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arXiv · 1404.7707

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

Lynn Heller·Sebastian Heller
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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.

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Lynn Heller, Sebastian Heller. 2014-04-30. Abelianization of Fuchsian Systems on a 4-punctured sphere and applications. https://arxiv.org/abs/1404.7707

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