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 ...
which satisfies
::
: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
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.
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 ...
s.
References
Books
*
*
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
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