On algebraic equations satisfied by hypergeometric solutions of the qKZ equation

TL;DR

This paper studies the algebraic equations satisfied by hypergeometric solutions of the qKZ equation at resonance steps, revealing their values lie in a special invariant subbundle linked to quantum groups.

Contribution

It demonstrates that at resonance steps, hypergeometric solutions of the qKZ equation are confined to an invariant subbundle and characterizes this space using quantum group theory.

Findings

Hypergeometric solutions lie in the subbundle of quantized conformal blocks at resonance.

The solutions span the entire subbundle under certain conditions.

The space of solutions is described via the quantum group U_q(sl(2)).

Abstract

We consider the $sl(2)$ quantized Knizhnik-Zamolodchikov equation (qKZ), defined in terms of rational R-matrices. The properties of the equation change when the step of the equation takes a resonance value. In this case the discrete connection defined by the qKZ equation has a invariant subbundle which we call the subbundle of quantized conformal blocks. Solutions of the qKZ equation were constructed in [TV1], [MV1] in terms of multidimensional hypergeometric integrals. In this paper we show that for a resonance step all hypergeometric solutions take values in the subbundle of quantized conformal blocks, moreover the values span the subbundle of quantized conformal blocks under certain conditions. We describe the space of hypergeometric solutions in terms of the quantum group $U_qsl(2)$.