  We study equivariant localization formulas for phase space path integrals
when the phase space is a multiply connected compact Riemann surface. We
consider the Hamiltonian systems to which the localization formulas are
applicable and show that the localized partition function for such systems is a
topological invariant which represents the non-trivial homology classes of the
phase space. We explicitly construct the coherent states in the canonical
quantum theory and show that the Hilbert space is finite dimensional with the
wavefunctions carrying a projective representation of the discrete homology
group of the phase space. The corresponding coherent state path integral then
describes the quantum dynamics of a novel spin system given by the quantization
of a non-symmetric coadjoint Lie group orbit. We also briefly discuss the
geometric structure of these quantum systems.
