Solving 3d Gravity with Virasoro TQFT

Abstract

We propose a precise reformulation of 3d quantum gravity with negative cosmological constant in terms of a topological quantum field theory based on the quantization of the Teichmüller space of Riemann surfaces that we refer to as ``Virasoro TQFT.'' This TQFT is similar, but importantly not equivalent, to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory. This sharpens the folklore that 3d gravity is related to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory into a precise correspondence, and resolves some well-known issues with this lore at the quantum level. Our proposal is computationally very useful and provides a powerful tool for the further study of 3d gravity. In particular, we explain how together with standard TQFT surgery techniques this leads to a fully algorithmic procedure for the computation of the gravity partition function on a fixed topology exactly in the central charge. Mathematically, the relation leads to many nontrivial conjectures for hyperbolic 3-manifolds, Virasoro conformal blocks and crossing kernels.

72 pages

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• A microscopic realization of dS${}_3$
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• 3d Gravity as a random ensemble
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• TQFT gravity and ensemble holography
• Multipartite edge modes and tensor networks
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• Black hole wavefunctions and microcanonical states
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• Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble
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