Geometrical aspects of localization theory

• Main Author: Szabo, Richard Joseph
• Format: Others
• Language: English
• Published: 2009
• Online Access: http://hdl.handle.net/2429/7480

Some recent developments in topological quantum field theory have focused on localization techniques using equivariant cohomology to reduce functional integrals to finite-dimensional expressions from which physical and mathematical characteristics are readily deduced. In this thesis we examine the applicability of these localization techniques by analysing in detail the geometric constraints that these methods assume. After an extensive review of the relevant background material, we focus on the applications of equivariant localization techniques to phase space path integrals and classify the 2-dimensional Hamiltonian systems with simply-connected phase spaces to which these formalisms can be applied using their fundamental geometric constraints. We show that for maximally symmetric phase spaces the localizable Hamiltonian systems all appear in harmonic oscillator forms, while for non-homogeneous spaces the possibilities are more numerous. In the latter cases the Riemannian structures become rather complicated. We show that these systems all share the common property that their quantum dynamics can be described using coherent states, usually associated with coadjoint Lie group orbits, and we evaluate the associated character formulas. We then show how these results generalize to the case where the phase space is a multiply-connected compact Riemann surface. After discussing how the previous formalisms should be appropriately modified in this case, we show that the partition function for the localizable Hamiltonian systems describes a rich topological field theory which represents the first homology of the phase space. The coherent states in this case are also constructed and it is shown that the Hubert space is finite-dimensional. The wavefunctions carry a projective representation of the phase space homology group and describe modular invariants of the quantum theory. Finally, we discuss some geometric methods for analysing corrections to the semi-classical approximation for dynamical systems whose path integrals do not localize. We show that the usual isometric symmetry needed for localization can be replaced by a weaker conformal symmetry requirement. We then introduce an alternative method to the loop expansion for obtaining corrections to the semi-classical approximation which expresses the correction terms as Poincaré dual forms of homology cycles of the phase space. === Science, Faculty of === Physics and Astronomy, Department of === Graduate