A Vertex Operator Approach for Form Factors of Belavin's $(\mathbb{Z}/n\mathbb{Z})$-Symmetric Model and Its Application

A vertex operator approach for form factors of Belavin's $(\mathbb{Z}/n\mathbb{Z})$-symmetric model is constructed on the basis of bosonization of vertex operators in the $A^{(1)}_{n-1}$ model and vertex-face transformation. As simple application for $n=2$, we obtain expressions for $2m$-point form factors related to the $\sigma^z$ and $\sigma^x$ operators in the eight-vertex model.


Introduction
In [1] and [2] we derived the integral formulae for correlation functions and form factors, respectively, of Belavin's (Z/nZ)-symmetric model [3,4] on the basis of vertex operator approach [5]. Belavin's (Z/nZ)-symmetric model is an n-state generalization of Baxter's eight-vertex model [6], which has (Z/2Z)-symmetries. As for the eight-vertex model, the integral formulae for correlation functions and form factors were derived by Lashkevich and Pugai [7] and by Lashkevich [8], respectively.
It was found in [7] that the correlation functions of the eight-vertex model can be obtained by using the free field realization of the vertex operators in the eight-vertex SOS model [9], with insertion of the nonlocal operator Λ, called 'the tail operator'. The vertex operator approach for higher spin generalization of the eight-vertex model was presented in [10]. The vertex operator approach for higher rank generalization was presented in [1]. The expression of the spontaneous polarization of the (Z/nZ)-symmetric model [11] was also reproduced in [1], on the basis of vertex operator approach. Concerning form factors, the bosonization scheme for the eight-vertex model was constructed in [8]. The higher rank generalization of [8] was presented in [2]. It was shown in [12,13] that the elliptic algebra U q,p ( sl N ) relevant to the (Z/nZ)-symmetric model provides the Drinfeld realization of the face type elliptic quantum group B q,λ ( sl N ) tensored by a Heisenberg algebra.
The present paper is organized as follows. In Section 2 we review the basic definitions of the (Z/nZ)-symmetric model [3], the corresponding dual face model A (1) n−1 model [14], and the vertex-face correspondence. In Section 3 we summarize the vertex operator algebras relevant to the (Z/nZ)-symmetric model and the A (1) n−1 model [1,2]. In Section 4 we construct the free field representations of the tail operators, in terms of those of the basic operators for the type I [15] and the type II [16] vertex operators in the A (1) n−1 model. Note that in the present paper we use a different convention from the one used in [1,2]. In Section 5 we calculate 2m-point form factors of the σ z -operator and σ x -operator in the eight-vertex model, as simple application for n = 2. In Section 6 we give some concluding remarks. Useful operator product expansion (OPE) formulae and commutation relations for basic bosons are given in Appendix A.

Basic def initions
The present section aims to formulate the problem, thereby fixing the notation.

Theta functions
The Jacobi theta function with two pseudo-periods 1 and τ (Im τ > 0) are defined as follows: for a, b ∈ R. Let n ∈ Z 2 and r ∈ R >1 , and also fix the parameter x such that 0 < x < 1. We will use the abbreviations, Note that For later conveniences we also introduce the following symbols: where z = x 2v , 1 j n and In particular we denote χ(v) = χ 1 (v). These factors will appear in the commutation relations among the type I and type II vertex operators. The integral kernel for the type I and the type II vertex operators will be given as the products of the following elliptic functions:

Belavin's (Z/nZ)-symmetric model
Let V = C n and {ε µ } 0 µ n−1 be the standard orthonormal basis with the inner product ε µ , ε ν = δ µν . Belavin's (Z/nZ)-symmetric model [3] is a vertex model on a two-dimensional square lattice L such that the state variables take the values of (Z/nZ)-spin. The model is (Z/nZ)-symmetric in a sense that the R-matrix satisfies the following conditions: The definition of the R-matrix in the principal regime can be found in [2]. The present R-matrix has three parameters v, ǫ and r, which lie in the following region:

The A
The weight lattice P and the root lattice Q of A (1) n−1 are usually defined. For a ∈ h * , we set An ordered pair (a, b) ∈ h * 2 is called admissible if b = a +ε µ , for a certain µ (0 µ n − 1).
For (a, b, c, d) ∈ h * 4 , let W c d b a v be the Boltzmann weight of the A and (d, c) are admissible. Non-zero Boltzmann weights are parametrized in terms of the elliptic theta function of the spectral parameter v. The explicit expressions of W can be found in [2]. We consider the so-called Regime III in the model, i.e., 0 < v < 1.

Vertex-face correspondence
Let t(v) a a−εµ be the intertwining vectors in C n , whose elements are expressed in terms of theta functions. As for the definitions see [2]. Then t(v) a a−εµ 's relate the R-matrix of the (Z/nZ)symmetric model in the principal regime and Boltzmann weights W of the A Let us introduce the dual intertwining vectors satisfying From (2.4) and (2.5), we have For fixed r > 1, let and z = x 2v . Then we have 3 Vertex operator algebra

Vertex operators for the (Z/nZ)-symmetric model
Let H (i) be the C-vector space spanned by the half-infinite pure tensor vectors of the forms The type I vertex operator Φ µ (v) can be defined as a half-infinite transfer matrix. The opera- , satisfying the following commutation relation: When we consider an operator related to 'creation-annihilation' process, we need another type of vertex operators, the type II vertex operators that satisfy the following commutation relations: where H CTM is the CTM Hamiltonian defined as follows: Then we have the homogeneity relations

Vertex operators for the
can be defined as a half-infinite transfer matrix. The l,k+εµ , satisfying the following commutation relation: The free field realization of Φ(v 2 ) b a was constructed in [15]. See Section 4.2.
The type II vertex operators should satisfy the following commutation relations: n−1 model in regime III is given as follows: and H v (ν, µ) is the same one as (3.1). Then we have the homogeneity relations The free field realization of Ψ * (v) ξ ′ ξ was constructed in [16]. See Section 4.3.

Tail operators and commutation relations
In [1] we introduced the intertwining operators between H (i) and H which satisfy In order to obtain the form factors of the (Z/nZ)-symmetric model, we need the free field representations of the tail operator which is offdiagonal with respect to the boundary conditions: where k = a + ρ, l = ξ + ρ, k ′ = a ′ + ρ, and Then we have From the invertibility of the intertwining vector and its dual vector, we have Note that the tail operator (3.3) satisfies the following intertwining relations [1,2]: where We should find a representation of Λ(u) ξ ′ a ′ ξ a and fix the constant ∆u that solves (3.5) and (3.6).
We will deal with the bosonic Fock spaces F l,k , (l, k ∈ h * ) generated by B j −m (m > 0) over the vacuum vectors |l, k :

Type I vertex operators
Let us define the basic operators for j = 1, . . . , n − 1 where β 1 = − r−1 r and z = x 2v as usual. The normal product operation places P α 's to the right of Q β 's, as well as B m 's (m > 0) to the right of B −m 's. For some useful OPE formulae and commutation relations, see Appendix A.

Type II vertex operators
Let us define the basic operators for j = 1, . . . , n − 1

Free field realization of tail operators
In order to construct free field realization of the tail operators, we also need another type of basic operators: Concerning useful OPE formulae and commutation relations, see Appendix A.

Free field realization of CTM Hamiltonian
Let be the CTM Hamiltonian on the Fock space F l,k [19]. Then we have the homogeneity relation and the trace formula Let ρ

Form factors for n = 2
In this section we would like to find explicit expressions of form factors for n = 2 case, i.e., the eight-vertex model form factors. Here, we adopt the convention that the components 0 and 1 for n = 2 are denoted by + and −. Form factors of the eight-vertex model are defined as matrix elements of some local operators. For simplicity, we choose σ z as a local operator: µµ ′ is the matrix unit on the j-th site. The free field representation of σ z is given by Here, Φ * ε (u) is the dual type I vertex operator, whose free filed representation can be found in [1,2].
The corresponding form factors with 2m 'charged' particles are given by where In this section we denote the spectral parameters by z j = x 2u j , and denote integral variables by w a = x 2va . By using the vertex-face transformation, we can rewrite (5.1) as follows: where H F is the CTM Hamiltonian defined by (4.3). Let Then we have For simplicity, let l j = l − j for 1 j 2m. Then from the relation (3.4), Λ(u 0 ) l 2m k 2 l k vanishes unless k 2 = k − 2. Thus, the sum over k 1 and k 2 on (5.2) reduces to only one non-vanishing term. Furthermore, we note the formula Here, we use k − l ≡ i (mod 2). The sum with respect to k for the trace over the zero-modes parts can be calculated as follows: where Thus, F (i) m (σ z ; u 1 , . . . , u 2m ) ll−1···l−2m can be obtained as follows: where f ′ (v) is defined by (2.6) for n = 2, a scalar function F ψ * ψ * (z) and a scalar β m are On (5.3), the integral contour C should be chosen such that all integral variables w a lie in the convergence domain x 3 |z j | < |w a | < x|z j |.
Gathering phase factors on (5.3), we have e −π √ −1 3mr 2(r−1) . Redefining f ′ (v) by a scalar factor, we thus obtain the equality: By comparing the transformation properties with respect to l for both sides on (5.4), we conclude that F (i) m (σ z ; u 1 , . . . , u 2m ) ν 1 ···ν 2m are independent of l, and also that as expected. When m = 1, we have where C (z) is a constant. This is a same result obtained by Lashkevich in [8], up to a scalar factor 1 . Next, let us choose σ x as a local operator: Then the relation for F and with {0} in β m replaced by [[0]], respectively. Here, The transformation properties with respect to l implies that F (i) m (σ x ; u 1 , . . . , u 2m ) ν 1 ···ν 2m are independent of l, and also that as expected. Furthermore, 2-point form factors for σ x -operator can be obtained as follows: where C (x) is a constant. The expressions (5.5) and (5.6) are essentially same as the results obtained by Lukyanov and Terras [20] 2 .

Concluding remarks
In this paper we present a vertex operator approach for form factors of the (Z/nZ)-symmetric model. For that purpose we constructed the free field representations of the tail operators Λ ξ ′ a ′ ξ a , the nonlocal operators which relate the physical quantities of the (Z/nZ)-symmetric model and the A (1) n−1 model. As a result, we can obtain the integral formulae for form factors of the (Z/nZ)symmetric model, in principle.
Our approach is based on some assumptions. We assumed that the vertex operator algebra defined by (3.2) and (3.5), (3.6) correctly describes the intertwining relation between the (Z/nZ)-symmetric model and the A (1) n−1 model. We also assumed that the free field representations (4.1), (4.2) provide relevant representations of the vertex operator algebra.
As a consistency check of our bosonization scheme, we presented the integral formulae for form factors which are related to the σ z -operator and σ x -operator in the eight-vertex model, i.e., the (Z/2Z)-symmetric model. The expressions (5.3) and (5.4) for σ z form factors and σ x analogues remind us of the determinant structure of sine-Gordon form factors found by Smirnov [21]. In Smirnov's approach form factors in integrable models can be obtained by solving matrix Riemann-Hilbert problems. We wish to find form factors formulae in the eight-vertex model on the basis of Smirnov's approach in a separate paper.
where g * j (z) and ρ j (z) are defined by (2.2) and (2.3), respectively. From (A.1) and (A.2), we obtain the following commutation relations: where the δ-function is defined by the following formal power series δ(z) = n∈Z z n .
Finally, we list the OPE formulae for W −α j (v) and other basic operators: From these, we obtain