Lagrangian fillings for Legendrian links of finite or affine Dynkin type

TL;DR

This paper establishes a correspondence between seeds in cluster structures of certain Dynkin types and the existence of multiple exact Lagrangian fillings for Legendrian links, expanding known classifications beyond type A and D.

Contribution

It introduces new families of Legendrian links with infinitely many Lagrangian fillings associated to various Dynkin and affine types, linking cluster mutations to Legendrian loops.

Findings

At least as many Lagrangian fillings as seeds for finite and affine Dynkin types.

Constructs Lagrangian fillings with specific symmetries for various seed types.

Shows the connection between Coxeter mutations and Legendrian loops.

Abstract

We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type $\mathsf{ADE}$ or affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with rotational symmetry as seeds of type $\mathsf{B}$, $\mathsf{G}_2$, $\tilde{\mathsf{G}}_2$, $\tilde{\mathsf{B}}$, or $\tilde{\mathsf{C}}_2$, and with conjugation symmetry as seeds of type $\mathsf{F}_4$, $\mathsf{C}$, $\mathsf{E}_6^{(2)}$, $\tilde{\mathsf{F}}_4$, or $\mathsf{A}_5^{(2)}$. These families are the first known Legendrian links with (infinitely many) exact Lagrangian fillings (with symmetry) that exhaust all seeds in the corresponding cluster structures beyond type $\mathsf{A} \mathsf{D}$. Furthermore, we show that the $N$-graph realization of (twice of) Coxeter mutation of type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$ corresponds to a Legendrian loop of the corresponding Legendrian links. Especially, the loop of type $\tilde{\mathsf{D}}$ coincides with the one considered by Casals and Ng.