Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $\overline{\rho}$ of the Lie group $\mathrm{Diff}_c(M)$ 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 $\mathrm{Diff}_c(M)_0$. As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology $H^2_{\mathrm{ct}}(\mathcal{X}_c(M), \mathbb{R})$ of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand--Fuks cohomology in view of the compact support condition.