  This work studies Legendrian loop actions on exact Lagrangian fillings of
Legendrian links in $(\R^3, \xi_{\st})$. By identifying the induced action of
Legendrian loops as generators of cluster modular groups, we establish the
existence of faithful group actions on the exact Lagrangian fillings of several
families of Legendrian positive braid closures, including all positive torus
links. In addition, we leverage a Nielsen-Thurston-like classification of
cluster automorphisms to provide new combinatorial and algebraic tools for
proving that a Legendrian loop action has infinite order.
