EQUIVARIANT LOCALIZATION, SPIN SYSTEMS AND TOPOLOGICAL QUANTUM-THEORYON RIEMANN SURFACES

We study equivariant localization formulas for phase space path-integr als when the phase space is a multiply connected compact Riemann surfa ce. We consider the Hamiltonian systems to which the localization form ulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the nontrivi al homology classes of the phase space. We explicitly construct the co herent states in the canonical quantum theory and show that the Hilber t space is finite-dimensional with the wave functions carrying a proje ctive representation of the discrete homology group of the phase space . The corresponding coherent state path-integral then describes the qu antum dynamics of a novel spin system given by the quantization of a n onsymmetric coadjoint Lie group orbit. We also briefly discuss the geo metric structure of these quantum systems.