Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 127, 46 pages      arXiv:1809.05738
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

Victoria Hoskins
Freie Universität Berlin, Arnimallee 3, Raum 011, 14195 Berlin, Germany

Received September 25, 2018, in final form November 18, 2018; Published online December 04, 2018

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.

Key words: algebraic moduli problems; geometric invariant theory; representation theory of quivers; vector bundles and Higgs bundles on curves.

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