Anti-symplectic involution and Floer cohomology

Details Anti-symplectic involution and Floer cohomology Anti-symplectic involution and Floer cohomology

2009-12-14

arxiv.org/abs/0912.2646v2

Mathematics

Symplectic Geometry

51 pages, 5 figures

Scientific paper

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 $\R P^{2n+1}$ 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 symplectic manifold in complete generality.

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