JT gravity and the asymptotic Weil-Petersson volume

TL;DR

This paper investigates the asymptotic behavior of Weil-Petersson volumes in JT gravity for large genus, using PDEs and conjecturing formulas for general boundary cases, advancing understanding of moduli space integrals in quantum gravity.

Contribution

It introduces new asymptotic estimates for Weil-Petersson volumes in JT gravity and proposes a conjecture for their general behavior at large genus.

Findings

Asymptotic estimates for $V_{g,2}$ and $V_{g,3}$ at large $g$

Use of PDEs to analyze Weil-Petersson volumes

Conjecture on the general form of $V_{g,n}$ for large $g$

Abstract

A path integral in Jackiw-Teitelboim (JT) gravity is given by integrating over the volume of the moduli of Riemann surfaces with boundaries, known as the "Weil-Petersson volume," together with integrals over wiggles along the boundaries. The exact computation of the Weil-Petersson volume $V_{g,n}(b_1, \ldots, b_n)$ is difficult when the genus $g$ becomes large. We utilize two partial differential equations known to hold on the Weil-Petersson volumes to estimate asymptotic behaviors of the volume with two boundaries $V_{g,2}(b_1, b_2)$ and the volume with three boundaries $V_{g,3}(b_1, b_2, b_3)$ when the genus $g$ is large. Furthermore, we present a conjecture on the asymptotic expression for general $V_{g,n}(b_1, \ldots, b_n)$ with $n$ boundaries when $g$ is large.