Universal character, phase model and topological strings on $\pmb {\mathbb {C}^3}$

Abstract

In this paper, we consider two different subjects: the algebra of universal characters $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ (a generalization of Schur functions) and the phase model of strongly correlated bosons. We find that the two-site generalized phase model can be realized in the algebra of universal characters, and the entries in the monodromy matrix of the phase model can be represented by the vertex operators $\Gamma _i^\pm (z) (i=1,2)$ which generate universal characters. Meanwhile, we find that these vertex operators can also be used to obtain the A-model topological string partition function on $\mathbb {C}^3$.

1 Introduction

Symmetric functions were used to determine irreducible characters of highest weight representations of the classical groups [1]. The universal character, as a generalization of Schur function, describes the character of an irreducible rational representation of GL(n) [2], which upgrades that Schur function is the character of an irreducible polynomial representation of GL(n). Symmetric functions also appear in mathematical physics, especially in integrable models. The group in the Kyoto school uses Schur functions in a remarkable way to understand the KP and KdV hierarchies [3]. Tsuda defined the UC hierarchy which is a generalization of KP hierarchy and obtained that the tau functions of UC hierarchy can be realized in terms of the universal characters. He also proved that the UC hierarchy has relations with Painlevé equations [4] by similar reductions. In this paper, we consider two different subjects: the algebra of universal characters and the phase model of strongly correlated bosons.

One purpose of this paper is to give the representation of the two-site generalized phase model on the algebra of universal characters $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$. The phase model, which is the so-called crystal limit of the quantum group [5], is an integrable model and can be solved in the formulism of the quantum inverse scattering method [6]. Our results will show that the limit of the quantum inverse scattering method has an interpretation in terms of the algebra of universal characters. The crucial elements in our discussion are vertex operators $\Gamma _i^\pm (z),\ (i=1,2)$. The Fermions can be defined from these vertex operators. In the special case $\mu =\emptyset$, the universal character $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ will be reduced to the Schur function $S_{\lambda }(\mathbf{x})$, and the correspondence between vertex operators and fermions in this special case is a part of the well-known Boson-Fermion correspondence.

The relation between the algebra of Schur functions and the phase model is known from [7, 8]. There is the following isometry between states in the phase model and Schur functions

$$\begin{aligned} \bigotimes _{i=0}^{M} |n_i\rangle _i\mapsto S_{\lambda }(\mathbf{x}),\quad \lambda =1^{n_1}2^{n_2}\ldots . \end{aligned}$$

The actions of the entries in monodromy matrix T(u) on Schur function are obtained from the truncated expansions of the vertex operators

$$\begin{aligned} \tilde{\Gamma }^+(z)=e^{\xi (\mathbf{x},z)},\quad \tilde{\Gamma }^-(z)=e^{\xi (\tilde{\partial }_\mathbf{x},z^{-1})} \end{aligned}$$

where $\xi (\mathbf{x},z)=\sum _{n=1}^\infty x_n z^n$ and

$$\begin{aligned} \tilde{\partial }_\mathbf{x}=\left( \partial _{x_1}, \frac{1}{2}\partial _{x_2},\frac{1}{3}\partial _{x_3},\ldots \right) , \quad \partial _{x_i}=\frac{\partial }{\partial _{x_i}}. \end{aligned}$$

In this paper, we generalize vertex operators to the type

$$\begin{aligned} \Gamma _1^-(z)= & {} e^{\xi (\mathbf{x}-\tilde{\partial }_\mathbf{y}, z)},\quad \Gamma _1^+(z)=e^{\xi (\tilde{\partial }_\mathbf{x}, z^{-1})},\\ \Gamma _2^-(z)= & {} e^{\xi (\mathbf{y}-\tilde{\partial }_\mathbf{x}, z)},\quad \Gamma _2^+(z)=e^{\xi (\tilde{\partial }_\mathbf{y}, z^{-1})}, \end{aligned}$$

these vertex operators can generate universal characters and fermions can be defined from them. We define the map from the state in the two-site generalized phase model to the universal character by

$$\begin{aligned} \bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)}\mapsto S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y}) \end{aligned}$$

with

$$\begin{aligned} \lambda =1^{n_1}2^{n_2}\ldots ,\quad \mu =1^{m_1}2^{m_2}\ldots . \end{aligned}$$

The actions of creation operators $B_1(u)$ and $B_2(u)$, which are entries in the monodromy matrix T(u) of the phase model, on universal characters are determined by the truncated expansions of $\Gamma _1^-(z)$ and $\Gamma _2^-(z)$ respectively, and the annihilation operators $C_1(u)$ and $C_2(u)$ (multiplied by a coefficient) are adjoint to the operators $B_1(u)$ and $B_2(u)$, and determined by the truncated expansions of $\Gamma _1^+(z)$ and $\Gamma _2^+(z)$ respectively.

Another purpose of this paper is to discuss the relations between the vertex operators $\Gamma _i^-(z)$, $\Gamma _i^+(z)$ $(i=1,2)$ and the MacMahon functions. It is known that the A-model topological string partition function $Z_{\mathbb {C}^{3}}^{top}$ on $\mathbb {C}^3$ can be written as a Fermionic correlator involving the vertex operators $\tilde{\Gamma }^+(z)$ and $\tilde{\Gamma }^-(z)$ with a particular specialization of the values of $z=q^{\pm 1/2},q^{\pm 3/2},q^{\pm 5/2}\cdots$. In this paper, we will give that $Z_{\mathbb {C}^{3}}^{top}$ can also be obtained from the vertex operators $\Gamma _i^-(z)$ and $\Gamma _i^+(z)$ $(i=1,2)$ with the same specialization of the values of z.

The paper is organized as follows. In Sect. 2, we recall the definition of universal character and its vertex operator realization, then we give the actions of the vertex operators on $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ which is helpful for our discussion. In Sect. 3, we recall the phase model. In Sect. 4, we define the representation of the two-site generalized phase model on the algebra of universal characters, and we find that the actions of the entries in monodromy matrix on universal characters are obtained from the truncated expansions of the vertex operators discussed in Sect. 2. In Sect. 5, we give that the MacMahon function can be obtained from these vertex operators.

2 Universal characters and vertex operators

In this section, we review the definition of universal character and its vertex operator realization as in papers [2, 4], then we review the definition of the UC hierarchy as in paper  [4]. We also give a new proposition in this section which is helpful for the discussion in Sect. 4. Let $\mathbf{x}=(x_1,x_2,\ldots )$ and $\mathbf{y}=(y_1,y_2,\ldots )$. The operators $h_n(\mathbf{x})$ are determined by the generating function:

$$\begin{aligned} \sum _{n=0}^\infty h_n(\mathbf{x})z^n=e^{\xi (\mathbf{x},z)},\quad \xi (\mathbf{x},z)=\sum _{n=1}^\infty x_n z^n \end{aligned}$$

(1)

and set $h_n(\mathbf{x})=0$ for $n<0$. The operators $h_n(\mathbf{x})$ can be explicitly written as

$$\begin{aligned} h_n(\mathbf{x})=\sum _{k_1+2k_2+\cdots nk_n=n}\frac{x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}}{k_1!k_2!\cdots k_n!}. \end{aligned}$$

Note that $h_n(\mathbf{x})$ is the complete homogeneous symmetric function of $t_1,t_2,\ldots$ if we replace $ix_i$ with the power sum $p_i=\sum _{j=1}^\infty t_j^i$.

For a pair of Young diagrams $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and $\mu =(\mu _1,\mu _2,\ldots ,\mu _{l'})$, the universal character $S_{[\lambda ,\mu ]}=S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ is a polynomial of variables $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb {C}[\mathbf{x},\mathbf{y}]$ defined by the twisted Jacobi–Trudi formula [2]:

$$\begin{aligned}&S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})\nonumber \\&\quad =\text {det}\left( \begin{array}{ll} h_{\mu _{l'-i+1}+i-j}(\mathbf{y}), &{}\quad 1\le i\le l' \\ h_{\lambda _{i-l'}-i+j}(\mathbf{x}), &{}\quad l'+1\le i\le l+l' \end{array} \right) _{1\le i,j\le l+l'}. \end{aligned}$$

(2)

Define the degree of each variables $x_n, y_n, \ n=1,2,\ldots$ by

$$\begin{aligned} \text {deg }x_n=n,\quad \text {deg }y_n=-n \end{aligned}$$

then $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ is a homogeneous polynomial of degree $|\lambda |-|\mu |$, where $|\lambda |=\lambda _1+\lambda _2+\cdots +\lambda _l$ is called the weight of $\lambda$. Note that $S_\lambda (\mathbf{x})$ is a special case of the universal character: $S_\lambda (\mathbf{x})=\det (h_{\lambda _i-i+j}(\mathbf{x}))=S_{[\lambda ,\emptyset ]}(\mathbf{x},\mathbf{y})$.

Introduce the following vertex operators

$$\begin{aligned} \Gamma _1^-(z)= & {} e^{\xi (\mathbf{x}-\tilde{\partial }_\mathbf{y}, z)},\quad \Gamma _1^+(z)=e^{\xi (\tilde{\partial }_\mathbf{x}, z^{-1})}, \end{aligned}$$

(3)

$$\begin{aligned} \Gamma _2^-(z)= & {} e^{\xi (\mathbf{y}-\tilde{\partial }_\mathbf{x}, z)},\quad \Gamma _2^+(z)=e^{\xi (\tilde{\partial }_\mathbf{y}, z^{-1})}. \end{aligned}$$

(4)

Define

$$\begin{aligned} X^\pm (z)= & {} \sum _{n\in \mathbb {Z}}X^\pm _nz^n=e^{\pm \xi (\mathbf{x}-\tilde{\partial }_\mathbf{y}, z)}e^{\mp \xi (\tilde{\partial }_\mathbf{x}, z^{-1})}, \end{aligned}$$

(5)

$$\begin{aligned} Y^\pm (z^{-1})= & {} \sum _{n\in \mathbb {Z}}Y^\pm _nz^{-n}=e^{\pm \xi (\mathbf{y}-\tilde{\partial }_\mathbf{x}, z^{-1})}e^{\mp \xi (\tilde{\partial }_\mathbf{y}, z)}. \end{aligned}$$

(6)

The operators $X_i^\pm$ satisfy the following Fermionic relations:

$$\begin{aligned} X_i^\pm X_j^\pm +X_{j-1}^\pm X_{i+1}^{\pm }= & {} 0,\\ X_i^+X_j^-+X_{j+1}^{-}X_{i-1}^+= & {} \delta _{i+j,0}. \end{aligned}$$

The same relations hold also for $Y_i^\pm$, and $X_i^\pm$ and $Y_i^\pm$ are commutative. The operators $X_i^+$ and $Y_i^+$ are raising operators for the universal characters such that

$$\begin{aligned} S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})=X_{\lambda _1}^+\cdots X_{\lambda _l}^+Y_{\mu _1}^+\cdots X_{\mu _{l'}}^+\cdot 1 \end{aligned}$$

(7)

where the Young diagrams $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _l)$ and $\mu =(\mu _1,\mu _2,\ldots ,\mu _{l'})$.

For an unknown function $\tau =\tau (\mathbf{x},\mathbf{y})$, the bilinear relations [4]

$$\begin{aligned} \sum _{i+j=-1}X_i^-\tau \otimes X_j^+\tau =\sum _{i+j=-1}Y_i^-\tau \otimes Y_j^+\tau =0 \end{aligned}$$

(8)

is called the UC hierarchy.

It turns out that $\tau (\mathbf{x},\mathbf{y})$ equals

$$\begin{aligned} \tau (\mathbf{x},\mathbf{y})=\tau _1(\mathbf{x}-\tilde{\partial }_\mathbf{y})\tau _2(\mathbf{y}-\tilde{\partial }_\mathbf{x})\cdot 1 \end{aligned}$$

where $\tau _1(\mathbf{x})$ and $\tau _2(\mathbf{x})$ are tau functions of KP hierarchy. In special case, the universal character $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ is also the solution of UC hierarchy, and

$$\begin{aligned} S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})=S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y})S_\mu (\mathbf{y}-\tilde{\partial }_\mathbf{x})\cdot 1. \end{aligned}$$

(9)

From this, we get the conclusion which is helpful for the following discussion.

Proposition 2.1

The vertex operators in (3) and (4) act on the universal characters $S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})$ in the following way,

$$\begin{aligned} \Gamma _1^-(z)S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})= & {} \sum _{n=0}^\infty z^n S_{[\lambda \cdot (n),\mu ]}(\mathbf{x},\mathbf{y})\nonumber \\= & {} \sum _{\nu \succ \lambda }z^{|\nu |-|\lambda |} S_{[\nu ,\mu ]}(\mathbf{x},\mathbf{y}), \end{aligned}$$

(10)

$$\begin{aligned} \Gamma _2^-(z)S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})= & {} \sum _{n=0}^\infty z^n S_{[\lambda ,\mu \cdot (n)]}(\mathbf{x},\mathbf{y})\nonumber \\= & {} \sum _{\nu \succ \mu }z^{|\nu |-|\mu |} S_{[\lambda ,\nu ]}(\mathbf{x},\mathbf{y}), \end{aligned}$$

(11)

where the multiplication of two Young diagram $\lambda$ and (n) satisfies the Pieri formula [9, 10], and $\lambda \succ \mu$ means that the Young diagrams $\lambda$ and $\mu$ are interlaced, in the sense of $\lambda _1\ge \mu _1\ge \lambda _2\ge \mu _2\ge \cdots$. The operators $\Gamma _1^+(z)$ and $\Gamma _2^+(z)$ are adjoint to $\Gamma _1^-(z)$ and $\Gamma _2^-(z)$ respectively, that is,

$$\begin{aligned} \Gamma _1^+(z)S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})= & {} \sum _{n=0}^\infty z^n S_{[\lambda / (n),\mu ]}(\mathbf{x},\mathbf{y})\nonumber \\= & {} \sum _{\lambda \succ \nu }z^{|\lambda |-|\nu |} S_{[\nu ,\mu ]}(\mathbf{x},\mathbf{y}), \end{aligned}$$

(12)

$$\begin{aligned} \Gamma _2^+(z)S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})= & {} \sum _{n=0}^\infty z^n S_{[\lambda ,\mu /(n)]}(\mathbf{x},\mathbf{y})\nonumber \\= & {} \sum _{\mu \succ \nu }z^{|\mu |-|\nu |} S_{[\lambda ,\nu ]}(\mathbf{x},\mathbf{y}), \end{aligned}$$

(13)

where $\lambda /\mu$ denotes a skew diagram.

3 Phase model

In this section, we review the definition of phase model as in papers [7, 8] and no new result is given. We begin with a bosonic system based on the following algebra [7, 8]

$$\begin{aligned}{}[N,\phi ] = - \phi ,\quad [N,\phi ^{\dag }] = \phi ^{\dag },\quad [\phi ,\phi ^{\dag }] = \pi , \end{aligned}$$

(14)

where $\pi =|0\rangle \langle 0|$ is the vacuum projection. The operator $\phi$ is one-sided isometry

$$\begin{aligned} \phi \phi ^{\dag } = 1,\qquad \phi ^{\dag } \phi = 1-\pi . \end{aligned}$$

This algebra can be represented in the Fock space $\mathcal {F}$ consisting of n-particle states $|n\rangle$, the operators $\phi$, $\phi ^\dag$ and N acting as the phase operators and the number of particles operator, respectively,

$$\begin{aligned}&\phi ^{\dag }|n\rangle =|n+1\rangle ,\quad \phi |n\rangle = |n-1\rangle , \quad \\&\phi |0\rangle = 0, \quad N|n\rangle = n|n\rangle , \end{aligned}$$

where $|0\rangle$ is the vacuum state, the special case $n=0$ of the n particle state.

Let the tensor product

$$\begin{aligned} \mathcal {F} = \bigotimes _{i=0}^M \mathcal {F}_i, \end{aligned}$$

(15)

be $M+1$ copies of the Fock space. Denote by $\phi _i, \phi ^{\dag }_i, N_i$ the operators that act as $\phi , \phi ^{\dag }, N$ in (14), respectively, in the ith space and identically in the other spaces.

The phase model is a model of a periodic chain with the hamiltonian  [11,12,13]

$$\begin{aligned} H = -\frac{1}{2} \sum _{i=0}^M \big (\phi ^{\dag }_i \phi _{i+1} + \phi _i \phi ^{\dag }_{i+1} -2N_i \big ). \end{aligned}$$

(16)

Define the operator of the total number of particles by

$$\begin{aligned} \hat{N} = \sum _{i=0}^M N_i. \end{aligned}$$

Then the N-particle vectors in this space are of the form

$$\begin{aligned} \bigotimes _{i=0}^M |n_i\rangle _i,\qquad \text {where} \ |n_i\rangle _i = (\phi _i^{\dag })^{n_i}|0\rangle _i,\quad N=\sum _{i=0}^M n_i, \end{aligned}$$

(17)

the numbers $n_i$ are called the occupation numbers of the state (17).

From [6], we know that the phase model is integrable. Introduce the L-matrix

$$\begin{aligned} L_i(u) = \left( \begin{array}{ll} u^{-1} &{}\quad \phi ^{\dag }_i \\ \phi _i &{}\quad u \end{array} \right) , \qquad i=0,\ldots , M, \end{aligned}$$

where u is a scalar parameter, here we treat u as uI with I being the identity operator in $\mathcal {F}$. For every $i=0,\ldots , M$, the L-matrix satisfies the bilinear equation

$$\begin{aligned} R(u,v)\big (L_i(u)\otimes L_i(v) \big ) = \big (L_i(v)\otimes L_i(u) \big ) R(u,v), \end{aligned}$$

(18)

where R-matrix R(u, v) is a $4\times 4$ matrix given by

$$\begin{aligned} R(u,v) = \left( \begin{array}{cccc} f(v,u) &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad g(v,u) &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad g(v,u) &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad f(v,u) \end{array} \right) , \end{aligned}$$

(19)

with

$$\begin{aligned} f(v,u) = \frac{u^2}{u^2-v^2},\quad g(v,u) = \frac{uv}{u^2-v^2}. \end{aligned}$$

Define the monodromy matrix by

$$\begin{aligned} T(u) = L_M(u) L_{M-1}(u)\cdots L_0(u), \end{aligned}$$

which gives the solution of the phase model. It also satisfies the bilinear equation

$$\begin{aligned} R(u,v)\big (T(u)\otimes T(v) \big ) = \big (T(v)\otimes T(u) \big ) R(u,v). \end{aligned}$$

(20)

Let

$$\begin{aligned} T(u) = \left( \begin{array}{cc} A(u) &{}\quad B(u) \\ C(u) &{}\quad D(u) \end{array} \right) , \end{aligned}$$

(21)

we have

$$\begin{aligned} \hat{N} B(u) = B(u) (\hat{N}+1), \qquad \hat{N} C(u) = C(u) (\hat{N}-1). \end{aligned}$$

(22)

Therefore, we call B(u) the creation operator and C(u) the annihilation operator. The operators A(u) and D(u) do not change the total number of particles.

Denote by $|0\rangle _j$ the vacuum vector in $\mathcal {F}_j$ and by $|0\rangle =\otimes _{i=0}^M |0\rangle _i$. Let

$$\begin{aligned} |\Psi (u_1,\ldots ,u_N) \rangle = \prod _{i=1}^N B(u_j) |0\rangle , \end{aligned}$$

(23)

which is a N particle state.

According to [7, 8], there is the following isometry between the states (17) and the Schur functions

$$\begin{aligned} \bigotimes _{i=0}^{M} |n_i\rangle _i\mapsto S_{\lambda }(\mathbf{x}),\quad \lambda =1^{n_1}2^{n_2}\ldots , \end{aligned}$$

(24)

and the operator B(u) acts on Schur functions as the operator of multiplication by $u^MH_M(u^2)$, where $H_M(t)=\sum _{k=0}^Mt^kh_k$ is the truncated generating function of the complete homogeneous symmetric functions $h_k$. Then the state $|\Psi (u_1,\ldots ,u_N) \rangle$ has the following expansion

$$\begin{aligned}&|\Psi (u_1,\ldots ,u_N) \rangle =\sum _\lambda S_\lambda (u_1^2,\ldots ,u_N^2)\bigotimes _{i=0}^M |n_i\rangle _i,\nonumber \\&\quad \lambda =1^{n_1}2^{n_2}\ldots . \end{aligned}$$

(25)

In the following, we will generalize this work [7, 8] to realize the two-site generalized phase model in the algebra of universal characters, and the entries in the monodromy matrix can be obtained from the vertex operators which generate the universal characters.

4 Two-site generalized phase model and universal characters

In this section, we will give the representation of the two-site generalized phase model on the algebra of universal characters. All the results in this section are new. Now fix two positive integers $M_1,M_2$ and consider the tensor products

$$\begin{aligned} \mathcal {F}^{(1)} = \bigotimes _{i=0}^{M_1} \mathcal {F}_i^{(1)}\quad \quad \mathcal {F}^{(2)} = \bigotimes _{i=0}^{M_2} \mathcal {F}_i^{(2)}, \end{aligned}$$

(26)

which $\mathcal {F}^{(1)}$ and $\mathcal {F}^{(2)}$ are $M_1+1$ and $M_2+1$ copies of the Fock space respectively. Denote by $\phi _i^{(1)}, \phi ^{(1)\dag }_i, N_i^{(1)}$ the operators that act as $\phi , \phi ^{\dag }, N$ in (14), respectively, in the ith space $\mathcal {F}_i^{(1)}$ and identically in the other spaces of $\mathcal {F}^{(1)}$ and in all spaces of $\mathcal {F}^{(2)}$, and denote by $\phi _i^{(2)}, \phi ^{(2)\dag }_i, N_i^{(2)}$ the operators that act as $\phi , \phi ^{\dag }, N$, respectively, in the ith space $\mathcal {F}_i^{(2)}$ and identically in the other spaces of $\mathcal {F}^{(2)}$ and in all spaces of $\mathcal {F}^{(1)}$. Let

$$\begin{aligned} \mathcal {F}=\mathcal {F}^{(1)}\bigotimes \mathcal {F}^{(2)} \end{aligned}$$

(27)

so that $\mathcal {F}$ is $M_1+M_2+2$ copies of the Fock space. Denote by $|0\rangle _j^{(i)}$ the vacuum vector in $\mathcal {F}_j^{(i)}$ with $i=1,2.$

Define the operators

$$\begin{aligned} \hat{N}_1 = \sum _{i=0}^{M_1} N_i^{(1)}, \quad \hat{N}_2 = \sum _{i=0}^{M_2} N_i^{(2)},\quad \text {and} \quad \hat{N}=\hat{N}_1+\hat{N}_2. \end{aligned}$$

The $(N_1,N_2)$-particle vectors in space $\mathcal {F}$ are of the form

$$\begin{aligned}&\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)},\qquad \text {with}\quad N_1=\sum _{i=0}^{M_1} n_i, \nonumber \\&\quad N_2=\sum _{i=0}^{M_2} m_i, \end{aligned}$$

(28)

where

$$\begin{aligned} |n_i\rangle _i^{(1)}= (\phi _i^{(1)\dag })^{(n_i)}|0\rangle _j^{(1)},\quad |m_i\rangle _i^{(2)}= (\phi _i^{(2)\dag })^{(m_i)}|0\rangle _j^{(2)}. \end{aligned}$$

Define the map $\jmath : \mathcal {F}\rightarrow \mathbb {C}[\mathbf{x}, \mathbf{y}]$ by

$$\begin{aligned} \jmath (\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)})= S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y}) \end{aligned}$$

(29)

with

$$\begin{aligned} \lambda =1^{n_1}2^{n_2}\ldots ,\quad \mu =1^{m_1}2^{m_2}\ldots . \end{aligned}$$

(30)

In fact, this association is not quite unique: partitions $\lambda ,\ \mu$ themselves do not know about numbers $n_0$ and $m_0$ of particles. Nonetheless, if we fix the total numbers of particles $N_1$ and $N_2$, we can deduce $n_0=N_1-l(\lambda )$ and $m_0=N_2-l(\mu )$, where $l(\lambda )$ is the length of partition $\lambda$, that is, the number of rows in $\lambda$. Note that the correspondence (7) in [8] is a special case of the map $\jmath$ defined above.

The L-matrices are

$$\begin{aligned} L^{(1)}_i(u)= & {} \left( \begin{array}{cc} u^{-1} &{}\quad \phi ^{(1)\dag }_i \\ \phi _i^{(1)} &{}\quad u \end{array} \right) , \qquad i=0,\ldots , M_1,\\ L^{(2)}_i(u)= & {} \left( \begin{array}{cc} u^{-1} &{}\quad \phi ^{(2)\dag }_i \\ \phi _i^{(2)} &{}\quad u \end{array} \right) , \qquad i=0,\ldots , M_2, \end{aligned}$$

and the monodromy matrix is

$$\begin{aligned} T(u)= & {} L^{(2)}_{M_2}(u) \cdots L^{(2)}_0(u)L^{(1)}_{M_1}(u) \cdots L^{(1)}_0(u). \end{aligned}$$

Each L-matrix, as well as the monodromy matrix, satisfies the bilinear equation again

$$\begin{aligned} R(u,v)\big (L^{(j)}_i(u)\otimes L^{(j)}_i(v) \big )= & {} \big (L^{(j)}_i(v)\otimes L^{(j)}_i(u) \big ) R(u,v),\nonumber \\&j=1,2,\nonumber \\ R(u,v)\big (T(u)\otimes T(v) \big )= & {} \big (T(v)\otimes T(u) \big ) R(u,v), \end{aligned}$$

(31)

with the same R-matrix in (19).

Let

$$\begin{aligned} T_i(u)= & {} L^{(i)}_{M_i}(u)L^{(i)}_{M_i-1}(u) \cdots L^{(1)}_0(u)\end{aligned}$$

(32)

$$\begin{aligned}= & {} \left( \begin{array}{cc} A_i(u) &{}\quad B_i(u) \\ C_i(u) &{}\quad D_i(u) \end{array} \right) ,\quad i=1,2. \end{aligned}$$

(33)

The operators $B_1(u)$ and $B_2(u)$ are called the creation operators and $C_1(u)$ and $C_2(u)$ the annihilation operators, in the sense that they increase and decrease the total numbers of particles

$$\begin{aligned} \hat{N}_i B_i(u)&= B_i(u) (\hat{N}_i+1), \qquad \hat{N}_i C_i(u) = C_i(u) (\hat{N}_i-1) \nonumber \\&\text {for}\quad i=1,2. \end{aligned}$$

(34)

The operators $A_i(u)$ and $D_i(u)\ (i=1,2)$ do not change the total number of particles.

We call ith Fock space $\mathcal {F}_i^{(j)}\ (j=1,2)$ the i-energy space. Note that the correspondence (29) does not take into account the numbers $n_0,\ m_0$ of zero-energy particles, Therefore (29) gives a representation of the positive-energy space

$$\begin{aligned} \hat{\mathcal {F}}=\bigotimes _{i=1}^{M_1} \mathcal {F}_i^{(1)}\bigotimes \bigotimes _{i=1}^{M_2} \mathcal {F}_i^{(2)}, \end{aligned}$$

in the algebra of symmetric functions $\mathbb {C}[\mathbf{x},\mathbf{y}]$, in fact in its subspace $\mathbb {C}_{M_1,M_2}[\mathbf{x},\mathbf{y}]$ generated by universal characters $S_{[\lambda ,\mu ]}$ where the Young diagrams $\lambda$ have at most $M_1$ columns and $\mu$ at most $M_2$ columns. From the definition of the twisted Jacobi–Trudi formula, we can define this subspace by supposing

$$\begin{aligned}&h_{M_1+1}(\mathbf{x})=h_{M_1+2}(\mathbf{x})=\cdots =0,\quad \nonumber \\&h_{M_2+1}(\mathbf{y})=h_{M_2+2}(\mathbf{y})=\cdots =0. \end{aligned}$$

By the definition of $\hat{N}_1,\ \hat{N}_2$, the space $\mathcal {F}$ has a decomposition into $(N_1,N_2)$-particle subspaces $\mathcal {F}^{N_1,N_2}$, i.e.,

$$\begin{aligned} \mathcal {F}=\bigoplus _{N_1,N_2\ge 0}\mathcal {F}^{N_1,N_2}=\bigoplus _{N_1,N_2\ge 0}\mathcal {F}_1^{N_1}\otimes \mathcal {F}_2^{N_2}. \end{aligned}$$

(35)

Under the map (29), the subspace $\mathcal {F}^{N_1,N_2}$ corresponds to $\mathbb {C}_{M_1,M_2}^{N_1,N_2}[\mathbf{x},\mathbf{y}]$ which is spanned by universal characters $S_{[\lambda ,\mu ]}$ whose diagrams $\lambda$ lie in the $N_1\times M_1$ box and $\mu$ lie in the $N_2\times M_2$ box. That Young diagram $\lambda$ lies in the $N\times M$ box means $\lambda$ has at most N rows and at most M columns.

Define the projection $P:\mathcal {F}\rightarrow \hat{\mathcal {F}}$ by forgetting the zero energy states, and define the operator $\mathcal {B}_i(u):=PB_i(u)P$ for $i=1,2$. Then $\mathcal {B}_i(u)$ are operators acting on the space $\hat{\mathcal {F}}\cong \mathbb {C}_{M_1,M_2}[\mathbf{x},\mathbf{y}]$. Since $B_i(u)$ are creation operators, then $\mathcal {B}_1(u)$ sends $\mathbb {C}_{M_1,M_2}^{N_1,N_2}[\mathbf{x},\mathbf{y}]$ to $\mathbb {C}_{M_1,M_2}^{N_1+1,N_2}[\mathbf{x},\mathbf{y}]$ and $\mathcal {B}_2(u)$ sends $\mathbb {C}_{M_1,M_2}^{N_1,N_2}[\mathbf{x},\mathbf{y}]$ to $\mathbb {C}_{M_1,M_2}^{N_1,N_2+1}[\mathbf{x},\mathbf{y}]$. In the following, we discuss the actions of $\mathcal {B}_i(u)$ on $\mathbb {C}_{M_1,M_2}[\mathbf{x},\mathbf{y}]$. Define $\tilde{\mathcal {B}}_i(u)$ by $\mathcal {B}_i(u)=u^{-M_i}\tilde{\mathcal {B}}_i(u)$ for $i=1,2$. Then we can derive the following proposition.

Proposition 4.1

In the space $\mathbb {C}_{M_1,M_2}[\mathbf{x},\mathbf{y}]$,

$$\begin{aligned} \tilde{\mathcal {B}}_1(u)= & {} H_{M_1}(\mathbf{x}-\tilde{\partial }_\mathbf{y},u^2), \end{aligned}$$

(36)

$$\begin{aligned} \tilde{\mathcal {B}}_2(u)= & {} H_{M_2}(\mathbf{y}-\tilde{\partial }_\mathbf{x},u^2), \end{aligned}$$

(37)

where $H_n(\mathbf{x},t)=\sum _{k=0}^n t^kh_k(\mathbf{x})$, and $h_k(\mathbf{x})$ is defined in (1), which in fact is the complete homogeneous symmetric function.

To prove this proposition, we need the following lemma.

Lemma 4.2

For any Schur functions $S_\lambda$ and complete symmetric function $h_k$, we have

$$\begin{aligned} h_k(\mathbf{y}-\tilde{\partial }_\mathbf{x})S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y})=S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y})h_k(\mathbf{y}-\tilde{\partial }_\mathbf{x}). \end{aligned}$$

(38)

Proof

This holds since $y_n-\frac{1}{n}\partial _{x_n}$ and $x_m-\frac{1}{m}\partial _{y_m}$ are commutative. $\square$

The proof of Proposition 4.1.

Proof

We know that

$$\begin{aligned} S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y})=S_{\lambda }(\mathbf{x}-\tilde{\partial }_\mathbf{y})S_\mu (\mathbf{y}-\tilde{\partial }_\mathbf{x})\cdot 1, \end{aligned}$$

and in [7, 8]

$$\begin{aligned} \tilde{\mathcal {B}}(u)\bigotimes _{i=0}^M |n_i\rangle _i=\sum _{k=0}^Mu^{2k}h_k S_\lambda , \quad \lambda =1^{n_1}2^{n_2}\ldots . \end{aligned}$$

Here the mapping sign $\jmath$ is omitted at the left of the equation and we will do the same in the following. Hence,

$$\begin{aligned}&\tilde{\mathcal {B}}_1(u)\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)}\\&\quad = \left( \tilde{\mathcal {B}}_1(u)\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)} \right) \bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)}\\&\quad = \sum _{k=0}^{M_1}u^{2k}h_k(\mathbf{x}-\tilde{\partial }_\mathbf{y}) S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y})S_\mu (\mathbf{y}-\tilde{\partial }_\mathbf{x})\cdot 1, \end{aligned}$$

and

$$\begin{aligned}&\tilde{\mathcal {B}}_2(u)\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)}\\&\quad =\bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \left( \tilde{\mathcal {B}}_2(u)\bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)} \right) \\&\quad =S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y}) \left( \sum _{k=0}^{M_2}u^{2k}h_k(\mathbf{y}-\tilde{\partial }_\mathbf{x})S_\mu (\mathbf{y}-\tilde{\partial }_\mathbf{x}) \right) \cdot 1\\&\quad = \sum _{k=0}^{M_2}u^{2k}h_k(\mathbf{y}-\tilde{\partial }_\mathbf{x}) S_\lambda (\mathbf{x}-\tilde{\partial }_\mathbf{y})S_\mu (\mathbf{y}-\tilde{\partial }_\mathbf{x})\cdot 1. \end{aligned}$$

$\square$

Note that the truncated generating function $H_n(\mathbf{x},t)=\sum _{k=0}^n t^kh_k(\mathbf{x})$ can also be regarded as the full generating function $H(\mathbf{x},t)=\sum _{k=0}^\infty t^kh_k(\mathbf{x})$ under the specialization: $H_n(\mathbf{x},t)=H(\mathbf{x},t)|_{h_{n+1}(\mathbf{x})=h_{n+2}(\mathbf{x})=\ldots =0}$. Then we get the following corollary.

Corollary 4.3

In the $M_1,M_2\rightarrow \infty$ limit, the actions of the creation operators $\tilde{\mathcal {B}}_1(u)$ and $\tilde{\mathcal {B}}_2(u)$ on the space $\hat{\mathcal {F}}$ correspond to the multiplications of $H(\mathbf{x}-\tilde{\partial }_\mathbf{y},u^2)$ and $H(\mathbf{y}-\tilde{\partial }_\mathbf{x},u^2)$ on the space $\mathbb {C}[\mathbf{x},\mathbf{y}]$ respectively.

Define

$$\begin{aligned} |\tilde{\Psi }_N(u_1,\ldots ,u_N)\rangle :=\prod _{j=1}^NB_2(u_j)B_1(u_j)|0\rangle \end{aligned}$$

(39)

then we obtain the following proposition.

Proposition 4.4

The expansion of the (N, N)-particle vector (39) in terms of basis vector (28) is given by the formula

$$\begin{aligned}&|\tilde{\Psi }_N(u_1,\ldots ,u_N)\rangle \\&\quad =(u_1\cdots u_N)^{-M_1-M_2}\sum _{\lambda ,\mu }S_\lambda (u_1^2,\ldots ,u_N^2)S_\mu (u_1^2,\ldots ,u_N^2)\\&\qquad \bigotimes _{i=0}^{M_1} |n_i\rangle _i^{(1)}\bigotimes \bigotimes _{i=0}^{M_2} |m_i\rangle _i^{(2)}\\&\quad =(u_1\cdots u_N)^{-M_1-M_2}\sum _{\lambda ,\mu }S_\lambda (u_1^2,\ldots ,u_N^2)S_\mu (u_1^2,\ldots ,u_N^2)\\&\qquad S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y}) \end{aligned}$$

where the sum is over Young diagrams $\lambda$ with at most N rows and at most $M_1$ columns, Young diagrams $\mu$ with at most N rows and at most $M_2$ columns.

Proof

The operators $B_1(u)$ and $B_2(u)$ are commutative.

$$\begin{aligned} \prod _{j=1}^NB_1(u_j)= & {} (u_1\cdots u_N)^{-M_1}\sum _{\lambda }S_{\lambda }(u_1^2,\ldots ,u_N^2)S_\lambda (\mathbf{x}-\tilde{ \partial }_\mathbf{y}) \nonumber \\\end{aligned}$$

(40)

$$\begin{aligned} \prod _{j=1}^NB_2(u_j)= & {} (u_1\cdots u_N)^{-M_2}\sum _{\mu }S_{\mu }(u_1^2,\ldots ,u_N^2)S_\mu (\mathbf{y}-\tilde{ \partial }_\mathbf{x})\nonumber \\ \end{aligned}$$

(41)

then we have

$$\begin{aligned}&|\tilde{\Psi }_N(u_1,\ldots ,u_N)\rangle \nonumber \\&\quad =(u_1\cdots u_N)^{-M_1-M_2}\sum _{\lambda ,\mu }S_\lambda (u_1^2,\ldots ,u_N^2)S_\mu (u_1^2,\ldots ,u_N^2)\nonumber \\&\qquad S_\lambda (\mathbf{x}-\tilde{ \partial }_\mathbf{y}) S_\mu (\mathbf{y}-\tilde{ \partial }_\mathbf{x})\cdot 1\nonumber \\&\quad = (u_1\cdots u_N)^{-M_1-M_2}\sum _{\lambda ,\mu }S_\lambda (u_1^2,\ldots ,u_N^2)S_\mu (u_1^2,\ldots ,u_N^2)\nonumber \\&\qquad S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y}). \end{aligned}$$

(42)

By the restrictions on $\lambda$ and $\mu$, we obtain the conclusion. $\square$

Recall that

$$\begin{aligned} T(u)= & {} \left( \begin{array}{cc} A(u) &{}\quad B(u) \\ C(u) &{}\quad D(u) \end{array} \right) \\= & {} T_2(u)T_1(u)=\left( \begin{array}{cc} A_2(u) &{}\quad B_2(u) \\ C_2(u) &{}\quad D_2(u) \end{array} \right) \left( \begin{array}{cc} A_1(u) &{}\quad B_1(u) \\ C_1(u) &{}\quad D_1(u) \end{array} \right) \end{aligned}$$

then

$$\begin{aligned} B(u)=A_2(u)B_1(u)+ B_2(u) D_1(u). \end{aligned}$$

The matrix entries of $T_i(u),i=1,2$ are related by the following formulas:

$$\begin{aligned}&B_i(u)=uA_i(u)\phi _0^{(i)\dag },\quad C_i(u)=u^{-1}\phi _0^{(i)}A^{\dag }_i(u^{-1}), \\&D_i(u)=\phi _0^{(i)}A^{\dag }_i(u^{-1})\phi _0^{(i)\dag }. \end{aligned}$$

Let $\mathcal {A}_i(u)=PA_i(u)P,\ \mathcal {C}_i(u)=PC_i(u)P,\ \mathcal {D}_i(u)=PD_i(u)P$, where P is the projection $\mathcal {F}\rightarrow \hat{\mathcal {F}}$, then we have

Lemma 4.5

The operators $\mathcal {A}_i(u),\ \mathcal {B}_i(u),\ \mathcal {C}_i(u),\ \mathcal {D}_i(u), (i=1,2)$ are related by the following formulas

$$\begin{aligned} \mathcal {A}_i(u)=u^{-1}\mathcal {B}_i(u),\ \mathcal {C}_i(u)=\mathcal {B}^{\dag }_i(u^{-1}),\ \mathcal {D}_i(u)=u\mathcal {B}^{\dag }_i(u^{-1}). \end{aligned}$$

Since $\mathcal {B}_1(u)=u^{-M_1}H_{M_1}(\mathbf{x}-\tilde{ \partial }_\mathbf{y}, u^2)$, we obtain the following lemma.

Lemma 4.6

Let $\tilde{\mathcal {C}}_1(u)=u^{-M_1}\mathcal {C}_1(u)$, we have

$$\begin{aligned} \tilde{\mathcal {C}}_1(u)= H_{M_1}^\bot (\mathbf{x}-\tilde{\partial }_\mathbf{y},u^{-2})= H_{M_1}^\bot (\mathbf{x},u^{-2}) \end{aligned}$$

where $H_{M_1}^\bot (\mathbf{x},t)=\sum _{k=0}^{M_1}t^kh_k^{\bot ,M_1}(\mathbf{x})$, and $h_k^{\bot ,M_1}(\mathbf{x})$ is the adjoint to the operator of multiplication by $h_k(\mathbf{x})$.

Define $\mathcal {B}(u)=PB(u)P$, we have

$$\begin{aligned} \mathcal {B}(u)= & {} \mathcal {B}_2(u)(u^{-1}\mathcal {B}_1(u)+u\mathcal {B}_1^\dag (u^{-1}))\\= & {} u^{-M_2}H_{M_2}(\mathbf{y}-\tilde{\partial }_\mathbf{x},u^2)(u^{-M_1-1}H_{M_1}(\mathbf{x}-\tilde{\partial }_\mathbf{y},u^2)\\&+\,u^{M_1+1}H_{M_1}^\bot (\mathbf{x}-\tilde{\partial }_\mathbf{y},u^{-2})). \end{aligned}$$

We write $H_{M_1}(\mathbf{x}-\tilde{\partial }_\mathbf{y},u^2)$ by $H_1(u^2)$ for short. From the bilinear equation (20), we have

$$\begin{aligned}&u_1^{M_1+1}u_2^{-M_1-1}H_1^\bot (u_1^{-2})H_1(u_2^{2})\\&\quad =u_1^{M_1+1}u_2^{-M_1-1}\frac{u_1^2}{u_1^2-u_2^2}H_1(u_2^{2})H_1^\bot (u_1^{-2})\\&\qquad -\,u_1^{-M_1-1}u_2^{M_1+1}\frac{u_1^2}{u_1^2-u_2^2}H_1(u_1^{2})H_1^\bot (u_2^{-2}). \end{aligned}$$

Recall that

$$\begin{aligned} |\Psi _N(u_1,\ldots ,u_N)\rangle =\prod _{j=1}^N B(u_j)|0\rangle . \end{aligned}$$

Then by calculation, we get

Proposition 4.7

The operators $u_1^{-1}\mathcal {B}_1(u_1)+u_1\mathcal {B}_1^\dag (u_1^{-1})$ and $u_2^{-1}\mathcal {B}_1(u_2)+u_2\mathcal {B}_1^\dag (u_2^{-1})$ are commutative, which tells us that the coefficients, in the expansion $|\Psi _N(u_1,\ldots ,u_N)\rangle$ in terms of basis vector (28), are symmetric functions of variables $u_1^2,\ldots ,u_N^2$.

Since

$$\begin{aligned}&\prod _{j=1}^N \mathcal {B}(u_j)|0\rangle \\&\quad =\prod _{j=1}^N \mathcal {B}_2(u_j)\left( u_j^{-1}\mathcal {B}_1(u_j)+u_j\mathcal {B}_1^\dag (u_j^{-1})\right) |0\rangle \\&\quad =\prod _{j=1}^N \mathcal {B}_2(u_j)\prod _{j=1}^N\left( u_j^{-1}\mathcal {B}_1(u_j)+u_j\mathcal {B}_1^\dag (u_j^{-1})\right) |0\rangle \end{aligned}$$

and

$$\begin{aligned} \prod _{j=1}^N \mathcal {B}_2(u_j)=(u_1\cdots u_N)^{-M_2}\sum _{\mu }S_\mu (u_1^2,\ldots ,u_N^2)S_\mu (\mathbf{y}-\tilde{ \partial }_\mathbf{x}) \end{aligned}$$

where $\mu$ is a Young diagram with at most N rows and at most $M_2$ columns. Hence, in the following, we consider the expansion of $\prod _{j=1}^N(u_j^{-1}\mathcal {B}_1(u_j)+u_j\mathcal {B}_1^\dag (u_j^{-1}))|0\rangle$.

Proposition 4.8

Let $k_1,k_2\cdots ,k_i$ be in the set $\{1,2,\ldots , N\}$ and satisfy $k_1<k_2<\cdots <k_i$. We denote $u_{k_1}u_{k_2}\cdots u_{k_i}$ by $u_{\{k\}}$ for short. Then we have

$$\begin{aligned}&\prod _{j=1}^N\left( u_j^{-1}\mathcal {B}_1(u_j)+u_j\mathcal {B}_1^\dag (u_j^{-1})\right) |0\rangle \\&\quad =(u_1\cdots u_N)^{M_1+1}\sum _{i=0}^N\sum _{k_1,\ldots ,k_i}(u_{\{k\}})^{-2M_1-2}\\&\qquad \times \,\prod _{j\ne k_i}\frac{u_j^2}{u_j^2-u_{k_i}^2}H_1(u_{k_1}^2)\cdots H_1(u_{k_i}^2)\cdot 1 \end{aligned}$$

where

$$\begin{aligned}&H_1(u_{k_1}^2)\cdots H_1(u_{k_i}^2)\cdot 1=\sum _{\lambda }S_{\lambda }(u_{k_1}^2,\ldots ,u_{k_i}^2)S_{\lambda }(\mathbf{x}\\&\qquad -\tilde{\partial }_\mathbf{y})\cdot 1. \end{aligned}$$

Proof

One can prove it by inductions. $\square$

From the discussion above, we get the expansion of $|\Psi _N(u_1,\ldots ,u_N)\rangle$.

Proposition 4.9

The N-particle vector $|\Psi _N(u_1,\ldots ,u_N)\rangle$ can be written as

$$\begin{aligned} |\Psi _N(u_1,\ldots ,u_N)\rangle= & {} (u_1\cdots u_N)^{-M_2+M_1+1}\\&\times \,\sum _{i=0}^N\sum _{k_1,\ldots ,k_i}(u_{\{k\}})^{-2M_1-2}\prod _{j\ne k_i}\frac{u_j^2}{u_j^2-u_{k_i}^2}\\&\times \,\sum _{\lambda ,\mu }S_{\lambda }(u_{k_1}^2,\ldots ,u_{k_i}^2)\\&\times \, S_\mu (u_1^2,\ldots ,u_N^2)S_{[\lambda ,\mu ]}(\mathbf{x},\mathbf{y}) \end{aligned}$$

where $\lambda$ is a Young diagram with at most i rows and $M_1$ columns, and $\mu$ a Young diagram with at most N rows and $M_2$ columns.

In a special case, we have $S_{[\lambda ,\emptyset ]}(\mathbf{x},\mathbf{y})=S_{\lambda }(\mathbf{x})$. Let $M_2=\emptyset$,

$$\begin{aligned} |\Psi _N(u_1,\ldots ,u_N)\rangle= & {} \prod _{j=1}^N B_1(u_j)|0\rangle \\= & {} (u_1\cdots u_N)^{-M_1}\prod _{j=1}^N \tilde{\mathcal {B}}_1(u_j)|0\rangle \\= & {} \sum _{\lambda }S_\lambda (u_1^2,\ldots ,u_N^2)S_\lambda (\mathbf{x}) \end{aligned}$$

which is the same as in [7, 8].

5 Vertex operators and topological strings on $\mathbb {C}^3$

The A-model topological string partition function on $\mathbb {C}^3$ is given by the MacMahon function

$$\begin{aligned} Z_{\mathbb {C}^3}^{top}=M(q)=\prod _{n=1}^\infty \frac{1}{(1-q^n)^n}=\sum _{n=0}^\infty P(n)q^n. \end{aligned}$$

(43)

It is related to the topological vertex and P(n) counts the number of plane partition whose total boxes number equals n. It is known that the generating function of plane partitions can be written as a fermionic correlator involving the standard vertex operators

$$\begin{aligned} \tilde{\Gamma }^+(z)=e^{\xi (\mathbf{x},z)},\quad \tilde{\Gamma }^-(z)=e^{\xi (\tilde{\partial }_\mathbf{x},z^{-1})} \end{aligned}$$

(44)

with a particular specialization of the values of $z=q^{\pm 1/2},q^{\pm 3/2}, q^{\pm 5/2},\ldots$.

In the following, we will give that $Z_{\mathbb {C}^3}^{top}$ can also be obtained from vertex operators $\Gamma ^{\pm }_i(t)$, $i=1,2$. We first give the relations between vertex operators $\Gamma ^{\pm }_i(t)$, $i=1,2$ and operators $\tilde{\mathcal {B}}_i(u)$ and $\tilde{\mathcal {C}}_i(u)$.

Proposition 5.1

In the $M_1,M_2\rightarrow \infty$ limit, operators $\tilde{\mathcal {B}}_i(u)$ and $\tilde{\mathcal {C}}_i(u)$ have the following vertex operator representations

$$\begin{aligned} \tilde{\mathcal {B}}_1(u)= & {} e^{\xi (\mathbf{x}-\tilde{\partial }_\mathbf{y},u^2)}=\Gamma _1^+(u^2), \end{aligned}$$

(45)

$$\begin{aligned} \tilde{\mathcal {C}}_1(u)= & {} e^{\xi (\tilde{\partial }_\mathbf{x},u^{-2})}\ =\Gamma _1^-(u^2),\end{aligned}$$

(46)

$$\begin{aligned} \tilde{\mathcal {B}}_2(u)= & {} e^{\xi (\mathbf{y}-\tilde{\partial }_\mathbf{x},u^2)}=\Gamma _2^+(u^2),\end{aligned}$$

(47)

$$\begin{aligned} \tilde{\mathcal {C}}_2(u)= & {} e^{\xi (\tilde{\partial }_\mathbf{y},u^{-2})}\ =\Gamma _2^-(u^2), \end{aligned}$$

(48)

where the vertex operators $\Gamma _i^\pm (t),i=1,2$ are defined in (3) and (4).

To obtain the form of $Z_{\mathbb {C}^3}^{top}$ written by a fermionic correlator involving vertex operators $\Gamma ^{\pm }_i(t)$, $i=1,2$, we need the following lemma.

Lemma 5.2

The following relation holds

$$\begin{aligned} \langle 0|\prod _{m=1}^\infty \Gamma _2^-(q^{m-1/2})\Gamma _1^-(q^{m-1/2})|0\rangle =\prod _{n\ge 1}(1-q^n)^n. \end{aligned}$$

(49)

Proof

Since

$$\begin{aligned} \Gamma _1^-(w)= & {} e^{\xi (\mathbf{x}, w)}e^{-\xi (\tilde{\partial }_\mathbf{y}, w)},\\ \Gamma _2^-(z)= & {} e^{\xi (\mathbf{y}, z)}e^{-\xi (\tilde{\partial }_\mathbf{x}, z)}, \end{aligned}$$

and

$$\begin{aligned} e^{-\xi (\tilde{\partial }_\mathbf{x}, z)}e^{\xi (\mathbf{x}, w)}=(1-zw)e^{\xi (\mathbf{x}, w)}e^{-\xi (\tilde{\partial }_\mathbf{x}, z)} \end{aligned}$$

then we get

$$\begin{aligned} \Gamma _2^-(z)\Gamma _1^-(w)=(1-zw):\Gamma _2^-(z)\Gamma _1^-(w): \end{aligned}$$

where the normal order is defined as usual. Using this formula step by step, we get the conclusion. $\square$

Then we obtain the following result.

Proposition 5.3

The A-model topological string partition function on $\mathbb {C}^3$ (the MacMahon function) equals

$$\begin{aligned} Z_{\mathbb {C}^3}^{top}= & {} \langle 0|\prod _{m=1}^\infty \Gamma _2^+(q^{-m+1/2})\Gamma _1^+(q^{-m+1/2})\nonumber \\&\times \,\prod _{m=1}^\infty \Gamma _2^-(q^{m-1/2})\Gamma _1^-(q^{m-1/2})|0\rangle . \end{aligned}$$

(50)

Proof

Since

$$\begin{aligned} \Gamma _i^+(z)\Gamma _i^-(w)= & {} \frac{1}{1-w/z}\Gamma _i^-(w)\Gamma _i^+(z),\quad i=1,2,\\ \Gamma _i^+(z)\Gamma _j^-(w)= & {} \Gamma _i^j(w)\Gamma _i^+(z),\quad i,j=1,2,\ i\ne j. \end{aligned}$$

Then the right hand side of (50) equals

$$\begin{aligned} \frac{1}{(1-q^n)^n}\frac{1}{(1-q^n)^n}\langle 0|\prod _{m=1}^\infty \Gamma _2^-(q^{m-1/2})\Gamma _1^-(q^{m-1/2})|0\rangle . \end{aligned}$$

By the Lemma 5.2, we get the conclusion. $\square$