Yang-Mills equations I would like that someone explain to me the Yang-Mills equations as defined in some books:
$$ \begin{cases} d_D F = 0 \\ *d_D *F = J \end{cases} $$
What is $ d_D $? What is $ F $? What are $ *d $ and $ *F $? What is $ J $?
Can we represent those quantities with matrix to simplify the explanation?
Thank you very much all of you.
 A: In case this is the standard electrodynamics in 4d, the meaning of these symbols is the following $\rm d$ is exterior derivative, $F$ is the field strength 2-form (obtained as $F = {\rm d} A$ from the potential 1-form $A$) and $J$ is the electromagnetic current. The equations you have written are called Maxwell equations. The symbol $*$ is the Hodge dual, a standard operation in exterior algebra that takes a $k$-form to a $n-k$-form when we are in an $n$-dimensional setting.
Now, for your more general questions. General Yang-Mills theory is not the simple $U(1)$ abelian gauge as above ($U(1)$ gauge symmetry means that we are free to recalibrate the potential to $A + {\rm d} \xi$ without affecting $F$ since $\rm d^2 = 0$) but has a bigger (usually non-abelian) group $G$ of symmetries. Associated to this group is connection (related to $A$ above) and its curvature $F$ that live on certain bundles related to the group $G$ above. The symbol $d_D$ is the covariant exterior derivative. It combines the usual exterior derivative with the covariant $\nabla$ operator associated with the connection. This is a pleasant formalism that makes the very hard Yang-Mills theory look almost like a Maxwell theory. Finally, $J$ is some kind of current, depending on the theory.
