Notes on the Verlinde formula in nonrational conformal field theories

Abstract

We review and extend evidence for the validity of a generalized Verlinde formula, in particular, nonrational conformal field theories. We identify a subset of representations of the chiral algebra in nonrational 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 ${H}_{3}^{+}$. We analyze the three-point functions of Liouville theory and of ${H}_{3}^{+}$ in detail to directly identify the fusion coefficients from the operator product expansion. Moreover, we check the validity of a proposed generic formula for localized brane one-point functions in nonrational conformal field theories.