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

TL;DR

This work develops a non-perturbative extension of the Gel’fand–Dikii ODE method to extract Weil–Petersson volumes and intersection-theory data from double-scaled matrix models, incorporating resurgence to form a full transseries. It systematically builds ZZ, FZZT, and mixed ZZ–FZZT sectors for the one-point function via an extended transseries for the resolvent and confirms consistency with non-perturbative topological recursion and WKB results in the (2,3) minimal string. The authors derive general large-order growth predictions for $V_{g,1}(b)$, validating them in JT gravity and various JT supergravity theories, including a proof of a Stanford–Witten conjecture for $ ext{N}=1$ JT supergravity and new growth formulae for $ ext{N}=2$ and $ ext{N}=4$ cases. The framework thus provides a compact, recursive route to full non-perturbative data for moduli-space volumes and related intersection numbers, with broad implications for non-perturbative gravity models and their matrix-model realizations.

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 of Eynard et.al. 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.

Figures (8)

• Figure 1: Illustration of a genus three hyperbolic Riemann surface with a single boundary of length $\beta$.
• Figure 2: On the left, a pictorial representation of the cycle $B \subset \Sigma$ and on the right, of the contour $\gamma(z)\subset \Sigma$. The cut connecting $\infty$ to $E_0$ is denoted by a green line while the saddle $E^\star$ (red dot) is represented by a pinch.
• Figure 3: Pictorial representation of the contour $\gamma(z_1,z_2) \subset \Sigma$.
• Figure 4: On the left, a pictorial representation of the contour $\gamma(z) \subset \Sigma$ and on the right, of the contour $\gamma(z_1,z_2)\subset \Sigma$.
• Figure 5: Pictorial representation of the contour $\mathcal{C}_{\text{res}}$.