4D Edge Currents from 5D Chern-Simons Theory

A class of two dimensional conformal field theories is known to correspond to three dimensional Chern-Simons theory. Here we claim that there is an analogous class of four dimensional field theories corresponding to five dimensional Chern-Simons theory. The four dimensional theories give a coupling between a scalar field and an external divergenceless vector field and they may have some application in magnetohydrodynamics. Like in conformal theories they possess a diffeomorphism symmetry, which for us is along the direction of the vector field, and their generators are analogous to Virasoro generators. Our analysis of the abelian Chern-Simons system uses elementary canonical methods for the quantization of field theories defined on manifolds with boundaries. Edge states appear for these systems and they yield a four dimensional current algebra. We examine the quantization of these algebras in several special cases and claim that a renormalization of the $5D$ Chern-Simons coupling is necessary for removing divergences.