Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings

TL;DR

This paper investigates nonperturbative effects in matrix models and topological strings, providing explicit instanton calculations and predicting large-order behavior, supported by extensive numerical validation across multiple models.

Contribution

It offers explicit instanton amplitude results and connects nonperturbative effects with large-order behavior in matrix models and topological strings, including new predictions for Hurwitz numbers and Gromov-Witten invariants.

Findings

Explicit one-instanton amplitude calculations

Predictions for large-order behavior of genus expansion

Numerical validation across various models

Abstract

This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.