@Danu Yeah, so there is some symmetry transformation that generates it, meaning either one can phrase the duality transformation in terms of $A$, or this current is not actually associated to the duality transformation
@Danu Yes...but I don't even understand why that thing they call a conserved current is...a conserved current! Look at that weird "total derivative operator", how does that give us a "normal" conserved current?
@Danu The "proper" way to get the symmetry corresponding to a conserved quantity is to take the Noether charge and calculate the canonical transformation it generates in the Hamiltonian formulation, but I'm not sure that actually works in a gauge theory
It might be that there is no symmetry corresponding to this conserved current.
@EmilioPisanty It has for non-degenerate Lagrangian systems, i.e. those that have a unconstrained Hamiltonian formulation, because then the Noether charge just generates the symmetry. But for a gauge theory, you can't pass to the Hamiltonian formulation in a straightforward manner, in particular, you can't express the velocities in terms of the momenta
I'm not aware of an inverse Noether theorem for such degenerate systems - there might be one, but it is not obvious to me how one would prove it
@Danu By that method of characteristics and "adjoint symmetries" in section 3 of Anco, I guess
Note that they seem to really only work on the level of the equations
Symmetries of the equations of motion can fail to lift to symmetries of the action, I think