Abelianization of Fuchsian systems on a $4$-punctured sphere and applications

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 construct 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.

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Published 10 January 2017