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We prove a functorial correspondence between a category of logarithmic$\mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and acategory of abelian logarithmic connections on an appropriate spectral doublecover $\pi : \Sigma \to X$. The proof is by constructing a pair of inversefunctors $\pi^{\text{ab}}, \pi_{\text{ab}}$, and the key is the construction ofa certain canonical cocycle valued in the automorphisms of the direct imagefunctor $\pi_\ast$.

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