# 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?

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.