nLab
Killing spinor

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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

vψ=κγ vψ \nabla_v \psi = \kappa \gamma_v \psi

for some constant κ\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

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