$q$-Difference Kac–Schwarz Operators in Topological String Theory

$\boldsymbol{q}$-Difference Kac–Schwarz Operators
in Topological String Theory††This paper is a contribution to the Special
Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology,
Quantisation, and Beyond. The full collection is available at
http://www.emis.de/journals/SIGMA/moduli-spaces-2016.html

Kanehisa TAKASAKI † and Toshio NAKATSU ‡

K. Takasaki and T. Nakatsu

† Department of Mathematics, Kinki University,
† 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
<EMAIL_ADDRESS>
‡ Institute of Fundamental Sciences, Setsunan University,
‡ 17-8 Ikeda Nakamachi, Neyagawa, Osaka 572-8508, Japan
<EMAIL_ADDRESS>
Received September 08, 2016, in final form February 17, 2017; Published online
February 21, 2017

The perspective of Kac–Schwarz operators is introduced to the authors’
previous work on the quantum mirror curves of topological string theory in
strip geometry and closed topological vertex. Open string amplitudes on each
leg of the web diagram of such geometry can be packed into a multi-variate
generating function. This generating function turns out to be a tau function
of the KP hierarchy. The tau function has a fermionic expression, from which
one finds a vector $|W\rangle$ in the fermionic Fock space that represents a
point $W$ of the Sato Grassmannian. $|W\rangle$ is generated from the vacuum
vector $|0\rangle$ by an operator $g$ on the Fock space. $g$ determines an
operator $G$ on the space $V=\mathbb{C}((x))$ of Laurent series in which $W$
is realized as a linear subspace. $G$ generates an admissible basis
$\\{\Phi_{j}(x)\\}_{j=0}^{\infty}$ of $W$. $q$-difference analogues $A$, $B$
of Kac–Schwarz operators are defined with the aid of $G$. $\Phi_{j}(x)$’s
satisfy the linear equations $A\Phi_{j}(x)=q^{j}\Phi_{j}(x)$,
$B\Phi_{j}(x)=\Phi_{j+1}(x)$. The lowest equation $A\Phi_{0}(x)=\Phi_{0}(x)$
reproduces the quantum mirror curve in the authors’ previous work.

topological vertex; mirror symmetry; quantum curve; $q$-difference equation;
KP hierarchy; Kac–Schwarz operator

37K10; 39A13; 81T30

##  1 Introduction

It is argued [1, 11, 10, 31] that a “quantum mirror curve” emerges on the
B-model (Kodaira–Spencer theory) side of A-model topological string theory
when a brane is inserted as a probe. This quantum curve is represented by a
quantum mechanical equation of the form

$\displaystyle\hat{H}(\hat{x},\hat{y})\Psi=0.$ (1.1)

$\hat{H}(\hat{x},\hat{y})$ is quantization of a function $H(x,y)$ on a 2D
phase space, and the curve

$\displaystyle H(x,y)=0$

is called the mirror curve. In the case of local toric Calabi–Yau threefolds,
$H(x,y)$ is the function with which the mirror manifold is defined by the
equation

$\displaystyle zw=H(x,y).$

The phase space therein has the logarithmic symplectic form $d\log x\wedge
d\log y$, and one can choose $\hat{x}$ and $\hat{y}$ as

$\displaystyle\hat{x}=x,\qquad\hat{y}=q^{D},$

where

$\displaystyle q=e^{-\hbar},\qquad D=x\frac{d}{dx}.$

(1.1) thereby becomes a $q$-difference equation for the wave function
$\Psi=\Psi(x)$ of the probe.

It is also pointed out [26] that quantum curves are closely related to the
Eynard–Orantin topological recursion [18]. The notion of topological recursion
was introduced to topological string theory through a matrix model description
[12, 41]. This new computational tool led to the BKMS remodeling conjecture
[7] as well as a new conjectural recursion formula for simple Hurwitz numbers
[8] (both of which were solved by mathematicians [17, 21, 45]). The mirror
curve and a 2-point kernel $B(p,q)$ on this curve are the inputs of
topological recursion. The outputs are a set of multi-point kernels $W_{g,n}$,
$g\geq 0$, $n\geq 1$, that amount to multi-point resolvents of a matrix model.
Amplitudes (or partition functions) of strings are constructed from these
multi-point kernels. Gukov and Sułkowski’s observation [26] (based on Eynard
and Orantin’s interpretation of the notion of Baker–Akhiezer functions [18])
is that a WKB expansion of the wave function $\Psi(x)$, too, can be
constructed from $W_{g,n}$’s. Gukov and Sułkowski show, along with many other
examples, a partial result on computation of the WKB expansion in the case of
$\mathbb{C}^{3}$ and the resolved conifold. A complete derivation of the
quantum mirror curve for these two cases is achieved by Zhou [66].

We are interested in deriving the quantum mirror curve from the A-model side,
namely, in the topological vertex formalism [2]. The wave function $\Psi(x)$
is defined as a generating function of open string amplitudes. Topological
vertex offers a combinatorial description of these amplitudes. If one can
compute this generating function explicitly, one should be able to verify the
equation (1.1) of the quantum mirror curve by any means. This is indeed done
for $\mathbb{C}^{3}$ [1, 2] and the resolved conifold [32, 35, 36]. The wave
functions for these cases are quantum dilogarithmic functions [19, 20] and
$q$-hypergeometric (or basic hypergeometric) functions [23], hence satisfy
equations of the form (1.1). Some other examples of computations, all falling
into “strip geometry” [33], can be found in the literature on different
subjects [6, 14, 39, 62]. A unified expression of the quantum mirror curves
for general strip geometry is presented in our previous work [60]. Moreover,
we extended these results to the simplest example of “off-strip geometry”
called closed topological vertex [61].

The aim of this paper is to re-examine our previous results [60, 61] in the
perspective of “Kac–Schwarz operators”. This enables us to highlight a role of
the Sato Grassmannian and the KP hierarchy [53, 56] that was hidden in the
previous derivation of quantum mirror curves of topological string theory.
Moreover, Kac–Schwarz operators for topological string theory exhibit some new
features, which are significant in the context of integrable hierarchies.

The notion of Kac–Schwarz operators [34, 54] originates in the string
equations of 2D quantum gravity [13, 15, 22, 43] (see also the recent paper
[55] focused on quantum curves). The Kac–Schwarz operators for the $(a,b)$
minimal model of 2D quantum gravity are a pair of ordinary differential
operators $A$, $B$ of the form

$\displaystyle A=\frac{1}{bz^{b-1}}\frac{d}{dz}+z^{a}+\text{terms of lower
degrees},\qquad B=z^{b}$

that satisfy the canonical commutation relation

$\displaystyle[A,B]=1.$

They characterize a point $W$ of the Sato Grassmannian as a linear subspace of
the space $V=\mathbb{C}((z^{-1}))$ of formal Laurent series by the condition

$\displaystyle AW\subset W,\qquad BW\subset W.$

The Witten–Kontsevich theory of intersection numbers on a moduli space of
curves [38, 63], which amounts to the special case where $(a,b)=(1,2)$, can be
treated in the same way.

A variant of this conventional formulation of Kac–Schwarz operators is used to
derive quantum spectral curves of various Hurwitz numbers [3, 4] and quantum
mirror curves of a few cases of topological string theory [67]. The
Kac–Schwarz operators therein are differential operators of infinite order (in
the Hurwitz case) or $q$-difference operators (in the topological string
case), and satisfy the modified commutation relation

$\displaystyle[A,B]=A$

in the Hurwitz case and

$\displaystyle AB=qBA$ (1.2)

in the topological string case. Moreover, it is pointed out that a particular
element $\Psi$ of $W$ (more precisely, the first element of an admissible
basis of $W$) satisfies the equation

$\displaystyle A\Psi=0$

in the Hurwitz case and

$\displaystyle A\Psi=\Psi$

in the topological string case. These equations are interpreted as quantum
spectral/mirror curves.

To capture the quantum mirror curves of general strip geometry and closed
topological vertex along these lines, we borrow the idea of generating
operator developed by Alexandrov et al. [3, 4]. This is an invertible operator
$G$ on $V=\mathbb{C}((x))$, $x=z^{-1}$, that generates an admissible basis
$\\{\Phi_{j}(x)\\}_{j=0}^{\infty}$ of the relevant point $W$ of the Sato
Grassmannian. The Kac–Schwarz operators in this case are $q$-difference
operators, and defined as

$\displaystyle A=G\cdot q^{-D}\cdot G^{-1},\qquad B=G\cdot x^{-1}\cdot
G^{-1}.$ (1.3)

The basis functions $\Phi_{j}(x)$ satisfy the linear equations

$\displaystyle A\Phi_{j}(x)=q^{j}\Phi_{j}(x),\qquad
B\Phi_{j}(x)=\Phi_{j+1}(x).$ (1.4)

Since $\Phi_{0}(x)$ turns out to be the generating function of open string
amplitudes, the lowest equation $A\Phi_{0}(x)=\Phi_{0}(x)$ of (1.4) can be
identified with the quantum mirror curve of topological string theory. As the
algebraic relation (1.2) implies, the other equations
$A\Phi_{j}(x)=q^{j}\Phi_{j}(x)$ can be derived from the lowest equation by
applying $B$ iteratively. Thus $B$ plays the role of a “ladder operator”
connecting these equations, though we do not know what physical meaning this
tower of extra linear equations have.

We derive the generating operator $G$ from a fermionic expression of the KP
tau function that is defined as a multi-variate generating function of open
string amplitudes. Our computational technique developed in the previous work
[60, 61] yield a fermionic expression of the tau function almost
automatically. This expression contains the vector $|W\rangle$ of the
fermionic Fock space representing $W$. $|W\rangle$ is generated from the
vacuum vector $|0\rangle$ by an operator $g$ on the Fock space. In the case of
topological string theory of strip geometry and closed topological vertex, $g$
is made from building blocks that are familiar in the operator formalism of
topological vertex [50, 64]. We can thereby find an explicit expression of
$G$. This enables us to find the admissible basis and the $q$-difference
Kac–Schwarz operators as well.

Viewing this machinery on the whole, one may say that the KP hierarchy plays
the role of a “mirror map”. The inputs of this map are open string amplitudes
in the topological vertex formalism. They are manufactured into a KP tau
function. This tau function determines a point $W$ of the Sato Grassmannian,
hence a vector $|W\rangle$ of the fermionic Fock space. From an admissible
basis of $W$ and associated Kac–Schwarz operators, one obtains a quantum curve
as the output. Such a mapping from the A-model side to the B-model side is
commonly called a mirror map.

This paper is organized as follows. In Section 2, the notion of generating
operators and Kac–Schwarz operators are introduced and illustrated for
topological string theory on $\mathbb{C}^{3}$ and the resolved conifold. These
two cases are presented as a prototype of the subsequent case studies.
Relation to “hypergeometric tau functions” is briefly mentioned in the end of
this section. Section 3 deals the case of general strip geometry. A multi-
variate generating function of open string amplitudes is constructed for each
external line, or “leg”, of the web diagram, and shown to be a KP tau function
with a fermionic expression. Actually, the tau functions on the vertical legs
and those on the non-vertical legs turn out to have distinct structures. This
difference is reflected in the types of functions forming the admissible
bases. The generating operator $G$ and the $q$-difference operators $A$, $B$
are computed explicitly. Results of the previous work [60] are reproduced
successfully. Section 4 presents similar results in the case of closed
topological vertex. The relevant tau functions are shown to have a slightly
more complicated structure. This characteristic is inherited by the functions
forming the associated admissible basis. Nevertheless, computations are mostly
parallel to the case of strip geometry. Actually, the previous work [61] can
be reproduced in a more systematic way in the language of the operators $G$,
$A$, $B$.

##  2 $\boldsymbol{q}$-difference Kac–Schwarz operators

###  2.1 Generating operator of admissible basis

A point $W$ of the big cell of the Sato Grassmannian is realized by a linear
subspace

$\displaystyle W=\operatorname{Span}\\{\Phi_{j}(x)\\}_{j=0}^{\infty}$

of the space $V=\mathbb{C}((x))$ of formal Laurent series with an admissible
basis

$\displaystyle\Phi_{j}(x)=\sum_{i=-j}^{\infty}\phi_{ij}x^{i},\qquad\phi_{-j,j}\not=0,\qquad
j=0,1,\ldots.$

For convenience, we allow the leading coefficients $\phi_{-j,j}$ to be
unnormalized, though they can be normalized to $1$. The variable $x$ amounts
to the inverse of the usual spectral parameter $z$ of the KP hierarchy:

$\displaystyle x=z^{-1}.$

If the admissible basis is generated by a single invertible $q$-difference
operator $G=G(x,q^{D})$ as

$\displaystyle\Phi_{j}(x)=Gx^{-j},\qquad j=0,1,\dots,$ (2.1)

the basis elements satisfy the linear $q$-difference equations (1.4) with
respect to the operators $A$, $B$ defined by (1.3). In particular, the first
basis element $\Phi_{0}(x)$ satisfies the equation

$\displaystyle A\Phi_{0}(x)=\Phi_{0}(x),$ (2.2)

which we identify with the quantum mirror curve of topological string theory.

(1.3) are $q$-difference analogues of the Kac–Schwarz operators for Hurwitz
numbers of various types [3, 4, 67]. In those cases, $G$ is a differential
operator of infinite order, $G=G(x,D)$. The Kac–Schwarz operators $A$, $B$ are
defined as

$\displaystyle A=-G\cdot D\cdot G^{-1},\qquad B=G\cdot x^{-1}\cdot G^{-1}$

and satisfy the twisted canonical commutation relation (1.2). The admissible
basis generated by $G$, in turn, satisfies the linear differential equations

$\displaystyle A\Phi_{j}(x)=j\Phi_{j}(x),\qquad B\Phi_{j}(x)=\Phi_{j+1}(x).$

The lowest equation

$\displaystyle A\Phi_{0}(x)=0$

gives the quantum spectral curve for Hurwitz numbers.

$q$-difference Kac–Schwarz operators are presented by Zhou [67] for open
string amplitudes of $\mathbb{C}^{3}$ and the resolved conifold with a general
framing parameter $f$. We review these results (in the case where $f=0$) as
prototypes of our subsequent consideration. Unlike Zhou’s formulation, we make
full use of the theory of Schur functions [40] and the fermionic formalism of
integrable hierarchies [42].

Figure 1: Web diagram of $\mathbb{C}^{3}$ (left) and resolved conifold
(right).

###  2.2 Example: $\boldsymbol{\mathbb{C}^{3}}$

The open string amplitudes $Z_{\beta}$,
$\beta=(\beta_{1},\beta_{2},\ldots)\in\mathcal{P}$ (the set of all
partitions), of topological string theory on $\mathbb{C}^{3}$ in the presence
of a stack of branes (Fig. 1, left) are given by special values of the
infinite-variate Schur functions $s_{\lambda}$ as

$\displaystyle
Z_{\beta}=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}\big{(}q^{-\rho}\big{)},\qquad
q^{-\rho}=\big{(}q^{1/2},q^{3/2},\ldots,q^{i-1/2},\ldots\big{)},$

where $\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta$ denotes the conjugate
partition of $\beta$. $Z_{\beta}$ is nothing but the special case
$C_{\varnothing\varnothing\beta}$ of the topological vertex.

We use the Schur functions
$s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})$ of auxiliary
variables $\boldsymbol{x}=(x_{1},x_{2},\ldots)$ to define a generating
function $\tau(\boldsymbol{x})$ of $Z_{\beta}$’s:

$\displaystyle\tau(\boldsymbol{x})=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})Z_{\beta}.$

This sum can be readily computed by the Cauchy identity of Schur functions:

$\displaystyle\tau(\boldsymbol{x})=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}\big{(}q^{-\rho}\big{)}=\prod_{i,j=1}^{\infty}\big{(}1-q^{i-1/2}x_{j}\big{)}^{-1}.$

By the so called Miwa transformation

$\displaystyle t_{k}=\frac{1}{k}\sum_{i=1}^{\infty}x_{i}^{k}$ (2.3)

to the time variables $\boldsymbol{t}=(t_{1},t_{2},\ldots)$ of the KP
hierarchy, this infinite product turns into an exponential function:

$\displaystyle\tau(\boldsymbol{x})=\exp\left(\sum_{k=1}^{\infty}\frac{q^{k/2}}{1-q^{k}}t_{k}\right).$

This is an exponential tau function of the KP hierarchy.

To find an associated generating operator $G$, it is convenient to start from
the fermionic expression

$\displaystyle\tau(\boldsymbol{x})=\big{\langle}0|\Gamma_{+}(\boldsymbol{x})\Gamma_{-}\big{(}q^{-\rho}\big{)}|0\big{\rangle}$
(2.4)

of $\tau(\boldsymbol{x})$. $\langle 0|$ and $|0\rangle$ are the vacuum vectors
of the fermionic Fock spaces on which the creation-annihilation operators
$\psi_{n}$, $\psi^{*}_{n}$, $n\in\mathbb{Z}$, of a 2D charged free fermions
act. $\Gamma_{\pm}(\boldsymbol{x})$ are multi-variate extensions

$\displaystyle\Gamma_{\pm}(\boldsymbol{x})=\prod_{i=1}^{\infty}\Gamma_{\pm}(x_{i})$

of the vertex operators [50, 64]

$\displaystyle\Gamma_{\pm}(z)=\exp\left(\sum_{k=1}^{\infty}\frac{z^{k}}{k}J_{\pm
k}\right),$

where $J_{k}$’s are basis elements of the $U(1)$ current algebra:

$\displaystyle J_{k}=\sum_{n\in\mathbb{Z}}{:}\psi_{k-n}\psi^{*}_{n}{:}.$

Note that the Schur functions can be expressed as

$\displaystyle s_{\lambda}(\boldsymbol{x})=\langle
0|\Gamma_{+}(\boldsymbol{x})|\lambda\rangle=\langle\lambda|\Gamma_{-}(\boldsymbol{x})|0\rangle.$

Assembling these building blocks, one obtains the fermionic expression (2.4)
of $\tau(\boldsymbol{x})$. Moreover, by the Miwa transformation (2.3),
$\Gamma_{+}(\boldsymbol{x})$ turns into the generator

$\displaystyle\gamma_{+}(\boldsymbol{t})=\exp\left(\sum_{k=1}^{\infty}t_{k}J_{k}\right)$

of time evolutions of the KP hierarchy. (2.4) thereby becomes the standard
expression

$\displaystyle\tau(\boldsymbol{x})=\big{\langle}0|\gamma_{+}(\boldsymbol{t})\Gamma_{-}\big{(}q^{-\rho}\big{)}|0\big{\rangle}$

of a KP tau function. The vector

$\displaystyle|W\rangle=\Gamma_{-}\big{(}q^{-\rho}\big{)}|0\rangle$

in the Fock space represents the associated point $W$ of the Sato
Grassmannian. This vector is generated from $|0\rangle$ by the operator
$g=\Gamma_{-}(q^{-\rho})$.

The generating operator $G$ can be obtained by translating the operator $g$ on
the fermionic Fock space to an operator on the space $V$ of Laurent series.
The rule of translation is based on the following well known correspondences
[42, 53, 56] (see also Alexandrov’s detailed account [3]):

  * 1.

The correspondence

$\displaystyle\hat{A}=\sum_{i,j\in\mathbb{Z}}a_{ij}{:}\psi_{-i}\psi^{*}_{j}{:}\longleftrightarrow
A=(a_{i,j})_{i,j\in\mathbb{Z}}$

between fermion bilinears and $\mathbb{Z}\times\mathbb{Z}$ matrices based on
the commutation relations

$\displaystyle[\hat{A},\hat{B}]=\widehat{[A,B]}+\text{$c$-number term}.$

  * 2.

The correspondence

$\displaystyle D\longleftrightarrow\Delta=(i\delta_{ij})_{i,j\in\mathbb{Z}},$
$\displaystyle q^{D}\longleftrightarrow
q^{\Delta}=(q^{i}\delta_{ij})_{i,j\in\mathbb{Z}},$ $\displaystyle
x^{-k}\longleftrightarrow\Lambda^{k}=(\delta_{i+k,j})_{i,j\in\mathbb{Z}}$

between differential/difference operators and $\mathbb{Z}\times\mathbb{Z}$
matrices based on the relations

$\displaystyle
D\sum_{i\in\mathbb{Z}}\phi_{i}x^{i}=\sum_{i\in\mathbb{Z}}i\phi_{i}x^{i},$
$\displaystyle
q^{D}\sum_{i\in\mathbb{Z}}\phi_{i}x^{i}=\sum_{i\in\mathbb{Z}}q^{i}\phi_{i}x^{i},$
$\displaystyle
x^{-k}\sum_{i\in\mathbb{Z}}\phi_{i}x^{i}=\sum_{i\in\mathbb{Z}}\phi_{i+k}x^{i}.$

By this rule, the operator

$\displaystyle
g=\Gamma_{-}(q^{-\rho})=\exp\left(\sum_{i,k=1}^{\infty}\frac{q^{k(i-1/2)}}{k}J_{-k}\right)$

on the fermionic Fock space turns into the multiplication operator

$\displaystyle
G=\exp\left(\sum_{i,k=1}^{\infty}\frac{q^{k(i-1/2)}}{k}x^{k}\right)=\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}$
(2.5)

on $V$. The multiplier is a quantum dilogarithmic function. The admissible
basis (2.1) consists of the functions

$\displaystyle\Phi_{j}(x)=x^{-j}\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}=\sum_{k=0}^{\infty}\frac{q^{k/2}}{(q;q)_{k}}x^{k-j},$
(2.6)

where we have used the $q$-Pochhammer symbol

$\displaystyle(a;q)_{k}=\begin{cases}(1-a)(1-aq)\cdots\big{(}1-aq^{k-1}\big{)}&\text{if
$k>0$},\\\ 1&\text{if $k=0$}.\end{cases}$

It is straightforward to compute the $q$-difference Kac–Schwarz operators
(1.3) from (2.5). The operator $A=G\cdot q^{-D}\cdot G^{-1}$ can be computed
with the aid of the operator identity

$\displaystyle q^{D}f(x)q^{-D}=f(qx)$ (2.7)

as

$\displaystyle
A=q^{-D}\prod_{i=1}^{\infty}\big{(}1-q^{i+1/2}x\big{)}^{-1}\cdot\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}=q^{-D}\big{(}1-q^{1/2}x\big{)}.$
(2.8)

The operator $B=G\cdot x^{-1}\cdot G^{-1}$ becomes a trivial one:

$\displaystyle B=x^{-1}.$

The $q$-difference equation (2.2) for $\Phi_{0}(x)$ reads

$\displaystyle q^{-D}\big{(}1-q^{1/2}x\big{)}\Phi_{0}(x)=\Phi_{0}(x)$

or, equivalently,

$\displaystyle\big{(}1-q^{1/2}x-q^{D}\big{)}\Phi_{0}(x)=0,$

which agrees with the well known equation of the quantum mirror curve of
$\mathbb{C}^{3}$ [1, 2].

###  2.3 Example: Resolved conifold

The foregoing construction of the operators $G$, $A$, $B$ can be extended to
the resolved conifold. The unnormalized open string amplitudes $Z_{\beta}$
with a stack of branes on one of the external lines of the web diagram (Fig.
1, right) can be expressed as

$\displaystyle
Z_{\beta}=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}\big{(}q^{-\rho}\big{)}\prod_{i,j=1}^{\infty}\big{(}1-Qq^{-\beta_{i}+i+j-1}\big{)},$

where $Q$ is the Kähler parameter of the internal line of the web diagram. We
consider the generating function

$\displaystyle\tau(\boldsymbol{x})=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})Z_{\beta}/Z_{\varnothing}=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}\big{(}q^{-\rho}\big{)}\prod_{i,j=1}^{\infty}\dfrac{1-Qq^{-\beta_{i}+i+j-1}}{1-Qq^{i+j-1}}$

of the normalized amplitudes $Z_{\beta}/Z_{\varnothing}$.

To derive a fermionic expression of $\tau(\boldsymbol{x})$, we recall the
following formula, $\lambda=(\lambda_{1},\lambda_{2},\ldots)\in\mathcal{P}$,
from the previous work [61]:

$\displaystyle\prod_{i,j=1}^{\infty}\dfrac{1-Qq^{-\lambda_{i}+i+j-1}}{1-Qq^{i+j-1}}=\left\langle\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda\,|\,\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)|\,\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda\right\rangle.$
(2.9)

$\langle\lambda|$ and $|\lambda\rangle$ denote the excited states

$\displaystyle\langle\lambda|=\langle-\infty\,|\,\cdots\psi^{*}_{\lambda_{i}-i+1}\cdots\psi^{*}_{\lambda_{2}-1}\psi^{*}_{\lambda_{1}},\qquad|\lambda\rangle=\psi_{-\lambda_{1}}\psi_{-\lambda_{2}+1}\cdots\psi_{-\lambda_{i}+i-1}\cdots\,|\,-\infty\rangle$

in the charge-$0$ sector of the Fock spaces. The ground states
$\langle\varnothing|,|\varnothing\rangle$ labeled by the zero partition are
nothing but the vacuum vectors $\langle 0|,|0\rangle$. $V^{(k)}_{0}$’s are the
$m=0$ part of the basis

$\displaystyle
V^{(k)}_{m}=q^{-km/2}\sum_{n\in\mathbb{Z}}q^{kn}{:}\psi_{m-n}\psi^{*}_{n}{:},\qquad
k,m\in\mathbb{Z},$

of the quantum torus algebra [47, 49]. Since $V^{(k)}_{0}$’s do not mix the
excited states, the fermionic expression (2.9) of the $Q$-dependent factor in
the definition of $\tau(\boldsymbol{x})$ can be merged to the fermionic
expression

$\displaystyle
s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(q^{-\rho})=\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\,|\,\Gamma_{-}(q^{-\rho})\,|\,0\big{\rangle}$

of $s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(q^{-\rho})$ as

$\displaystyle\prod_{i,j=1}^{\infty}\dfrac{1-Qq^{-\beta_{i}+i+j-1}}{1-Qq^{i+j-1}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}\big{(}q^{-\rho}\big{)}=\left\langle\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\,|\,\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)\Gamma_{-}\big{(}q^{-\rho}\big{)}\,|\,0\right\rangle.$

Thus $\tau(\boldsymbol{x})$ can be expressed in the fermionic form

$\displaystyle\tau(\boldsymbol{x})=\left\langle
0\,|\,\Gamma_{+}(\boldsymbol{x})\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)\Gamma_{-}\big{(}q^{-\rho}\big{)}\,|\,0\right\rangle.$
(2.10)

The associated point $W$ of the Sato Grassmannian is determined by the vector

$\displaystyle|W\rangle=\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)\Gamma_{-}\big{(}q^{-\rho}\big{)}|0\rangle$

in the fermionic Fock space.

The generating operator $G$ of an admissible basis of $W$ can be obtained by
translating the generating operator

$\displaystyle
g=\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)\Gamma_{-}\big{(}q^{-\rho}\big{)}$

to an operator on $V$. We have seen that $\Gamma_{-}(q^{-\rho})$ amounts to
the multiplicative operator (2.5). As regard the first factor, since

$\displaystyle
V^{(k)}_{0}=\sum_{n\in\mathbb{Z}}q^{kn}{:}\psi_{-n}\psi^{*}_{n}{:}$

corresponds to the $q$-shift operator $q^{kD}$, one can compute its avatar on
$V$ as

$\displaystyle\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}q^{kD}\right)=\exp\left(\sum_{i,k=1}^{\infty}\frac{(Qq^{i-1+D})^{k}}{k}\right)=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}.$

$G$ is a product of these two operators:

$\displaystyle
G=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\cdot\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}.$
(2.11)

To compute the admissible basis (2.1), one can partly make use of the results
for $\mathbb{C}^{3}$. Namely, applying $G$ to $x^{-j}$’s amount to applying
$\prod\limits_{i=1}^{\infty}(1-Qq^{i-1+D})^{-1}$ to the admissible basis (2.6)
of $\mathbb{C}^{3}$. Thus the admissible basis in this case can be computed as

$\displaystyle\Phi_{j}(x)=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\left(x^{-j}\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}\right)$
$\displaystyle\hphantom{\Phi_{j}(x)}{}=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\left(\sum_{k=0}^{\infty}\frac{q^{k/2}}{(q;q)_{k}}x^{k-j}\right)=\sum_{k=0}^{\infty}\frac{q^{k/2}}{(q;q)_{k}}x^{k-j}\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+k-j}\big{)}^{-1}.$

Rewriting this expression slightly, one finds that

$\displaystyle\Phi_{j}(x)=\frac{1}{\prod\limits_{i=1}^{\infty}(1-q^{i-j-1})}\sum_{k=0}^{\infty}\frac{(Qq^{-j};q)_{k}q^{k/2}}{(q;q)_{k}}x^{k-j}.$
(2.12)

Thus the essential part of $\Phi_{j}(x)$’s turn out to be $q$-hypergeometric
functions.

One can similarly cut unnecessary computation for the $q$-difference
Kac–Schwarz operators (1.3). Namely, one can use (2.8) to rewrite $A=G\cdot
q^{-D}\cdot G^{-1}$ as

$\displaystyle A=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\cdot
q^{-D}\big{(}1-q^{1/2}x\big{)}\cdot\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}$
$\displaystyle\hphantom{A=}{}=q^{-D}-q^{-D}q^{1/2}\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\cdot
x\cdot\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}.$

By the operator identity

$\displaystyle x^{-1}f(D)x=f(D+1),$ (2.13)

the main part of the second term on the right side can be computed as

$\displaystyle\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\cdot
x\cdot\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}=x\prod_{i=1}^{\infty}\big{(}1-Qq^{i+D}\big{)}^{-1}\cdot\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}$
$\displaystyle\hphantom{\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}^{-1}\cdot
x\cdot\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+D}\big{)}}{}=x\big{(}1-Qq^{D}\big{)}.$

Thus $A$ can be expressed as

$\displaystyle A=q^{-D}\big{(}1-q^{1/2}x\big{(}1-Qq^{D}\big{)}\big{)}.$ (2.14)

Moreover, this computation shows that $B$ can be obtained as an inverse of
$x(1-Qq^{D})$:

$\displaystyle B=\big{(}1-Qq^{D}\big{)}^{-1}x^{-1}.$ (2.15)

The $q$-difference equation (2.2) for $\Phi_{0}(x)$ reads

$\displaystyle
q^{-D}\big{(}1-q^{1/2}x\big{(}1-Qq^{D}\big{)}\big{)}\Phi_{0}(x)=\Phi_{0}(x).$

This equation can be converted to the equation

$\displaystyle\big{(}1-q^{1/2}x\big{)}\Phi_{0}(x)=\big{(}1-Qq^{1/2}x\big{)}\Phi_{0}(qx),$
(2.16)

known in the literature [26, 32, 35, 36, 66].

###  2.4 Relation to hypergeometric tau functions

(2.4) and (2.10) belong to a class of KP tau functions that can be expressed
in the $\boldsymbol{x}$-variables as

$\displaystyle\tau(\boldsymbol{x})=\big{\langle}0\,|\,\Gamma_{+}(\boldsymbol{x})h\Gamma_{-}\big{(}q^{-\rho}\big{)}\,|\,0\big{\rangle},$
(2.17)

where $h$ is an operator of the form

$\displaystyle h=\exp\left(\sum_{n\in\mathbb{Z}}\log
h_{n}{:}\psi_{-n}\psi^{*}_{n}{:}\right).$ (2.18)

In the correspondence with $\mathbb{Z}\times\mathbb{Z}$ matrices, $h$ amounts
to a diagonal matrix, and the parameters $h_{n}$ are nothing but the diagonal
matrix elements. The generating operator $G$ on $V$ corresponds to the
generating operator $g=h\Gamma_{-}(q^{-\rho})$ on the fermionic Fock space.

These KP tau functions can be derived from tau functions of the 2D Toda
hierarchy of the form

$\displaystyle\tau(s,\boldsymbol{t},\tilde{\boldsymbol{t}})=\big{\langle}s\,|\,\gamma_{+}(\boldsymbol{t})h\gamma_{-}(\tilde{\boldsymbol{t}})\,|\,s\big{\rangle},$
(2.19)

where

$\displaystyle\gamma_{-}(\tilde{\boldsymbol{t}})=\exp\left(\sum_{k=1}^{\infty}\tilde{t}_{k}J_{-k}\right),$

by specialization to

$\displaystyle s=0,\qquad
t_{k}=\frac{1}{k}\sum_{i=1}^{\infty}x_{i}^{k},\qquad\tilde{t}_{k}=\frac{q^{k/2}}{k(1-q^{k})}.$

Toda tau functions of the form (2.19), called “hypergeometric tau functions”,
were introduced by Orlov and Scherbin in their study on multi-variate
hypergeometric functions [51, 52]. As pointed out therein (see also the work
of Kharchev, Marshakov, Mironov and Morozov [37] in a different context), $h$
is diagonal with respect to the basis
$\\{|\lambda\rangle\\}_{\lambda\in\mathcal{P}}$ of the charge-$0$ sector, and
the diagonal matrix elements have the contents-product form

$\displaystyle\langle\lambda\,|\,h\,|\,\lambda\rangle=\prod_{(i,j)\in\lambda}r_{j-i+1},$

where

$\displaystyle r_{n}=\frac{h_{n}}{h_{n-1}}.$

Since the work Okounkov [48], hypergeometric tau functions have played a
significant role as generating functions of Hurwitz numbers and many other
combinatorial notions [5, 24, 25, 27, 28, 30, 44, 59, 65] (see also Harnad’s
comprehensive review [29]).

In the case of the resolved conifold, see (2.10), the operator $h$ can be
expressed as

$\displaystyle
h=\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}\right).$

Since

$\displaystyle\sum_{k=1}^{\infty}\frac{Q^{k}}{k(1-q^{k})}V^{(k)}_{0}=\sum_{n\in\mathbb{Z}}\sum_{k=1}^{\infty}\frac{Q^{k}q^{kn}}{k(1-q^{k})}{:}\psi_{-n}\psi^{*}_{n}{:},$

the parameters $h_{n}$, $r_{n}$ can be computed as

$\displaystyle
h_{n}=\exp\left(\sum_{k=1}^{\infty}\frac{Q^{k}q^{kn}}{k(1-q^{k})}\right)=\prod_{i=1}^{\infty}\big{(}1-Qq^{i-1+n}\big{)}^{-1}$

and

$\displaystyle r_{n}=1-Qq^{n-1}.$

This belongs to a case considered by Orlov and Scherbin in the context of
$q$-hypergeometric functions. As we show in the next section, tau functions in
more general strip geometry, too, have a similar interpretation.

##  3 Strip geometry

###  3.1 Generalities

The toric diagram of strip geometry (Fig. 2) is a strip of height $1$ divided
to triangles of area $1/2$. If the toric diagram consists of $N$ triangles,
the associated web diagram is acyclic and has $N$ vertices, $N-1$ internal
lines and $N+2$ external lines (called “legs” for short). The legs other than
the leftmost and rightmost ones are vertical. Let $Q_{1},\ldots,Q_{N-1}$
denote the Kähler parameters assigned to the internal lines.

Figure 2: Web diagram (solid) and toric diagram (gray) of strip geometry.

Let $Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}$ denote the open
string amplitudes of this geometry. $\alpha_{0}$, $\alpha_{N}$ are partitions
on the leftmost and rightmost legs, and $\beta_{1},\ldots,\beta_{N}$ are
partitions on the vertical legs. The method of topological vertex is known to
work particularly well for these amplitudes.

In the case where $\alpha_{0}=\alpha_{N}=\varnothing$, these amplitudes are
computed by Iqbal and Kashani-Poor [33] in a closed form. To state their
result, we define the sign $\sigma_{n}=\pm 1$ of the $n$-th vertex in
accordance with the direction of the vertical leg emanating from the vertex:

  * (a)

$\sigma_{n}=+1$ if the vertical leg points up,

  * (b)

$\sigma_{n}=-1$ if the vertical leg points down.

We further introduce the auxiliary notations

$\displaystyle\beta^{(n)}=\begin{cases}\beta_{n}&\text{if $\sigma_{n}=+1$},\\\
\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}&\text{if
$\sigma_{n}=-1$},\end{cases}\qquad Q_{mn}=Q_{m}Q_{m+1}\cdots Q_{n-1}.$

The result of Iqbal and Kashani-Poor thereby reads

$\displaystyle
Z^{\varnothing\varnothing}_{\beta_{1}\cdots\beta_{N}}=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{1}}\big{(}q^{-\rho}\big{)}\cdots
s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{N}}\big{(}q^{-\rho}\big{)}\\!\prod_{1\leq
m<n\leq
N}\prod_{i,j=1}^{\infty}\\!\big{(}1-Q_{mn}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(m)}_{i}-\beta^{(n)}_{j}+i+j-1}\big{)}^{-\sigma_{m}\sigma_{n}},\\!\\!\\!$
(3.1)

where $\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(n)}_{i}$ and
$\beta^{(n)}_{i}$ denote the $i$-th parts of
$\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(n)}$ and $\beta^{(n)}$.

The amplitudes not restricted to $\alpha_{0}=\alpha_{N}=\varnothing$ are known
to have the following fermionic expression [16, 46, 58]:

$\displaystyle
Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}=q^{(1-\sigma_{1})\kappa(\alpha_{0})/4}q^{(1+\sigma_{N})\kappa(\alpha_{N})/4}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{1}}\big{(}q^{-\rho}\big{)}\cdots
s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{N}}\big{(}q^{-\rho}\big{)}$
$\displaystyle\hphantom{Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}=}{}\times\big{\langle}\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0}\,|\,\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\beta^{(1)}-\rho}\big{)}\Gamma^{\sigma_{1}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(1)}-\rho}\big{)}(\sigma_{1}Q_{1}\sigma_{2})^{L_{0}}\cdots$
$\displaystyle\hphantom{Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}=}{}\times\Gamma^{\sigma_{N-1}}_{-}\big{(}q^{-\beta^{(N-1)}-\rho}\big{)}\Gamma^{\sigma_{N-1}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(N-1)}-\rho}\big{)}\big{(}\sigma_{N-1}Q_{N-1}\sigma_{N}\big{)}^{L_{0}}$
$\displaystyle\hphantom{Z^{\alpha_{0}\alpha_{N}}_{\beta_{1}\cdots\beta_{N}}=}{}\times\Gamma^{\sigma_{N}}_{-}\big{(}q^{-\beta^{(N)}-\rho}\big{)}\Gamma^{\sigma_{N}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(N)}-\rho}\big{)}\,|\,\alpha_{N}\big{\rangle},$
(3.2)

where we have used the common notations

$\displaystyle\kappa(\lambda)=\sum_{i=1}^{\infty}\lambda_{i}(\lambda_{i}-2i+1),\qquad
L_{0}=\sum_{n\in\mathbb{Z}}n{:}\psi_{-n}\psi^{*}_{n}{:},$

and $\Gamma^{\sigma}_{\pm}(\boldsymbol{x})$ denote either
$\Gamma_{\pm}(\boldsymbol{x})$ or the vertex operators

$\displaystyle\Gamma^{\prime}_{\pm}(\boldsymbol{x})=\prod_{i=1}^{\infty}\Gamma^{\prime}_{\pm}(x_{i}),\qquad\Gamma^{\prime}_{\pm}(z)=\exp\left(-\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k}J_{\pm
k}\right)$

of another type [64] as

$\displaystyle\Gamma^{\sigma}_{\pm}(\boldsymbol{x})=\begin{cases}\Gamma_{\pm}(\boldsymbol{x})&\text{if
$\sigma=+1$},\\\ \Gamma^{\prime}_{\pm}(\boldsymbol{x})&\text{if
$\sigma=-1$}.\end{cases}$

The amplitude for $N=1$ (with $\sigma_{1}=1$) agrees with the single
topological vertex $C_{\lambda\mu\nu}$:

$\displaystyle
Z^{\lambda\mu}_{\nu}=C_{\lambda\mu\nu}=q^{\kappa(\mu)/2}s_{\,{\vphantom{\nu}}^{\mathrm{t}}\\!\,\nu}\big{(}q^{-\rho}\big{)}\sum_{\eta\in\mathcal{P}}s_{\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda/\eta}\big{(}q^{-\nu-\rho}\big{)}s_{\mu/\eta}\big{(}q^{-\,{\vphantom{\nu}}^{\mathrm{t}}\\!\,\nu-\rho}\big{)}.$

The formula (3.1) of Iqbal and Kashani-Poor can be recovered from (3.2) as
well. If $\alpha_{0}=\alpha_{N}=\varnothing$, (3.2) simplifies as

$\displaystyle
Z^{\varnothing\varnothing}_{\beta_{1}\cdots\beta_{N}}=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{1}}\big{(}q^{-\rho}\big{)}\cdots
s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{N}}\big{(}q^{-\rho}\big{)}\big{\langle}0\,|\,\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\beta^{(1)}-\rho}\big{)}\Gamma^{\sigma_{1}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(1)}-\rho}\big{)}(\sigma_{1}Q_{1}\sigma_{2})^{L_{0}}\cdots$
$\displaystyle\hphantom{Z^{\varnothing\varnothing}_{\beta_{1}\cdots\beta_{N}}=}{}\times\Gamma^{\sigma_{N-1}}_{-}\big{(}q^{-\beta^{(N-1)}-\rho}\big{)}\Gamma^{\sigma_{N-1}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(N-1)}-\rho}\big{)}(\sigma_{N-1}Q_{N-1}\sigma_{N})^{L_{0}}$
$\displaystyle\hphantom{Z^{\varnothing\varnothing}_{\beta_{1}\cdots\beta_{N}}=}{}\times\Gamma^{\sigma_{N}}_{-}\big{(}q^{-\beta^{(N)}-\rho}\big{)}\Gamma^{\sigma_{N}}_{+}\big{(}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(N)}-\rho}\big{)}\,|\,0\big{\rangle}.$

One can use the commutation relations [50, 64]

$\displaystyle\Gamma_{+}(\boldsymbol{x})\Gamma_{-}(\boldsymbol{y})=\prod_{i,j=1}^{\infty}(1-x_{i}y_{j})^{-1}\cdot\Gamma_{-}(\boldsymbol{y})\Gamma_{+}(\boldsymbol{x}),$
$\displaystyle\Gamma^{\prime}_{+}(\boldsymbol{x})\Gamma^{\prime}_{-}(\boldsymbol{y})=\prod_{i,j=1}^{\infty}(1-x_{i}y_{j})^{-1}\cdot\Gamma^{\prime}_{-}(\boldsymbol{y})\Gamma^{\prime}_{+}(\boldsymbol{x}),$
$\displaystyle\Gamma_{+}(\boldsymbol{x})\Gamma^{\prime}_{-}(\boldsymbol{y})=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\cdot\Gamma^{\prime}_{-}(\boldsymbol{y})\Gamma_{+}(\boldsymbol{x}),$
$\displaystyle\Gamma^{\prime}_{+}(\boldsymbol{x})\Gamma_{-}(\boldsymbol{y})=\prod_{i,j=1}^{\infty}(1+x_{i}y_{j})\cdot\Gamma_{-}(\boldsymbol{y})\Gamma^{\prime}_{+}(\boldsymbol{x})$
(3.3)

to move $\Gamma^{\sigma}_{+}$’s to the right until they hit $|0\rangle$ and
disappears. What remains are $\Gamma^{\sigma}_{-}$’s, which hit $\langle 0|$
and disappear. Contributions of $c$-number factors in (3.3) give rise to the
infinite products in (3.1).

###  3.2 Generating functions for vertical legs

Let us consider the generating function

$\displaystyle\tau_{n}(\boldsymbol{x})=\sum_{\beta_{n}\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}}(\boldsymbol{x})Z_{n,\beta_{n}}/Z,$

where

$\displaystyle
Z_{n,\beta_{n}}=Z^{\varnothing\varnothing}_{\cdots\varnothing\beta_{n}\varnothing\cdots},\qquad
Z=Z^{\varnothing\varnothing}_{\varnothing\cdots\varnothing},$

of the normalized open string amplitudes with respect to the partition
$\beta_{n}$ on the $n$-th vertical leg. As we show below,
$\tau_{n}(\boldsymbol{x})$ has a fermionic expression that implies that
$\tau_{n}(\boldsymbol{x})$ is a special KP tau function of the type shown in
(2.17).

The unnormalized amplitudes are a specialization of (3.1):

$\displaystyle
Z_{n,\beta_{n}}=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}}\big{(}q^{-\rho}\big{)}\prod_{m<n}\prod_{i,j=1}^{\infty}\big{(}1-Q_{mn}q^{-\beta^{(n)}_{i}+i+j-1}\big{)}^{-\sigma_{m}\sigma_{n}}$
$\displaystyle\hphantom{Z_{n,\beta_{n}}=}{}\times\prod_{m>n}\prod_{i,j=1}^{\infty}\big{(}1-Q_{nm}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta^{(n)}_{i}+i+j-1}\big{)}^{-\sigma_{n}\sigma_{m}}.$

The normalized amplitudes $Z_{n,\beta_{n}}/Z$ can be cast into a fermionic
expression with the aid of (2.9) and its variant [61]

$\displaystyle\prod_{i,j=1}^{\infty}\dfrac{1-Qq^{-\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda_{i}+i+j-1}}{1-Qq^{i+j-1}}=\left\langle\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda\,|\,\exp\left(-\sum_{k=1}^{\infty}\frac{Q^{k}q^{k}}{k(1-q^{k})}V^{(-k)}_{0}\right)\,|\,\,{\vphantom{\lambda}}^{\mathrm{t}}\\!\,\lambda\right\rangle$
(3.4)

as follows.

######  Lemma 3.1.

The normalized amplitudes can be expressed as

$\displaystyle
Z_{n,\beta_{n}}/Z=\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}\,|\,h\Gamma_{-}(q^{-\rho})\,|\,0\big{\rangle},$
(3.5)

where

$\displaystyle
h=\exp\left(-\sum_{m<n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{mn}^{k}}{k(1-q^{k})}V^{(k)}_{0}+\sum_{m>n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{nm}^{k}q^{k}}{k(1-q^{k})}V^{(-k)}_{0}\right)$
(3.6)

in the case where $\sigma_{n}=+1$, and

$\displaystyle
h=\exp\left(-\sum_{m<n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{mn}^{k}q^{k}}{k(1-q^{k})}V^{(-k)}_{0}+\sum_{m>n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{nm}^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)$
(3.7)

in the case where $\sigma_{n}=-1$.

###### Proof.

First consider the case where $\sigma_{n}=+1$. The normalized amplitudes in
this case can be expressed as

$\displaystyle
Z_{n,\beta_{n}}/Z=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}}(q^{-\rho})\prod_{m<n}\prod_{i,j=1}^{\infty}\frac{(1-Q_{mn}q^{-\beta_{n,i}+i+j-1})^{-\sigma_{m}}}{(1-Q_{mn}q^{i+j-1})^{-\sigma_{m}}}$
$\displaystyle\hphantom{Z_{n,\beta_{n}}/Z=}{}\times\prod_{m>n}\prod_{i,j=1}^{\infty}\frac{(1-Q_{nm}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n,i}+i+j-1})^{-\sigma_{m}}}{(1-Q_{nm}q^{i+j-1})^{-\sigma_{m}}}.$

One can apply (2.9) to the quotients in the product over $m<n$ and (3.4) to
the quotient in the product over $m>n$. This yields an expression of the form
(3.5) with $h$ defined as (3.6). Next examine the case where $\sigma_{n}=-1$.
The normalized amplitudes changes as

$\displaystyle
Z_{n,\beta_{n}}/Z=s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n}}(q^{-\rho})\prod_{m<n}\prod_{i,j=1}^{\infty}\frac{(1-Q_{mn}q^{-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{n,i}+i+j-1})^{\sigma_{m}}}{(1-Q_{mn}q^{i+j-1})^{\sigma_{m}}}$
$\displaystyle\hphantom{Z_{n,\beta_{n}}/Z=}{}\times\prod_{m>n}\prod_{i,j=1}^{\infty}\frac{(1-Q_{nm}q^{-\beta_{n,i}+i+j-1})^{\sigma_{m}}}{(1-Q_{nm}q^{i+j-1})^{\sigma_{m}}}.$

One can derive an expression of the form (3.5) in the same way with the
operator $h$ defined as (3.7). ∎

The fermionic expression (3.5) of the normalized amplitudes imply the
following conclusion.

######  Theorem 3.2.

$\tau_{n}(\boldsymbol{x})$ is a tau function of the KP hierarchy with the
fermionic expression

$\displaystyle\tau_{n}(\boldsymbol{x})=\big{\langle}0\,|\,\Gamma_{+}(\boldsymbol{x})h\Gamma_{-}(q^{-\rho})\,|\,0\rangle,$
(3.8)

where $h$ is the operator shown in (3.6) and (3.7).

######  Remark 3.3.

$h$ is an operator of the form (2.18). The parameters $h_{j}$, $r_{j}$ can be
computed as follows:

  * (a)

In the case where $\sigma_{n}=+1$,

$\displaystyle
h_{j}=\exp\left(-\sum_{m<n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{mn}^{k}}{k(1-q^{k})}q^{kj}+\sum_{m>n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{nm}^{k}q^{k}}{k(1-q^{k})}q^{-kj}\right)$
$\displaystyle\hphantom{h_{j}}{}=\prod_{m<n}\exp\left(-\sigma_{m}\sum_{i,k=1}\frac{(Q_{mn}q^{i-1+j})^{k}}{k}\right)\cdot\prod_{m>n}\exp\left(\sigma_{m}\sum_{i,k=1}^{\infty}\frac{(Q_{nm}q^{i-j})^{k}}{k}\right)$
$\displaystyle\hphantom{h_{j}}{}=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-1+j}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-j}\big{)}^{-\sigma_{m}}$
(3.9)

and

$\displaystyle
r_{j}=\prod_{m<n}\big{(}1-Q_{mn}q^{j-1}\big{)}^{-\sigma_{m}}\cdot\prod_{m>n}\big{(}1-Q_{nm}q^{1-j}\big{)}^{-\sigma_{m}}.$

  * (b)

In the case where $\sigma_{n}=-1$,

$\displaystyle
h_{j}=\exp\left(-\sum_{m<n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{mn}^{k}q^{k}}{k(1-q^{k})}q^{-kj}+\sum_{m>n}\sigma_{m}\sum_{k=1}^{\infty}\frac{Q_{nm}^{k}}{k(1-q^{k})}q^{kj}\right)$
$\displaystyle\hphantom{h_{j}}{}=\prod_{m<n}\exp\left(-\sigma_{m}\sum_{i,k=1}\frac{(Q_{mn}q^{i-j})^{k}}{k}\right)\cdot\prod_{m>n}\exp\left(\sigma_{m}\sum_{i,k=1}^{\infty}\frac{(Q_{nm}q^{i-1+j})^{k}}{k}\right)$
$\displaystyle\hphantom{h_{j}}{}=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-j}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-1+j}\big{)}^{-\sigma_{m}}$
(3.10)

and

$\displaystyle
r_{j}=\prod_{m<n}\big{(}1-Q_{mn}q^{1-j}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\big{(}1-Q_{nm}q^{j-1}\big{)}^{\sigma_{m}}.$

This result shows that the tau functions $\tau_{n}(\boldsymbol{x})$ are
examples of Orlov and Scherbin’s $q$-hypergeometric tau functions [51, 52].

###  3.3 Admissible basis and Kac–Schwarz operators

(3.8) is a somewhat complicated, but rather straightforward generalization of
the tau function (2.10) for the resolved conifold. One can obtain the
generating operator $G$ of an admissible basis of $W$ and the associated
$q$-difference Kac–Schwarz operators $A$, $B$ in much the same way as the case
of the resolved conifold.

One can derive the generating operator $G$ from $g=h\Gamma_{-}(q^{-\rho})$ by
replacing

$\displaystyle V^{(k)}_{0}\to
q^{kD},\qquad\Gamma_{-}(q^{-\rho})\to\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}.$

Thus the generating operator $G$, like (2.11), takes the factorized form

$\displaystyle G=H\cdot\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1},$
(3.11)

where $H$ is the following avatar of $h$:

  * (a)

In the case where $\sigma_{n}=+1$,

$\displaystyle
H=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-1+D}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-D}\big{)}^{-\sigma_{m}}.$
(3.12)

  * (b)

In the case where $\sigma_{n}=-1$,

$\displaystyle
H=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-D}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-1+D}\big{)}^{-\sigma_{m}}.$
(3.13)

Note that (3.12) and (3.13) have the same structure as the expressions (3.9)
and (3.10) of the parameters $h_{j}$ of the operator $h$.

Let $\\{\Phi_{j}(x)\\}_{j=0}^{\infty}$ denote the admissible basis generated
by $G$:

$\displaystyle\Phi_{j}(x)=H\left(x^{-j}\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}\right)=\sum_{k=1}^{\infty}\frac{q^{k/2}}{(q;q)_{k}}Hx^{k-j}.$

Rewriting this expression further, we obtain the following result.

######  Theorem 3.4.

The basis elements $\Phi_{j}(x)$ are functions of the following form:

  * (a)

In the case where $\sigma_{n}=+1$,

$\displaystyle\Phi_{j}(x)=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-1-j}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i+j}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{\Phi_{j}(x)}{}\times\sum_{k=0}^{\infty}\frac{\prod\limits_{m<n}(Q_{mn}q^{-j};q)_{k}^{-\sigma_{m}}\prod\limits_{m>n}(Q_{nm}q^{j};q^{-1})_{k}^{-\sigma_{m}}q^{k/2}}{(q;q)_{k}}x^{k-j}.$

  * (b)

In the case where $\sigma_{n}=-1$,

$\displaystyle\Phi_{j}(x)=\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i+j}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-1-j}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{\Phi_{j}(x)}{}\times\sum_{k=0}^{\infty}\frac{\prod\limits_{m<n}(Q_{mn}q^{j};q^{-1})_{k}^{\sigma_{m}}\prod\limits_{m>n}(Q_{nm}q^{-j};q)_{k}^{\sigma_{m}}q^{k/2}}{(q;q)_{k}}x^{k-j}.$

As remarked in the case of the resolved conifold, see (2.12), the essential
part of $\Phi_{j}(x)$’s are $q$-hypergeometric functions. In particular,
$\Phi_{0}(x)$ agrees, up to a constant multiplier, with the generating
function

$\displaystyle\Psi(x)=\sum_{k=0}^{\infty}x^{k}Z_{n,(1^{k})}/Z$ (3.14)

of the normalized open string amplitudes considered in our previous work [60].

Let us proceed to computation of the $q$-difference Kac–Schwarz operators $A$,
$B$. Since $G$ has the partially factorized form (3.11) and the second factor
has the property

$\displaystyle\prod_{i=1}^{\infty}\big{(}1-q^{i-1/}x\big{)}^{-1}\cdot
q^{-D}\cdot\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}=q^{-D}\big{(}1-q^{1/2}x\big{)},$

see (2.8), we have the following preliminary expressions of $A$ and $B$:

$\displaystyle A=q^{-D}H\cdot\big{(}1-q^{1/2}x\big{)}\cdot H^{-1},\qquad
B=H\cdot x^{-1}\cdot H^{-1}.$

Thus our main task is to compute $H\cdot x^{\pm 1}\cdot H^{-1}$.

######  Lemma 3.5.

$\displaystyle H\cdot x\cdot H^{-1}=xR,$

where

$\displaystyle
R=\prod_{m<n}\big{(}1-Q_{mn}q^{\sigma_{n}D}\big{)}^{-\sigma_{m}\sigma_{n}}\cdot\prod_{m>n}\big{(}1-Q_{nm}q^{-\sigma_{n}D}\big{)}^{-\sigma_{n}\sigma_{m}}.$
(3.15)

###### Proof.

By the identity (2.13), one can move $x$ to the left. In the case where
$\sigma_{n}=+1$, this computation proceeds as follows:

$\displaystyle H\cdot x\cdot
H^{-1}=x\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i+D}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-1-D}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{H\cdot x\cdot
H^{-1}=}{}\times\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-D}\big{)}^{\sigma_{m}}\cdot\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-1+D}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{H\cdot x\cdot
H^{-1}}{}=x\prod_{m<n}\big{(}1-Q_{mn}q^{D}\big{)}^{-\sigma_{m}}\cdot\prod_{m>n}\big{(}1-Q_{nm}q^{-D}\big{)}^{-\sigma_{m}}.$

The case where $\sigma_{n}=-1$ can be treated in the same way:

$\displaystyle H\cdot x\cdot
H^{-1}=x\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-1-D}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i+D}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{H\cdot x\cdot
H^{-1}=}{}\times\prod_{m>n}\prod_{i=1}^{\infty}\big{(}1-Q_{nm}q^{i-1+D}\big{)}^{\sigma_{m}}\cdot\prod_{m<n}\prod_{i=1}^{\infty}\big{(}1-Q_{mn}q^{i-D}\big{)}^{-\sigma_{m}}$
$\displaystyle\hphantom{H\cdot x\cdot
H^{-1}}{}=x\prod_{m<n}\big{(}1-Q_{mn}q^{-D}\big{)}^{\sigma_{m}}\cdot\prod_{m>n}\big{(}1-Q_{nm}q^{D}\big{)}^{\sigma_{m}}.$
∎

By this lemma applied to the preliminary expression, we find the following
explicit form of $A$ and $B$.

######  Theorem 3.6.

The $q$-difference Kac–Schwarz operators can be expressed as

$\displaystyle A=q^{-D}\big{(}1-q^{1/2}xR\big{)},\qquad
B=(xR)^{-1}=R^{-1}\cdot x^{-1},$

where $R$ is the operator defined in (3.15).

This is a generalization of the result for the resolved conifold, see (2.14)
and (2.15), for which $R=1-Qq^{D}$. The $q$-difference equation (2.16) for
$\Phi_{0}(x)$ is generalized as

$\displaystyle\big{(}1-q^{1/2}xR\big{)}\Phi_{0}(x)=\Phi_{0}(qx).$ (3.16)

$R$ can be expressed as

$\displaystyle R=\frac{Q(q^{D})}{P(q^{D})},$

where $P(q^{D})$ and $Q(q^{D})$ are Laurent polynomials $q^{D}$:

$\displaystyle
P(q^{D})=\prod_{m<n,\sigma_{m}\sigma_{n}>0}\big{(}1-Q_{mn}q^{\sigma_{n}D}\big{)}\cdot\prod_{m>n,\sigma_{m}\sigma_{n}>0}\big{(}1-Q_{nm}q^{-\sigma_{n}D}\big{)},$
$\displaystyle
Q(q^{D})=\prod_{m<n,\sigma_{m}\sigma_{n}<0}\big{(}1-Q_{mn}q^{\sigma_{n}D}\big{)}\cdot\prod_{m>n,\sigma_{m}\sigma_{n}<0}\big{(}1-Q_{nm}q^{-\sigma_{n}D}\big{)}.$

(3.16) agrees with the $q$-difference equation derived for the generating
function (3.14) in our previous work [60].

###  3.4 Generating functions for non-vertical legs

Let us now consider the generating functions

$\displaystyle\tau_{0}(\boldsymbol{x})=\sum_{\alpha_{0}\in\mathcal{P}}s_{\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0}}(\boldsymbol{x})Z_{0,\alpha_{0}}/Z,\qquad
Z_{0,\alpha_{0}}=Z^{\alpha_{0}\varnothing}_{\varnothing\cdots\varnothing},$
$\displaystyle\tau_{N+1}(\boldsymbol{x})=\sum_{\alpha_{N}\in\mathcal{P}}s_{\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{N}}(\boldsymbol{x})Z_{N+1,\alpha_{N}}/Z,\qquad
Z_{N+1,\alpha_{N}}=Z^{\varnothing\alpha_{N}}_{\varnothing\cdots\varnothing},$

of the normalized open string amplitudes for the leftmost and rightmost legs.
As it turns out, their structure is quite distinct from the foregoing
generating functions
$\tau_{1}(\boldsymbol{x}),\ldots,\tau_{N}(\boldsymbol{x})$.

The unnormalized amplitudes are a specialization of (3.2):

$\displaystyle
Z_{0,\alpha_{0}}=q^{(1-\sigma_{1})\kappa(\alpha_{0})/4}\big{\langle}\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0}\,|\,\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\sigma_{1}}_{+}\big{(}q^{-\rho}\big{)}(\sigma_{1}Q_{1}\sigma_{2})^{L_{0}}\cdots$
$\displaystyle\hphantom{Z_{0,\alpha_{0}}=}{}\times(\sigma_{N-1}Q_{N-1}\sigma_{N})^{L_{0}}\Gamma^{\sigma_{N}}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\sigma_{N}}_{+}\big{(}q^{-\rho}\big{)}\,|\,0\big{\rangle},$
(3.17) $\displaystyle
Z_{N+1,\alpha_{N}}=q^{(1+\sigma_{N})\kappa(\alpha_{N})/4}\big{\langle}0\,|\,\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\sigma_{1}}_{+}\big{(}q^{-\rho}\big{)}(\sigma_{1}Q_{1}\sigma_{2})^{L_{0}}\cdots$
$\displaystyle\hphantom{Z_{N+1,\alpha_{N}}=}{}\times(\sigma_{N-1}Q_{N-1}\sigma_{N})^{L_{0}}\Gamma^{\sigma_{N}}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\sigma_{N}}_{+}(q^{-\rho})|\alpha_{N}\rangle.$
(3.18)

These amplitudes can be converted to a reduced form by the following
procedure:

  * 1.

$\kappa(\lambda)$ is the matrix elements of the diagonal operator

$\displaystyle K=\sum_{n\in\mathbb{Z}}(n-1/2)^{2}{:}\psi_{-n}\psi^{*}_{n}{:}$

on the fermionic Fock space in the sense that

$\displaystyle\kappa(\lambda)=\langle\lambda|K|\lambda\rangle.$

Consequently, the $c$-number prefactors
$q^{(1-\sigma_{1})\kappa(\alpha_{0})/4}$ and
$q^{(1+\sigma_{N})\kappa(\alpha_{N})/4}$ can be moved inside
$\langle\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0}|$ and
$|\alpha_{N}\rangle$ as the operators $q^{-(1-\sigma_{1})K/4}$ and
$q^{(1+\sigma_{N})K/4}$. Note that the sign of the exponent of the first
factor is flipped because of the identity
$\kappa(\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0})=-\kappa(\alpha_{0})$.

  * 2.

Operators of the form $Q^{L_{0}}$ in (3.17) can be moved to the right until
they hit the vacuum vector and disappear. Those in (3.18) can be eliminated in
the same way. In the course of this procedure, the vertex operators are
modified as

$\displaystyle
Q^{L_{0}}\Gamma^{\sigma}_{-}(\boldsymbol{x})=\Gamma^{\sigma}_{-}(Q\boldsymbol{x})Q^{L_{0}},\qquad\Gamma^{\sigma}_{+}(\boldsymbol{x})Q^{L_{0}}=Q^{L_{0}}\Gamma^{\sigma}_{+}(Q\boldsymbol{x}).$

  * 3.

$\Gamma^{\sigma}_{-}$’s in (3.17) and $\Gamma^{\sigma}_{+}$’s in (3.18) can be
moved until they hit the vacuum vectors and disappear. Exchanging the order of
vertex operators yields $c$-number factors as the commutation relations (3.3)
show. These $c$-number factors do not depend on $\alpha_{,}\alpha_{N}$ and
cancel out in the normalized amplitudes.

Thus the following reduced expressions of the normalized amplitudes are
obtained:

$\displaystyle
Z_{0,\alpha_{0}}/Z=\left\langle\,{\vphantom{\alpha}}^{\mathrm{t}}\\!\,\alpha_{0}\,|\,q^{-(1-\sigma_{1})K/4}\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\rho}\big{)}\prod_{n=2}^{N}\Gamma^{\sigma_{n}}_{-}\big{(}\sigma_{1}Q_{1n}\sigma_{n}q^{-\rho}\big{)}\,|\,0\right\rangle,$
$\displaystyle Z_{N+1,\alpha_{N}}/Z=\left\langle
0\,|\prod_{n=1}^{N-1}\Gamma^{\sigma_{n}}_{+}\big{(}\sigma_{n}Q_{nN}\sigma_{N}q^{-\rho}\big{)}\cdot\Gamma^{\sigma_{N}}_{+}\big{(}q^{-\rho}\big{)}q^{(1+\sigma_{N})K/4}\,|\,\alpha_{N}\right\rangle.$

This implies that the generating functions $\tau_{0}(\boldsymbol{x})$,
$\tau_{N+1}(\boldsymbol{x})$ of these normalized amplitudes are tau functions
of the KP hierarchy:

######  Theorem 3.7.

$\tau_{0}(\boldsymbol{x})$ and $\tau_{N+1}(\boldsymbol{x})$ are tau functions
of the KP hierarchy with the following fermionic expression:

$\displaystyle\tau_{0}(\boldsymbol{x})=\left\langle
0\,|\,\Gamma_{+}(\boldsymbol{x})q^{-(1-\sigma_{1})K/4}\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\rho}\big{)}\prod_{n=2}^{N}\Gamma^{\sigma_{n}}_{-}\big{(}\sigma_{1}Q_{1n}\sigma_{n}q^{-\rho}\big{)}\,|\,0\right\rangle,$
(3.19) $\displaystyle\tau_{N+1}(\boldsymbol{x})=\left\langle
0\,|\prod_{n=1}^{N-1}\Gamma^{\sigma_{n}}_{+}\big{(}\sigma_{n}Q_{nN}\sigma_{N}q^{-\rho}\big{)}\cdot\Gamma^{\sigma_{N}}_{+}\big{(}q^{-\rho}\big{)}q^{(1+\sigma_{N})K/4}\Gamma^{\prime}_{-}(\boldsymbol{x})\,|\,0\right\rangle.$
(3.20)

######  Remark 3.8.

One can rewrite the fermionic expression of $\tau_{N+1}(\boldsymbol{x})$ as

$\displaystyle\tau_{N+1}(\boldsymbol{x})=\left\langle
0\,|\,\Gamma_{+}(\boldsymbol{x})q^{-(1+\sigma_{N})K/4}\Gamma^{-\sigma_{N}}_{-}\big{(}q^{-\rho}\big{)}\prod_{n=1}^{N-1}\Gamma^{-\sigma_{n}}_{-}\big{(}\sigma_{N}Q_{Nn}\sigma_{n}q^{-\rho}\big{)}\,|\,0\right\rangle,$
(3.21)

where $Q_{nm}$ for $n>m$ denotes $Q_{n}Q_{n-1}\cdots Q_{m-1}$. This shows that
$\tau_{N+1}(\boldsymbol{x})$ may be thought of as the generating function for
the leftmost leg of the web diagram rotated 180 degrees.

Although the structure of (3.19), (3.20) and (3.21) are distinct from the tau
functions (3.8) for the vertical legs, $q$-difference Kac–Schwarz operators
can be derived in much the same way. We here present the result for
$\tau_{0}(\boldsymbol{x})$ only. $\tau_{N+1}(\boldsymbol{x})$ can be treated
in the same way.

The generating operator $G$ for $\tau_{0}(\boldsymbol{x})$ can be obtained as
an avatar of the operator

$\displaystyle
g=q^{-(1-\sigma_{1})K/4}\Gamma^{\sigma_{1}}_{-}\big{(}q^{-\rho}\big{)}\prod_{n=2}^{N}\Gamma^{\sigma_{n}}_{-}\big{(}\sigma_{1}Q_{1n}\sigma_{n}q^{-\rho}\big{)}$

on the fermionic Fock space. In view of the correspondence

$\displaystyle
K\longleftrightarrow(\Delta-1/2)^{2}\longleftrightarrow(D-1/2)^{2},$

the first factor $q^{-(1-\sigma_{1})K/4}$ corresponds to the operator
$q^{-(1-\sigma_{1})(D-1/2)^{2}/4}$ on $V$. On the other hand, just as
$\Gamma_{-}(q^{-\rho})$ is translated to (2.5), vertex operators of the form
$\Gamma^{\sigma}_{-}(Qq^{-\rho})$ corresponds to the multiplication operators

$\displaystyle\exp\left(\sigma\sum_{k=1}^{\infty}\frac{(\sigma
Qq^{1/2})^{k}}{k(1-q^{k})}x^{k}\right)=\prod_{i=1}^{\infty}\big{(}1-\sigma
Qq^{i-1/2}\big{)}^{-\sigma}.$

Thus $G$ turns out to take the following form:

$\displaystyle
G=q^{-(1-\sigma_{1})(D-1/2)^{2}/4}\prod_{i=1}^{\infty}\big{(}1-\sigma_{1}q^{i-1/2}x\big{)}^{-\sigma_{1}}\cdot\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1-\sigma_{1}Q_{1n}q^{i-1/2}x\big{)}^{-\sigma_{n}}.$

The associated admissible basis of $W$ consists of the following functions:

  * (a)

In the case where $\sigma_{1}=+1$,

$\displaystyle\Phi_{j}(x)=x^{-j}\prod_{i=1}^{\infty}\big{(}1-q^{i-1/2}x\big{)}^{-1}\cdot\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1-Q_{1n}q^{i-1/2}x\big{)}^{-\sigma_{n}},$

  * (b)

in the case where $\sigma_{1}=-1$,

$\displaystyle\Phi_{j}(x)=q^{-(D-1/2)^{2}/2}\left(x^{-j}\prod_{i=1}^{\infty}\big{(}1+q^{i-1/2}x\big{)}\cdot\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1+Q_{1n}q^{i-1/2}x\big{)}^{-\sigma_{n}}\right).$

These results show that the structure of the generating operator and the
admissible basis, too, is entirely different from the case of vertical legs
shown in (3.11). In particular, the essential part of $\Phi_{j}(x)$’s are
composite quantum dilogarithmic functions. In this sense, the situation is
rather similar to the case of $\mathbb{C}^{3}$, see (2.6).

Computation of the $q$-difference Kac–Schwarz operators is more or less
parallel to the case of $\mathbb{C}^{3}$ except that the avatar
$q^{(D-1/2)^{2}/2}$ of $q^{K/2}$ joins the game.

######  Theorem 3.9.

The $q$-difference Kac–Schwarz operators $A$, $B$ for
$\tau_{0}(\boldsymbol{x})$ can be expressed as

$\displaystyle
A=q^{-D}\big{(}1-q^{1/2}x\big{)}\prod_{n=2}^{N}\big{(}1-Q_{1n}q^{1/2}x\big{)}^{\sigma_{n}},\qquad
B=x^{-1}$

in the case where $\sigma_{1}=+1$, and

$\displaystyle
A=q^{-D}\big{(}1+q^{1/2}xq^{-D}\big{)}^{-1}\prod_{n=2}^{N}\big{(}1+Q_{1n}q^{1/2}xq^{-D}\big{)}^{\sigma_{n}},\qquad
B=\big{(}xq^{-D}\big{)}^{-1}=q^{D}x^{-1}$

in the case where $\sigma_{1}=-1$.

###### Proof.

Since the case where $\sigma_{1}=+1$ is simpler, let us explain the case where
$\sigma_{1}=-1$. In this case, $G$ is a product of two operators as

$\displaystyle G=q^{-(D-1/2)^{2}/2}G^{\prime},$

where

$\displaystyle
G^{\prime}=\prod_{i=1}^{\infty}\big{(}1+q^{i-1/2}x\big{)}\cdot\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1+Q_{1n}q^{i-1/2}x\big{)}^{-\sigma_{n}}.$

As an intermediate step, one can compute $A^{\prime}=G^{\prime}\cdot
q^{-D}\cdot G^{\prime-1}$ with the aid of (2.7) as

$\displaystyle
A^{\prime}=q^{-D}\prod_{i=1}^{\infty}\big{(}1+q^{i+1/2}x\big{)}\cdot\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1+Q_{1n}q^{i+1/2}x\big{)}^{-\sigma_{n}}$
$\displaystyle\hphantom{A^{\prime}=}{}\times\prod_{n=2}^{N}\prod_{i=1}^{\infty}\big{(}1+Q_{1n}q^{i-1/2}x\big{)}^{\sigma_{n}}\cdot\prod_{i=1}^{\infty}\big{(}1+q^{i-1/2}x\big{)}^{-1}$
$\displaystyle\hphantom{A^{\prime}}{}=q^{-D}\big{(}1+q^{1/2}x\big{)}^{-1}\prod_{n=2}^{N}\big{(}1+Q_{1n}q^{1/2}x\big{)}^{\sigma_{n}}.$

Thus $A=G\cdot q^{-D}\cdot G^{-1}$ can be expressed as

$\displaystyle
A=q^{-(D-1/2)^{2}/2}\cdot\big{(}1+q^{1/2}x\big{)}^{-1}\prod_{n=2}^{N}\big{(}1+Q_{1n}q^{1/2}x\big{)}^{\sigma_{n}}\cdot
q^{(D-1/2)^{2}/2}.$

This turns into the final form by the operator identity

$\displaystyle
q^{-(D-1/2)^{2}/2}f(x)q^{(D-1/2)^{2}/2}=f\big{(}xq^{-D}\big{)}.$ ∎

##  4 Closed topological vertex

Let $Z_{\beta_{1}\beta_{2}}$ denote the open string amplitude of closed
topological vertex with two stacks of branes as shown in Fig. 3. $Q_{1}$,
$Q_{2}$, $Q_{3}$ are the Kähler parameters assigned to the internal lines. The
amplitude $Z_{\varnothing\varnothing}$ in the closed sector was computed by
Bryan and Karp [9] and Sułkowski [57]. In our previous work [61], we computed
$Z_{\beta_{1},\beta_{2}}$ and obtained the following fermionic expression:

$\displaystyle
Z_{\beta_{1}\beta_{2}}=q^{\kappa(\beta_{2})/2}\prod_{i,j=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{-\beta_{1,i}-\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{2,j}+i+j-1}\big{)}^{-1}$
$\displaystyle\hphantom{Z_{\beta_{1}\beta_{2}}=}{}\times\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{1}\,|\,\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma_{+}\big{(}q^{-\rho}\big{)}(-Q_{1})^{L_{0}}\Gamma^{\prime}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\prime}_{+}\big{(}q^{-\rho}\big{)}(-Q_{3})^{L_{0}}$
$\displaystyle\hphantom{Z_{\beta_{1}\beta_{2}}=}{}\times\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma_{+}\big{(}q^{-\rho}\big{)}(-Q_{2})^{L_{0}}\Gamma^{\prime}_{-}\big{(}q^{-\rho})\Gamma^{\prime}_{+}\big{(}q^{-\rho}\big{)}\,|\,\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{2}\big{\rangle}.$
(4.1)

Figure 3: Closed topological vertex with two stacks of branes.

The main part
$\langle\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{1}|\cdots|\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta_{2}\rangle$
of (4.1) happens to be essentially the same as the open string amplitude of
strip geometry shown in Fig. 4. This part is corrected by the first two
factors. In the previous work, we made use of this particular structure to
derive $q$-difference equations for the generating functions

$\displaystyle\Psi_{1}(x)=\sum_{k=0}^{\infty}x^{k}Z_{(1^{k})\varnothing}/Z,\qquad\Psi_{2}(x)=\sum_{k=0}^{\infty}x^{k}Z_{\varnothing(1^{k})}/Z,\qquad
Z=Z_{\varnothing\varnothing},$ (4.2)

of normalized open string amplitudes.

Figure 4: New strip geometry emerging from $Z_{\beta_{1}\beta_{2}}$.

###  4.1 Generating functions as tau functions

We can construct two multi-variate generating functions:

$\displaystyle\tau_{1}(\boldsymbol{x})=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})Z_{1,\beta}/Z,\qquad
Z_{1,\beta}=Z_{\beta\varnothing},$
$\displaystyle\tau_{2}(\boldsymbol{x})=\sum_{\beta\in\mathcal{P}}s_{\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta}(\boldsymbol{x})Z_{2,\beta}/Z,\qquad
Z_{2,\beta}=Z_{\varnothing\beta}.$

As we show below, both $\tau_{1}(\boldsymbol{x})$ and
$\tau_{2}(\boldsymbol{x})$ are tau functions of the KP hierarchy, and the
associated points of the Sato Grassmannian have admissible bases that are
generated by $q$-difference operators. The reasoning is similar to the case
the of generating functions of $Z_{0,\alpha_{0}}/Z$ and $Z_{N+1,\alpha_{N}}/Z$
in the last section.

Let us examine $Z_{1,\beta}/Z$ in detail. As explained in the case of strip
geometry, one can eliminate the operators of the form $(-Q)^{L_{0}}$ and
$\Gamma^{\sigma}_{-}$ from the unnormalized amplitude

$\displaystyle
Z_{1,\beta}=\prod_{i,j=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{-\beta_{i}+i+j-1}\big{)}^{-1}\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\,|\,\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma_{+}\big{(}q^{-\rho}\big{)}(-Q_{1})^{L_{0}}\Gamma^{\prime}_{-}\big{(}q^{-\rho}\big{)}$
$\displaystyle\hphantom{Z_{1,\beta}=}{}\times\Gamma^{\prime}_{+}\big{(}q^{-\rho}\big{)}(-Q_{3})^{L_{0}}\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma_{+}\big{(}q^{-\rho}\big{)}(-Q_{2})^{L_{0}}\Gamma^{\prime}_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\prime}_{+}\big{(}q^{-\rho}\big{)}\,|\,0\big{\rangle}$

as

$\displaystyle Z_{1,\beta}=(\text{constant independent of
$\beta$})\cdot\prod_{i,j=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{-\beta_{i}+i+j-1}\big{)}^{-1}$
$\displaystyle\hphantom{Z_{1,\beta}=}{}\times\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\,|\,\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\prime}_{-}\big{(}{-}Q_{1}q^{-\rho}\big{)}\Gamma_{-}\big{(}Q_{1}Q_{3}q^{-\rho}\big{)}\Gamma^{\prime}_{-}\big{(}{-}Q_{1}Q_{2}Q_{3}q^{-\rho}\big{)}\,|\,0\big{\rangle},$

The first constant factor disappears in the normalized amplitude. The
contribution of the infinite product can be computed by the formula (2.9).
Thus the normalized amplitude can be expressed as

$\displaystyle
Z_{1,\beta}/Z=\big{\langle}\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\,|\,g_{1}\,|\,0\big{\rangle},$

where

$\displaystyle
g_{1}=\exp\left(-\sum_{k=1}^{\infty}\frac{(Q_{1}Q_{2})^{k}}{k(1-q^{k})}V^{(k)}_{0}\right)$
$\displaystyle\hphantom{g_{1}=}{}\times\Gamma_{-}\big{(}q^{-\rho}\big{)}\Gamma^{\prime}_{-}\big{(}{-}Q_{1}q^{-\rho}\big{)}\Gamma_{-}\big{(}Q_{1}Q_{3}q^{-\rho}\big{)}\Gamma^{\prime}_{-}\big{(}{-}Q_{1}Q_{2}Q_{3}q^{-\rho}\big{)}.$
(4.3)

In much the same way, $Z_{2,\beta}/Z$ turns out to be expressed as

$\displaystyle
Z_{2,\beta}/Z=\big{\langle}0\,|\,g_{2}\,|\,\,{\vphantom{\beta}}^{\mathrm{t}}\\!\,\beta\big{\rangle},$

where

$\displaystyle
g_{2}=\Gamma_{+}\big{(}{-}Q_{1}Q_{2}Q_{3}q^{-\rho}\big{)}\Gamma^{\prime}_{+}\big{(}Q_{2}Q_{3}q^{-\rho}\big{)}\Gamma_{+}\big{(}{-}Q_{2}q^{-\rho}\big{)}\Gamma^{\prime}_{+}\big{(}q^{-\rho}\big{)}$
$\displaystyle\hphantom{g_{2}=}{}\times\exp\left(\sum_{k=1}^{\infty}\frac{(Q_{1}Q_{2}q)^{k}}{k(1-q^{k})}V^{(-k)}_{0}\right)q^{-K/2}.$
(4.4)

These expressions of the normalized amplitudes lead us to the following
conclusion.

######  Theorem 4.1.

$\tau_{1}(\boldsymbol{x})$ and $\tau_{2}(\boldsymbol{x})$ are tau functions of
the KP hierarchy with the fermionic expression

$\displaystyle\tau_{1}(\boldsymbol{x})=\big{\langle}0\,|\,\Gamma_{+}(\boldsymbol{x})g_{1}\,|\,0\big{\rangle},\qquad\tau_{2}(\boldsymbol{x})=\big{\langle}0\,|\,g_{2}\Gamma_{-}(\boldsymbol{x})\,|\,0\big{\rangle},$

where $g_{1}$ and $g_{2}$ are the operators defined in (4.3) and (4.4).

######  Remark 4.2.

The fermionic expression of $\tau_{2}(\boldsymbol{x})$ can be rewritten as

$\displaystyle\tau_{2}(\boldsymbol{x})=\big{\langle}0\,|\,\Gamma_{+}(\boldsymbol{x})\,{\vphantom{g_{2}}}^{\mathrm{t}}\\!\,g_{2}\,|\,0\big{\rangle},$

where

$\displaystyle\,{\vphantom{g_{2}}}^{\mathrm{t}}\\!\,g_{2}=q^{-K/2}\exp\left(\sum_{k=1}^{\infty}\frac{(Q_{1}Q_{2}q)^{k}}{k(1-q^{k})}V^{(-k)}_{0}\right)$
$\displaystyle\hphantom{\,{\vphantom{g_{2}}}^{\mathrm{t}}\\!\,g_{2}=}{}\times\Gamma^{\prime}_{-}\big{(}q^{-\rho}\big{)}\Gamma_{-}\big{(}{-}Q_{2}q^{-\rho}\big{)}\Gamma^{\prime}_{-}\big{(}Q_{2}Q_{3}q^{-\rho}\big{)}\Gamma_{-}\big{(}{-}Q_{1}Q_{2}Q_{3}q^{-\rho}\big{)}.$

###  4.2 Admissible basis and Kac–Schwarz operators

Since all building blocks of $\tau_{1}(\boldsymbol{x})$ and
$\tau_{2}(\boldsymbol{x})$ are now familiar, one can derive the associated
admissible bases and $q$-difference Kac–Schwarz operators in almost the same
way as we have done in the preceding sections. Actually, the computations are
partly more complicated, and exhibit a new aspect that is characteristic of
this case. Let us consider $\tau_{1}(\boldsymbol{x})$ to illustrate the
computations.

The generating operator $G$ of an admissible basis
$\\{\Phi_{j}(x)\\}_{j=0}^{\infty}$ for $\tau_{1}(\boldsymbol{x})$ takes such a
form as

$\displaystyle
G=\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\cdot\Phi(x),$

where $\Phi(x)$ is the composite dilogarithmic function

$\displaystyle\Phi(x)=\prod_{i=1}^{\infty}\frac{(1-Q_{1}q^{i-1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{i-1/2}x)}{(1-q^{i-1/2}x)(1-Q_{1}q_{3}q^{i-1/2}x)}.$

The admissible basis is generated from $\Phi(x)$ as

$\displaystyle\Phi_{j}(x)=\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\big{(}x^{-j}\Phi(x)\big{)}.$

$\Phi_{j}(x)$’s are neither quantum dilogarithmic nor $q$-hypergeometric.
Power series expansion of these functions take such a form as

$\displaystyle\Phi_{j}(x)=\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1-j}\big{)}\cdot\sum_{k=0}^{\infty}\frac{b_{k}}{(Q_{1}Q_{2}q^{-j};q)_{k}}x^{k-j},$
(4.5)

where $b_{k}$’s are coefficients of the power series expansion

$\displaystyle\Phi(x)=\sum_{k=0}^{\infty}b_{k}x^{k}$

of $\Phi(x)$. Comparing the expansion of $\Phi_{0}(x)$ with the expansion of
the generating function $\Psi_{1}(x)$ presented in our previous work [61], one
can confirm that $\Phi_{0}(x)$ agrees with $\Psi_{1}(x)$ except for the
constant multiplier $\prod\limits_{i=1}^{\infty}(1-Q_{1}Q_{2}q^{i-1})$.

In the previous work, we derived a $q$-difference equation for $\Psi_{1}(x)$
from the $q$-difference equation

$\displaystyle\Phi(qx)=\frac{(1-q^{1/2}x)(1-Q_{1}Q_{3}q^{1/2}x)}{(1-Q_{1}q^{1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{1/2}x)}\Phi(x)$
(4.6)

for $\Phi(x)$ by a quite primitive method. The notion of Kac–Schwarz operators
leads us to a more systematic derivation of the $q$-difference equation.

Let us proceed to the $q$-difference Kac–Schwarz operators. It is easy to
compute $B=G\cdot x^{-1}\cdot G^{-1}$:

$\displaystyle B=\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\cdot
x^{-1}\cdot\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}^{-1}$
$\displaystyle=\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\cdot\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i+D}\big{)}^{-1}\cdot
x^{-1}=\big{(}1-Q_{1}Q_{2}q^{D}\big{)}\cdot x^{-1}.$ (4.7)

To compute $A=G\cdot q^{-D}\cdot G^{-1}$, one can start from the algebraic
relation

$\displaystyle\Phi(x)\cdot
q^{-D}\cdot\Phi(x)^{-1}=q^{-D}\frac{(1-q^{1/2}x)(1-Q_{1}Q_{3}q^{1/2}x)}{(1-Q_{1}q^{1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{1/2}x)},$

which is the $q$-difference equation (4.6) in disguise. This relation yields
the following intermediate expression of $A$:

$\displaystyle
A=q^{-D}\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\cdot\frac{(1-q^{1/2}x)(1-Q_{1}Q_{3}q^{1/2}x)}{(1-Q_{1}q^{1/2}x)(1-Q_{1}Q_{2}Q_{3}q^{1/2}x)}$
$\displaystyle\hphantom{A=}{}\times\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}^{-1}.$

Since (4.7) implies that

$\displaystyle\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}\cdot
x\cdot\prod_{i=1}^{\infty}\big{(}1-Q_{1}Q_{2}q^{i-1+D}\big{)}^{-1}=x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1},$

one can rewrite the foregoing intermediate expression of $A$ as

$\displaystyle
A=q^{-D}\frac{\left(1-q^{1/2}x(1-Q_{1}Q_{2}q^{D})^{-1}\right)\left(1-Q_{1}Q_{3}q^{1/2}x(1-Q_{1}Q_{2}q^{D})^{-1}\right)}{\left(1-Q_{1}q^{1/2}x(1-Q_{1}Q_{2}q^{D})^{-1}\right)\left(1-Q_{1}Q_{2}Q_{3}q^{1/2}x(1-Q_{1}Q_{2}q^{D})^{-1}\right)}.$
(4.8)

Let us summarize these computations. Although we omit details, similar results
hold for $\tau_{2}(\boldsymbol{x})$ as well.

######  Theorem 4.3.

(4.7) and (4.8) are $q$-difference Kac–Schwarz operators for
$\tau_{1}(\boldsymbol{x})$. (4.5) give an admissible basis of the associated
point of the Sato Grassmannian.

###  4.3 Remarks on $\boldsymbol{q}$-difference equations

Let us rewrite the equation $A\Phi_{0}(x)=\Phi_{0}(x)$ for $\Phi_{0}(x)$ as

$\displaystyle\big{(}1-q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}\big{(}1-Q_{1}Q_{3}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}\Phi_{0}(x)$
$\displaystyle\qquad{}=\big{(}1-Q_{1}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}\big{(}1-Q_{1}Q_{2}Q_{3}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}q^{D}\Phi_{0}(x).\\!\\!\\!$
(4.9)

Although this equation looks considerably different from the $q$-difference
equation presented in our previous work [61], they are equivalent as we show
below.

######  Lemma 4.4.

For any constants $P_{1}$, $P_{2}$ and $Q$,

$\displaystyle\big{(}1-P_{1}x\big{(}1-Qq^{D}\big{)}^{-1}\big{)}\big{(}1-P_{2}x\big{(}1-Qq^{D}\big{)}^{-1}\big{)}$
$\displaystyle\qquad{}=\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-Qq^{D-2}\big{)}^{-1}\big{(}1-Qq^{D-2}-P_{1}x\big{)}\big{(}1-Qq^{D-1}-P_{2}x\big{)}.$

###### Proof.

Do straightforward computations using (2.13) as follows:

$\displaystyle\big{(}1-P_{1}x\big{(}1-Qq^{D}\big{)}^{-1}\big{)}\big{(}1-P_{2}x\big{(}1-Qq^{D}\big{)}^{-1}\big{)}$
$\displaystyle\qquad{}=\big{(}1-P_{1}\big{(}1-Qq^{D-1}\big{)}^{-1}x\big{)}\big{(}1-P_{2}\big{(}1-Qq^{D-1}\big{)}^{-1}x\big{)}$
$\displaystyle\qquad{}=\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-Qq^{D-1}-P_{1}x\big{)}\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-Qq^{D-1}-P_{2}x\big{)}$
$\displaystyle\qquad{}=\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-P_{1}x\big{(}1-Qq^{D-1}\big{)}^{-1}\big{)}\big{(}1-Qq^{D-1}-P_{2}x\big{)}$
$\displaystyle\qquad{}=\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-P_{1}\big{(}1-Qq^{D-2}x\big{)}^{-1}x\big{)}\big{(}1-Qq^{D-1}-P_{2}x\big{)}$
$\displaystyle\qquad{}=\big{(}1-Qq^{D-1}\big{)}^{-1}\big{(}1-Qq^{D-2}x\big{)}^{-1}\big{(}1-Qq^{D-2}-P_{1}x\big{)}\big{(}1-Qq^{D-1}-P_{2}x\big{)}.$
∎

One can use this lemma to rewrite both sides of (4.9) and collect terms to the
left side. The outcome reads

$\displaystyle\big{(}1-Q_{1}Q_{2}q^{D-1}\big{)}^{-1}\big{(}1-Q_{1}Q_{2}q^{D-2}\big{)}^{-1}\hat{H}\Phi_{0}(x)=0,$
(4.10)

where

$\displaystyle\hat{H}=\big{(}1-Q_{1}Q_{2}q^{D-2}-q^{1/2}x\big{)}\big{(}1-Q_{1}Q_{2}q^{D-1}-Q_{1}Q_{3}q^{1/2}x\big{)}$
$\displaystyle\hphantom{\hat{H}=}{}-\big{(}1-Q_{1}Q_{2}q^{D-2}-Q_{1}q^{1/2}x\big{)}\big{(}1-Q_{1}Q_{2}q^{D-1}-Q_{1}Q_{2}Q_{3}q^{1/2}x\big{)}q^{D}.$

We are thus substantially in the same situation as encountered in the previous
work. Namely, as far as $Q_{1}Q_{2}$ is not equal to integral powers of $q$
(and this condition is assumed implicitly in the foregoing consideration), the
two operators in front of $\hat{H}$ in (4.10) are invertible on
$V=\mathbb{C}((x))$. Therefore one can remove these operators and obtain the
equation

$\displaystyle\hat{H}\Phi_{0}(x)=0,$ (4.11)

which is one of the equivalent forms of the $q$-difference equations derived
in the previous work. Moreover, as pointed out therein, $\hat{H}$ itself can
be factorized as

$\displaystyle\hat{H}=\big{(}1-Q_{1}Q_{2}q^{D-2}\big{)}\hat{K},$

where $\hat{K}$ is another $q$-difference operator. Therefore (4.11) can be
further reduced to

$\displaystyle\hat{K}\Phi_{0}(x)=0.$ (4.12)

It is this equation (4.12) that is proposed in our previous work as a
candidate of the quantum mirror curve of close topological vertex. Actually,
the foregoing computations show that $\hat{K}$ is related to the operators on
both sides of (4.9) as

$\displaystyle\big{(}1-q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}\big{(}1-Q_{1}Q_{3}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}$
$\displaystyle\qquad\quad{}-\big{(}1-Q_{1}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}\big{(}1-Q_{1}Q_{2}Q_{3}q^{1/2}x\big{(}1-Q_{1}Q_{2}q^{D}\big{)}^{-1}\big{)}q^{D}$
$\displaystyle\qquad{}=\big{(}1-Q_{1}Q_{2}q^{D-1}\big{)}^{-1}\hat{K}.$

### Acknowledgements

The authors are grateful to Motohico Mulase for discussion and encouragement.
We owe him the idea that an integrable hierarchy may be thought of as a mirror
map. This work is partly supported by JSPS Kakenhi Grant No. 25400111 and No.
15K04912.

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