  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 $\pi_\ast$.
