Gauged Hamiltonian Floer homology I: definition of the Floer homology groups

TL;DR

This paper introduces a new vortex Floer homology group for aspherical Hamiltonian G-manifolds, providing a more accessible alternative to traditional Floer homology by avoiding virtual techniques.

Contribution

It defines vortex Floer homology using classical perturbation methods, enabling homology over integers and broadening the scope of Floer theory for Hamiltonian G-manifolds.

Findings

Constructed vortex Floer homology group $VHF(M, ;H)$

Achieved transversality via classical perturbation

Defined homology over ${\u2115}$ and ${\u2115}_2$

Abstract

We construct the vortex Floer homology group $VHF (M,\mu;H)$ for an aspherical Hamiltonian $G$-manifold $(M, \omega)$ with moment map $\mu$ and a class of $G$-invariant Hamiltonian loop $H_t$, following the proposal of [3]. This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of $M$. We achieve the transversality of the moduli space by the classical perturbation argument instead of the virtual technique, so the homology can be defined over ${\mathbb Z}$ or ${\mathbb Z}_2$.