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? 



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


