  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.
