Anti-symplectic involution and Floer cohomology

Abstract

The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions $\tau$ on a symplectic manifold. We introduce the notion of $\tau$-relatively spin structure for an anti-symplectic 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. And we also do explicit calculation of Floer cohomology of $\mathbb{R}{P}^{2n+1}$ 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 symplectic manifold in complete generality.

85 pages, final version, to appear in Geometry and Topology

References in corpus (4)

• Lagrangian Floer theory on compact toric manifolds I
• Opening Mirror Symmetry on the Quintic
• Disk enumeration on the quintic 3-fold
• Symmetries of Lagrangian fibrations

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• Versality of the relative Fukaya category
• Real Lagrangians in Calabi-Yau Threefolds
• Real loci in (log-) Calabi-Yau manifolds via Kato-Nakayama spaces of toric degenerations
• Relative quantum cohomology of the Chiang Lagrangian
• Twisted Sectors for Lagrangian Floer Theory on Symplectic Orbifolds
• On the formality problem for manifolds with special holonomy
• An iterated graph construction and periodic orbits of Hamiltonian delay equations