 Polar-frame parallel spinor

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.