## 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

(1)$\left[X,\mathrm{BG}\right]$

of maps from $X$ to a topological space called $\mathrm{BG}$, the classifying space of $\mathrm{BG}$-bundles. The claim is that this construction can be generalized to 1-nonabelian gerbes/2-bundles by replacing $\mathrm{BG}$ 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 $\mathrm{String}\left(n\right)$-bundles over $M$ are ‘the same’ as 1-gerbes/2-bundles with structure 2-group ${P}_{1}\mathrm{Spin}\left(n\right)$.

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 $•\stackrel{g}{\to }•$ for each $g\in G$
-a 2-simplex

(2)$•\stackrel{{g}_{1}}{\to }•\stackrel{{g}_{2}}{\to }•⇒•\stackrel{{g}_{1}{g}_{2}}{\to }$

for all ${g}_{1},{g}_{2}\in G$
- a 3-simplex for each four composable such 2-simplices
- and so on .

Next, consider some manifold $M$ with a good covering $U=\left\{{U}_{i}{\right\}}_{i\in I}$ by open, contractible subsets ${U}_{i}\subset M$. The nerve ${U}_{•}$ of this covering has

- a 0-simplex ${U}_{i}$ for each $i\in I$
- a 1-simplex ${U}_{i}\to {U}_{j}$ for every nonempty double intersection ${U}_{i}\cap {U}_{j}$
- 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 $\left[{U}_{•},{G}_{•}\right]$ of such simplicial maps have to do with equivalence classes of $G$-bundles over $M$.

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

(3)$\left[{U}_{•},{G}_{•}\right]\simeq \left[\mid {U}_{•}\mid ,\mid {G}_{•}\mid \right]\phantom{\rule{thinmathspace}{0ex}}.$

Moreover, $\mid {U}_{•}\mid$ is homopy equivalent to $M$ itself, so that finally one has that

(4)$\left[M,\mid {G}_{•}\mid \right]:=\left[M,\mathrm{BG}\right]$

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)$\left[M,\mid G{2}_{•}\mid \right]$

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 ${\mathrm{hol}}_{\cdot }$ with $•$.

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

Next, Brano Jurčo notes a certain relation between the nerve $G{2}_{•}$ of a 2-group $G2$ regarded as a 2-category with a single object, and the nerve $N\left(G2\right)$ 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 $\overline{W}Q$. The claim is that

(6)$G{2}_{•}=\overline{W}N\left(G2\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 $\mid N\left(G2\right)\mid$-bundles.

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

(7)$G2={P}_{1}\mathrm{Spin}\left(n\right)$

such that

(8)$\mid N\left(G2\right)\mid \simeq \mathrm{String}\left(n\right)$

is the $\mathrm{String}\left(n\right)$-group. So in this case the above would say that $\mathrm{String}\left(n\right)$-bundles over $M$ have the same classification as ${P}_{1}\mathrm{Spin}\left(n\right)$-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 $\mathrm{BG}$ for $G$ nonabelian (a conjecture really):

It is well known that for an abelian group $A$, $\mathrm{BA}$ is again an abelian group, and hence ${B}^{n}A$ is well defined. But for nonabelian groups, things get tricky fast.

Taking the analogy with abelian gerbes, where $\mathrm{BBA}$ is the classifying space for $A$-gerbes, we would like to have $\mathrm{BBG}$ the classifying space for $G$-gerbes. But what is $\mathrm{BBG}$? If we think of the 2-bundle interpretation of things, $G$-gerbes are equivalent to $\mathrm{AUT}\left(G\right)=\left(G\to \mathrm{Aut}\left(G\right)\right)$-2-bundles (yet to be proved, I suppose, but no reason to doubt). Thus we could possibly interpret $\mathrm{BBG}$ as a sort of classifying space $B\mathrm{AUT}\left(G\right)$. Now $\mathrm{AUT}\left(G\right)$ 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 $\mathrm{BG}$ and $\mathrm{AUT}\left(G\right)$? Can we think of ${B}^{n}G$ 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.

(break)

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

Of course, $\mathrm{Tors}\left(G\right)$ is a torsor (again, not Baez’s defn) under the gr-stack/2-group $\mathrm{Bitors}\left(G\right)\sim \mathrm{AUT}\left(G\right)$. 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 ‘$\mathrm{BBG}$’ does not literally make sense for nonabelian $G$, we expect ‘$B{B}^{n}G$’ to be the nerve (in Duskin’s $n$-categorical sense) of an $\left(n+1\right)$-group which has $G$ as its space of $n+1$-morphisms.

- This is strictly true for $n=0$: Here $\mathrm{BG}$ 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 $\mathrm{BBG}$ 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 $G\to 1$.

- It is morally true for $n=1$ with nonabelian $G$ and the heuristic relation ‘$\mathrm{BBG}↔\mathrm{Aut}\left(G\right)$’, because $\mathrm{Aut}\left(G\right)$ 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 $\left(m-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
Read the post Equivariant Structures on Gerbes
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