The symplectic vortex equations and invariants of Hamiltonian group actions

Abstract

In this paper we define invariants of Hamiltonian group actions\nfor central regular values of the moment map. The key hypotheses\nare that the moment map is proper and that the ambient\nmanifold is symplectically aspherical. The invariants are based\non the symplectic vortex equations. Applications include an\nexistence theorem for relative periodic orbits, a computation for\ncircle actions on a complex vector space, and a theorem\nabout the relaton between the invariants introduced here and the\nSeiberg-Witten invariants of a product of a Riemann surface\nwith a two-sphere.