Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

Lynn Heller, Sebastian Heller, Claudio Meneses

TL;DR

This work analyzes the semiclassical degeneration of the Hitchin hyperkähler metric for strongly parabolic $\mathfrak{sl}(2,\mathbb{C})$-Higgs bundles on the $n$-punctured sphere as parabolic weights are scaled to zero ($t\to0$). It constructs real holomorphic sections (twistor lines) of the parabolic Deligne–Hitchin moduli space by solving a loop-group monodromy problem, linking them to the twistor lines of hyperpolygon spaces via a precise limiting process. It proves that, after rescaling by $t^{-1}$, the Hitchin metric on energy-bounded subsets of $\mathcal{M}_{\mathrm{Higgs}}(t\alpha)$ converges to the hyperkähler metric on the hyperpolygon space $\mathcal{X}_\alpha$, providing a finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections in the semiclassical regime are expressed explicitly through iterated integrals of logarithmic differentials on the punctured sphere, enabling systematic computation of metric expansions. Overall, the hyperpolygon spaces emerge as the natural finite-dimensional limits governing the degeneration of Hitchin moduli spaces, with twistor theory giving a cohesive, computable framework for the asymptotics.

Abstract

We investigate the Hitchin hyperkähler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $tα$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $λ$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperkähler metric on the hyperpolygon space $\mathcal X_α$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.

Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces

TL;DR

This work analyzes the semiclassical degeneration of the Hitchin hyperkähler metric for strongly parabolic -Higgs bundles on the -punctured sphere as parabolic weights are scaled to zero (). It constructs real holomorphic sections (twistor lines) of the parabolic Deligne–Hitchin moduli space by solving a loop-group monodromy problem, linking them to the twistor lines of hyperpolygon spaces via a precise limiting process. It proves that, after rescaling by , the Hitchin metric on energy-bounded subsets of converges to the hyperkähler metric on the hyperpolygon space , providing a finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections in the semiclassical regime are expressed explicitly through iterated integrals of logarithmic differentials on the punctured sphere, enabling systematic computation of metric expansions. Overall, the hyperpolygon spaces emerge as the natural finite-dimensional limits governing the degeneration of Hitchin moduli spaces, with twistor theory giving a cohesive, computable framework for the asymptotics.

Abstract

We investigate the Hitchin hyperkähler metric on the moduli space of strongly parabolic -Higgs bundles on the -punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights as . Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of -connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperkähler metric on the hyperpolygon space , which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.
Paper Structure (26 sections, 25 theorems, 211 equations)

This paper contains 26 sections, 25 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Twistor theory for strongly parabolic Higgs bundles
  3. Strongly Parabolic Higgs Bundles
  4. Simpson's Non-Abelian Hodge Correspondence
  5. Parabolic Deligne-Hitchin Moduli Spaces
  6. Twistor Lines
  7. Twisted holomorphic symplectic form for Fuchsian $\lambda$-connections
  8. Hyperpolygon spaces
  9. The Eguchi--Hanson space
  10. Twistor lines
  11. Bottom Component
  12. The hyperpolygon space
  13. Complex orbits, stability, and the hyperkähler metric
  14. Hyperpolygons and strongly parabolic Higgs bundles
  15. Construction of twistor lines
  16. ...and 11 more sections

Key Result

Theorem 2.1

(Parabolic non-abelian Hodge correspondence Sim2). Let $\mathcal{M}_{\mathrm{Higgs}}$ be the moduli space of polystable strongly parabolic $\mathfrak{sl}(2,\mathbb C)$-Higgs bundles on $(\Sigma,{\bf D})$ with fixed parabolic weights $0<\alpha_j<\tfrac{1}{2}$, and let $\mathcal{M}_{\mathrm{dR}}$ be t which maps a strongly parabolic Higgs bundle $(\mathcal{P},\Phi)$ to the equivalence class of a fla

Theorems & Definitions (51)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • Definition 3.5
  • Proposition 3.6
  • Remark 3.7
  • Remark 3.8
  • ...and 41 more