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.
Bibliographical noteFunding Information:
We have benefited very much from Paul Seidel’s work , . We also thank him for helpful comments to our work. We are indebted to Kenji Fukaya for Lemma 9.8 and valuable advices. We are grateful to Kaoru Ono for the kind explanations of orientations in , and to Siu-Cheong Lau for several helpful discussions especially on character group actions. We would like to express our gratitude to Yong-Geun Oh, Kwokwai Chan, Hsian-hua Tseng, Mainak Poddar, Nick Sheridan, Hiroshige Kajiura, Gyoung-Seog Lee, Hyung-Seok Shin, Sangwook Lee for their help. We thank an anonymous referee pointing out an error in the original version of this manuscript. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST)Ministry of Education, Science and Technology (No. 2012R1A1A2003117)
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All Science Journal Classification (ASJC) codes
- Geometry and Topology