What do gerbes and complex powers of line bundles have to do with each other?

If L is any line bundle on a space (scheme, whatever) X, A is any (additive) abelian group, and a an element of A, there is a natural construction of an A-gerbe $L^a$ as follows. By definition, $L^a$ should be a "sheaf of categories", or stack (not algebraic) on X, and here are its categories of sections. Identify L with its total space, which is a $mathbb{G}_m$-bundle on X, and for any open set U in X, let $L^a(U)$ be the category of all A-torsors on $L|_U$ whose monodromy about each fiber of $L|_U to U$ is a.

One can check that this really is a gerbe: it is locally nonempty, since if L is trivial over U, you can write $L|_U = mathbb{G}_m times U$ and then pull back the unique A-torsor on $mathbb{G}_m$ with monodromy a. It has a natural action of A-torsors on X, given by pulling up along the bundle map $L to X$ and tensoring. And this action is free and transitive, since the difference of two a-monodromic torsors on $L|_U$ has trivial monodromy on each fiber and therefore descends to X.

Why do I call this $L^a$? Suppose that $L = mathcal{O}_X(D)$ for a divisor D, where for simplicity let's say that D is irreducible of degree n; then L gets a natural trivialization on $U = X setminus D$ having a pole of order n along D. As shown above, this induces a trivialization $phi$ of $L^a$ on U, and if we pick a small open set V intersecting D and such that D is actually defined by an equation f of degree n, then we get a second (noncanonical) trivialization $psi$ of $L^a$ on V. You can check that the difference $psi^{-1} phi$, which is an automorphism of the trivial gerbe on $U cap V$, is in fact described by the A-torsor $mathcal{T} = f^{-1}(mathcal{L}_a)$, where $f colon U cap V to mathbb{G}_m$ and $mathcal{L}_a$ is the A-torsor of monodromy a. Since f has degree n, $mathcal{T}$ has monodromy na about D. Thus, it is only reasonable to say that the natural trivialization $phi$ has a pole of order na, which is consistent with the behavior of the trivialization of L itself on U, when raising to integer powers.

What does this have to do with twisting of differential operators? Suppose we have some kind of sheaves (D-modules, locally constant sheaves, perverse sheaves; technically, they should form a stack admitting an action of A-torsors). On the one hand, one could mimic the above construction of $L^a$ to describe a-monodromic sheaves on L, and this is what is often called twisting. On the other hand, there is a natural way to directly twist sheaves by the gerbe $L^a$ without mentioning L at all (that is, you can twist by any A-gerbe). The procedure is as follows: a twisted sheaf is the assignment, to every open set U in X, of a collection of sheaves on U parameterized by the sections of $L^a(U)$, and compatible with tensoring by A-torsors. Of course, since if $L^a(U)$ is nonempty this is the same as giving just one sheaf, this is sort of overkill, but the choice of just one such sheaf is noncanonical whereas this description is canonical. These collections should be compatible with the restriction functors $L^a(U) to L^a(V)$ when $V subset U$. It is an exercise to reader to check that this is the same as the other definition of twisting :)

Man, you asked the right question at the right time. My thesis is all about this stuff.