Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations ρ¯ of the Lie group Diffc(M) of compactly supported diffeomorphisms of a smooth manifold M that satisfy a so-called generalized positive energy condition. In particular, this captures representations that are in a suitable sense compatible with a KMS state on the von Neumann algebra generated by ρ¯. We show that if M is connected and dim(M)>1, then any such representation is necessarily trivial on the identity component Diffc(M)0. As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology Hct2(Xc(M),R) of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.