

An even simpler object is the simple moduli space $\mathcal{M}_{simp}$ [34],
which differs from $\mathcal{M}_{poly}$ in that not the entire automorphism
group is considered, but only its toric subgroup.

$\mathcal{M}_{simp}=\mathbb{P}(\mathbf{L}(\Delta)_{0})/T,$ (53)

where $\mathbf{L}(\Delta)$ is the set of Laurent polynomials $L$ with support
$\mathrm{Supp}(L)\subset\Delta\cap N$ such that the interior integral points
of the facets of $\Delta$ do not lie in $\mathrm{Supp}(L)$.

It is known [34] that $\mathcal{M}_{simp}$ is a finite covering of
$\mathcal{M}_{poly}$. For us, the following statement [26] will be important:

The compactification of the simple moduli space
$\overline{\mathcal{M}}_{simp}$ is the toric variety associated with the
secondary polytope of the polytope $\Delta$.

This means that the simple moduli space can again be described in
combinatorial terms. Since each face of the polyhedron corresponds to a closed
orbit of the torus in the toric variety, and the set of such orbits forms a
stratification of the toric variety, then the set of all regular subdivisions
of the polytope $\Delta$ will correspond to the stratification of the moduli
space $\overline{\mathcal{M}}_{simp}$. Therefore, it would be extremely
interesting to modify the matrix model (26) or tensor model (41) in such a way
that they enumerate precisely regular convex subdivisions. In this sense, it
would resemble Kontsevich’s matrix model, which enumerates in a certain sense
the cellular structure of the decorated moduli space of complex structures.

##  5 Discussion

###  5.1 Main results

We constructed the matrix model of triangulations (22) for two-dimensional
integral polyhedra. Correlators of the special form (19) in this model are
generating functions for the numbers $\ell^{*}(\Delta)$ and the characteristic
functions of triangulations in the sense of formula (31). Then a generalized
model (26) is constructed, which enumerates already arbitrary subdivisions of
integral polygones in a similar way (Theorem 1’). Due to the special structure
of the potential in this model, a certain family of Ward identities can be
written as operators (32), which form the Virasoro algebra.

Next, we gave considerations about a natural generalization to the case of
integral polytopes of an arbitrary dimension. In this case, we have to
consider the tensor models described in subsection 3.4. Despite the fact that
we do not give an explicit form of the potential in such a model, it can be
defined recursively (Appendix B). Generalization of the Theorem 1’ is given:
correlators of the tensor model enumerate subdivisions of polytopes in the
same way as it the matrix model so. But only subdivisions of special type
arise here (Proposition 1). Finally, the Ward identities are generalized to
higher dimensional case, which in turn no longer form the Virasoro algebra,
but lead to similar relations. Such a deformation of the Virasoro algebra
seems rather interesting, but we leave it for further research.

###  5.2 Open questions and further research

The matrix models approach for studying the combinatorics of integral
polytopes and the connection of this combinatorics with the geometry of toric
varieties and hypersurfaces allows us to formulate many interesting questions
and directions for further research. We list here an incomplete list of some
of the most amusing questions in our opinion:

  * •

As mentioned above, the existence of the Virasoro algebra indicates the
existence of “complete solvability” of the matrix model. So the next natural
step would be to obtain explicit recurrent relations on the correlators of the
matrix model (26) for two-dimensional polytopes and, possibly, for the tensor
model (41). We do not think, however, that in our case it will be possible to
reconstruct an arbitrary correlator from a small dataset. The fact is that the
“complete solvability” of a matrix model (for example, the Hermitian one)
means the possibility to calculate an arbitrary invariant (with respect to
conjugation by unitary matrices) correlator. In our case, there is no unitary
invariance, and we are interested in rather special correlators (19). We
assume that the recurrence relations will allow us to recover the generating
function (31) for an arbitrary polyhedron from the same generating function
for an arbitrary simplex.However, calculating such a generating function even
for a simplex is a difficult task.

  * •

We have already said that our models, strictly speaking, are not matrix or
tensor, they do not have unitary invariance. However, this can be fixed by
adding more tensor “fields”. The question is whether these fields can be given
a natural combinatorial/geometric meaning in terms of the diagram technique.
It is also not clear what happens to the Ward identities in this case.

  * •

Recall that our global goal is to describe the Hodge numbers and the moduli
space of toric hypersurfaces using our matrix/tensor model. Despite the fact
that the Hodge numbers are expressed in terms of the numbers $\ell(\Delta)$
(51), which are reproduced in our matrix model, we would like to obtain
operators that, when acting on correlators (19), would give exactly the Hodge
numbers.

  * •

As for the moduli spaces, first we would like to describe the stratification
of the space $\overline{\mathcal{M}}_{simp}$. For this, as mentioned above, it
is necessary to consider regular subdivisions into convex polyhedra.
Therefore, we would like to see if it is possible to modify our matrix/tensor
models so that they reproduce only such subdivisions.

  * •

Another question is how to describe reflexive pairs of polytopes in terms of
matrix models. Is it possible to write an operator that, when acting on
correlators (19), does not vanish only the correlators of reflexive polyhedra?
Maybe such a result can be obtained in a certain limit of our models?

  * •

We do not know how to describe trees which gives the same polytope (or
corresponding term in potential). It would be interesting to understand
whether there exist such elementary tree transformations that preserve
associated polytopes and generate all of such transformations.

##  6 Acknowledgements

Our work is supported in part by the grant of the Foundation for the
Advancement of Theoretical Physics “BASIS” and by RFBR grants 20-01-00644 and
21-51-46010. We are grateful to Alexey Sleptsov and Alexandr Popolitov for
very useful discussions and remarks.

##  7 Appendix A. Proof of the Theorem 1

Proof of the Theorem 1 and it’s Corollary

As mentioned in subsection 3.1, the diagram technique corresponds to gluing
triangles with vertices at integral points of the bounded lattice $I^{2}_{N}$.
Indeed, to each factor $\omega^{ijk}H_{ij}H_{jk}H_{ki}$ under
$\langle\dots\rangle_{0}$ we associate a trivalent vertex of the ribbon graph
with weight $\omega^{ijk}$ , whose three boundaries correspond to indices
$i,j,k$. The multiplier $H_{i_{1}i_{2}}\dots H_{i_{n}i_{1}}$ is associated
with a vertex of valency $n$ with indices $i_{1},\dots,i_{n}$ on the
boundaries. Averaging with the Gaussian weight $\langle\dots\rangle_{0}$
according to Wick’s rule yields all possible gluings of such vertices into an
orientable ribbon graph. After that, we can continue the ribbon graph to a
two-dimensional oriented surface by gluing two-dimensional disks along all
boundaries of it. Therefore, we obtain the sum over the genus $g$ of the
surface and all graphs $\Gamma(g)$ on them with one vertex of valence $n$ and
$k$ vertices of valence 3 (where $k$ is the degree of $\beta$):

$\left\langle H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots
H_{i_{n}i_{1}}(W(H))^{k}\right\rangle_{0}=\sum\limits_{g=0}^{\infty}\sum\limits_{\gamma\in\Gamma(g)}\sum\limits_{\\{i_{f}\in
I^{2}_{N},\>f\in F(\gamma)\\}}\prod\limits_{v\in
V^{\prime}(\gamma)}\omega_{v}^{i_{f_{1}(v)}i_{f_{2}(v)}i_{f_{3}(v)}}$ (54)

where $F(\gamma)$ is the set of faces of the graph $\gamma$ on the
corresponding surface, $V^{\prime}(\gamma)$ is the set of vertices, except for
the vertex of valency $n$, $f_{j}(v)$ are graph faces on whose boundary lies
the vertex $v$ ($j\in\\{1,2,3\\}$).

We now recall that the dual graph can be associated with the graph
$\gamma\in\Gamma(g)$. All its faces will be triangles, except for one selected
one, which is an $n$-gon. Therefore, we can rewrite the previous expansion as
a sum over triangulations of a surface of genus $g$ with the boundary which is
the $n$-gon:

$\left\langle H_{i_{1}i_{2}}H_{i_{2}i_{3}}\dots
H_{i_{n}i_{1}}(W(H))^{k}\right\rangle_{0}=\sum\limits_{g=0}^{\infty}\sum\limits_{\\{\tau\in\tilde{\Gamma}_{n}(g)\\}}\sum\limits_{i_{v}\in
I^{2}_{N},\>v\in V(\tau)}\prod\limits_{t\in
T(\tau)}\omega_{t}^{i_{v_{1}(t)}i_{v_{2}(t)}i_{v_{3}(t)}}$ (55)

where $\tilde{\Gamma}_{n}(g)$ is the set of triangulations of a surface of
genus $g$ with boundary (on which $n$ triangulation vertices lie), $V(\tau)$
is the set of $\tau$ vertices, $T(\tau)$ is the set of triangles in $\tau$,
$v_{j}(t)$ are vertices of triangle $t$.

The decomposition of (55) does not depend on the explicit form of
$\omega_{ijk}$ and on the index set $I^{2}_{N}$. The fact that the indices
belong to a bounded lattice gives us a mapping of the vertices of each
triangulation $\tau$ with fixed indices $i_{v}$ to the lattice $I^{2}_{N}$.
This mapping extends linearly to both edges and vertices. Therefore, the sum
$\sum\limits_{i_{v}}$ has the meaning of the sum over all triangulation
mappings in $I^{2}_{N}$, such that the vertices of the boundary map
bijectively and order-preserving into the set $\\{i_{1},\dots,i_{n}\\}\subset
I^{2}_{N}$.

Now let us fix the particular formula for $\omega_{ijk}$ as in (23). An
important property here is that only triangulations make a nonzero
contribution, in which, when mapped onto a lattice, all triangles are oriented
consistently and are non-degenerate
($\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})>0$). What this really means is that
the triangulation mapping has to be a covering. This implies, firstly, that
the image of a mapping can only be the interior of the region bounded by the
polygon $i_{1}i_{2}\dots i_{n}$. Secondly, of the entire sum $\sum_{g}$ only
the genus $g=0$ is non-zero, that is, the covering by the disk. Finally, the
sum over the graphs is finite, since the number of vertices in a triangulation
is limited due to the non-degeneracy of triangles (and, consequently, the
surjectivity of the mapping of the set of vertices, since the covering is
univalent and has no branches).

We note here that if we chose the vertices $i_{1},\dots,i_{n}$ so that they do
not form the boundary of the polygon, but, for example, are a polyline that
wraps around zero twice, then branched coverings with non-surjective mapping
of vertices could also be admissible.

Thus, we obtain the sum over all triangulations of the $\tau$ of $n$-gon
$i_{1}\dots i_{n}$ with weights:

$\prod\limits_{t\in
T(\tau)}e^{2\textrm{Vol}(t)(x^{v_{1}(t)}+x^{v_{2}(t)}+x^{v_{3}(t)})},$ (56)

where $2\textrm{Vol}(t)$ is twice the volume of the triangle $t$ arising from
the factor $\epsilon^{ab}(i_{a}-j_{a})(k_{b}-j_{b})$. The product over all
triangles can now be raised to the exponent and get the sum, which, after
reducing similar terms, has the form:

$2\sum\limits_{a\in V(\tau)}x^{a}\sum\limits_{t\in T(\tau):\>a\in
t}\textrm{Vol}(t)=2\sum\limits_{a\in V(\tau)}x^{a}\phi_{a}(\tau).$ (57)

Gathering everything together now, we obtain the required statement.

To prove the corollary, we note that the set of vertices in any maximal
triangulation coincides with the set of interior points of the polygon and its
vertices. Further, knowing the Euler characteristic of a surface with a
boundary and the relation between the number of edges and triangles in a
triangulation (3/2 of the number of triangles is equal to the number of edges
+ a correction to this formula taking into account the boundary), one can
easily relate the number of interior points of the polygon $\ell^{*}(\Delta)$
with the number of triangles $|\tau|$.

##  8 Appendix B. Virasoro-like algebra of trees

Let us describe in more detail the structure of the potential of the tensor
model (41) for polytopes of dimension $d$. It has the following form

$\sum\limits_{n=1}^{\infty}\sum\limits_{\alpha\in\mathcal{T}_{n}}t_{(n,\alpha)}W^{n}_{\alpha}(T)$
(58)

where we grade the contributions to the potential with respect to the powers
$(d-1)n$ of the tensor $T$. We define sets $\mathcal{T}_{n}$ and, accordingly,
the potentials $W^{n}_{\alpha}(T)$ by recursion relations.

Let’s start with the definition of $\mathcal{T}_{1}$. This is a set of one
element corresponding to a $d$-dimensional simplex. The corresponding
potential has already been written above (40):

$W^{1}(T)=\sum\limits_{i_{0}\dots i_{d}}\Omega^{d+1}_{i_{0}\dots
i_{d}}\prod\limits_{k=0}^{d}T_{i_{0}\dots\hat{i}_{k}\dots i_{d}}$ (59)

where $\Omega^{d+1}_{i_{0}\dots i_{d}}$ defined as

$\Omega^{d+1}_{i_{0}\dots i_{d}}\sim\begin{cases}1,\text{ if
}\epsilon^{m_{1}\dots
m_{d}}((i_{1})_{m_{1}}-(i_{0})_{m_{1}})\dots((i_{d})_{m_{d}}-(i_{0})_{m_{d}})>0\\\
0,\text{ if }\epsilon^{m_{1}\dots
m_{d}}((i_{1})_{m_{1}}-(i_{0})_{m_{1}})\dots((i_{d})_{m_{d}}-(i_{0})_{m_{d}})\leq
0\end{cases}.$ (60)

Note that the index set of $\Omega^{d+1}$ has the following structure:

  * •

given $d$-element subsets $\\{i_{0},\dots,\hat{i}_{k},\dots,i_{n}\\}$
corresponding to $T_{i_{0}\dots\hat{i}_{k}\dots i_{d}}$ tensors in potential
$W^{1}(T)$,

  * •

any two such subsets intersect in a $d-1$-element subset.

It is obvious that this corresponds to the structure of the simplicial complex
of the boundary of a $d$-dimensional simplex (if we add all possible subsets
to the $d$-element subsets). In higher powers of the potential in $T$, we want
to have a similar structure of the simplicial complex of the boundary of some
polyhedron, which we will denote by $\Delta(\alpha)$. To each such $d$-element
subset we associate one factor $T$ in the corresponding term of the potential.

Now let the set $\mathcal{T}_{n}$ and the potentials
$W^{n}_{\alpha},\>\alpha\in\mathcal{T}_{n}$ with the corresponding structure
on $\Omega^{n}_{\ alpha}$ are given. Then the set $\mathcal{T}_{n+1}$ consists
of all $d$-element subsets of all $\Omega^{n}$. In turn, each $W^{n}$ of the
form:

$W^{n}_{\alpha}(T)=\Omega_{i_{1}\dots i_{l_{1}}\dots i_{l_{2}}\dots
i_{l_{d}}\dots i_{q}}(T_{\dots}\dots T_{i_{l_{1}}\dots i_{l_{k}}}\dots
T_{\dots})$ (61)

together with the $d$-element subset $\\{i_{l_{1}},\dots,i_{l_{d}}\\}$ we will
associate the potential

$W^{n+1}_{\alpha^{\prime}}(T)=\Omega_{i_{1}\dots i_{l_{1}}i_{l_{0}}\dots
i_{l_{2}}\dots i_{l_{d}}\dots
i_{n}}^{\prime}(T_{\dots}\dots\hat{T}_{i_{l_{1}}\dots i_{l_{k}}}\dots
T_{\dots})\prod\limits_{j=1}^{d}T_{i_{l_{0}}\dots i_{l_{j}}\dots i_{l_{d}}}$
(62)

where $\Omega^{\prime}$ is defined by the formula

$\Omega_{i_{1}\dots i_{l_{1}}i_{l_{0}}\dots i_{l_{2}}\dots i_{l_{d}}\dots
i_{n}}^{\prime}=\begin{cases}\Omega_{i_{1}\dots i_{l_{1}}\dots i_{l_{2}}\dots
i_{l_{d}}\dots i_{n}}\Omega^{d+1}_{i_{l_{0}}\dots i_{l_{d}}},\text{ if
}\Delta(i_{1}\dots i_{l_{1}}i_{l_{0}}\dots i_{l_{2}}\dots i_{l_{d}}\dots
i_{n})\text{ is convex}\\\ 0,\text{otherwise}\end{cases}$ (63)

Essentially, the recurrence relation (63) simply corresponds to gluing an
additional $d$-simplex to the polytope corresponding to $W^{n}_{\alpha}(T)$.

The resulting potential can be described in other, perhaps more descriptive
terms. We will associate a tree whose set of edges for each vertex is ordered
with each $W^{n}_{\alpha}(T)$. Namely, we will consider trees with fixed
valence $d+1$ and open edges (having a beginning but no the end), where
outgoing edges at each vertex are numbered from 0 to $d$. We will do it in the
following way. Let us associate the potential $W^{1}(T)$ with the simplest
tree, which is the point from which $d+1$ open edges emanate. Now let the
potential $W^{n}_{\alpha}$ be associated with a tree. The (63) recursion
relation is equivalent to adding the simplest tree and gluing one of its open
edges to an already existing open edge. The number of open edges is equal to
the degree of the corresponding term of the potential in $T$. Thus, such trees
correspond one-to-one to the elements of the sets $\mathcal{T}_{n}$.

Therefore, the potential of the tensor model can be rewritten as the sum over
the trees described above. Let’s denote the set of all such trees as
$\mathcal{T}$.

Ward identities and Virasoro-like algebra

Now, in order to obtain the Ward identities (44), we just need to insert an
operator of the following form under the integral of our tensor model:

$\hat{L}_{(\alpha,v)}=W^{n}_{\tau}(T,\dots,\frac{\partial}{\partial
T},\dots,T)$ (64)

where $\frac{\partial}{\partial T}$ is in the place corresponding to the open
edge $v$. So operators (64) labeled by pairs $(\alpha,v)$ of some tree and a
marked point of this tree.

It is easy to show that these operators, when acting on the
$\langle\Delta\rangle$ correlator, can be rewritten as:

$\hat{L}_{(\alpha,v)}=-\mu\frac{\partial}{\partial
t_{\alpha}}+\sum\limits_{\alpha^{\prime}\in\mathcal{T}}t_{\alpha^{\prime}}\sum\limits_{\tilde{\alpha}\in(\alpha^{\prime}\\#(\alpha,v))}\frac{\partial}{\partial
t_{\tilde{\alpha}}}+\hat{I}_{(\alpha,v)}$ (65)

where the operator $\hat{I}_{\alpha}$ acts on $\langle\Delta\rangle$ by
changing $\Delta$ in the analogous way as it was for the matrix model.
Geometrically it glue the boundary of $\Delta(\alpha)$ to the boundary of
$\Delta$ along the $d-1$-simplex associated with $v$.

Commuting such operators, we obtain the relations (45).

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