Cyclic group actions on Fukaya categories and mirror symmetry

Let $(X,\omega)$ be a compact symplectic manifold whose first Chern class $c_1(X)$ is divisible by a positive integer $n$. We construct a $\mathbb{Z}_{2n}$-action on its Fukaya category $Fuk(X)$ and a $\mathbb{Z}_n$-action on the local models of its moduli of Lagrangian branes. We show that this action is compatible with the gluing functions for different local models.