Killing spinor is a term used in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and physics. By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistor spinors which are also

eigenspinor In quantum mechanics, eigenspinors are thought of as basis vectors representing the general spin state of a particle. Strictly speaking, they are not vectors at all, but in fact spinors. For a single spin 1/2 particle, they can be defined as th ...

s of the Dirac operator. The term is named after Wilhelm Killing. Another equivalent definition is that Killing spinors are the solutions to the

Killing equation In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal genera ...

for a so-called Killing number. More formally: :A Killing spinor on a Riemannian

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

''M'' is a spinor field

\psi

which satisfies ::

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

:for all tangent vectors ''X'', where

\nabla

is the spinor

covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...

\cdot

Clifford multiplication In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomp ...

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 and superstring theory, in particular for finding solutions which preserve some

supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...

. They are a special kind of spinor field related to

Killing vector field In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...

s and Killing tensors.

References

Books

* *

External links

"Twistor and Killing spinors in Lorentzian geometry,"
by Helga Baum (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