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Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

Bas Janssens, Milan Niestijl

TL;DR

The paper investigates representations of the group of compactly supported diffeomorphisms $\mathrm{Diff}_c(M)$ that are compatible with generalized positive energy (GPE) and, in KMS-type settings, with thermal states. The authors show that for connected manifolds with $\dim(M)>1$, every smooth GPE representation of $\mathrm{Diff}_c(M)\rtimes_\upsilon \mathbb{R}$ is trivial on the identity component, i.e. $\mathrm{Diff}_c(M)_0$ lies in the kernel, highlighting a strong representation-theoretic localization at infinity. A central technical achievement is the complete computation of the continuous second Lie algebra cohomology $H^2_{\mathrm{ct}}(\mathcal{X}_c(M),\mathbb{R})$: it vanishes when $\dim(M)>1$, and in dimension one it matches $H^0_{\mathrm{dR}}(M)$, reflecting a Virasoro-type obstruction in the noncompact, one-dimensional case. The paper also clarifies how this cohomological picture compares with Gelfand–Fuks cohomology, proving injectivity of the natural map $H^2_{\mathrm{ct}}(\mathcal{X}(M),\mathbb{R}) \to H^2_{\mathrm{ct}}(\mathcal{X}_c(M),\mathbb{R})$ while typically not being surjective in the dimension-one setting. Collectively, these results constrain the possible generalized positive energy and KMS representations of diffeomorphism groups, with implications for asymptotic symmetry groups and their quantum representations.

Abstract

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $\overlineρ$ 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 $\overlineρ$. 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.

Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms

TL;DR

The paper investigates representations of the group of compactly supported diffeomorphisms that are compatible with generalized positive energy (GPE) and, in KMS-type settings, with thermal states. The authors show that for connected manifolds with , every smooth GPE representation of is trivial on the identity component, i.e. lies in the kernel, highlighting a strong representation-theoretic localization at infinity. A central technical achievement is the complete computation of the continuous second Lie algebra cohomology : it vanishes when , and in dimension one it matches , reflecting a Virasoro-type obstruction in the noncompact, one-dimensional case. The paper also clarifies how this cohomological picture compares with Gelfand–Fuks cohomology, proving injectivity of the natural map while typically not being surjective in the dimension-one setting. Collectively, these results constrain the possible generalized positive energy and KMS representations of diffeomorphism groups, with implications for asymptotic symmetry groups and their quantum representations.

Abstract

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations of the Lie group of compactly supported diffeomorphisms of a smooth manifold 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 . We show that if is connected and , then any such representation is necessarily trivial on the identity component . As an intermediate step towards this result, we determine the continuous second Lie algebra cohomology of the Lie algebra of compactly supported vector fields. This is subtly different from Gelfand--Fuks cohomology in view of the compact support condition.
Paper Structure (22 sections, 21 theorems, 46 equations)

This paper contains 22 sections, 21 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projective unitary representations
  4. Cohomology of Lie algebras and Lie groups
  5. Generalized positive energy (GPE) representations
  6. GPE representations of $\mathop{\mathrm{Diff}}\nolimits_c(M) \rtimes_\upsilon \mathbb{R}$
  7. Ideals of the Lie algebra $\mathcal{X}_c(M)$
  8. The proof of thm: gpe_vfields
  9. The proof of Theorem A
  10. Continuous second Lie algebra cohomology
  11. Manifolds $M$ of dimension $\dim(M) > 1$
  12. The local setting
  13. The case $k=-1$.
  14. The case $0 \leq k \leq n$.
  15. The case $k = n+1$.
  16. ...and 7 more sections

Key Result

Proposition 2.12

Let $\overline{\pi} : \mathfrak{g} \to \mathfrak{pu}(\mathcal{D})$ be a projective unitary representation of $\mathfrak{g}$ that is of generalized positive energy at $\xi \in \mathfrak{g}$. Let $\omega \in C_{\mathrm{ct}}^2(\mathfrak{g}, \mathbb{R})$ be a 2-cocycle whose class $[\omega]$ in $H^2_\ma In particular, if $[\omega] = 0$ in $H^2_\mathrm{ct}(\mathfrak{g}, \mathbb{R})$, then

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Lie algebra cohomology
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 50 more