Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

<jats:title>Abstract</jats:title> <jats:p>Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\overline{\rho }$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> </jats:alternatives> </jats:inline-formula> of the Lie group <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${{\,\textrm{Diff}\,}}_c(M)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Diff</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of compactly supported diffeomorphisms of a smooth manifold <jats:italic>M</jats:italic> 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 <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\overline{\rho }$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> </jats:alternatives> </jats:inline-formula>. We show that if <jats:italic>M</jats:italic> is connected and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\dim (M) &gt; 1$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>, then any such representation is necessarily trivial on the identity component <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${{\,\textrm{Diff}\,}}_c(M)_0$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>Diff</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mi>c</mml:mi> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula>. As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$H^2_\textrm{ct}(\mathcal {X}_c(M), \mathbb {R})$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mtext>ct</mml:mtext> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand–Fuks cohomology in view of the compact support condition.</jats:p>