  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.
