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.
Actually,the principal bundle connections which minimize an energy functional are the only ones
of physical interest. For example, the Yang-Mills
connections minimize the Yang-Mills functional.
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 
.
