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

We construct the vortex Floer homology group VHF(M,μ;H)VHF\left ( M, \mu ; H\right ) for an aspherical Hamiltonian GG-manifold (M,ω,μ)(M, \omega , \mu ) and a class of GG-invariant Hamiltonian loops HtH_t, following a proposal of Cieliebak, Gaio, and Salamon (2000). This is a substitute for the ordinary Hamiltonian Floer homology of the symplectic quotient of MM. The equation for connecting orbits is a perturbed symplectic vortex equation on the cylinder R×S1\mathbb {R} \times S^1. 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 Z\mathbb {Z}.