Antisymplectic involution and Floer cohomology
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.