3d gravity from Virasoro TQFT: Holography, wormholes and knots

TL;DR

This work deepens the Virasoro TQFT program as a precise bulk description of 3D gravity with negative cosmological constant, and substantiates its holographic interpretation by matching gravity partition functions to Virasoro and Teichmüller TQFT constructions. It develops the refined volume conjecture, connects handlebody and non-handlebody saddles to Liouville/CFT data, and demonstrates how multi-boundary wormholes encode higher moments and non-Gaussian corrections in the boundary ensemble. By detailed analysis of the figure eight knot complement, including Dehn surgery, the paper provides strong evidence for the equivalence of Virasoro and Teichmüller TQFTs and showcases a practical toolkit for computing gravity path integrals across a wide class of topologies. Together, these results establish a robust, top-down framework for holographic statistics in AdS3 gravity that unifies bulk geometric data with ensemble-averaged boundary CFT observables.

Abstract

We further develop the description of three-dimensional quantum gravity with negative cosmological constant in terms of Virasoro TQFT formulated in our previous paper arXiv:2304.13650. We compare the partition functions computed in the Virasoro TQFT formalism to the semiclassical evaluation of Euclidean gravity partition functions. This matching is highly non-trivial, but can be checked directly in some examples. We then showcase the formalism in action, by computing the gravity partition functions of many relevant topologies. For holographic applications, we focus on the partition functions of Euclidean multi-boundary wormholes with three-punctured spheres as boundaries. This precisely quantifies the higher moments of the structure constants in the proposed ensemble boundary dual and subjects the proposal to thorough checks. Finally, we investigate in detail the example of the figure eight knot complement as a hyperbolic 3-manifold. We show that the Virasoro TQFT partition function is identical to the partition function computed in Teichmüller theory, thus giving strong evidence for the equivalence of these TQFTs. We also show how to produce a large class of manifolds via Dehn surgery on the figure eight knot.

Figures (12)

• Figure 1: Tetrahedron with dihedral angles specified.
• Figure 2: The Conway knot and its mutant, the Kinoshita-Teresaka knot. They are not equivalent, but the value of $Z_\text{Vir}$ is the same.
• Figure 3: The network of Wilson lines used to derive the relation between the modular crossing kernel $\mathbb{S}$ and the sphere crossing kernel $\mathbb{F}$.
• Figure 4: The Euclidean wormhole with the topology of a three-punctured sphere times an interval that contributes to the variance of the structure constants in the ensemble description of the dual of AdS$_3$ gravity.
• Figure 5: The three-boundary sphere four-point wormhole $M_3$.