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

We construct the vortex Floer homology group $VHF (M,μ;H)$ for an aspherical Hamiltonian $G$-manifold $(M, ω)$ with moment map $μ$ 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 ${\mathbb Z}_2$.