On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories

TL;DR

This paper compares two methods for calculating fusion rules in c_{p,1} logarithmic conformal field theories, extending the Verlinde formula to include indecomposable representations and providing explicit fusion rules.

Contribution

It develops a non-semisimple Verlinde formula extension and demonstrates the equivalence of two fusion rule calculation methods for c_{p,1} theories.

Findings

Both methods yield equivalent fusion rules.

Explicit fusion rules are derived for all p > 2.

The extended Verlinde formula accounts for indecomposable representations.

Abstract

Two different approaches to calculate the fusion rules of the c_{p,1} series of logarithmic conformal field theories are discussed. Both are based on the modular transformation properties of a basis of chiral vacuum torus amplitudes, which contains the characters of the irreducible representations. One of these is an extension, which we develop here for a non-semisimple generalisation of the Verlinde formula introduced by Fuchs et al., to include fusion products with indecomposable representations. The other uses the Verlinde formula in its usual form and gets the fusion coefficients in the limit, in which the basis of torus amplitudes degenerates to the linear dependent set of characters of irreducible and indecomposable representations. We discuss the effects, which this linear dependence has on any result for fusion rules, which are calculated from these character's modular transformation properties. We show that the two presented methods are equivalent. Furthermore we calculate explicit BPZ-like expressions for the resulting fusion rules for all p larger than 2.