Refined curve counting with descendants and quantum mirrors

TL;DR

This paper provides a formula linking the structure constants of a quantized mirror algebra for log Calabi--Yau surfaces to higher genus descendant logarithmic Gromov--Witten invariants, extending the weak Frobenius structure conjecture.

Contribution

It generalizes the weak Frobenius structure conjecture to the q-refined setting by relating invariants to counts of quantum broken lines in quantum scattering diagrams.

Findings

Derived a formula for quantization structure constants in terms of Gromov--Witten invariants.

Extended the weak Frobenius structure conjecture to a q-refined context.

Connected invariants to quantum broken line counts in scattering diagrams.

Abstract

Given a log Calabi--Yau surface $(Y,D)$, Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher genus descendant logarithmic Gromov--Witten invariants of $(Y,D)$. Our result generalises the weak Frobenius structure conjecture for surfaces to the $q$-refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.