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

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.

v2: 55 pages, revised version, published in Compositio Math