Maxwell equations with a source ending somewhere Consider a 1-dimensional object in 4-dimensional spacetime ($t,x,y,z)$ that carries a charge $q$ under a gauge symmetry of some 2-form gauge field with field strength $F$ (a 3-form). Let's take the object to lie along the $x$ direction (i.e. $y=z=0$), from $x=0$ to infinity. Maxwell's equation is then $$d\star F=q\theta(x)\delta^{(2)}(y,z)$$
where $\theta(x)$ is the Heaviside function and the delta function just sets $z$ and $y$ to zero. Differentiating both sides, I find that $$q\delta(x)=0$$
whose solution I guess is $q=0$ -- the object was never charged to begin with. Does this mean a source cannot have a finite extent? Or rather that if it is to end somewhere, there must be something else "carrying away" the charge? Is there a more formal way of saying this? Thanks
 A: I think your 2-form charge (or rather flux) is not valid since it is not conserved.
In your Lagrangian you have a coupling like $B^{\mu\nu}J_{\mu\nu}$ where $J$ is your 2-form current on the right side of your equation. Gauge invariance in $B$ means $J$ must be a conserved current,
$$\nabla_\mu J^{\mu\nu}=0$$
and this in turn implies that the dual $\star J$ must be an exact form, $d\star J=0$. Thus if you integrate $\star J$ over a closed surface you must get zero.
In your case $\star J$ should be understood as $q \theta(x)\delta^{(2)}(y,z)dy\wedge dz$. (Think by analogy about how ordinary current is in the $dt$ direction and the dual is in $dx\wedge dy\wedge dz$ to understand this). But this is not exact since its integral on a sphere doesn't vanish.
The situation is very similar to how in electromagnetism on a compact space you must have vanishing net charge.
A: First look at the electromagnetic field. Let $\mathbf{A}$ be the EM 1-form and $\mathbf{F}=\mathrm{d}\mathbf{A}$ its field 2-form. Then $\star\mathbf{F}$ is a 2-form and $\mathbf{J}:=\mathrm{d}\star\mathbf{F}$ a 3-form. This requires that $\mathrm{d}\mathbf{J} = \mathrm{d}^2\star\mathbf{F} = 0,$ which is the equation of charge conservation $\left( \partial_t\rho + \nabla\cdot\mathbf{j} \right) dt \wedge dx \wedge dy \wedge dz = 0.$
In the same way, if $\mathbf{A}$ is a 2-form and $\mathbf{F}=\mathrm{d}\mathbf{A}$ its field 3-form. Then $\star\mathbf{F}$ is a 1-form and $\mathbf{J}:=\mathrm{d}\star\mathbf{F}$ a 2-form which must satisfy an equation of charge conservation $\mathrm{d}\mathbf{J}=0.$ The source that you have given isn't conserved.
