q-Wakimoto Modules and Integral Formulae of the Quantum Knizhnik-Zamolodchikov Equations

Matrix elements of intertwining operators between $q$-Wakimoto modules associated to the tensor product of representations of $U_q(\widehat{sl_2})$ with arbitrary spins are studied. It is shown that they coincide with the Tarasov-Varchenko's formulae of the solutions of the qKZ equations. The result generalizes that of the previous paper [Kuroki K., Nakayashiki A., SIGMA 4 (2008), 049, 13 pages, arXiv:0802.1776].


Introduction
In [8] the integral formulae of the quantum Knizhnik-Zamolodchikov (qKZ) equations [3] for the tensor product of spin 1/2 representation of U q (sl 2 ) arising from q-Wakimoto modules have been studied. The formulae are identified with those of Tarasov-Varchenko's formulae. The aim of this paper is to generalize the results to the case of tensor product of representations with arbitrary spins.
It is known that certain matrix elements of intertwining operators between q-Wakimoto modules satisfy the qKZ equation [3,10]. Thus it is interesting to compute those matrix elements explicitly. In [5] two kinds of intertwining operators were introduced, type I and type II. They were defined according as the position of evaluation representations. In the application to the study of solvable lattice models two types of operators have their own roles. Type I and type II operators correspond to states and particles respectively. The properties of traces exhibit very different structure. However as far as the matrix elements are concerned they are not expected to be very different [5].
In [8] a computation of matrix elements has been carried out in the case of type I operator and the tensor product of 2-dimensional vector representation of U q (sl 2 ) generalizing the result of [10] (see the previous paper [8]). In this paper we compute matrix elements for the composition of the type I intertwining operators [5] associated to finite dimensional irreducible representations of U q (sl 2 ). We perform certain multidimensional integrals and sums explicitly. It is shown that the formulae thus obtained coincide with those of Matsuo [9], Tarasov and Varchenko [13] without the term corresponding to the deformed cycles.
To obtain actual matrix elements of intertwining operators it is necessary to specify certain contours of integration associated to screening operators. We do not consider this problem in this paper. To find integration contours describing each composition of intertwining operators is an important open problem. We also remark that the formulae for type II intertwining operators are not obtained in this paper. The computation of them looks quite different from that for type I case as opposed to the expectation. It is interesting to find the way to get a similar result for matrix elements in the case of type II operators.
The paper is organized in the following manner. The construction of the solutions of the qKZ equations due to Tarasov and Varchenko is reviewed in Section 2. In Section 3 a free field construction of intertwining operator is reviewed. The formulae for the matrix elements of some operators are calculated in Section 4. The main theorem of this paper is stated in this section. In Section 5 the proof of the main theorem is given. The evaluation representation of U q ( sl 2 ) is explicitly described in Appendix A. Appendix B gives the explicit form of the R-matrix in special cases. The explicit forms of the operators which appear in Section 3 are given in Appendix C. Appendix D contains the list of OPE's which is necessary to derive the integral formulae.

Tarasov-Varchenko's formulae
We review Tarasov-Varchenko's formula for solutions of the qKZ equations. In this paper we assume that q is a complex number such that |q| < 1. We mainly follow the notation of [13]. For a nonnegative integer l let is given in Appendix A. Let l 1 and l 2 be nonnegative integers and R l 1 ,l 2 (z) ∈ End(V (l 1 ) ⊗ V (l 2 ) ) the trigonometric quantum R-matrix uniquely determined by the following conditions: The explicit form of the R-matrix is given in Appendix B in case l 1 = 1 or l 2 = 1. We set where for a complex number a with |a| < 1 Let k be a complex number. We set We assume that p satisfies |p| < 1. Let T j denote the p-shift operator of z j , T j f (z 1 , . . . , z n ) = f (z 1 , . . . , pz j , . . . z n ).
Let l 1 , . . . , l n and N be nonnegative integers. The qKZ equation for a V l 1 ⊗ · · · ⊗ V ln -valued function Ψ(z 1 , . . . , z n ) is where κ is a complex parameter, R i,j (z) signifies that R l i ,l j (z) acts on the i-th and j-th components of the tensor product and κ h j acts on j-th component as We set Consider a sequence (ν) = (ν 1 , . . . , ν n ) satisfying 0 ≤ ν i ≤ l i for all i and N = n i=1 ν i . Let satisfying the following conditions: (ii) Θ(t, z) is holomorphic on (C * ) n+N in t 1 , . . . , z n and symmetric in t 1 , . . . , t N ; . . . , pt a , . . . , t N , z) and T z j W = W (t, z 1 , . . . , pz j , . . . , z n ).
Define the phase function Φ(t, z) by For W ∈ F ell let where T N is a suitable deformation of the torus specified as follows. The integrand has simple poles at The contour of integration in t a is a simple closed curve which rounds the origin in the counterclockwise direction and separates the following two sets Let L be a complex number and is a solution of the qKZ equation (1) for any W ∈ F ell where (−ǫ) = (l 1 − ǫ 1 , . . . , l n − ǫ n ) and

Free field realizations
We briefly review the free field construction of the representation of the U q ( sl 2 ) of level k [1,10,11] and intertwining operators [2,6,7]. We mainly follow the notation of [6]. We set Let k be a complex number and {a n , b n , c n , Other combinations of elements are supposed to commute. Set N ± = C[a n , b n , c n | ± n > 0].
Let r be a complex number and s an integer. The Fock module F r,s is defined to be the free N − module of rank one generated by the vector |r, s satisfying N + |r, s = 0,ã 0 |r, s = r|r, s ,b 0 |r, s = −2s|r, s ,c 0 |r, s = −2s|r, s .

q-Wakimoto Modules and Integral Formulae
We set The right Fock module F † r,s and F † r are similarly defined using the vector r, s| satisfying the conditions r, s|N − = 0, r, s|ã 0 = r r, s|, r, s|b 0 = −2s r, s|, r, s|c 0 = −2s r, s|.
Notice that F r and F † r have left and right U q ( sl 2 )-module structure respectively [10,11]. Let They become left and right highest weight vectors of U q ( sl 2 ) with the weight LΛ 1 + (k − L)Λ 0 respectively, where Λ 0 and Λ 1 are fundamental weights of sl 2 . We consider operators the explicit forms of which are given in Appendix C. We set where W r is a certain submodule of F r called q-Wakimoto module [10]. The operator J − (u) is a generating function of a part of generators of the Drinfeld realization for U q ( sl 2 ) at level k.
The operator S(t) commutes with U q ( sl 2 ) modulo total differences. Here modulo total differences means modulo functions of the form Then the function F (t, z) satisfies qKZ equation (1)

Integral formulae
Define the components of F (t, z) by where (ν) = (ν 1 , . . . , ν n ). By the conditions on weights is satisfied. We assume this condition once for all. Let The main result of this paper is The formula for F (ν) (t, z) is of the form of (2), (3). More precisely in Tarasov-Varchenko's formula (2), (3), W can be written as This W ′ specifies an intertwiner. In this paper we don't consider the problem on specifying W ′ . To prove Theorem 1 let us begin by writing down the formula obtained by the free field description of operators φ l (z), J − (u), S(t) given in Appendix C. Let (ǫ) = (ǫ 1 , . . . , ǫ N ), (µ) = (µ 1,1 , . . . , µ 1,n 1 , . . . , µ r,nr ) ∈ {0, 1} N . Then F (ν) (t, z) can be written as and the integrand in the right hand side signifies to take the coefficient of and C N is a suitable deformation of the torus T N specified as follows. We introduce the lexicographical order For a given (m) = (m 1 , . . . , m r ), 1 ≤ m i ≤ n i , we define The contour for the integration variable u i 1 ,i 2 is a simple closed curve rounding the origin in the counterclockwise direction such that q l j +k+2 z j ((i 1 , See the beginning of the next section for the notation of the q-binomial coefficient Using J (ν) (a) , F (ν) (t, z) can be written as Theorem 1 straightforwardly follows from the following proposition.
This proposition is proved by performing integrals in the variables u i,j in the next section.

Proof of Proposition 1
We set for nonnegative integers n, m (n ≥ m). To prove Proposition 1, we have to calculate J (ν) (ǫ)(a) . We need the following lemmas. (ii) Proof . By the q-binomial theorem we have the equation The assertions (i) and (ii) easily follow from this equation.
Proof . Set It is easy to see that F (t) is an antisymmetric polynomial. So we can write where S(t) is a symmetric polynomial. Moreover S(t) is a homogeneous polynomial of degree m and deg t i S(t) = 1 for all i ∈ {1, . . . , n}. Hence we have for some constant c.
The number (−1) We can show where ℓ(σ) is the inversion number of σ.
Here we have used the q-binomial theorem.

For a given sequence (m
Lemma 4. We have Proof . We integrate with respect the variables u i,j , (i, j) ∈ A + µ in the order u ℓ + is calculated by taking the residue at ∞. After this integration the integrand as a function of u ℓ + 1,2 has a similar structure. Then the integral with respect to u ℓ + 1,2 is calculated by taking residue at ∞ and so on. Finally we get i is the resulting contour for (u ℓ 1,1 , . . . , u ℓr,a r ). We set q-Wakimoto Modules and Integral Formulae 13 Next we perform integrations with respect to the remaining variables u i,j , (i, j) ∈ A µ in the order u ℓr,a r , . . . , u ℓ r,1 , u ℓ r−1,a r−1 , . . . , u ℓ 1,1 . The poles of the integrand inside C ℓr,a r are 0 and q k+2 t b , b = 1, . . . , N . Thus we have C ℓr,a r du ℓr,a r 2πi I The integrand in u ℓ r,ar −1 has the poles at 0 and q k+2 t b inside C ℓ r,ar −1 and so on. Finally we get Res u ℓ 1,1 =w ℓ 1,1 · · · Res u ℓr,a r =w ℓr,a r Proof . Using Lemma 4 we have Then the right hand side of (4) is equal to (as−γs) .
Here we have used Lemma 1 (i). By (ii) of Lemma 1 we have It is easy to show a−γ+s λ=s n m=0 Lemma 5. If a i = n i for some i, Proof . It is enough to show the following equation.
If a i = n i for all i, then (z k(i 1 ) − q l k(i 1 ) t b i 1 ,i 2 ) where By Lemma 3 the right hand side of (7) becomes