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 ${\mathrm{Diff}}_c(M)$ 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 $H_{\mathrm{ct}}^2({\mathcal{X}}_c(M),\mathbb{R})$ of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand--Fuks cohomology in view of the compact support condition.

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