Twistor spinor (${\nabla }_aϵ=\frac{1}{n}{\gamma }_a{\gamma }^b{\nabla }_bϵ$)

A twistor spinor has vanishing gamma-trace-free part of its spinor covariant derivative. In $n$ dimensions the displayed equation uses gamma matrices satisfying $\{{\gamma }_a,{\gamma }_b\}=2g_{ab}$. It is weaker than being a parallel spinor. In four dimensions it implies $[{\gamma }_a{\nabla }_b+{\gamma }_b{\nabla }_a-g_{ab}{\gamma }^c{\nabla }_c/2]ϵ=0$.

In the conformal frame for $g={\Omega }^2\eta$, identify the spinor fibres using the chosen vierbein. If ${ϵ}_0$ is a flat constant spinor field, the spin connection of a conformally flat metric gives ${\nabla }_aϵ={\hat{\gamma }}_a{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Contracting with curved ${\gamma }^a={\Omega }^{-1}{\hat{\gamma }}^a$ yields ${\gamma }^a{\nabla }_aϵ=n{\Omega }^{-1}{\hat{\gamma }}^c({\partial }_c\log \Omega )ϵ/2$. Thus ${\nabla }_aϵ={\gamma }_a{\gamma }^b{\nabla }_bϵ/n$: the rescaled parallel spinor is a twistor spinor.