We know the classical Maxwell equation of motion (eom) with both electric and magnetic source can be written as:

(1) Explicit form

or more schematically as:

(2) Differential form
$$
d * F = * J_e
$$
$$
dF =* J_m
$$

My question is that do we have such classical Yang-Mills equation of motion with both electric and magnetic source in
both

(1) Explicit form?

(2) Differential form? Naively, we may write
$$
D * F = * J_e
$$
$$
D F =* J_m
$$
where $F= dA + A \wedge A$ and $D=d + [A, ]$ as the covariant derivative version of exterior derivative $d$.

But: To be aware that for example, the $SU(2)$ Yang-Mills and $SO(3)$ Yang-Mills theory may have distinct constraint on the magnetic monopole (or the t Hooft loop). It does not seem to me that $J_e$ or $J_m$ contain such information?

This post imported from StackExchange Physics at 2020-11-09 09:44 (UTC), posted by SE-user annie marie heart