  In this paper we define invariants of Hamiltonian group actions for central
regular values of the moment map. The key hypotheses are that the moment map is
proper and that the ambient manifold is symplectically aspherical. The
invariants are based on the symplectic vortex equations. Applications include
an existence theorem for relative periodic orbits, a computation for circle
actions on a complex vector space, and a theorem about the relation between the
invariants introduced here and the Seiberg--Witten invariants of a product of a
Riemann surface with a two-sphere.
