Equivariant Localization, Spin Systems and Topological Quantum Theory on Riemann Surfaces

TL;DR

This paper develops equivariant localization formulas for phase space path integrals on Riemann surfaces, revealing topological invariants and a finite-dimensional quantum Hilbert space with novel spin system dynamics.

Contribution

It introduces a new localization approach for systems on Riemann surfaces, constructing explicit coherent states and linking quantum dynamics to topological invariants.

Findings

Localized partition function is a topological invariant.

Hilbert space is finite dimensional with projective homology representations.

Quantum dynamics described by a novel spin system from non-symmetric coadjoint orbit quantization.

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.