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Solving 3d gravity with Virasoro TQFT

Scott Collier, Lorenz Eberhardt, Mengyang Zhang

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SciPost Phys. 15, 151 (2023) · published 11 October 2023


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 Teichm├╝ller space of Riemann surfaces that we refer to as "Virasoro TQFT". This TQFT is similar, but importantly not equivalent, to SL(2, $\mathbb{R}$) Chern-Simons theory. This sharpens the folklore that 3d gravity is related to 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.

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