On the rational solutions of the Knizhnik-Zamolodchikov equation

Abstract:

We present some new results on the rational solutions of the Knizhnik-Zamolodchikov (KZ) equation for the four-point conformal blocks of isospin I primary fields in the SU (2) _k Wess-Zumino-Novikov-Witten (WZNW) model. The rational solutions corresponding to integrable representations of the affine algebra (2) _k have been classified in [#!MST!#,#!ST!#]; 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 introduced in [#!STH!#] and display a new series of rational solutions for isospins I = k + 1,  k ∈ N corresponding to non-integrable representations of (2) _k .