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Para-hyper-Kähler Geometry of the Deformation Space of Maximal Globally Hyperbolic Anti-de Sitter Three-Manifolds
About this Title
Filippo Mazzoli, Andrea Seppi and Andrea Tamburelli
Publication: Memoirs of the American Mathematical Society
Publication Year:
2025; Volume 306, Number 1546
ISBNs: 978-1-4704-7303-7 (print); 978-1-4704-8054-7 (online)
DOI: https://doi.org/10.1090/memo/1546
Published electronically: January 27, 2025
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries on Anti-de Sitter geometry
- 3. The toy model: Genus 1
- 4. The general case: Genus $\geq 2$
- 5. Geometric interpretations
- 6. Symplectic reduction
- A. Para-complex geometry
Abstract
In this paper we study the para-hyper-Kähler geometry of the deformation space of MGHC anti-de Sitter structures on $\Sigma \times \mathbb R$, for $\Sigma$ 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 $T^*\mathcal T(\Sigma )$, 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 $\mathbb {P}\mathrm {SL}(2,\mathbb {B})$-character variety, where $\mathbb {B}$ is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
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