# Killing spinor

Killing spinor

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 twistorspinors which are also eigenspinors of the Dirac operator. 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:

:A Killing spinor on a manifold "M" is a spinor field $phi$ which satisfies

::$abla_Xphi=lambda Xcdotphi$

:for all tangent vectors "X", where $abla$ is the spinor covariant derivative, $cdot$ is Clifford 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 and superstring theory, in particular for finding solutions which preserve some supersymmetry. They are a special kind of spinor field related to Killing vector fields and Killing tensors.

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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