Abelianisation of Logarithmic sl_2-Connections

We prove a functorial correspondence between a category of logarithmic $\frak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $\pi : \sf{\Sigma} \to X$. The proof is by constructing a pair of inverse functors $\pi^\ab, \pi_\ab$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $\pi_\ast$.