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

29 pages, 6 figures, v2: minor corrections, PRD version

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• D-branes in non-critical superstrings and duality in N=1 gauge theories with flavor
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• On modular properties of the AdS3 CFT
• Modular invariant partition functions for non-compact G/Ad(H) models