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Antisymplectic involution and Floer cohomology

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta and Kaoru Ono

Geometry & Topology 21 (2017) 1–106

The main purpose of the present paper is a study of orientations of the moduli spaces of pseudoholomorphic discs with boundary lying on a real Lagrangian submanifold, ie the fixed point set of an antisymplectic involution τ on a symplectic manifold. We introduce the notion of τ–relative spin structure for an antisymplectic involution τ and study how the orientations on the moduli space behave under the involution τ. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the τ–fixed point set of symplectic manifolds and, in particular, prove its unobstructedness in the case of Calabi–Yau manifolds. We also do explicit calculation of Floer cohomology of P2n+1 over Λ0,nov, which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of a symplectic manifold in complete generality.

symplectic Geometry, Lagrangian submanifold, pseudoholomorphic curve, Floer cohomology, antisymplectic involution, orientation, quantum cohomology, unobstructed Lagrangian submanifolds
Mathematical Subject Classification 2010
Primary: 53D40, 53D45
Secondary: 14J33
Received: 25 January 2010
Revised: 27 November 2015
Accepted: 3 January 2016
Published: 10 February 2017
Proposed: Gang Tian
Seconded: Yasha Eliashberg, Richard Thomas
Kenji Fukaya
Simons Center for Geometry and Physics
State University of New York Stony Brook
Stony Brook, NY 11794-3636
United States
Yong-Geun Oh
Center for Geometry and Physics
Institute for Basic Sciences
Gyungbuk, 37673
South Korea
Hiroshi Ohta
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602
Kaoru Ono
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502