Off-shell Partition Functions in 3d Gravity

TL;DR

This work develops a canonical quantization approach to 3d gravity with negative cosmological constant, focusing on chiral gravity whose phase space is the moduli space of Riemann surfaces. Partition functions on Σ×S^1 are computed via geometric quantization and an index theorem, with equivariant localization reducing off-shell integrals to finite moduli-space integrals; a fake partition function is isolated to capture non-oscillatory contributions. The authors prove a dual topological recursion that computes these fake partition functions for arbitrary Σ and demonstrate a JT gravity limit through a double-scaling procedure, connecting to known holographic and Mirzakhani-type results. They also discuss the limitations, such as divergences in non-chiral gravity and the sensitivity to compactification choices, and outline rich avenues for extending the framework to twisted sectors, conical defects, and supersymmetric generalizations.

Abstract

We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of $\mathrm{PSL}(2,\mathbb{R})$ Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface $Σ$ is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form $Σ\times \mathrm{S}^1$, where $Σ$ can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of $n$ asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over $\overline{\mathcal{M}}_{g,n}$, which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton's constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces $Σ$. There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.

Figures (4)

• Figure 1: A typical surface in $\mathscr{M}_2^{(1)}$. The red wiggly surface is schematically the cutoff surface and the black denotes the marked point on it.
• Figure 2: The fixed point set in $\overline{\mathscr{M}}_{g,m}^{(n)}$ is given by $\overline{\mathcal{M}}_{g,n+m}$. Here we drew a $g=1$ surface with three asymptotic boundaries and two marked points. In order for the surface to be invariant under the rotations of the three boundaries, it has to be nodal and the boundary cut offs (that we drew in red) are round. Thus the fixed point set is naturally isomorphic to $\overline{\mathcal{M}}_{1,5}$ where two of the punctures are the punctures that we denoted by a cross in the figure and three punctures correspond to the nodes in the drawing.
• Figure 3: This picture illustrates the reduction from the full partition function to the primary partition function and the reduction of the associated moduli spaces. Here we drew the case $g=1$ and $n=2$.
• Figure 5: A pictorial depiction of the dilaton equation for $g=1$ and $n=1$.