#### Volume 21, issue 1 (2017)

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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
##### Abstract

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 $\tau$ on a symplectic manifold. We introduce the notion of $\tau$–relative spin structure for an antisymplectic involution $\tau$ and study how the orientations on the moduli space behave under the involution $\tau$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $\tau$–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 $ℝ{P}^{2n+1}$ over ${\Lambda }_{0,nov}^{ℤ}$, 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.

##### Keywords
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
##### Publication
Revised: 27 November 2015
Accepted: 3 January 2016
Published: 10 February 2017
Proposed: Gang Tian
Seconded: Yasha Eliashberg, Richard Thomas
##### Authors
 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 Pohang Gyungbuk, 37673 South Korea Hiroshi Ohta Graduate School of Mathematics Nagoya University Nagoya 464-8602 Japan Kaoru Ono Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan