A non-rational Verlinde formula from Virasoro TQFT

Abstract

We use the Virasoro TQFT to derive an integral identity that we view as a non-rational generalization of the Verlinde formula for the Virasoro algebra with central charge $c\geq 25$. The identity expresses the Virasoro fusion kernel as an integral over a ratio of modular S-kernels on the (punctured) torus. In particular, it shows that the one-point S-kernel diagonalizes the Virasoro $6j$ symbol. After carefully studying the analytic properties of this `Virasoro-Verlinde formula', we present three applications. In boundary Liouville CFT, the formula ensures the open-closed duality of the boundary one-point function on the annulus. In pure 3d gravity, it provides an essential step in computing the partition function on hyperbolic 3-manifolds that fiber over the circle. Lastly, in AdS$_3$/CFT$_2$, the formula computes a three-boundary torus wormhole, which leads to a prediction for the statistical correlation between the density of states and two OPE coefficients in the dual large-$c$ CFT ensemble. We conclude by discussing the implications of our result for the fusion rules in generic non-rational 2d CFTs.