Lagrangian fillings for Legendrian links of finite or affine Dynkin type

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 $\mathsf{AD}$. 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.

85 pages with many figures. This is a combined version of arXiv:2101.01943 and arXiv:2107.04283. The part of Lagrangian fillings with rotational or conjugation symmetry has been completely revised. Relative cycles of Lagrangian fillings are investigated concerning frozen variables in the A-cluster structure