# Killing spinor

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Type of Dirac operator eigenspinor

**Killing [spinor](/source/Spinor)** is a term used in [mathematics](/source/Mathematics) and [physics](/source/Physics).

## Definition

By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those [twistor](/source/Twistor_theory) spinors which are also [eigenspinors](/source/Eigenspinor) of the [Dirac operator](/source/Dirac_operator).[1][2][3] The term is named after [Wilhelm Killing](/source/Wilhelm_Killing).

Another equivalent definition is that Killing spinors are the solutions to the [Killing equation](/source/Killing_equation) for a so-called Killing number.

More formally:[4]

- A **Killing spinor** on a [Riemannian](/source/Riemannian_manifold) [spin](/source/Spin_structure) [manifold](/source/Manifold) *M* is a [spinor field](/source/Spinor_field) ψ {\displaystyle \psi } which satisfies

- - ∇ X ψ = λ X ⋅ ψ {\displaystyle \nabla _{X}\psi =\lambda X\cdot \psi }

- for all [tangent vectors](/source/Tangent_space) *X*, where ∇ {\displaystyle \nabla } is the spinor [covariant derivative](/source/Covariant_derivative), ⋅ {\displaystyle \cdot } is [Clifford multiplication](/source/Clifford_multiplication) and λ ∈ C {\displaystyle \lambda \in \mathbb {C} } is a constant, called the **Killing number** of ψ {\displaystyle \psi } . If λ = 0 {\displaystyle \lambda =0} then the spinor is called a parallel spinor.

## Applications

In physics, Killing spinors are used in [supergravity](/source/Supergravity) and [superstring theory](/source/Superstring_theory), in particular for finding solutions which preserve some [supersymmetry](/source/Supersymmetry). They are a special kind of spinor field related to [Killing vector fields](/source/Killing_vector_field) and [Killing tensors](/source/Killing_tensor).

## Properties

If M {\displaystyle {\mathcal {M}}} is a manifold with a Killing spinor, then M {\displaystyle {\mathcal {M}}} is an [Einstein manifold](/source/Einstein_manifold) with [Ricci curvature](/source/Ricci_curvature) R i c = 4 ( n − 1 ) α 2 {\displaystyle Ric=4(n-1)\alpha ^{2}} , where α {\displaystyle \alpha } is the Killing constant.[5]

### Types of Killing spinor fields

If α {\displaystyle \alpha } is purely imaginary, then M {\displaystyle {\mathcal {M}}} is a [noncompact manifold](/source/Noncompact); if α {\displaystyle \alpha } is 0, then the spinor field is parallel; finally, if α {\displaystyle \alpha } is real, then M {\displaystyle {\mathcal {M}}} is compact, and the spinor field is called a "real spinor field".

## References

1. **[^](#cite_ref-1)** Th. Friedrich (1980). "Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung". *[Mathematische Nachrichten](/source/Mathematische_Nachrichten)*. **97**: 117–146. [doi](/source/Doi_(identifier)):[10.1002/mana.19800970111](https://doi.org/10.1002%2Fmana.19800970111).

1. **[^](#cite_ref-2)** Th. Friedrich (1989). "On the conformal relation between twistors and Killing spinors". *Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II*. **22**: 59–75.

1. **[^](#cite_ref-3)** [A. Lichnerowicz](/source/Andr%C3%A9_Lichnerowicz) (1987). "Spin manifolds, Killing spinors and the universality of Hijazi inequality". *Lett. Math. Phys*. **13** (4): 331–334. [Bibcode](/source/Bibcode_(identifier)):[1987LMaPh..13..331L](https://ui.adsabs.harvard.edu/abs/1987LMaPh..13..331L). [doi](/source/Doi_(identifier)):[10.1007/bf00401162](https://doi.org/10.1007%2Fbf00401162). [S2CID](/source/S2CID_(identifier)) [121971999](https://api.semanticscholar.org/CorpusID:121971999).

1. **[^](#cite_ref-4)** Friedrich, Thomas (2000), *Dirac Operators in Riemannian Geometry*, [American Mathematical Society](/source/American_Mathematical_Society), pp. 116–117, [ISBN](/source/ISBN_(identifier)) [978-0-8218-2055-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-2055-1)

1. **[^](#cite_ref-5)** Bär, Christian (1993-06-01). ["Real Killing spinors and holonomy"](https://doi.org/10.1007/BF02102106). *Communications in Mathematical Physics*. **154** (3): 509–521. [Bibcode](/source/Bibcode_(identifier)):[1993CMaPh.154..509B](https://ui.adsabs.harvard.edu/abs/1993CMaPh.154..509B). [doi](/source/Doi_(identifier)):[10.1007/BF02102106](https://doi.org/10.1007%2FBF02102106). [ISSN](/source/ISSN_(identifier)) [1432-0916](https://search.worldcat.org/issn/1432-0916).

## Books

- Lawson, H. Blaine; [Michelsohn, Marie-Louise](/source/Marie-Louise_Michelsohn) (1989). *Spin Geometry*. [Princeton University Press](/source/Princeton_University_Press). [ISBN](/source/ISBN_(identifier)) [978-0-691-08542-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08542-5).

- Friedrich, Thomas (2000), *Dirac Operators in Riemannian Geometry*, [American Mathematical Society](/source/American_Mathematical_Society), [ISBN](/source/ISBN_(identifier)) [978-0-8218-2055-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-2055-1)

## External links

- ["Twistor and Killing spinors in Lorentzian geometry,"](http://www.emis.de/journals/SC/2000/4/pdf/smf_sem-cong_4_35-52.pdf) by [Helga Baum](/source/Helga_Baum) (PDF format)

- [*Dirac Operator* From MathWorld](http://mathworld.wolfram.com/DiracOperator.html)

- [*Killing's Equation* From MathWorld](http://mathworld.wolfram.com/KillingsEquation.html)

- [*Killing and Twistor Spinors on Lorentzian Manifolds,* (paper by Christoph Bohle) (postscript format)](https://web.archive.org/web/20041107222740/http://www.math.tu-berlin.de/~bohle/pub/dipl.ps)

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