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 ≥ 25. The identity expresses the Virasoro fusion kernel as an integral over a ratio of modular S-kernels on the (punct...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-04-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP04(2025)015 |
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| Summary: | 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 ≥ 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 AdS3/CFT2, 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. |
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| ISSN: | 1029-8479 |