On Asymptotics and Resurgent Structures of Enumerative Gromov-Witten Invariants

TL;DR

This paper explores the asymptotic growth and resurgent structures of Gromov-Witten invariants across various Calabi-Yau threefolds, revealing factorial growth and nonperturbative sectors linked to mirror symmetry and enumerative geometry.

Contribution

It demonstrates the factorial asymptotics of Gromov-Witten invariants at large genus and degree and connects these to resurgent transseries and mirror symmetry, providing new insights into their nonperturbative structure.

Findings

Gromov-Witten invariants grow factorially at large genus and degree.

Resurgent analysis reveals nonperturbative sectors controlling asymptotics.

Examples include resolved conifold, local P^2, local P^1 x P^1, local curves, and the quintic.

Abstract

Making use of large-order techniques in asymptotics and resurgent analysis, this work addresses the growth of enumerative Gromov-Witten invariants---in their dependence upon genus and degree of the embedded curve---for several different threefold Calabi-Yau varieties. In particular, while the leading asymptotics of these invariants at large genus or at large degree is exponential, at combined large genus and degree it turns out to be factorial. This factorial growth has a resurgent nature, originating via mirror symmetry from the resurgent-transseries description of the B-model free energy. This implies the existence of nonperturbative sectors controlling the asymptotics of the Gromov-Witten invariants, which could themselves have an enumerative-geometry interpretation. The examples addressed include: the resolved conifold; the local surfaces local P^2 and local P^1 x P^1; the local curves and Hurwitz theory; and the compact quintic. All examples suggest very rich interplays between resurgent asymptotics and enumerative problems in algebraic geometry.