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 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 H2(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.
All Science Journal Classification (ASJC) codes
- Geometry and Topology