Parallels between moduli of quiver representations and vector bundles over curves

TL;DR

This review explores the deep connections between moduli spaces of quiver representations and vector bundles over curves, highlighting their geometric structures, hyperk"ahler analogues, and surprising links to counting problems over finite fields.

Contribution

It provides a comprehensive comparison of moduli spaces for quivers and vector bundles, including their hyperk"ahler structures and related counting formulas, unifying different geometric and algebraic perspectives.

Findings

Hyperk"ahler structures unify quiver and bundle moduli spaces.

Counting absolutely indecomposable objects relates to Betti cohomology.

Surprising links between finite field counts and complex geometric invariants.

Abstract

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 hyperk\"ahler 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) hyperk\"ahler 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.