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Disk potential functions for polygon spaces

Yoosik Kim, Siu-Cheong Lau, Xiao Zheng

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.

Disk potential functions for polygon spaces

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.
Paper Structure (20 sections, 40 theorems, 166 equations, 9 figures)

This paper contains 20 sections, 40 theorems, 166 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Polygon spaces and bending systems
  3. Polygon spaces and Grassmannians
  4. Bending systems and Gelfand--Zeitlin systems
  5. Potential functions of polygon spaces
  6. Disk potential functions
  7. Equivariant Floer theory and disk potential functions of reduced spaces
  8. Realization of cluster varieties via SYZ mirror symmetry
  9. Caterpillar bending toric varieties
  10. The toric variety associated to the Gelfand--Zeitlin polytope
  11. The toric variety associated to the caterpillar bending polytope
  12. Classification of effective disk classes
  13. Toric degenerations of caterpillar bending systems
  14. Computation of homotopy groups
  15. Effective disk classes
  16. ...and 5 more sections

Key Result

Theorem A

Suppose that $\mathbf{r} = (r_1, r_2, \cdots, r_{n+3})$ is equilateral and generic (and hence $n$ is even). Let in $\mathbb{C}P(\wedge^2 \mathbb{C}^{n+3})$. Passing $\check{X}$ to a subvariety of the affine chart given by $p_{1, 2} \neq 0$, regard the variable $p_{i,j}$ as affine coordinate $p_{i,j}/p_{1,2}$ by setting $p_{i,j} \coloneqq p_{i,j}/p_{1,2}$. Then a Floer theoretical SYZ mirror of th

Figures (9)

  • Figure 1: Bending superpotential and caterpillar bending systems
  • Figure 2: Caterpillar bending system
  • Figure 3: Singular fiber $(\mathbb{S}^1)^3 \times \mathbb{S}^2 \times \mathrm{SO}(3)^{2-1}$.
  • Figure 4: Whitehead move
  • Figure 5: The ladder diagram and dual ladder diagram
  • ...and 4 more figures

Theorems & Definitions (77)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Theorem 2.2: HausmannKnutson
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: ThimmGuilleminSternbergGC
  • Theorem 2.6: KlyachkoKapovichMillson
  • Theorem 2.7: Theorem 4.4 in Bouloc
  • Theorem 2.8: HausmannKnutson
  • ...and 67 more