How an action that stabilizes a bundle gerbe gives rise to a Lie group extension
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911763833815040 |
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| author | Janssens, Bas Kristel, Peter |
| author_facet | Janssens, Bas Kristel, Peter |
| contents | Let $\mathcal{G}$ be a bundle gerbe with connection on a smooth manifold $M$, and let $ρ: G \rightarrow \operatorname{Diff}(M)$ be a smooth action of a Fréchet--Lie group $G$ on $M$ that preserves the isomorphism class of $\mathcal{G}$. In this setting, we obtain an abelian extension of $G$ that consists of pairs $(g,A)$, where $g \in G$, and $A$ is an isomorphism from $ρ_{g}^{*}\mathcal{G}$ to $\mathcal{G}$. We equip this group with a natural structure of abelian Fréchet--Lie group extension of $G$, under the assumption that the first integral homology of $M$ is finitely generated. As an application, we construct the universal central extension (in the category of Fréchet--Lie groups) of the group of Hamiltonian diffeomorphisms of a symplectic surface. As an intermediate step, we obtain a central extension of the group of exact volume-preserving diffeomorphisms of a 3-manifold whose corresponding Lie algebra extension is conjectured to be universal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | How an action that stabilizes a bundle gerbe gives rise to a Lie group extension Janssens, Bas Kristel, Peter Differential Geometry Symplectic Geometry 22E65 (Primary) 22F50, 53C08, 58D05 (Secondary) Let $\mathcal{G}$ be a bundle gerbe with connection on a smooth manifold $M$, and let $ρ: G \rightarrow \operatorname{Diff}(M)$ be a smooth action of a Fréchet--Lie group $G$ on $M$ that preserves the isomorphism class of $\mathcal{G}$. In this setting, we obtain an abelian extension of $G$ that consists of pairs $(g,A)$, where $g \in G$, and $A$ is an isomorphism from $ρ_{g}^{*}\mathcal{G}$ to $\mathcal{G}$. We equip this group with a natural structure of abelian Fréchet--Lie group extension of $G$, under the assumption that the first integral homology of $M$ is finitely generated. As an application, we construct the universal central extension (in the category of Fréchet--Lie groups) of the group of Hamiltonian diffeomorphisms of a symplectic surface. As an intermediate step, we obtain a central extension of the group of exact volume-preserving diffeomorphisms of a 3-manifold whose corresponding Lie algebra extension is conjectured to be universal. |
| title | How an action that stabilizes a bundle gerbe gives rise to a Lie group extension |
| topic | Differential Geometry Symplectic Geometry 22E65 (Primary) 22F50, 53C08, 58D05 (Secondary) |
| url | https://arxiv.org/abs/2401.13453 |