Solving 3d Gravity with Virasoro TQFT

TL;DR

This paper introduces Virasoro TQFT, a topological quantum field theory reformulation of 3d quantum gravity with negative cosmological constant, enabling exact computations of partition functions and revealing deep mathematical conjectures.

Contribution

It provides a precise reformulation of 3d gravity as Virasoro TQFT, distinct from but related to SL(2,R) Chern-Simons theory, and offers an algorithmic method for computing gravity partition functions.

Findings

Enables exact computation of gravity partition functions on fixed topologies.

Establishes connections to hyperbolic 3-manifolds and Virasoro conformal blocks.

Resolves quantum-level issues in the folklore relating 3d gravity to SL(2,R) Chern-Simons theory.

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 Teichm\"uller 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.