Hyperkähler Degenerations from Parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs Bundles Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces

TL;DR

This work proves a precise hyperkähler degeneration connecting ALG-$D_4$ Hitchin moduli spaces on a punctured sphere to ALE-$D_4$ hyperpolygon spaces via a tuned limiting process: as the scaling parameter $R\to0$ and the parabolic weights approach $\frac{1}{2}$ according to $\alpha_i(R)=\frac{1}{2}-R\beta_i$, the pullback of ALG metrics converges to twice the hyperpolygon metric. Central to the argument is a gauge-theoretic embedding $\mathcal T$ from hyperpolygon spaces $\mathcal X(\vec\beta)$ to Higgs-moduli spaces $\mathcal M^{\mathrm Higgs}(\vec\alpha)$, preserving holomorphic symplectic structures up to a factor $2\pi$, and the demonstration that the $U(1)$-moment maps align in the limit (up to $2\pi$). The analysis blends finite-dimensional hyperkähler quotient geometry with infinite-dimensional Hitchin theory, employing local harmonic-model metrics, a gluing construction, and a detailed study of tangent deformations to establish pointwise metric convergence and Torelli-number compatibility. The results extend beyond the $n=4$ case to arbitrary finite $n$, illuminating a pathway toward a broader modular degeneration framework for gravitational instantons via gauge-theoretic correspondences. The approach provides a robust bridge between Nakajima quiver varieties and Higgs-bundle moduli, with potential implications for Coulomb-Higgs branch interactions and higher-rank generalizations.

Abstract

Complete hyperkähler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled by certain moduli spaces of strongly parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on the Riemann sphere with $n=4$ punctures. Meanwhile, a family of ALE-$D_4$ spaces are modeled by certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG-$D_4$ metrics parameterized by $R$ converges pointwise to the ALE-$D_4$ metric as $R \to 0$. While the connection to gravitational instantons occurs in the $n=4$ case, we prove our result for any finite $n$.

Figures (6)

• Figure 1: The Nakajima quiver $\widetilde{Q}_n$
• Figure 2: A hyperpolygon represented as vectors in $\mathfrak{su}(2)\cong\mathbb{R}^3$
• Figure 3: The chamber structures for $\vec{\alpha}$ (left) and $\vec{\beta}$ (right).
• Figure 4: The nilpotent cone of $\mathcal{M}(\vec{\alpha})$. The three exterior spheres are labelled by the three short subsets $I_1,I_2,I_3\in\mathcal{I}$. The $\mathbb{C}^\times$-fixed points are shown in red.
• Figure 5: ALG Degenerate Limit as $R\to0$

Theorems & Definitions (100)

• Theorem : (c.f. Theorem \ref{['thm: HK degeneration']})
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Definition 2.1.4
• Definition 2.1.5
• Remark 2.1.6: Geometric interpretation of unitary hyperpolygons
• Proposition 2.1.7
• proof
• Corollary 2.1.8