On random matrix statistics of 3d gravity

TL;DR

This paper demonstrates that 3d gravity on certain manifolds can be described by a random matrix model, specifically the Virasoro minimal string, with explicit calculations confirming the correspondence.

Contribution

It establishes a connection between 3d gravity on Riemann surface times interval manifolds and the Virasoro minimal string, providing explicit path integral evaluations.

Findings

Path integrals match universal random matrix expressions for specific cases.

Spectral correlators of open strings emerge from the gravitational path integrals.

Explicit evaluation of the path integral as a gravitational inner product for certain surfaces.

Abstract

We show that 3d gravity on manifolds that are topologically a Riemann surface times an interval $\Sigma_{g,n}\times I$ with end-of-the-world branes at the ends of the interval is described by a random matrix model, namely the Virasoro minimal string. Because these manifolds have $n$ annular asymptotic boundaries, the path integrals naturally correspond to spectral correlators of open strings upon inverse Laplace transforms. For $g=0$ and $n=2$, we carry out an explicit path integration and find precise agreement with the universal random matrix expression. For Riemann surfaces with negative Euler characteristic, we evaluate the path integral as a gravitational inner product between states prepared by two copies of Virasoro TQFT. Along the way, we clarify the effects of gauging the mapping class group and the connection to chiral 3d gravity.