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

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.