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
RP2nC1 over Z
0;nov , 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.
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