  We study Kohn-Dirac operators $D_\theta$ on strictly pseudoconvex CR
manifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbb
Z}$. Certain components of $D_\theta$ are CR invariants. We also derive CR
invariant twistor operators of weight $\ell$. Harmonic spinors correspond to
cohomology classes of some twisted Kohn-Rossi complex. Applying a
Schr\"odinger-Lichnerowicz-type formula, we prove vanishing theorems for
harmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructions
to positive Webster curvature.
