On the rational solutions of the su(2)_k Knizhnik-Zamolodchikov equation

TL;DR

This paper investigates rational solutions of the Knizhnik-Zamolodchikov equation in the SU(2)_k Wess-Zumino-Novikov-Witten model, introducing new solutions for non-integrable representations and providing a new perspective on known solutions.

Contribution

It offers a novel description of rational solutions as braid-invariant combinations and presents new solutions for non-integrable isospin representations.

Findings

Classified rational solutions for integrable representations.

Introduced new solutions for non-integrable isospins I = k+1.

Provided a braid-invariant formulation of solutions.

Abstract

We present some new results on the rational solutions of the Knizhnik-Zamolodchikov equation for the four-point conformal blocks of isospin I primary fields in the SU(2)_k Wess-Zumino-Novikov-Witten model. The rational solutions corresponding to integrable representations of the affine algebra su(2)_k have been classified by Michel, Stanev and Todorov; provided that the conformal dimension is an integer, they are in one-to-one correspondence with the local extensions of the chiral algebra. Here we give another description of these solutions as specific braid-invariant combinations of the so called regular basis and display a new series of rational solutions for isospins I = k+1 corresponding to non-integrable representations of the affine algebra.