Twisted cyclic group actions on Fukaya categories and mirror symmetry (Q2692693)

Fix \((X, \omega )\) a compact symplectic manifold such that its first Chern class \(c_1(X)\in H^2(X; \mathbb{Z})\) is divisible by a positive integer \(n\) i.e. there exists \(\alpha \in H^2(X; \mathbb{Z})\) with \(c_1(X)=n\alpha \). The paper under review establishes very interesting properties of two associated objects: 1) the Fukaya category of \(X\); 2) the moduli \(\mathcal{M}\) of Lagrangian branes on \(X\) with the superpotential \(W\). The main tools are some cyclic group actions arising from the divisibility of \(c_1(X)\). One main result is the Theorem 1.4. giving a \(\mathbb{Z}_n\)-action on \((\mathcal{M}, W)\). As example, a Kähler \(X\) is considered with an anticanonical divisor \(D\) since then \((\mathcal{M}, W)\) is exactly the uncompactified SYZ mirror of the pair \((X, D)\).