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.

arXiv commentv2: 55 pages, revised version, published in Compositio Math

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Cited by in corpus (6)

• The quantum tropical vertex
• The canonical wall structure and intrinsic mirror symmetry
• Stable maps to Looijenga pairs
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Scattering diagrams, theta functions, and refined tropical curve counts
• Stable maps to Looijenga pairs: orbifold examples