Killing spinor is a term used in mathematics and physics.

Definition

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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.[1][2][3] 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:[4]

A Killing spinor on a Riemannian spin manifold M is a spinor field   which satisfies
 
for all tangent vectors X, where   is the spinor covariant derivative,   is Clifford multiplication and   is a constant, called the Killing number of  . If   then the spinor is called a parallel spinor.

Applications

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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.

Properties

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If   is a manifold with a Killing spinor, then   is an Einstein manifold with Ricci curvature  , where   is the Killing constant.[5]

Types of Killing spinor fields

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If   is purely imaginary, then   is a noncompact manifold; if   is 0, then the spinor field is parallel; finally, if   is real, then   is compact, and the spinor field is called a ``real spinor field."

References

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  1. ^ Th. Friedrich (1980). "Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung". Mathematische Nachrichten. 97: 117–146. doi:10.1002/mana.19800970111.
  2. ^ Th. Friedrich (1989). "On the conformal relation between twistors and Killing spinors". Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II. 22: 59–75.
  3. ^ A. Lichnerowicz (1987). "Spin manifolds, Killing spinors and the universality of Hijazi inequality". Lett. Math. Phys. 13: 331–334. Bibcode:1987LMaPh..13..331L. doi:10.1007/bf00401162. S2CID 121971999.
  4. ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, pp. 116–117, ISBN 978-0-8218-2055-1
  5. ^ Bär, Christian (1993-06-01). "Real Killing spinors and holonomy". Communications in Mathematical Physics. 154 (3): 509–521. doi:10.1007/BF02102106. ISSN 1432-0916.

Books

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