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We study Kohn-Dirac operators $D_\theta$ on strictly pseudoconvex CRmanifolds with ${\rm spin}^{\mathbb C}$ structure of weight $\ell\in{\mathbbZ}$. Certain components of $D_\theta$ are CR invariants. We also derive CRinvariant twistor operators of weight $\ell$. Harmonic spinors correspond tocohomology classes of some twisted Kohn-Rossi complex. Applying aSchr\"odinger-Lichnerowicz-type formula, we prove vanishing theorems forharmonic spinors and (twisted) Kohn-Rossi groups. We also derive obstructionsto positive Webster curvature.

Invariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry