Editing Killing spinor

{{Short description|Type of Dirac operator eigenspinor}}
'''Killing [[spinor]]''' is a term used in [[mathematics]] and [[physics]].  

== Definition ==
By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those [[Twistor theory|twistor]]
spinors which are also [[eigenspinor]]s of the [[Dirac operator]].<ref>{{cite journal|title=Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung|author=Th. Friedrich|journal=[[Mathematische Nachrichten]]|volume=97|year=1980|pages=117–146|doi=10.1002/mana.19800970111}}</ref><ref>{{cite journal|title=On the conformal relation between twistors and Killing spinors|author=Th. Friedrich|journal=Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II|volume=22|year=1989|pages=59–75}}</ref><ref>{{cite journal|title=Spin manifolds, Killing spinors and the universality of Hijazi inequality|author=A. Lichnerowicz|author-link=André Lichnerowicz|journal=Lett. Math. Phys.|volume=13|year=1987|issue=4 |pages=331–334|doi=10.1007/bf00401162|bibcode = 1987LMaPh..13..331L |s2cid=121971999}}</ref> The term is named after [[Wilhelm Killing]].

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

More formally:<ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] |pages= 116–117| year=2000|isbn=978-0-8218-2055-1}}
</ref>

:A '''Killing spinor''' on a  [[Riemannian manifold|Riemannian]] [[Spin structure|spin]] [[manifold]] ''M'' is a [[spinor field]] <math>\psi</math> which satisfies

::<math>\nabla_X\psi=\lambda X\cdot\psi</math>

:for all [[tangent space|tangent vectors]] ''X'', where <math>\nabla</math> is the spinor [[covariant derivative]], <math>\cdot</math> is [[Clifford multiplication]] and <math>\lambda \in \mathbb{C}</math> is a constant, called the '''Killing number''' of <math>\psi</math>. If <math>\lambda=0</math> then the spinor is called a parallel spinor.

== Applications ==
In physics, Killing spinors are used in [[supergravity]] and [[superstring theory]], in particular for finding solutions which preserve some [[supersymmetry]].  They are a special kind of spinor field related to [[Killing vector field]]s and [[Killing tensor]]s.

== Properties ==
If <math>\mathcal{M}</math> is a manifold with a Killing spinor, then <math>\mathcal{M}</math> is an [[Einstein manifold]] with [[Ricci curvature]] <math>Ric=4(n-1)\alpha^2 </math>, where <math>\alpha</math> is the Killing constant.<ref>{{Cite journal |last=Bär |first=Christian |date=1993-06-01 |title=Real Killing spinors and holonomy |url=https://doi.org/10.1007/BF02102106 |journal=Communications in Mathematical Physics |language=en |volume=154 |issue=3 |pages=509–521 |doi=10.1007/BF02102106 |bibcode=1993CMaPh.154..509B |issn=1432-0916}}</ref> 
===Types of Killing spinor fields===
If <math>\alpha</math> is purely imaginary, then <math>\mathcal{M}</math> is a [[noncompact|noncompact manifold]]; if <math>\alpha</math> is 0, then the spinor field is parallel; finally, if <math>\alpha</math> is real, then <math>\mathcal{M}</math> is compact, and the spinor field is called a ``real spinor field."

==References==
{{Reflist}}

==Books==
* {{Cite book | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise |author2-link=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 }}
* {{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}}

==External links==
*[http://www.emis.de/journals/SC/2000/4/pdf/smf_sem-cong_4_35-52.pdf "Twistor and Killing spinors in Lorentzian geometry,"] by [[Helga Baum]] (PDF format)
*[http://mathworld.wolfram.com/DiracOperator.html ''Dirac Operator'' From MathWorld]
*[http://mathworld.wolfram.com/KillingsEquation.html ''Killing's Equation'' From MathWorld]
*[https://web.archive.org/web/20041107222740/http://www.math.tu-berlin.de/~bohle/pub/dipl.ps ''Killing and Twistor Spinors on Lorentzian Manifolds,'' (paper by Christoph Bohle) (postscript format) ]

[[Category:Riemannian geometry]]
[[Category:Structures on manifolds]]
[[Category:Supersymmetry]]
[[Category:Spinors]]


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