Resurgent Structure of the Topological String and the First Painlev\'e Equation

TL;DR

This paper derives an explicit formula for the Stokes automorphism in topological string theory, linking it to the Painlevé I equation's non-linear Stokes phenomenon and resurgent structures, applicable to all Calabi-Yau threefolds.

Contribution

It provides a new explicit formula for the Stokes automorphism in topological strings, connecting quantum periods, Painlevé I, and resurgent structures across Calabi-Yau threefolds.

Findings

Formula derived from Painlevé I non-linear Stokes phenomenon

Connection established between topological strings and elliptic curves

Resurgent structure formula valid for all Calabi-Yau threefolds

Abstract

We present an explicit formula for the Stokes automorphism acting on the topological string partition function. When written in terms of the dual partition function, our formula implies that flat coordinates in topological string theory transform as quantum periods, and according to the Delabaere-Dillinger-Pham formula. We first show how the formula follows from the non-linear Stokes phenomenon of the Painlev\'e I equation, together with the connection between its $\tau$-function and topological strings on elliptic curves. Then, we show that this formula is also a consequence of a recent conjecture on the resurgent structure of the topological string, based on the holomorphic anomaly equations, and it is in fact valid for arbitrary Calabi-Yau threefolds.