Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Source abstract

Abstract Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $$\overline{\rho }$$ ρ ¯ of the Lie group $${{\,\textrm{Diff}\,}}_c(M)$$ 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 $$\overline{\rho }$$ ρ ¯ . We show that if M is connected and $$\dim (M) > 1$$ dim ( M ) > 1 , then any such representation is necessarily trivial on the identity component $${{\,\textrm{Diff}\,}}_c(M)_0$$ Diff c ( M ) 0 . As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology $$H^2_\textrm{ct}(\mathcal {X}_c(M), \mathbb {R})$$ H ct 2 ( X c ( 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.