- Killing spinor
**Killing**is a term used inspinor mathematics andphysics . By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates thosetwistor spinors which are also eigenspinors of theDirac operator . The term is named afterWilhelm Killing .Another equivalent definition is that Killing spinors are the solutions to the

Killing equation for a so-calledKilling number .More formally:

:A

**Killing spinor**on amanifold "M" is aspinor field $phi$ which satisfies::$abla\_Xphi=lambda\; Xcdotphi$

:for all tangent vectors "X", where $abla$ is the spinor

covariant derivative , $cdot$ isClifford multiplication and $lambda$ is a constant, called the Killing number. If $lambda=0$ then the spinor is called a parallel spinor.In physics, Killing spinors are used in

supergravity andsuperstring theory , in particular for finding solutions which preserve somesupersymmetry . They are a special kind of spinor field related toKilling vector field s andKilling tensor s.**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)*]

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