The Arnold-Givental conjecture and moment Floer homology

TL;DR

This paper proves the Arnold-Givental conjecture for certain Lagrangian submanifolds in symplectic quotients by employing moment Floer homology, which uses vortex equations to address bubbling issues in Floer theory.

Contribution

It introduces a novel approach using moment Floer homology with vortex equations to prove the conjecture for Lagrangians fixed by antisymplectic involutions.

Findings

Proved the Arnold-Givental conjecture for specific Lagrangians in Marsden-Weinstein quotients.

Developed a new Floer homology framework using symplectic vortex equations.

Addressed bubbling phenomena that hindered standard Floer homology methods.

Abstract

We prove the Arnold-Givental conjecture for a class of Lagrangian submanifolds in Marsden-Weinstein quotients which are fixpoint sets of some antisymplectic involution. For these Lagrangians the Floer homology cannot in general be defined by standard means due to the bubbling phenomenon. To overcome this difficulty we consider moment Floer homology whose boundary operator is defined by counting solutions of the symplectic vortex equations on the strip which have better compactness properties than the original Floer equations.