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
on a symplectic manifold. We introduce the notion of
–relative spin structure for an
antisymplectic involution
and study how the orientations on the moduli space behave under the
involution .
We also apply this to the study of Lagrangian Floer theory of real
Lagrangian submanifolds. In particular, we study unobstructedness of the
–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
over ,
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.