Solving 3d Gravity with Virasoro TQFT

TL;DR

The paper formulates 3d gravity with negative cosmological constant as a Virasoro TQFT obtained from quantizing Teichmüller space, thereby replacing the traditional SL(2,ℝ) Chern–Simons viewpoint with a framework built on Virasoro conformal blocks and Liouville theory. It provides a concrete, algorithmic method to compute gravity partition functions on a fixed hyperbolic topology by sewing Virasoro TQFT blocks and summing over boundary mapping-class images, while carefully handling the noncompact Hilbert space and framing anomalies. A central result is the gravity–Virasoro correspondence Z_grav(M) = 1/|Map0(M,∂M)| ∑γ |Z_Vir(M^γ)|^2, with detailed constructions for inner products, wormholes, and Heegaard splittings, giving a toolkit to explore hyperbolic 3-manifolds, volume conjectures, and Liouville structure constants in a gravitational setting. The work also clarifies the relationship to JT gravity, discusses the prospects and limitations of naive surgery, and sketches extensions to higher spin, supersymmetric, and boundary-CFT ensemble contexts, positioning Virasoro TQFT as a powerful computational bridge between bulk gravity and boundary conformal data.

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.

Figures (10)

• Figure 1: Pair of pants decomposition of a genus 1 surface with two punctures. We also draw the dual graph in red.
• Figure 2: The mapping class group of thermal AdS$_3$ is generated by Dehn twists around the contractible disk drawn in red. This means that we cut the solid torus along the red disk, twist it by 360 degrees and then glue it back.
• Figure 3: The once-punctured torus fibered over the circle. The mapping torus is specified by the strength of the puncture and the mapping class group element $\gamma\in \text{PSL}(2,\mathbb{Z})$.
• Figure 4: A genus-2 Heegaard splitting.
• Figure 5: The Heegaard splitting of $\text{S}^3$ into two solid tori. We view $\text{S}^3$ as the one-point compactification of $\mathbb{R}^3$. One solid torus is drawn in black and the other complementary torus in blue, which in particular includes the compactification point.