Moduli Spaces of Generalized Hyperpolygons

  title={Moduli Spaces of Generalized Hyperpolygons},
  author={Steven Rayan and Laura P. Schaposnik},
  journal={arXiv: Algebraic Geometry},
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… 

Figures from this paper

Hyperbolic band theory through Higgs bundles
. Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation and which, to date, have been engineered artificially. A corresponding
The Kapustin–Witten equations and nonabelian Hodge theory
Arising from a topological twist of $\mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by
All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us
Geometric models and variation of weights on moduli of parabolic Higgs bundles over the Riemann sphere: a case study
We construct explicit geometric models for moduli spaces of stable parabolic Higgs bundles on the Riemann sphere, in the case of rank two, four marked points, any degree, and arbitrary weights. The


Construction of Instantons
In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean
The construction of ALE spaces as hyper-Kähler quotients
On decrit la construction d'une famille particuliere de 4-varietes hyper-Kahler: les espaces asymptotiquement localement euclidiens ALE. On decrit une 4-variete de Riemann avec juste une extremite
Electric-Magnetic Duality And The Geometric Langlands Program
The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N=4 super Yang-Mills theory in four dimensions. The key ingredients are
Asymptotic geometry of the moduli space of parabolic SL(2,C)-higgs bundles
  • arXiv preprint arXiv :
  • 2001
Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles
Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E}, t
Generalized B-Opers
Opers were introduced by Beilinson-Drinfeld [arXiv:math.AG/0501398]. In [J. Math. Pures Appl. 82 (2003), 1-42] a higher rank analog was considered, where the successive quotients of the oper
On the cohomology ring of the hyperKähler analogue of the polygon spaces
Quasi-parabolic Higgs bundles and null hyperpolygon spaces
The moduli space of quasi-parabolic Higgs bundles over a compact Riemann surface is introduced and a natural involution is considered, studying its fixed point locus when theinline-formula content-type is considered.
. In “Quantization of Hitchin’s Integrable System and Hecke Eigen-sheaves”, Beilinson and Drinfeld introduced the “very good” property for a smooth complex equidimensional stack. They prove that for