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

2024/01/24 by Bas Janssens, Janssens, Bas, Peter Kristel +1
Mathematics · #22E65 (Primary) 22F50 #53C08 #58D05 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2401.13453

openalex publication_date 2024/01/24 · openalex created_date 2024/01/26 · openalex updated_date 2026/07/28

Abstract

Let G be a bundle gerbe with connection on a smooth manifold M, and let ρ: G → Diff(M) be a smooth action of a Fréchet--Lie group G on M that preserves the isomorphism class of G. In this setting, we obtain an abelian extension of G that consists of pairs (g,A), where g ∈ G, and A is an isomorphism from ρ_g^*G to 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.

Related

• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
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