Log Calabi-Yau mirror symmetry and non-archimedean disks

Abstract

We give a uniform construction that includes the mirror algebra to a smooth log Calabi-Yau variety with maximal boundary, proper over an affine variety, or to a one parameter maximal compact Calabi-Yau degeneration, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are counts of non-archimedean analytic disks. More generally, we study 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.