THE EDGE STATES OF THE BF SYSTEM AND THE LONDON EQUATIONS

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 w1+ ∞ 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 ∧ ∗ F and dB ∧ ∗ 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