Hyperpolygons and Hitchin systems

Abstract

We study the hyperkähler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperkähler 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.

arXiv comment25 pages, 9 figures, comments welcome

References in corpus (7)

• Mirrors of 3d Sicilian theories
• Moduli Spaces of Parabolic Higgs Bundles and Parabolic K(D) Pairs over Smooth Curves: I
• Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform
• Quivers and difference Painleve equations
• Cohomology of large semiprojective hyperkaehler varieties
• Morse decomposition for D-module categories on stacks
• Symplectic form on hyperpolygon spaces

Cited by in corpus (11)

• T-Branes at the Limits of Geometry
• Kirwan surjectivity for quiver varieties
• Hyperbolic band theory through Higgs bundles
• Moduli spaces of generalized hyperpolygons
• Spectral curves for the triple reduced product of coadjoint orbits for SU(3)
• All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
• Moduli Spaces of Homologically Trivial Parabolic (Higgs) Bundles on the Projective Line and an Application to the Deligne-Simpson Problem
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Counterexamples to hyperkahler Kirwan surjectivity
• Moduli spaces of parabolic bundles over ${\mathbb{P}}^1$ with five marked points
• Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized hyperkähler reduction