Finite group actions on Lagrangian Floer theory

Abstract: We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a {\em spin profile} as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in $H^2(G;\Z/2)$. For a class of Lagrangian submanifolds which have the same spin profiles, we define a finite group action on their Fukaya category. In consequence, we obtain the $s$-equivariant Fukaya category as well as the $s$-orbifolded Fukaya category for each group cohomology class $s$. We also develop a version with $G$-equivariant bundles on Lagrangian submanifolds, and explain how character group of $G$ acts on the theory. As an application, we define an orbifolded Fukaya-Seidel category of a $G$-invariant Lefschetz fibration, and also discuss homological mirror symmetry conjectures with group actions.