Resurgent Stokes Data for Painleve Equations and Two-Dimensional Quantum (Super) Gravity

TL;DR

This paper computes the complete resurgent Stokes data for Painleve I and II equations, revealing their structures and implications for quantum gravity and string theories, using a novel 'closed-form asymptotics' method.

Contribution

It introduces a new analytical method to determine the full Stokes data for nonlinear, resonant Painleve equations, connecting resurgent analysis with quantum gravity models.

Findings

Complete analytical Stokes data for Painleve I and II

Explicit connection-formulae for Stokes phenomena

Validation through high-precision numerical tests

Abstract

Resurgent-transseries solutions to Painleve equations may be recursively constructed out of these nonlinear differential-equations -- but require Stokes data to be globally defined over the complex plane. Stokes data explicitly construct connection-formulae which describe the nonlinear Stokes phenomena associated to these solutions, via implementation of Stokes transitions acting on the transseries. Nonlinear resurgent Stokes data lack, however, a first-principle computational approach, hence are hard to determine generically. In the Painleve I and Painleve II contexts, nonlinear Stokes data get further hindered as these equations are resonant, with non-trivial consequences for the interconnections between transseries sectors, bridge equations, and associated Stokes coefficients. In parallel to this, the Painleve I and Painleve II equations are string-equations for two-dimensional quantum (super) gravity and minimal string theories, where Stokes data have natural ZZ-brane interpretations. This work computes for the first time the complete, analytical, resurgent Stokes data for the first two Painleve equations, alongside their quantum gravity or minimal string incarnations. The method developed herein, dubbed "closed-form asymptotics", makes sole use of resurgent large-order asymptotics of transseries solutions -- alongside a careful analysis of the role resonance plays. Given its generality, it may be applicable to other distinct (nonlinear, resonant) problems. Results for analytical Stokes coefficients have natural structures, which are described, and extensive high-precision numerical tests corroborate all analytical predictions. Connection-formulae are explicitly constructed, with rather simple and compact final results encoding the full Stokes data, and further allowing for exact monodromy checks -- hence for an analytical proof of our results.