Killing

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

is a term used in mathematics and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

. 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 In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise forma ...

. The term is named after

Wilhelm Killing Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. Life Killing studied at the University of Mü ...

. Another equivalent definition is that Killing spinors are the solutions to the Killing equation for a so-called Killing number. More formally: :A Killing spinor on a Riemannian spin manifold ''M'' is a

spinor field In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\col ...

\psi

which satisfies ::

\nabla_X\psi=\lambda X\cdot\psi

:for all

tangent vectors In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...

''X'', where

\nabla

is the spinor covariant derivative,

\cdot

is Clifford multiplication and

\lambda \in \mathbb

is a constant, called the Killing number of

\psi

. If

\lambda=0

then the spinor is called a parallel spinor. In physics, Killing spinors are used in

supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...

and

superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t ...

, in particular for finding solutions which preserve some supersymmetry. They are a special kind of spinor field related to Killing vector fields and

Killing tensor In mathematics, a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept in pseudo-Riemannian geometry, and is mainly used in the theory of gener ...

References

Books

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External links

"Twistor and Killing spinors in Lorentzian geometry,"
by

Helga Baum Helga Baum (née Dlubek, born 1954) is a German mathematician. She is professor for differential geometry and global analysis in the Institute for Mathematics of the Humboldt University of Berlin Humboldt-Universität zu Berlin (german: H ...

(PDF format)
''Dirac Operator'' From MathWorld''Killing and Twistor Spinors on Lorentzian Manifolds,'' (paper by Christoph Bohle) (postscript format)
Riemannian geometry Structures on manifolds Supersymmetry Spinors {{Riemannian-geometry-stub