Killing spinor

By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistor

spinors which are also eigenspinors of the Dirac operator.{{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}}{{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}}{{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}} 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:{{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}}

:A Killing spinor on a Riemannian spin manifold M is a spinor field $\backslash psi$ which satisfies

::$\backslash nabla\_X\backslash psi=\backslash lambda\; X\backslash cdot\backslash psi$

:for all tangent vectors X, where $\backslash nabla$ is the spinor covariant derivative, $\backslash cdot$ is Clifford multiplication and $\backslash lambda\; \backslash in\; \backslash mathbb\{C\}$ is a constant, called the Killing number of $\backslash psi$. If $\backslash lambda=0$ 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.

If $\backslash mathcal\{M\}$ is a manifold with a Killing spinor, then $\backslash mathcal\{M\}$ is an Einstein manifold with Ricci curvature $Ric=4(n-1)\backslash alpha^2$, where $\backslash alpha$ is the Killing constant.{{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}}

If $\backslash alpha$ is purely imaginary, then $\backslash mathcal\{M\}$ is a noncompact manifold; if $\backslash alpha$ is 0, then the spinor field is parallel; finally, if $\backslash alpha$ is real, then $\backslash mathcal\{M\}$ is compact, and the spinor field is called a ``real spinor field."

{{Reflist}}

• {{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}}

• [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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