Perhaps this is a bit naïve, but I was wondering if it possible to (at least formally) represent [Bundle Gerbes](http://en.wikipedia.org/wiki/Bundle_gerbe) as Characteristic Classes. Disclaimer: My understanding of Bundle Gerbes is limited to [this](http://arxiv.org/abs/math.dg/0508618) 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



Bundle Gerbes as Characteristic Classes
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*.




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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.




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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.


