Distinguished bases in the wrapped Floer cohomology of tropical Lagrangian surfaces

TL;DR

This paper introduces a new construction for tropical Lagrangian surfaces in $(\mathbb{C}^*)^2$, enabling the identification of distinguished bases in their wrapped Floer cohomology and mirror functions.

Contribution

It presents a novel method to construct special Lagrangian tropical surfaces that facilitate explicit computation of Floer cohomology bases.

Findings

Identified distinguished bases in wrapped Floer cohomology for various examples.

Established correspondence between Floer cohomology bases and functions on mirror curves.

Enhanced understanding of the asymptotic behavior of holomorphic disks near cylindrical ends.

Abstract

We introduce a new construction for tropical Lagrangian surfaces in $(\mathbb{C}^*)^2$. This construction makes the surfaces special Lagrangian, which gives a strong control over the asymptotic behavior of holomorphic disks near each cylindrical end. As a result, we find a distinguished basis of the wrapped Floer cohomology ring $HW^\bullet(L_f, L_f)$ for several different examples and find the corresponding distinguished bases of functions on the mirror curves.