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Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

Lynn Heller, Sebastian Heller, Martin Traizet

TL;DR

This work develops a complex-analytic program to realize the non-abelian Hodge correspondence for strongly parabolic SL(2,C) Higgs fields on a 4-punctured sphere via loop group methods and the Deligne–Hitchin framework. The authors construct real holomorphic sections (twistor lines) near the trivial limit by solving a Monodromy Problem with Fuchsian potentials, enabling explicit Taylor expansions of geometric structures in the parabolic weight parameter t. In the t→0 limit, the rescaled hyper-Kähler metric on the moduli space is identified with the Eguchi–Hanson metric on the blown-up nilpotent orbit, and the limit NAHC provides a bridge between parabolic Higgs data and flat connections on the corresponding surface. The approach yields first-order (and higher) deformations of the non-abelian Hodge correspondence and the twisted symplectic form, and reveals MPL-based identities arising from the higher-order terms, highlighting deep connections between hyper-Kähler geometry, parabolic structures, and loop-group techniques with potential implications for related moduli problems.

Abstract

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using complex analytic methods. We identify the rescaled limit hyper-Kähler moduli space $\mathcal M_t$ at $t=0$ to be the completion of the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ modulo a $\mathbb Z_2\times\mathbb Z_2$ action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's $λ$-connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form $\varpi_t$ on $\mathcal M_t$ at $t=0$ which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of $\mathcal M_t$ lead to some identities of certain MPLs which we believe deserve further investigations.

Loop group methods for the non-abelian Hodge correspondence on a 4-punctured sphere

TL;DR

This work develops a complex-analytic program to realize the non-abelian Hodge correspondence for strongly parabolic SL(2,C) Higgs fields on a 4-punctured sphere via loop group methods and the Deligne–Hitchin framework. The authors construct real holomorphic sections (twistor lines) near the trivial limit by solving a Monodromy Problem with Fuchsian potentials, enabling explicit Taylor expansions of geometric structures in the parabolic weight parameter t. In the t→0 limit, the rescaled hyper-Kähler metric on the moduli space is identified with the Eguchi–Hanson metric on the blown-up nilpotent orbit, and the limit NAHC provides a bridge between parabolic Higgs data and flat connections on the corresponding surface. The approach yields first-order (and higher) deformations of the non-abelian Hodge correspondence and the twisted symplectic form, and reveals MPL-based identities arising from the higher-order terms, highlighting deep connections between hyper-Kähler geometry, parabolic structures, and loop-group techniques with potential implications for related moduli problems.

Abstract

The non-abelian Hodge correspondence is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions to the self-duality equations. In this paper we construct self-duality solutions for strongly parabolic Higgs fields on a -punctured sphere with parabolic weights using complex analytic methods. We identify the rescaled limit hyper-Kähler moduli space at to be the completion of the nilpotent orbit in modulo a action, equipped with the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's -connections interpretation. By construction we can compute the Taylor expansions of the holomorphic symplectic form on at which turn out to have closed form expressions in terms of multiple polylogarithms (MPLs). The geometric properties of lead to some identities of certain MPLs which we believe deserve further investigations.
Paper Structure (39 sections, 45 theorems, 392 equations, 1 figure)

This paper contains 39 sections, 45 theorems, 392 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The hyper-Kähler structures and their twistor spaces
  4. The Deligne-Hitchin moduli space $\mathcal{M}_{DH}$
  5. Automorphisms of the Deligne-Hitchin moduli space
  6. Real sections
  7. Loop groups
  8. Reconstruction of self-duality solutions from admissible negative real sections
  9. Goldman's symplectic form on the moduli space of $\lambda$-connections
  10. Strongly parabolic Higgs bundles
  11. Parabolic structures and logarithmic connections
  12. Parabolic structures on the 4-punctured sphere
  13. Strongly parabolic Higgs fields
  14. Non-abelian Hodge correspondence in the strongly parabolic case
  15. Rational weights
  16. ...and 24 more sections

Key Result

Theorem 1

For $t\sim 0$ fixed let $C>0$ and consider Higgs fields with $\mathcal{E}(d, \Phi) <C.$ Then, there exists $\varepsilon >0,$ depending only on the constant $C$, such that twistor lines $s_t$ can be constructed with $s_t(\lambda = 0) = t \Phi$ using complex analytic methods only. In particular, the m

Figures (1)

  • Figure 1: Two views on an equivariant minimal surface in $\mathbb H^3$ obtained from a strongly parabolic nilpotent Higgs field on ${\mathbb C} P^1$ with four singular points and parabolic weight $t=\alpha_j=\tfrac{1}{6}$. Image by Nick Schmitt.

Theorems & Definitions (123)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Remark 2
  • Example 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 113 more