Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations

Abstract

In the last Chapter we deduced the general features of the localization formalism on a simply-connected symplectic manifold. We found general forms for the Hamiltonian functions in terms of the underlying phase space Riemannian geometry which is required for their Feynman path integrals to manifestly localize. This feature is quite interesting from the point of view that, as the quantum theory is always ab initio metric-independent, this analysis probes the role that the geometry and topology plays towards the understanding of quantum integrability. For instance, we saw that the classical trajectories of a harmonic oscillator must be embedded into a rotationally-invariant geometry and that as such its orbits were always circular trajectories. For more complicated systems these quantum geometries are less familiar and endow the phase space with unusual Riemannian structures (i.e. complicated forms of the localization supersymmetries). In any case, all the localizable Hamiltonians were essentially harmonic oscillators (e.g. the height function for a spherical phase space geometry) in some form or another, and their quantum partition functions could be represented naturally using coherent state formalisms associated with the Poisson-Lie group actions of the isometry groups of the phase space. In the non-homogeneous cases we saw, in particular, that to investigate equivariant localization in general one needs to determine if a Riemannian geometry can possess certain symmetries imposed by some rather ad-hoc restrictions from the dynamical system.