### Connections on Nonabelian Gerbes and their Holonomy

#### Posted by Urs Schreiber

Finally this sees the light of day:

U.S. and Konrad Waldorf
*Connections on non-abelian Gerbes and their Holonomy*

arXiv:0808.1923

**Abstract:**
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 bundles gerbes with connection, as well as further concepts arise as particular cases of transport 2-functors for appropriate choices of 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.

This builds on

Smooth functors vs. differential forms - which establishes the relation between smooth 2-functors with values in Lie 2-groups and differential $L_\infty$-algebraic connection data;

Parallel transport and functors - which establishes the relation between transport $n$-functors and $n$-bundles/($n-1$)-gerbes with connection for $n=1$

and realizes the construction I did with John in Higher gauge theory as a cocycle in second nonabelian *differential* cohomology which represents a globally defined transport 2-functor.

$n$-Café regulars will remember some discussion of the development of the notion of locally trivializable transport $n$-functors and their classification in nonabelian differential cohomology in early Café entries such as

On Transport, Part I

On Transport, Part II

and many other ones, to some extent summarized in this big set of slides. You might enjoy Konrad’s more readable slides .

For instance you can read about 2-vector transport for associated String 2-bundles (discussed for instance here), twisted vector bundles with connection as quasi-trivializations of 2-vector transport (which I talked about at the Fields institute), local formulas for abelian and nonabelian surface holonomy as vaguely conceived before here, see all this now related to Street’s theory of codescent in the context of differential cohomology which is the right formalization of those *2-paths in the Cech 2-groupoid* which I kept talking about once upon a time.

Alas, a couple of things didn’t quite make it into this file or are only indicated. But if you could wait that long you probably can also wait a little longer still…

## Re: Connections on Nonabelian Gerbes and their Holonomy

Would it be asking too much if someone can open an entry on Gerbes on ncatlab? I would like to learn about them…