Abelianization of Fuchsian Systems on a 4-punctured sphere and applications

TL;DR

This paper establishes a new abelianization method for rank 2 Fuchsian systems on a 4-punctured sphere, creating a correspondence with flat line bundle connections on a torus and providing a novel proof of Witten's volume formula.

Contribution

It introduces an explicit abelianization procedure that links Fuchsian systems to line bundle connections, enriching the structure of the moduli space with new Darboux coordinates.

Findings

Established a 2-to-1 correspondence between flat line bundle connections and Fuchsian systems.

Provided a complex analytic proof of Witten's formula for the moduli space volume.

Enhanced understanding of the moduli space structure through new Darboux coordinates.

Abstract

In this paper we consider special linear Fuchsian systems of rank $2$ on a $4-$punctured sphere and the corresponding parabolic structures. Through an explicit abelianization procedure we obtain a $2-$to$-1$ correspondence between flat line bundle connections on a torus and these Fuchsian systems. This naturally equips the moduli space of flat $SL(2,\mathbb C)-$connections on a $4-$punctured sphere with a new set of Darboux coordinates. Furthermore, we apply our theory to give a complex analytic proof of Witten's formula for the symplectic volume of the moduli space of unitary flat connections on the $4-$punctured sphere.