Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations

Abstract

The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{τ,η}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in terms of elliptic hypergeometric functions and decribe the monodromy properties of solutions. It turns out that the monodromy transformations of solutions are described in terms of elliptic R-matrices associated with pairs of representations of the "dual" elliptic quantum group $E_{p,η}(sl_2)$, where $p$ is the step of the difference equations. Our description of the monodromy is analogous to the Kohno-Drinfeld description the monodromy group of solutions of the KZ differential equations associated to a simple Lie algebra in terms of the corresponding quantum group.