Renyi Entropies, the Analytic Bootstrap, and 3D Quantum Gravity at Higher Genus

Abstract

We compute the contribution of the vacuum Virasoro representation to the genus-two partition function of an arbitrary CFT with central charge $c>1$. This is the perturbative pure gravity partition function in three dimensions. We employ a sewing construction, in which the partition function is expressed as a sum of sphere four-point functions of Virasoro vacuum descendants. For this purpose, we develop techniques to efficiently compute correlation functions of holomorphic operators, which by crossing symmetry are determined exactly by a finite number of OPE coefficients; this is an analytic implementation of the conformal bootstrap. Expanding the results in $1/c$, corresponding to the semiclassical bulk gravity expansion, we find that---unlike at genus one---the result does not truncate at finite loop order. Our results also allow us to extend earlier work on multiple-interval Renyi entropies and on the partition function in the separating degeneration limit.

63 pages + refs

References in corpus (8)

• Three-Dimensional Gravity Revisited
• Virasoro Conformal Blocks and Thermality from Classical Background Fields
• Entanglement Entropy at Large Central Charge
• String Universality for Permutation Orbifolds
• Single Interval Rényi Entropy At Low Temperature
• A universal correction to higher spin entanglement entropy
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• The $\mathcal{W}$-algebra bootstrap of 6d $\mathcal{N}=(2,0)$ theories
• A universal sum over topologies in 3d gravity