Resurgence Matches Quantization

TL;DR

This paper demonstrates that the nonperturbative topological string partition function obtained through mirror curve quantization matches the resurgent transseries approach, revealing a deep underlying connection between two distinct frameworks.

Contribution

It shows the equivalence of nonperturbative results from quantization and resurgence methods in topological string theory for the local P2 Calabi-Yau.

Findings

Resurgent transseries can be resummed to match the quantized mirror curve results.

Stokes phenomena play a crucial role in the matching process.

The match indicates a unified underlying nonperturbative structure.

Abstract

The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi-Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local P2 toric Calabi-Yau threefold, the present work shows how the Borel-Pade-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure.