Hyperpolygons and Hitchin systems (Q2801966)

The authors consider hyperpolygon spaces which arise from moduli of representations of doubled star-shaped quivers. Hyperpolygon spaces are hyper-Kähler analogues of moduli spaces of semistable \(n\)-gons in complex projective space. They prove that the natural hyper-Kähler Kirwan map is surjective. The authors use equivariant Morse theory with the norm-square of a certain moment map to produce a recursive formula that calculates the Betti numbers of these spaces for all ranks. The hyper-Kähler geometry of hyperpolygon spaces is naturally related to the theory of Higgs bundles, and hence to Hitchin systems. The authors use this to establish the complete integrability of hyperpolygon spaces for ranks up to and including 3.