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Perhaps this is a bit naïve, but I was wondering if it possible to (at least formally) represent Bundle Gerbes as Characteristic Classes. Disclaimer: My understanding of Bundle Gerbes is limited to this paper of Hitchin so perhaps I'm not thinking of this correctly. Just for reference, a Bundle Gerbe is defined by specifying an open cover $\{U_i\}$ of a manifold $M$ that has associated to it maps $g_{ijk} : U_i \cap U_j \cap U_k \rightarrow S^1$ that satisfy certain cocycle-like conditions, $g_{jkl} g_{ikl}^{-1} g_{ijl} g^{-1}_{ijk}$. One can define connective structures with $3$-form curvatures $H$ on Bundle Gerbes that define principle circle bundles on the Loop Space of $M$ (See Hitchin, Page 4). These connective structures are classified by their curvatures, $[H / 2\pi] \in H^3(M,\mathbb{Z})$ just like the curvature $2$-form of a line bundle generates the first Chern Class. Explicitly, my question is the following:

Can we expand the definition of a bundle gerbe on a manifold $M$ to an arbitrary compact, finite-dimensional Lie Group $G$ by considering a Bundle Gerbe to instead be the set of maps $g_{ijk} : U_i \cap U_j \cap U_k \rightarrow G$? If $\dim G = n$, will $H^{n}(M,\mathbb{Z})$ classify the Principle $G$-bundles on $\Omega M$?

Again, my understand of gerbes is quite insufficient so perhaps this is "obvious" in some other literature. If this is the case, could you please cite a reference?

Thanks!

PS: I'm not sure if the compactness is truly necessary, I just added it with the hope that its more likely in the compact case

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So is your question whether there is a notion of nonabelian gerbe? There is an obvious problem with this, because although one can make sense of $H^1(M,G)$ for nonabelian $G$, but not of $H^{n>1}(M,G)$. On the other hand, there are reasons (coming from Physics) to believe that nonabelian gerbes have to exist. In some talks at the Newton Institute in 1996, Edward Witten asked that very question. I am not sure if there has been any progress since then. –  José Figueroa-O'Farrill Apr 13 '11 at 12:26
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There is a notion of nonabelian bundle gerbe. It can be found in "Nonabelian bundle gerbes, their differential geometry and gauge theory" by Aschieri, Cantini and Jurco. There is also a notion of nonabelian cohomology with values in a crossed module. –  Ulrich Pennig Apr 13 '11 at 13:41
    
Thanks for the responses. I suppose that if $G$ is non-abelian it definitely fit the bill and I think that Ulrich's answer (at least from a cursory reading) looks pretty good. However, is the classification still cohomological? It seems as if we now need K-Theory to classify non-Abelian gerbes –  Tarun Chitra Apr 13 '11 at 15:42

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up vote 6 down vote accepted

The fact that you are dealing with compact and/or finite dimensional Lie groups is completely irrelevant. The fact that these group are Lie is also partially irrelevant (unless you care about putting connections on your bundle gerbes, in which case it becomes very relevant). More relevant is whether the groups abelian or not. A priori, the cocycle relation only makes sense for abelian groups.

But there is also a theory of non-abelian (bundle) gerbes, where you allow non-abelian groups. The cocycles have two kinds of data: Maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Inn}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where $\mathrm{Inn}(G)$ denotes the group of inner automorphisms of $G$.

These non-abelian gerbes are classified by $H^2(-,Z(G))$, the second Cech cohomology group with coefficients in the sheaf of $Z(G)$-valued functions. [that's a non-trivial theorem]

That was the case of a trivial band.


A band is the same thing as an $\mathrm{Out}(G)$-principal bundle. Say you are given an $\mathrm{Out}(G)$ principal bundle $P$, described by transition functions $b_{ij}:U_i\cap U_j\to \mathrm{Out}(G)$. Then you can twist the above definition as follows: The cocycles now consist of maps
$\alpha_{ij}:U_i\cap U_j\to \mathrm{Aut}(G)$ and maps
$g_{ijk}:U_i\cap U_j\cap U_k \to G$,
where the $\alpha_{ij}$ are lifts of the $b_{ij}$.

The gerbes with band $P$ are classified by a set that is either ♦ empty, or ♦ isomorphic to $H^2(-,Z(G)\times_{\mathrm{Out}(G)} P)$, the second Cech cohomology group with coefficients in the sheaf of sections of $Z(G)\times_{\mathrm{Out}(G)} P$.

Whether or not that set is empty depends on the value of an obstruction class that lives in $H^3(-,Z(G)\times_{\mathrm{Out}(G)} P)$. It's non-empty iff that obstruction vanishes.


Finally, to answer your last question. If $G$ is a Lie group and you have a bundle gerbe with connection (trivialized over the base point), then you get a $G$-principal bundle, but only on a subspace of the based loop space $\Omega M$. It's the subspace consisting of those loops over which the band $P$ and its connection trivialize.

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Thanks a lot! Is there any chance you know of a reference where I could find the proofs of these facts? –  Tarun Chitra Apr 13 '11 at 16:15
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The standard reference is Giraud's book Cohomologie non-abelienne. This book is unreadable in the strongest possible meaning of the word "unreadable". So... no. –  André Henriques Apr 13 '11 at 16:37
    
Depends on taste, André, I find most of the contemporary articles in this area, which are often nonsystematic in terminology and notation, plus wave hands and use jargon on most issues, much less readable than Giraud's book. –  Zoran Skoda Apr 13 '11 at 17:03
    
Actually, is there any chance that either of you know of an English book? Or is there an English translation around of <i>Cohomologie non-abelienne</i>? –  Tarun Chitra Apr 13 '11 at 17:14
    
I'm currently reading Brylisnki's Loop Spaces, Characteristic Classes, and Geometric Quantization which might be helpful. –  cheyne Jun 27 '12 at 22:40

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