Modular Data and Verlinde Formulae for Fractional Level WZW Models I

TL;DR

This paper advances the understanding of modular properties and Verlinde formula applications in fractional level affine sl(2) theories, resolving previous issues with fusion coefficients and constructing modular invariants.

Contribution

It completes the modular analysis for certain fractional levels, deriving transformations, verifying fusion rules, and constructing invariant partition functions.

Findings

Derived modular transformations for admissible irreducible representations at levels -1/2 and -4/3.

Confirmed the Verlinde formula matches known fusion rules.

Constructed infinite sets of modular invariant partition functions.

Abstract

The modular properties of fractional level affine sl(2)-theories and, in particular, the application of the Verlinde formula, have a long and checkered history in conformal field theory. Recent advances in logarithmic conformal field theory have led to the realisation that problems with fractional level models stem from trying to build the theory with an insufficiently rich category of representations. In particular, the appearance of negative fusion coefficients for admissible highest weight representations is now completely understood. Here, the modular story for certain fractional level theories is completed. Modular transformations are derived for the complete set of admissible irreducible representations when the level is k=-1/2 or k=-4/3. The S-matrix data and Verlinde formula are then checked against the known fusion rules with complete agreement. Finally, an infinite set of modular invariant partition functions is constructed in each case.