Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting

TL;DR

This paper develops a quantum deformation framework for mirrors of log Calabi-Yau surfaces using higher genus curve counts, leading to new noncommutative algebra structures.

Contribution

It introduces q-deformed scattering diagrams based on higher genus invariants to construct deformation quantizations of mirror spaces.

Findings

Constructed quantum mirrors via q-deformed scattering diagrams.

Produced canonical bases for noncommutative algebras.

Extended mirror symmetry to include higher genus curve counts.

Abstract

Gross, Hacking, and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding noncommutative algebras of functions.