Antisymplectic involution and Floer cohomology

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

doi:10.2140/gt.2017.21.1

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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

DOI: 10.2140/gt.2017.21.1

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 $τ$ 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 $ℝ P^{2 n + 1}$ over $Λ_{0, n o v}^{ℤ}$ , 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

References

Publication

Received: 25 January 2010

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

Volume 21, issue 1 (2017)