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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SpringerOpen
2025-04-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP04(2025)015 |
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| author | Boris Post Ioannis Tsiares |
| author_facet | Boris Post Ioannis Tsiares |
| author_sort | Boris Post |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-19b691e0f404425d881dfd6697ffec21 |
| institution | OA Journals |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | SpringerOpen |
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| series | Journal of High Energy Physics |
| spelling | doaj-art-19b691e0f404425d881dfd6697ffec212025-08-20T01:49:48ZengSpringerOpenJournal of High Energy Physics1029-84792025-04-012025416110.1007/JHEP04(2025)015A non-rational Verlinde formula from Virasoro TQFTBoris Post0Ioannis Tsiares1Institute for Theoretical Physics, University of AmsterdamInstitut de Physique Théorique, Université Paris-Saclay, CNRS, CEAAbstract 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.https://doi.org/10.1007/JHEP04(2025)015Scale and Conformal SymmetriesTopological Field TheoriesAdS-CFT CorrespondenceQuantum Groups |
| spellingShingle | Boris Post Ioannis Tsiares A non-rational Verlinde formula from Virasoro TQFT Journal of High Energy Physics Scale and Conformal Symmetries Topological Field Theories AdS-CFT Correspondence Quantum Groups |
| title | A non-rational Verlinde formula from Virasoro TQFT |
| title_full | A non-rational Verlinde formula from Virasoro TQFT |
| title_fullStr | A non-rational Verlinde formula from Virasoro TQFT |
| title_full_unstemmed | A non-rational Verlinde formula from Virasoro TQFT |
| title_short | A non-rational Verlinde formula from Virasoro TQFT |
| title_sort | non rational verlinde formula from virasoro tqft |
| topic | Scale and Conformal Symmetries Topological Field Theories AdS-CFT Correspondence Quantum Groups |
| url | https://doi.org/10.1007/JHEP04(2025)015 |
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