Connections on non-abelian Gerbes and their Holonomy

Details Connections on non-abelian Gerbes and their Holonomy Connections on non-abelian Gerbes and their Holonomy

2008-08-14

arxiv.org/abs/0808.1923v1

Mathematics

Differential Geometry

87 pages + 17 pages appendix, preliminary version

Scientific paper

We introduce transport 2-functors as a new way to describe connections on gerbes with arbitrary strict structure 2-groups. On the one hand, transport 2-functors provide a manifest notion of parallel transport and holonomy along surfaces. On the other hand, they have a concrete local description in terms of differential forms and smooth functions. We prove that Breen-Messing gerbes, abelian and non-abelian bundle gerbes with connection, as well as further concepts arise as particular cases of transport 2-functors, for appropriate choices of the structure 2-group. Via such identifications transport 2-functors induce well-defined notions of parallel transport and holonomy for all these gerbes. For abelian bundle gerbes with connection, this induced holonomy coincides with the existing definition. In all other cases, finding an appropriate definition of holonomy is an interesting open problem to which our induced notion offers a systematical solution.

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