Fusion Rules in WZW Models

Abstract

This chapter is mainly concerned with the calculation of fusion rules in WZW models. As in any rational conformal field theory, fusion rules can be calculated from the Verlinde formula, in terms of the modular transformation matrices S. For WZW models, all these matrices are known explicitly. However, working out a few examples is convincing enough that the Verlinde formula is not very convenient for explicit calculations. Indeed, the dimension of the matrix S grows very quickly with the level and the rank of the algebra. Nevertheless, the formula itself allows us to derive useful identities (which is done in Sect. 16.1), in addition to being the starting point for a more efficient approach: the affine extension of the character method used for the calculation of tensor products in finite Lie algebra. This method is described in Sect. 16.2. It leads to a very nice relation between tensor-product coefficients and fusion coefficients. The concept of quantum dimension, naturally related to the character method, is introduced in Sect. 16.3.KeywordsFusion RuleQuantum DimensionOuter AutomorphismElementary CouplingFusion CoefficientThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.