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.