Multiplicative vector fields on bundle gerbes

Abstract

This paper makes use of multiplicative vector fields on Lie groupoids to model infinitesimal symmetries of S 1 -bundle gerbes. It is shown that a connective structure on a bundle gerbe gives rise to a natural horizontal lift of multiplicative vector fields to the bundle gerbe, and that the 3-curvature presents the obstruction to the horizontal lift being a morphism of Lie 2-algebras. Connectionpreserving multiplicative vector fields on a bundle gerbe with connective structure are shown to inherit a natural Lie 2-algebra structure; moreover, this Lie 2-algebra is quasi-isomorphic to the Poisson-Lie 2-algebra of the 2-plectic base manifold (M, χ), where χ is the 3-curvature of the connective structure. As an application of this result, we give an analogue of a formula of Kostant in the 2-plectic and quasi-Hamiltonian context.

On the prequantisation map for 2-plectic manifolds

“…The results and proofs of this article will appear in part in the PhD thesis of the first named author. They are independent and different from the research of Krepski and Vaughan in [13], and of Schreiber and co-workers (see, e.g., [7]). The main difference stems from our objective to always give a global geometric picture (including a P U (H)-bundle geometrising the cohomology class of the 2-plectic form) and explicit formulae, notably for the arising Lie 2-algebra homomorphisms.…”

“…Our definition asks for a more complete set-up for the connection to arise, notably a principal P U (H)-bundle and a gerbe (or equivalently, a central S 1 -extension of groupoids) on its total space having the "right" 3-curvature. In [13], the notion of a quantisation used is closer to ours, but the analogon of our P U (H)-bundle projection Π is there only a surjective submersion.…”

“…Remark 2.13. The space of weak infinitesimal symmetries is also considered in [13], where they are referred to as vector fields that preserve (A, θ), but the above crucial characterisation does not appear there. Our Proposition 2.13 below corresponds to Proposition 3.17 and Corollary 3.18 in [13].…”

“…Remark 3.7. Theorem 5.1 of [13] shows that L • (M, ω) and WSym(P, Y ) are quasi-isomorphic via the existence of an "invertible butterfly". Our techniques are completely different and yield an explicit map of Lie 2-algebras that turns out to be a quasi-isomorphism.…”