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

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

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  • Lynn Heller
  • Sebastian Heller

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FachzeitschriftJournal of Symplectic Geometry, Vol.
PublikationsstatusVeröffentlicht - 30 Apr. 2014

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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Abelianization of Fuchsian Systems on a 4-punctured sphere and applications. / Heller, Lynn; Heller, Sebastian.
in: Journal of Symplectic Geometry, Vol., 30.04.2014.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

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N2 - 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.

AB - 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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JF - Journal of Symplectic Geometry, Vol.

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