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

Abstract

Let $\mathcal{G}$ be a bundle gerbe with connection on a smooth manifold $M$, and let $\rho: G \rightarrow \operatorname{Diff}(M)$ be a smooth action of a Fr\'echet--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 $\rho_{g}^{*}\mathcal{G}$ to $\mathcal{G}$. We equip this group with a natural structure of abelian Fr\'{e}chet--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 Fr\'echet--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.