The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation

TL;DR

This paper explores nonperturbative solutions in matrix models and string theories using resurgent transseries, focusing on multi-cut solutions, Stokes phases, and the Painleve II equation, revealing detailed large-order behaviors and explicit formulas.

Contribution

It provides the first explicit nonperturbative solutions for multi-cut matrix models' free energies and analyzes their large-order asymptotics, especially for the Painleve II equation.

Findings

Explicit formulas for Z_2 symmetric two-cut models

Analysis of large-order behavior in multi-instanton sectors

Nonperturbative solutions for Painleve II and related models

Abstract

Resurgent transseries have recently been shown to be a very powerful construction in order to completely describe nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where an 't Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study of Stokes phases associated to multi-cut solutions of generic matrix models, constructing nonperturbative solutions for their free energies and exploring the asymptotic large-order behavior around distinct multi-instanton sectors. Explicit formulae are presented for the Z_2 symmetric two-cut set-up, addressing the cases of the quartic matrix model in its two-cut Stokes phase; the "triple" Penner potential which yields four-point correlation functions in the AGT framework; and the Painleve II equation describing minimal superstrings.