Quantum Theory on Manifolds and its Application to Gauge Theory

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I considered quantization on a homogeneous space G/H as a first step toward quantizing on a manifold having a non-trivial topology. I showed that the inequivalent quantizations known to be allowed on the space G/H can be reproduced from the quantum theory on the group G by regarding the system as a constraint system and using Dirac's procedure applicable to such systems. I then showed that the induced gauge potential that appears on the space G/H is the canonical connection, and that it is essentially identical to the gauge potential which arises in the standard setting of Berry's phase. Next, as another class of models possessing a non-trivial manifold, I considered SL (n) Toda lattice models obtained by Hamiltonian reduction from the WZNW model. I classified all possible types of phases spaces obtained this way for n=2,3,4, and, in particular for the simplest case n=2, constructed the quantum theory explicitly where it is found that the theory is characterized by an angle parameter rheta.Quite independently of the above line of research, I also studied the path-integral approach to quantizing on G/H.I found that, by generalizing the approach for multiply-connected spaces, it is possible to recover the inequivalent quantizations if we add a weight factor given by irreducible representations of the subgroup H,and that this leads precisely to the system with the constraints mentioned above. Implication of this result to gauge theories is also examined in this path-integral framework.