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

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 ${\mathbb{Z}}_2$.

arXiv commentv2. 48 pages, submitted version, corrected a mistake in the first arXiv version pointed by Woodward. v1. 47 pages

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