Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Abstract

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations \(\overline{\rho }\) of the Lie group \({{\,\textrm{Diff}\,}}_c(M)\) 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 \(\overline{\rho }\). We show that if M is connected and \(\dim (M) > 1\), then any such representation is necessarily trivial on the identity component \({{\,\textrm{Diff}\,}}_c(M)_0\). As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology \(H^2_\textrm{ct}(\mathcal {X}_c(M), \mathbb {R})\) of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.