The crossing matrices of WZW SU(2) model and minimal models with the quantum 6 j symbols

Abstract

We directly derive the crossing (fusion and braiding) matrices for arbitrary isospins of the WZW SU(2) model in the Feigin-Fuchs integral representation and prove explicitly that they are quantum Racah-Wigner 6 j matrices under proper normalization conditions. For integer k of the Kac-Moody algebra, the matrices may be truncated and the admissible states are closed under analytic continuation. The non-admissible terms will not appear in the other channels after braiding. The consistency of crossing matrices with the locality condition is also proven. Similar results are obtained for the minimal models, which are in agreement with the results obtained by G. Felder, J. Frölich and G. Keller.