Universal Character, Phase Model and Topological Strings on $\mathbb C^3$

In this paper, we consider two different subjects: the algebra of universal characters $S_{[\lambda,\mu]}({\bf x},{\bf y})$ (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 $\mathbb C^3$.


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]. T. 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 [λ,µ] (x, 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 Γ ± i (z), (i = 1, 2). The Fermions can be defined from these vertex operators. In the special case µ = ∅, the universal character S [λ,µ] (x, y) will be reduced to the Schur function S λ (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 The actions of the entries in monodromy matrix T (u) on Schur function are obtained from the truncated expansions of the vertex operators In this paper, we generalize vertex operators to the type , these vertex operators can generate universal characters and fermions can be defined from them. We define the map from the state in the phase model to the universal character by 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 Γ − 1 (z) and Γ − 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 Γ + 1 (z) and Γ + 2 (z) respectively. Another purpose of this paper is to discuss the relations between the vertex operators Γ − i (z), Γ + i (z) (i = 1, 2) and the MacMahon functions. It is known that the A-model topological string partition function Z top C 3 on C 3 can be written as a Fermionic correlator involving the vertex operatorsΓ + (z) andΓ − (z) with a particular specialization of the values of z = q ±1/2 , q ±3/2 , q ±5/2 · · · . In this paper, we will give that Z top C 3 can also be obtained from the vertex operators Γ − i (z) and Γ + i (z) (i = 1, 2) with the same specialization of the values of z.
The paper is organized as follows. In section 2, we recall the definition of universal character and its vertex operator realization, then we give the actions of the vertex operators on S [λ,µ] (x, y) which is helpful for our discussion. In section 3, we recall the phase model. In section 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 section 2. In section 5, we give that the MacMahon function can be obtained from these vertex operators.

Universal characters and vertex operators
Let x = (x 1 , x 2 , · · · ) and y = (y 1 , y 2 , · · · ). The operators h n (x) are determined by the generating function: and set h n (x) = 0 for n < 0. The operators h n (x) can be explicitly written as Note that h n (x) is the complete homogeneous symmetric function if we replace ix i with the power sum p i . For a pair of Young diagrams λ = (λ 1 , λ 2 , · · · , λ l ) and µ = (µ 1 , µ 2 , · · · , µ l ′ ), the universal character S [λ,µ] = S [λ,µ] (x, y) is a polynomial of variables x and y in C[x, y] defined by the twisted Jacobi-Trudi formula [2]: ( Define the degree of each variables x n , y n , n = 1, 2, · · · by deg x n = n, deg y n = −n then S [λ,µ] (x, y) is a homogeneous polynomial of degree |λ| − |µ|, where |λ| = λ 1 + λ 2 + · · · + λ l is called the weight of λ. Note that S λ (x) is a special case of the universal character: Introduce the following vertex operators Define The operators X ± i satisfy the following Fermionic relations: The same relations hold also for Y ± i , and X ± i and Y ± i are commutative. The operators X + i and Y + i are raising operators for the universal characters such that where the Young diagrams λ = (λ 1 , λ 2 , · · · , λ l ) and µ = (µ 1 , µ 2 , · · · , µ l ′ ).
For an unknown function τ = τ (x, y), the bilinear relations [4] i+j=−1 is called the UC hierarchy. It turns out that τ (x, y) equals where τ 1 (x) and τ 2 (x) are tau functions of KP hierarchy. In special case, the universal character S [λ,µ] (x, y) is also the solution of UC hierarchy, and 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 [λ,µ] (x, y) in the following way, where the multiplication of two Young diagram λ and (n) satisfies the Pieri formula [9,10], and λ ≻ µ means that the Young diagrams λ and µ are interlaced, in the sense of where λ/µ denotes a skew diagram.

Phase model
We begin this section with a bosonic system based on the following algebra [7,8] [ where π = |0 0| is the vacuum projection. The operator φ is one-sided isometry This algebra can be represented in the Fock space F consisting of n-particle states |n , the operators φ, φ † and N acting as the phase operators and the number of particles operator, respectively, where |0 is the vacuum state, the special case n = 0 of the n particle state. Let the tensor product be M +1 copies of the Fock space. Denote by φ i , φ † i , N i the operators that act as φ, φ † , 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] Define the operator of the total number of particles bŷ Then the N -particle vectors in this space are of the form 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 where u is a scalar parameter, here we treat u as uI with I being the identity operator in F. For every i = 0, . . . , M , the L-matrix satisfies the bilinear equation where R-matrix R(u, v) is a 4 × 4 matrix given by with Define the monodromy matrix by which gives the solution of the phase model. It also satisfies the bilinear equation Let we haveN B(u) = B(u)(N + 1),N C(u) = C(u)(N − 1).
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 j the vacuum vector in F j and by which is a N particle state. According to [7,8], there is the following isometry between the states (17) and the Schur functions In the following, we will generalize this work [7,8] to realize the 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.

Two-site generalized phase model and universal characters
Now fix two positive integers M 1 , M 2 and consider the tensor products which F (1) and F (2) are M 1 + 1 and M 2 + 1 copies of the Fock space respectively. Denote by φ the operators that act as φ, φ † , N in (14), respectively, in the ith space F (1) i and identically in the other spaces of F (1) and in all spaces of F (2) , and denote by φ the operators that act as φ, φ † , N , respectively, in the ith space F (2) i and identically in the other spaces of F (2) and in all spaces of F (1) . Let so that F is M 1 + M 2 + 2 copies of the Fock space. Denote by |0 The (N 1 , N 2 )-particle vectors in space F are of the form where with λ = 1 n 1 2 n 2 . . . , µ = 1 m 1 2 m 2 . . . .
In fact, this association is not quite unique: partitions λ, µ 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(λ) and m 0 = N 2 − l(µ), where l(λ) is the length of partition λ, that is, the number of rows in λ. Note that the correspondence (7) in [8] is a special case of the map  defined above. The L-matrices are and the monodromy matrix is Each L-matrix, as well as the monodromy matrix, satisfies the bilinear equation again with the same R-matrix in (19). Let 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 particleŝ 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 F (j) i (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 spacê in the algebra of symmetric functions C[x, y], in fact in its subspace C M 1 ,M 2 [x, y] generated by universal characters S [λ,µ] where the Young diagrams λ have at most M 1 columns and µ at most M 2 columns. From the definition of the twisted Jacobi-Trudi formula, we can define this subspace by supposing By the definition ofN 1 ,N 2 , the space F has a decomposition into (N 1 , N 2 )-particle subspaces F N 1 ,N 2 , i.e., Under the map (29), the subspace F N 1 ,N 2 corresponds to C N 1 ,N 2 M 1 ,M 2 [x, y] which is spanned by universal characters S [λ,µ] whose diagrams λ lie in the N 1 × M 1 box and µ lie in the N 2 × M 2 box. That Young diagram λ lies in the N × M box means λ has at most N rows and at most M columns.
Define the projection P : F →F by forgetting the zero energy states, and define the operator B i (u) := P B i (u)P for i = 1, 2. Then B i (u) are operators acting on the spacê In the following, we discuss the actions of Then we can derive the following proposition.
where H n (x, t) = n k=0 t k h k (x), and h k (x) is defined in (1), which in fact is the complete homogeneous symmetric function.
To prove this proposition, we need the following lemma.
Proof. this holds since y n − 1 n ∂ xn and x m − 1 m ∂ ym are commutative. The proof of Proposition 4.1.
Proof. We know that Here the mapping sign  is omitted at the left of the equation and we will do the same in the following. Hence, Note that the truncated generating function H n (x, t) = n k=0 t k h k (x) can also be regarded as the full generating function H(x, t) = ∞ k=0 t k h k (x) under the specialization: H n (x, t) = H(x, t)| h n+1 (x)=h n+2 (x)=...=0 . Then we get the following corollary. Corollary 4.3. In the M 1 , M 2 → ∞ limit, the actions of the creation operatorsB 1 (u) andB 2 (u) on the spaceF correspond to the multiplications of H(x −∂ y , u 2 ) and H(y − ∂ x , u 2 ) on the space C[x, y] respectively.
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 where the sum is over Young diagrams λ with at most N rows and at most M 1 columns, Young diagrams µ with at most N rows and at most M 2 columns.
Proof. The operators B 1 (u) and B 2 (u) are commutative.
then we have By the restrictions on λ and µ, we obtain the conclusion.
Recall that The matrix entries of T i (u), i = 1, 2 are related by the following formulas: Let A i (u) = P A i (u)P, C i (u) = P C i (u)P, D i (u) = P D i (u)P , where P is the projection F →F, then we have , D i (u), (i = 1, 2) are related by the following formulas , we obtain the following lemma.
is the adjoint to the operator of multiplication by h k (x).
Define B(u) = P B(u)P , we have We write H M 1 (x −∂ y , u 2 ) by H 1 (u 2 ) for short. From the bilinear equation (20), we have Recall that Then by calculation, we get Proposition 4.7. The operators u −1 1 B 1 (u 1 ) + u 1 B † 1 (u −1 1 ) and u −1 2 B 1 (u 2 ) + u 2 B † 1 (u −1 2 ) are commutative, which tells us that the coefficients, in the expansion |Ψ N (u 1 , · · · , u N ) in terms of basis vector (28), are symmetric functions of variables u 2 1 , · · · , u 2 N . Since where µ is a Young diagram with at most N rows and at most M 2 columns. Hence, in the following, we consider the expansion of N j=1 (u −1 j B 1 (u j ) + u j B † 1 (u −1 j ))|0 .
Proof. One can prove it by inductions.
From the discussion above, we get the expansion of |Ψ N (u 1 , · · · , u N ) .
Proposition 4.9. The N -particle vector |Ψ N (u 1 , · · · , u N ) can be written as where λ is a Young diagram with at most i rows and M 1 columns, and µ a Young diagram with at most N rows and M 2 columns.
5 Vertex operators and topological strings on C 3 Let p k denote the power sum, it is known that the following relation holds Then we have the following proposition.
The A-model topological string partition function on C 3 is given by the MacMahon function 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 opera-torsΓ