The electromagnetic action in the language of differential geometry is given by
$$\displaystyle{S \sim \int F \wedge \star F},$$
where $A$ is the one-form potential and $F={\rm d}A$ is the two-form field strength.
At the extremum of the action $S$, $F$ is constrained by ${\rm d}F=0$ and ${\rm d}\star F=0$.
Now, generalise the above action to
$$\displaystyle{S \sim \int H \wedge \star H}$$
where $B$ is the two-form potential and $H={\rm d}B$ is the three-form field strength.
At the extremum of the action $S$, $H$ is constrained by ${\rm d}H=0$ and ${\rm d}\star H=0$.
Are there any qualitative differences between the two sets of equations in $d+1$-dimensions?
