Communications in Number Theory and Physics

Volume 17 (2023)

Number 2

Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super) gravity

Pages: 385 – 552

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n2.a5

Authors

Salvatore Baldino (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Ricardo Schiappa (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Maximilian Schwick (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Roberto Vega (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Abstract

Resurgent-transseries solutions to Painlevé 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 Painlevé I and Painlevé 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 Painlevé I and Painlevé 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 conjectures for the first time the complete, analytical, resurgent Stokes data for the first two Painlevé 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 Painlevé I results.

Keywords

resurgence, transseries, resonance, Painlevé I, Painlevé II, 2D quantum gravity, 2D supergravity, minimal strings, resurgent Stokes data, Stokes phenomena, connection formulae, monodromy, large-order behavior, resurgent asymptotics, Borel analysis

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Received 15 April 2022

Accepted 31 March 2023

Published 4 May 2023