Communications in Number Theory and Physics

Volume 11 (2017)

Number 4

On asymptotics and resurgent structures of enumerative Gromov–Witten invariants

Pages: 707 – 790

DOI: http://dx.doi.org/10.4310/CNTP.2017.v11.n4.a1

Authors

Ricardo Couso-Santamaría (CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Ricardo Schiappa (CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal; and Département de Physique Théorique & Section de Mathématiques, Université de Genève, Switzerland)

Ricardo Vaz (CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal; and DESY Theory Group, DESY, Hamburg, Germany)

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 $\mathbb{P}^2$ and local $\mathbb{P}^1 \times \mathbb{P}^1$; 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.

Keywords

asymptotics, resurgent analysis, enumerative geometry, algebraic geometry, topological strings, Gromov–Witten invariants, Gopakumar–Vafa invariants

Full Text (PDF format)

Received 31 May 2016

Accepted 17 March 2017

Published 29 November 2017