Classical Yang-Mills equation of motion with both electric and magnetic sources?

Question

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

Comments

Please clarify your question. Are you asking why the same equations can be written in such different forms?

This post imported from StackExchange Physics at 2020-11-09 09:44 (UTC), posted by SE-user G. Smith

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No. But I am interested in knowing both (1) Explicit form and (2) Differential form for YM with e and m source

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

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OK, that makes it clearer. Thank you. Do you really want the Yang-Mills equations written in terms of $\mathbf{E}$ and $\mathbf{B}$? They ate usually written in terms of $F^{\mu\nu}$.

This post imported from StackExchange Physics at 2020-11-09 09:44 (UTC), posted by SE-user G. Smith

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written as $E^a$ and $B^a$ are preferred. But you can convert to $F$ first

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