The Edge States of the BF System and the London Equations

Abstract

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.