# 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

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