TITLE: is there an anyon structure analogous to spin structure for rank 2 bundle?
QUESTION [7 upvotes]: A spin structure on a Riemannian bundle of rank >2 is the lift of the structure group from $\text{SO}(n)$ to its universal cover $\text{Spin}(n).$ It may also be defined in the case $n=2$ as the lift of the structure group to a double cover of $\text{SO}(2)$, which is of course not the universal cover. 
So what about lifts to other covers of $\text{SO}(2)$. Does the lift to a three-to-one cover of $\text{SO}(2)$ have a topological obstruction living in $H^2(B,\mathbb{Z}/3)$? Is the concept of a lift to the universal cover of $\text{SO}(2)$ affected by the unusual statistics of anyons?
Searching the literature for spin structures, I found only the double cover case. And searching the archive for anyons yields nothing but solid state physics articles. So I would be happy just to have a reference to any work discussing this case.

REPLY [3 votes]: Here is a different explanation, in terms of exact sequences. There is an exact sequences:
$0 \to \mathbb Z/n \to SO(2) \to SO(2) \to 0 $
since $SO(2)$ is the $n$-fold cover of $SO(2)$. Taking cohomology, we get:
$H^1(B, SO(2) ) \to H^1(B, SO(2) ) \to H^2(B, \mathbb Z/n)$
Since $H^1(B,G)$ is the group of principal $G$-bundles for $G$ an abelian group, the exact sequence property means that the bundle lifts if and only if the class in $H^2(B,\mathbb Z/n)$ is zero. 
It is a bit more difficult to see why this method why the cohomology class is just the first Chern class of the associated complex line bundle modulo $n$.