Killing spinor (original) (raw)

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Type of Dirac operator eigenspinor

Killing spinor is a term used in mathematics and physics.

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:[4]

A Killing spinor on a Riemannian spin manifold M is a spinor field ψ {\displaystyle \psi } {\displaystyle \psi } which satisfies

∇ X ψ = λ X ⋅ ψ {\displaystyle \nabla _{X}\psi =\lambda X\cdot \psi } {\displaystyle \nabla _{X}\psi =\lambda X\cdot \psi }

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

Types of Killing spinor fields

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

  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 (4): 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. Bibcode:1993CMaPh.154..509B. doi:10.1007/BF02102106. ISSN 1432-0916.