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

TL;DR

This paper investigates whether a Verlinde-type relation between modular data and fusion extends to non-rational CFTs. By analyzing bosonic Liouville theory, the $H_3^+$ model, and the supersymmetric $SL(2,\mathbb{R})/U(1)$ coset, the authors show that a generalized Verlinde formula applies to degenerate representations via analytic continuation, with fusion coefficients emerging as residues of a complexified Fourier transform of modular data. They relate these results to boundary CFT via Cardy-type brane calculations and demonstrate that the same kernel structure underlying rational theories can encode fusion in non-rational settings. These findings illuminate the algebraic structure of non-rational CFTs and provide tools for constructing localized branes in non-compact backgrounds, with potential extensions to broader classes of theories, including higher supersymmetry.

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.

Figures (6)

• Figure 1: Integration contour and poles for the operator product expansion of a degenerate field and a non-degenerate field.
• Figure 2: Integration contour and poles for the operator product expansion of a degenerate field and a non-degenerate field.
• Figure 3: Integration contour and poles for the operator product expansion of two degenerate fields.
• Figure 4: Integration contour and poles for the operator product expansion of two non-degenerate fields.
• Figure 5: Integration contour and poles for the operator product expansion of a degenerate field and a non-degenerate field.