Moduli spaces of generalized hyperpolygons

Abstract

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-$g$ Riemann surface, where $g$ is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-$g$ Riemann surface (generally with positive codimension). We show that, under certain assumptions on flag types, the space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system of Gelfand-Tsetlin type, inherited from the reduction of partial flag varieties. In the case where all flags are complete, we present the Hamiltonians explictly. We also remark upon the discretization of the Hitchin equations given by hyperpolygons, the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

arXiv comment24 pages, 7 figures, to appear in Q. J. Math

References in corpus (17)

• Quasi-Hamiltonian Geometry of Meromorphic Connections
• Arithmetic harmonic analysis on character and quiver varieties
• Higgs bundles and (A,B,A)-branes
• T-Branes at the Limits of Geometry
• Kirwan surjectivity for quiver varieties
• Global Weyl groups and a new theory of multiplicative quiver varieties
• Real structures on moduli spaces of Higgs bundles
• Branes through finite group actions
• Hyperpolygons and Hitchin systems
• On the image of the parabolic Hitchin map
• Aspects of the topology and combinatorics of Higgs bundle moduli spaces
• Branes in the moduli space of framed instantons
• Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
• Spectral curves for the triple reduced product of coadjoint orbits for SU(3)
• Generalized B-Opers
• Group actions on quiver varieties and applications
• Symplectic form on hyperpolygon spaces

Cited by in corpus (7)

• Hyperbolic band theory through Higgs bundles
• Infinitely many 4d N=1 SCFTs with a=c
• The Kapustin-Witten equations and nonabelian Hodge theory
• All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study
• Moduli spaces of parabolic bundles over ${\mathbb{P}}^1$ with five marked points