December 30, 2008

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:

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 $\infty$-groupoids 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 corresponding associated higher 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.

Posted at December 30, 2008 3:40 PM UTC

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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.

Posted by: Jonathan Vos Post on July 31, 2009 1:50 AM | Permalink | Reply to this

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

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

In principle, yes. A right thread. But notice some terminological subtelties:

As we have discussed before, the term “groupoidification” as used by John Baez has a slightly different connotation than what other people might mean by it.

John’s notion is a special case of what, following David Ben-Zvi, we started calling geometric $\infty$-function theory:

this is about the operation of pull-pushing entities through spans/correspondences.

What is in other circles, such as in the article that you point to, understood under “groupoidification” is “turning a category freely into a groupoid”.

This last statement is conveniently discussed at the level of nerves , where it is the special case of a more general operation known as Kan fibrant replacement.

This reads in the nerve of an $\infty$-category and spits out the “$\infty$-groupoid obtained from it by inverting all cells”.

But there is a close relation between this latter notion of groupoidification and the former one: both involve spans.

Indeed, if you look at the entry Kan fibrant replacement you’ll see that one nice way to form the $\infty$-groupoid associated to an $\infty$-category $C$ is to:

- let its 1-morphisms be all spans in $C$

- let its 2-morphisms be all spans of spans in $C$

- let its 3-morphisms be all spans of spans of spans in $C$, etc.

(Right at the moment this is spelled out at that entry only for $n=1$. But either you take it as an exercise to check that the prescription for the simplicial set called $sd \Delta^n$ there produces precisely this picture, or else you wait until some good soul spells it out on the $n$Lab in more detail).

So what John calls groupoidification is (the low dimensional case of) what in this context one would call groupoidification, namely passing to spans, followed by a next step where these spans are fed into a pull-push functor of sorts.

Finally, concerning that particular article that you point to: this presents a variation of the folk model structure on categories. This is the kind of structure that allows us to say the equivalent of “Kan fibrant replacement” without first passing to nerves by using the notion of fibrant replacement given by the model structure on the collection of categories itself.

So this particular article is not really relevant for the kind of “groupoidification” application that the above entry is about. Even though there is some kind of general relation. But the aim and purpose of that article is really a different one.

Posted by: Urs Schreiber on July 31, 2009 10:32 AM | Permalink | Reply to this

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

Thank you, Urs. You are a very good teacher to be able to transform a bad teacher’s “you’re wrong” to a student, into “You’re right in another way, so let me teach you something to positively reinforce your interest.”

I’ve been wrestling with enumerations of certain polyhedral complexes (with polyhedra other than just simplexes) up to isometry, and rigidity theorems on these, and that the Lie algebra of the automorphisms of these complexes is a semidirect product of the Lie algebras which don’t have nilpotent ideals. It all seems to be a decategorified version of something that I don’t know.

I’ve read in Bull.Math’s obituary issue that Chern did the calculations in his head for S.-S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944) 747-752.

Posted by: Jonathan Vos Post on July 31, 2009 11:21 PM | Permalink | Reply to this

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