Maxwell and Bianchi equations

Question

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 $$\partial_{\mu}[(ImZ)F^{\mu\nu}+i(ReZ)\tilde{F}^{\mu\nu}]=0?$$

Answer 1

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.