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'''Killing [[spinor]]''' is a term used in [[mathematics]] and [[physics]].  By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those [[twistor]]
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spinors which are also [[eigenspinor]]s of the [[Dirac operator]].<ref>{{cite journal|title=Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung|author=Th. Friedrich|journal=[[Mathematische Nachrichten]]|volume=97|year=1980|pages=117-146}}</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=[[André Lichnerowicz|A. Lichnerowicz]]|journal=Lett. Math. Phys.|volume=13|year=1987|pages=331-334}}</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.  
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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</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.
 
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.
 
==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 | postscript=<!--None-->}}
* {{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]
*[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]]
 
 
{{math-stub}}

Revision as of 20:30, 11 February 2014

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