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Pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form

Nicholas Rungi, Andrea Tamburelli

TL;DR

The paper develops a neighborhood of Teichmüller space inside the SL(3,R)-Hitchin component Hit_3(Σ) endowed with a mapping class group invariant pseudo-Kähler structure (g,I,ω). This is achieved via an infinite-dimensional symplectic reduction, first by the Hamiltonian group Ham(Σ,ρ) and then by the symplectomorphism quotient, with a circle action rotating cubic differentials providing a Hamiltonian perspective; the construction ties the Hitchin data to hyperbolic affine spheres through Wang’s equation and Labourie–Loftin’s cubic-differential parametrization. A key outcome is the comparison with Goldman’s symplectic form ω_G, which coincides with ω_f on the Fuchsian locus in both tangent and transverse directions, implying that (ω_G,I) cannot define a Kähler structure on Hit_3(Σ). Additionally, the work situates Hit_3(Σ) as a finite-dimensional quotient of an infinite-dimensional space, clarifying the role of the moment map and the reduction process in the Hitchin geometry and linking the results to the geometry of Teichmüller space via the Weil–Petersson metric. Overall, the paper provides a principled, reduction-based framework for pseudo-Kähler geometry on the SL(3,R) Hitchin component and reveals intrinsic obstructions to a Kähler pairing with Goldman’s form.

Abstract

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-Kähler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman's $\boldsymbolω_G$, we prove that the pair $(\boldsymbolω_G, \mathbf{I})$ cannot define a Kähler structure on the Hitchin component.

Pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form

TL;DR

The paper develops a neighborhood of Teichmüller space inside the SL(3,R)-Hitchin component Hit_3(Σ) endowed with a mapping class group invariant pseudo-Kähler structure (g,I,ω). This is achieved via an infinite-dimensional symplectic reduction, first by the Hamiltonian group Ham(Σ,ρ) and then by the symplectomorphism quotient, with a circle action rotating cubic differentials providing a Hamiltonian perspective; the construction ties the Hitchin data to hyperbolic affine spheres through Wang’s equation and Labourie–Loftin’s cubic-differential parametrization. A key outcome is the comparison with Goldman’s symplectic form ω_G, which coincides with ω_f on the Fuchsian locus in both tangent and transverse directions, implying that (ω_G,I) cannot define a Kähler structure on Hit_3(Σ). Additionally, the work situates Hit_3(Σ) as a finite-dimensional quotient of an infinite-dimensional space, clarifying the role of the moment map and the reduction process in the Hitchin geometry and linking the results to the geometry of Teichmüller space via the Weil–Petersson metric. Overall, the paper provides a principled, reduction-based framework for pseudo-Kähler geometry on the SL(3,R) Hitchin component and reveals intrinsic obstructions to a Kähler pairing with Goldman’s form.

Abstract

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the -Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-Kähler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman's , we prove that the pair cannot define a Kähler structure on the Hitchin component.
Paper Structure (27 sections, 70 theorems, 319 equations)

This paper contains 27 sections, 70 theorems, 319 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proofs
  3. Comparison with the anti-de Sitter case
  4. Preliminaries
  5. The PSL(3,R)-Hitchin component
  6. Hyperbolic affine spheres
  7. Wang's equation
  8. Labourie and Loftin's parametrization
  9. A conformal change of metric
  10. The Weil-Petersson Kähler metric on Teichmüller space
  11. The Lie algebra of the group of (Hamiltonian) symplectomorphisms
  12. Teichmüller space as a symplectic reduction
  13. A formula for the differential of the curvature
  14. The pseudo-Kähler metric on the Hitchin component
  15. Construction of D^3(S)
  16. ...and 12 more sections

Key Result

Theorem 1

A There exists a bi-linear, alternating and closed $2$-tensor $\boldsymbol{\omega}$ on $\mathrm{Hit}_3(\Sigma)$ such that $\mathbf{g}(\cdot,\cdot):=\boldsymbol{\omega}(\mathbf{I} \cdot,\cdot)$ defines an indefinite symmetric bi-linear form. Moreover, the triple $(\mathbf{g},\boldsymbol{\omega},\math

Theorems & Definitions (122)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 2.1: Hitchin hitchin1992lie
  • Theorem 2.2: goldman1990convex, choi1993convex
  • ...and 112 more