Parallel spinor ($\nabla ϵ=0$)

A parallel spinor field is annihilated by the spinor covariant derivative. Its value at one point determines it on a connected domain by parallel transport. The spinor curvature identity implies $R_{\mu \nu AB}{\gamma }^{AB}ϵ=0$, but this local condition alone need not settle global existence: the spin structure and holonomy also matter. Lorentzian parallel spinors need not force the whole Ricci tensor to vanish, unlike the positive-definite case.

In flat three-dimensional spacetime with polar spatial orthonormal coframe $(dT,d\rho ,\rho d\varphi )$, the spin connection has ${\omega }_{12}=-d\varphi$. Thus every parallel spinor has the displayed components. Since $({\gamma }^1{\gamma }^2{)}^2=-1$, these components change sign under $\varphi \mapsto \varphi +2\pi$. This is the lift of a full rotation to the spin structure extending over the disk; the polar frame itself does not extend over the origin. In a Cartesian frame the same parallel spinors have constant components.