Modular data and Verlinde formulae for fractional level WZW models II

This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac–Moody algebra image at an arbitrary admissible level k. Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce non-negative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck fusion rules are determined explicitly. These rules reproduce the well-known “fusion rules” of Koh and Sorba, negative coefficients included, upon quotienting the Grothendieck fusion ring by a certain ideal.