Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

Symmetry Integrability and Geometry-Methods and Applications

article

Victoria Hoskins¹ 

[1] Freie Universität Berlin

关键词: algebraic moduli problems;    geometric invariant theory;    representation theory of quivers;    vector bundles and Higgs bundles on curves;   

DOI  :  10.3842/SIGMA.2018.127

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown