Question

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

Answer 1

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.