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Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

Filippo Mazzoli, Andrea Seppi, Andrea Tamburelli

TL;DR

This work constructs and analyzes a para-hyperKähler structure on the deformation space $\mathcal{MGH}(\Sigma)$ of maximal globally hyperbolic AdS 3-manifolds. It shows the structure consists of a neutral metric $\mathbf{g}$ together with a complex structure $\mathbf{I}$ and two para-complex structures $\mathbf{J},\mathbf{K}$ satisfying para-quaternionic relations, with the symplectic forms $\omega_I,\omega_J,\omega_K$ encoding AdS-geometric data. The authors connect these structures to three different parameterizations of the MGHC space—Mess's left-right hyperbolic data, Krasnov–Schlenker's cotangent bundle picture, and $PSL(2,\mathbb{B})$-character varieties—via explicit maps and a genus-1 toy model that guides the higher-genus analysis. A key technical achievement is the introduction of an invariant distribution $V_{(J,\sigma)}$ on an infinite-dimensional model, enabling a rigorous reduction to the MGHC deformation space and establishing integrability and nondegeneracy of the para-hyperKähler data. The results elucidate the interplay between Teichmüller theory, Goldman's symplectic form in the para-complex setting, and circle actions, offering a coherent geometric framework for MGHC AdS manifolds and their moduli.

Abstract

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on $Σ\times\mathbb R$, for $Σ$ a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on $K$-surfaces, the identification with the cotangent bundle $T^*\mathcal{T}(Σ)$, and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the $\mathrm{PSL}(2,\mathbb{B})$-character variety, where $\mathbb{B}$ is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.

Para-hyperKähler geometry of the deformation space of maximal globally hyperbolic anti-de Sitter three-manifolds

TL;DR

This work constructs and analyzes a para-hyperKähler structure on the deformation space of maximal globally hyperbolic AdS 3-manifolds. It shows the structure consists of a neutral metric together with a complex structure and two para-complex structures satisfying para-quaternionic relations, with the symplectic forms encoding AdS-geometric data. The authors connect these structures to three different parameterizations of the MGHC space—Mess's left-right hyperbolic data, Krasnov–Schlenker's cotangent bundle picture, and -character varieties—via explicit maps and a genus-1 toy model that guides the higher-genus analysis. A key technical achievement is the introduction of an invariant distribution on an infinite-dimensional model, enabling a rigorous reduction to the MGHC deformation space and establishing integrability and nondegeneracy of the para-hyperKähler data. The results elucidate the interplay between Teichmüller theory, Goldman's symplectic form in the para-complex setting, and circle actions, offering a coherent geometric framework for MGHC AdS manifolds and their moduli.

Abstract

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on , for a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on -surfaces, the identification with the cotangent bundle , and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the -character variety, where is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.

Paper Structure

This paper contains 67 sections, 78 theorems, 466 equations.

Table of Contents

  1. Introduction
  2. Motivation and state-of-the-art
  3. Deformation space of MGHC AdS manifolds
  4. Para-hyperKähler structures
  5. Parameterizations of MGH(S)
  6. The circle action
  7. The character variety
  8. Outline of techniques and proofs
  9. Preliminaries on Anti-de Sitter geometry
  10. Maximal surfaces
  11. Cotangent bundle of Teichmüller space
  12. Mess' parameterization
  13. Constant curvature surfaces and circle action
  14. An equivalent model for MS(S)
  15. The toy model: genus 1
  16. ...and 52 more sections

Key Result

Theorem 1

Let $\Sigma$ be a closed oriented surface of genus $\geq 1$. Then $\mathcal{MGH}(\Sigma)$ admits a $MCG(\Sigma)$-invariant para-hyperKähler structure $(\mathbf{g},\mathbf{I},\mathbf{J},\mathbf{K})$. When $\Sigma$ has genus $\geq 2$, the Fuchsian locus $\mathcal{F}(\Sigma)$ is totally geodesic and $(

Theorems & Definitions (167)

  • Definition
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Theorem 9
  • ...and 157 more