Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2

TL;DR

This paper advances the understanding of nonperturbative topological string theory by applying resurgent transseries to the holomorphic anomaly equations, specifically analyzing the local CP2 Calabi-Yau background and uncovering intricate instanton structures.

Contribution

It extends the holomorphic anomaly framework to include resurgent transseries, providing detailed analysis of instanton sectors and their resurgence relations in local CP2 geometry.

Findings

Resurgent transseries accurately describe nonperturbative effects.

Instanton actions exhibit Z_3 symmetry and resonance.

Numerical checks confirm the transseries predictions.

Abstract

The holomorphic anomaly equations describe B-model closed topological strings in Calabi-Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local CP2 toric Calabi-Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by Z_3 symmetry, alongside another action related to the Kahler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.