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