Modular Data and Verlinde Formulae for Fractional Level WZW Models II

Abstract

This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl(2) 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.

30 pages, 1 figure, several tables; v2 fixed minor typos

References in corpus (4)

• sl^(2)_{-1/2}: A Case Study
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Free fermion resolution of supergroup WZNW models
• The Extended Algebra of the SU(2) Wess-Zumino-Witten Models

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