What are Killing spinors?

Question

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is greatly appreciated but a less formal one would be preferred.

Answer 1

The term is used slightly differently in mathematics and physics, in particular in supergravity. Supergravity extends general relativity with local supersymmetry and includes a spin-3/2 particle called the gravitino.

In GR the parameter of an infinitesimal diffeomorphism is a vector. If the diffeomorphism leaves the metric invariant it is an isometry and the parameter is a Killing vector.

In supergravity the parameter of a supersymmetry is a spinor. If it leaves all the fields invariant it is called a Killing spinor.

Here's an excerpt from https://arxiv.org/abs/hep-th/9803116 by K.S Stelle.

Here $\epsilon$ is the supersymmetry parameter, $\psi$ is the gravitino, $F$ is an antisymmetric 4th rank tensor, $e$ is a vielbein, $\omega$ is the spin connection and the $\Gamma^{AB...}$ are products of Dirac gamma matrices. This makes $D_A$ the covariant derivative of a spinor.

The condition for a Killing spinor is that the right hand side of (4.17) vanishes. Note that when $F=0$ this reduces to $D_A\epsilon=0$, i.e. $\epsilon$ is covariantly constant. Mathematicians call this a parallel spinor. For them a Killing spinor satisfies $D_A\epsilon = \lambda\Gamma_A\epsilon$ for some $\lambda$.