EQUIVARIANT LOCALIZATION, SPIN SYSTEMS AND TOPOLOGICAL QUANTUM THEORY ON RIEMANN SURFACES

Abstract

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 wave functions 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 nonsymmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems.