Killing spinor
Killing spinor is a term used in mathematics and physics.
Definition
[edit]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
[edit]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
[edit]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
[edit]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
[edit]- ^ 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.
- ^ 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.
- ^ 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.
- ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, pp. 116–117, ISBN 978-0-8218-2055-1
- ^ 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
[edit]- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
- Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1
External links
[edit]- "Twistor and Killing spinors in Lorentzian geometry," by Helga Baum (PDF format)
- Dirac Operator From MathWorld
- Killing's Equation From MathWorld
- Killing and Twistor Spinors on Lorentzian Manifolds, (paper by Christoph Bohle) (postscript format)