4D limit of melting crystal model and its integrable structure

TL;DR

This paper investigates the 4D limit of the melting crystal model related to 5D SUSY $U(1)$ Yang-Mills theory, revealing its integrable structure and connection to Gromov-Witten invariants via quantum spectral curves.

Contribution

It introduces a systematic 4D limit procedure for the melting crystal model, linking it to Gromov-Witten theory and establishing its tau function properties within integrable hierarchies.

Findings

The 4D limit yields a difference equation matching the quantum spectral curve of Gromov-Witten theory.

The 4D counterpart of the partition function is a tau function of the KP hierarchy.

Extensions to deformations produce tau functions of the 1D Toda hierarchy.

Abstract

This paper addresses the problems of quantum spectral curves and 4D limit for the melting crystal model of 5D SUSY $U(1)$ Yang-Mills theory on $\mathbb{R}^4\times S^1$. The partition function $Z(\mathbf{t})$ deformed by an infinite number of external potentials is a tau function of the KP hierarchy with respect to the coupling constants $\mathbf{t} = (t_1,t_2,\ldots)$. A single-variate specialization $Z(x)$ of $Z(\mathbf{t})$ satisfies a $q$-difference equation representing the quantum spectral curve of the melting crystal model. In the limit as the radius $R$ of $S^1$ in $\mathbb{R}^4\times S^1$ tends to $0$, it turns into a difference equation for a 4D counterpart $Z_{\mathrm{4D}}(X)$ of $Z(x)$. This difference equation reproduces the quantum spectral curve of Gromov-Witten theory of $\mathbb{CP}^1$. $Z_{\mathrm{4D}}(X)$ is obtained from $Z(x)$ by letting $R \to 0$ under an $R$-dependent transformation $x = x(X,R)$ of $x$ to $X$. A similar prescription of 4D limit can be formulated for $Z(\mathbf{t})$ with an $R$-dependent transformation $\mathbf{t} = \mathbf{t}(\mathbf{T},R)$ of $\mathbf{t}$ to $\mathbf{T} = (T_1,T_2,\ldots)$. This yields a 4D counterpart $Z_{\mathrm{4D}}(\mathbf{T})$ of $Z(\mathbf{t})$. $Z_{\mathrm{4D}}(\mathbf{T})$ agrees with a generating function of all-genus Gromov-Witten invariants of $\mathbb{CP}^1$. Fay-type bilinear equations for $Z_{\mathrm{4D}}(\mathbf{T})$ can be derived from similar equations satisfied by $Z(\mathbf{t})$. The bilinear equations imply that $Z_{\mathrm{4D}}(\mathbf{T})$, too, is a tau function of the KP hierarchy. These results are further extended to deformations $Z(\mathbf{t},s)$ and $Z_{\mathrm{4D}}(\mathbf{T},s)$ by a discrete variable $s \in \mathbb{Z}$, which are shown to be tau functions of the 1D Toda hierarchy.

Equations

$$Z=\sum _{\pi \in \mathcal{PP}}{q}^{|\pi |}$$

$$|\pi |=\sum _{i,j=1}^{\infty }{\pi }_{ij}.$$

$$Z=\sum _{\lambda \in \mathcal{P}}{s}_{\lambda }(q^{-\rho }{)}^2,$$

$$\mathbf{x}=q^{-\rho }=(q^{1/2},q^{3/2},\dots ,q^{i-1/2},\dots ).$$

$$Z=\prod _{i,j=1}^{\infty }(1-q^{i+j-1}{)}^{-1}=\prod _{n=1}^{\infty }(1-q^n{)}^{-n}.$$

$$s_{\lambda }(q^{-\rho })=\frac{q^{-\kappa (\lambda )/4}}{{\prod }_{(i,j)\in \lambda }(q^{-h(i,j)/2}-q^{h(i,j)/2})},$$

$$\kappa (\lambda )=2\sum _{(i,j)\in \lambda }(j-i)=\sum _{i=1}^{\infty }\left(({\lambda }_i-i+1/2{)}^2-(-i+1/2{)}^2\right),$$

$$h(i,j)=({\lambda }_i-j)+({}^{\mathrm{t}}{\lambda }_j-i)+1$$

$$\frac{\dim \lambda }{|\lambda |!}=\frac{1}{{\prod }_{(i,j)\in \lambda }{h}_{(i,j)}},$$

$$Z(\mathbf{t})=\sum _{\lambda \in \mathcal{P}}{s}_{\lambda }(q^{-\rho }{)}^2{Q}^{|\lambda |}{e}^{\varphi (\mathbf{t},\lambda )},$$

$$\varphi (\mathbf{t},\lambda )=\sum _{k=1}^{\infty }{t}_k{\varphi }_k(\lambda ).$$

$${\varphi }_k(\lambda )=\sum _{i=1}^{\infty }\left(q^{k({\lambda }_i-i+1)}-q^{k(-i+1)}\right).$$

$$Z(\mathbf{t},s)=\sum _{\lambda \in \mathcal{P}}{s}_{\lambda }(q^{-\rho }{)}^2{Q}^{|\lambda |+s(s+1)/2}{e}^{\varphi (\mathbf{t},s,\lambda )},$$

$$\varphi (\mathbf{t},s,\lambda )=\sum _{k=1}^{\infty }{t}_k{\varphi }_k(s,\lambda ),$$

$${\varphi }_k(s,\lambda )=\sum _{k=1}^{\infty }\left(q^{k({\lambda }_i-i+1+s)}-q^{k(-i+1+s)}\right)+\frac{1-q^{ks}}{1-q^k}{q}^k.$$

$${\varphi }_k(s,\lambda )=\sum _{i=1}^{\infty }{q}^{k({\lambda }_i-i+1+s)}-\sum _{i=1}^{\infty }{q}^{k(-i+1)}.$$

$$\frac{1-q^{ks}}{1-q^k}{q}^k$$

$$=\{\begin{array}{ll}{q}^k+\cdots +q^{ks}& \text{for s>0},\\ 0& \text{for s=0},\\ -1-q^{-k}-\cdots -q^{k(s+1)}& \text{for s<0}.\end{array}$$

$${\psi }_m{\psi }_n^{\ast }+{\psi }_n^{\ast }{\psi }_m={\delta }_{m+n,0},{\psi }_m{\psi }_n+{\psi }_n{\psi }_m=0,{\psi }_m^{\ast }{\psi }_n^{\ast }+{\psi }_n^{\ast }{\psi }_m^{\ast }=0,$$

$${\psi }_n^{\ast }|0⟩=0\text{for n>0},{\psi }_n|0⟩=0\text{for n≥0},$$

$$⟨0|{\psi }_n=0\text{for n<0},⟨0|{\psi }_n^{\ast }=0\text{for n≤0}.$$

$$|\lambda ⟩$$

$$⟨\lambda |$$

$$L_0=\sum _{n\in \mathbb{Z}}n:{\psi }_{-n}{\psi }_n^{\ast }:,K=\sum _{n\in \mathbb{Z}}(n-1/2{)}^2:{\psi }_{-n}{\psi }_n^{\ast }:,$$

$$H_k=\sum _{n\in \mathbb{Z}}{q}^{kn}:{\psi }_{-n}{\psi }_n^{\ast }:,J_k=\sum _{n\in \mathbb{Z}}:{\psi }_{k-n}{\psi }_n^{\ast }:,$$

$$:{\psi }_m{\psi }_n^{\ast }:={\psi }_m{\psi }_n^{\ast }-⟨0|{\psi }_m{\psi }_n^{\ast }|0⟩,$$

$${\Gamma }_{\pm k}(x)=\exp \left(\sum _{k=1}^{\infty }\frac{x^k}{k}{J}_{\pm k}\right),{\Gamma }_{\pm k}^{\prime }(x)=\exp \left(-\sum _{k=1}^{\infty }\frac{(-x{)}^k}{k}{J}_{\pm k}\right),$$

$${\Gamma }_{\pm k}(x_1,x_2,\dots )=\prod _{i\ge 1}{\Gamma }_{\pm k}(x_i),{\Gamma }_{\pm k}^{\prime }(x_1,x_2,\dots )=\prod _{i\ge 1}{\Gamma }_{\pm k}^{\prime }(x_i).$$

$$⟨\lambda |L_0|\mu ⟩=|\lambda |{\delta }_{\lambda \mu },⟨\lambda |K|\mu ⟩=\kappa (\lambda ){\delta }_{\lambda \mu },⟨\lambda |H_k|\mu ⟩={\varphi }_k(\lambda ){\delta }_{\lambda \mu }.$$

$$⟨\lambda |{\Gamma }_{-}(\mathbf{x})|\mu ⟩=⟨\mu |{\Gamma }_{+}(\mathbf{x})|\lambda ⟩=s_{\lambda /\mu }(\mathbf{x}),$$

$$⟨\lambda |{\Gamma }_{-}^{\prime }(\mathbf{x})|\mu ⟩=⟨\mu |{\Gamma }_{+}^{\prime }(\mathbf{x})|\lambda ⟩=s_{{}^{\mathrm{t}}\lambda /{}^{\mathrm{t}}\mu }(\mathbf{x}).$$

$$Z(\mathbf{t})=⟨0|{\Gamma }_{+}(q^{-\rho })Q^{L_0}{e}^{H(\mathbf{t})}{\Gamma }_{-}(q^{-\rho })|0⟩,$$

$$H(\mathbf{t})=\sum _{k=1}^{\infty }{t}_k{H}_k.$$

$$Z(\mathbf{t})=\exp \left(\sum _{k=1}^{\infty }\frac{q^k{t}_k}{1-q^k}\right)⟨0|\exp \left(\sum _{k=1}^{\infty }(-1{)}^k{q}^{k/2}{t}_k{J}_k\right)g_1|0⟩,$$

Full text

2010 Mathematics Subject Classification: 14N35, 37K10, 39A13

Key words: melting crystal model, quantum curve, KP hierarchy, Toda hierarchy, bilinear equation, Gromov-Witten theory

1 Introduction

The melting crystal model [15] is a statistical model of 5D SUSY Yang-Mills theory on $\mathbb{R}^{4}\times S^{1}$ [19] in the self-dual background [20, 21]. The partition function is a sum over all possible shapes (represented by plane partitions) of 3D Young diagrams. The name of the model originates in the physical interpretation of the complement of a 3D Young diagram in the positive octant of $\mathbb{R}^{3}$ as a melting crystal corner. By the method of diagonal slicing [26], the partition function can be converted to a sum over ordinary partitions. This sum reproduces the Nekrasov partition function of instantons in 5D SUSY Yang-Mills theory.

In the previous work [18], we studied the simplest case that amounts to $U(1)$ gauge theory. The main subject was an integrable structure of the partition function deformed by an infinite number of external potentials. The deformed partition function $Z(\boldsymbol{t},s)$ depends on the coupling constants $\boldsymbol{t}=(t_{1},t_{2},\ldots)$ of those potentials and a discrete variable $s\in\mathbb{Z}$. We proved, with the aid of symmetries of a quantum torus algebra, that $Z(\boldsymbol{t},s)$ is essentially a tau function of the 1D Toda hierarchy [37]. This result has been extended to some other types of melting crystal models [32, 33].

An open problem raised therein is to find an appropriate prescription for the 4D limit as the radius $R$ of $S^{1}$ in $\mathbb{R}^{4}\times S^{1}$ tends to [math]. The melting crystal model of $U(1)$ gauge theory has two parameters $q,Q$. By setting these parameters in a particular $R$-dependent form and letting $R\to 0$, the undeformed partition function $Z=Z(\boldsymbol{0},0)$ converges to the 4D Nekrasov function $Z_{\mathrm{4D}}$ [20, 21]. It is not so straightforward to achieve the 4D limit of the deformed partition function $Z(\boldsymbol{t},s)$. In a naive prescription [18], all coupling constants other than $t_{1}$ decouple from $Z(\boldsymbol{t},s)$ in the limit as $R\to 0$. On the other hand, a deformation $Z_{\mathrm{4D}}(\boldsymbol{t},s)$ of $Z_{\mathrm{4D}}$ by an infinite number of external potentials is proposed in the literature [14]. What we need is an $R$-dependent transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ of $\boldsymbol{t}$ to a new set of coupling constants $\boldsymbol{T}=(T_{1},T_{2},\ldots)$ such that $Z(\boldsymbol{t}(\boldsymbol{T},R),s)$ converges to $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ as $R\to 0$.

As a warm-up for tackling this problem, we consider the so called quantum spectral curves. This is inspired by the work of Dunin-Barkowski et al. [6] on Gromov-Witten theory of $\mathbb{C}\mathbb{P}^{1}$. They derived a quantum spectral curve of $\mathbb{C}\mathbb{P}^{1}$ in the perspective of topological recursion [7, 8, 22]. Since the deformed 4D Nekrasov function $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ of $U(1)$ gauge theory coincides with a generating function of all genus Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ [12, 24], it will be natural to reconsider this issue from the point of view of the melting crystal model.

Recently, we proposed a new approach to quantum mirror curves in topological string theory [34]. This approach is based on the notions of Kac–Schwarz operators [11, 30] and generating operators [2, 3] in the KP hierarchy [28, 29]. $Z(\boldsymbol{t},s)$ may be thought of as a set of KP tau functions labelled by $s\in\mathbb{Z}$. In particular, $Z(\boldsymbol{t})=Z(\boldsymbol{t},0)$ resembles the tau functions in topological string theory. Our method developed for topological string theory can be applied to $Z(\boldsymbol{t})$ to derive a quantum spectral curve. This quantum curve is represented by a $q$-difference equation for a single-variate specialization $Z(x)$ of $Z(\boldsymbol{t})$.

We show that this $q$-difference equation turns into the quantum spectral curve of Dunin-Barkowski et al. as $R\to 0$. To this end, we choose an $R$-dependent transformation $x=x(X,R)$ of $x$ to a new variable $X$. $Z(x(X,R))$ thereby converges to a function $Z_{\mathrm{4D}}(X)$ as $R\to 0$. $Z_{\mathrm{4D}}(X)$ is exactly the function considered by Dunin-Barkowski et al. and shown to satisfy a difference equation that represents their quantum spectral curve. Although being rather simple, the transformation $x=x(X,R)$ is indispensable to achieve this limit.

This result can be further extended to the multi-variate partition function $Z(\boldsymbol{t})$. We construct an $R$-dependent transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ of the coupling constants $\boldsymbol{t}$, and show that $Z(\boldsymbol{t}(\boldsymbol{T},R))$ does converge to the correct 4D counterpart $Z_{\mathrm{4D}}(\boldsymbol{T})=Z_{\mathrm{4D}}(\boldsymbol{T},0)$ as $R\to 0$. The same transformation of the coupling constants can be used to derive $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ from $Z(\boldsymbol{t},s)$ as well. The construction of this transformation is far more complicated than the transformation $x=x(X,R)$ of the variable $x$. The complexity stems from a structural difference in the external potentials of $Z(\boldsymbol{t})$ and $Z_{\mathrm{4D}}(\boldsymbol{T})$, hence being unavoidable. We believe that our choice is nevertheless the most natural among all possible prescriptions.

As a byproduct of this prescription of 4D limit, we can show that $Z_{\mathrm{4D}}(\boldsymbol{T})$ satisfies a set of Fay-type bilinear equations. These bilinear equations characterize all tau functions of the KP hierarchy [1, 28, 35]. This implies that $Z_{\mathrm{4D}}(\boldsymbol{T})$, too, is a tau function of the KP hierarchy. Moreover, by a similar characterization of tau functions of the Toda hierarchy [31, 36], we can deduce that $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ is a tau function of the 1D Toda hierarchy. Actually, these functions are known as generating functions of all-genus Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ [12, 24]. The Toda conjecture on Gromov-Witten theory of $\mathbb{C}\mathbb{P}^{1}$ [5, 9, 16, 27] can be thus explained in a different perspective.

This paper is organized as follows. Section 2 is a brief review of the melting crystal model. Combinatorial and fermionic expressions of the deformed partition function $Z(\boldsymbol{t})$ are introduced. The fermionic expression is further converted to a form that fits into the method of our work on quantum mirror curves of topological string theory [34]. Section 3 presents the quantum spectral curve of the melting crystal model. The single-variate specialization $Z(x)$ of $Z(\boldsymbol{t})$ is introduced, and shown to satisfy a $q$-difference equation representing the quantum spectral curve. The computations are mostly parallel to the case of topological string theory. Section 4 deals with the issue of 4D limit. The $R$-dependent transformations $x=x(X,R)$ and $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ are introduced, and $Z(x(X,R))$ and $Z(\boldsymbol{t}(\boldsymbol{T},R))$ are shown to converge as $R\to 0$. The functions $Z_{\mathrm{4D}}(X)$ and $Z_{\mathrm{4D}}(\boldsymbol{T})$ obtained in this limit are computed explicitly. The difference equation for $Z_{\mathrm{4D}}(X)$ is derived, and confirmed to agree with the result of Dunin-Barkowski et al. [6]. Section 5 is devoted to Fay-type bilinear equations. A three-term bilinear equation plays a central role here. The bilinear equation for $Z(\boldsymbol{t})$ is shown to turn into a similar bilinear equation for $Z_{\mathrm{4D}}(\boldsymbol{T})$ as $R\to 0$. The corresponding results for $Z(\boldsymbol{t},s)$ and $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ are presented in Appendix. Section 6 concludes this paper.

2 Melting crystal model

2.1 Partition function of 3D Young diagrams

The partition function of the simplest melting crystal model with a single parameter $q$ is the sum

[TABLE]

of the Boltzmann weight $q^{|\pi|}$ over the set $\mathcal{P}\mathcal{P}$ of all plane partitions. The plane partition $\pi=(\pi_{ij})_{i,j=1}^{\infty}$ represents a 3D Young diagram that consists of stacks of unit cubes of height $\pi_{ij}$ put on the unit squares $[i-1,i]\times[j-1,j]$ of the plane. $|\pi|$ denotes the volume of the 3D Young diagram:

[TABLE]

By the method of diagonal slicing [26], one can convert the sum (2.1) over $\mathcal{P}\mathcal{P}$ to a sum over the set $\mathcal{P}$ of all ordinary partitions $\lambda=(\lambda_{i})_{i=1}^{\infty}$ as

[TABLE]

where $s_{\lambda}(q^{-\rho})$ is the special value (a kind of principal specialization) of the infinite-variate Schur function $s_{\lambda}(\boldsymbol{x})$, $\boldsymbol{x}=(x_{1},x_{2},\ldots)$, at

[TABLE]

Moreover, by the Cauchy identities of Schur functions [13], one can rewrite the sum (2.2) into an infinite product:

[TABLE]

This infinite product is known as the MacMahon function.

The special value $s_{\lambda}(q^{-\rho})$ has the hook-length formula [13]

[TABLE]

where $\kappa(\lambda)$ is the commonly used notation

[TABLE]

and $h(i,j)$ denote the the hook length

[TABLE]

of the cell $(i,j)$ in the Young diagram of shape $\lambda$. $\,{\vphantom{\lambda}}^{\mathrm{t}}\!\,\lambda_{i}$’s are the parts of the conjugate partition $\,{\vphantom{\lambda}}^{\mathrm{t}}\!\,\lambda$ that represents the transposed Young diagram. Thus $s_{\lambda}(q^{-\rho})$ is a $q$-deformation of the number

[TABLE]

where $|\lambda|=\sum_{i=1}^{\infty}\lambda_{i}$, and $\dim\lambda$ is the number of standard tableau of shape $\lambda$. (2.4) plays a central role in Gromov-Witten/Hurwitz theory of $\mathbb{C}\mathbb{P}^{1}$ [23, 24, 27].

2.2 Deformation by external potentials

We now introduce a parameter $Q$ and an infinite set of coupling constants $\boldsymbol{t}=(t_{1},t_{2},\ldots)$, and deform (2.2) as

[TABLE]

where

[TABLE]

The external potentials $\phi_{k}(\lambda)$ are defined as

[TABLE]

The sum on the right hand side of (2.6) is a finite sum because only a finite number of $\lambda_{i}$’s are non-zero. These potentials are $q$-analogues of the so called Casimir invariants of the infinite symmetric group $S_{\infty}$, which we shall encounter in the 4D limit.

The following fact is a consequence of our previous work on the melting crystal model [18]. This fact is by no means obvious from the definition of $Z(\boldsymbol{t})$, and explained with the aid of algebraic structures hidden behind a fermionic expression of this partition function. Moreover, we shall need further refinements of this statement to consider an associated quantum spectral curve.

Theorem 1**.**

$Z(\boldsymbol{t})$* is a tau function of the KP hierarchy with time variables $\boldsymbol{t}=(t_{1},t_{2},\ldots)$.*

Actually, $Z(\boldsymbol{t})$ is a member of a set of functions $Z(\boldsymbol{t},s)$, $s\in\mathbb{Z}$, considered in our previous work:

[TABLE]

where

[TABLE]

The $s$-dependent formulation stems from a fermionic interpretation of $Z(\boldsymbol{t})$ that we shall review below. The external potentials $\phi_{k}(s,\lambda)$ are obtained by rearrangement of terms of the formal expression

[TABLE]

This explains the origin of the last term of $\phi(s,\lambda)$:

[TABLE]

It is shown in our previous work [18] that $Z(\boldsymbol{t},s)$ is essentially (i.e., up to a simple multiplier and rescaling of the time variables) a tau function of the 1D Toda hierarchy. This implies, in particular, that $Z(\boldsymbol{t},s)$ is also a collection of tau functions of the KP hierarchy labelled by $s$ [37].

2.3 Fermionic expression of partition function

Let $\psi_{n},\psi^{*}_{n}$, $n\in\mathbb{Z}$, be the creation–annihilation operators111For the sake of convenience, as in our previous work [18], we label these operators with integers rather than half-integers. The free fermion fields are defined as $\psi(z)=\sum_{n\in\mathbb{Z}}\psi_{n}z^{-n-1}$ and $\psi^{*}(z)=\sum_{n\in\mathbb{Z}}\psi^{*}_{n}z^{-n}$. of 2D charged free fermion theory with the anti-commutation relations

[TABLE]

and $|0\rangle$, $\langle 0|$, $s\in\mathbb{Z}$, the vacuum vectors of the fermionic Fock and dual Fock spaces that satisfy the vacuum conditions

[TABLE]

The charge-[math] sectors of the Fock spaces are spanned by the excited states $|\lambda\rangle$, $\langle\lambda|$, $\lambda\in\mathcal{P}$:

[TABLE]

where $n$ is chosen so that $\lambda_{i}=0$ for $i>n$. In particular, $|\emptyset\rangle$ and $\langle\emptyset|$ agree with the vacuum states. The charge-$s$ sectors of the Fock spaces are spanned by similar vectors $|s,\lambda\rangle$, $\langle s,\lambda|$, $\lambda\in\mathcal{P}$.

The fermionic expression of the aforementioned partition functions employs the normally ordered fermion bilinears

[TABLE]

the vertex operators [26, 38]

[TABLE]

and their multi-variate extensions

[TABLE]

The action of these operators on the fermionic Fock space leaves the charge-[math] sector invariant. $L_{0}$, $K$ and $H_{k}$ are diagonal with respect to $|\lambda\rangle$’s:

[TABLE]

The matrix elements of the vertex operators are the skew Schur functions $s_{\lambda/\mu}(\boldsymbol{x})$, $\boldsymbol{x}=(x_{1},x_{2},\ldots)$:

[TABLE]

One can use these building blocks to rewrite the combinatorial definition (2.5) of $Z(\boldsymbol{t})$ as

[TABLE]

where

[TABLE]

As shown in our previous work with the aid of symmetries of a quantum torus algebra [18] , (2.12) can be converted to the following form. This implies that $Z(\boldsymbol{t})$ is a tau function of the KP hierarchy.

Theorem 2**.**

[TABLE]

where

[TABLE]

One can rewrite (2.13) further to clarify its characteristic as a tau function of the KP hierarchy. Firstly, the exponential prefactor on the right hand side can be taken inside the vev as

[TABLE]

with

[TABLE]

This is a consequence of the commutation relation

[TABLE]

among $J_{n}$’s that span the $U(1)$ current (or Heisenberg) algebra. Remarkably, the operator generated in front of $g_{1}$, too, is related to a vertex operator as

[TABLE]

Thus $g_{2}$ can be expressed as

[TABLE]

Secondly, the multipliers $(-1)^{k}q^{k/2}$ of $t_{k}$’s can be removed by the scaling relation

[TABLE]

(2.15) thereby turns into the more standard expression

[TABLE]

as a tau function of the KP hierarchy [10, 17].

Remark 1*.*

As we shall see in the next section, one can simplify the operator $g_{2}$ to

[TABLE]

without changing the associated tau function of the KP hierarchy. $g_{1}$ is defined in the somewhat complicated form (2.14) to enjoy the algebraic relations

[TABLE]

These relations ensure that the associated tau function of the 2D Toda hierarchy reduces to a tau function of the 1D Toda hierarchy [18].

Remark 2*.*

In the previous work [18], we used the operator

[TABLE]

in place of $K$. Accordingly, the fermionic expression of the partition functions presented therein takes a slightly different form. This does not affect the essential part of the fermionic expression.

3 Quantum spectral curve

3.1 Single-variate specialization

Let $Z(x)$ denote the single-variate specialization of $Z(\boldsymbol{t})$ obtained by substituting

[TABLE]

The combinatorial definition (2.5) of $Z(\boldsymbol{t})$ and its fermionic expressions (2.13) and (2.15) are accordingly specialized as follows.

Lemma 1**.**

[TABLE]

Proof.

Substituting (3.1) for $\phi(\boldsymbol{t},\lambda)$ yields

[TABLE]

hence

[TABLE]

∎

Lemma 2**.**

[TABLE]

Proof.

Substituting (3.1) in (2.13) and (2.15) yields

[TABLE]

(cf. the computation in the proof of the previous lemma) and

[TABLE]

∎

As we shall see in the next section, the combinatorial expression (3.2) of $Z(x)$ has a desirable form from which one can derive the equation of quantum curve of Dunin-Barkowski et al. [6]. To apply the method of our previous work [34], however, it is more convenient to have $\Gamma_{+}(x)$ rather than $\Gamma^{\prime}_{+}(x)$ in the fermionic expression (3.3) of $Z(x)$. This problem can be settled by the following transformation rule of matrix elements of fermionic operators under conjugation of partitions [38]:

Lemma 3**.**

[TABLE]

Proof.

These identities are consequences of (2.9), (2.10), (2.11) and the following property of $\kappa(\lambda)$:

[TABLE]

∎

We can apply this rule to $\Gamma^{\prime}_{+}(x)$ and the building blocks of $g_{2}$ to rewrite (3.3) as

[TABLE]

where

[TABLE]

Having obtained the fermionic expression (3.4) containing $\Gamma_{+}(x)$, we now remove the other $\Gamma_{+}$’s from (3.4). This is the last step for applying the method of out previous work [34].

Lemma 4**.**

[TABLE]

where

[TABLE]

Proof.

Since the rightmost two factors $\Gamma^{\prime}_{+}(q^{-\rho})q^{-K/2}$ of (3.5) act on the vacuum vector trivially as

[TABLE]

one can remove these operators from (3.4). Moreover, one can use the scaling relation

[TABLE]

and the commutation relation

[TABLE]

of the vertex operators [26, 38] to rewrite the product of the three operators in the middle of (3.5) as

[TABLE]

The two factors $Q^{L_{0}}\Gamma^{\prime}_{+}(Qq^{-\rho})$ in the last line hit the vacuum vector $|0\rangle$ and disappear. What remains are the constant $\prod_{n=1}^{\infty}(1-Qq^{n})^{-n}$ and the operator $g^{\prime}$. ∎

Remark 3*.*

Note that the operator $g^{\prime}$ amounts to the operator $g$ defined in (3.7). Also note that the foregoing computations are actually a proof of the identity

[TABLE]

of vectors in the fermionic Fock space.

3.2 Generating operator of admissible basis

We now borrow the idea of generating operators from the work of Alexandrov et al. [2, 3]. A point of the Sato Grassmannian can be represented by a linear subspace $W$ of the space $V=\mathbb{C}((x))$ of formal Laurent series [28, 29]. The generating operator is a linear automorphism $G$ of $V$ that maps $W_{0}=\mathrm{Span}\{x^{-j}\}_{j=0}^{\infty}$ to $W$, so that an admissible basis $\{\Phi_{j}(x)\}_{j=0}^{\infty}$ of $W$ can be expressed as

[TABLE]

In the fermionic formalism of the KP hierarchy [10, 17], $W$ corresponds to a vector $|W\rangle$ of the fermionic Fock space. The associated tau function can be defined as

[TABLE]

Its special value at

[TABLE]

is related to the first member $\Phi_{0}(x)$ of an admissible basis of $W$ as

[TABLE]

where $C$ is a nonzero constant.

If $|W\rangle$ is generated from the vacuum vector $|0\rangle$ by an operator $g$ as

[TABLE]

and $g$ is a special operator, such as a product of vertex operators and particular diagonal operators, then one can find $G$ rather easily from $g$ by the correspondence

[TABLE]

between fermion bilinears and differential operators. This is the way how Alexandrov et al. derived the generating operator for various types of Hurwitz numbers [3]. We did similar computations for tau functions in topological string theory [34]. We apply the same method to the operator $g^{\prime}$ of (3.7).

It is now straightforward to find the generating operator $G$ of the subspace $W\subset V$ determined by the operator $g^{\prime}$ of (3.7). According to (3.12), $q^{-K/2}$ corresponds to a differential operator of infinite order:

[TABLE]

The three vertex operators in $g^{\prime}$ amount to multiplication operators:

[TABLE]

The generating operator is given by a product of these operators as follows.

Theorem 3**.**

The generating operator $G$ for the subspace $W\subset V$ determined by the operator $g^{\prime}$ of (3.7) can be expressed as

[TABLE]

3.3 Derivation of quantum spectral curve

Since the structure of the generating operator (3.13) resembles those of tau functions in topological string theory [34], we define the Kac–Schwarz operator $A$ in essentially the same form, 222This operator amounts to the inverse $A^{-1}$ of the Kac–Schwarz operator $A$ considered therein. namely,

[TABLE]

The members $\Phi_{j}(x)$ of the admissible basis (3.9) thereby satisfy the linear equations

[TABLE]

In particular, the equation

[TABLE]

for $\Phi_{0}(x)$ (equivalently, $\tau([x])$) represents the quantum spectral curve. As we show below, $A$ is a $q$-difference operator of finite order.

Lemma 5**.**

[TABLE]

Proof.

One can compute $A=G\cdot q^{D}\cdot G^{-1}$ step by step. The first step is to apply the last infinite product of (3.13) and its inverse to $q^{D}$. This can be carried out with the aid of the operator identity

[TABLE]

as follows:

[TABLE]

The next step is to apply $q^{-(D-1/2)^{2}/2}$ and its inverse to the last operator. This can be achieved by the identity

[TABLE]

as follows:

[TABLE]

The first infinite product of (3.13) and its inverse transform $q^{-D}$ and $q^{D}$ in this operator product as

[TABLE]

Thus one obtains the result shown in (3.14). ∎

Let us expand (3.14) and move $q^{\pm D}$ in each term to the right end. The outcome reads

[TABLE]

We are thus led to the following final expression of the quantum spectral curve of the melting crystal model.

Theorem 4**.**

$Z(x)$* satisfies the equation*

[TABLE]

with respect to the $q$-difference operator (3.15).

4 Prescription for 4D limit

The 4D limit of the partition function $Z(\boldsymbol{0})$ at $\boldsymbol{t}=\boldsymbol{0}$ is achieved by setting the parameters as

[TABLE]

and letting $R\to 0$ [18]. $R$ is the radius of the fifth dimension of $\mathbb{R}^{4}\times S^{1}$ in which SUSY Yang-Mills theory lives [19], $\hbar$ is a parameter of the self-dual $\Omega$ background, and $\Lambda$ is an energy scale of 4D $\mathcal{N}=2$ SUSY Yang-Mills theory [20, 21]. The definition of 4D limit of $Z(x)$ and $Z(\boldsymbol{t})$ needs $R$-dependent transformations of $x$ and $\boldsymbol{t}$.

4.1 4D limit of $Z(x)$ and quantum spectral curve

Alongside the substitution (4.1) of parameters, we transform the variable $x$ to a new variable $X$ as

[TABLE]

As it turns out below, both the combinatorial expression (3.2) and the $q$-difference equation (3.16) of $Z(x)$ behave nicely as $R\to 0$ under this $R$-dependent transformation of $x$.

Lemma 6**.**

[TABLE]

where

[TABLE]

Proof.

As $R\to 0$ under the $R$-dependent transformations (4.1) and (4.2), the building blocks of (3.2) behave as

[TABLE]

Note that the hook-length formulae (2.3) and (2.4) are used in the derivation of the first line above. ∎

Lemma 7**.**

[TABLE]

Proof.

(3.15) implies that $A-1$ can be expressed as

[TABLE]

As $R\to 0$ under the transformations (4.1) and (4.2), each term of this expression behaves as follows:

[TABLE]

∎

As a consequence of the foregoing two facts, we obtain the following difference equation for $Z_{\mathrm{4D}}(X)$.

Theorem 5**.**

$Z_{\mathrm{4D}}(X)$* satisfies the difference equation*

[TABLE]

By the shift$X\to X+\hbar$ of $X$, (4.4) turns into the equation

[TABLE]

which agrees with the equation derived by Dunin-Barkowski et al. [6]. Moreover, as they found, this equation can be converted to the simpler form

[TABLE]

by the gauge transformation

[TABLE]

where $B(t)$ is the generating function

[TABLE]

of the Bernoulli numbers. It is this equation (4.5) that is identified by Dunin-Barkowski et al. [6] as the equation of quantum spectral curve for Gromov-Witten theory of $\mathbb{C}\mathbb{P}^{1}$. Its classical limit

[TABLE]

as $\hbar\to 0$ (with $\Lambda$ normalized to $1$) is the spectral curve of topological recursion in this case [7, 8, 22]. We have thus re-derived the quantum spectral curve of $\mathbb{C}\mathbb{P}^{1}$ from the 4D limit of the melting crystal model.

4.2 4D limit of $Z(\boldsymbol{t})$

As shown in the proof of Lemma 6, the deformed Boltzmann weight $s_{\lambda}(q^{-\rho})^{2}Q^{|\lambda|}$ behaves nicely in the limit as $R\to 0$. To achieve the 4D limit of $Z(\boldsymbol{t})$, we have only to find an appropriate $R$-dependent transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ to the coupling constants $\boldsymbol{T}=(T_{1},T_{2},\ldots)$ of 4D external potentials $\phi^{\mathrm{4D}}_{k}(\lambda)$ for which the identity

[TABLE]

holds. In accordance with the commonly adopted setting in the literature [14], we wish to tune the transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ so that $\phi^{\mathrm{4D}}_{k}(\lambda)$’s take the polynomial form

[TABLE]

Thus the problem is how to derive these polynomial potentials from the exponential potentials (2.6). The following is a clue to this problem.

Lemma 8**.**

As $R\to 0$ under the transformation (4.1) of the parameters,

[TABLE]

Proof.

The difference of the two identities

[TABLE]

yields the identity

[TABLE]

One can derive (4.8) by specializing this identity to $u=\lambda_{i}-i+1$ and $v=-i+1$ and summing the outcome over $i=1,2,\ldots$. ∎

(4.8) implies the identity

[TABLE]

for $\phi_{k}(\lambda)$’s and the potentials shown in (4.7). Since

[TABLE]

one can conclude that the identity (4.6) holds if $t_{k}$’s and $T_{k}$’s are related by the linear relations

[TABLE]

This gives an $R$-dependent transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ that we have sought for. Note that this is a triangular (hence invertible) linear transformation between $\boldsymbol{t}$ and $\boldsymbol{T}$.

Let $Z_{\mathrm{4D}}(\boldsymbol{T})$ denote the deformed partition function

[TABLE]

with the external potentials (4.7). We are thus led to the following conclusion.

Theorem 6**.**

As $R\to 0$ under the $R$-dependent transformation $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{T},R)$ of the coupling constants defined by (4.10), $Z(\boldsymbol{t})$ converges to $Z_{\mathrm{4D}}(\boldsymbol{T})$:

[TABLE]

Remark 4*.*

It is easy to see that $Z_{\mathrm{4D}}(X)$ and $Z_{\mathrm{4D}}(\boldsymbol{T})$ are connected by the substitution

[TABLE]

as

[TABLE]

This fact plays a role in the next section.

Remark 5*.*

Although appears to be somewhat ad hoc, our prescription is essentially the only way to implement the 4D limit. This prescription is based on the natural relation (4.9) that connects the exponential and polynomial functions in the external potentials. The $R$-dependent transformation (4.10) of the coupling constants is reminiscent of the method of various scaling limits in statistical mechanics and quantum field theory.

5 Bilinear equations

5.1 Fay-type bilinear equations for KP hierarchy

Let us recall the notion of Fay-type bilinear equations in the theory of the KP hierarchy [1, 28, 35].

Given a general tau function $\tau(\boldsymbol{t})$, one can consider an $N$-variate generalization of (3.11):

[TABLE]

Its product with the Vandermonde determinant

[TABLE]

is the $N$-point function of the fermion field $\psi^{*}(x^{-1})$ in the background state $|W\rangle$ [10, 17].

Actually, it is more convenient to leave $\boldsymbol{t}$ as well. Let $\tau(\boldsymbol{t},x_{1},\ldots,x_{N})$ denote the function thus obtained:

[TABLE]

By virtue of the aforementioned interpretation as the $N$-point function of a fermion field, the product

[TABLE]

with the Vandermonde determinant satisfies the bilinear equations

[TABLE]

where $\widehat{x_{j}}$ means removing $x_{j}$ from the list of variables therein. As pointed out by Sato and Sato [28], these equations are avatars of the Plücker relations among the Plücker coordinates of a Grassmann manifold.

The simplest ($N=2$) case

[TABLE]

of (5.2), referred to as a Fay-type bilinear equation, is known to play a particular role. Specialized to $x_{4}=0$, this equation turns into the so called Hirota–Miwa equation

[TABLE]

Moreover, dividing this equation by $x_{3}$ and letting $x_{3}\to 0$ yield the differential Fay identity [1]

[TABLE]

where $\tau_{t_{1}}(\boldsymbol{t})$ denotes the $t_{1}$-derivative of $\tau(\boldsymbol{t})$. It is known [35] that the differential Fay identity characterizes a general tau function of the KP hierarchy in the following sense.

Theorem 7**.**

A function $\tau(\boldsymbol{t})$ of $\boldsymbol{t}=(t_{1},t_{2},\ldots)$ is a tau function of the KP hierarchy if and only if it satisfies (5.5).

As a corollary, it turns out that each of (5.3) and (5.4), too, is a necessary and sufficient condition for a function $\tau(\boldsymbol{t})$ to be a KP tau function. This fact is a clue to the subsequent consideration.

Remark 6*.*

One can rewrite (5.3) and (5.4) to the equivalent forms

[TABLE]

and

[TABLE]

5.2 Bilinear equations in melting crystal model

Let $Z(\boldsymbol{t},x_{1},\ldots,x_{N})$ denote the function

[TABLE]

This function can be obtained from $Z(\boldsymbol{t})$ by shifting $\boldsymbol{t}$ as

[TABLE]

Note that the single-variate specialization $Z(x)$ of $Z(\boldsymbol{t})$ for the quantum spectral curve coincides with $Z(\boldsymbol{0},q^{1/2}x)$. Since $Z(\boldsymbol{t})$ is a tau function of the KP hierarchy, the aforementioned bilinear equations imply, in particular, the three-term equation

[TABLE]

as a consequence of (5.6).

These bilinear equations turn out to survive the 4D limit. Let us set the parameters $q,Q$ and the coupling constants $\boldsymbol{t}$ to the $R$-dependent form shown in (4.1) and (4.10), and transform the variables $x_{1},\ldots,x_{N}$ to new variables $X_{1},\ldots,X_{N}$ as

[TABLE]

Note that this transformation takes essentially the same form as the relation (4.2) between $x$ and $X$ in the 4D limit of the quantum spectral curve. As $R\to 0$ under these $R$-dependent transformations, $Z(\boldsymbol{t},x_{1},\ldots,x_{N})$ converges to a function of the form

[TABLE]

Since the differences $x_{i}-x_{j}$ in $\Delta(x_{1},\ldots,x_{N})$ behave as

[TABLE]

the three-term bilinear equation (5.9), divided by $R^{2}$ before letting $R\to 0$, turns into the equation

[TABLE]

for $Z_{\mathrm{4D}}(\boldsymbol{T},X_{i},X_{j})$’s. The more general bilinear equations (5.2), too, have 4D counterparts.

Let us note here that $Z_{\mathrm{4D}}(\boldsymbol{T},X_{1},\ldots,X_{N})$ can be obtained from $Z_{\mathrm{4D}}(\boldsymbol{T})$ by shifting $\boldsymbol{T}$ as

[TABLE]

just as $Z_{\mathrm{4D}}(X)$ and $Z_{\mathrm{4D}}(\boldsymbol{T})$ are connected by the substitution shown in (4.13). This is the same relation as $\tau(\boldsymbol{t},x_{1},\ldots,x_{N})$ is derived from $\tau(\boldsymbol{t})$ except that $x_{j}$’s are replaced by $h/X_{j}$’s. For more precise comparison with the bilinear equations for KP tau functions, one should rewrite (5.12) as

[TABLE]

This equation literally corresponds to (5.6). According to Theorem 7, this is enough to deduce the following conclusion.

Theorem 8**.**

$Z_{\mathrm{4D}}(\boldsymbol{T})$* is a tau function of the KP hierarchy.*

5.3 Extension to Toda hierarchy

$Z_{\mathrm{4D}}(\boldsymbol{T})$ has a fermionic expression, analogous to (2.12), of the form

[TABLE]

where

[TABLE]

It is natural to extend this function to a set of functions $Z_{\mathrm{4D}}(\boldsymbol{T},s)$, $s\in\mathbb{Z}$, as

[TABLE]

where $|s\rangle$ and $\langle s|$ are the ground states of the charge-$s$ sector of the Fock spaces. These partition functions have the combinatorial expression

[TABLE]

where

[TABLE]

The external potentials are obtained by rearrangement of terms in the formal expression

[TABLE]

hence

[TABLE]

$Z_{\mathrm{4D}}(\boldsymbol{T},s)$ is known as a generating function of all-genus Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ [24], and proven to be a tau function of the 1D Toda hierarchy [5, 9, 16].

The foregoing approach to the integrable structure of $Z_{\mathrm{4D}}(\boldsymbol{T})$ can be carried over to $Z_{\mathrm{4D}}(\boldsymbol{T},s)$. Namely, one can use a Toda version [31, 36] of Fay-type bilinear equations to prove the following 333A full generating function of all-genus Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ [24] depends on another set of variables $s_{1},s_{2},\ldots$ (the decedents of $s$) as well. These variables are now set to [math], though the Toda conjecture is proven in the presence of these variables and the associated time evolutions (the extended Toda hierarchy [4]). In this sense, our proof is incomplete as an alternative proof of the Toda conjecture.:

Theorem 9**.**

$Z_{\mathrm{4D}}(\boldsymbol{T},s)$* is a tau function of the 1D Toda hierarchy.*

The proof is far more complicated than the case of $Z_{\mathrm{4D}}(\boldsymbol{T})$. We present its detail in Appendix.

6 Conclusion

Primary motivation of this work was to understand the result of Dunin-Barkowski et al. [6] in the language of the quantum spectral curve of the melting crystal model. In the course of solving this problem, we have found how to achieve the 4D limit of the deformed partition function $Z(\boldsymbol{t})$ itself. As a byproduct, this prescription for 4D limit has turned out to transfer Fay-type bilinear equations from $Z(\boldsymbol{t})$ to its 4D limit $Z_{\mathrm{4D}}(\boldsymbol{T})$.

It will be better to summarize these results from two aspects, namely, quantum curves and bilinear equations:

Quantum curves: One can derive a quantum spectral curve of the melting crystal model by the method of our work on quantum mirror curves in topological string theory [34]. This quantum curve is formulated as the $q$-difference equation (3.16) for the single-variate specialization $Z(x)$ of $Z(\boldsymbol{t})$. Its 4D limit is achieved by transforming the variable $x$ to a new variable $X$ as shown in (4.2) and letting $R\to 0$. (3.16) thereby turns into the difference equation (4.4) for the 4D version $Z_{\mathrm{4D}}(X)$ of $Z(x)$. (4.4) can be further converted to the quantum spectral curve (4.5) of Gromov-Witten theory of $\mathbb{C}\mathbb{P}^{1}$. (4.4) and (4.5) are derived by Dunin-Barkowski et al. [6] by genuinely combinatorial computations. Our approach highlights a role of the KP hierarchy that underlies these quantum curves.

• 2.

Bilinear equations: According to our previous work on the melting crystal model [18], $Z(\boldsymbol{t})$ is a tau function of the KP hierarchy. As $R\to 0$ under the $R$-dependent transformation (4.10) of the coupling constants, $Z(\boldsymbol{t})$ converges to the 4D version $Z_{\mathrm{4D}}(\boldsymbol{T})$. In this limit, the three-term bilinear equation (5.9) for $Z(\boldsymbol{t})$ turns into its counterpart (5.12) for $Z_{\mathrm{4D}}(\boldsymbol{T})$. This implies that $Z_{\mathrm{4D}}(\boldsymbol{T})$, too, is a tau function of the KP hierarchy. Thus we have obtained a new approach to the integrable structure in Okounkov and Pandharipande’s generating function of all-genus Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ [24].

As explained in Appendix, the method of 4D limit for bilinear equations can be extend to the $s$-deformed partition functions $Z(\boldsymbol{t},s)$ and $Z_{\mathrm{4D}}(\boldsymbol{T},s)$. This is a yet another proof of the fact [5, 9, 16] that $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ is a tau function of the 1D Toda hierarchy.

Let us stress that the integrable structure of $Z_{\mathrm{4D}}(\boldsymbol{T})$ still remains to be fully elucidated. Its 5D (or K-theoretic) lift $Z(\boldsymbol{t})$ has a fermionic expression, such as (2.17), that shows manifestly that $Z(\boldsymbol{t})$ is a tau function of the KP hierarchy. Moreover, since the generating operator in the fermionic expression is rather simple, one can even find the associated generating operator $G$ in $V=\mathbb{C}((x))$ explicitly. In contrast, no similar fermionic expression of $Z_{\mathrm{4D}}(\boldsymbol{T})$ is currently known. The preliminary fermionic expression (5.13) of $Z_{\mathrm{4D}}(\boldsymbol{T})$ cannot be converted to such a form by the method of our previous work [18]. The limiting procedure from $Z(\boldsymbol{t})$ is a way to overcome this difficulty, but this will not be a final answer.

An alternative approach will be the route from the equivariant Gromov-Witten theory. A generating function of the equivariant Gromov-Witten invariants of $\mathbb{C}\mathbb{P}^{1}$ is known to be a tau function of the 2D Toda hierarchy [25]. This generating function should reproduce $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ in the limit as the equivariant parameter tends to [math]. We hope to address this issue elsewhere.

Acknowledgements

The author is grateful to Toshio Nakatsu for valuable comments. This work is partly supported by the JSPS Kakenhi Grant JP25400111, JP15K04912 and JP18K03350.

Appendix A Derivation of Toda hierarchy

We prove Theorem 9 in this appendix. This proof is based on a 1D Toda version of Theorem 7. This theorem characterizes all tau functions of the 1D Toda hierarchy by two bilinear equations. The 5D partition function itself is not a genuine tau function, and satisfies slightly modified bilinear equations. In the 4D limit, these equations turn into bilinear equations for the 4D partition function. These equations agree with the bilinear equations for 1D Toda tau functions. Thus we can conclude that the 4D partition function is a tau function of the 1D Toda hierarchy.

A.1 Toda tau function in 5D partition function

Let us consider the $s$-deformed 5D partition function $Z(\boldsymbol{t},s)$ written in the fermionic form

[TABLE]

Just as the fermionic expressions (2.12) of $Z(\boldsymbol{t})$ is converted to (2.13), one can rewrite $Z(\boldsymbol{t},s)$ [18] as

[TABLE]

Moreover, one can remove the multipliers $(-1)^{k}q^{k/2}$ of $t_{k}$’s as

[TABLE]

Note that the $s$-dependent prefactors of the vevs originate in the relations

[TABLE]

Let $\tau(\boldsymbol{t},s)$ denote the vev in (A.3):

[TABLE]

As a consequence of (2.19), one can move the exponential operator of $J_{k}$’s to the right as

[TABLE]

This means that $\tau(\boldsymbol{t},s)$ is a tau function of the 1D Toda hierarchy [37]. The product of this function with the exponential function of $t_{k}$’s in (A.3), too, is a tau function of the 1D Toda hierarchy. This is, however, not the case for further multiplication by $q^{-(4s^{3}-s)/12}q^{-s(s+1)/2}$. As we show below, these two $s$-dependent prefactors modify the bilinear equations.

A.2 Bilinear equations for 5D partition function

The 1D Toda hierarchy can be characterized by the two bilinear equations shown below. This is a consequence of the characterization of the 2D Toda hierarchy by three Fay-type identities [31, 36].

Theorem 10**.**

A function $\tau(\boldsymbol{t},s)$ of $\boldsymbol{t}=(t_{1},t_{2},\ldots)$ and $s$ is a tau functions of the 1D Toda hierarchy if and only if it satisfies the following bilinear equations:

[TABLE]

The tau function (A.4) and its product with the exponential function of $t_{k}$’s in (A.3) satisfy these equations. We convert these equations to bilinear equations for $Z(\boldsymbol{t},s)$. It is convenient to separate some simple factors from $Z(\boldsymbol{t},s)$ as

[TABLE]

where

[TABLE]

Note that $\phi(\boldsymbol{t},s,\emptyset)$ consists of the “correction terms” (2.8):

[TABLE]

The relation between the tau function and the 5D partition function thereby takes such a form as

[TABLE]

Plugging this expression into (A.6) and (A.7), we obtain bilinear equations for $\tilde{Z}(\boldsymbol{t},s)$. The prefactors $q^{(4s^{3}-s)/12}$, $(qQ)^{s(s+1)/12}$ and $e^{\phi(\boldsymbol{t},s,\emptyset)}$ of $\tilde{Z}(\boldsymbol{t},s)$ yield extra factors in the bilinear equations, e.g.,

[TABLE]

In the same sense, we have the following extra factors:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus the bilinear equations for $\tilde{Z}(\boldsymbol{t},s)$ take the following form:

[TABLE]

A.3 4D limit of 5D partition function

The prescription of 4D limit for $Z(\boldsymbol{t})$ can be carried over to $\tilde{Z}(\boldsymbol{t},s)$. Note that the external potentials $\tilde{\phi}_{k}(s,\lambda)$ in $\tilde{Z}(\boldsymbol{t},s)$ are $s$-dependent analogues of $\phi_{k}(\lambda)$’s:

[TABLE]

Consequently, as $R\to 0$ under the $R$-dependent setting (4.1) of $q$ and $Q$, we have the relation

[TABLE]

where

[TABLE]

hence a meaningful limit

[TABLE]

under the same $R$-dependent parametrization (4.10) of the coupling constants as used for the 4D limit of $Z(\boldsymbol{t})$. $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ is an $s$-dependent analogue of $Z_{\mathrm{4D}}(\boldsymbol{T})$ of the form

[TABLE]

with the external potential

[TABLE]

Let us recall that the correction term (5.15) of $\phi^{\mathrm{4D}}_{k}(s,\lambda)$ is nothing but $\phi^{\mathrm{4D}}_{k}(s,\emptyset)$. In other words, we have the relation

[TABLE]

Thus $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ is related to the full 4D partition function (5.14) as

[TABLE]

just like the relation (A.8) between $Z(\boldsymbol{t},s)$ and $\tilde{Z}(\boldsymbol{t},s)$.

A.4 Bilinear equations for 4D partition function

We can derive bilinear equations for $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ from (A.10) and (A.11) by setting

[TABLE]

just like (4.2) and (5.10), and letting $R\to 0$ under the $R$-dependent parametrization (4.1), (4.10) of $q,Q$ and $\boldsymbol{t}$.

Deriving a 4D counterpart of (A.10) is straightforward. In the limit as $R\to 0$, the shifted partition functions in (A.10) turn into shifted partition functions of $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ as

[TABLE]

just as shown in the derivation of (5.11). Other simple factors in (A.10) behave as

[TABLE]

Thus we obtain the following bilinear equation:

[TABLE]

To derive a 4D counterpart of (A.11), we have to cope with the strange exponential factor therein. By its origin, this factor is related to $\phi(\boldsymbol{t},s,\emptyset)$ and $\phi(\boldsymbol{t},s\pm 1,\emptyset)$ as

[TABLE]

Consequently, upon substituting $q=e^{-R\hbar}$ and $t_{k}=t_{k}(\boldsymbol{T},R)$, we can take the limit as $R\to 0$:

[TABLE]

Thus (A.11) turns into the following bilinear equation:

[TABLE]

Since $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ and $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ are linearly related as shown in (A.15), we can translate the foregoing bilinear equations (A.16) and (A.17) for $\tilde{Z}_{\mathrm{4D}}(\boldsymbol{T},s)$ to equations for $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ in much the same way as the derivation of (A.10) and (A.11). Thus we obtain the bilinear equations

[TABLE]

and

[TABLE]

for $Z_{\mathrm{4D}}(\boldsymbol{T},s)$. By replacing $\hbar/X\to x$ and $\hbar/Y\to y$, these equations turn into a form that is identical to (A.6) and (A.7). This implies that $Z_{\mathrm{4D}}(\boldsymbol{T},s)$ is a tau function of the 1D Toda hierarchy.