Notes on the Verlinde formula in non-rational conformal field theories

TL;DR

This paper investigates the Verlinde formula in non-rational conformal field theories, extending its applicability by identifying relevant representations and analyzing three-point functions in Liouville theory and H3+.

Contribution

It introduces a generalized Verlinde formula for non-rational CFTs and identifies a subset of representations that relate modular S-matrices to fusion coefficients.

Findings

Extended the Verlinde formula to non-rational CFTs.

Identified representations that relate S-matrices and fusion coefficients.

Analyzed three-point functions in Liouville and H3+ theories.

Abstract

We review and extend evidence for the validity of a generalized Verlinde formula in particular non-rational conformal field theories. We identify a subset of representations of the chiral algebra in non-rational conformal field theories that give rise to an analogue of the relation between modular S-matrices and fusion coefficients in rational conformal field theories. To that end we review and extend the Cardy-type brane calculations in bosonic and supersymmetric Liouville theory (and its duals) as well as in the hyperbolic three-plane H3+. We analyze the three-point functions of Liouville theory and of H3+ in detail to directly identify the fusion coefficients from the operator product expansion.