3d Gravity as a random ensemble

TL;DR

This paper provides evidence that a matrix-tensor model is dual to 3D gravity including topological sums, extending the duality between random matrix models and gravitational theories to three dimensions with connections to CFT data and topological quantum field theory.

Contribution

It demonstrates how the matrix-tensor ensemble encodes 3D gravity with topology sum, relating Feynman rules to three-manifold surgery and connecting to Virasoro TQFT.

Findings

Partition functions match VTQFT on hyperbolic manifolds.

Differences arise on non-hyperbolic geometries affecting eigenvalue statistics.

Schwinger-Dyson equations relate to three-manifold combinatorics.

Abstract

We give further evidence that the matrix-tensor model studied in \cite{belin2023} is dual to AdS$_{3}$ gravity including the sum over topologies. This provides a 3D version of the duality between JT gravity and an ensemble of random Hamiltonians, in which the matrix and tensor provide random CFT$_2$ data subject to a potential that incorporates the bootstrap constraints. We show how the Feynman rules of the ensemble produce a sum over all three-manifolds and how surgery is implemented by the matrix integral. The partition functions of the resulting 3d gravity theory agree with Virasoro TQFT (VTQFT) on a fixed, hyperbolic manifold. However, on non-hyperbolic geometries, our 3d gravity theory differs from VTQFT, leading to a difference in the eigenvalue statistics of the associated ensemble. As explained in \cite{belin2023}, the Schwinger-Dyson (SD) equations of the matrix-tensor integral play a crucial role in understanding how gravity emerges in the limit that the ensemble localizes to exact CFT's. We show how the SD equations can be translated into a combinatorial problem about three-manifolds.