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.

39 pages

References in corpus (2)

• The elliptic gamma function and SL(3,Z) x Z^3
• Special functions, conformal blocks, Bethe ansatz, and SL(3,Z)

Cited by in corpus (10)

• Hypergeometric theta functions and elliptic Macdonald polynomials
• Basic Properties of Non-Stationary Ruijsenaars Functions
• A gerbe for the elliptic gamma function
• Multiple sine, multiple elliptic gamma functions and rational cones
• Non-stationary difference equation for q-Virasoro conformal blocks
• Traces of intertwiners for quantum affine sl_2 and Felder-Varchenko functions
• Affine Macdonald conjectures and special values of Felder-Varchenko functions
• Traces of intertwiners for quantum affine algebras and difference equations (after Etingof-Schiffmann-Varchenko)
• The orthogonality and qKZB-heat equation for traces of U_q(g)-intertwiners
• Elliptic gamma functions, triptic curves and SL_3(Z)