Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Author: Hoskins, V.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209877
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Parallels between Moduli of Quiver Representations and Vector Bundles over Curves / V. Hoskins // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 65 назв. — англ.

This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperkähler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperkähler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.