Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

TL;DR

This paper develops methods to compute and relate multi-instanton corrections in matrix models and topological strings, providing a nonperturbative framework that connects formal series to physical definitions.

Contribution

It introduces trans-series solutions for recursion relations in matrix models and applies Borel resummation to establish nonperturbative definitions, extending to topological string theories.

Findings

Derived multi-instanton corrections in specific matrix models.

Linked formal series to nonperturbative definitions via resurgent analysis.

Highlighted the relevance of trans-series in topological string theory.

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