Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations

TL;DR

This paper develops integral representations for solutions to boundary quantum Knizhnik-Zamolodchikov equations, involving elliptic and trigonometric weight functions, providing a basis of solutions in tensor product spaces.

Contribution

It introduces a novel integral construction of solutions to boundary qKZ equations using elliptic and trigonometric weights, expanding the analytical tools for these equations.

Findings

Constructed integral solutions form a basis over quasi-constant meromorphic functions.

Solutions involve scalar elliptic and vector trigonometric weight functions.

Provides a new analytical framework for boundary qKZ equations.

Abstract

We construct integral representations of solutions to the boundary quantum Knizhnik-Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine $\mathfrak{sl}_2$, with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.