Modular Data and Verlinde Formulae for Fractional Level WZW Models II

TL;DR

This paper thoroughly analyzes the modular properties and Verlinde formula for fractional level WZW models based on affine Kac-Moody algebra sl(2), providing explicit fusion rules and character transformations.

Contribution

It offers a complete account of the modular transformations and Verlinde formula for admissible levels, including explicit fusion rules and Grothendieck fusion coefficients.

Findings

Derived modular transformations for all irreducible admissible modules.

Applied a continuous Verlinde formula to obtain non-negative integer fusion coefficients.

Explicitly determined Grothendieck fusion rules matching known fusion rules with negative coefficients.

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.