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

TL;DR

The paper advances the nonperturbative analysis of the large $N$ limit by extending resurgent transseries to multi-cut Stokes phases in matrix models. It develops a two-parameter transseries framework and validates it across explicit ${ m Z}_2$-symmetric two-cut quartic and triple Penner models, as well as the Painlevé II double-scaling limit, linking eigenvalue tunneling, spectral-curve data, and Stokes coefficients. By combining saddle-point, orthogonal-polynomial, and spectral-geometry methods, the authors derive explicit instanton actions, one-loop factors, and complete nonperturbative free energies, then corroborate them with high-precision large-order data using Richardson extrapolation. The Painlevé II analysis provides a concrete realization of generalized multi-instanton sectors, their logarithmic resonant contributions, and the connection to 2d supergravity/type 0B string theory free energy. Overall, the work strengthens the role of resurgence in decoding the full nonperturbative content of multi-cut gauge/string theories and charts directions toward broader AGT-related applications and higher-cut generalizations.

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.