Communications in Number Theory and Physics

Volume 17 (2023)

Number 2

On quasi-tame Looijenga pairs

Pages: 313 – 341

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n2.a3

Authors

Andrea Brini (School of Mathematics and Statistics, University of Sheffield, United Kingdom; and CNRS, DR 13, Montpellier, France)

Yannik Schüler (School of Mathematics and Statistics, University of Sheffield, United Kingdom)

Abstract

We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact $\log$ Gromov–Witten invariants of Looijenga pairs to other curve counting invariants of Gromov–Witten/Gopakumar–Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the $\log$ invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.

Received 12 January 2022

Accepted 13 March 2023

Published 4 May 2023