  The (Berry-Aharonov-Anandan) geometric phase acquired during a cyclic quantum
evolution of finite-dimensional quantum systems is studied. It is shown that a
pure quantum state in a (2J+1)-dimensional Hilbert space (or, equivalently, of
a spin-J system) can be mapped onto the partition function of a gas of
independent Dirac strings moving on a sphere and subject to the Coulomb
repulsion of 2J fixed test charges (the Majorana stars) characterizing the
quantum state. The geometric phase may be viewed as the Aharonov-Bohm phase
acquired by the Majorana stars as they move through the gas of Dirac strings.
Expressions for the geometric connection and curvature, for the metric tensor,
as well as for the multipole moments (dipole, quadrupole, etc.), are given in
terms of the Majorana stars. Finally, the geometric formulation of the quantum
dynamics is presented and its application to systems with exotic ordering such
as spin nematics is outlined.
