Disk potential functions for polygon spaces

TL;DR

This paper constructs a Floer-theoretic mirror for polygon spaces by computing disk potential functions of certain Lagrangian fibers, advancing the understanding of mirror symmetry in symplectic geometry.

Contribution

It introduces a new method to compute disk potential functions for polygon spaces using open Gromov--Witten invariants and integrable systems, leading to explicit mirror cluster varieties.

Findings

Derived the Floer-theoretic SYZ mirror for polygon spaces.

Computed disk potential functions for monotone torus fibers.

Produced a mirror cluster variety of type A without frozen variables.

Abstract

We derive a Floer theoretical SYZ mirror for an equilateral and generic polygon space. The disk potential function of the monotone torus fiber of the caterpillar bending system is calculated by computing non-trivial open Gromov--Witten invariants from the structural result of the monotone Fukaya category, the topology of fibers of completely integrable systems, and toric degenerations. Then, combining the result with the work of Nohara--Ueda [NU20] and Marsh--Rietsch [MR20], we obtain the disk potential functions of bending systems and produce a mirror cluster variety of type A without frozen variables via Lagrangian Floer theory.