Distinguished bases in the wrapped Floer cohomology of tropical Lagrangian surfaces

Abstract

Homological mirror symmetry implies a correspondence between Lagrangian Floer chain complexes and corresponding Hom spaces in the category of coherent sheaves. One notable aspect of studying chain complexes and their homology groups is that the methods for calculation and their properties can vary significantly. For example, while there are vast studies of canonical bases of the cohomology rings on the algebraic side, the descriptions of Lagrangian Floer cohomology rings often fail to give a distinguished basis. In this paper, we present a model for tropical Lagrangians in (C^∗)^2 that gives rise to distinguished bases in their wrapped Floer cohomology. The main ingredient is the “argument constraint,” which imposes a nontrivial geometric restriction on holomorphic disks bounded by these special Lagrangians. As a model example, we consider a 1-family of Lagrangians that are generically non-exact. Under mirror symmetry, this will also give distinguished bases of the ring of functions on P^1 minus 4 points.