Journal of Symplectic Geometry

Volume 15 (2017)

Number 2

Finite group actions on Lagrangian Floer theory

Pages: 307 – 420



Cheol-Hyun Cho (Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, Seoul, Korea)

Hansol Hong (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)


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 $H^2(G; \mathbb{Z}/2)$. 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.

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Received 22 March 2015

Published 26 July 2017