Gauge theory studies principal bundle connections, called gauge fields, on a principal bundle. These
connections correspond to fields, in physics, such as an electromagnetic field, and
the Lie group of the principal
bundle corresponds to the symmetries of the physical
system. The base manifold to the principal
bundle is usually a four-dimensional manifold
which corresponds to space-time. In the case of an electromagnetic field, the symmetry group is the unitary
group 
.
The other two groups that arise in physical theories are
the special unitary groups 
and 
. Also, a group representation 
of the symmetry group, called internal space, gives rise to an associated
vector bundle.
The principal bundle connections of interest include critical points of an energy functional.
For example, Yang-Mills connections are critical points
of the Yang-Mills functional and need not minimize
it. These connections are useful in low-dimensional
topology. In fact, in Donaldson theory, the
collection of Yang-Mills connections gives topological invariants
of the base manifold 
.
