The Verlinde formula in logarithmic CFT

TL;DR

This paper reviews and refines a recent formalism for applying the Verlinde formula to logarithmic conformal field theories, addressing challenges in computing fusion rules in these complex models.

Contribution

It introduces improvements to the modular formalism for logarithmic CFTs and discusses methods for determining fusion rules in models with simple current extensions.

Findings

Refined the modular formalism for certain logarithmic theories

Applied the formalism to simple examples for validation

Explored fusion rules in simple current extensions

Abstract

In rational conformal field theory, the Verlinde formula computes the fusion coefficients from the modular S-transformations of the characters of the chiral algebra's representations. Generalising this formula to logarithmic models has proven rather difficult for a variety of reasons. Here, a recently proposed formalism (arXiv:1303.0847 [hep-th]) for the modular properties of certain classes of logarithmic theories is reviewed, and refined, using simple examples. A formalism addressing fusion rules in simple current extensions is also reviewed as a means to tackle logarithmic theories to which the proposed modular formalism does not directly apply.