Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929688367071232 |
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| author | Janssens, Bas Niestijl, Milan |
| author_facet | Janssens, Bas Niestijl, Milan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms Janssens, Bas Niestijl, Milan Mathematical Physics Differential Geometry Representation Theory 22E66, 17B65, 17B66, 17B56, 17B81 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. |
| title | Generalized Positive Energy Representations of the Group of Compactly Supported Diffeomorphisms |
| topic | Mathematical Physics Differential Geometry Representation Theory 22E66, 17B65, 17B66, 17B56, 17B81 |
| url | https://arxiv.org/abs/2404.06110 |