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Conformal blocks for the WZW model on tori can be represented by vectorvalued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfyingvanishing conditions on root hyperplanes. We introduce a quantum version ofthese vanishing conditions in the sl(2) case. They are compatible with the qKZBequations and are obeyed by the hypergeometric solutions as well as by theircritical level counterpart, which are Bethe eigenfunctions of IRF row-to-rowtransfer matrices. In the language of IRF models the vanishing conditions turnout to be equivalent to the sl(2) fusion rules defining restricted models.

communications in contemporary mathematics

RESONANCE RELATIONS FOR SOLUTIONS OF THE ELLIPTIC QKZB EQUATIONS, FUSION RULES, AND EIGENVECTORS OF TRANSFER MATRICES OF RESTRICTED INTERACTION-ROUND-A-FACE MODELS