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 sym-metry 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 Rényi entropies and on the partition function in the separating degeneration limit.
Highlights
Three-dimensional gravity has proven to be a fruitful testing ground for our ideas about holography
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
The result is a weighted sum over four-point functions of local operators on the sphere
Summary
Three-dimensional gravity has proven to be a fruitful testing ground for our ideas about holography. Perhaps the most interesting question is whether pure general relativity — a theory with only metric degrees of freedom — with a negative cosmological constant exists as a quantum theory in its own right. If this were the case, one should be able to find its holographic dual for a given value of GN /RAdS, the Newton constant in AdS units. From the boundary point of view, it captures the dynamics of the Virasoro sector of any two-dimensional CFT with central charge c = 3RAdS/2GN. This semi-microscopic interpretation is unavailable in higher-dimensional AdS/CFT, where the stress tensor does not generate a closed symmetry algebra
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