Non Abelian Differentiable Gerbes

Abstract

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for $G$-gerbes over stacks. We also introduce $G$-central extensions of groupoids, generalizing the standard groupoid $S^1$-central extensions. As an example, we apply our theory to study the differential geometry of $G$-gerbes over a manifold.

67 pages, references added and updated, final version to appear in Adv. Math

References in corpus (12)

• Higher Yang-Mills Theory
• Topological and Smooth Stacks
• Higher gauge theory I: 2-Bundles
• Introduction to the language of stacks and gerbes
• Differentiable Stacks and Gerbes
• G-gerbes, principal 2-group bundles and characteristic classes
• Deformation quantization modules I:Finiteness and duality
• Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes
• Deformation quantization modules II. Hochschild class
• Deformations of gerbes on smooth manifolds
• Equivariant Dixmier-Douady Classes
• Central Extensions of Gerbes

Cited by in corpus (19)

• Principal 2-bundles and their gauge 2-groups
• G-gerbes, principal 2-group bundles and characteristic classes
• Four Equivalent Versions of Non-Abelian Gerbes
• Non-abelian higher gauge theory and categorical bundle
• Regular Poisson manifolds of compact types (PMCT 2)
• Higher Lie algebra actions on Lie algebroids
• Geometry of Maurer-Cartan Elements on Complex Manifolds
• Higher Gauge Theory
• Double Principal Bundles
• Principal actions of stacky Lie groupoids
• Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid
• On 2-Holonomy
• Cosymplectic groupoids
• On two notions of a Gerbe over a stack
• Atiyah sequences and connections on principal bundles over differentiable stacks
• Shifted coisotropic structures for differentiable stacks
• The integration problem for principal connections
• Extension of topological groupoids and Serre, Hurewicz morphisms
• On fibrations of Lie groupoids