Statistics of three-dimensional black holes from Liouville line defects

Abstract: Black holes and wormholes in the gravitational path integral can be used to calculate the statistics of heavy operators. An explicit example in higher dimensions is provided by thin shells of matter. We study these solutions in 3D gravity, and reproduce the behavior of black holes and wormholes from the dual CFT using the large-$c$ conformal bootstrap. The CFT operator that creates a thin shell black hole is a line defect, so we begin by using the bootstrap to study the statistics of line defects, both at finite $c$ and in the holographic large-$c$ limit. The crossing equation leads to a universal formula for the average high-energy matrix elements of the line defect in any compact, unitary 2d CFT with $c>1$. The asymptotics are controlled by a line defect in Liouville CFT at the same value of the central charge. At large $c$, three distinct quantities are related: The statistics of line defects in holographic CFTs, the individual matrix elements of a line defect in Liouville CFT, and the on-shell action of black holes and wormholes in 3D gravity. The three calculations match for black holes, and if the statistics of the line defects are assumed to be approximately Gaussian, then a class of wormholes is also reproduced by the dual CFT.