How are sources described in gauge theory? Let's only assume the case of electromagnetism. If one varies the Yang-Mills-Functional one gets the Yang-Mills-equation $*d*F=0$. The whole theory can be geometrically described on principal bundles. On the associated bundle, the Klein-Gordon- or Dirac-equation also has a natural geometrical interpretation.
But how does one deal with sources in the formalism of gauge theory on bundles? What is the geometrical/topological interpretation of the full Maxwell equations $*d*F=j$, $dF=0$, including the source term? Can the source be viewed as e.g. monopoles, giving rise to non trivial bundles, imitating the source term?
 A: Because gauge invariance  requires  covariant "conservation"
$$
0=\nabla_\mu J^\mu\equiv \partial_\mu J^\mu +[A_\mu, J^\mu],
$$
the  sources $J^\mu$ in non-abelian theory cannot be $c$-numbers. Instead they are Wilson Loops. You can also make them classical but with internal dynamics, as in the Wong equations.
Notice that the "charge" of a source in Yang-Mills theory is the representation of the gauge group in which the Wilson line belongs. For example in the SU(3) colour theory quarks have charge "${\bf 3}$" because they have 3-d representation and gluons have charge "${\bf 8}$" because they are  in the 8-dimensional adjoint representation. "Addition" of the charges is the Clebsch-Gordan decomposition of the tensor product of the representations such as
$$
{\bf 8}\times {\bf 8}= {\bf 1}+ {\bf 8}+{\bf 8}+ {\bf 10}+\overline {\bf 10}+{\bf 27}.
$$
In particular gauge "charge" must come in discrete lumps. You cannot have a smooth charge distribution.
Working out the effect of the sources on the gauge bundle is non trivial. For an abelian theory they just induce a holonomy. For non-abelian theories the result is highly non-trivial. The simplest case that I know of results in the Knizhnik-Zamolodchikov equations of conformal field theory, and its spacetime interpretation as Witten's knot invariants.
