Differentiable Stacks and Gerbes

2006/05/27 by Kai Behrend, Ping Xu, Behrend, Kai +1 · 8 citations
Mathematics · Medicine · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Ophthalmology and Eye Disorders

paper · pdf · doi:10.48550/arxiv.math/0605694

openalex publication_date 2006/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹-bundles and S¹-gerbes over differentiable stacks. In particular, we establish the relationship between S¹-gerbes and groupoid S¹-central extensions. We define connections and curvings for groupoid S¹-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹-bundles and S¹-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S¹-central extensions with prescribed curvature-like data.

Citations

• Foundations of Topological Stacks I
• Topological and Smooth Stacks
• Gerbes on complex reductive Lie groups
• Bundle Gerbes: Stable Isomorphism and Local Theory
• Introduction to the language of stacks and gerbes

Cited by

• Categorifying isomonodromic deformations via Lie groupoids I: Logarithmic singularities
• Lyapunov 1-forms on orbifolds
• The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe
• Simplicial sheaves of modules and Morita invariance of groupoid cohomology
• Higher Gauge Flow Models
• Topology of singular foliations of closed 1-forms on orbifolds
• Morita equivalence of shifted symplectic Lie n-groupoids
• Yang-Mills theory for multiplicative Ehresmann connections

Related

• S¹-bundles and gerbes over differentiable stacks
• Geometric structures on Lie groupoids and differentiable stacks
• Equivariant gerbes over compact simple Lie groups
• Reduction of strongly equivariant bundle gerbes with connection and curving
• Bundle Gerbes