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

## TL;DR

This paper establishes a functorial correspondence between logarithmic sl_2 connections on a curve and abelian logarithmic connections on a spectral cover, using inverse functors and a canonical cocycle construction.

## Contribution

It introduces a new functorial equivalence between categories of sl_2 and abelian logarithmic connections via spectral covers and inverse functors.

## Key findings

- Constructed inverse functors pi^{ab} and pi_{ab} for the correspondence.
- Developed a canonical cocycle in automorphisms of the direct image functor.
- Proved the functorial equivalence between the categories.

## Abstract

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 $\pi_\ast$.