数理解析研究所講究録
第 1913 巻 2014 年 182-201

182

Remarks on partition functions of
topological string theory on
generalized conifolds
Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University
Yoshida, Sakyo, Kyoto 606-8501, Japan
takasaki@math.h.kyoto-u.ac.jp

Abstract
The notion of topological vertex and the construction of topological string
partition functions on local toric Calabi-Yau 3-folds are reviewed. Implications of an explicit formula of partition functions for the generalized conifolds
are considered. Generating functions of part of the partition functions areshown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer
functions play the role of wave functions, and satisfy -diﬀerence equations.
These -diﬀerence equations represent the quantum mirror curves conjectured
by Gukov and Sulkowski.
$q$

$q$

Mathematics Subject Classification 2010: $05E05,$ $37K10,$ $81T30$
Key words: string theory, partition function, topological vertex, generalized
conifold, mirror curve, Schur function, integrable hierarchy

1

Introduction

Topological string theory is a simplified version of string theory in which topologicalproperties of the target space are captured by dynamics of strings [1]. Some ten years
ago, the method of (topological vertex” was introduced as a technique for calculating
the (amplitudes” of topological string theory on toric (or, more precisely, local
toric) Calabi-Yau 3-folds [2]. This method enables one to describe the amplitudes
in the language of purely combinatorial notions such as partitions and (skew) Schurfunctions.
Local toric Calabi-Yau 3-folds are characterized by graphical data called “webThe vertices ofdiagrams”’ (which are dual to two-dimensional “toric diagrams. Thesea web diagram represent copies of the simplest Calabi-Yau 3-fold$C^{3}$$C^{3\prime}s$

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are glued together to form the 3-fold $X$ in question. The web (or toric) diagram
encodes the gluing data.
The topological vertex is the amplitude
of topological string theory on
with boundary conditions imposed on string world sheets. The indicesareinteger partitions that specify the boundary conditions. These partitions are placedon the three edges emanating from the vertex. (Web diagrams are trivalent.) Whenthe copies ofare glued together, their vertex weights are multiplied along withedges weights, and summed over all possible configurations of the partitions on theinternal edges. The vertex weightitself is a somewhat complicated combinationof special values of skew Schur functions. Thus the amplitude of topological string
theory on $X$ (also called the “partition function”’ from the point of view of statisticalmechanics) is a sum of combinatorial quantities with respect to the partitions on
the inner edges.
When the 3-fold $X$ is the resolved conifold or its generalizations called “generalized conifolds one can calculate the sum over partitions explicitly with the aidof the Cauchy identities for skew Schur functions. In this paper, we review this result and consider its implications in the context of “integrable hierarchies”’ [4, 5, 6]
and “quantum mirror curves”’ [7, 8]. In particular, we present an explicit form of-diﬀerence equations for “wave functions”’ [9] of the generalized conifold. This is a
generalization of the known result on the resolved conifold. Although free fermionsand vertex operators are very convenient tools [10, 11], we dare not use them andresort to the Cauchy identities.
$C^{3}$$C_{\alpha\beta\gamma}$$\alpha,$$\beta,$$\gamma$$C^{3}$$C_{\alpha\beta\gamma}$$q$2

Partitions and Schur functions

In this section, we recall some relevant notions from combinatorics of integer partitions and Schur functions [3].

2.1

Partitions

In this paper, partitions are understood to be decreasing sequence of non-negative$k=1$ , 2, . . ., in which only a finite number are non-zero:
integers$\lambda_{i},$$\lambda=(\lambda_{1},$$\lambda_{2},$..

$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq 0,$$\exists N\forall i>N\lambda_{i}=0.$

The standard notations$l( \lambda)=\max\{i|\lambda_{i}\neq 0\}, |\lambda|=\sum_{i\geq 1}\lambda_{i}, \kappa(\lambda)=\sum_{i\geq 1}\lambda_{i}(\lambda_{i}-2i+1)$

for the length, the weight and the second Casimir value are used throughout thispaper. Partitions are in one-to-one correspondence with Young diagrams by identi$i=1$ , 2, . . . with the lengths of rows. Let
fying the partsdenote the conjugate
partition of . The parts$j=1$ , 2, . . ., of
are the lengths of columns of the$t\lambda$$\lambda_{i},$$\lambda$$t\lambda_{j},$$t\lambda$

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same Young diagram. The length $h(i,j)$ of the hook cornered at $(i,j)\in\lambda$ can be
thereby expressed as$h(i,j)=\lambda_{i}-i+t\lambda_{j}-j+1.$

if the correare said to satisfy the inclusion relation
Two partitions
sponding Young diagrams satisfy the inclusion relation (equivalently, if their parts
for $i\geq 1$ ). The diﬀerence of those Youngsatisfy the inequalities
and called a skew Young diagram of shape
diagram is denoted by$\lambda,$$\lambda\supset\mu$$\mu$$\lambda_{i}\geq\mu_{i}$$\lambda_{i},$$\mu_{i}$$\lambda/\mu.$$\lambda/\mu$

Schur and skew Schur functions

2.2

and the skew Schur functions
These partitions label the Schur functionsof a finite or infinite number of variables $x=(x_{1}, x_{2}, \cdots)$ . In a combiis a sum of the form
natorial definition,$s_{\lambda}(x)$$s_{\lambda/\mu}(x)$$s_{\lambda}(x)$$s_{\lambda}(x)= \sum_{T\in SSTab(\lambda)}x^{T}$(1),

where SSTab ( ) denotes the set of all semi-standard tableaux of shape . By definition, a semi-standard tableau of shape is an array of positive integers $T(i,j)$that are put on the cells $(i,j)\in\lambda$ of the Young diagram and increasing in the rows
and strictly increasing in the columns:$\lambda$$\lambda$$\lambda$$T$$T(i+1,j)>T(i,j)\leq T(i,j+1)$ .

The summand$x^{T}$

is the monomial$x^{T}= \prod_{(i,j)\in\lambda}x_{T(i,j)}$determined by the entries of . In the same sense, the skew Schur function
is defined by the sum
$T$

(2)

$s_{\lambda/\mu}(x)= \sum_{T\in SSTab(\lambda/\mu)}x^{T}$of all semi-standard tableaux of shapeover the set SSTabis again a monomial of the form$(\lambda/\mu)$$x^{T}= \prod_{(i,j)\in\lambda/\mu}x_{T(i,j)}.$When$\lambda\not\supset\mu$, we define$\mathcal{S}_{\lambda/\mu}(X)=0.$$s_{\lambda/\mu}(x)$$\lambda/\mu$. The summand

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In the case of $N$ -variables, the entries of tableaux are restricted to $[N]=$$\{1, 2, . . . , N\}$ .
Consequently, the Schur functions for partitions of length greater$N$vanish,than$\mathcal{S}_{\lambda}(x_{1}, \ldots, x_{N})=0$(3)

if $l(\lambda)>N$ ,and the non-vanishing ones are given by the finite sum(4)$s_{\lambda}(x_{1}, \ldots, x_{N})=s_{\lambda}(x_{1}, \ldots, x_{N}, 0,0, \ldots)=\sum_{T\in SSTab(\lambda,[N])}x^{T}$of semi-standard tableaux of shape with entries in $[N].$over the set SSTabThe Schur and skew Schur functions are symmetric functions. This fact, farfrom being obvious in the foregoing definitions (1) and (2), becomes manifest in the
Jacobi-Trudi formulae$\lambda$$(\lambda, [N])$$s_{\lambda/\mu}(x)=\det(h_{\lambda_{i}-\mu_{j}-i+j}(x))_{i,j=1}^{n}=\det(e_{t\lambda_{i}-t\mu j^{-i+j}}(x))_{i,j=1}^{m}$,(5)

and $m\geq l(t\lambda)$ . The entries
and $m$ are chosen to be such that$h_{k}(x)$ and $e_{k}(x)$ , $k=0$ , 1, 2, . . ., of the determinants are the complete and elementary
symmetric functions
where$n\geq l(\lambda)$$n$$h_{k}(x)= \sum_{i_{1}\leq\cdots\leq i_{k}}x_{i_{1}}\cdots x_{i_{k}},$$e_{k}(x)= \sum_{i_{1}<\cdots<i_{k}}x_{i_{1}}\cdots x_{i_{k}}$for $k\geq 1,$$h_{0}(x)=e_{0}(x)=1.$

2.3

Cauchy identities

The Schur functions satisfy the Cauchy identities$\sum_{\lambda\in \mathcal{P}}s_{\lambda}(x)_{\mathcal{S}_{\lambda}}(y)=\prod_{i,j\geq 1}(1-x_{i}y_{j})^{-1},$(6)$\sum_{\lambda\in \mathcal{P}}s_{\lambdat\lambda}(x)s(y)=\prod_{i,j\geq 1}(1+x_{i}y_{j})$

,

where
denotes the set of all partitions. These identities are generalized to theskew Schur functions as$\mathcal{P}$$\sum_{\lambda\in \mathcal{P}}s_{\lambda/\mu}(x)\mathcal{S}_{\lambda/\nu}(y)=\prod_{i,j\geq 1}(1-x_{i}y_{j})^{-1}\sum_{\lambda\in \mathcal{P}}s_{\mu/\lambda}(y)s_{\nu/\lambda}(x)$,(7)$\sum_{\lambda\in \mathcal{P}}\mathcal{S}_{\lambda/\mu}x)_{\mathcal{S}}(y)=\prod_{i,j\geq 1}(1+x_{i}y_{jt\mu/t\lambda})\sum_{\lambda\in \mathcal{P}}s(y)s_{\nu/\lambda}(x)$.

When $\mu=\nu=\emptyset$ , they reduce to (6). Moreover, since the Schur and skew Schurfunctions have the homogeneity$s_{\lambda}(Qx)=Q^{|\lambda|}s_{\lambda}(x) , s_{\lambda/\mu}(Qx)=Q^{|\lambda|-|\mu|}s_{\lambda/\mu}(x)$

,

(8)

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one can slightly generalize (6) and (7) as$\sum_{\lambda\in \mathcal{P}}Q^{|\lambda|}s_{\lambda}(x)s_{\lambda}(y)=\prod_{i,j\geq 1}(1-Qx_{i}y_{j})^{-1},$(9)$\sum_{\lambda\in \mathcal{P}}Q^{|\lambda|}s_{\lambdat\lambda}(x)s(y)=\prod_{i,j\geq 1}(1+Qx_{i}y_{j})$,and$\sum_{\lambda\in \mathcal{P}}Q^{|\lambda|}s_{\lambda/\mu}(x)s_{\lambda/\nu}(y)=\prod_{i,j\geq 1}(1-Qx_{i}y_{j})^{-1}\sum_{\lambda\in \mathcal{P}}Q^{|\lambda|}s_{\mu/\lambda}(Qy)s_{\nu/\lambda}(Qx)$,(10)$\sum_{\lambda\in \mathcal{P}}Q|\lambda|_{S_{\lambda/t\lambda/t\nu}\mu}(x)s(y)=\prod_{i,j\geq 1}(1+Qx_{i}y_{jt\mu/t\lambda})\sum_{\lambda\in \mathcal{P}}Q^{|\lambda|}s(Qy)_{\mathcal{S}_{\nu/\lambda}}(Qx)$.

Topological vertex and partition functions

3

In this section, we review the notion of topological vertex and the construction of theamplitudes (or partition functions) of topological string theory on local toric CalabiYau 3-folds. We refer details to Marino’s book [1] and references cited therein, inparticular, the original paper [2] of Aganagic et al.

3.1

Topological vertex

The topological vertex depends on a common parameter . We consider this parameter to be a complex number with $|q|>1$ . Relevant generating functions are
expanded in negative powers of
The topological vertex has the combinatorial expression
$q$

$q.$

$C_{\alpha\beta\gamma}=s_{\beta}(q^{\rho})q^{\kappa(\gamma)/2} \sum_{\nu\in \mathcal{P}}s\nu(q^{t\beta+\rho})_{\mathcal{S}}(q^{\beta+\rho})$

,

(11)

where $a,$are partitions on the three (clockwise ordered) legs emanating fromthe vertex (Figure 1) and is the infinite-dimensional vector$\beta,$$\gamma$$\rho$$\rho=(-\frac{1}{2}, -\frac{3}{2}, \ldots, -i+\frac{1}{2}, \ldots)$

The definition of the topological vertex thus contains special values of Schur andskew Schur function at$q^{\rho}=(q^{-i+1/2})_{i=1}^{\infty},$$q^{\beta+\rho}=(q^{\beta_{i}-i+1/2})_{i=1}^{\infty},$$q^{t\beta+\rho}=(q^{t\beta\dot{.}-i+1/2})_{i=1}^{\infty}$

These special values, primarily being power series ofof . In particular,
has the hook formula$q$$q^{-1}$, become rational functions$\mathcal{S}_{\beta}(q^{\rho})$$s_{\beta}(q^{\rho})= \frac{q^{\kappa(\beta)/4}}{\prod_{(i,j)\in\beta}(q^{h(i,j)/2}-q^{-h(i,j)/2})}$

.

(12)

187
$\beta$Figure 1: Partitions on legs of topological vertex
Unfortunately, no hook-like formula seems to be known for the special values.
andAn extremely nontrivial property of the topological vertex is the cyclic symmetry$s_{\alpha/\nu}(q^{t\beta+\rho})$$st\gamma/v(q^{\beta+\rho})$$C_{\alpha\beta\gamma}=C_{\beta\gamma\alpha}=C_{\gamma\alpha\beta}$

(13)

.

This property can be derived from a “crystal model”’ [10] of the topological vertex. It
seems diﬃcult to prove it directly from the conventional knowledge [3] on the Schur, the three quantities inand skew Schur functions. In the special case where(13) can be expressed as$\gamma=\emptyset$$C_{\alpha\beta\emptyset}=s_{\beta}(q^{\rho})s_{\alpha}(q^{t\beta+\rho}) , C_{\emptyset\alpha\beta}=s_{\alpha}(q^{\rho})q^{\kappa(\beta)/2}s_{t\beta}(q^{\alpha+\rho})$$C_{\beta\emptyset\alpha}=q^{\kappa(\alpha)/2} \sum_{\nu\in \mathcal{P}}\mathcal{S}_{\beta/\nu}(q^{\rho})s_{t\alpha/\nu}(q^{\rho})$

,

(14)

.

and using the hook formula (12), one can reduce the cyclic
Replacing
symmetry in this case to the identities$\betaarrow$$t\beta$$s_{\alpha}(q^{\rho})s_{\betat\alpha/v}(q^{\alpha+\rho})=q^{(\kappa(\alpha)+\kappa(\beta))/2} \sum_{\nu\in \mathcal{P}}\mathcal{S}(q^{\rho})s_{t\beta/\nu}(q^{\rho})=\mathcal{S}_{\beta}(q^{\rho})s_{\alpha}(q^{\beta+\rho})$

.

(15)

A direct }roof of (15) can be found in Zhou’s paper [12].
$I$

Gluing topological vertices

3.2

The amplitude of topological string theory on a local toric Calabi-Yau 3-fold $X$ isobtained by “gluing” the vertex weights along the internal edges of the web diagram.
itself is the amplitude of the simplest 3-fold
The vertex weight
The internal lines, too, are also weighted. The n-th internal line of the web
is the so called “K\"ahler parameter, wherediagram given the weightis the partition assigned to one of the two vertices connected to this internalandon the same line, because its leg along this line isline. The other vertex have$C^{3}.$$C_{\alpha\beta\gamma}$$is$$(-Q_{n})^{|\alpha_{n}|}$$\alpha_{n}$$t_{O_{n}}$$Q_{n}$

188
given an opposite orientation therein. The negative sign in the weight is also related
to this inversion of orientation.
Thus the n-th internal line and the two vertices on both ends altogether have the. This weight is further modified by the “framing
weight, where is an integer determined by directional vectors
factor”’:of the legs carrying$C_{\alpha_{n}\beta_{\mathfrak{n}}\gamma_{n}}(-Q_{n})^{|\alpha_{\mathfrak{n}}|}C_{t\alpha_{n}\beta_{\mathfrak{n}}’\gamma_{n}’}$$(-1)^{r_{n}|\alpha_{n}|}q^{-r_{\mathfrak{n}}\kappa(\alpha_{n})/2}$$v_{n},$$\beta_{n},$$v_{n}’$$r_{n}$$\beta_{n}’$$r_{n}=v_{n}\wedge v_{n}’=\det(v_{n}, v_{n}’)$.

The total amplitude , which is referred to as “partition function”’ in the following,
is obtained by summing the product of these weights over all possible configurationof partitions on the internal lines:
$Z$

$a$

$Z= \sum_{1\alpha,\cdots,\alpha_{N\in \mathcal{P}}}\cdots C_{\alpha_{n}\beta_{n}\gamma_{n}}(-Q_{n})^{|\alpha_{n}|}(-1)^{r_{n}|\alpha_{n}|}q^{-r_{n}\kappa(\alpha_{n})/2}C_{t\alpha_{n}\beta_{n}’\gamma_{n}’}\cdots$(16)

Let us illustrate the construction of the partition function in the case of theresolved conifold $X=\mathcal{O}(-1)\oplus \mathcal{O}(-1)arrow CP^{1}$ . The web diagram has two vertices
to the external legs andas shown in Figure 2. We assign the partitions
to the internal line. The partition
the partition
and the K\"ahler parameter
form
function
is a sum of the$\alpha_{0},$$\alpha_{1}$$\beta_{1},$$\beta_{2},$$\alpha_{2}$$Q$$Z=Z_{\beta_{1}\beta_{2}}^{\alpha_{0}\alpha 2}$$Z_{\beta_{1}\beta_{2}}^{\alpha\alpha}02= \sum_{\alpha 1\in \mathcal{P}}C_{1}(-Q)^{|\alpha|}1C_{t\alpha\beta_{2}\alpha 2}$(The framing factor vanishes in this case.) Whenare simplified to those shown in (14):.

$\alpha_{0}=\alpha_{2}=\emptyset$(17), the vertex weight$C_{\alpha_{1}\beta_{1}\emptyset}=s_{\beta_{1}}(q^{\rho})s_{\alpha_{1}}(q^{t\beta_{1}+\rho}) , c_{\iota_{\alpha_{1}\beta_{2}\emptyset}}=s_{\beta_{1}}(q^{\rho})s_{t\alpha_{1}}(q^{t\beta_{2}+\rho})$.

The partition function can be thereby written as$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}= \mathcal{S}_{\beta_{1}}(q^{\rho})s_{\beta_{21}}(q^{\rho})\sum_{\alpha 1\in \mathcal{P}}(-Q)^{|\alpha_{1}|}s_{\alpha}(q^{t\beta_{1}+\rho})s_{t\alpha_{1}}(q^{t\beta_{2}+\rho})$,and, by the Cauchy identities (9), boils down to the well known product formula$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}=s_{\beta_{1}}(q^{\rho})_{\mathcal{S}_{\beta_{2}}}(q^{\rho}) \prod_{i,j=1}^{\infty}(1-Qq^{t\beta_{1,i}+t\beta_{2,j}-i-j+1})$

3.3

.

(18)

Implications of cyclic symmetry

Let us consider implications of the cyclic symmetry (13) of the vertex weights in thecase of the resolved conifold.

189$\beta_{1}$Figure 2: Partitions assigned to web diagram of resolved conifold

According to the consequences (14) of the cyclic symmetry, the vertex weights
in the definition (17) of
have another expression:$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}$$C_{\alpha_{1}\beta_{1}\emptyset}=q^{\kappa(\alpha_{1})/2} \sum_{1\nu\in \mathcal{P}}s_{\beta_{1}/\nu_{1}}(q^{\rho})s_{t\alpha 1/\nu_{1}}(q^{\rho})$,$C_{t\alpha_{1}\beta_{2}\emptyset}=q^{\kappa(t\alpha_{1})/2} \sum_{2\nu\in \mathcal{P}}s_{\beta_{2}/\nu_{2}}(q^{\rho})s_{\alpha 1/\nu_{2}}(q^{\rho})$.

Substituting this expression in (17) and noting the general property$\kappa(t\lambda)=-\kappa(\lambda)$of the second Casimir value, one can rewrite$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}$to the the following triple sum:

$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}= \sum_{2\alpha_{1},\nu_{1},v\in \mathcal{P}}(-Q)^{|\alpha|}1s_{\beta_{1}/\nu_{1}}(q^{\rho})s_{t\alpha_{1}/\nu_{1}}(q^{\rho})s_{\beta_{2}/\nu_{2}}(q^{\rho})_{\mathcal{S}_{\alpha_{1}/\nu_{2}}}(q^{\rho})$.

By the Cauchy identities (10) for skew Schur functions, the partial sum overbe expressed as$\alpha_{1}$can$\alpha\in \mathcal{P}\sum_{1}(-Q)^{|\alpha_{1}|}s_{t\alpha_{1/1}}\nu(q^{\rho})_{\mathcal{S}_{\alpha_{1}/\nu_{2}}}(q^{\rho})$$= \prod_{i,j=1}^{\infty}(1-Qq^{-i-j+1})\sum_{\mu\in \mathcal{P}}t\nu_{1}/t\nu\cdot$

Thus, up to the pre-factor
another triple sum:$\prod_{i,j=1}^{\infty}(1-Qq^{-i^{j}+1})$, the partition function turns into

$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}= \prod_{i,j=1}^{\infty}(1-Qq^{-i-j+1})\cross$$\cross\sum_{2\mu,\nu_{1},\nu\in \mathcal{P}}(-Q)^{|\alpha_{1}|}s_{\beta_{1}/\nu_{11/t\mu}}(q^{\rho})s_{t\nu}(-Qq^{\rho})s_{\beta_{2}/\nu_{2}}(q^{\rho})s_{t\nu_{2}/\mu}(-Qq^{\rho})$.

190
Let us note that the pre-factor is essentially the inverse of the MacMahon function(19)$M(Q, q)= \prod_{n=1}^{\infty}(1-Qq^{n})^{-n}=\prod_{i,j=1}^{\infty}(1-Qq^{i+j-1})^{-1}$

with

$q$

replaced by . It is well known that the MacMahon function is a generating$q$function for weighted enumeration of $3D$ Young diagrams [10].
The last triple sum is also somewhat remarkable, because the partial sums over
and are special values of the so called $(\langle supersymmetric$ skew Schur functions”$v_{1}$$\nu_{2}$$s_{\lambda/\mu}(x|y)= \sum_{\nu\in \mathcal{P}}s_{\lambda/\nu}(x)s_{t\nu/t\mu}(y)$Thus we find another expression of$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}$

.

(20)

:

$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}= \prod_{i,j=1}^{\infty}(1-Qq^{-i-j+1})\sum_{\mu\in \mathcal{P}}(-Q)^{|\mu|}s_{\beta_{1}/\mu}(q^{\rho}|-Qq^{\rho})s_{\beta_{2}/t\mu}(q^{\rho}|-Qq^{\rho})$

.

(21)

The existence of the two expressions (18) and (21) of the partition function
is a special feature of the resolved conifold. In particular, this implies the non-trivial$Z_{\beta_{1}\beta_{2}}^{\emptyset\emptyset}$

identities$\mathcal{S}_{\beta_{1}}(q^{\rho})_{\mathcal{S}_{\beta_{2}}}(q^{\rho})\prod_{i,j=1}^{\infty}\frac{1-Qq^{t\beta_{1,i}+t\beta_{2,j}-i-j+1}}{1-Qq^{-i-j+1}}$$= \sum_{\mu\in \mathcal{P}}(-Q)^{|\mu|}s_{\beta_{1}/\mu}(q^{\rho}|-Qq^{\rho})_{\mathcal{S}_{\beta_{2}/t\mu}}(q^{\rho}|-Qq^{\rho})$

3.4

.

(22)

Generalized conifolds

The product formula (18) is extended by Iqbal and Kashani-Poor [13] to generalized conifolds namely, local toric Calabi-Yau 3-folds whose toric diagrams aretriangulations of a strip” (see Figure 3). The associated web diagram is acyclic,
and each vertex has a vertical external leg. We assign the partitions , . . . ,to the non-vertical legs of the leftmost
to these vertical legs, the partitions,...,
and the K\"ahler parameters
and rightmost vertices, and the partitions
is thus given by apartitionfunction, . . . , QN to the internal lines. The
sum with respect to , . . . ,Following the notations of Nagao [14] and Sulkowski [15], we now introduce the$n=1$ , . . . , $N$ , that represent the (type” of the vertices:
indices$\langle$$\langle$$\beta_{1}$$\alpha_{0},$$\alpha_{N}$$\alpha_{1}$$Q_{1}$$\alpha_{N}$$Z_{\beta_{1}..\beta_{N}}^{\alpha 0.\alpha}N$$\alpha_{1}$$\alpha_{N}.$$\sigma_{n}=\pm 1,$$\bullet$$\sigma_{n}=+1$

if the vertical leg of the n-th vertex is “down”$\bullet$$\sigma_{n}=-1$

if the vertical leg of the n-th vertex is “up”$\beta_{N}$191$\beta_{1}$$\beta_{5}$$\beta_{4}$Figure 3: Web diagram on triangulated strip

For example, for the web diagram of Figure 3,$\sigma_{1}=+1, \sigma_{2}=-1, \sigma_{3}=-1, \sigma_{4}=+1, \sigma_{5}=+1.$

With these notations, one can summarize the result of Iqbal and Kashani-Poor[13] into the following beautiful formula:$Z_{\beta_{1}\cdots\beta_{N}}^{\emptyset\emptyset}$$=s_{\beta_{1}}(q^{\rho}) \cdots s_{\beta_{N}}(q^{\rho})\prod_{1\leq 7n<n\leq N}\prod_{i,j=1}^{\infty}(1-Q_{m,n-1}q^{t\beta_{i}^{(m)}+\beta_{j}^{(n)}-i-j+1})^{-\sigma_{m}\sigma_{n}}$(23)Here we have introduced the abbreviation$Q_{m,n}=Q_{m}Q_{m+1}\cdots Q_{n}$and define$\beta^{(n)}$as$\beta^{(n)}=\{\begin{array}{ll}\beta_{n} if \sigma_{n}=+1,t\beta_{n} if \sigma_{n}=-1.\end{array}$

The method of Iqbal and Kashani-Poor is based on a nested use of the Cauchy
identities. On the other hand, Nagao [14] and Sulkowski [15], generalizing the workin terms of free fermionsof Eguchi and Kanno [16], presented a formula ofand vertex operators [10, 11]. (23) can be derived from the fermionic formula as
well. This is also a place where integrable hierarchies come into the game, because
the same fermions and vertex operators are also fundamental tools for integrable
hierarchies [17].
$Z_{\beta_{1}\cdots\beta_{N}}^{\alpha\alpha}0N$

4

Generating functions of partition functions

The partition functions,...,
The partitions$Z_{\beta_{1}\cdot\cdot\beta_{N}}^{\alpha 0.\alpha}N$$\alpha_{0},$$\alpha_{N},$$\beta_{1}$are amplitudes of (open” topological string theory.
represent “boundary conditions”’ to string world$\beta_{N}$192
sheets. One can construct amplitudes of “closed” topological string theory by multiplying these open string amplitudes with auxiliary Schur functions and summingover all possible configurations of partitions on the external legs. These generating
functions are closely related to integrable hierarchies [4, 5, 6] and quantum mirror
curves [7, 8].

4.1

Various generating functions

The partition functions$Z_{\beta_{1},\ldots,\beta_{N}}^{\emptyset\emptyset}$can be packed into the the generating function(24)$Z^{\emptyset\emptyset}(x_{1}, \ldots, x_{N})=\sum_{\beta_{1},\ldots,\beta_{N}\in \mathcal{P}}Z_{\beta_{1},\ldots,\beta_{N}^{\mathcal{S}_{\beta_{1}}}}^{\emptyset\emptyset}(x^{(1)})\cdots s_{\beta_{N}}(x^{(N)})$of the $N$-tuple of variables $x^{(n)}=(x_{1}^{(1)},$be specialized to the generating functions$x_{2}^{(2)},$..

$n=1$ , . . . , $N$ .

This function can(25)$Z_{n}(x)( \ldots,,x, 0, \ldots)=\sum_{\beta_{n}\in \mathcal{P}}Z^{\emptyset\emptyset_{\emptyset,\beta_{\mathfrak{n}},\emptyset}},,\ldots s_{\beta_{n}}(x)$of a single set of variables $x=(x_{1}, x_{2}, . . As we$ shall $see$ below, $Z_{n}(x)$ ’s are taufunctions of the KP hierarchy [17] with respect to the “time variables”’$t_{k}= \frac{1}{k}\sum_{i\geq 1}x_{i}^{k},$$k=1$ , 2, . . . ,(26)which are nothing but the so called “power sums”’ divided by the degree . Presumwill be a tau function of the $N$ -component”’ KP hierarchy
ably,with respect to the $N$ -tuple of time variables$k$$Z^{\emptyset\emptyset}(t^{(1)}, \ldots, x^{(N)})$$t_{k}^{(n)}= \frac{1}{k}\sum_{i\geq 1}x_{i}^{(n)k},$$k=1$ , 2, . . . ,$n=1$ , . . . , $N,$though we do not have a proof.
These are generating functions with the leftmost and rightmost partitions
being suppressed to . If these partitions are turned on, a new generating function
can be obtained:$\alpha_{0},$$\alpha_{N}$$\emptyset$$Z_{\beta_{1}\cdots\beta_{N}}(y, z)= \sum_{N\alpha 0,\alpha\in \mathcal{P}}Z_{\beta_{1}\cdots\beta_{N}}^{\alpha\alpha}0Ns_{\alpha 0}(y)s_{\alpha_{N}}(z)$

.

(27)

From the fermionic representation of Nagao [14] and Sulkowski [15], one can deducethat this is a tau function of the 2-component KP hierarchy or, rather, the Toda
hierarchy (with the lattice coordinate fixed to $s=0$ ). Let us mention that thisgenerating function is similar to the tau function in the melting crystal model ofsupersymmetric $5DU(1)$ gauge theory [18, 19].
$s$

193
The most general generating function is, of course, obtained by turning on allpartitions:$Z(y, x^{(1)}, \ldots, x^{(N)}, z)$$= \sum_{\alpha 0,\beta_{1},\ldots,\beta_{N}\alpha_{N\in \mathcal{P}}},Z_{\beta_{1},\ldots,\beta_{N}}^{\alpha_{0}\alpha_{N}}s_{\alpha 0}(y)s_{\beta_{1}}(x^{(1)})\cdots s_{\beta_{N}}(x^{(N)})s_{\alpha_{N}}(z)$

.

(28)

It is natural to expect that this function, too, becomes a tau function of the multicomponent KP hierarchy.

Generating functions as tau functions

4.2

Let us explain why $Z_{n}(x)$ ’s may be thought of as tau functions of the KP hierarchy.
This is based on the fact that the coeﬃcients of $Z_{n}(x)$ take the factorized form
(29)

$Z^{\emptyset\emptyset_{\emptyset,\beta_{n},\emptyset}},\ldots=s_{\beta_{n}}(q^{\rho)\prod_{i=1}^{\infty}f_{\beta_{n,i}-i+1}\prod_{i=1}^{\infty}9^{t}\beta_{n,i}-i+1},$

where$f_{k}= \prod_{m=1}^{n-1}\prod_{j=1}^{\infty}(1-Q_{m,n-1}q^{k-j})^{-\sigma_{m}},$$g_{k}= \prod_{m=n+1}^{N}\prod_{j=1}^{\infty}(1-Q_{n,m-1}q^{k-j})^{-\sigma_{m}}$if $\sigma_{n}=+1$ and$f_{k}= \prod_{m=n+1}^{N}\prod_{j=1}^{\infty}(1-Q_{m,n-1}q^{k-j})^{\sigma_{m}},$$g_{k}= \prod_{m=1}^{n-1}\prod_{j=1}^{\infty}(1-Q_{n,m-1}q^{k-j})^{\sigma_{m}}$if $\sigma_{n}=-1.$
We first consider the simplified generating function$Z(x)= \sum_{\beta\in \mathcal{P}}s_{\beta}(q^{\rho})s_{\beta}(x)$

.

(30)

This amounts to letting $Q_{m}=0$ for $m=1$ , . . . , $N$ , in $Z_{n}(x)$ . Since
this is nothing but the generating function for. By the simplest Cauchy identities(6), one can rewrite $Z(x)$ to an infinite product of the form
$s_{\beta_{n}}(q^{\rho})=C_{\emptyset\beta_{n}\emptyset},$$C^{3}$$Z(x)= \prod_{i,j=1}^{\infty}(1-x_{i}q^{-j+1/2})^{-1}=\prod_{i=1}^{\infty}\Phi_{q}-1(x_{i})$where$\Phi_{q}(x)$

(31)

,

is the the quantum dilogarithmic function$\Phi_{q}(x)=\prod_{j=1}^{\infty}(1-xq^{j-/2})^{-1}=\exp(\sum_{k=1}^{\infty}\frac{q^{k/2}x^{k}}{k(1-q^{k})})$

.

(32)

194
From the exponential form of the quantum dilogarithm, one can see that $Z(x)$ is anexponential function of a linear combination of the KP time variables:(33)$Z(x)= \exp(\sum_{k=1}^{\infty}\frac{q^{-k/2}t_{k}}{k(1-q^{-k})})=\exp(\sum_{k=1}^{\infty}\frac{t_{k}}{k[k]})$Here we have introduced the notation(34)$[k]=q^{k/2}-q^{-k/2}$

that is commonly used in the literature on the topological vertex. Since any exponential function of a linear form of the time variables is (trivial) tau function,$Z(x)$ is indeed a tau function.
We now return to $Z_{n}(x)$ . The coeﬃcients (29) of its Schur function expansion are. It is known in theobtained from those of $Z(x)$ by multiplying
theory of integrable hierarchies [17] that this is a very special type of transformations
on the space of tau functions. The tau functions of the KP hierarchy in general have
Schur function expansions$a$$\prod_{i=1}^{\infty}f_{\lambda_{i}-i+1gt\lambda_{i}-i+1}$$\tau(x)=\sum_{\lambda\in \mathcal{P}}a_{\lambda}s_{\lambda}(x)$are Pl\"ucker coordinates of an infinite dimensional Grassin which the coeﬃcientsmann manifold (the so called “Sato Grassmannian The action of $GL(\infty)$ on this$a_{\lambda}$

manifold induces transformations of tau functions. In particular, diagonal transformations$a_{\lambda} \mapsto a_{\lambda}\prod_{i=1}^{\infty}f_{\lambda_{i}-i+1}\prod_{i=1}^{\infty}gt\lambda_{i}-i+1$of the Pl\"ucker coordinates are realized by the action of diagonal matrices. Being
derived from the (trivial) tau function $Z(x)$ by these transformations, $Z_{n}(x)$ , too,is a tau function.
, we are
As regards the more universal generating function$N$ -component KP hierarchy.
of
the
tau
function
a
this
is
prove
that
still unble toIn this respect, the case of the resolved conifold is rather special and resemble the. Let us specify this case.
generating function (30) for
Recall that the partition function of the resolved conifold has another expression(21). By a nested use of the Cauchy identities (10) for skew Schur functions, one
can derive from this expression the following product formula:$Z^{\emptyset\emptyset}(x^{(1)}, \ldots, x^{(N)})$$C^{3}$$Z^{\emptyset\emptyset}(x^{(1)}, x^{(2)})$$= \prod_{i_{\Gamma-}1}^{\infty}\frac{(1-Qq^{-i-j+1})(1-Qx_{i}^{(1)}q^{-j+1/2})(1-Qx_{j}^{(2)}q^{-i+1/2})(1-Qx_{i}^{(1)}x_{j}^{(2)})}{(1-x_{i}^{(1)}q^{-j+1/2}\}(1-x_{j}^{(2)}q^{-i+1/2})}$

.

(35)

195
This formula can be further converted to the exponential form
$Z^{\emptyset\emptyset}(x^{(1)}, x^{(2)})$$= \prod_{i,j=1}^{\infty}(1-Qq^{-i-j+1})\exp(\sum_{k=1}^{\infty}\frac{(1-Q^{k})(t_{k}^{(1)}+t_{k}^{(2)})}{k[k]}-\sum_{k=1}^{\infty}kQ^{k}t_{k}^{(1)}t_{k}^{(2)})$

.

(36)

This expression shows thatis a tau function of the Toda hierarchy(hence, of the 2-component KP hierarchy) that is independent of the lattice coordinate . Actually, this is an “almost trivial”’ tau function of the Toda hierarchy, justas (30) is a trivial tau function of the KP hierarchy. It is remarkable that such tau
functions have a non-trivial structure in the variables.
$Z^{\emptyset\emptyset}(x^{(1)}, x^{(2)})$

$s$

$x$

4.3

Wave functions as Baker-Akhiezer function

We now specialize the variables in $Z_{n}(x)$ to $x=(x,$ $0,0,$ . .
$s_{\lambda}(x, 0,0, \ldots)=\{\begin{array}{ll}x^{k} if \lambda=(k) , k=0, 1, 2, .

the partition$\beta_{n}$

.

Since

.,0 otherwise,\end{array}$(37)

in the definition (25) of $Z_{n}(x)$ is restricted to$\beta_{n}=(k)$,$k=0$ , 1, 2, . . . .

Thus $Z_{n}(x)$ reduces to$Z_{n}(x)= \sum_{k=0}^{\infty}Z^{\emptyset\emptyset_{\emptyset_{)}(k),\emptyset}},\ldots x^{k}.$Let$\Phi_{n}(x)$denote the normalized generating function $Z_{n}(x)/Z_{n}(0)$ , namely,$\Phi_{n}(x)=1+\sum_{k=1}^{\infty}a_{k^{X^{k}}}, a_{k}=\frac{z\emptyset.\emptyset_{\emptyset,(k),\emptyset}}{z\emptyset\emptyset_{\emptyset,\emptyset,\emptyset}}$

.

(38)

This function is studied by Kashani-Poor [9] as a “wave function” in topologicalstring theory. Actually, this function amounts to the “dual Baker-Akhiezer function of the KP hierarchy. The genuine Baker-Akhiezer function corresponds to thegenerating function$\Psi_{n}(x)=1+\sum_{k=1}^{\infty}b_{k}(-x)^{k}, b_{k}=\frac{z\emptyset\emptyset_{\emptyset,(1^{k}),\emptyset}}{Z^{\emptyset\emptyset_{\emptyset,\emptyset,\emptyset}}}$of the partition functions restricted to$\beta_{n}=(1^{k})=(1, .., 1)\tilde{k}..$

.

(39)

196
The sign factor $(-1)^{k}$ is inserted for matching with the usual definition of the BakercorrespondandAkhiezer function [17]. In the language of free fermions,. To be more precise, the usual Baker-Akhiezer
andto the fermion fieldsof the KP hierarchy; thesefunctions depend on the time variables$t=0.$
tospecialized
time variables are now$\Psi_{n}(x)$$\Phi_{n}(x)$$\psi^{*}(x)$$\psi(x)$$t=(t_{1_{\rangle}}t_{2}, \ldots)$-diﬀerence equations for wave functions
Kashani-Poor [9] pointed out, in the case of the resolved conifold, that these “wave
4.4

$q$

functions”’ satisfy linear -diﬀerence equations. Gukov and Sulkowski [7] interpreted
these equations as a realization of the “quantum mirror curve”’ of the resolved conifold.
One can derive those -diﬀerence equations for the generalized conifolds as well.
are parallel, let us explain the derivation forandSince the cases ofin detail.
The first thing to do is to express the coeﬃcients of this power series in an explicit
form. To this end, apply the formula (23) to the case where
$q$

$q$

$\Phi_{n}(x)$$\Psi_{n}(x)$$\Phi_{n}(x)$$\beta_{n}=(k)$,The hook formula (12) implies that$\beta_{m}=\emptyset$

for $m\neq n.$$s_{\beta_{\mathfrak{n}}}(q^{\rho})=s_{(k)}(q^{\rho})$can be expressed as$s_{(k)}(q^{\rho})= \frac{q^{k(k-1)./4}}{[1][2]\cdot\cdot[k]},$recall the definition (34) of $[k]$ . Thus, after some more algebra, the following expresfor $k>0$ can be obtained:
sion of the coeﬃcients$a_{k}$$a_{k}= \frac{q^{k(k-1)/4}}{[1]\cdots[k]}\prod_{1\leq m<n}\prod_{i=1}^{k}(1-Q_{m,n-1}q^{\sigma_{\mathfrak{n}}(k-i)})^{-\sigma_{n}\sigma_{n}}\cross$$\cross\prod_{n<m\leq N}\prod_{i=1}^{k}(1-Q_{n,m-1}q^{-\sigma_{n}(k-i)})^{-\sigma_{n}\sigma_{m}}$.

One can rewrite this somewhat complicated expression of$a_{k}= \frac{q^{k(k-1)/4}}{[1]\cdots[k]}\frac{C_{n}(1)C_{n}(q)\cdot.\cdot.\cdot.C_{n}(q^{k-1})}{B_{n}(1)B_{n}(q)B_{n}(q^{k-1})}$(40)$a_{k}$

’s as
(41)

,

where $B_{n}(y)$ and $C_{n}(y)$ are Laurent polynomials of :$y$

$B_{n}(y)= \prod_{1\leq m<n,\sigma_{m}\sigma_{\mathfrak{n}}>0}(1-Q_{m,n-1}y^{\sigma_{n}})\prod_{n<m\leq N,\sigma_{m}\sigma_{n}>0}(1-Q_{n,m-1}y^{-\sigma_{n}})$$C_{n}(y)= \prod_{1\leq m<n,\sigma_{m}\sigma_{n}<0}(1-Q_{m,n-1}y^{\sigma_{n}})_{n<m\leq N}n_{\sigma_{m}\sigma_{n}<0}(1-Q_{n,m-1}y^{-\sigma_{\mathfrak{n}}})$

,

.

197
Thus

$a_{k}$’s turn out to satisfy the recurrence relations$a_{k}=a_{k-1} \frac{q^{(k-1)/2}C_{n}(q^{k-1})}{[k]B_{n}(q^{k-1})}$

.

(42)

One can thereby derive the -diﬀerence equation$q$$[x \partial_{x}]\Phi_{n}(x)=x\frac{C_{n}(q^{x\partial_{x}})}{B_{n}(q^{x\partial_{x}})}q^{x\partial_{x}/2}\Phi_{n}(x)$Note that$q^{x\partial_{x}}$and$q^{x\partial_{x}}f(x)=f(qx)$,$[x\partial_{x}]$, where$\partial_{x}=\partial/\partial x$

.

(43)

, act as -shift and -diﬀerence operators:
$q$

$q$

$[x\partial_{x}\exists f(x)=(q^{x\partial_{x}/2}-q^{-x\partial_{x}/2})f(x)=f(q^{1/2}x)-f(q^{-1/2}x)$.

(43) shows a precise form of the -diﬀerence equations conjectured by Gukov andSulkowski in a vague form.
In much the same way, the following -diﬀerence equation for
can bederived:$q$$\Psi_{n}(x)$$q$$[-x \partial_{x}]\Psi_{n}(x)=x\frac{C_{n}(q^{-x\partial_{x}})}{B_{n}(q-x\partial_{x})}q^{-x\partial_{x}/2}\Psi_{n}(x)$

.

(44)

Note that this equation is formally related to (43) by the inversion $qarrow q^{-1}$ of theparameter
Let us illustrate the -diﬀerence equations in the case of the resolved conifold.
andin this case are identical and become a power series of the form
$q.$

$q$

$\Phi_{1}(x)$$\Phi_{2}(x)$$\Phi(x)=1+\sum_{k=1}^{\infty}\frac{q^{k(k-1)/4}(1-Q)(1-Qq^{-1})\cdots(1-Qq^{1-k})}{[1][2]\cdots[k]}x^{k}$

.

(45)

This power series satisfies the -diﬀerence equation$q$$\Phi(q^{1/2}x)-\Phi(q^{-1/2}x)=x(1-Qq^{-x\partial_{x}})\Phi(q^{1/2}x)$

,

(46)

which can be rewritten as$\Phi(x)=\frac{1-Qq^{-1/2_{X}}}{1-q^{-1/2_{X}}}\Phi(q^{-1}x)$

The last equation implies that that$\Phi(x)$.

is an infinite product of the form$\Phi(x)=\prod_{n=1}^{\infty}\frac{1-Qq^{-n+1/2_{X}}}{1-q^{-n+1/2_{X}}}$

,

(47)

hence a quotient of two quantum dilogarithmic functions defined in (32) with being
replaced by$q$$q^{-1}.$

198
as the “decoupling” limit letting $Qarrow 0$ . The
One can consider the case of
foregoing results (45), (46) and (47) thereby reduce to the wave function$C^{3}$$\Phi(x)=1+\sum_{k=1}^{\infty}\frac{q^{k(k-1)./4}}{[1][2]\cdot\cdot[k]}x^{k}$

(48)

,

the -diﬀerence equation$q$$\Phi(q^{1/2}x)-\Phi(q^{-1/2}x)=x\Phi(q^{1/2}x)$

(49)

,

and the infinite product formula(50)$\Phi(x)=\prod_{n=1}^{\infty}(1-q^{-n+1/2}x)^{-1}$. Thus the wave function in this case is the quantum dilogarithmic functionforitself.
Let us stress that the infinite product formulae (47) and (50) of the wave functions, which are well known to experts, are a special feature of the resolved conifold. This feature stems from the simple structure of the -diﬀerence equations
and(46) and (49). The -diﬀerence equation (44) for other generalized conifolds are more
complicated, and presumably do not imply an infinite product formula of solutions.
turns into
In the classical $(qarrow 1)$ limit, the -diﬀerence equation (43) forthe equation$C^{3}$$C^{3}$$q$$q$$\Phi_{n}(x)$

$q$

(51)

$y^{1/2}-y^{-1/2}=x \frac{C_{n}(y)}{B_{n}(y)}y^{1/2}$

yields the sameof the “mirror curve” The -diﬀerence equation (44) forform
equation
in
the
this
rewrite.
canOneequation with replaced by$\Psi_{n}(x)$

$q$

$y$

$y^{-1}$$x= \frac{(1-y^{-1})B_{n}(y)}{C_{n}(y)}$

,

(52)

which almost agrees with the one conjectured by Gukov and Sulkowski. For example,
this equation for the resolved conifold reads$x= \frac{1-y^{-1}}{1-Qy-1}$

.

(53)

Although this equations is slightly diﬀerent from the usual mirror curve, the discrepancy can be resolved by “framing transformations”’ [7, 8].

199

5

Conclusion

The partition functionsof topological string theory on a generalized conifold
have rich mathematical contents. In this paper, we have mostly considered the casewhere the partitions
on the leftmost and rightmost external legs of the web
diagram are specialized toArmed with the explicit formula (23) of these partition functions and the Cauchy
identities, we have shown the following facts:$Z_{\beta_{1}\cdots\beta_{N}}^{\alpha_{0}\alpha_{N}}$$\alpha_{0},$$\alpha_{N}$$\alpha_{0}=\alpha_{N}=\emptyset.$$\bullet$

The generating functions (or closed string partition function) $Z_{n}(x)$ defined in(25) are tau functions of the KP hierarchy. The variables are linked with the
KP time variables $t=(t_{1}, t_{2}, \ldots)$ as shown in (26). This is a piece of evidence
indicating that the generating functionofwill be a$N$tau function of the -component KP hierarchy.
$x$$Z^{\emptyset\emptyset}(x^{(1)}, \ldots, x^{(N)})$$\bullet$$Z_{\beta_{1}\cdots\beta_{N}}^{\emptyset\emptyset}$

The wave functions
anddefined in (38) and (39) are the dual pairof Baker-Akhiezer functions of the KP hierarchy specialized to $t=0$ . Thesewave functions satisfy the -diﬀerence equations (43) and (44). In the classicallimit, these -diﬀerence equations turn into the equation (51) or, equivalently,(52), of the mirror curve of the generalized conifold.
$\Phi_{n}(x)$$\Psi_{n}(x)$

$q$

$q$

$\bullet$The case of the resolved conifold is very special. Because of the special form of
the -diﬀerence equation (46), the wave function has an infinite product form
(47). Moreover, the generating function, too, can be factorized
as shown in (35). From this factorized form,turns out to bea very special tau function of the 2-component KP hierarchy. This reasoning
cannot be applied to generalized conifolds.
$q$$Z^{\emptyset\emptyset}(x^{(1)}, x^{(2)})$$Z^{\emptyset\emptyset}(x^{(1)}, x^{(2)})$andare turned on, we can use the fermionic representation of, 15 to show that the generating function
defined by (27) isa tau function of the Toda hierarchy. This link with the Toda hierarchy will lead toa new perspective of quantum mirror curves. This issue will be reported elsewhere.

When$\alpha_{N}$$\alpha_{0}$$Z_{\beta_{1}\cdots\beta_{N}}^{\alpha_{0}\alpha_{N}}[14$$]$$Z_{\beta_{1}\cdots\beta_{N}}(y, z)$Acknowledgements
This work is partly supported by JSPS Grants-in-Aid for Scientific Research No.
21540218 and No. 22540186 from the Japan Society for the Promotion of Science.

References[1] M. Marino, Chern-Simons theory, matrix models, and topological strings, Oxford University Press, 2005.
[2] M. Aganagic, A. KIemm, M. Marino and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005), 425-478, $arXiv:hep-th/03051\mathfrak{X}.$200[3] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University
Press, 1995.

[4] M. Aganagic, R. Dijkgraaf, A. Klemm, M. Marino and C. Vafa, Topologicalstrings and integrable hierarchies, Commun. Math. Phys. 261 (2006), 451-516,
arXiv: hep-th/0312085[5] J. Zhou, Hodge integrals and integrable hierarchies, Lett. Math. Phys. 93(2010), 55C7l, $arXiv:math/0310408$ [math.AG][6] F. Deng and J. Zhou, On fermionic representation of the Gromov-Witten invariants of the resolved conifold, arXiv:1201.2500v1 [math.AG][7] S. Gukov and P. Sulkowski,
arXiv: 1108.0002vl [hep-th].

$A$-polynomial,[8] J. Zhou, Quantum mirror curves forarXiv: 1207.0598 [math.AG].

$\mathbb{C}^{3}$$B$-model, and quantization,and the resolved conifold,[9] A. K Kashani-Poor, The wave function behavior of the open topological string
partition function on the conifold, JHEP 04 (2007), 004, arXiv:hep-th/0606112[10] A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classicalcrystals, in: P. Etingof, V. Retakh and I.M. Singer (eds.), The unity of mathematics, Progr. Math. vol. 244 (Birkh\"auser, 2006), pp. 597-618, arXiv:hep$th/0309208$[11] B. Young and J. Bryan, Generating functions for colored 3D Young diagramsand the Donaldson-Thomas invariants of orbifolds, arXiv:0802.3948 [math.CO].
[12] J. Zhou, A conjecture on Hodge integrals, arXiv:math. $AG/0310282.$[13] A. Iqbal and A.-K. Kashani-Poor, The vertex on a strip, Adv. Theor. Math.
Phys. 10 (2006), 317-343, arXiv:hep-th/0410174.

[14] K. Nagao, Non-commutative Donaldson-Thomas theory and vertex operators,
Geometry and Topology 15 (2011) 1509-1543, arXiv:0910.5477 [math.AG].
[15] P. Sullkowski, Wall-crossing, free fermions and crystal melting, Commun. Math.
Phys. 301 (2011), 517-562, arXiv:0910.5485 [hep-th].

[16] T. Eguchi and H. Kanno, Geometric transitions, Chern-Simons gauge theory
and Veneziano type amplitudes, Phys. Lett. B585 (2004) 163-172, arXiv:hep$th/0312223.$[17] T. Miwa, M. Jimbo and E. Date, Solitons (Cambridge University Press, 2000).

201[18] T. Nakatsu and K. Takasaki, Melting crystal, quantum torus and Toda hierarchy, Commun. Math. Phys. 285 (2009), 445-468, arXiv:0710.5339 [hep-th].

[19] T. Nakatsu and K. Takasaki, Integrable structure of melting crystal model withexternal potentials, Advanced Studies in Pure Mathematics vol. 59 (Mathematical Society of Japan, 2010), pp. 201-223, arXiv:0807.4970 [math-ph]