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August 20, 2005

CFT, Gerbes and K-Theory in Oberwolfach, VI

Posted by urs

With kind permission, here are more details on Brano Jurčo’s talk:

Branislav Jurčo
On the Classification of Nonabelian Bundle Gerbes with Connections
(in preparation)

Update 5th October 2005: The corresponding preprint has now appeared as math.DG/0510078.

Principal G-bundles over X are classified by homotopy equivalence classes


of maps from X to a topological space called BG, the classifying space of BG-bundles. The claim is that this construction can be generalized to 1-nonabelian gerbes/2-bundles by replacing BG with the nerve of the structure 2-group (regarded as a 2-category with a single object).

In fact, the method would directly generalize to n-gerbes/n+1 -bundles for arbitrary n.

Using this result it should be possible to prove the conjecture from the end of math.QA/0504123, which says that String(n)-bundles over M are ‘the same’ as 1-gerbes/2-bundles with structure 2-group P 1 Spin(n).

The construction is based on the fact (see for instance the first few pages of Duskin for a review) that categories can equivalently be regarded as simplicial sets with functors equivalently regarded as simplicial maps, and that in the presence of a topology these simplicial objects can be turned into topological spaces.

To see how the construction works, reconsider the familiar case of a principal G-bundle. Regard the structure group G as a category with a single object and one morphism for each group element. The nerve G of that category is obtained by including

- a single 0-simplex
- a 1-simplex g for each gG
-a 2-simplex

(2)g 1 g 2 g 1 g 2

for all g 1 ,g 2 G
- a 3-simplex for each four composable such 2-simplices
- and so on .

Next, consider some manifold M with a good covering U={U i} iI by open, contractible subsets U iM. The nerve U of this covering has

- a 0-simplex U i for each iI
- a 1-simplex U iU j for every nonempty double intersection U iU j
- a 2-simplex for every triple intersection
- and so on.

Assume the covering is very fine and consider a simplicial map from the nerve of the covering to the nerve of G, as defined above. This evidently looks a lot like a (consistent!) assignment of transition functions of a G-bundle to the covering U. One can easily see that a gauge transformation of such transitions is nothing but a homotopy of the above simplicial maps. Hence homotopy equivalence classes [U ,G ] of such simplicial maps have to do with equivalence classes of G-bundles over M.

One can take the ‘geometric realization’ of this simplicial construction. If everything works right one has

(3)[U ,G ][U ,G ].

Moreover, U is homopy equivalent to M itself, so that finally one has that

(4)[M,G ]:=[M,BG]

is seen to classify G-bundles on M.

Now generalize this idea. It was noted in hep-th/0412325 (see here for a revised version) that the transition functions of a nonabelian gerbe/principal 2-bundle over M with structure 2-group G2 similarly come from simplices in the 2-group G2 , regarded as a 2-category with a single object.

By the same reasoning as above, it should follow that homotopy classes of simplicial maps

(5)[M,G2 ]

from M to the geometric realization of the nerve of the 2-group G2 (in the 2-category-sense) classify nonabelian G2 -gerbes/2-bundles over M.

I cannot draw diagrams right now, but in order to visualize this have a look at the diagram which I reproduced in a previous entry and just replace all appearances of hol with .

Clearly, if this is true at the level of 2-bundles, it should go through for Gn-bundles for all n.

Next, Brano Jurčo notes a certain relation between the nerve G2 of a 2-group G2 regarded as a 2-category with a single object, and the nerve N(G2 ) of the same 2-group but regarded as a 1-category. There is a general notion of classifying space for simplicial groups Q, which is a simplicial space called W¯Q. The claim is that

(6)G2 =W¯N(G2 ).

This again should mean that the classifying space for G2 -2-bundles (the geometric realization of the left hand side) is the same as that for N(G2 )-bundles.

Now, in math.QA/0504123 it was shown that there is a certain 2-group

(7)G2 =P 1 Spin(n)

such that

(8)N(G2 )String(n)

is the String(n)-group. So in this case the above would say that String(n)-bundles over M have the same classification as P 1 Spin(n)-2-bundles over M, as conjectured. (I have sketched another argument why this should be true in my thesis (see section 10.7.2 and 12.3.1).)

Brano also does something for the classification of gerbes with 1-connection, but has not yet included the curving 2-form. From the argument mentioned in the previous entry it seems that the same line of reasoning should apply for p-bundles with p-connection and p-holonomy, too.

Posted at August 20, 2005 1:43 PM UTC

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Re: CFT, Gerbes and K-Theory in Oberwolfach, VI

Hi Urs

It’s wonderful to see such excellent reporting. Have you heard any more about that 2-topology thesis?

Posted by: Kea on August 23, 2005 1:49 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VI

You mean Igor Bacović’s work on bisites, which I had mentioned recently? No, no more details yet. It’s supposed to appear by the end of this year, sometime, as far as I understand.

Posted by: Urs on August 23, 2005 9:41 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VI

An aside here on BG for G nonabelian (a conjecture really):

It is well known that for an abelian group A, BA is again an abelian group, and hence B nA is well defined. But for nonabelian groups, things get tricky fast.

Taking the analogy with abelian gerbes, where BBA is the classifying space for A-gerbes, we would like to have BBG the classifying space for G-gerbes. But what is BBG? If we think of the 2-bundle interpretation of things, G-gerbes are equivalent to AUT(G)=(GAut(G))-2-bundles (yet to be proved, I suppose, but no reason to doubt). Thus we could possibly interpret BBG as a sort of classifying space BAUT(G). Now AUT(G) is really a gr-stack (thinking here a bit more general than 2-bundles, we can replace the group G by a sheaf of groups).

Is there some equivalence here between BG and AUT(G)? Can we think of B nG as a (possibly degenerate at some stage) gr-n-stack (or an n-group, if G is just a group)?

As a disclaimer, I had a thought that some of this might be in Breen’s book on classifying 2-gerbes, but I wanted to get it down before having a look. People with more experience than I can perhaps do something with the above naive approach.


I’ve just had a peek at “On the classification of 2-gerbes…” and just rediscovered the interesting fact that Tors(G) (or GBund, if you like - I don’t use Baez’s definition of torsor) is the associated stack to BG, where BG=NG, thinking of G as a groupoid.

Of course, Tors(G) is a torsor (again, not Baez’s defn) under the gr-stack/2-group Bitors(G)AUT(G). Hmmm

(possibly with foot in mouth), D

Posted by: David Roberts on August 24, 2005 3:20 AM | Permalink | Reply to this

Re: CFT, Gerbes and K-Theory in Oberwolfach, VI

I’d agree that this looks like the right inuition. While ‘BBG’ does not literally make sense for nonabelian G, we expect ‘BB nG’ to be the nerve (in Duskin’s n-categorical sense) of an (n+1 )-group which has G as its space of n+1 -morphisms.

- This is strictly true for n=0 : Here BG is the nerve of the 1-group which has G as its space of 1-morphisms (on a single object).

- It is also strictly true for n=1 and G abelian: Here BBG is the same as the realization of the 2-category nerve of the 2-group which has G as its space of 2-morphisms (on a single 1-morphism on a single object), namely the 2-group coming from the crossed module G1 .

- It is morally true for n=1 with nonabelian G and the heuristic relation ‘BBGAut(G)’, because Aut(G) is a 2-group with G as its space of 2-morphisms (on a given source 1-morphism).

So every application of B sort of lifts G from n-morphisms to n+1 -morphisms, in a sense. For G abelian we can assume all (m<n+1 )-morphisms to be trivial and are done. But for G nonabelian there is more information in these lower-m-morphisms, which is why the naïve relation breaks down.

I am sure the true cognoscenti could say this in a much better and deeper way.

Posted by: Urs Schreiber on August 24, 2005 9:37 AM | Permalink | Reply to this
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