Abelianisation of logarithmic $$\mathfrak {sl}_2$$-connections

We prove a functorial correspondence between a category of logarithmic $$\mathfrak {sl}_2$$ sl 2 -connections on a curve $${\mathsf {X}}$$ X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover "Equation missing". The proof is by constructing a pair of inverse functors $$\pi ^\text {ab}, \pi _\text {ab}$$ π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $$\pi _*$$ π ∗ .