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Journal of Physics A Mathematical and Theoretical

Theta function solutions of the quantum Knizhnik–Zamolodchikov–Bernard equation for a face model

Peter E. Finch, Robert Weston, Paul Zinn-Justin

Published Jan 5, 2016• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/49/6/064001

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Asymptotic solutions to the quantized Knizhnik-Zamolodchikov equation associated with glN+1\frak{gl}_{N+1}glN+1​ are constructed. The leading term of an asymptotic solution is the Bethe vector -- an eigenvector of the transfer-matrix of a quantum spin chain model. We show that the norm of the Bethe vector is equal to the product of the Hessian of a suitable function and an explicitly written rational function. This formula is an analogue of the Gaudin-Korepin formula for the norm of the Bethe vector. It is shown that, generically, the Bethe vectors form a base for the gl2\frak{gl}_2gl2​ case.