Twisted cyclic quiver varieties on curves

Abstract

We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the number of points at which the cyclic moduli space intersects a Hitchin fibre; and we describe explicitly certain \(\mathbb {C}^\times \)-flows into the nilpotent cone. We also extend this description to moduli of certain twisted cyclic quivers whose rank vector has components larger than 1. We show that, for certain choices of underlying bundle, such moduli spaces decompose as a product of cyclic quiver varieties in which each node is a line bundle.

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Fig. 1

Notes

  1. 1.

    On \({\mathbb {P}}^1\), fixing \(\deg E=d\) forces , and so up to isomorphism there is no freedom in the choice of P. Also, is trivial. Hence, from the point of view of E, there is no difference between a \(\mathrm{GL}\), an \(\mathrm{SL}\), and a \(\mathrm{PGL}\) bundle. Strictly speaking, \(\Phi \) must be trace-free in the \(\mathrm{SL}\) and \(\mathrm{PGL}\) case, while the \(\mathrm{GL}\) case has no such restriction. This will make no impact on the analysis for \(g=0\), however, given that twisted cyclic Higgs bundles will always naturally have trace 0.

  2. 2.

    This also covers the case \(\mu _{\mathrm{tot}}>a_i\) for all i, simply by considering the dual quiver representation, which will have \(\mu _{\mathrm{tot}}<a_i\) for all i.

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Acknowledgements

We thank Peter Gothen and Qiongling Li for useful discussions. We also thank the anonymous referee for helpful clarifications and corrections on a prior version of this manuscript.

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Correspondence to Steven Rayan.

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The first named author was supported during this work by an NSERC Discovery Grant. The second named author was supported by an NSERC Alexander Graham Bell Scholarship. We thank the GEAR Network (NSF Grants DMS 1107452, 1107263, 1107367 RNMS: Geometric Structures and Representation Varieties) for supporting the workshop “Geometry and Physics of Gauge Theories at Infinity” (August 2018) co-organized by the first author, where formative steps in this work were taken. Parts of this work were completed during the Mathematisches Forschungsinstitut Oberwolfach Workshop “Geometry and Physics of Higgs Bundles” and during the “Geometry and Physics of Hitchin Systems” Thematic Program at the Simons Center for Geometry and Physics (both in May 2019). The first named author thanks the organizers of the former and both authors thank the organizers of the latter.

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Rayan, S., Sundbo, E. Twisted cyclic quiver varieties on curves. European Journal of Mathematics 7, 205–225 (2021). https://doi.org/10.1007/s40879-019-00365-0

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Keywords

  • Cyclic Higgs bundle
  • Twisted Higgs bundle
  • Twisted quiver variety
  • Hitchin section
  • Moduli space
  • Stability

Mathematics Subject Classification

  • 14D20
  • 14H60
  • 16G20