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How an action that stabilizes a bundle gerbe gives rise to a Lie group extension

Bas Janssens, Peter Kristel

Abstract

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.

How an action that stabilizes a bundle gerbe gives rise to a Lie group extension

Abstract

Let be a bundle gerbe with connection on a smooth manifold , and let be a smooth action of a Fréchet--Lie group on that preserves the isomorphism class of . In this setting, we obtain an abelian extension of that consists of pairs , where , and is an isomorphism from to . We equip this group with a natural structure of abelian Fréchet--Lie group extension of , under the assumption that the first integral homology of 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.
Paper Structure (30 sections, 30 theorems, 139 equations, 4 figures)

This paper contains 30 sections, 30 theorems, 139 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The KKS extension.
  3. KKS extensions for bundle gerbes.
  4. Universal central extension of the Hamiltonian diffeomorphism group of a surface.
  5. Higher bundle gerbes.
  6. Acknowledgements
  7. Preliminaries: Bundle gerbes and transgression
  8. The strict monoidal 2-groupoid of bundle gerbes with connection
  9. Holonomy
  10. Transgression of bundle gerbes to the loop space
  11. The group of isomorphisms of a gerbe
  12. The abelian extension as a group.
  13. Functoriality of the group extension
  14. Interpretation in terms of the transgressed bundle
  15. Lie group structures
  16. ...and 15 more sections

Key Result

Theorem 1

If $M$ is an orientable, finite-dimensional manifold for which $H_1(M,\mathbb{Z})$ is finitely generated, then eq:H1extIntro is an abelian extension of Fréchet--Lie groups.

Figures (4)

  • Figure 1: A morphism $\Pi = (\pi_{21}, f_{12}, \Lambda_{12})$ from $G_1 \curvearrowright (M_1, \mathcal{G}_{1})$ to ${G_2 \curvearrowright (M_2, \mathcal{G}_{2})}$.
  • Figure 2: The composition of $(\pi_{21}, f_{12}, \Lambda_{12})$ with $(\pi_{32}, f_{23}, \Lambda_{23})$.
  • Figure 3: The morphism $\Lambda^{-1} \circ f^{*}A \circ \rho^{*}_{\pi(g)}\Lambda$ from $\rho_{\pi(g)}^{*}\mathcal{G}_{N}$ to $\mathcal{G}_{N}$.
  • Figure 4: Commutative diagram for $(g, \Phi) \in \operatorname{Aut}_{G}(\mathscr{T}(\mathcal{G}), \nabla, \mu)$.

Theorems & Definitions (62)

  • Theorem : A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • proof : Proof of Lemma \ref{['Lemma:HolonomyCurvature']}
  • Definition 2.7: Fusion structures
  • ...and 52 more