Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

Abstract

We study bundle gerbes on manifolds $M$ that carry an action of a connected Lie group $G$. We show that these data give rise to a smooth 2-group extension of $G$ by the smooth 2-group of hermitean line bundles on $M$. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev-Mickelsson-Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group $G$, we prove that the smooth 2-group extensions of $G$ arising from our construction provide new models for the string group of $G$.

79 pages, 2 figures; v2: minor corrections, comments and references added; Final version to be published in Communications in Mathematical Physics

References in corpus (9)

• Higher Topos Theory
• Higher-Dimensional Algebra V: 2-Groups
• Introduction to the language of stacks and gerbes
• More Morphisms between Bundle Gerbes
• Global gauge anomalies in two-dimensional bosonic sigma models
• Infinitesimal Symmetries of Dixmier-Douady Gerbes
• Principal $\infty$-Bundles and Smooth String Group Models
• Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation
• Non-associative magnetic translations: A QFT construction

Cited by in corpus (9)

• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Quantum symmetries in orbifolds and decomposition
• Principal $\infty$-Bundles and Smooth String Group Models
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• Chern correspondence for higher principal bundles
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Infinite Dimensional Topological-Holomorphic Symmetry in Three-Dimensions
• Gerbes in Geometry, Field Theory, and Quantisation
• Extended Field Theories as higher Kaluza-Klein theories