Lagrangian fillings for Legendrian links of affine type

Abstract: We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with certain symmetries as seeds of type $\tilde{\mathsf{B}}_n$, $\tilde{\mathsf{F}}_4$, $\tilde{\mathsf{G}}_2$, and $\mathsf{E}_6^{(2)}$. These families are the first known Legendrian links with infinitely many fillings that exhaust all seeds in the corresponding cluster structures. Furthermore, we show that Legendrian realization of Coxeter mutation of type $\tilde{\mathsf{D}}$ corresponds to the Legendrian loop considered by Casals and Ng.