Theta function solutions of the qKZB equation for a face model

TL;DR

This paper provides explicit theta function solutions to the qKZB equation for an elliptic face model, demonstrating their role as eigenvectors of the transfer matrix in a specific combinatorial case.

Contribution

It introduces explicit theta function solutions to the qKZB equation for the SOS face model and links them to eigenvectors of the transfer matrix on the combinatorial line.

Findings

Solutions are expressed as theta functions.

On the combinatorial line, solutions are eigenvectors of the transfer matrix.

Establishes a connection between the qKZB solutions and the three-colour model.

Abstract

We consider the quantum Knizhnik-Zamolodchikov-Bernard equation for a face model with elliptic weights, the SOS model. We provide explicit solutions as theta functions. On the so-called combinatorial line, in which the model is equivalent to the three-colour model, these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions.