Hyperpolygons and Hitchin systems

TL;DR

This paper explores hyperpolygon spaces as hyperk"ahler analogues of moduli spaces, proving surjectivity of the hyperk"ahler Kirwan map, deriving Betti number formulas, and establishing integrability for ranks up to 3.

Contribution

It introduces a recursive Betti number formula for hyperpolygon spaces and links them to degenerate Hitchin systems, demonstrating their integrability.

Findings

Hyperk"ahler Kirwan map is surjective.

Betti numbers of hyperpolygon spaces can be calculated recursively.

Hyperpolygon spaces are completely integrable systems for ranks up to 3.

Abstract

We study the hyperk\"ahler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperk\"ahler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.