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Augmentations, Fillings, and Clusters

Honghao Gao, Linhui Shen, Daping Weng

Abstract

We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K2 (aka. A-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

Augmentations, Fillings, and Clusters

Abstract

We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K2 (aka. A-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in [EHK16], we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.

Paper Structure

This paper contains 34 sections, 58 theorems, 207 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Context
  3. From Fillings to Cluster Seeds
  4. Finite Type Classifications
  5. Infinitely Many Exact Lagrangian Fillings
  6. Cluster
  7. Positive braid Legendrian Links
  8. Augmentation Varieties for Positive Braid Legendrian Links
  9. From fillings to clusters
  10. Exact Lagrangian Cobordisms and Enhanced EHK Functors
  11. EHK Functorial Morphisms for Admissible Cobordisms
  12. Cluster Charts from Admissible Fillings
  13. Computing Cluster Seeds associated with Admissible Fillings
  14. Infinitely Many Fillings
  15. Full Cyclic Rotation and Donaldson-Thomas Transformation
  16. ...and 19 more sections

Key Result

Theorem 1.2

The augmentation variety $\mathrm{Aug}(\Lambda_\beta)$ is a cluster $\mathrm{K}_2$ variety associated to the quiver $Q_\beta$. The degree zero Reeb chords of $\Lambda_\beta$ are cluster variables that generate the coordinate ring of $\mathrm{Aug}(\Lambda_\beta)$.

Figures (7)

  • Figure 1: An $\mathrm{E}_9$ quiver with two frozen vertices
  • Figure 2: Ng's resolution
  • Figure 3: Saddle Cobordism
  • Figure 4: Dipping and Saddle Cobordism in the Lagrangian Projection
  • Figure 5: Cyclic Rotation
  • ...and 2 more figures

Theorems & Definitions (142)

  • Example 1.1
  • Theorem 1.2: Theorem \ref{['3.6']}, Corollary \ref{['all Reeb chords are cluster variables']}, and Proposition \ref{['prop initial seed']}
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: Theorem \ref{['thm correspondence']} and Corollary \ref{['cor 3.40']}
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 132 more