Hyperpolygon Space: A Geometric Overview

• Hyperpolygon space is a hyperkähler analogue of polygon moduli spaces, defined via Nakajima quiver varieties and moment map constraints.
• It is isomorphic to the moduli space of parabolic Higgs bundles on the Riemann sphere, linking symplectic geometry and geometric representation theory.
• The structure exhibits complete integrability and complex birational transformations, with applications in Hamiltonian dynamics and mirror symmetry.

A hyperpolygon space is a hyperkähler analogue of the moduli space of closed polygons in Euclidean space, realized as a Nakajima quiver variety associated to a star-shaped quiver, and is also naturally isomorphic to the moduli space of parabolic Higgs bundles on the Riemann sphere with prescribed weights and residues. These spaces provide a rich interplay between symplectic/Hamiltonian geometry, geometric representation theory, and the theory of integrable systems.

1. Quiver Variety Construction and Moment Maps

Let $n \geq 3$ and fix positive real parameters (the "lengths" or "weights") $$\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbb{R}_{>0}^n$$. The hyperpolygon space $X(\alpha)$ is defined as a hyperkähler quotient of $T^*\mathbb{C}^{2n}$ by a compact group $K = (SU(2) \times U(1)^n)/(\mathbb{Z}/2\mathbb{Z})$: $$X(\alpha) = \left\{ (q,p) \in T^*\mathbb{C}^{2n}\ \big|\ \mu_{\mathbb{C}}(q,p) = 0,\ \mu_{\mathbb{R}}(q,p) = (a_i) \right\} \bigg/ K.$$ Here,

• $q_i \in \mathbb{C}^2$ and $p_i \in (\mathbb{C}^2)^*$ for $i=1,\ldots,n$;
• The real moment map

$$\mu_\mathbb{R}(q,p) = \left( \frac{i}{2} \sum_{i=1}^n (q_i q_i^* - p_i^* p_i) \in \mathfrak{u}(2)^*,\quad (\lVert q_i \rVert^2 - \lVert p_i \rVert^2)_{i=1}^n \in \mathbb{R}^n \right);$$

• The complex moment map

$$\mu_\mathbb{C}(q,p) = \left( \sum_{i=1}^n q_i \otimes p_i \in \mathfrak{sl}_2(\mathbb{C})^*,\quad (p_i(q_i))_{i=1}^n \in \mathbb{C}^n \right).$$

$X(\alpha)$ is of real dimension $4n-12$ (complex dimension $2n-6$) for generic $\alpha$ (Biswas et al., 2013). The GIT (Geometric Invariant Theory) perspective gives $$X(\alpha) = \mu_\mathbb{C}^{-1}(0)//_\alpha K_\mathbb{C}$$, where $\alpha$ is a stability parameter and $$K_\mathbb{C} = (SL(2,\mathbb{C}) \times (\mathbb{C}^*)^n)/(\mathbb{Z}/2\mathbb{Z})$$.

2. Relation to Parabolic Higgs Bundles

Biswas–Florentino–Godinho–Mandini established a canonical isomorphism between $X(\alpha)$ and the moduli space $H(\alpha)$ of stable rank-2 holomorphically trivial parabolic Higgs bundles on $\mathbb{CP}^1$ with parabolic divisor $D = \{x_1, \ldots, x_n\}$, trivial underlying bundle, and weights $\{(\alpha_i, 0)\}$ at each point. In this identification, the Higgs field $\Phi$ on $\mathbb{CP}^1$ is a meromorphic endomorphism with first-order poles at each $x_i$ and nilpotent residue $(q_i p_i)_0$ (the trace-free part) at each marked point. The open stable locus in the quiver perspective corresponds exactly to the stable locus of parabolic Higgs bundles. This isomorphism is a complex symplectomorphism, intertwining the canonical (Liouville) form on $T^*U$ with the Hitchin symplectic form on the Higgs bundle moduli (Biswas et al., 2013, Godinho et al., 2011).

3. Symplectic and Integrable Structure

The holomorphic symplectic structure on $X(\alpha)$ comes from the exterior derivative of the tautological 1-form on the cotangent bundle (the Liouville form), and this corresponds exactly to the Hitchin–Bottacin–Ramanan holomorphic symplectic form under the isomorphism with Higgs bundles.

Hyperpolygon spaces are algebraically completely integrable systems. For higher rank and generalized spaces, the Gelfand–Tsetlin Hamiltonians descend from the partial flag reductions on the quiver arms, and their commutativity realizes the algebraic complete integrability: there are exactly $1/2 \dim_\mathbb{C} X$ independent Poisson-commuting Hamiltonians (Rayan et al., 2020, Fisher et al., 2014). The Hitchin map for the bundle $(E, \Phi)$ assigns the characteristic polynomial invariants, giving a Lagrangian fibration structure over a vector space of differentials.

4. Wall-Crossing, Flops, and Birational Geometry

The stability condition defining $X(\alpha)$ changes when the parameters $\alpha$ cross specific hyperplanes, called "walls," given by $\sum_{i\in S} \alpha_i = \sum_{i\notin S} \alpha_i$ for $S \subset \{1, \ldots, n\}$. As $\alpha$ crosses a wall, the GIT stability locus and therefore the birational type of $X(\alpha)$ changes via a Mukai flop. This manifests in both the structure of $X(\alpha)$ and the moduli of parabolic Higgs bundles, where the structural transformation matches a Thaddeus flip (Godinho et al., 2011, Bellamy et al., 2021). These wall crossings organize the chamber structure in the space of parameters and govern the birational geometry of the crepant resolutions of the singular hyperpolygon space $X_n(0)$ (Hubbard, 2024, Bellamy et al., 2021).

Projective and non-projective crepant resolutions of $X_n(0)$ are classified combinatorially by maximally biconnected simplicial complexes on $\{1,\dots,n\}$; the number of such resolutions equals the Hoşten-Morris number $\lambda(n)$ (Hubbard, 2024).

5. Topology and Cohomology

The cohomology ring of $X(\alpha)$ admits a uniform presentation in terms of generators $c_j \in H^2$ subject to relations encoding both the circle-action Morse theory and the symplectic reduction. Betti numbers are explicitly given: $b_{2k}$ equals the number of subsets $I \subset \{1, \ldots, n\}$ of size $k+1$ with $|I| \leq n/2$, and odd cohomology vanishes (Fisher et al., 2014). The Kirwan map in hyperkähler equivariant cohomology is surjective except possibly in the middle degree.

Core components of $X(\alpha)$ (fixed-point loci under natural circle actions) arise as Lagrangian subvarieties and their cohomology rings and intersection numbers can be computed via explicit recursion relations (Godinho et al., 2011).

6. Null Hyperpolygon Spaces and Minkowski Polygon Correspondence

Setting the real moment map parameter to zero yields singular (noncompact) "null hyperpolygon spaces" $X_{\mathrm{null}}$; the fixed points of a natural involution in the moduli of quasi-parabolic Higgs bundles correspond to moduli spaces of null polygons in Minkowski 3-space. The involution on the Higgs side $(E, \Phi) \mapsto (E, -\Phi)$ translates to $(p,q) \mapsto (-p, q)$ in the hyperpolygon model, and the fixed-point locus $X(\alpha)^\sigma$ is stratified by configuration types, each corresponding to such moduli (Godinho et al., 2019, Biswas et al., 2012).

7. Extensions: Higher Rank, Generalized Hyperpolygons, and Mirror Correspondence

More generally, hyperpolygon spaces are special cases of Nakajima quiver varieties associated to comet-shaped quivers with arbitrary rank and genus. The associated moduli of generalized hyperpolygons can be described as data of pairs of closed polygons in $\mathfrak{su}(r)$ and $\mathfrak{sl}(r,\mathbb{C})$ satisfying moment map constraints, with additional structure encoded by loops and flag data on the quiver (Rayan et al., 2020).

Hyperpolygons act as discretizations of the Hitchin equations; the correspondence with Higgs bundles extends to mirror symmetry: specific involutions lift to branes of type (B,A,A) in the sense of Kapustin–Witten. The framework accommodates dualities between tame (parabolic) and wild (irregular) Hitchin systems, linked to Painlevé transcendents and the Riemann–Hilbert correspondence.

• (Biswas et al., 2013) Symplectic form on hyperpolygon spaces
• (Godinho et al., 2011) Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles
• (Biswas et al., 2012) Polygons in Minkowski three space and parabolic Higgs bundles of rank two on CP^1
• (Fisher et al., 2014) Surjectivity of the hyperkähler Kirwan map
• (Fisher et al., 2014) Hyperpolygons and Hitchin systems
• (Godinho et al., 2019) Quasi-parabolic Higgs bundles and null hyperpolygon spaces
• (Bellamy et al., 2021) All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
• (Hubbard, 2024) All crepant resolutions of hyperpolygon spaces via their Cox rings
• (Heller et al., 30 Dec 2025) Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces
• (Rayan et al., 2020) Moduli spaces of generalized hyperpolygons