Charge dynamics on $U(1)$ abundles

The quantum dynamics of a particle on $(2d-2)$ universal $U(1)$ fiber bundles is considered. The analysis on these bundles generalizes, in particular, the description of a system consisting of a charge and a Dirac magnetic monopole. The spectral properties of Hamiltonians of a charge in a connection field of monopole type are studied, and a representation in which they have a simple form is found. The complete set of states in the global spaces of the $U(1)$ bundles graded by the topological number $n$ is constructed. It is shown that in each sector of the state space corresponding to the number $n\ne0$ spatial parity is violated. The violation is topological in nature, i.e., it does not depend on the choice of the connection in the considered fiber bundles.