Traces of intertwiners for quantum groups and difference equations, II

TL;DR

This paper investigates twisted traces of intertwining operators for quantum affine algebras, revealing they satisfy key difference equations and exhibit symmetries, thus extending previous results to more general settings.

Contribution

It demonstrates that after renormalization, these traces satisfy important difference equations and symmetries, generalizing earlier work to quantum and infinite-dimensional cases.

Findings

Traces satisfy Macdonald-Ruijsenaars and q-KZB difference equations.

Trace functions are symmetric under parameter exchange when twisting automorphism is trivial.

Results extend previous q=1 and finite-dimensional cases to quantum affine algebras.

Abstract

In this paper we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights lambda, mu, three scalar parameters q, omega, k, and spectral parameters z_1,...,z_N, which may be regarded as q-analogs of conformal blocks of the Wess-Zumino-Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined in math.QA/0110081. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations -- the Macdonald-Ruijsenaars equation, the q-Knizhnik-Zamolodchikov-Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation lambda <--> mu, k <--> omega. Thus, our results here generalize our previous results, dealing with the case q = 1 and the finite dimensional case.