We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but they have some geometric life. They have Chern classes, and one can twist differential operators by them. How should I think about these? What do they have to do with gerbes?
[Math] What do gerbes and complex powers of line bundles have to do with each other
gerbesline-bundlestensor-powers
Related Solutions
I have a couple things to say.
First, believe your definition of gerbe is slightly incorrect. When you say that your stack is locally isomorphic to $U \times B\mathbb{G}_m$, this isomorphism needs to preserve some additional structure. It might be okay for $\mathbb{G}_m$-gerbes, by accident, but for general non-abelian gerbes you will run into trouble. (It might still be okay for $\mu$-gerbes, where $\mu$ is a sheaf of abelian groups over X).
There are several ways to add this extra structure, but I think the most common are not necessarily the most enlightening. The fact of the matter is that $B\mathbb{G}_m$ is a group object in stacks and it "acts" on the gerbe over $X$. The local isomorphism to $U \times B\mathbb{G}_m$ needs to respect this action. Morally, you should think of a gerbe as a principal bundle with structure "group" $B\mathbb{G}_m$.
The reason that this isn't the most common way to explain what a gerbe is, is that making this precise requires a certain comfortability with 2-categories and coherence equations that most people don't seem to have. Times are changing though. Just as for ordinary principal bundles, you can (in nice settings, say noetherian separated) classify them in terms of Cech data. When you do this you see that the only important part is the coherence data, which gives a 2-cocycle. For non-abelian gerbes you get non-trivial stuff which mixes together parts which look like data for a 1-cocycle and a 2-cocycle. I agree with Kevin that, at this point, if you really want to understand this stuff you should fill-in the rest of the details on your own. It is a good exercise!
Alternatively, if higher categories make you uncomfortable, you can be cleaver. You can still make a definition along the lines of the one you outline precise without venturing into the world of higher categories and "coherent group objects in stacks". I recommend Anton's course notes on Stacks as taught by Martin Olsson. Section 31 has a definition of $\mu$-gerbes which is equivalent to the one I sketched above but avoids the higher categorical aspects. There is also a proof that such gerbes are classified by $H^2(X; \mu)$. Enjoy!
Just to reiterate. In a gerbe you are not patching together classifying spaces, you are patching together classifying stacks. Despite the common notation, there is a difference. A stack is fundamentally an object in a 2-category. This means that you need to deal with 2-morphisms and that they can be just as important as the 1-morphisms. For $B\mathbb{G}_m$, the 1-morphisms (which preserve the multiplication action of the stack $B \mathbb{G}_m$ !!) are all equivalent, so there is no Cech 1-cocycle data at all. All you get are the coherence data, which form a 2-cocycle.
This is one reason that I prefer the notation $[pt/\mathbb{G}_m]$ to denote the stack $B\mathbb{G}_m$. This is particularly important in the topological setting where these are truly different objects.
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.
Best Answer
Complex powers of line bundles are classes in $H^{1,1}$, or equivalently sheaves of twisted differential operators (TDO) (let's work in the complex topology). This maps to $H^2$ with $\mathbb{C}$ coefficients, or modding out by $\mathbb{Z}$-cohomology, to $H^2$ with $\mathbb{C}^\times$ coefficients. The latter classifies $\mathbb{C}^\times$ gerbes, ie gerbes with a flat connection (usual gerbes can be described by $H^2(X,\mathcal{O}^\times)$). Note that honest line bundles give the trivial gerbe.
In fact the category of modules over a TDO only depends on the TDO up to tensoring with line bundles --- ie it only depends on the underlying gerbe, and can be described as ordinary $\mathcal{D}$-modules on the gerbe. Or if you prefer, regular holonomic modules over a TDO are the same as perverse sheaves on the underlying gerbe. This is explained eg in the encyclopedic Chapter 7 of Beilinson-Drinfeld's Quantization of Hitchin Hamiltonians document, or I think also in a paper of Kashiwara eg in the 3-volume Asterisque on singularities and rep theory (and maybe even his recent $\mathcal{D}$-modules book). B&D talk in terms of crystalline $\mathcal{O}^\times$ gerbes rather than $\mathbb{C}^\times$ gerbes but the story is the same.