Antisymplectic involution and Floer cohomology

Abstract

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 tau on a symplectic manifold. We introduce the notion of tau-relative spin structure for an antisymplectic 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. We also do explicit calculation of Floer cohomology of RP2n+1 over Lambda(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.