Holonomy and Parallel Transport for Abelian Gerbes

✍ Scribed by Marco Mackaay; Roger Picken

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 445 KB
• Volume: 170
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2002.2085

✦ Synopsis

In this paper, we establish a one-to-one correspondence between Uð1Þ-gerbes with connections, on the one hand, and their holonomies, for simply connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with group Uð1Þ on a simply connected manifold M is a group morphism from the thin second homotopy group to Uð1Þ; satisfying a smoothness condition, where a homotopy between maps from ½0; 1 2 to M is thin when its derivative is of rank 42: For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case.