v 1 5 J un 1 99 6 SOLUTIONS OF THE ELLIPTIC QKZB EQUATIONS AND BETHE

Abstract

In this paper we give integral representations for solutions of the system of elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equations in the case of sl2. The qKZB equations [F] are a quantum deformation of the KZB differential equations obeyed by correlation functions of the Wess–Zumino–Witten model on tori. They have the form Ψ(z1, . . . , zj + p, . . . , zn) = Kj(z1, . . . , zn; τ, η, p)Ψ(z1, . . . , zn). The unknown function Ψ takes values in a space of vector valued functions of a complex variable λ, and the Kj are difference operators in λ. The parameters of this system of equations are τ (the period of the elliptic curve), η (“Planck’s constant”), p (the step) and n “highest weights” Λ1, . . . ,Λn ∈ C. The operators Kj are expressed in terms of R-matrices of the elliptic quantum group Eτ,η(sl2). In the trigonometric limit τ → i∞, the qKZB equations reduce to the trigonometric qKZ equations [FR] obeyed by correlation functions of statistical models and form factors of integrable quantum field theories in 1+1 dimensions. The KZB equations can be obtained in the semiclassical limit: η → 0, p → 0, p/η finite. When the step p of the qKZB equations goes to zero (with the other parameters fixed) our construction gives common eigenfunctions of the n commuting operators Kj(z1, . . . , zn; τ, η, 0) in the form of the Bethe ansatz. These difference operators are closely related to the transfer matrices of IRF models of statistical mechanics. Our results follow from the main theme of this paper: a geometric construction of tensor products of evaluation Verma modules over the elliptic quantum group Eτ,η(sl2). In particular, we obtain some formulae given in [FV1] for the action of generators on these modules.

Cite this paper

@inproceedings{Felder1996v15, title={v 1 5 J un 1 99 6 SOLUTIONS OF THE ELLIPTIC QKZB EQUATIONS AND BETHE}, author={Giovanni Felder and V. Tarasov and A . VARCHENKO}, year={1996} }