Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

Abstract

We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the socalled combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions. IntroductionThe Knizhnik-Zamolodchikov (KZ) equations [29] are a set of compatible differential equations satisfied by conformal blocks in Conformal Field Theory. The original work on KZ equations was concerned with the theory on a sphere, but Bernard generalised it to the torus [10] and to higher genus Riemann surfaces [11], leading to the so-called Knizhnik-Zamolodchikov-Bernard (KZB) equations. These differential equations come from a flat connection over the moduli space of Riemann surfaces with L marked points; in the present work, we shall only be concerned with the variation of the marked points, and not with the variation of the underlying Riemann surface, whose genus will always be one.The KZ equations are intimately related to the representation theory of affine algebras. In [23], difference equations, now customarily called qKZ equations, were introduced as analogues of the KZ equations for quantised affine algebras. These equations were formulated in terms of the Rmatrix of the associated quantum integrable system, which satisfies the Yang-Baxter equation. For a review of qKZ equations see [16] and references contained there.Generalising both the KZB and qKZ equations, Felder introduced the qKZB equations [17]. An important new ingredient was the use of an elliptic solution of the dynamical Yang-Baxter equation. The qKZB equations can be viewed as related to a representation of an (extended) affine Weyl group, which will always be the affine symmetric group for us, on an appropriate functional space, dynamical R-matrices being "generalised R-matrices" in the sense of Cherednik [12]. In the latter work, two types of solutions of the qKZ(B) equation were considered: ordinary solutions, corresponding to looking for eigenvectors of the commutative subgroup of the affine Weyl group, and symmetric solutions, which are invariant under the whole of the affine Weyl group. In the second case, we call the set of equations that these satisfy, the qKZ(B) system.In fact, the qKZ system predated the qKZ equation -see [42] where form factors of a quantum integrable model are shown to satisfy the qKZ system, as well as [25] for a similar connection to correlation functions. In the context of dynamical R-matrices associated to so-called ABF models, the qKZB system was considered in [20] using the vertex operator approach [25]. This approach was developed further for a range of elliptic face and vertex models in [34,2,24,26,31,30]. Our approach in this paper is different, and the rationale for developing this alternative to the vertex operator approach is discussed in Section 5.In an unrelated development, Stroganov [43] and Razumov and Stroganov [36,37] not...

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