Local mirror symmetry in the tropics

Abstract

We discuss how the Gross-Siebert reconstruction theorem applies to the local mirror symmetry of Chiang, Klemm, Yau and Zaslow. The reconstruction theorem associates to certain combinatorial data a degeneration of (log) Calabi-Yau varieties. While in this case most of the subtleties of the construction are absent, an important normalization condition already introduces rich geometry. This condition guarantees the parameters of the construction are canonical coordinates in the sense of mirror symmetry. The normalization condition is also related to a count of holomorphic disks and cylinders, as conjectured in our work and partially proved in various works of Chan, Cho, Lau, Leung and Tseng. We sketch a possible alternative proof of these counts via logarithmic Gromov-Witten theory. There is also a surprisingly simple interpretation via rooted trees marked by monomials, which points to an underlying rich algebraic structure both in the relevant period integrals and the counting of holomorphic disks.