Hodge dual and QED I was studying the paper Topological massive gauge theories in three dimensions by Deser, Jackiw and Templeton. In the paper, they use Hodge dual for some reason which I don't understand at all. So I asked some people, they said: if $dF = 0$ then $F = ^* d\phi$ for some form $\phi$, where F is the Maxwell electromagnetic tensor in differential form. They also added: $F=dA$ hence, $A= d\psi + ^*d\chi $, again for some suitable forms. 
I don't see how and "don't even remember" correctly, can someone shed some light on this. I don't know as how $F = ^* d\phi$  and $A= d\psi + ^*d\chi $ are obtained. What is going on? 
Also how are these used in understanding or simplifying something in QED?
 A: I'm assuming that, if you're reading this paper, then you're at least familiar with the basic manipulations surrounding differential forms. Let us denote $\Omega_p(\mathcal{M})$ the set of $p$-forms on some manifold $\mathcal{M}$, and equip this with an exterior derivative $\mathrm{d}:\Omega_p(\mathcal{M})\to\Omega_{p+1}(\mathcal{M})$.
The Hodge dual (or Hodge star) is an operator $\star$ that defines a canonical isomorphism from $\Omega_{p}(\mathcal{M})$ to $\Omega_{n-p}(\mathcal{M})$, where $n=\dim(\mathcal{M})$. If, given two $p$-forms $v$ and $w$, their inner product is $\langle v,w\rangle$ (for QED this would be something like $\langle F,F\rangle\equiv F_{\mu\nu}F^{\mu\nu}$ for the field-strength 2-form $F$), then the Hodge star is defined to satisfy
$$w\wedge\star v\equiv \langle w,v\rangle\,V_n,$$
where $V_n$ is the volume $n$-form (typically $\mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n$ in a coordinate basis).
To learn more about the Hodge star and get more of a handle on it, I highly recommend Nakahara's book Geometry, Topology, and Physics, as well as basically any standard text on differential geometry.
Now, I don't know about the exact question you're asking, but the Hodge dual is a very useful tool in electrodynamics. In particular, the vacuum Maxwell equations take the particularly simple form (at least in four dimensions)
$$\mathrm{d}F=0,\hspace{0.5cm}\mathrm{d}\star F=0.$$
The QED action takes the simple form
$$S\propto\int F\wedge\star F,$$
which generalizes to non-abelian theories. The $\theta$ term takes the form
$$S_{\theta}\propto\theta\int F\wedge F.$$
Furthermore, in a non-vacuum solution, one can find the electric and magnetic charges bounded by a surface $\Sigma$ are given by
$$Q_e\propto\int_{\Sigma}\star F,\hspace{0.5cm}Q_m\propto\int_{\Sigma}F.$$
They also allow for generalizations of QED to extended objects, such as strings and membranes that arise in string theory, and allow one to define the magnetic and electric charges on these objects.
In summary, the Hodge dual, as an integral part of our modern language in differential geometry, finds a natural place in QED.
(I can't answer the actual details of your question -- namely why $F=\star\mathrm{d}\phi$ or $A=\mathrm{d}\psi+\star\mathrm{d}\chi$ -- without actually reading the paper in question. Sorry about that.)
Anyway, I hope this helps!
A: You can find this in many mathematical books about differential forms and de-Rham cohomology. 
$F$ is a $2$-form. If $F$ is closed, i.e. $dF=0$, then by Poincare lemma, locally $F$ can be written as $F=dA$. If you perform a gauge transformation $A\rightarrow A+d\chi$, where $\chi$ is a smooth function, then $F$ is invariant.
Why don't you read de-Rham cohomology and Hodge star duality from Nakahara first?
