The diffeomorphism group and flat principal bundles

Abstract

Representations of the semidirect product group D■Diff(M) in the context of fiber bundle theory are studied. Here, D is the group of compactly supported functions and Diff(M) is the group of compactly supported diffeomorphisms of a manifold M. The carrier space is taken as the space of equivariant functions on a flat principal bundle over M where M is multiply connected. Two principal bundles are taken as equivalent if they are related by a gauge transformation. For the U(1) case it is found that the representations are irreducible, and the equivalence classes of representations are in one to one correspondence with the equivalence classes of bundles. Simple examples are discussed.