Abelianisation of Logarithmic $mathfrak{sl}_2$-Connections

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