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