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.
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.
