On quasi-tame Looijenga pairs

TL;DR

This paper proves a conjecture linking higher genus log Gromov-Witten invariants of Looijenga pairs to other curve counting invariants, using a novel $q$-hypergeometric resummation technique.

Contribution

It introduces a new $q$-hypergeometric resummation method to relate complex invariants of Looijenga pairs, advancing understanding of their enumerative geometry.

Findings

Established a new identity of $q$-series with combinatorial significance.

Connected higher genus invariants to Gromov-Witten and Gopakumar-Vafa type invariants.

Provided a closed-form resummation of quantum tropical vertex calculations.

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.