Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

TL;DR

This work defines a trans-series framework to extract nonperturbative, multi-instanton corrections in matrix models with orthogonal-polynomial methods, linking formal expansions to genuine nonperturbative definitions via Borel resummation and resurgence. It demonstrates the approach in the Hermitian quartic and the Gross–Witten–Wadia unitary models, showing how single- and multi-instanton sectors reproduce and constrain large-order behavior, double-scaling limits, and Painlevé structures. The results illuminate how nonperturbative effects encode holographic, topological-string data, including background-independent sums over instanton sectors and the Hastings–McLeod solution in Painlevé II, providing a blueprint for extending nonperturbative completions to broader string-theoretic contexts. The analysis offers concrete computational schemes for instanton actions, Stokes data, and median resummations, and it clarifies how perturbative ambiguities cancel against nonperturbative contributions in physically meaningful, real solutions.

Abstract

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix model