[Paper Review] Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors

Andree Lischewski|arXiv (Cornell University)|Mar 28, 2013

Geometric Analysis and Curvature Flows20 references3 citations

TL;DR

This paper classifies pseudo-Riemannian geometries admitting twistor spinors by linking their conformal holonomy to the existence of Ricci-isotropic pseudo-Walker metrics and fully characterizes the local structure of twistor spinor zero sets. It shows that twistor spinors are locally equivalent to parallel spinors off their zero sets and provides a complete geometric description in split signatures (3,2) and (3,3), revealing integrable distributions in the non-generic, non-zero case.

ABSTRACT

We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geometries. We can completely describe its local structure, construct a natural projective structure on it, and show that locally every twistor spinor with zero is equivalent to a parallel spinor off the zero set. An application of these results in low-dimensional split-signatures leads to a complete geometric description of local geometries admitting non-generic twistor spinors in signatures (3,2) and (3,3) which complements the well-known description of the generic case.

Motivation & Objective

• To classify pseudo-Riemannian geometries admitting nontrivial twistor spinors, especially in non-Lorentzian signatures.
• To understand the geometric structure of the zero set of a twistor spinor and its role in determining local geometry.
• To clarify when twistor spinors are locally equivalent to parallel spinors, particularly off their zero sets.
• To extend results from Lorentzian and Riemannian cases to low-dimensional split signatures (3,2) and (3,3), including non-generic cases.
• To characterize the conformal holonomy representation and its relation to the existence of Ricci-isotropic metrics in the conformal class.

Proposed method

• Uses the curved orbit decomposition for parabolic geometries to analyze the zero set of twistor spinors.
• Applies the conformal holonomy representation to show that a totally lightlike subspace fixed by holonomy implies a Ricci-isotropic pseudo-Walker metric in the conformal class.
• Employs the twistor operator and its kernel to study conformally covariant spinor fields, relating them to parallel spinors via the spinor covariant derivative and Clifford multiplication.
• Analyzes the real spinor modules Δ_{p,q}^ℝ and their stabilizers under Spin⁺(p,q) to classify possible orbit types and kernel dimensions.
• Uses the relation between the Dirac operator and the twistor equation to distinguish cases where ⟨φ, D^g φ⟩ ≠ 0 (Einstein metric) or = 0 (Ricci-flat or pure spinor cases).
• Applies known classification results for parallel spinors and Killing spinors to deduce local holonomy and metric structure in low signatures.

Experimental results

Research questions

• RQ1 Which pseudo-Riemannian geometries admit nontrivial twistor spinors, and how are they classified via conformal holonomy?
• RQ2 What is the local geometric structure of the zero set of a twistor spinor, and how does it support a natural projective structure?
• RQ3 In which cases is a twistor spinor locally equivalent to a parallel spinor off its zero set?
• RQ4 How do the geometric properties of twistor spinors differ in non-generic cases (e.g., signature (3,2) and (3,3)) compared to the generic case?
• RQ5 What role do integrable distributions play in the geometry of non-generic twistor spinors without zeroes?

Key findings

• A totally lightlike subspace fixed by the conformal holonomy representation implies the existence of a Ricci-isotropic pseudo-Walker metric in the conformal class.
• Every twistor spinor is locally equivalent to a parallel spinor off its zero set, with the zero set consisting of isolated points or totally lightlike submanifolds.
• In signature (4,3), the zero set of a real twistor spinor consists locally of isolated points (with G₂ holonomy off the set) or 3-dimensional totally lightlike planes (with parallel pure spinor structure off the set).
• In signature (4,2), the zero set consists of isolated points (with Ricci-flat pseudo-Kaehler metric off the set) or 2-dimensional totally lightlike nullplanes (with parallel pure spinor structure off the set).
• When ⟨φ, D^g φ⟩ ≠ 0, the conformal class contains an Einstein metric with nonzero scalar curvature, and the spinor cannot be rescaled to a parallel spinor.
• In the case ⟨φ, D^g φ⟩ = 0 and ⟨φ, φ⟩ ≡ 0, the twistor spinor is locally conformally equivalent to a parallel spinor, with a nontrivial kernel in the associated spinor representation.