Q-deformed KZB heat equation: Completeness, modular properties and SL(3,ℤ)

Abstract

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. © 2002 Elsevier Science (USA).