# Central charges in local mirror symmetry via hypergeometric duality

Zengrui Han  Department of Mathematics
Rutgers University
Piscataway, NJ 08854<EMAIL_ADDRESS>
###### Abstract.

We apply the better-behaved GKZ hypergeometric systems to study toric Calabi-
Yau Deligne-Mumford stacks and their Hori-Vafa mirrors given by affine
hypersurfaces in $({\mathbb{C}}^{*})^{d}$. We show the equality between
A-model and B-model central charges, in terms of period integrals and
hypergeometric series respectively. This settles a conjecture of Hosono, which
could also be considered as a generalization of the Gamma conjecture for local
mirror symmetry.

###### Contents

  1. 1 Introduction
  2. 2 Central charges as solutions to better-behaved GKZ systems
  3. 3 Asymptotic behavior of period integrals via tropical geometry
  4. 4 Equality of A-model and B-model central charges
  5. A Residual volume and orbifold cohomology
  6. B An identity of Beta function

##  1\. Introduction

###  1.1. Toric Calabi-Yau Deligne-Mumford stacks and their Landau-Ginzburg
mirrors

Let $N$ be a lattice of rank $d+1$, and $\Delta$ be a lattice polytope of
dimension $d$ which lies on a primitive hyperplane $\deg(-)=1$ in
$N_{{\mathbb{R}}}:=N\otimes{\mathbb{R}}$ where $\deg:N\rightarrow{\mathbb{Z}}$
is a linear function. Let $C$ be the $(d+1)$-dimensional finite rational
polyhedral cone in $N_{{\mathbb{R}}}$ over the polytope $\Delta$, i.e.,
$C:={\mathbb{R}}_{\geq 0}(\Delta\oplus 1)$. This data encodes an affine toric
variety with Gorenstein singularities
$X=\operatorname{Spec}{\mathbb{C}}[C^{\vee}\cap N^{\vee}]$, in the sense that
its dualizing sheaf is trivial.

Let $\\{v_{i}\\}_{i=1}^{n}$ be a subset of lattice points in the polytope
$\Delta$ that includes all the vertices. For each regular triangulation of the
polytope $\Delta$ that involves some points in $\\{v_{i}\\}_{i=1}^{n}$, the
corresponding subdivision $\Sigma$ of the cone $C$ defines a smooth toric
Deligne-Mumford stack ${\mathbb{P}}_{\mathbf{\Sigma}}$ in the sense of
Borisov-Chen-Smith [5], where $\mathbf{\Sigma}:=(\Sigma,\\{v_{i}\\})$ is the
stacky fan whose distinguished lattice points on the rays are chosen to be the
primitive ray generators $v_{i}$’s, providing a crepant resolution of the
affine Gorenstein toric variety $X$. The toric stack
${\mathbb{P}}_{\mathbf{\Sigma}}$ is Calabi-Yau in the sense that its canonical
class is trivial.

Consider the hypersurface defined as
$Z_{f}:=(f=0)\subseteq({\mathbb{C}}^{*})^{d}$ where $f$ is a Laurent
polynomial defined as

$\displaystyle
f(z_{1},\cdots,z_{d})=\sum_{i=1}^{n}x_{i}z^{v_{i}}=\sum_{i=1}^{n}x_{i}\cdot
z_{1}^{v_{i1}}\cdots z_{d}^{v_{id}}$

By an abuse of notation, here we denote the $d$-dimensional vector obtained by
deleting the last coordinate (which is equal to $1$) of $v_{i}$ using the same
notation. In this paper, we adopt the idea that the family of hypersurfaces
$\\{Z_{f}\\}$ should be viewed as the mirror family of the smooth toric
Calabi-Yau Deligne Mumford stack ${\mathbb{P}}_{\mathbf{\Sigma}}$. Note that
this is a generalization of the Hori-Vafa mirror [17], where each cone in
$\Sigma$ is assumed to be unimodular, thus the corresponding toric stack is a
smooth variety.

According to Kontsevich’s homological mirror symmetry conjecture, the bounded
derived category $D^{b}({\mathbb{P}}_{\mathbf{\Sigma}})$ of coherent sheaves
(category of B-branes) on the toric stack ${\mathbb{P}}_{\mathbf{\Sigma}}$
should be equivalent to some appropriately defined Fukaya-type category of the
Landau-Ginzburg model $(({\mathbb{C}}^{*})^{d},f)$ (category of A-branes):

$\displaystyle
D^{b}({\mathbb{P}}_{\mathbf{\Sigma}})\xrightarrow{\sim}Fuk(({\mathbb{C}}^{*})^{d},f)$
(1.1)

While the mathematical meaning of the former is clear, the definition of the
latter seems to be subtle. However, in any sense, the objects in a reasonably
defined Fukaya-type category of $(({\mathbb{C}}^{*})^{d},f)$ should be
Lagrangian submanifolds of the torus $({\mathbb{C}}^{*})^{d}$ which satisfies
certain admissibility condition that take the potential $f$ into account.

Another mathematical formulation of mirror symmetry is provided by the SYZ
conjecture. Roughly speaking, it predicts that two spaces that are mirror to
each other should arise from dual special Lagrangian torus fibrations over
some base space. In our setting, the fibration on the mirror side is given by
the logarithmic map

$\displaystyle\operatorname{Log}:({\mathbb{C}}^{*})^{d}\rightarrow{\mathbb{R}}^{d},\
(z_{1},\cdots,z_{d})\mapsto(\log|z_{1}|,\cdots,\log|z_{d}|)$ (1.2)

Furthermore, the SYZ conjecture predicts that under the equivalence (1.1), the
mirror cycles of line bundles (resp. skyscraper sheaves) on
${\mathbb{P}}_{\mathbf{\Sigma}}$ should be Lagrangian sections (fibers) of the
fibration (1.2). In particular, when the coefficients of the Laurent
polynomial $f$ are all positive real numbers, the mirror cycle of the
structure sheaf $\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$ should be the
positive real section
$({\mathbb{R}}_{>0})^{d}\subseteq({\mathbb{C}}^{*})^{d}$, and those of
arbitrary line bundles corresponding to torus-invariant divisors $\sum
a_{i}D_{i}$ are obtained by parallel transport of $({\mathbb{R}}_{>0})^{d}$.

###  1.2. Central charges as solutions to better-behaved GKZ systems

In this paper, instead of studying the HMS statement of the Hori-Vafa mirror
pair in such generality, we focus on the more classical version of mirror
symmetry, namely the equality between the central charges of the A-branes and
the B-branes.

The notion of the central charge appears in physics literatures and has played
an important role in Douglas’s $\Pi$-stability [14] and its mathematical
formalism [12] due to Bridgeland. By definition the central charge of A-branes
(B-branes, respectively) is a linear function defined on the Grothendieck
group of the derived category of A-branes (B-branes, respectively). In the
setting of local mirror symmetry, Hosono [18] interpreted the central charges
of the A-branes and B-branes as certain period integrals and hypergeometric
series, and conjectured the equality between these two types of central
charges, see [18, Conjecture 2.2]. It is also observed that both of the
A-model and B-model central charges can be viewed as solutions to certain GKZ
hypergeometric systems, therefore GKZ systems play an important role in the
study of local mirror symmetry. However, we believe the better-behaved GKZ
systems, introduced by Borisov-Horja [9] and studied in a series of papers
[10, 7, 6, 16], are more suitable for this consideration:

######  Definition 1.1.

Consider the system of partial differential equations on the collection of
functions $\\{\Phi_{c}(x_{1},\ldots,x_{n})\\}$ in complex variables
$x_{1},\ldots,x_{n}$, indexed by the lattice points in $C$:

$\displaystyle\partial_{i}\Phi_{c}=\Phi_{c+v_{i}},\quad\sum_{i=1}^{n}\langle\mu,v_{i}\rangle
x_{i}\partial_{i}\Phi_{c}+\langle\mu,c\rangle\Phi_{c}=0$

for all $\mu\in N^{\vee}$, $c\in C$ and $i=1,\ldots,n$. We denote this system
by $\mathrm{bbGKZ}(C,0)$. Similarly by considering lattice points in the
interior $C^{\circ}$ only, we can define $\mathrm{bbGKZ}(C^{\circ},0)$.

It is proved in [6] that the two systems $\mathrm{bbGKZ}(C,0)$ and
$\mathrm{bbGKZ}(C^{\circ},0)$ are dual to each other by constructing an
explicit non-degenerate pairing between the solution spaces. The key
observation in this paper is that the A-model central charges (i.e., the
period integrals) are naturally solutions to the dual system
$\mathrm{bbGKZ}(C^{\circ},0)$ rather than $\mathrm{bbGKZ}(C,0)$ (which is
equivalent to the usual GKZ systems in normal cases). Motivated by this, we
introduce our version of the A-model central charges, which is a slight
modification of the original definition of Hosono.

######  Definition 1.2.

For each lattice point $c$ in the interior $C^{\circ}$ of the cone $C$, we
define the following holomorphic form

$\displaystyle\omega_{c}:=(-1)^{\deg c-1}(\deg c-1)!\frac{z^{c}}{f^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{d_{d}}$

on the complement $({\mathbb{C}}^{*})^{d}\backslash Z_{f}$, where $f=\sum
x_{i}z^{v_{i}}$. The A-model central charge associated to a Lagrangian
submanifold $L$ (with certain admissibility conditions) is defined to be the
collection of period integrals

$\displaystyle Z^{A,L}(x)=(Z^{A,L}_{c}(x))_{c\in
C^{\circ}}=\left(\int_{L}\omega_{c}\right)_{c\in C^{\circ}}.$

where each $\int_{L}\omega_{c}$ is viewed as a holomorphic function with the
coefficients $x_{i}$ of $f$ as variables.

Following the arguments of Batyrev [3] and Borisov-Horja [9], it is easy to
verify that the A-model central charges are solutions to the dual system
$\mathrm{bbGKZ}(C^{\circ},0)$, see Proposition 2.4.

On the other hand, a more natural candidate for the B-model central charges is
the hypergeometric series $\Gamma^{\circ}$ with values in the cohomology with
compact support, instead of the original cohomology-valued hypergeometric
series of Hosono (which is closer to the series $\Gamma$ taking values in the
usual cohomology). For precise definitions of $\Gamma$ and $\Gamma^{\circ}$,
see Definition 2.12.

######  Definition 1.3.

Let $N$, $C$, and $\mathbf{\Sigma}=(\Sigma,\\{v_{i}\\})$ be as previous. For
any class $\mathcal{E}\in K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ in the
compactly supported K-theory of ${\mathbb{P}}_{\mathbf{\Sigma}}$, we define
its B-model central charge by

$\displaystyle Z^{B,\mathcal{E}}(x)=(Z^{B,\mathcal{E}}_{c}(x))_{c\in
C^{\circ}}=(\chi(-,\mathcal{E})\circ\Gamma^{\circ}_{c})_{c\in C^{\circ}}$

The main theorem of this paper is that the B-model central charge associated
to any line bundle $\mathcal{L}=\sum_{i}a_{i}D_{i}$ over
${\mathbb{P}}_{\mathbf{\Sigma}}$ is equal to the A-model central charge
associated to the mirror cycle $\operatorname{mir}(\mathcal{L})$ of
$\mathcal{L}$ predicted by the SYZ conjecture.

######  Theorem 1.4.

For any line bundle $\mathcal{L}\in K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$,
the A- and B-model central charges coincide:

$\displaystyle Z^{A,\mathrm{mir}(\mathcal{L})}=Z^{B,\mathcal{L}}.$

Since $K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ is generated by classes of line
bundles, this equality is extended to the whole K-group
$K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$.

###  1.3. Relations to Gamma conjecture and hypergeometric duality

In [1], Abouzaid, Ganatra, Iritani and Sheridan proposed a new approach to
study the asymptotic behavior of period integrals in the setting of Batyrev
mirror pairs (i.e., Calabi-Yau hypersurfaces in Fano toric varieties). Roughly
speaking, the idea is to cut the cycles into pieces according to the tropical
geometry of the Laurent polynomial $f$, then approximate each piece by the
volume of certain polytopes which could again be related to the gamma classes
on the mirror side. They utilized this method to give an alternative proof of
the mirror-symmetric Gamma conjecture for the Batyrev mirror pairs.

The first step to prove our main result Theorem 1.4 is to apply their tropical
method to our local mirror symmetry setting. More precisely, in §3 we prove
the following asymptotic formula for the period integral of the affine
hypersurface $Z_{f}$ in $({\mathbb{C}}^{*})^{d}$ corresponding to the positive
real locus ${\mathbb{R}}_{>0}^{d}$.

######  Theorem 1.5 (=Theorem 3.1).

Fix a triangulation $\Sigma$ of the cone $C$, and denote the corresponding
convex piecewise linear function by $\psi$. Then the asymptotic behavior of
$\Psi_{c}(t^{-\psi(v_{1})},...)$ when $t\rightarrow+\infty$ is given by

$\displaystyle Z^{A,{\mathbb{R}}_{>0}^{d}}_{c}(t^{-\psi(v_{1})},...)=$
$\displaystyle t^{\psi(c)}\frac{(-1)^{{\operatorname{rk}}N-\deg c}}{(2\pi{\rm
i})^{|\sigma(c)|}|\operatorname{Box}(\sigma(\gamma))|}\cdot\int_{\gamma(c)}t^{\omega}\cdot\hat{\Gamma}_{\gamma(c)}F_{I_{c}}+o(t^{\psi(c)})$

For precise definitions of the symbols, see §3.

This could be viewed as the Gamma conjecture for local mirror symmetry. Our
computation could be seen as a generalization of the one in [1] because they
assumed the polytopes $\Delta$ are reflexive, while ours are arbitrary.
Moreover, we compute the asymptotics of period integrals corresponding to
arbitrary interior points in the cone $C$, while they essentially dealt with
the one corresponding to the unique interior point of the (reflexive) polytope
of degree equals to 1. Finally, we work with smooth toric Deligne-Mumford
stacks, while they worked with varieties.

The second essential ingredient of the proof is the hypergeometric duality
established by Borisov and the author in [6]. More precisely, an explicit
formula for the pairing between the solution spaces to $\mathrm{bbGKZ}(C,0)$
and $\mathrm{bbGKZ}(C^{\circ},0)$ is constructed:

$\displaystyle\langle\Phi,\Psi\rangle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}\left(\prod_{i\in I}x_{i}\right)\Phi_{c}\Psi_{d}$

where $\Phi$ and $\Psi$ are solutions to $\mathrm{bbGKZ}(C,0)$ and
$\mathrm{bbGKZ}(C^{\circ},0)$ respectively, and $\xi_{c,d,I}$’s are certain
constant coefficients which only depends on the combinatorics of the cone. The
non-degeneracy of this pairing allows us to lift the equality of leading terms
to a genuine equality between central charges on the two sides. An immediate
benefit of this is the following. In [1], the authors has to construct
explicitly tropical approximations of cycles mirror to arbitrary torus-
invariant divisors $\sum a_{i}D_{i}$ and repeat the computation of asymptotics
for them (because generally asymptotic expansion does not commute with taking
leading terms). While in principle, we can do the same thing in our setting,
it is no longer necessary since the equality on the level of functions
themselves (rather than just the leading terms) allows us to conclude simply
by looking at the monodromy on both sides, see Corollary 4.5.

###  1.4. Organization of the paper

The paper is organized as follows. In section 2, we formulate the precise
definitions of A-model and B-model central charges in our setting, and
interpret them as solutions to certain better-behaved GKZ systems. We also
review the basic knowledge on smooth toric DM stacks and their orbifold
cohomology in section 2. In section 3, we compute the asymptotic behavior of
the period integrals (A-model central charges) when approaching a large radius
limit point corresponding to a regular triangulation $\Sigma$. In section 4,
we apply the hypergeometric duality to prove the equality of A-model and
B-model central charges. Appendices A and B contain auxiliary results that are
used in the computation of section 3.

###  1.5. Acknowledgements

The author would like to thank his advisor Lev Borisov for consistent support,
helpful discussions and useful comments throughout the preparation of this
paper. The author thanks Bohan Fang for answering a question on [18] and [1].
The author also thanks Shaozong Wang for helpful discussions.

##  2\. Central charges as solutions to better-behaved GKZ systems

In this section, we give precise definitions of the A-model and B-model
central charges for Hori-Vafa mirror pairs. Our definitions differ slightly
from the ones in [18]. Along the lines we briefly recall the basic
preliminaries needed to understand the results and arguments of the paper.

###  2.1. A-model central charges

Denote the coordinates on $T:=({\mathbb{C}}^{*})^{d}$ by
$z=(z_{1},\cdots,z_{d})$. Consider the Laurent polynomial
$f=\sum_{i=1}^{n}x_{i}z^{v_{i}}$, where $z^{v_{i}}$ is defined as
$z_{1}^{v_{i,1}}\cdots z_{d}^{v_{i,d}}$ (Note that the last coordinate of
$v_{i}$, which is 1, is omitted). It defines a hypersurface $Z_{f}:=(f=0)$ in
the algebraic torus $T$.

######  Definition 2.1.

For each lattice point $c$ in the interior $C^{\circ}$ of the cone $C$, we
define the following holomorphic form

$\displaystyle\omega_{c}:=(-1)^{\deg c-1}(\deg c-1)!\frac{z^{c}}{f^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{d_{d}}$

on the complement $({\mathbb{C}}^{*})^{d}\backslash Z_{f}$, where $f=\sum
x_{i}z^{v_{i}}$. The A-model central charge associated to a Lagrangian
submanifold $L$ (with certain admissibility conditions) is defined to be the
collection of period integrals111 Note that the constant factor
$\frac{(-1)^{d}}{(2\pi{\rm i})^{d+1}}$ plays no essential role.

$\displaystyle Z^{A,L}(x)=(Z^{A,L}_{c}(x))_{c\in
C^{\circ}}=\left(\frac{(-1)^{d}}{(2\pi{\rm
i})^{d+1}}\int_{L}\omega_{c}\right)_{c\in C^{\circ}}$

where each $\int_{L}\omega_{c}$ is viewed as a holomorphic function with the
coefficients $x_{i}$ of $f$ as variables.

Since we are mostly interested in the mirror cycles of line bundles in this
paper, from now on we will assume $L$ to be the Lagrangian sections of the
fibration $\pi:({\mathbb{C}}^{*})^{d}\rightarrow{\mathbb{R}}^{d}$ defined by
$z\mapsto\log|z|$, see the discussion prior to §1.2. The following result
explains the reason why it is more natural to think of the period integrals
over Lagrangian sections as solutions to the dual system
$\operatorname{bbGKZ}(C^{\circ},0)$ rather than the usual system
$\operatorname{bbGKZ}(C,0)$. Similar results can be found in [15] and [4].

######  Lemma 2.2.

The period integral

$\displaystyle\int_{{\mathbb{R}}_{\geq 0}^{d}}\frac{z^{c}}{f(z)^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{z_{d}}$

is absolutely convergent if and only if $c\in C^{\circ}$.

###### Proof.

We make the coordinate change $z_{i}=e^{y_{i}}$, then the integral becomes

$\displaystyle\int_{{\mathbb{R}}^{d}}\frac{e^{c\cdot y}}{f(e^{y})^{\deg
c}}dy_{1}\wedge\cdots\wedge dy_{d}=\int_{{\mathbb{R}}^{d}}\frac{e^{c\cdot
y}}{(\sum_{j\in\Delta}x_{j}e^{v_{j}\cdot y})^{\deg c}}dy_{1}\wedge\cdots\wedge
dy_{d}$

Now we divide the space ${\mathbb{R}}^{d}$ into cone regions according to the
normal fan $\Sigma$ of the polytope $\Delta$. More precisely, we divide
${\mathbb{R}}^{d}$ as the union of the following cone regions

$\displaystyle\\{\sigma_{v_{k}}:=-\left({\mathbb{R}}_{\geq
0}(\Delta-v)\right)^{\vee}:v_{k}\in\Delta\\}$

note that this differs from the usual definition of the normal fan by a
negative sign. Fix a cone region $\sigma_{v_{k}}$, it’s easy to see that over
$\sigma_{v_{k}}$ the dominant term in the denominator
$\sum_{j\in\Delta}x_{j}e^{v_{j}\cdot y}$ is exactly the monomial
$x_{k}e^{v_{k}\cdot y}$. Therefore it suffices to consider the absolute
convergence of

$\displaystyle\int_{\sigma_{v_{k}}}e^{(c-(\deg c)v_{k})\cdot
y}dy_{1}\wedge\cdots\wedge dy_{d}$

which is again equivalent to the condition that

$\displaystyle(c-(\deg c)v_{k})\cdot y<0,\quad\forall\text{ ray generators
}y\text{ of }\sigma_{v_{k}}\text{ and }\forall v_{k}\in\Delta$ (2.1)

Recall that the polytope $\Delta$ could be defined as the intersection of
half-spaces (i.e., the facet representation of $\Delta$):

$\displaystyle\Delta=\bigcap_{F\text{ facet of }\Delta}(h_{F}\geq 0)$

where $h_{F}$ is the defining equation of the support hyperplane spanned by
the facet $F$. It is then straightforward to observe that the condition (2.1)
is equivalent to that $c/(\deg c)$ is an interior point of the polytope
$\Delta$, which is again equivalent to that $c$ is in the interior of the cone
$C$. ∎

Now we consider a general Lagrangian section $L$. Additional restrictions are
required to ensure the absolute convergence of the period integral. We write
the section $L:{\mathbb{R}}^{d}\rightarrow({\mathbb{C}}^{*})^{d}$ as

$\displaystyle(y_{1},\cdots,y_{d})\mapsto(e^{y_{1}}\cdot e^{{\rm
i}\theta_{1}(y_{1},\cdots,y_{d})},\cdots,e^{y_{d}}\cdot e^{{\rm
i}\theta_{d}(y_{1},\cdots,y_{d})})$

######  Proposition 2.3.

Suppose $L:{\mathbb{R}}^{d}\rightarrow({\mathbb{C}}^{*})^{d}$ is a section of
the fibration $T\rightarrow{\mathbb{R}}^{d}$ such that
$\det(I_{d}+\left(\frac{\partial\theta_{i}}{\partial y_{j}}\right)_{i,j})$ is
bounded, then the integral $\int_{L}\omega_{c}$ is absolutely convergent for
all $c\in C^{\circ}$.

###### Proof.

Follows directly from the last lemma and the observation that
$\det(I_{d}+\left(\frac{\partial\theta_{i}}{\partial y_{j}}\right)_{i,j})$ is
the determinant of the Jacobian of the change of variables. ∎

######  Proposition 2.4.

Suppose $\gamma$ satisfies the condition in Proposition 2.3, then
$\Psi=(\Psi_{c})_{c\in C^{\circ}}$ where

$\displaystyle\Psi_{c}(x_{1},\cdots,x_{n}):=\int_{L}\omega_{c}$

gives a solution to the system $\operatorname{bbGKZ}(C^{\circ},0)$.

###### Proof.

The idea of the proof comes from Batyrev [3] and Borisov-Horja [9]. However,
since the cycles we are integrating over are non-compact, some additional
cares must be taken.

To prove the equation $\partial_{i}\Psi_{c}=\Psi_{c+v_{i}}$ for any $i$, note
that we have

$\displaystyle\partial_{i}\left(\frac{z^{c}}{f(z)^{\deg c}}\right)=(-\deg
c)\frac{z^{c+v_{i}}}{f(z)^{\deg(c+v_{i})}}$

which gives $\partial_{i}\omega_{c}=\omega_{c+v_{i}}$, and the absolute
convergence of the integral ensures that differentiation commutes with
integration.

To prove the second equation

$\displaystyle\sum_{i=1}^{n}\langle\mu,v_{i}\rangle
x_{i}\partial_{i}\Psi_{c}+\langle\mu,c\rangle\Psi_{c}=0,\quad\forall\mu\in
N^{\vee}$

we look at the standard basis $\mu_{1},\cdots,\mu_{d+1}$ of $N^{\vee}$. For
$1\leq k\leq d$, an elementary computation shows that
$\sum_{i=1}^{n}\langle\mu_{k},v_{i}\rangle
x_{i}\omega_{c+v_{i}}+\langle\mu_{k},c\rangle\omega_{c}$ is equal to

$\displaystyle(-1)^{\deg c-1}(\deg
c-1)!z_{k}\partial_{z_{k}}\left(\frac{z^{c}}{f^{\deg
c}}\right)\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{z_{d}}$

Note that

$\displaystyle z_{k}\partial_{z_{k}}\left(\frac{z^{c}}{f^{\deg
c}}\right)\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{z_{d}}=d\left(\frac{z^{c}}{f^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\widehat{\frac{dz_{k}}{z_{k}}}\wedge\cdots\frac{dz_{d}}{z_{d}}\right)$

Take a chain of compact subsets $B_{1}\subseteq B_{2}\subseteq\cdots\subseteq
T$ such that $\cup B_{m}=T$ (e.g., take $B_{m}$ to be the box defined by
$e^{-m}\leq|z_{j}|\leq e^{m}$). By Stokes’ theorem the integration of
$z_{k}\partial_{z_{k}}\left(\frac{z^{c}}{f^{\deg c}}\right)$ over $B_{i}\cap
L$ is equal to

$\displaystyle\int_{\partial(B_{m}\cap L)}\frac{z^{c}}{f^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\widehat{\frac{dz_{k}}{z_{k}}}\wedge\cdots\frac{dz_{d}}{z_{d}}$

which tends to 0 when $m\rightarrow+\infty$ due to the absolute convergence of

$\displaystyle\int_{L}\frac{z^{c}}{f^{\deg
c}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{d}}{z_{d}}.$

On the other hand, by dominated convergence theorem the sequence of integrals
converges to the integration of
$z_{k}\partial_{z_{k}}\left(\frac{z^{c}}{f^{\deg c}}\right)$ over $L$. This
finishes the proof of the case when $1\leq k\leq d$. Finally, if $k=d+1$,
i.e., $\mu_{k}=\deg$, then an elementary computation shows that
$\sum_{i=1}^{n}\langle\mu_{k},v_{i}\rangle
x_{i}\omega_{c+v_{i}}+\langle\mu_{k},c\rangle\omega_{c}$ is zero. ∎

###  2.2. B-model central charges

Now we define central charges on the B-model, i.e., the toric Deligne-Mumford
stack ${\mathbb{P}}_{\mathbf{\Sigma}}$, in terms of certain cohomology-valued
Gamma series. We will be using the same notation as §1.1.

######  Remark 2.5.

We emphasize here that throughout this paper, we denote the smooth toric
Deligne-Mumford stack associated to a stacky fan
$\mathbf{\Sigma}=(\Sigma,\\{v_{i}\\})$ by ${\mathbb{P}}_{\mathbf{\Sigma}}$,
and denote its coarse moduli space (i.e., the toric variety associated to the
usual fan $\Sigma$) by ${\mathbb{P}}_{\Sigma}$.

We first recall the definition of orbifold cohomology of the toric Deligne-
Mumford stack ${\mathbb{P}}_{\mathbf{\Sigma}}$. Recall that the orbifold
cohomology of a Deligne-Mumford stack is defined as the usual cohomology of
its inertia stack (with a shift of grading which will be ignored here since it
plays no role in our setting). Due to the non-compactness of
${\mathbb{P}}_{\mathbf{\Sigma}}$, there are two types of orbifold cohomology
theory associated to it, namely the usual orbifold cohomology and the orbifold
cohomology with compact support.

######  Definition 2.6.

For each cone $\sigma\in\Sigma$ we define $\operatorname{Box}(\sigma)$ to be
the set of lattice points $\gamma$ which can be written as
$\gamma=\sum_{i\in\sigma}\gamma_{i}v_{i}$ with $0\leq\gamma_{i}<1$. We denote
the union of all $\operatorname{Box}(\sigma)$ by $\operatorname{Box}(\Sigma)$.
To each element $\gamma\in\operatorname{Box}(\Sigma)$ we associate a twisted
sector222Abusing the notation slightly, we will denote the closed substack
${\mathbb{P}}_{\mathbf{\Sigma}/\sigma(\gamma)}$ by $\gamma$ when there is no
confusion. ${\mathbb{P}}_{\mathbf{\Sigma}/\sigma(\gamma)}$ of
${\mathbb{P}}_{\mathbf{\Sigma}}$ corresponding to the minimal cone
$\sigma(\gamma)$ in $\Sigma$ containing $\gamma$, which is the closed substack
of ${\mathbb{P}}_{\mathbf{\Sigma}}$ defined by the quotient fan
$\Sigma/\sigma(\gamma)$. We define the dual of a twisted sector
$\gamma=\sum\gamma_{i}v_{i}$ by

$\displaystyle\gamma^{\vee}=\sum_{\gamma_{i}\not=0}(1-\gamma_{i})v_{i}.$

or equivalently, the unique element in $\operatorname{Box}(\sigma(\gamma))$
that satisfies

$\displaystyle\gamma^{\vee}=-\gamma\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$

The inertia stack of ${\mathbb{P}}_{\mathbf{\Sigma}}$ is the disjoint union of
all the twisted sectors.

######  Proposition 2.7.

As usual, $\operatorname{Star}(\sigma(\gamma))$ denotes the set of cones in
$\Sigma$ that contain $\sigma(\gamma)$. Cohomology space $H_{\gamma}$ of the
twisted sector $\gamma$ is naturally isomorphic to the quotient of the
polynomial ring
${\mathbb{C}}[D_{i}:i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)]$
by the ideal generated by the relations

$\displaystyle\prod_{j\in J}D_{j},\
J\not\in\operatorname{Star}(\sigma(\gamma)),\quad\text{and
}\sum_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\mu(v_{i})D_{i},\
\mu\in\operatorname{Ann}(v_{i},i\in\sigma(\gamma)).$

There is a ${\mathbb{C}}[D_{1},\ldots,D_{n}]$-module structure on $H_{\gamma}$
defined by declaring $D_{i}=0$ for
$i\not\in\operatorname{Star}(\sigma(\gamma))$ and solving (uniquely) for
$D_{i},i\in\sigma(\gamma)$ to satisfy the linear relations
$\sum_{i=1}^{n}\mu(v_{i})D_{i}=0$ for all $\mu\in N^{\vee}$.

######  Proposition 2.8.

Cohomology space with compact support $H_{\gamma}^{c}$ is generated by $F_{I}$
for $I\in\operatorname{Star}(\sigma(\gamma))$ such that
$\sigma_{I}^{\circ}\subseteq C^{\circ}$ with relations

$\displaystyle\quad D_{i}F_{I}-F_{I\cup\\{i\\}}\text{ for }i\not\in
I,I\cup\\{i\\}\in\operatorname{Star}(\sigma(\gamma))$ $\displaystyle\text{and
}D_{i}F_{I}\text{ for }i\not\in
I,I\cup\\{i\\}\not\in\operatorname{Star}(\sigma(\gamma))$

as a module over $H_{\gamma}$.

######  Definition 2.9.

The orbifold cohomology $H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$
of the smooth toric DM stack ${\mathbb{P}}_{\mathbf{\Sigma}}$ is defined as
the direct sum $\bigoplus_{\gamma}H_{\gamma}$ over all twisted sectors.
Similarly, the orbifold cohomology with compact support
$H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$ is defined as
$\bigoplus_{\gamma}H_{\gamma}^{c}$. We denote by $1_{\gamma}$ the generator of
$H_{\gamma}$.

We introduce a special type of characteristic classes of the smooth toric DM
stack ${\mathbb{P}}_{\mathbf{\Sigma}}$, called Gamma classes, which play an
essential role in toric mirror symmetry.

######  Definition 2.10.

For each twisted sector $\gamma$ of ${\mathbb{P}}_{\mathbf{\Sigma}}$, we
define its Gamma class by

$\displaystyle\widehat{\Gamma}_{\gamma}=\prod_{i\in\sigma(\gamma)}\Gamma(\gamma_{i}+\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})$

which is a cohomology class in $H_{\gamma}^{*}$. We define the Gamma class of
${\mathbb{P}}_{\mathbf{\Sigma}}$ to be the direct sum of Gamma classes of all
of its twisted sectors.

There is a natural non-degenerate pairing between
$H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$ and
$H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$ called _Euler
characteristic pairing_. The origin is the eponymous pairing on the
$K$-theories, which is then translated to the orbifold cohomology via the
Chern character, for details see appendix A and references therein. We will
not be using the original definition in section 2-4, but rather the following
formula proved in [7].

######  Proposition 2.11.

The Euler characteristic pairing
$\chi:H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})\otimes
H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})\rightarrow{\mathbb{C}}$
on the toric DM stack $\mathbb{P}_{\Sigma}$ is given by

$\displaystyle\chi(a,b)=\chi(\oplus_{\gamma}a_{\gamma},\oplus_{\gamma}b_{\gamma})=\sum_{\gamma}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\operatorname{Td}(\gamma^{\vee})a_{\gamma}^{*}b_{\gamma^{\vee}}$

Here $*$ is an involution map on the orbifold cohomology
$H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$, given by
$(1_{\gamma})^{*}=1_{\gamma^{\vee}}$, and $(D_{i})^{*}=-D_{i}$. Moreover,
$\operatorname{Td}(\gamma)$ is the Todd class of the twisted sector $\gamma$,
defined as

$\displaystyle\operatorname{Td}(\gamma)=\frac{\prod_{i\in\operatorname{Star}\sigma(\gamma)\backslash\sigma(\gamma)}D_{i}}{\prod_{i\in\operatorname{Star}\sigma(\gamma)}(1-e^{-D_{i}})}.$

The integration map $\int\colon H_{\gamma}^{c}\rightarrow{\mathbb{C}}$ takes
values $\frac{1}{{\rm Vol}_{\overline{I}}}$ on each generator $F_{I}$, where
${\rm Vol}_{\overline{I}}$ denotes the volume of the cone
$\overline{\sigma_{I}}$ in the quotient fan $\Sigma/\sigma(\gamma)$. It takes
value zero on all elements of $H_{\gamma}^{c}$ of lower degree.

Now we are ready to define the cohomology-valued Gamma series solutions to the
systems $\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$.

######  Definition 2.12.

We define the cohomology-valued Gamma series $\Gamma$ and $\Gamma^{\circ}$ as

$\displaystyle\Gamma_{c}(x_{1},\ldots,x_{n})=\bigoplus_{\gamma}\sum_{l\in
L_{c,\gamma}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm i}})}$ (2.2)

for lattice point $c\in{\mathbb{C}}$ and

$\displaystyle\Gamma_{c}^{\circ}(x_{1},\ldots,x_{n})=\bigoplus_{\gamma}\sum_{l\in
L_{c,\gamma}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm
i}})}\left(\prod_{i\in\sigma}D_{i}^{-1}\right)F_{\sigma}$

for lattice point $c\in{\mathbb{C}}^{\circ}$, where both direct sums are taken
over twisted sectors $\gamma=\sum_{j\in\sigma(\gamma)}\gamma_{j}v_{j}$ and the
set $L_{c,\gamma}$ is the set of solutions to $\sum_{i=1}^{n}l_{i}v_{i}=-c$
with $l_{i}-\gamma_{i}\in{\mathbb{Z}}$ for all $i$, and $\sigma$ is the set of
$i$ with $l_{i}\in{\mathbb{Z}}_{<0}$. The numerator is defined by picking a
branch of $\log(x_{i})$.

It is proved in [6] that these series converge absolutely and uniformly on
compacts in a neighborhood of the large radius limit point corresponding to
the triangulation $\Sigma$. After composing them with linear functions on the
orbifold cohomology spaces, we get holomorphic functions with values in
${\mathbb{C}}$. It is proved that all solutions to the systems
$\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$ are
obtained by composing $\Gamma$ and $\Gamma^{\circ}$ with linear functions on
$H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$ and
$H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$.

Now we are able to define our version of B-model central charge.

######  Definition 2.13.

Let $N$, $C$, and $\mathbf{\Sigma}=(\Sigma,\\{v_{i}\\})$ be as previous. For
any class $\mathcal{E}\in K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ in the
compactly supported K-theory of ${\mathbb{P}}_{\mathbf{\Sigma}}$, we define
its B-model central charge by

$\displaystyle Z^{B,\mathcal{E}}(x)=(Z^{B,\mathcal{E}}_{c}(x))_{c\in
C^{\circ}}=(\chi(-,\mathcal{E})\circ\Gamma^{\circ}_{c})_{c\in C^{\circ}}.$

##  3\. Asymptotic behavior of period integrals via tropical geometry

The goal of this section is to study the asymptotic behavior of the A-model
central charge associated to the real positive locus
$({\mathbb{R}}_{>0})^{d}$.

We first set up the notations that will be used throughout this section. We
will be using the same notations as in §2. Furthermore, let $\Sigma$ be a
regular triangulation of the cone $C$, and denote the corresponding convex
piecewise linear function by $\psi$. For each lattice point $c$ in the
interior $C^{\circ}$, we denote the minimal cone in $\Sigma$ containing $c$ by
$\sigma(c)$, and write $c=\sum_{i\in\sigma(c)}c_{i}v_{i}$. Then there is a
unique twisted sector $\gamma(c)\in\operatorname{Box}(\Sigma)$ given by
$\sum_{i\in\sigma(c)}\\{c_{i}\\}v_{i}$, where $\\{c_{i}\\}$ is the fractional
part of $c_{i}$. Finally, we denote the set of indices $i$ such that $c_{i}=1$
by $I_{c}$. Note that $\sigma(c)$ is a disjoint union of $I_{c}$ and
$\sigma(\gamma(c))$.

The main result of this section is an asymptotic formula for the A-model
central charges of the positive real Lagrangian ${\mathbb{R}}_{>0}^{d}$ when
the parameter approaches the large radius limit corresponding to the
triangulation $\Sigma$, which should correspond to the structure sheaf
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$ on the B-model according to the
prediction of SYZ conjecture. For simplicity, we introduce an extra variable
$t\in{\mathbb{R}}$ and consider the 1-parameter family of Laurent polynomials
$\\{f_{t}\\}$ where $f_{t}=\sum t^{-\psi(v_{i})}z^{v_{i}}$. Then the large
radius limit is achieved by taking $t\rightarrow+\infty$. Moreover, to
emphasize the dependence of the form $\omega_{c}$ on the parameter $t$, we use
the notation $\omega_{t,c}$ instead.

######  Theorem 3.1.

The asymptotic behavior of

$\displaystyle
Z^{A,{\mathbb{R}}_{>0}^{d}}_{c}(t^{-\psi(v_{1})},...)=\frac{(-1)^{d}}{(2\pi{\rm
i})^{d+1}}\int_{{\mathbb{R}}_{>0}^{d}}\omega_{t,c}$

when $t\rightarrow+\infty$ is given by

$\displaystyle Z^{A,{\mathbb{R}}_{>0}^{d}}_{c}(t^{-\psi(v_{1})},...)=$
$\displaystyle t^{\psi(c)}\frac{(-1)^{{\operatorname{rk}}N-\deg c}}{(2\pi{\rm
i})^{|\sigma(c)|}|\operatorname{Box}(\sigma(\gamma))|}\cdot\int_{\gamma(c)}t^{\omega}\cdot\hat{\Gamma}_{\gamma(c)}F_{I_{c}}+o(t^{\psi(c)})$

where $\omega=\frac{1}{2\pi{\rm i}}\sum_{i=1}^{n}\psi(v_{i})D_{i}$, and
$\hat{\Gamma}_{\gamma(c)}$ is the Gamma class of $\gamma(c)$ as defined in
Definition 2.10.

The proof of this theorem occupies the rest of this section. To begin with, we
make a change of coordinates. We denote the moment map on
$({\mathbb{C}}^{*})^{d}$ by

$\displaystyle\operatorname{Log}_{t}:({\mathbb{C}}^{*})^{d}\rightarrow{\mathbb{R}}^{d},(z_{1},\cdots,z_{d})\mapsto(\log_{t}|z_{1}|,\cdots,\log_{t}|z_{d}|)$

and its right-inverse by

$\displaystyle
i_{t}:{\mathbb{R}}^{d}\rightarrow({\mathbb{C}}^{*})^{d},(y_{1},\cdots,y_{d})\mapsto(t^{y_{1}},\cdots,t^{y_{d}}).$

Note that the positive real locus is identified with ${\mathbb{R}}^{d}$ under
the map $i_{t}$. The original integration now becomes

$\displaystyle\int_{{\mathbb{R}}_{>0}^{d}}\omega_{t,c}=\int_{{\mathbb{R}}^{d}}i_{t}^{*}\omega_{t,c}$

with the new coordinates $\\{y_{i}\\}$.

We divide the proof into two steps. The first step of the proof (§3.1) is to
cut the domain ${\mathbb{R}}^{d}$ into pieces according to the tropicalization
of the Laurent polynomial $f_{t}$, then relate the integration over each piece
to the volume of certain polytopes. In the second step (§3.2) we relate them
to the integral of Gamma classes on the toric stack
${\mathbb{P}}_{\mathbf{\Sigma}}$. A key ingredient of the second step is a
Duistermaat-Heckman type lemma adapted to our setting that is stated and
proved in Appendix A.

###  3.1. Subdivision of the domain

For each $i\in\Delta$, we consider the tropicalization $\beta_{i}$ of the
monomial $t^{-\psi(v_{i})}z^{v_{i}}$ defined as

$\displaystyle\beta_{i}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}},\quad
p\mapsto\langle v_{i},p\rangle-\psi(v_{i})$

which is an affine function on ${\mathbb{R}}^{d}$. Following the idea of [1],
we define333Our definition differs from the one in [1] by reversing the
direction of the inequality, due to the difference between
$t\rightarrow+\infty$ and $t\rightarrow 0^{+}$.

$\displaystyle
U^{q}:=\\{p\in{\mathbb{R}}^{d}:\beta_{i}(p)\leq\beta_{q}(p),\forall
i\in\Delta\\}$

for any lattice point $q\in\Delta$. Furthermore, for any $q\in\Delta$ and any
subset $q\not\in K\subseteq\Delta$ we define a subset $U^{q,K}$ of $U^{q}$ by

$\displaystyle
U^{q,K}:=\\{p\in{\mathbb{R}}^{d}:\beta_{q}(p)-\beta_{i}(p)\in[0,\epsilon],\forall
i\in K,\beta_{q}(p)-\beta_{i}(p)\in[\epsilon,+\infty),\forall i\not\in q\sqcup
K\\}$

for some fixed small positive number $\epsilon>0$. Intuitively, $U^{q,K}$ is
the region where the tropical monomial $\beta_{q}$ is the largest (hence
dominates the asymptotics) and the tropical monomials $\\{\beta_{k}\\}_{k\in
K}$ are not far behind.

######  Remark 3.2.

By the standard argument of tropical geometry, we have the following facts
about $U^{q}$ and $U^{q,K}$. First of all, the set $U^{q,K}$ is non-empty if
and only if $q\sqcup K$ is a cone in $\Sigma$. Moreover, $U^{q,K}$ is
unbounded if and only if $q\sqcup K$ is a cone on the boundary, i.e.,
$\operatorname{relint}(q\sqcup K)\subseteq\partial\Delta$. We will not use the
latter fact in rest of the paper so we omit its proof.

Hence the original integral can be written as a sum

$\displaystyle\int_{{\mathbb{R}}^{d}}i_{t}^{*}\omega_{t,c}=\sum_{q,K}\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}$

The first observation is the following lemma which states that only the region
$U^{q,K}$ with $\sigma(c)\subseteq q\sqcup K\in\Sigma$ contributes to the
leading term when $t\rightarrow+\infty$. Otherwise the growth of the integral
over the piece will be $O(t^{\psi(c)-\epsilon})$ for some $\epsilon>0$.

######  Lemma 3.3.

As $t\rightarrow+\infty$, for $q$ and $K$ with $\sigma(c)\not\subseteq q\sqcup
K$ we have $\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}=O(t^{\psi(c)-\epsilon})$ for
some $\epsilon>0$. If $\sigma(c)\subseteq q\sqcup K\in\Sigma$, then
$\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}$ is

$\displaystyle(-1)^{\deg c-1}$ $\displaystyle(\deg c-1)!t^{\psi(c)}(\log
t)^{d}\int_{U^{q,K}}\frac{\prod_{i\in
K}(t^{\beta_{i}-\beta_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{\beta_{i}-\beta_{q}}\right)^{\deg c}}\prod_{i}dy_{i}$
$\displaystyle+O(t^{\psi(c)-\epsilon}(\log t)^{d}).$

###### Proof.

First we suppose $\sigma(c)\not\subseteq q\sqcup K$. We have

$\displaystyle z^{c}$
$\displaystyle=z^{\sum_{i\in\sigma(c)}c_{i}v_{i}}=z^{c_{q}v_{q}}\cdot\prod_{i\in\sigma(c)\cup
K\backslash q}z^{c_{i}v_{i}}$
$\displaystyle=t^{\sum_{i\in\sigma(c)}c_{i}\psi(v_{i})}(t^{-\psi(v_{q})}z^{v_{q}})^{c_{q}}\cdot\prod_{i\in\sigma(c)\cup
K\backslash q}(t^{-\psi(v_{i})}z^{v_{i}})^{c_{i}}$
$\displaystyle=t^{\sum_{i\in\sigma(c)}c_{i}\psi(v_{i})}(t^{-\psi(v_{q})}z^{v_{q}})^{\sum_{i\in
q\sqcup K\cup\sigma(c)}c_{i}}\cdot\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}})^{c_{i}}$
$\displaystyle=t^{\psi(c)}(t^{-\psi(v_{q})}z^{v_{q}})^{\deg
c}\cdot\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}})^{c_{i}}$

Similarly we can compute

$\displaystyle f_{t}(z)^{\deg c}$
$\displaystyle=(\sum_{i\in\Delta}t^{-\psi(v_{i})}z^{v_{i}})^{\deg c}$
$\displaystyle=(t^{-\psi(v_{q})}z^{v_{q}})^{\deg c}\cdot(1+\sum_{i\in
K}t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}}+\sum_{j\not\in q\sqcup
K}t^{-\psi(v_{j})+\psi(v_{q})}z^{v_{j}-v_{q}})^{\deg c}$
$\displaystyle=(t^{-\psi(v_{q})}z^{v_{q}})^{\deg c}\cdot\left(1+\sum_{i\in
K}t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}}+O(t^{-\epsilon})\right)^{\deg
c}$

Therefore the form $\omega_{t,c}$ is

$\displaystyle\omega_{t,c}$ $\displaystyle=(-1)^{\deg c-1}(\deg
c-1)!t^{\psi(c)}\frac{\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}}+O(t^{-\epsilon})\right)^{\deg
c}}\cdot\prod_{i}\frac{dz_{i}}{z_{i}}$ $\displaystyle=(-1)^{\deg c-1}(\deg
c-1)!t^{\psi(c)}\left(\frac{\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{-\psi(v_{i})+\psi(v_{q})}z^{v_{i}-v_{q}}\right)^{\deg
c}}+O(t^{-\epsilon})\right)\cdot\prod_{i}\frac{dz_{i}}{z_{i}}$

hence the pullback $i_{t}^{*}\omega_{t,c}$ is

$\displaystyle i_{t}^{*}\omega_{t,c}$ $\displaystyle=(-1)^{\deg c-1}(\deg
c-1)!t^{\psi(c)}\left(\frac{\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{\beta_{i}-\beta_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{\beta_{i}-\beta_{q}}\right)^{\deg c}}+O(t^{-\epsilon})\right)\cdot(\log
t)^{d}\prod_{i}dy_{i}$

So the integration $\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}$ is equal to

$\displaystyle(-1)^{\deg c-1}$ $\displaystyle(\deg c-1)!t^{\psi(c)}(\log
t)^{d}\int_{U^{q,K}}\frac{\prod_{i\in\sigma(c)\cup K\backslash
q}(t^{\beta_{i}-\beta_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{\beta_{i}-\beta_{q}}\right)^{\deg c}}\prod_{i}dy_{i}$
$\displaystyle+O(t^{\psi(c)-\epsilon}(\log t)^{d}).$

Notice that if $\sigma(c)\not\subseteq q\sqcup K$, then there exists
$i\in\sigma(c)$ such that $\beta_{q}-\beta_{i}\geq\epsilon$ where $\epsilon>0$
is the constant used in the definition of $U^{q,K}$, i.e., the nominator of
the integrand will contribute a factor of $t^{-\epsilon}$. Therefore the first
term in the above expression is also $O(t^{\psi(c)-\epsilon}(\log t)^{d})$. By
changing $\epsilon$ to a smaller positive number we can adsorb the logarithmic
term and get $O(t^{\psi(c)-\epsilon})$.

Now suppose $\sigma(c)\subseteq q\sqcup K$. Then the same computation as above
shows that the integral $\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}$ is equal to

$\displaystyle(-1)^{\deg c-1}$ $\displaystyle(\deg c-1)!t^{\psi(c)}(\log
t)^{d}\int_{U^{q,K}}\frac{\prod_{i\in
K}(t^{\beta_{i}-\beta_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{\beta_{i}-\beta_{q}}\right)^{\deg c}}\prod_{i}dy_{i}$
$\displaystyle+O(t^{\psi(c)-\epsilon}(\log t)^{d}).$

∎

By this lemma we can disregard integrals over $U^{q,K}$ with
$\sigma(c)\not\subseteq q\sqcup K$ when computing the leading term of the
asymptotic behavior.

We now fix a pair $(q,K)$ with $\sigma(c)\subseteq q\sqcup K$ and look at the
integral

$\displaystyle\int_{U^{q,K}}\frac{\prod_{i\in
K}(t^{\beta_{i}-\beta_{q}})^{c_{i}}}{\left(1+\sum_{i\in
K}t^{\beta_{i}-\beta_{q}}\right)^{\deg c}}\prod_{i}dy_{i}$ (3.1)

that appears in the leading term of $\int_{U^{q,K}}i_{t}^{*}\omega_{t,c}$. We
make the following change of coordinate on the region $U^{q,K}$. We define

$\displaystyle b_{i}:=\beta_{q}-\beta_{i},\text{ for all }i\in K$

and complete $\\{b_{i}\\}_{i\in K}$ into an affine coordinate system on
${\mathbb{R}}^{d}$ by adding some covectors $\\{e_{j}\\}$. We then write the
standard affine volume form on ${\mathbb{R}}^{d}$ in terms of this new system
of coordinates:

$\displaystyle\prod_{i}dy_{i}=r_{q,K}\cdot\prod_{i\in
K}db_{i}\cdot\prod_{j}de_{j}$

So the original integral becomes

$\displaystyle\int_{U^{q,K}}\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}\cdot
r_{q,K}\cdot\prod_{i\in K}db_{i}\cdot\prod_{j}de_{j}$

Recall that the region $U^{q,K}$ is defined as

$\displaystyle
U^{q,K}=\left\\{p\in{\mathbb{R}}^{d}:b_{i}\in[0,\epsilon],\forall i\in K;\
\beta_{q}-\beta_{i}\in[\epsilon,\infty],\forall i\not\in q\sqcup K\right\\}$

We consider the projection $\pi_{b}:U^{q,K}\rightarrow[0,\epsilon]^{K}$ onto
the $(b_{i})_{i\in K}$-coordinate plane, and denote the fiber of a fixed
$(b_{i})_{i\in K}\in[0,\epsilon]^{K}$ by $F^{q,K}((b_{i})_{i\in K})$. Then the
integral above can be written as an iterated integral

$\displaystyle\int_{U^{q,K}}\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}\cdot
r_{q,K}\cdot\prod_{i\in K}db_{i}\cdot\prod_{j}de_{j}$ $\displaystyle=$
$\displaystyle\int_{[0,\epsilon]^{K}}\left(\int_{F^{q,K}((b_{i})_{i\in
K})}\frac{t^{-\sum_{i\in K}c_{i}b_{i}}}{\left(1+\sum_{i\in
K}t^{-b_{i}}\right)^{\deg c}}\cdot
r_{q,K}\cdot\prod_{j}de_{j}\right)\prod_{i\in K}db_{i}$ $\displaystyle=$
$\displaystyle\int_{[0,\epsilon]^{K}}\left(\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg
c}}\cdot\operatorname{vol}\left(F^{q,K}((b_{i})_{i\in
K})\right)\right)\prod_{i\in K}db_{i}$

By definition the fiber $F^{q,K}((b_{i})_{i\in K})$ is defined as

$\displaystyle F^{q,K}((b_{i})_{i\in
K})=\left\\{p\in{\mathbb{R}}^{d}:\beta_{q}-\beta_{i}=b_{i},\forall i\in K;\
\beta_{q}-\beta_{i}\in[\epsilon,\infty],\forall i\not\in q\sqcup K\right\\}$

To get rid of the $\epsilon$-dependence, we define a new polytope

$\displaystyle E^{q,K}((b_{i})_{i\in
K})=\left\\{p\in{\mathbb{R}}^{d}:\beta_{q}-\beta_{i}=b_{i},\forall i\in K;\
\beta_{q}-\beta_{i}\in[0,\infty],\forall i\not\in q\sqcup K\right\\}$

and express the fiber $F^{q,K}((b_{i})_{i\in K})$ in terms of these new
polytopes:

$\displaystyle F^{q,K}((b_{i})_{i\in K})=E^{q,K}((b_{i})_{i\in
K})\backslash\left(\bigcup_{j\not\in q\sqcup
K}\bigcup_{b_{j}\in[0,\epsilon]}E^{q,K\sqcup\\{j\\}}((b_{i})_{i\in
K},b_{j})\right)$

Then by inclusion-exclusion principle we have

$\displaystyle\operatorname{vol}(F^{q,K}((b_{i})_{i\in K}))=\sum_{J:J\supseteq
K,q\not\in J}(-1)^{|J\backslash K|}\int_{[0,\epsilon]^{J\backslash
K}}\operatorname{vol}(E^{q,J}((b_{i})_{i\in K},(b_{j}^{\prime})_{j\in
J\backslash K}))db^{\prime}$

Put all things together, the integral (3.1) becomes

$\displaystyle\int_{[0,\epsilon]^{K}}$ $\displaystyle\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}\cdot$
$\displaystyle\left(\sum_{J:J\supseteq K,q\not\in J}(-1)^{|J\backslash
K|}\int_{[0,\epsilon]^{J\backslash K}}\operatorname{vol}(E^{q,J}((b_{i})_{i\in
K},(b_{j}^{\prime})_{j\in J\backslash K}))db^{\prime}\right)db$

let $q$ and $K$ vary, we get

$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}(-1)^{|J\backslash
K|}\int_{[0,\epsilon]^{J}}$ $\displaystyle\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}\cdot$ (3.2)
$\displaystyle\operatorname{vol}(E^{q,J}((b_{i})_{i\in
K},(b_{j}^{\prime})_{j\in J\backslash K}))db^{\prime}db$

where the sum is taken over all triples $(J,K,q)$ such that $K\subseteq J$ and
$q\not\in J$. Note that the summand corresponding to $J$ is zero unless
$q\sqcup J$ is a cone in $\Sigma$.

###  3.2. Connection to Gamma classes

The goal of this subsection is to reveal the relationship between (3.2) with
the Gamma classes $\Gamma_{\gamma}$ of twisted sectors of the toric stack
${\mathbb{P}}_{\mathbf{\Sigma}}$. Again, we follow closely the idea of [1].

To start with, we apply the volume formula Proposition A.1 to the sum we
obtained at the end of the last subsection to rewrite it as

$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}$
$\displaystyle(-1)^{|J\backslash
K|}\int_{[0,\epsilon]^{J}}\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}\cdot$ (3.3)
$\displaystyle\left(\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}e^{D-\sum_{j\in
J}b_{j}D_{j}}\cdot\frac{D_{q\sqcup J}}{D_{\sigma(d)}}\cdot
F_{I_{c}}\right)db^{\prime}db$

where $D=\sum\psi(v_{i})D_{i}$, and $\gamma:=\gamma(c)$ is the unique twisted
sector corresponding to the lattice point $c\in C^{\circ}$. Now we consider
the following cohomology class in $H_{\gamma}^{*}$ obtained by scaling all
classes $D_{i}$ by a factor of $\frac{\log t}{2\pi{\rm i}}$:

$\displaystyle P_{t}=$ $\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in
J}(-1)^{|J\backslash K|}\left(\frac{\log t}{2\pi{\rm
i}}\right)^{|J|+1-|\sigma(c)|}\frac{D_{q\sqcup J}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\epsilon]^{J}}\left(\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in K}t^{-b_{i}}\right)^{\deg c}}e^{(\log
t)\frac{D}{2\pi{\rm i}}-\sum_{j\in J}b_{j}(\log t)\frac{D_{j}}{2\pi{\rm
i}}}\right)db^{\prime}db$

Since the integral over $\gamma$ is only relevant to the
$\deg=\dim\gamma=d+1-|\sigma(\gamma)|$ part, the expression (3.3) is equal to

$\displaystyle\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\left(\frac{\log
t}{2\pi{\rm i}}\right)^{-(d+1-|\sigma(c)|)}\int_{\gamma}P_{t}\cdot F_{I_{c}}$
(3.4)

note that $\deg F_{I_{c}}=|I_{c}|$ and $|\sigma(\gamma)|+|I_{c}|=|\sigma(c)|$.

The goal of the remaining part of this subsection is to prove the following
result which relates the cohomology class $P_{t}$ to the Gamma class
$\hat{\Gamma}_{\gamma}$ of the twisted sector $\gamma$:

######  Proposition 3.4.

The asymptotics of the class $P_{t}$ is given by

$\displaystyle P_{t}=\frac{(\log t)^{1-|\sigma(c)|}}{(\deg
c-1)!}t^{\omega}\widehat{\Gamma}_{\gamma}+O(t^{-\epsilon})$

where $\omega=\frac{1}{2\pi{\rm i}}\sum\psi(v_{i})D_{i}$.

To proceed, we consider the following analytic function in
$D_{1},\cdots,D_{n}$, where we think of the variables $D_{i}$’s as usual
complex numbers:

$\displaystyle Q_{t}(D_{1},\cdots,D_{n})=$
$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}(-1)^{|J\backslash
K|}\cdot\left(\frac{\log t}{2\pi{\rm
i}}\right)^{|J|+1-|\sigma(c)|}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{D_{q\sqcup J}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\epsilon]^{J}}\left(\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in
K}t^{-b_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}b_{j}(\log t)\frac{D_{j}}{2\pi{\rm
i}}}\right)db^{\prime}db$

The next proposition establishes the relationship between the function
$Q(D_{1},\cdots,D_{n})$ and the Gamma function $\Gamma$.

######  Proposition 3.5.

As functions in variables $D_{i}$’s we have the following identity:

$\displaystyle Q(D_{1},\cdots,D_{n})=(\log t)^{1-|\sigma(c)|}(2\pi{\rm
i})^{|\sigma(c)|}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{\prod_{i=1}^{n}\frac{D_{i}}{2\pi{\rm
i}}}{D_{\sigma(c)}}\cdot\frac{\prod_{i=1}^{n}\Gamma(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}{\Gamma(\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i}))}+O(t^{-\epsilon})$

###### Proof.

First we look at a single integral in the definition of $Q$:

$\displaystyle\int_{[0,\epsilon]^{J}}\left(\frac{t^{-\sum_{i\in
K}c_{i}b_{i}}}{\left(1+\sum_{i\in
K}t^{-b_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}b_{j}(\log t)\frac{D_{j}}{2\pi{\rm
i}}}\right)db^{\prime}db$

we make the change of variables $s_{i}:=(\log t)b_{i}$ to rewrite it as

$\displaystyle\int_{[0,\epsilon\log t]^{J}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}s_{j}\frac{D_{j}}{2\pi{\rm
i}}}\right)\frac{ds}{(\log t)^{|J|}}$

Now, we claim that we could replace the region $[0,\epsilon\log t]^{J}$ of the
integral by $[0,\infty)^{J}$ without changing the leading term of the
asymptotics. Namely, we have

$\displaystyle\int_{[0,\epsilon\log t]^{J}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}s_{j}\frac{D_{j}}{2\pi{\rm i}}}\right)ds$
$\displaystyle=\int_{[0,\infty)^{J}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}s_{j}\frac{D_{j}}{2\pi{\rm
i}}}\right)ds+O(t^{-\epsilon}).$

To see this, it suffices to observe that the integrand is controlled by

$\displaystyle e^{-\sum_{i\in K}c_{i}s_{i}}\cdot e^{-\sum_{j\in
J}s_{j}\frac{D_{j}}{2\pi{\rm i}}}$

then the claim follows from the fact that $\int_{\epsilon\log
t}^{+\infty}e^{-s}ds=O(t^{-\epsilon})$.

Therefore it suffices to look at

$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}(-1)^{|J\backslash
K|}\left(\frac{\log t}{2\pi{\rm i}}\right)^{|J|+1-|\sigma(c)|}e^{(\frac{\log
t}{2\pi{\rm i}})D}\frac{D_{q\sqcup J}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{J}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{j\in J}s_{j}\frac{D_{j}}{2\pi{\rm
i}}}\right)\frac{ds}{(\log t)^{|J|}}$

Note that $\int_{0}^{\infty}e^{-s_{j}\frac{D_{j}}{2\pi{\rm
i}}}ds_{j}=(2\pi{\rm i})/D_{j}$, integrating for all $j\in J\backslash K$, the
integral becomes

$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}(-1)^{|J\backslash
K|}\left(\frac{\log t}{2\pi{\rm i}}\right)^{|J|+1-|\sigma(c)|}e^{(\frac{\log
t}{2\pi{\rm i}})D}\frac{D_{q\sqcup J}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{K}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{i\in K}s_{i}\frac{D_{i}}{2\pi{\rm i}}}\prod_{j\in
J\backslash K}\frac{2\pi{\rm i}}{D_{j}}\right)\frac{ds}{(\log t)^{|J|}}$

which is equal to

$\displaystyle\sum_{(J,K,q):J\supseteq K,q\not\in J}$
$\displaystyle(-1)^{|J\backslash K|}\left(\frac{\log t}{2\pi{\rm
i}}\right)^{1-|\sigma(c)|}\left(\frac{1}{2\pi{\rm
i}}\right)^{|K|}e^{(\frac{\log t}{2\pi{\rm i}})D}\frac{D_{q\sqcup
K}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{K}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{i\in K}s_{i}\frac{D_{i}}{2\pi{\rm i}}}\right)ds$

We rewrite the sum as

$\displaystyle\sum_{(K,q):q\not\in K}$ $\displaystyle\left(\sum_{J:J\supseteq
K,q\not\in J}(-1)^{|J\backslash K|}\right)\left(\frac{\log t}{2\pi{\rm
i}}\right)^{1-|\sigma(c)|}\left(\frac{1}{2\pi{\rm
i}}\right)^{|K|}e^{(\frac{\log t}{2\pi{\rm i}})D}\frac{D_{q\sqcup
K}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{K}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{i\in K}s_{i}\frac{D_{i}}{2\pi{\rm i}}}\right)ds$

Note that for a fixed pair $(K,q)$, the sum

$\displaystyle\sum_{J:q\not\in J,K\subseteq J}(-1)^{|J\backslash
K|}=\sum_{P\subseteq\\{1,\cdots,N\\}\backslash(q\sqcup K)}(-1)^{|P|}$

which is equal to $(1+(-1))^{|\\{1,\cdots,n\\}\backslash(q\sqcup K)|}=0$ if
$\\{1,\cdots,n\\}\not=q\sqcup K$ and 1 otherwise. Therefore the only nonzero
term in the sum above corresponds to $K=\\{1,\cdots,n\\}\backslash q$. Hence
the sum above is equal to

$\displaystyle\sum_{q=1}^{n}$ $\displaystyle\left(\frac{\log t}{2\pi{\rm
i}}\right)^{1-|\sigma(c)|}\left(\frac{1}{2\pi{\rm
i}}\right)^{n-1}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{\prod_{i=1}^{n}D_{i}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{K}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{i\in K}s_{i}\frac{D_{i}}{2\pi{\rm i}}}\right)ds$

which is equal to

$\displaystyle(\log t)^{1-|\sigma(c)|}$ $\displaystyle(2\pi{\rm
i})^{|\sigma(c)|}\sum_{q=1}^{n}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{\prod_{i=1}^{n}\frac{D_{i}}{2\pi{\rm i}}}{D_{\sigma(c)}}$
$\displaystyle\cdot\int_{[0,\infty)^{K}}\left(\frac{e^{-\sum_{i\in
K}c_{i}s_{i}}}{\left(1+\sum_{i\in
K}e^{-s_{i}}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}e^{-\sum_{i\in K}s_{i}\frac{D_{i}}{2\pi{\rm i}}}\right)ds$

Make substitution $t_{i}=e^{-s_{i}}$, this is equal to

$\displaystyle(\log t)^{1-|\sigma(c)|}$ $\displaystyle(2\pi{\rm
i})^{|\sigma(c)|}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{\prod_{i=1}^{n}\frac{D_{i}}{2\pi{\rm i}}}{D_{\sigma(c)}}$
$\displaystyle\sum_{q\in\\{1,\cdots,n\\}}\int_{[0,1]^{n-1}}\frac{\prod_{i\not=q}t_{i}^{\frac{D_{i}}{2\pi{\rm
i}}+c_{i}}}{\left(1+\sum_{i\not=q}t_{i}\right)^{\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}}\frac{dt}{t}$

Apply Lemma B.1, this expression could be written as

$\displaystyle(\log t)^{1-|\sigma(c)|}(2\pi{\rm
i})^{|\sigma(c)|}e^{(\frac{\log t}{2\pi{\rm
i}})D}\frac{\prod_{i=1}^{n}\frac{D_{i}}{2\pi{\rm
i}}}{D_{\sigma(c)}}\cdot\frac{\prod_{i=1}^{n}\Gamma(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}{\Gamma(\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm i}}+c_{i}))}$

which finished the proof of the proposition. ∎

Now we have established the desired relationship between the function $Q_{t}$
and the Gamma functions. Applying them to the cohomology classes $D_{i}$
together with the following equality

$\displaystyle\frac{\prod_{i=1}^{n}\frac{D_{i}}{2\pi{\rm
i}}}{D_{\sigma(c)}}\cdot\frac{\prod_{i=1}^{n}\Gamma(\frac{D_{i}}{2\pi{\rm
i}}+c_{i})}{\Gamma(\sum_{i=1}^{n}(\frac{D_{i}}{2\pi{\rm i}}+c_{i}))}$
$\displaystyle=\frac{1}{(2\pi{\rm
i})^{|\sigma(c)|}}\frac{\prod_{i\not\in\sigma(c)}\frac{D_{i}}{2\pi{\rm
i}}}{(\deg c-1)!}\prod_{i\not\in\sigma(c)}\Gamma(\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\sigma(c)}\Gamma(\frac{D_{i}}{2\pi{\rm i}}+c_{i})$
$\displaystyle=\frac{1}{(2\pi{\rm i})^{|\sigma(c)|}(\deg
c-1)!}\prod_{i\not\in\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\sigma(\gamma)}\Gamma(\frac{D_{i}}{2\pi{\rm i}}+\gamma_{i})$
$\displaystyle=\frac{1}{(2\pi{\rm i})^{|\sigma(c)|}(\deg
c-1)!}\cdot\widehat{\Gamma}_{\gamma}$

finish the proof of Proposition 3.4.

Therefore the leading term of $\frac{(-1)^{d}}{(2\pi{\rm
i})^{d+1}}\int_{\mathbb{R}_{>0}^{d}}\omega_{t,c}$ is equal to (note that
$d+1-\deg c={\operatorname{rk}}N-\deg c$)

$\displaystyle t^{\psi(c)}\frac{(-1)^{{\operatorname{rk}}N-\deg c}}{(2\pi{\rm
i})^{|\sigma(c)|}|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}t^{\omega}\widehat{\Gamma}_{\gamma}\cdot
F_{I_{c}}$

which finishes the proof of Theorem 3.1.

##  4\. Equality of A-model and B-model central charges

The goal of this section is to establish the equality between A-model and
B-model central charges. The key ingredient is the hypergeometric duality [6].
More precisely, an explicit formula for the pairing between the solution
spaces to $\operatorname{bbGKZ}(C,0)$ and its dual
$\operatorname{bbGKZ}(C^{\circ},0)$ is given:

######  Definition 4.1.

For any pair of solutions $(\Phi_{c})$ and $(\Psi_{d})$ of the systems
$\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$, we define
a pairing

$\displaystyle\langle-,-\rangle:\operatorname{Sol}(\operatorname{bbGKZ}(C,0))\times\operatorname{Sol}(\operatorname{bbGKZ}(C^{\circ},0))\rightarrow{\mathbb{C}}$

by the following formula

$\displaystyle\langle\Phi,\Psi\rangle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}\left(\prod_{i\in I}x_{i}\right)\Phi_{c}\Psi_{d}$

where the coefficient $\xi_{c,d,I}$ is defined as follows. Fix a choice of a
generic vector $v\in C^{\circ}$. For a set $I$ of size ${\operatorname{rk}}N$
we consider the cone $\sigma_{I}=\sum_{i\in I}{\mathbb{R}}_{\geq 0}v_{i}$. The
coefficients $\xi_{c,d,I}$ for $c+d=v_{I}$ are defined as

$\displaystyle\xi_{c,d,I}=\left\\{\begin{array}[]{ll}&(-1)^{\operatorname{deg}(c)},\text{
if }\dim\sigma_{I}=\operatorname{rk}N\text{ and both }c+\varepsilon v\text{
and }d-\varepsilon v\in\sigma_{I}^{\circ}\\\ &0,\text{
otherwise}.\end{array}\right.$

Here the condition needs to hold for all sufficiently small positive number
$\varepsilon>0$.

The main result in [6] states that this pairing is non-degenerate.

######  Theorem 4.2.

For any pair of solutions $(\Phi_{c})$ and $(\Psi_{d})$ of the systems
$\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$, the
pairing $\langle\Phi,\Psi\rangle$ is a constant. Furthermore, the constant
pairing $\langle\Gamma,\Gamma^{\circ}\rangle$ of the cohomology-valued Gamma
series is equal to the inverse of the Euler characteristic pairing
$\chi:H_{\mathrm{orb}}^{*}\otimes
H_{\mathrm{orb},c}^{*}\rightarrow{\mathbb{C}}$ in the large radius limit. In
particular, $\langle-,-\rangle$ is non-degenerate.

We now combine the computation in section 3 together with the hypergeometric
duality to obtain the equality between A-model and B-model central charges. We
first prove the equality for the case of structure sheaf
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$ and its mirror cycle
${\mathbb{R}}_{\geq 0}^{d}$.

To start with, we recall the asymptotic behavior of the Gamma series $\Gamma$
that was computed in [6].

######  Lemma 4.3.

Let $t\rightarrow+\infty$, then for lattice point $c\in C$ and
$\gamma\not=\gamma^{\vee}(c)$, the summand of
$\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\cdots,t^{-\psi(v_{n})}x_{n})$ is
$o(t^{\psi(c)})$. For $\gamma=\gamma^{\vee}(c)$, we have

$\displaystyle\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots)=t^{\psi(c)}\prod_{i=1}^{n}{\rm
e}^{\frac{D_{i}}{2\pi{\rm i}}(\log x_{i}-\psi(v_{i})\log
t)}\prod_{i=1}^{n}\frac{x_{i}^{-c_{i}}}{\Gamma(1-c_{i}+\frac{D_{i}}{2\pi{\rm
i}})}(1+o(1)).$

###### Proof.

See [6, Lemma 3.10]. ∎

######  Theorem 4.4.

The A-model central charge associated to the positive real locus
$({\mathbb{R}}_{\geq 0})^{d}$ coincides with the B-model central charge
associated to the structure sheaf
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$.

###### Proof.

Throughout this proof, we will denote an interior lattice point by $d\in
C^{\circ}$ and denote the rank of the lattice $N$ by ${\operatorname{rk}}N$.

Consider the pairing

$\displaystyle\langle\Gamma,Z^{A,{\mathbb{R}}_{>0}^{d}}\rangle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}(\prod_{i\in I}x_{i})\Gamma_{c}\cdot
Z^{A,{\mathbb{R}}_{>0}^{d}}_{d}\in
H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$

we look at the component corresponding to a fixed twisted sector $\gamma$.
Combining Theorem 3.1, Lemma 4.3, by an argument similar to the proofs of [6,
Proposition 3.12, 3.13], the asymptotic behavior of

$\displaystyle\prod_{i=1}^{n}(t^{-\psi(v_{i})})\Gamma_{c,\gamma}(t^{-\psi(v_{i})})Z^{A,{\mathbb{R}}_{>0}^{d}}_{d}(t^{-\psi(v_{i})})$

is given by $o(1)$ unless $\gamma=\gamma^{\vee}(c)=\gamma(d)$ and both $I_{c}$
and $I_{d}$ are cones in $\Sigma$, in which case the asymptotic behavior is

$\displaystyle o(1)+\frac{1}{(2\pi{\rm
i})^{|I_{c}|}}\cdot\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\prod_{i=1}^{n}{\rm
e}^{\frac{D_{i}}{2\pi{\rm i}}(-\psi(v_{i})\log
t)}\frac{(-1)^{{\operatorname{rk}}N-\deg d}}{(2\pi{\rm
i})^{|\sigma(d)|}|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}t^{\omega}\hat{\Gamma}_{\gamma}F_{I_{d}}.$

Since $\langle\Gamma,\Psi\rangle$ is a constant, taking the constant term we
get

$\displaystyle\langle\Gamma,\Psi\rangle_{\gamma}$
$\displaystyle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\frac{(-1)^{{\operatorname{rk}}N-\deg
d}}{(2\pi{\rm
i})^{|I_{c}|+|\sigma(d)|}|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}\hat{\Gamma}_{\gamma}F_{I_{d}}$
$\displaystyle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\frac{(-1)^{{\operatorname{rk}}N-\deg
d}}{(2\pi{\rm
i})^{{\operatorname{rk}}N}|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}(-1)^{\dim\gamma^{\vee}-|I_{d}|}(\hat{\Gamma}_{\gamma})^{*}F_{I_{d}}$
$\displaystyle=\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\frac{(-1)^{{\operatorname{rk}}N-\deg
d}}{(2\pi{\rm i})^{{\operatorname{rk}}N}|\operatorname{Box}(\sigma(\gamma))|}$
$\displaystyle\quad\quad\quad\cdot\int_{\gamma^{\vee}}(2\pi{\rm
i})^{|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}+\dim\gamma^{\vee}-|I_{d}|}\frac{F_{I_{d}}}{\hat{\Gamma}_{\gamma^{\vee}}}\mathrm{Td}(\gamma^{\vee})$
$\displaystyle=\frac{1}{(2\pi{\rm
i})^{{\operatorname{rk}}N}}\sum_{c,d,I}\xi_{c,d,I}{\rm Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\frac{F_{I_{d}}}{\hat{\Gamma}_{\gamma^{\vee}}}\mathrm{Td}(\gamma^{\vee})$
$\displaystyle=\frac{1}{(2\pi{\rm
i})^{{\operatorname{rk}}N}}\sum_{c,d,I}\xi_{c,d,I}{\rm Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\cdot(1,\frac{F_{I_{d}}}{\hat{\Gamma}_{\gamma^{\vee}}})_{\mathrm{orb},\gamma}$

Here we used $\deg\gamma^{\vee}=|\sigma(\gamma)|+|I_{d}|-\deg
d=|\sigma(d)|-\deg d$, therefore

$\displaystyle(-1)^{{\operatorname{rk}}N-\deg
d+\deg\gamma^{\vee}+\dim\gamma^{\vee}-|I_{d}|}$
$\displaystyle=(-1)^{{\operatorname{rk}}N+|\sigma(d)|+{\operatorname{rk}}N-|\sigma(\gamma)|-|I_{d}|}$
$\displaystyle=(-1)^{{\operatorname{rk}}N+|\sigma(d)|+{\operatorname{rk}}N-|\sigma(d)|}=1$

By [6, Theorem 4.2] the following class in $H_{\mathrm{orb}}^{*}\otimes
H_{\mathrm{orb},c}^{*}$

$\displaystyle\frac{1}{(2\pi{\rm
i})^{{\operatorname{rk}}N}}\bigoplus_{\gamma}\sum_{c,d,I}\xi_{c,d,I}{\rm
Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\frac{D_{I_{c}}}{\hat{\Gamma}_{\gamma}}\otimes\frac{F_{I_{d}}}{\hat{\Gamma}_{\gamma^{\vee}}}$

is inverse to the Euler characteristic pairing, therefore for any $\gamma$ we
have

$\displaystyle\langle\Gamma,Z^{A,{\mathbb{R}}_{>0}^{d}}\rangle_{\gamma}=1_{\gamma}$

so
$\langle\Gamma,Z^{A,{\mathbb{R}}_{>0}^{d}}\rangle=\bigoplus_{\gamma}1_{\gamma}=ch(\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}})$,
i.e., $Z^{A,{\mathbb{R}}_{>0}^{d}}$ corresponds to the structure sheaf
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$. ∎

Now we finished the proof of Theorem 1.4 for the case of structure sheaf
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}$. Next we consider an arbitrary
line bundle $\mathcal{L}=\mathcal{O}(\sum_{i=1}^{n}a_{i}D_{i})$ corresponding
to a torus-invariant divisor $\sum_{i=1}^{n}a_{i}D_{i}$. The mirror cycle
$\operatorname{mir}(\mathcal{L})$ of $\mathcal{L}$ is constructed from
${\mathbb{R}}_{>0}^{d}$ by monodromy. More precisely, the divisor
$\sum_{i=1}^{n}a_{i}D_{i}$ defines a loop in the stringy Kähler moduli space
of ${\mathbb{P}}_{\mathbf{\Sigma}}$ by

$\displaystyle\phi:[0,1]\rightarrow{\mathbb{C}}^{n},\quad\theta\mapsto(e^{-2\pi{\rm
i}a_{1}\theta},\cdots,e^{-2\pi{\rm i}a_{n}\theta})$ (4.1)

We denote the Laurent polynomial corresponding to $\phi(\theta)$ by
$f^{(\theta)}$, then we have a family of hypersurfaces $Z_{f^{(\theta)}}$ in
$({\mathbb{C}}^{*})^{d}$, where $Z_{f^{(1)}}=Z_{f^{(0)}}=Z_{f}$. We then
define the mirror cycle of $\mathcal{L}$ to be the parallel transport of
${\mathbb{R}}_{>0}^{d}$ along this loop.

######  Corollary 4.5.

For any $\mathcal{L}=\mathcal{O}(\sum_{i=1}^{n}a_{i}D_{i})\in
K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$, the A- and B-model central charges
coincide:

$\displaystyle Z^{A,\mathrm{mir}(\mathcal{L})}=Z^{B,\mathcal{L}}.$

###### Proof.

It suffices to compare the monodromy along the loop (4.1) on both sides.
Recall that the Gamma series is given by

$\displaystyle\Gamma_{c}^{\circ}(x_{1},\ldots,x_{n})=\bigoplus_{\gamma}\sum_{l\in
L_{c,\gamma}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm
i}})}\left(\prod_{i\in\sigma}D_{i}^{-1}\right)F_{\sigma}$

the monodromy comes from the term
$\prod_{i=1}^{n}x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}=\prod_{i=1}^{n}e^{(l_{i}+\frac{D_{i}}{2\pi{\rm i}})\log x_{i}}$.

Fix $c$ and $\gamma$, when $\theta$ goes from 0 to 1, the $x_{i}$ goes around
the origin $a_{i}$ times clockwisely and hence the original $\log x_{i}$ now
becomes $\log x_{i}-2\pi{\rm i}a_{i}$, therefore contributes an extra factor
$e^{-a_{i}(2\pi{\rm i}l_{i}+D_{i})}$. Take product over all $i=1,\cdots,n$,
this is $e^{-\sum_{i}a_{i}(2\pi{\rm i}l_{i}+D_{i})}$. By definition of $l\in
L_{c,\gamma}$, we have $l_{i}\equiv\gamma_{i}\mod{\mathbb{Z}}$, therefore the
factor is equal to $e^{-\sum_{i}a_{i}(2\pi{\rm i}\gamma_{i}+D_{i})}$, which is
exactly the Chern character
$ch_{\gamma}(\mathcal{O}(-\sum_{i=1}^{n}a_{i}D_{i}))$.

Therefore the effect of the monodromy on the Gamma series is multiplying with
$ch(\mathcal{O}(-\sum_{i=1}^{n}a_{i}D_{i}))$. Then the B-model central charge
is obtained by composing with
$\chi(ch(\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}),-)$, which is

$\displaystyle\chi\left(1,ch(\mathcal{O}(-\sum_{i=1}^{n}a_{i}D_{i}))\cdot\Gamma_{c}^{\circ}\right)=\chi\left(ch(\mathcal{O}(\sum_{i=1}^{n}a_{i}D_{i})),\Gamma_{c}^{\circ}\right)$

by Proposition 2.11. This is exactly the central charge
$Z^{B,\mathcal{O}(\sum_{i=1}^{n}a_{i}D_{i})}$. It then follows immediately
from the construction of the mirror cycle of
$\mathcal{O}(\sum_{i=1}^{n}a_{i}D_{i})$ that the monodromy on the A-side
matches with the monodromy on the B-side. ∎

##  Appendix A Residual volume and orbifold cohomology

In this appendix we prove a result that relates the residual volume of the
polytopes $E^{q,J}((b_{j})_{j\in J})$ (for precise definitions of residual
volume and the polytopes, see section 3) with certain orbifold cohomology
classes with compact support of the toric Deligne-Mumford stack
${\mathbb{P}}_{\mathbf{\Sigma}}$. It could be viewed as a replacement of the
Duistermaat-Heckman lemma used in [1] adapted to our setting. We use the same
notations from section 3. Moreover, to avoid confusion, we will denote a
twisted sector corresponding to $\gamma\in\operatorname{Box}(\Sigma)$ by
${\mathbb{P}}_{\mathbf{\Sigma}/\gamma}$ throughout this appendix.

######  Proposition A.1.

The residual volume $\operatorname{vol}(E^{q,J}((b_{j})_{j\in J}))$ is equal
to

$\displaystyle\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{{\mathbb{P}}_{\mathbf{\Sigma}/\gamma}}e^{D-\sum_{j\in
J}b_{j}D_{j}}\frac{D_{q\sqcup J}}{D_{\sigma(c)}}F_{I_{c}}$

where $D:=\sum_{i}\psi(v_{i})D_{i}$, and $\gamma:=\gamma(c)$ is the unique
twisted sector corresponding to the interior lattice point $c\in C^{\circ}$.

We recall the (well-known) results on the relationship between line bundles on
toric varieties and the associated polytopes in §A.1, and review the
combinatorial formula for Euler characteristics of coherent sheaves on toric
Deligne-Mumford stacks from [10] in §A.2. Finally, we prove Proposition A.1 in
§A.3.

###  A.1. Line bundles on toric varieties and their associated polytopes

We briefly review the classical correspondence between line bundles on toric
varieties and their associated polytopes following [13].

Again, we denote by ${\mathbb{P}}_{\Sigma}$ the toric variety corresponding to
a fan $\Sigma$, and $D=\sum_{\rho}a_{\rho}D_{\rho}$ be a Cartier divisor on
${\mathbb{P}}_{\Sigma}$, where $D_{\rho}$’s are the torus-invariant divisors,
and we denote the primitive generators of the correponding rays in the fan by
$v_{\rho}$. The associated polytope $P_{D}$ of the line bundle
$\mathcal{O}_{{\mathbb{P}}_{\Sigma}}(D)$ is defined as444There is a difference
of signs in our definition with the one in [13].

$\displaystyle P_{D}:=\\{m\in M_{{\mathbb{R}}}:\langle
m,v_{\rho}\rangle+a_{\rho}\geq 0,\ \forall\rho\\}$

It is a well-known fact that the dimension of the global section of
$\mathcal{O}_{{\mathbb{P}}_{\Sigma}}(D)$ is equal to the number of lattice
points in the polytope $P_{D}$. In fact, we have

$\displaystyle\Gamma({\mathbb{P}}_{\Sigma},\mathcal{O}_{{\mathbb{P}}_{\Sigma}}(D))=\bigoplus_{m\in
P_{D}\cap M}{\mathbb{C}}\cdot\chi^{m}$

In this case the polytope $P_{D}$ is of full-dimension.

This correspondence could be generalized further to the case where the
polytope is not of full-dimension. In this case, the corresponding sheaf is
not a line bundle on ${\mathbb{P}}_{\Sigma}$, but the restriction of a line
bundle to a closed subvariety.

Finally, suppose we have a sheaf of the form $\mathcal{O}_{D^{\prime}}(D)$
where $D^{\prime}$ and $D$ are torus-invariant divisors and its associated
(non-full-dimensional) polytope $P$. We further assume that it is nef. Then by
Demazure vanishing theorem (see [13, Theorem 9.2.3]) all higher cohomology of
$\mathcal{O}_{D^{\prime}}(D)$ is zero, therefore we have

$\displaystyle\chi({\mathbb{P}}_{\Sigma},\mathcal{O}_{D^{\prime}}(D))=\chi(D^{\prime},\mathcal{O}_{D^{\prime}}(D))=\dim
H^{0}(D^{\prime},\mathcal{O}_{D^{\prime}}(D))=|P\cap M|$

i.e., the Euler characteristic of $\mathcal{O}_{D^{\prime}}(D)$ is equal to
the number of lattice points in the polytope $P$.

###  A.2. Euler characteristic of line bundles on toric Deligne-Mumford stacks

In this subsection, we review the combinatorial formula for Euler
characteristic of line bundles on toric Deligne-Mumford stacks.

In [10], two types of Grothendieck $K$-groups
$K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ and
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ associated to a toric Deligne-
Mumford stack ${\mathbb{P}}_{\mathbf{\Sigma}}$ are defined. The former is the
usual Grothendieck group of the bounded derived category
$D^{b}({\mathbb{P}}_{\mathbf{\Sigma}})$. The latter is defined as follows. We
denote by $D^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ the triangulated category of
bounded complexes of coherent sheaves on the toric Deligne-Mumford stack
${\mathbb{P}}_{\mathbf{\Sigma}}$ whose cohomology sheaves are supported on
$\pi^{-1}(0)$, where
$\pi:{\mathbb{P}}_{\mathbf{\Sigma}}\rightarrow\operatorname{Spec}{\mathbb{C}}[C^{\vee}\cap
N^{\vee}]$ is the structure map. Equivalently, $\pi^{-1}(0)$ is the union of
all torus orbits for interior cones $\sigma\in\Sigma$ (i.e.,
$C^{\circ}\subseteq C^{\circ}$). We call
$D^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ the derived category with compact
support of ${\mathbb{P}}_{\mathbf{\Sigma}}$, and
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ is defined to be its Grothendieck
group. We call $K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ (resp.
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$) the K-theory (resp. K-theory with
compact support) of ${\mathbb{P}}_{\mathbf{\Sigma}}$.

The following combinatorial descriptions of
$K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ and
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ are proved in [8].

######  Lemma A.2.

Let $C$, $v_{i}$ and $\Sigma$ be as before. We denote the class of the line
bundle $\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}(D_{i})$ corresponding to
the ray $v_{i}$ by $R_{i}$. Then $K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ is
isomorphic to the quotient of the ring ${\mathbb{C}}[R_{i}^{\pm 1}]$ by the
relations

$\displaystyle\prod_{i=1}^{n}R_{i}^{\mu(v_{i})}-1,\quad\mu\in N^{\vee},\text{
and }\prod_{i\in I}(1-R_{i}),\quad I\not\in\Sigma$

Furthermore, if we denote the class of the structure sheaf of the closed
substack corresponding to a cone $\sigma_{I}$ by $G_{I}$, then
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$ is a module over
$K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ generated by $G_{I}$ for all
$I\in\Sigma$ with $\sigma_{I}$ being an interior cone, with the relations
given by

$\displaystyle(1-R_{i}^{-1})G_{I}=G_{I\cup\\{i\\}}\text{ if
}I\cup\\{i\\}\in\Sigma\text{ and }0\text{ otherwise},\ \forall i$

The relations between $K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})$ (resp.
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$) and the orbifold cohomology
$H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$ (resp.
$H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})$) are given by the
following combinatorial Chern characters.

######  Lemma A.3.

There is a natural isomorphism

$\displaystyle
ch:K_{0}({\mathbb{P}}_{\mathbf{\Sigma}})\xrightarrow{\sim}H_{\mathrm{orb}}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})=\bigoplus_{\gamma\in\operatorname{Box}(\Sigma)}H_{\gamma}$

defined by

$\displaystyle
ch_{\gamma}(R_{i})=\begin{cases}1,\quad&i\not\in\operatorname{Star}(\sigma(\gamma))\\\
e^{D_{i}},\quad&i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)\\\
e^{2\pi{\rm
i}\gamma_{i}}\prod_{j\not\in\sigma(\gamma)}ch_{\gamma}(R_{j})^{\mu_{i}(v_{j})},\quad&i\in\sigma(\gamma)\end{cases}$

Similarly, there is a natural isomorphism

$\displaystyle
ch^{c}:K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})\xrightarrow{\sim}H_{\mathrm{orb},c}^{*}({\mathbb{P}}_{\mathbf{\Sigma}})=\bigoplus_{\gamma\in\operatorname{Box}(\Sigma)}H_{\gamma}^{c}$

defined by

$\displaystyle
ch_{\gamma}^{c}(\prod_{i=1}^{n}R_{i}^{l_{i}}G_{I})=\begin{cases}0,\quad&I\not\subseteq\operatorname{Star}(\sigma(\gamma))\\\
\prod_{i=1}^{n}ch_{\gamma}(R_{i}^{l_{i}})F_{I},\quad&I\subseteq\operatorname{Star}(\sigma(\gamma))\end{cases}$

Finally, we have the following combinatorial formula for the Euler
characteristic of sheaves in the K-theory with compact support
$K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})$.

######  Lemma A.4.

We define a linear function
$\chi:K_{0}^{c}({\mathbb{P}}_{\mathbf{\Sigma}})\rightarrow{\mathbb{Z}}$ by

$\displaystyle\chi(v)=\sum_{\gamma\in\operatorname{Box}(\Sigma)}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}ch_{\gamma}^{c}(v)\operatorname{Td}(\gamma)$

Then $\chi(v)$ is equal to the Euler characteristic of the sheaves represented
by the class $v$.

###### Proof.

The well-definedness is proved in [10, Prop.4.1, Prop.4.5]. The fact that it
computes the Euler characteristic of $v$ is proved in [10, Lem.5.3]. ∎

###  A.3. Proof of Proposition A.1

Before we start, we remark here that it suffices to prove the statement for
the case where $q\sqcup J$ is a cone in the fan $\Sigma$ because otherwise the
polytope $E^{q,J}((b_{j})_{j\in J})$ is empty and the right hand side of the
equality is zero due to the factor $D_{q\sqcup J}$.

We divide the proof into four steps.

Step 1. Recall that the polytope $E^{q,J}((b_{j})_{j\in J})$ is defined as

$\displaystyle E^{q,J}((b_{i})_{i\in
J})=\left\\{p\in{\mathbb{R}}^{d}:\beta_{q}-\beta_{i}=b_{i},\forall i\in J;\
\beta_{q}-\beta_{i}\in[0,\infty],\forall i\not\in q\sqcup J\right\\}$

where $\beta_{i}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ is a linear function
defined as $p\mapsto\langle v_{i},p\rangle-\psi(v_{i})$. Without loss of
generality we can assume all $b_{i}$’s are rational numbers. The defining
equalities and inequalities of $E^{q,J}((b_{j})_{j\in J})$ could be rewritten
as

$\displaystyle\langle v_{q}-v_{i},p\rangle+\psi(v_{i})-\psi(v_{q})-b_{i}=0$

for $i\in J$ and

$\displaystyle\langle v_{q}-v_{i},p\rangle+\psi(v_{i})-\psi(v_{q})\geq 0$

for $i\not\in q\sqcup J$. If we denote

$\displaystyle D:$ $\displaystyle=\sum_{i\in
J}(\psi(v_{i})-\psi(v_{q})-b_{i})D_{i}+\sum_{i\not\in q\sqcup
J}(\psi(v_{i})-\psi(v_{q}))D_{i}$
$\displaystyle=\sum_{i\in\operatorname{Star}(q)}\psi(v_{i})D_{i}-\sum_{i\in
J}b_{i}D_{i}$

then by the discussion in §A.1 we have

$\displaystyle\chi({\mathbb{P}}_{\Sigma/q},\mathcal{O}_{D_{J}}(lD))=|l\cdot
E^{q,J}((b_{i})_{i\in J})\cap M|$

where $l$ is any integer number that makes $l\cdot E^{q,J}((b_{i})_{i\in J})$
into a lattice polytope (the existence is due to the rationality of
$b_{i}$’s), and $D_{J}$ is the closed subvariety of ${\mathbb{P}}_{\Sigma/q}$
corresponding to the cone $J$ in the quotient fan $\Sigma/q$. We can then
further write this as

$\displaystyle|l\cdot E^{q,J}((b_{i})_{i\in J})\cap
M|=\chi({\mathbb{P}}_{\Sigma/(q\sqcup
J)},\mathcal{O}_{{\mathbb{P}}_{\Sigma/(q\sqcup J)}}(lD))$

Step 2. Denote the canonical map from the smooth toric Deligne-Mumford stack
${\mathbb{P}}_{\mathbf{\Sigma}}$ to its coarse moduli space (which is a
simplicial toric variety) ${\mathbb{P}}_{\Sigma}$ by $\pi$. We denote the line
bundle on ${\mathbb{P}}_{\mathbf{\Sigma}}$ defined by the same support
function with $D$ by $\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}(D)$. Since
in the definition of ${\mathbb{P}}_{\mathbf{\Sigma}}$ the additional data of a
vector on each ray of the stacky fan is chosen to be the primitive generator,
we know that the pushforward of
$\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}}}(D)$ is exactly
$\mathcal{O}_{{\mathbb{P}}_{\Sigma}}(D)$. On the other hand, it is a well-
known result (see e.g.,[2, Definition 4.1, Example 8.1]) for a tame Deligne-
Mumford stack $\mathcal{X}$ with coarse moduli space $X$, the canonical map
$\pi:\mathcal{X}\rightarrow X$ is cohomologically affine, which implies
$H^{i}(\mathcal{X},\mathcal{F})$ is equal to $H^{i}(X,\pi_{*}\mathcal{F})$ for
all $i>0$ and any coherent sheaf $\mathcal{F}$. Apply this fact to our
situation, we get $\chi({\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)},\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)}}(lD))=\chi({\mathbb{P}}_{\Sigma/(q\sqcup
J)},\mathcal{O}_{{\mathbb{P}}_{\Sigma/(q\sqcup J)}}(lD))$. So we have

$\displaystyle\chi({\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)},\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup J)}}(lD))=|l\cdot
E^{q,J}((b_{i})_{i\in J})\cap M|$

Step 3.

Then we apply the combinatorial formula of the Euler characteristic of line
bundles on toric Deligne-Mumford stacks, we get

$\displaystyle|l\cdot E^{q,J}((b_{i})_{i\in J})\cap M|$
$\displaystyle=\sum_{\gamma\in\operatorname{Box}(\Sigma/(q\sqcup
J))}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}ch_{\gamma}^{c}(l\cdot\mathcal{O}_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)}}(lD))\operatorname{Td}(\gamma)$
$\displaystyle=\sum_{\gamma\in\operatorname{Box}(\Sigma/(q\sqcup
J))}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma}e^{lD}\cdot\operatorname{Td}(\gamma)$

note that since $q\sqcup J$ is an interior cone (because it contains an
interior cone $\sigma(c)$ as a subcone), the quotient fan $\Sigma/(q\sqcup J)$
is complete, hence ${\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup J)}$ is compact
therefore $K_{0}$ and $K_{0}^{c}$ (and the corresponding Chern characters)
coincide.

The affine volume555Note that the affine volume differs with the residual
volume $\operatorname{vol}E^{q,J}((b_{i})_{i\in J})$ by a factor of the index
of $b_{i}$’s, namely the index of the sublattice spanned by $b_{i}$’s inside
the standard lattice ${\mathbb{Z}}^{d}$, see step 4.
$\operatorname{vol}_{\mathrm{aff}}E^{q,J}((b_{i})_{i\in J})$ is computed by

$\displaystyle\operatorname{vol}_{\mathrm{aff}}E^{q,J}((b_{i})_{i\in J})$
$\displaystyle=\lim_{l\rightarrow\infty}\frac{|l\cdot E^{q,J}((b_{i})_{i\in
J})\cap M|}{l^{\dim E^{q,J}((b_{i})_{i\in J})}}$
$\displaystyle=\sum_{\gamma\in\operatorname{Box}(\Sigma/(q\sqcup
J))}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\lim_{l\rightarrow\infty}\int_{\gamma}\frac{e^{lD}}{l^{{\operatorname{rk}}N-1-|J|}}\cdot\operatorname{Td}(\gamma)$

In the last step we used $\dim E^{q,J}((b_{i})_{i\in
J})={\operatorname{rk}}N-1-|J|$.

Now we claim that the only nonzero term in this sum is the $\gamma=0$ term. To
see this, we expand the $e^{lD}$ and the Todd class
$\operatorname{Td}(\gamma)$ as sums of homogeneous components:

$\displaystyle\int_{\gamma}\frac{e^{lD}}{l^{{\operatorname{rk}}N-1-|J|}}\cdot\operatorname{Td}(\gamma)=\int_{\gamma}\sum_{i,j=0}^{\infty}\frac{1}{l^{{\operatorname{rk}}N-1-|J|}}\frac{l^{i}D^{i}}{i!}\operatorname{Td}(\gamma)_{j}$

where $\operatorname{Td}(\gamma)_{j}$ denotes the degree $j$ part of the Todd
class of $\gamma$. By definition the only nonzero contribution comes from
terms with $(i,j)$ such that

$\displaystyle\deg(D^{i})+\deg(\operatorname{Td}(\gamma)_{j})=i+j$

is exactly equal to
$\dim(\gamma)={\operatorname{rk}}N-1-|J|-|\sigma(\gamma)|$. On the other hand,
if $i<{\operatorname{rk}}N-1-|J|$, the integral will be killed by taking limit
$l\rightarrow\infty$ due to the factor of
$\frac{1}{l^{{\operatorname{rk}}N-1-|J|}}$, so
$i\geq{\operatorname{rk}}N-1-|J|$. Combine these two observations together we
see that to have nonzero contribution, we must have

$\displaystyle{\operatorname{rk}}N-1-|\sigma(\gamma)|-|J|=i+j\geq{\operatorname{rk}}N-1-|J|+j$

that simplifies to $|\sigma(\gamma)|\leq-j$, which forces $j=0$,
$i={\operatorname{rk}}N-1-|J|$ and $\sigma(\gamma)=\emptyset$, i.e,
$\gamma=0$. Hence the claim is proved.

Therefore we have

$\displaystyle\operatorname{vol}_{\mathrm{aff}}E^{q,J}((b_{i})_{i\in
J})=\int_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)}}\frac{D^{{\operatorname{rk}}N-1-|J|}}{({\operatorname{rk}}N-1-|J|)!}=\int_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup
J)}}e^{D}$

Step 4. The affine and residual volume of the polytope $E^{q,J}((b_{i})_{i\in
J})$ satisfies

$\displaystyle\operatorname{vol}_{\mathrm{aff}}E^{q,J}((b_{i})_{i\in
J})=(\text{index of
}b_{i}^{\prime}s)\cdot\operatorname{vol}E^{q,J}((b_{i})_{i\in J})$

and the index of $b_{i}$’s is exactly equal to $|\operatorname{Box}(q\sqcup
J)|$, therefore

$\displaystyle\operatorname{vol}E^{q,J}((b_{i})_{i\in J})$
$\displaystyle=\frac{1}{|\operatorname{Box}(q\sqcup
J)|}\int_{{\mathbb{P}}_{\mathbf{\Sigma}/(q\sqcup J)}}e^{D}$
$\displaystyle=\int_{{\mathbb{P}}_{\mathbf{\Sigma}}}e^{D}\cdot F_{q\sqcup J}$
$\displaystyle=\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{{\mathbb{P}}_{\mathbf{\Sigma}/\sigma(\gamma)}}e^{D}\cdot
F_{q\sqcup J\backslash\sigma(\gamma)}$
$\displaystyle=\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{{\mathbb{P}}_{\mathbf{\Sigma}/\sigma(\gamma)}}e^{D}\cdot\frac{D_{q\sqcup
J}}{D_{\sigma(c)}}F_{I_{c}}$

where the second equality follows from the fact that the ratio between the
volume of a cone in the fan $\Sigma$ and that of the corresponding quotient
cone in the quotient fan $\Sigma/(q\sqcup J)$ is equal to
$|\operatorname{Box}(q\sqcup J)|$. The third equality holds for similar
reasons. The last equality is due to relations in the orbifold cohomology
space. The proof is now completed.

##  Appendix B An identity of Beta function

In this appendix, we prove an identity of multivariate beta functions that is
used in the computation in §3. Recall that the usual beta function $B(a,b)$ is
defined as

$\displaystyle B(a,b):=\int_{0}^{1}t^{a-1}(1-t)^{b-1}dt$

There is a well-known relation between beta function and gamma function
$B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$. There is an equivalent definition of
$B(a,b)$ given by

$\displaystyle B(a,b)=\int_{0}^{\infty}\frac{t^{a-1}}{(1+t)^{a+b}}dt$

Similarly, we can define the multivariable beta function by

$\displaystyle
B(a_{1},\cdots,a_{n})=\int_{[0,\infty)^{n-1}}\frac{t_{1}^{a_{1}}\cdots
t_{n-1}^{a_{n-1}}}{(1+t_{1}+\cdots+t_{n-1})^{a_{1}+\cdots+a_{n}}}dt_{1}\cdots
dt_{n-1}$

and there is an identity
$B(a_{1},\cdots,a_{n})=\Gamma(a_{1})\cdots\Gamma(a_{n})/\Gamma(a_{1}+\cdots+a_{n})$.

######  Lemma B.1.

We have the following identity:

$\displaystyle
B(a_{1},\cdots,a_{n})=\sum_{q=1}^{n}\int_{[0,1]^{n-1}}\frac{\prod_{i\not=q}t_{i}^{a_{i}-1}}{\left(1+\sum_{i\not=q}t_{i}\right)^{\sum_{i=1}^{n}a_{i}}}\prod_{i\not=q}dt_{i}$

###### Proof.

We treat the $q=n$ term and the other terms separately. The $q=n$ term could
be written as

$\displaystyle\int_{[0,1]^{n-1}}\frac{\prod_{i=1}^{n-1}t_{i}^{a_{i}-1}}{\left(1+\sum_{i=1}^{n-1}t_{i}\right)^{\sum_{i=1}^{n}a_{i}}}\prod_{i=1}^{n-1}dt_{i}=\sum_{q=1}^{n-1}\int_{A_{q}}\frac{\prod_{i=1}^{n-1}t_{i}^{a_{i}-1}}{\left(1+\sum_{i=1}^{n-1}t_{i}\right)^{\sum_{i=1}^{n}a_{i}}}\prod_{i=1}^{n-1}dt_{i}$

where $A_{q}$ is the region defined by $0\leq t_{q}\leq 1$ and $0\leq
t_{j}\leq t_{q}$ for $j\not=q$.

When $q\not=n$, we do the change of coordinate given by
$t_{i}\rightarrow\frac{t_{i}}{t_{n}}$ for $i\not=n$ and
$t_{n}\rightarrow\frac{1}{t_{n}}$, then rename the variable $t_{n}$ to
$t_{q}$. An elementary computation shows that the integral becomes

$\displaystyle\int_{B_{q}}\frac{\prod_{i=1}^{n-1}t_{i}^{a_{i}-1}}{\left(1+\sum_{i=1}^{n-1}t_{i}\right)^{\sum_{i=1}^{n}a_{i}}}\prod_{i=1}^{n-1}dt_{i}$

where $B_{q}$ is the region defined by $t_{q}\geq 1$ and $0\leq t_{j}\leq
t_{q}$ for $j\not=q$. Therefore the original sum can be written as the
integral over the union $\bigcup_{q=1}^{n-1}(A_{q}\cup B_{q})$. Now the result
follows from the observation that $A_{q}\cup B_{q}$ is the region defined by
$t_{q}\geq 0$ and $0\leq t_{j}\leq t_{q}$ for $j\not=q$, and
$\bigcup_{q=1}^{n-1}(A_{q}\cup B_{q})$ is exactly $[0,\infty)^{n-1}$. ∎

######  Remark B.2.

A similar identity was proved in [1] by using integration over certain
tropical projective spaces. The proof we provide here is purely elementary.

## References

  * [1] Mohammed Abouzaid, Sheel Ganatra, Hiroshi Iritani and Nick Sheridan, _The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods._ Geometry & Topology 24.5 (2020): 2547–2602.
  * [2] Jarod Alper, _Good moduli spaces for Artin stacks._ Annales de l’Institut Fourier. Vol. 63. No. 6. 2013.
  * [3] Victor Batyrev, _Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori_. Duke Math. J. 72.1 (1993): 349-409.
  * [4] Christine Berkesch, Jens Forsgard, and Mikael Passare. _Euler-Mellin integrals and A-hypergeometric functions._ Michigan Mathematical Journal 63.1 (2014): 101-123.
  * [5] Lev Borisov, Linda Chen, and Gregory Smith, _The orbifold Chow ring of toric Deligne-Mumford stacks._ Journal of the American Mathematical Society 18.1 (2005): 193-215.
  * [6] Lev Borisov, Zengrui Han, _On hypergeometric duality conjecture._ Advances in Mathematics, Volume 442 (2024):109582.
  * [7] Lev Borisov, Zengrui Han, Chengxi Wang, _On duality of certain GKZ hypergeometric systems._ Asian Journal of Mathematics 25.1 (2021): 65–88.
  * [8] Lev Borisov, R. Paul Horja, _On the K-theory of smooth toric DM stacks._ Snowbird Lectures on String Geometry, Contemp. Math., vol. 401, Amer. Math. Soc., Providence, RI (2006), pp. 21-42
  * [9] Lev Borisov, R. Paul Horja, _On the better behaved version of the GKZ hypergeometric system._ Mathematische Annalen 357.2 (2013): 585-603.
  * [10] Lev Borisov, R. Paul Horja, _Applications of homological mirror symmetry to hypergeometric systems: duality conjectures._ Advances in Mathematics 271 (2015): 153–187.
  * [11] Lev Borisov, R. Paul Horja, _Mellin-Barnes integrals as Fourier-Mukai transforms._ Advances in Mathematics 207.2 (2006): 876–927.
  * [12] Tom Bridgeland, _Stability conditions on triangulated categories._ Annals of Mathematics (2007): 317-345.
  * [13] David A. Cox, John B. Little, and Henry K. Schenck. _Toric varieties._ Vol. 124. American Mathematical Soc., 2011.
  * [14] Michael R. Douglas, _Dirichlet branes, homological mirror symmetry, and stability._ Proceedings of the International Congress of Mathematicians, Beijing, 2002. Higher Ed. Press, 2002.
  * [15] Lisa Nilsson and Mikael Passare. _Mellin transforms of multivariate rational functions._ Journal of Geometric Analysis 23.1 (2013): 24-46.
  * [16] Zengrui Han, _Analytic continuation of better-behaved GKZ systems and Fourier-Mukai transforms._ arXiv preprint arXiv:2305.12241 (2023).
  * [17] Kentaro Hori and Cumrun Vafa, _Mirror symmetry_. arXiv preprint hep-th/0002222 (2000).
  * [18] Shinobu Hosono, _Central charges, symplectic forms, and hypergeometric series in local mirror symmetry._ arXiv preprint hep-th/0404043 (2004).