Equivariant localization, spin systems and topological quantum theory on Riemann surfaces

DOI10.1142/S0217732394002550zbMATH Open1015.81528arXivhep-th/9406121OpenAlexW3103648159MaRDI QIDQ2727786

Publication date: 31 July 2001

Published in: Modern Physics Letters A (Search for Journal in Brave)

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 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.

Full work available at URL: https://arxiv.org/abs/hep-th/9406121

Mathematics Subject Classification ID

Quantization in field theory; cohomological methods (81T70) Applications of global analysis to the sciences (58Z05) Path integrals in quantum mechanics (81S40) Applications of manifolds of mappings to the sciences (58D30) Determinants and determinant bundles, analytic torsion (58J52)

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