Log Calabi-Yau mirror symmetry and non-archimedean disks

TL;DR

This paper constructs the mirror algebra for smooth affine log Calabi-Yau varieties using counts of non-archimedean disks, establishing a new link between algebraic geometry and non-archimedean analytic curves.

Contribution

It introduces a novel method to define mirror algebras via non-archimedean disk counts and proves the properness of related moduli spaces, advancing mirror symmetry theory.

Findings

Mirror algebra is finitely generated and commutative.

Moduli spaces of non-archimedean curves are proper.

Established a new enumerative approach for log Calabi-Yau varieties.

Abstract

We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of non-archimedean analytic disks. More generally, we studied the enumeration of non-archimedean analytic curves with boundaries, associated to a given transverse spine in the essential skeleton of the log Calabi-Yau variety. The moduli spaces of such curves are infinite dimensional. In order to obtain finite counts, we impose a boundary regularity condition so that the curves can be analytically continued into tori, that are unrelated to the given log Calabi-Yau variety. We prove the properness of the resulting moduli spaces, and show that the mirror algebra is a finitely generated commutative associative algebra, giving rise to a mirror family of log Calabi-Yau varieties.