Killing spinor: Difference between revisions
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== Properties == | == Properties == | ||
If <math>\mathcal{M}</math> is a manifold with a Killing spinor, then <math>\mathcal{M}</math> is an [[Einstein manifold]] with [[Ricci curvature]] <math>Ric=4(n-1)\alpha^2 </math>, where <math>\alpha</math> is the Killing constant.<ref>{{Cite journal |last=Bär |first=Christian |date=1993-06-01 |title=Real Killing spinors and holonomy |url=https://doi.org/10.1007/BF02102106 |journal=Communications in Mathematical Physics |language=en |volume=154 |issue=3 |pages=509–521 |doi=10.1007/BF02102106 |bibcode=1993CMaPh.154..509B |issn=1432-0916}}</ref> | If <math>\mathcal{M}</math> is a manifold with a Killing spinor, then <math>\mathcal{M}</math> is an [[Einstein manifold]] with [[Ricci curvature]] <math>Ric=4(n-1)\alpha^2 </math>, where <math>\alpha</math> is the Killing constant.<ref>{{Cite journal |last=Bär |first=Christian |date=1993-06-01 |title=Real Killing spinors and holonomy |url=https://doi.org/10.1007/BF02102106 |journal=Communications in Mathematical Physics |language=en |volume=154 |issue=3 |pages=509–521 |doi=10.1007/BF02102106 |bibcode=1993CMaPh.154..509B |issn=1432-0916|url-access=subscription }}</ref> | ||
===Types of Killing spinor fields=== | ===Types of Killing spinor fields=== | ||
If <math>\alpha</math> is purely imaginary, then <math>\mathcal{M}</math> is a [[noncompact|noncompact manifold]]; if <math>\alpha</math> is 0, then the spinor field is parallel; finally, if <math>\alpha</math> is real, then <math>\mathcal{M}</math> is compact, and the spinor field is called a ``real spinor field." | If <math>\alpha</math> is purely imaginary, then <math>\mathcal{M}</math> is a [[noncompact|noncompact manifold]]; if <math>\alpha</math> is 0, then the spinor field is parallel; finally, if <math>\alpha</math> is real, then <math>\mathcal{M}</math> is compact, and the spinor field is called a ``real spinor field." | ||
Latest revision as of 23:47, 19 June 2025
Template:Short description 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 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
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 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
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
Books
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