Killing spinor
Encyclopedia
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 twistor
spinors which are also eigenspinor
s of the Dirac operator
, see Th. Friedrich ,
"Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung", Math. Nachr. 97 (1980), 117-146. 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:
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 field
s and Killing tensors.
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
is a term used in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistor
spinors which are also eigenspinor
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...
s of the Dirac operator
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...
, see Th. Friedrich ,
"Der erste Eigenwert des Dirac Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung", Math. Nachr. 97 (1980), 117-146. The term is named after Wilhelm Killing
Wilhelm Killing
Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
.
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 manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
M is a spinorSpinorIn mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
field
which satisfies
- for all tangent vectorsTangent spaceIn mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
X, where
is the spinor covariant derivativeCovariant derivativeIn 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 differential operator, to be contrasted with the approach given...
,
is Clifford multiplication and 
is a constant, called the Killing number. If 
then the spinor is called a parallel spinor.
In physics, Killing spinors are used in supergravity
Supergravity
In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...
and superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
, in particular for finding solutions which preserve some supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
. They are a special kind of spinor field related to Killing vector field
Killing vector field
In mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
s and Killing tensors.