The word "holonomy" was first introduced into physic by Heinrich Hertz
(the first person to confirm experimentally Clerk Maxwell's
prediction of electromagnetic radiation). He used it to describe
constraints in classical mechanics. The idea has an immediate
application to Caratheodory's approach to classical thermodynamics
and has an amusing application to classical economics.
Geometrically it enters the theory of cosmic strings
and is related to the Gauss-Bonnet theorem. A simple extension
to quantum mechanics leads to the Berry phase and
and the Bohm-Anharonov effect. The extension to gravity
leads us to consider the phase (strictly a spin matrix)
acquired by a fermion freely falling in a
gravitational field, as is seen in neutron interference experiments.
Classifying the possible gravitational spin-matrix-phases leads to
special types of curved spacetimes with "metrics with special and
exceptional holonomy". They include Calabi-Yau spaces and metrics with
G_2 holonomy which arise in compactifications of super-string theory and
M-Theory.
The aim of the colloquium is to provide a pedagogic survey of these uses
of holonomy in physics accessible at the graduate student level.