Killing spinor explained

Killing spinor is a term used in mathematics and physics.

Definition

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

\psi which satisfies\nabla _{X\psi=λ} X ⋅ \psi

for all tangent vectors X, where

\nabla is the spinor covariant derivative, ⋅ is Clifford multiplication and λ \in C is a constant, called the Killing number of \psi . If λ=0 then the spinor is called a parallel spinor.

Applications

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

If

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

Types of Killing spinor fields

If

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

Books

• Book: Lawson . H. Blaine . Michelsohn . Marie-Louise . Marie-Louise Michelsohn. Spin Geometry . . 978-0-691-08542-5 . 1989 .

External links

• "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)

Notes and References

1. Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkei nichtnegativer Skalarkrümmung. Th. Friedrich. Mathematische Nachrichten. 97. 1980. 117–146. 10.1002/mana.19800970111.
2. On the conformal relation between twistors and Killing spinors. Th. Friedrich. Supplemento dei Rendiconti del Circolo Matematico di Palermo, Serie II. 22. 1989. 59–75.
3. Spin manifolds, Killing spinors and the universality of Hijazi inequality. A. Lichnerowicz. André Lichnerowicz. Lett. Math. Phys.. 13. 1987. 331–334. 10.1007/bf00401162. 1987LMaPh..13..331L . 121971999.
4. Bär . Christian . 1993-06-01 . Real Killing spinors and holonomy . Communications in Mathematical Physics . en . 154 . 3 . 509–521 . 10.1007/BF02102106 . 1432-0916.