Anti-symplectic involution and Floer cohomology

TL;DR

This paper investigates orientations of moduli spaces of pseudo-holomorphic discs with boundaries on real Lagrangian submanifolds, introduces $ au$-relatively spin structures, and explores their implications for Floer cohomology, including explicit calculations and applications to Calabi-Yau manifolds.

Contribution

It introduces $ au$-relatively spin structures for anti-symplectic involutions and applies them to study Floer cohomology of real Lagrangians, including explicit examples and general constructions.

Findings

Floer cohomology of $ P^{2n+1}$ differs from classical cohomology.

Unobstructedness of real Lagrangians in Calabi-Yau manifolds.

Rigorous construction of quantum Massey product.

Abstract

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.