Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Jasper V Stokman

doi:10.1007/s00220-015-2375-z

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.

Highlights

In Cherednik [10] associates to an abstract affine R-matrix {Rα}α, labelled by the roots α of an affine root system, a compatible system of equations called quantum affine KZ equations
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
Besides the quantum group approach, which always leads to quantum affine KZ equations of classical type, one can attach quantum affine KZ equations to affine Hecke algebra modules

Summary

Introduction

In Cherednik [10] associates to an abstract affine R-matrix {Rα}α, labelled by the roots α of an affine root system, a compatible system of equations called quantum affine KZ equations. Let f : Cn → C2 ⊗n be a C2 ⊗n-valued meromorphic function on Cn. We say that f is a solution of the boundary quantum KZ equations associated to the spin representation π sp = π(sξp) if f satisfies the difference equations

Results

Conclusion