### 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 $\mathrm{BG}$, 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 $\mathrm{String}(n)$-bundles over $M$ are ‘the same’ as 1-gerbes/2-bundles with structure 2-group ${P}_{1}\mathrm{Spin}(n)$.

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 ${G}_{\u2022}$ of that category is obtained by including

- a single 0-simplex $\u2022$

- a 1-simplex $\u2022\stackrel{g}{\to}\u2022$ for each $g\in G$

-a 2-simplex

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 $U=\{{U}_{i}{\}}_{i\in I}$ by open, contractible subsets ${U}_{i}\subset M$. The nerve ${U}_{\u2022}$ 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 $[{U}_{\u2022},{G}_{\u2022}]$ 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

Moreover, $\mid {U}_{\u2022}\mid $ is homopy equivalent to $M$ itself, so that finally one has that

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

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 $\u2022$.

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 $G{2}_{\u2022}$ 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 $\overline{W}Q$. The claim is that

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(G2)\mid $-bundles.

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

such that

is the $\mathrm{String}(n)$-group. So in this case the above would say that $\mathrm{String}(n)$-bundles over $M$ have the same classification as ${P}_{1}\mathrm{Spin}(n)$-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.

## 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?