Cyclic group actions on Fukaya categories and mirror symmetry

Abstract

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.

The construction of the action on Fukaya category is significantly simplified. A new section about the compatibility of the action on local mirrors with gluing functions is added. Comments are welcome