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 \((\overline{\rho}, \mathcal{H})\) 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 (which is subtly different from Gelfand--Fuks cohomology).