### Groupoidification from sigma-Models?

#### Posted by Urs Schreiber

The interest in groupoidification (see our recent discussion) is to a large extent motivated from the feeling that it illuminates general structural aspect of quantum field theory.

My motivation is this:

to every differential nonabelian cocycle describing an associated $\infty$-vector bundle with connection, there should canonically be associated the corresponding *$\sigma$-model* QFT, which, physically speaking, describes the worldvolume theory of a brane couopled to this $\infty$-bundle.

I have been thinking about this for quite a while now, starting with a series of posts on QFT of the charged $n$-particle. It took me a bit to get the required machinery into place, such as the interpretation of parallel transport $\infty$-functors for $\infty$-bundles with connection in homotopical cohomology theory, or the machinery of universal $\infty$-bundles.

Then John started teaching us about *groupoidification* and I noticed that this should naturally arise when forming the $\sigma$-model of a nonabelian cocycle. I chatted about the basic idea of this insight in An exercise in groupoidification: the path integral.

Now I found the time to expand on this in much more detail. Not that this is done yet, but a coherent closed picture seems to be emerging, which I describe in these notes:

$n$Lab/schreiber: Nonabelian cocycles and their $\sigma$-Model QFTs

I would like to draw the attention of those interested in groupoidification in particular to the considerations in section 3.3, page 12 *Sections and homotopies*, to section 4.2, page 16 *Branes and bibranes* and section 4.3, page 18 *Quantum propagation*, which aims to formalize the description of QFTs by Freed, Willerton and Bartlett in the $\infty$-functorial context and naturally runs into pull-push of bundles of groupoids.

There is a list of examples meant to illustrate how the abstract nonsense reproduces familiar structures in concrete examples. I need to spend more time on typing more details, but comparing section 5.1.6, p. 22 *The path integral* with the details given in An exercise in groupoidification: the path integral and then with the description of Dijkgraaf-Witten theory and the Yetter model as a $\sigma$-model at this entry should give the idea.

Here is the abstract:

**Abstract.**

Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to $\infty$-group

oids or even to general $\infty$-categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) – possibly equivariant, possibly with connection – as well as the correspondingassociatedhigher vector bundles.We propose, expanding on considerations in [Freed, Willerton, Bartlett], a systematic $\infty$-functorial formalization of the $\sigma$-model quantum field theory associated with a given nonabelian cocycle regarded as the background field for a brane coupled to it. We define propagation in these $\sigma$-model QFTs and recover central aspects of groupoidification of linear algebra.

In a series of examples we show how this formalization reproduces familiar structures in $\sigma$-models with finite target spaces such as Dijkgraaf-Witten theory and the Yetter model. The generalization to $\sigma$-models with smooth target spaces is developed elsewhere.

## Kohei Tanaka; Re: Groupoidification from sigma-Models?

Is this the right thread for a higher view of Groupoidification?

Kohei Tanaka, A model structure on the category of small categories for coverings, 30 July 2009.

Abstract: We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification.