nLab Killing spinor

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

A Killing spinor on a (pseudo-)Riemannian manifold is a spinor – a section of some spinor bundle $v \in \Gamma(S)$ that – that is taken by the covariant derivative of the corresponding Levi-Civita connection to a multiple of itself

$\nabla_v \psi = \kappa \gamma_v \psi$

for some constant $\kappa$.

If that constant is 0, hence if the spinor is covariant constant, then one also speaks of a covariant constant spinor or parallel spinor (with respect to the given metric structure).

Similarly a Killing vector is a covariantly constant vector field.

Pairing two covariant constant spinors to a vector yields a Killing vector.

References

Lecture notes include

• Parallel and Killing spinor fields (pdf)

A discussion with an eye towards applications in supersymmetry is around page 907 in volume II of

Discussion relating to special holonomy includes

Discussion relating to Killing vectors in supergeometry includes

Original articles include

• Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten Czechoslavakian-GDR-Polish scientific school on differential geometry Boszkowo/ Poland 1978, Sci. Comm., Part 1,2; 104-124 (1979)

• Thomas Friedrich, Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten, Colloquium Mathematicum vol. XLIV, Fasc. 2 (1981), 277-290.

Discussion relating to G2-structures includes

Revised on January 20, 2015 12:56:08 by Urs Schreiber (88.100.66.95)