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The quiver at the bottom of the twisted nilpotent cone on \(\mathbb P^1\)

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Abstract

For the moduli space of Higgs bundles on a Riemann surface of positive genus, critical points of the natural Morse–Bott function lie along the nilpotent cone of the Hitchin fibration and are representations of \(\text{ A }\)-type quivers in a twisted category of holomorphic bundles. The critical points that globally minimize the function are representations of \(\text{ A }_1\). For twisted Higgs bundles on the projective line, the quiver describing the bottom of the cone is more complicated. We determine it here. We show that the moduli space is topologically connected whenever the rank and degree are coprime, thereby verifying conjectural lowest Betti numbers coming from high-energy physics.

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Acknowledgements

This manuscript was started during the 2016 Symposium on Higgs Bundles in Geometry and Physics held at the Internationales Wissenschaftsforum Heidelberg. The author thanks the organizers for their hospitality and for providing a comfortable environment for discussion and work. The author is grateful to Steven Bradlow and Sergey Mozgovoy for useful discussions and comments on the manuscript, and also to Peter Gothen for pointing out a similar phenomenon for small weights in the parabolic case.

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  1. Department of Mathematics & Statistics, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, Saskatchewan, S7N 5E6, Canada

    Steven Rayan

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  1. Steven Rayan
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Correspondence to Steven Rayan.

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Rayan, S. The quiver at the bottom of the twisted nilpotent cone on \(\mathbb P^1\) . European Journal of Mathematics 3, 1–21 (2017). https://doi.org/10.1007/s40879-016-0120-6

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