Equivariant Floer homology is isomorphic to reduced Floer homology

TL;DR

This paper constructs a quantum model for equivariant Lagrangian Floer homology on symplectic manifolds with Hamiltonian group actions, proving it is isomorphic to the Floer homology of the reduced space under certain conditions.

Contribution

It extends equivariant Floer homology to the Novikov ring and proves its isomorphism to reduced Floer homology when the group action is free on the zero-level set.

Findings

Constructed a quantum model for equivariant Floer homology.

Proved the isomorphism between equivariant Floer homology and reduced Floer homology.

Validated Cazassus' conjecture under the free action condition.

Abstract

Given a symplectic manifold equipped with a Hamiltonian $G$-action and two $G$-invariant Lagrangians, we lift the construction of equivariant Lagrangian Floer homology of G.\@~Cazassus to the Novikov ring by constructing a ``quantum'' model in the vein of Biran and Cornea and Schm\"aschke. Using this refinement, we prove Cazassus' conjecture that, if the action on the zero-level of the moment map is free, then equivariant Floer homology agrees with the Floer homology of the Marsden-Weinstein reduction.