Non-perturbative data for Weil-Petersson volumes and intersection numbers using ordinary differential equations

TL;DR

This paper develops a non-perturbative method using ordinary differential equations to compute Weil-Petersson volumes and intersection numbers, including effects of branes, and applies it to various models of 2D gravity, confirming known results and making new predictions.

Contribution

It extends existing methods to extract non-perturbative data from ODEs for Weil-Petersson volumes, incorporating brane effects and providing new growth predictions for JT supergravity.

Findings

Computed non-perturbative quantities for Weil-Petersson volumes.

Validated predictions against topological recursion and JT gravity results.

Proved a conjecture of Stanford and Witten for N=1 JT supergravity.

Abstract

Recently, a new method was introduced for computing $V_{g,1}(b)$, the Weil-Petersson volumes of the moduli space of Riemann surfaces of genus $g$ with one geodesic boundary of length $b$, various supersymmetric generalizations of them, as well as analogous quantities in intersection theory. The physical setting is the computation of a certain one-point function in a variety of models of 2D gravity for which there is a double-scaled random matrix model (RMM) description. The method combines perturbative solutions of two ordinary differential equations (ODEs), the Gel'fand-Dikii resolvent equation, and the RMM's string equation. In this paper, we extend the method to extract non-perturbative information about the $V_{g,1}(b)$ (and their analogues) that is naturally contained in the full ODEs, providing an efficient prescription for computing the transseries coefficients of the one-point correlation function, fully incorporating ZZ-brane and FZZT-brane effects, and for the first time, mixed ZZ-FZZT-effects. We use as a case study the (2,3) minimal string, computing perturbative and non-perturbative quantities, comparing them to perturbative results from topological recursion, and to results from the recent non-perturbative topological recursion framework. As a particularly powerful further application we provide general predictions for the large order in $g$ growth of $V_{g,1}(b)$, and apply them to JT gravity, finding agreement with known results, and for analogous quantities in ${N} {=} 1$ JT supergravity, proving a conjecture of Stanford and Witten. Our predictions yield new growth formulae for the cases of ${N} {=} 2$ and ${N}{=}4$ JT supergravity.