q-deformed KZB heat equation: completeness, modular properties and SL(3,Z)

We study the properties of one-dimensional hypergeometric integral solutions of the q-difference ("quantum") analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SL(3,Z) transformation properties parallel to those of elliptic gamma functions.Comment: 39 page

Resonance Relations for Solutions of the Elliptic QKZB Equations, Fusion Rules, and Eigenvectors of Transfer Matrices of Restricted Interaction-round-a-face Models

Conformal blocks for the WZW model on tori can be represented by vector valued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfying vanishing conditions on root hyperplanes. We introduce a quantum version of these vanishing conditions in the sl(2) case. They are compatible with the qKZB equations and are obeyed by the hypergeometric solutions as well as by their critical level counterpart, which are Bethe eigenfunctions of IRF row-to-row transfer matrices. In the language of IRF models the vanishing conditions turn out to be equivalent to the sl(2) fusion rules defining restricted models.Comment: 64 pages, late

The q-deformed Knizhnik-Zamolodchikov-Bernard heat equation

We introduce a q-deformation of the genus one sl_2 Knizhnik-Zamolodchikov-Bernard heat equation. We show that this equation for the dependence on the moduli of elliptic curves is compatible with the qKZB equations, which give the dependence on the marked points.Comment: 18 pages, 1 figure, Latex. Revised and expanded version, with a new section on conformal block