Killing spinor

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 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 ${\displaystyle \psi }$ which satisfies

${\displaystyle \nabla _{X}\psi =\lambda X\cdot \psi }$

for all tangent vectors X, where ${\displaystyle \nabla }$ is the spinor covariant derivative, ${\displaystyle \cdot }$ is Clifford multiplication and ${\displaystyle \lambda \in \mathbb {C} }$ is a constant, called the Killing number of ${\displaystyle \psi }$. If ${\displaystyle \lambda =0}$ 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 ${\displaystyle {\mathcal {M}}}$ is a manifold with a Killing spinor, then ${\displaystyle {\mathcal {M}}}$ is an Einstein manifold with Ricci curvature ${\displaystyle Ric=4(n-1)\alpha ^{2}}$, where ${\displaystyle \alpha }$ is the Killing constant.^[5]

Types of Killing spinor fields

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

References

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

• 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

• "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)