Maxwell and Bianchi equations We frequently see: A certain action and then we are asked to solve for Bianchi identity and Maxwell equation.
I have often solved for them but I never knew what is the difference between the two? In more specific words, I know that Maxwell equation is derived from the action but what does a Bianchi identity mean?
EDIT: Now that in the comments we faced a new issue, I am editing this with a example.
Example: Lagragian is
$$L=-\frac{1}{4}ImZF_{\mu\nu}F{\mu\nu}-\frac{1}{8}ReZ\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$
So, the book (Van Proeyen's SUGRA book) says tat the equation of motion of this theory will be
$$\partial_{\mu}[(ImZ)F^{\mu\nu}+i(ReZ)\tilde{F}^{\mu\nu}]=0$$
How come this is the same as te usual Maxwell equation $d\star F=0$ (I believe this is a dual teory so maybe the equation Evan presented in the comments will be instead dF=0 but in all cases I could cut the story short and ask
 how did Van Proeyen reach this
$$\partial_{\mu}[(ImZ)F^{\mu\nu}+i(ReZ)\tilde{F}^{\mu\nu}]=0?$$
 A: The difference is perhaps illuminated best in the language of differential forms. Recall that the vector potential $A$ can be considered a 1-form field $A$, which we can act on with the exterior derivative operator $d$ to get the electromagnetic field strength tensor 
\begin{equation}
F=dA.
\end{equation}
Since $F$ is an exact form we know that it is closed. So
\begin{equation}
dF = d^2A=0.
\end{equation}
This is the Bianchi Identity $\nabla_{[\alpha}F_{\beta\gamma]}=0$, expressed in the language of differential forms. Now, another operation that we can apply to a differential form is the Hodge dual operator *, which maps $k$-forms to $(n-k)$-forms in $n$-dimensional space. So, in 4 spacetime dimensions the Hodge dual of the 2-form $F$ will be another 2-form, $*F$. Now, $*F$ is in general not an exact form and so it's exterior derivative need not vanish. Instead, we have
\begin{equation}
d*F=\mu_0 J,
\end{equation}
where $J$ is the current 3-form. This is Maxwell's equations $\partial_{\alpha}F^{\alpha\beta}=\mu_o J^{\beta}$ expressed in the language of differential forms.
