Floer cohomology of Dehn twists along real Lagrangian spheres

TL;DR

This paper investigates the Floer cohomology associated with Dehn twists along real Lagrangian spheres in symplectic manifolds with anti-symplectic involutions, revealing a fixed point element under the involution-induced automorphism.

Contribution

It introduces a new fixed point element in Floer cohomology for Dehn twists along real Lagrangian spheres, utilizing cobordism and Floer theory methods.

Findings

Existence of a distinguished fixed point element in Floer cohomology

Application of cobordism techniques to Floer theory

Insights into symmetries of Floer cohomology under involutions

Abstract

We study the Floer cohomology of the Dehn twist along a real Lagrangian sphere in a symplectic manifold endowed with an anti-symplectic involution. We prove that there exists a distinguished element in the Floer group that is a fixed point of the automorphism induced by the involution. Our methods of proof are based on Mak-Wu's cobordism and Floer-theoretic considerations.