Explaining Conformal Killing Spinors

[StatusX]

This could have gone in about 4 different forums, so I apologize if I picked the wrong one. I'm wondering if anyone can explain what (conformal) killing spinors are all about. All I can find is that they are sections of the spinor bundle of a spin manifold satisfying:

$$\nabla_\mu \epsilon = \lambda \gamma_\mu \epsilon$$

where $\nabla_\mu$ is the spinor covariant derivative, $\gamma_\mu$ are the dirac matrices, and $\lambda$ is a constant. I've also seen something like this pair of equations used to define them:

$$\nabla_\mu \epsilon = \gamma_\mu \epsilon'$$

$$\nabla_\mu \epsilon' = c R \gamma_\mu \epsilon$$

where R is the scalar curvature and c is some specific constant, possibly depending on dimension, that I can't remember right now.

First of all, which equation is correct, and in the case of the second, which is the killing spinor? Second, how should I think about these geometrically? Where do these equations come from? I've come across these working on supersymmetry, and I'd like to know why they are important there. If anyone could try to explain some of this, or point me to some good sources, I'd really appreciate it.

 

[StatusX]

Anyone? I'm still lost. I'm also aware there's a connection with Penrose's twistor spinors, so maybe someone could explain something about this.