  It is known that the 3d Chern-Simons interaction describes the scaling limit
of a quantum Hall system and predicts edge currents in a sample with boundary,
the currents generating a chiral $U(1)$ Kac-Moody algebra. It is no doubt also
recognized that in a somewhat similar way, the 4d $BF$ interaction (with $B$ a
two form, $dB$ the dual $^*j$ of the eletromagnetic current, and F the
electromagnetic field form) describes the scaling limit of a superconductor. We
show in this paper that there are edge excitations in this model as well for
manifolds with boundaries. They are the modes of a scalar field with invariance
under the group of diffeomorphisms (diffeos) of the bounding spatial
two-manifold. Not all of this group seem implementable by operators in quantum
theory, the implementable group being a subgroup of volume preserving diffeos.
The $BF$ system in this manner can lead to the $w_{1+\infty }$ algebra and its
variants. Lagrangians for fields on the bounding manifold which account for the
edge observables on quantization are also presented. They are the analogues of
the $1+1$ dimentional massless scalar field Lagrangian describing the edge
modes of an abelian Chern-Simons theory with a disk as the spatial manifold. We
argue that the addition of ``Maxwell'' terms constructed from $F\wedge ^*F$ and
$dB\wedge ^*dB$ do not affect the edge states, and that the augmented
Lagrangian has an infinite number of conserved charges- the aforementioned
scalar field modes- localized at the edges. This Lagrangian is known to
describe London equations and a massive vector field. A $(3+1)$ dimensional
generalization of the Hall effect involving vortices coupled to $B$ is also
proposed.
