\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
The line bundles that arise in the holonomy interpretations of the geometric
phase display curious similarities to those encountered in the statement of the
Borel-Weil-Bott theorem of the representation theory. The remarkable
relationship between the mathematical structure of the geometric phase and the
classification theorem for complex line bundles provides the necessary tools
for establishing the relevance of the Borel-Weil-Bott theorem to Berry's
adiabatic phase. This enables one to define a set of
topological charges for arbitrary compact connected semi-simple dynamical Lie
groups. These charges signify the topological content of the phase. They can be
explicitly computed. In this paper, the problem of the determination of the
parameter space of the Hamiltonian is also addressed. It is shown that, in
general, the parameter space is either a flag manifold or one of its
submanifolds. A simple topological argument is presented to indicate the
relation between the Riemannian structure on the parameter
space and Berry's connection. The results about the fiber bundles and gorup
theory are used to introduce a procedure to reduce the problem of the
nonadiabatic (geometric) phase to Berry's adiabatic phase for cranked
Hamiltonians. Finally, the possible relevance of the topological charges of the
geometric phase to those of the non-Abelian monopoles is pointed out.
\end{document}