Connection problems for quantum affine KZ equations and integrable lattice models

TL;DR

This paper constructs explicit solutions and analyzes the connection problems for quantum affine KZ equations related to affine Hecke algebras, with applications to integrable lattice models and elliptic solutions.

Contribution

It provides explicit bases of solutions, computes Weyl group cocycles, and links these to boundary qKZ equations and elliptic solutions in integrable models.

Findings

Constructed bases of solutions for quantum affine KZ equations.

Explicit computation of Weyl group cocycles in terms of theta functions.

Derived elliptic solutions for boundary qKZ equations and related models.

Abstract

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In case of a principal series module we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions. For the spin representation of the affine Hecke algebra of type C the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter's 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations.