Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations

Solutions of the qKZ equation associated with the quantum affine algebra $U_q(\hat{sl}_2)$ and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of $q$-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko.


Introduction
In 1992 I. Frenkel and N. Reshetikhin [7] had developed the theory of intertwining operators for quantum affine algebras and had shown that the matrix elements of intertwiners satisfy the quantized Knizhnik-Zamolodchikov (qKZ) equations.
The theory of intertwiners and qKZ equations was successfully applied to the study of solvable lattice models [9] (and references therein). As to the study of solutions of the qKZ equations, bases are constructed by Tarasov and Varchenko [22] in the form of multi-dimensional hypergeometric integrals in the case of U q ( sl 2 ). However solutions of the qKZ equations for other quantum affine algebras are not well studied [21].
The method of free fields is effective to compute correlation functions in conformal field theory (CFT) [3], in particular, solutions to the Knizhnik-Zamolodchikov (KZ) equations [18,19]. A similar role is expected for those of quantum affine algebras. Unfortunately it is difficult to say that this expectation is well realized, as we shall explain below.
Free field realizations of quantum affine algebras are constructed by Frenkel and Jing [6] for level one integrable representations of ADE type algebras and by Matsuo [15], Shiraishi [16] and Abada et al. [1] for representations with arbitrary level of U q ( sl 2 ). The latter results are extended to U q ( sl N ) in [2]. Free field realizations of intertwiners are constructed based on these representations in the case of U q ( sl 2 ) [9,10,12,15,4].
The simplicity of the Frenkel-Jing realizations makes it possible not only to compute matrix elements but also traces of intertwining operators [9], which are special solutions to the qKZ equations. The case of q-Wakimoto modules with an arbitrary level becomes more complex and the detailed study of the solutions of the qKZ equations making use of it is not well developed. In [15] Matsuo derived his integral formulae [14] from the formulae obtained by the free field calculation in the simplest case of one integration variable. However it is not known in general whether the integral formulae derived from the free field realizations recover those of [14,23,22] 1 .
The aim of this paper is to study this problem in the case of the qKZ equation with the value in the tensor product of two dimensional irreducible representations of U q ( sl 2 ). More general cases will be studied in a subsequent paper.
There are mainly two reasons why the comparison of two formulae is difficult. One is that the formulae derived from the free field calculations contain more integration variables than in Tarasov-Varchenko's (TV) formulae. This means that one has to carry out some integrals explicitly to compare two formulae. The second reason is that the formulae from free fields contains a certain sum. This stems from the fact that the current and screening operators are written as a sum which is absent in the non-quantum case. Since TV formulae have a similar structure to those for the solutions of the KZ equation [18,19], one needs to sum up certain terms explicitly for the comparison of two formulae. We carry out such calculations in the case we mentioned.
The plan of this paper is as follows. In Section 2 the construction of the hypergeometric solutions of the qKZ equation due to Tarasov and Varchenko is reviewed. The free field construction of intertwining operators is reviewed in Section 3. In Section 4 the formulae for the highest to highest matrix elements of some operators are calculated. The main theorem is also stated in this section. The transformation of the formulae from free fields to Tarasov-Varchenko's formulae is described in Section 5. In Section 6 the proof of the main theorem is given. Remaining problems are discussed in Section 7. The appendix contains the list of the operator product expansions which is necessary to derive the integral formula.

Tarasov-Varchenko's formula
Let V (1) = Cv 0 ⊕ Cv 1 be a two-dimensional irreducible representation of the algebra U q (sl 2 ), and R(z) ∈ End(V (1)⊗2 ) be a trigonometric quantum R-matrix given by Let p be a complex number such that |p| < 1 and T j denote the multiplicative p-shift operator of z j , The qKZ equation for the V (1)⊗n -valued function Ψ(z 1 , . . . , z n ) is where R ij (z) signifies that R(z) acts on the i-th and j-th components, κ is a complex parameter, acts on the j-th component as Let us briefly recall the construction of the hypergeometric solutions [22,20] of the equation (1).
In the remaining part of the paper we assume |q| < 1. We set Let n and l be non-negative integers satisfying l ≤ n. For a sequence (ǫ) = (ǫ 1 , . . . , ǫ n ) ∈ {0, 1} n satisfying ♯{i|ǫ i = 1} = l let satisfying the following conditions For W ∈ F ell let whereT l is a suitable deformation of the torus specified as follows [22]. Notice that the integrand has simple poles at The contour for the integration variable t a is a simple closed curve which rounds the origin in the counterclockwise direction and separates the following two sets, Then is a solution of the qKZ equation (1) for any W ∈ F ell .

Free field realizations
In this section we review the free field construction of the representation of the quantum affine algebra U q ( sl 2 ) of level k and intertwining operators. We mainly follow the notation in [10]. We set Let k be a complex number and Other combinations of elements are supposed to commute. Set Then the Fock module F r,s is defined to be the free N − -module of rank one generated by the vector which satisfies We set A representation of the quantum affine algebra U q ( sl 2 ) is constructed on F r for any r ∈ C in [16]. 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|.

Remark 1.
We change the definition of |r, s in [10]. Namely we use Let us introduce field operators which are relevant to our purpose. For x = a, b, c let The normal ordering is defined by specifying N + ,ã 0 ,b 0 ,c 0 as annihilation operators and N − , Q a , Q b , Q c as creation operators. With this notation let us define the operators l (z) =: exp a l; 2, k + 2|q k z; l−r (z) signifies to take the coefficient of (u 1 · · · u r ) −1 . The operator J − (z) is a generating function of a part of generators of the Drinfeld realization U q ( sl 2 ) at level k. While the operators φ (l) m (z) are conjectured to determine the intertwining operator for U q ( sl 2 ) modules [10,15] where W r is a certain submodule of F r specified as a kernel of a certain operator, called q-Wakimoto module [15,12,13,11,1], V (l) is the irreducible representation of U q (sl 2 ) with spin l/2 and V (l) z is the evaluation representation of U q ( sl 2 ) on V (l) . In this paper we exclusively consider the case l = 1 and set 1 .
The operator S(z) commutes with U q ( sl 2 ) modulo total difference. Here modulo total difference means modulo functions of the form Remark 2. The intertwining properties of φ (l) (z) for l ∈ Z are not proved in [10] as pointed out in [15]. However the fact that the matrix elements of compositions of φ (l) (z)'s and S(t)'s satisfy the qKZ equation modulo total difference can be proved in a similar way to Proposition 6.1 in [15] using the result of Konno [11] (see (4)).
They become left and right highest weight vectors of U q ( sl 2 ) with the weight mΛ 1 + (k − m)Λ 0 respectively, where Λ 0 , Λ 1 are fundamental weights of sl 2 . Consider which is a function taking the value in V (1)⊗n . Let Let the parameter p be defined from k by p = q 2(k+2) .
We assume |p| < 1 as before. Then the function F satisfies the following qKZ equation modulo total difference of a function [15,8,7,10,11] where

Integral formulae
Define the components of F (t, z) by where (ν) = (ν 1 , . . . , ν n ). By the weight condition F (ν) (t, z) = 0 unless the condition ♯{i|ν i = 0} = l is satisfied. We assume this condition once for all. Notice that Let Then F (ν) (t, z) can be written as where C l is a suitable deformation of the torus T l specified as follows.
The contour for the integration variable u i is a simple closed curve rounding the origin in the counterclockwise direction such that q k+3 z j (1 ≤ j ≤ n), q −2 u j (i < j), q −µ i (k+2) t a (1 ≤ a ≤ l) are inside and q k+1 z j (1 ≤ j ≤ n), q 2 u j (j < i) are outside.
By the operator product expansions (OPE) of the products of φ − (z), J − µ (u), S ǫ (t) in the appendix, one can compute the function F (ν) (ǫ)(µ) (t, z|u) explicitly. In order to write down the formula we need some notation. Set The main theorem in this paper is It follows that F (ν) (t, z) is given by

Transformation to Tarasov-Varchenko's formulae
We describe a transformation from F , which satisfies (4), to Ψ, which satisfies (1). The parameter κ is also determined as a function of l, m, n.
For a solution G of (4) let One can easily verify thatG satisfies (1) with κ = q 2l−2−n−2m using Let F be defined by (3) andF by (5). Theñ Then the condition (iii) for W is equivalent to the following conditions, To sum up we have is a solution of the qKZ equation (1), whereF is defined by (5) with F and F being given in (3) and (2) andW is defined by (6).

Proof of Theorem
Let A ± = {j|µ j = ±}. Suppose that the number of elements in A ± is r ± and write A ± = {l ± 1 < · · · < l ± r ± }. Set A = A − , r = r − and l i = l − i for simplicity. Let Lemma 1. We have Proof . We first integrate in the variables u j , j ∈ A + in the order u l + 1 , . . . , u l + r + . Let us consider the integration in u l + 1 . We denote the integration contour in u i by C i . The only singularity of the integrand outside C l + 1 is ∞. Thus the integral is calculated by taking residue at ∞. Since the integrand is of the form where H(u) is holomorphic at ∞. Then In this way the integral in u l + 1 is calculated. After this integration the integrand as a function of u l + 2 has a similar structure. Therefore the integration with respect to u l + 2 is carried out in a similar way and so on. Finally we get Here C n−r + is specified by similar conditions to C l , where u l + i 1 ≤ i ≤ r + are omitted. We denote the right hand side of this equation other than C n−r + j du j 2πiu j by I (ν)+ (ǫ)(µ) (t, z). Next we integrate with respect to the remaining variables u j , j ∈ A in the order u lr ,. . . ,u l 1 . Let us consider the integration with respect to u lr . The poles of the integrand inside C lr is Let us calculate the sum in {ǫ b i } assuming ǫ a i = +. Using we have Lemma 2. For N ≥ 1 we have Proof . Let Then the left hand side of (8) is equal to ǫ j det (a 1 (ǫ 1 ), . . . , a N (ǫ N )) = det Since the right hand side of (9) is zero.
By this lemma the right hand side of (7) becomes zero if r + > 0. Consequently G (ν) (µ) = 0 for r + > 0. Suppose that r + = 0. In this case r = l, l i = i (1 ≤ i ≤ l) and The theorem easily follows from this.

Concluding remarks
In this paper we study the solutions of the qKZ equation taking the value in the tensor product of the two dimensional evaluation representation of U q ( sl 2 ). The integral formulae are derived for the highest to highest matrix elements for certain intertwining operators by using free field realizations. The integrals with respect to u variables corresponding to the operator J − (u) are calculated and the sum arising from the expression of J − (u) and the screening operator S(t) is calculated. The formulae thus obtained coincide with those of Tarasov and Varchenko. The calculations in this paper can be extended to the case where the vector space V (1)⊗n is replaced by a tensor product of more general representations. It is an interesting problem to perform similar calculations for other quantum affine algebras [2] and the elliptic algebras [13]. In Tarasov-Varchenko's theory solutions of a qKZ equation are parametrized by elements of the elliptic hypergeometric space F ell while the matrix elements are specified by intertwiners. It is an interesting problem to establish a correspondence between intertwining operators and elements of F ell . With the results of the present paper one can begin to study this problem. Study in this direction will provide a new insight on the space of local fields and correlation functions of integrable field theories and solvable lattice models. The corresponding problem in CFT is studied in [5].