The Verlinde formula in logarithmic CFT

David Ridout,Simon Wood

doi:10.1088/1742-6596/597/1/012065

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 [1] 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.

Highlights

Logarithmic conformal field theory typically refers to a two-dimensional conformally invariant quantum field theory for which there exist correlation functions exhibiting logarithmic singularities
The determination of the fusion rules in rational theories is often significantly simplified through the use of the Verlinde formula [45]. This relies on computing the modular S-transformations of the characters of the irreducible representations. It has been known for quite some time [46] that the irreducible characters of a logarithmic conformal field theory need not carry a representation of the modular group, an observation which appears to completely invalidate the application of any Verlinde-type formula
Rather than try to compare and contrast these proposals, which would be a worthy goal in itself, we shall instead turn to a rather different formalism which has recently been introduced [1] for modular properties and Verlinde formulae in a more general class of logarithmic conformal field theories

Summary

Introduction

Logarithmic conformal field theory typically refers to a two-dimensional conformally invariant quantum field theory for which there exist correlation functions exhibiting logarithmic singularities. It has been known for quite some time [46] that the irreducible characters of a logarithmic conformal field theory need not carry a representation of the modular group, an observation which appears to completely invalidate the application of any Verlinde-type formula.

Results

Conclusion