Finite group actions on Lagrangian Floer theory

TL;DR

This paper develops a framework for incorporating finite group actions into Lagrangian Floer theory, including the Fukaya category, by introducing spin profiles and orbifolded categories, with applications to orbifolded Fukaya-Seidel categories and mirror symmetry.

Contribution

It introduces the notion of spin profiles to extend group actions to Lagrangian Floer theory and constructs equivariant and orbifolded Fukaya categories under finite group actions.

Findings

Defined finite group actions on Novikov-Morse theory.

Constructed s-equivariant and s-orbifolded Fukaya categories.

Applied to orbifolded Fukaya-Seidel categories and mirror symmetry.

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.