The origin of multiplets of chiral fields in SU(2)_k WZNW at rational level

TL;DR

This paper investigates the structure of solutions to the Knizhnik-Zamolodchikov equation in SU(2)_k WZNW models at rational levels, revealing multiplets of chiral fields with specific braid and quantum group properties.

Contribution

It demonstrates the existence of multiplets of chiral fields with SU(2) quantum numbers in rational level SU(2)_k WZNW models, clarifying their braid and quantum group representations.

Findings

Identifies solutions forming multiplets closed under braid matrices.

Shows these fields have integer or half-integer conformal dimensions.

Connects the structure to quantum group representations at roots of unity.

Abstract

We study solutions of the Knizhnik-Zamolodchikov equation for discrete representations of SU(2)_k at rational level k+2=p/q using a regular basis in which the braid matrices are well defined for all spins. We show that at spin J=(j+1)p-1 for half integer j there are always a subset of 2j+1 solutions closed under the action of the braid matrices. For integer j these fields have integer conformal dimension and all the 2j+1 solutions are monodromy free. The action of the braid matrices on these can be consistently accounted for by the existence of a multiplet of chiral fields with extra SU(2) quantum numbers (m=-j,...,j). In the quantum group SU_q(2), with q=e^{\f{-i \pi}{k+2}}, there is an analogous structure and the related representations are trivial with respect to the standard generators but transform in a spin j representation of SU(2) under the extended center.