Killing spinor

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.

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

- Pierre Deligne, P. Etingof, Dan Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and Edward Witten, (eds.)
*Quantum Fields and Strings, A course for mathematicians*, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

Revised on September 12, 2014 11:57:11
by Urs Schreiber
(185.26.182.37)