Antisymplectic involution and Floer cohomology

Fukaya, Kenji¹ ; Oh, Yong-Geun² ; Ohta, Hiroshi³ ; Ono, Kaoru⁴

¹ Simons Center for Geometry and Physics, State University of New York Stony Brook, Stony Brook, NY 11794-3636, United States
² Center for Geometry and Physics, Institute for Basic Sciences, Pohang, Gyungbuk, 37673, South Korea
³ Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
⁴ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Geometry & topology, Tome 21 (2017) no. 1, pp. 1-106.

Résumé

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 $ℝP^{2n+1}$ over ${\Lambda }_{0,nov}^{\mathbb{Z}}$, 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.

Détail

DOI : 10.2140/gt.2017.21.1

Classification : 53D40, 53D45, 14J33
Keywords: symplectic Geometry, Lagrangian submanifold, pseudoholomorphic curve, Floer cohomology, antisymplectic involution, orientation, quantum cohomology, unobstructed Lagrangian submanifolds

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