The String Coffee Table

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 $\mathrm{String}(n)$-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 $G_\bullet$ of that category is obtained by including

- a single 0-simplex $\bullet$
- a 1-simplex $\bullet \overset{g}{\to} \bullet$ 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 $U_\bullet$ 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 $[U_\bullet,G_\bullet]$ of such simplicial maps have to do with equivalence classes of $G$-bundles over $M$.

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

Moreover, $|U_\bullet|$ 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 $\bullet$.

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 $G2_\bullet$ 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 $\bar W Q$. 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 $|N(G2)|$-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.