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A non-rational Verlinde formula from Virasoro TQFT

Boris Post, Ioannis Tsiares

TL;DR

This work extends Verlinde-type fusion beyond rational CFTs by deriving a non-rational Verlinde-like formula for Virasoro symmetry at central charge $c\ge 25$ using Virasoro TQFT. The core result expresses the Virasoro fusion kernel as a convergent integral over the ratio of torus S-kernels, with a regulator $P_0$ yielding a well-behaved fusion density ${\mathcal N}_{P_0}$. The authors establish key properties (reality, positivity, degenerate vs non-degenerate behavior, and quantum-group connections) and show that the one-point S-kernel diagonalizes the Virasoro $6j$ symbol, linking to the modular double of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$. They illustrate three applications—open-closed duality in boundary Liouville BCFT, exact gravity partition functions for knot/link complements, and a three-boundary AdS$_3$ wormhole with predictions for universal CFT data—highlighting the formula’s potential to illuminate fusion rules in generic non-rational 2d CFTs. The results also provoke a generalized bootstrap perspective, proposing a framework to constrain the spectral density from fusion data and to explore fusion in non-diagonal Virasoro theories.

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.

A non-rational Verlinde formula from Virasoro TQFT

TL;DR

This work extends Verlinde-type fusion beyond rational CFTs by deriving a non-rational Verlinde-like formula for Virasoro symmetry at central charge using Virasoro TQFT. The core result expresses the Virasoro fusion kernel as a convergent integral over the ratio of torus S-kernels, with a regulator yielding a well-behaved fusion density . The authors establish key properties (reality, positivity, degenerate vs non-degenerate behavior, and quantum-group connections) and show that the one-point S-kernel diagonalizes the Virasoro symbol, linking to the modular double of . They illustrate three applications—open-closed duality in boundary Liouville BCFT, exact gravity partition functions for knot/link complements, and a three-boundary AdS wormhole with predictions for universal CFT data—highlighting the formula’s potential to illuminate fusion rules in generic non-rational 2d CFTs. The results also provoke a generalized bootstrap perspective, proposing a framework to constrain the spectral density from fusion data and to explore fusion in non-diagonal Virasoro theories.

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 . 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 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/CFT, 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- CFT ensemble. We conclude by discussing the implications of our result for the fusion rules in generic non-rational 2d CFTs.

Paper Structure

This paper contains 45 sections, 183 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. A Verlinde formula for the Virasoro algebra
  3. Rational CFT recap
  4. Virasoro TQFT proof
  5. Two ways of computing a 3d partition function.
  6. Properties of the fusion density
  7. Reality.
  8. Positivity.
  9. Identity selection rule.
  10. Degenerate representations.
  11. Non-degenerate representations.
  12. Quantum group symmetry and the Virasoro $6j$ symbol.
  13. Commutativity of the fusion algebra.
  14. Associativity of the fusion algebra.
  15. Boundedness of the fusion density.
  16. ...and 30 more sections

Figures (1)

  • Figure 1: Three applications of the Virasoro-Verlinde formula: $a)$ open-closed duality of the annulus one-point function in boundary Liouville CFT; $b)$ gravitational path integrals on knot and link complements in $S^3$ that fiber over the circle; $c)$ a three-boundary torus wormhole in AdS$_3$, with new implications for the statistical properties of universal CFT$_2$ data.