Harmonic spinors of the Dirac operator of connection with torsion in dimension four

It is shown that the existence of a parallel (-)-spinor with respect to a m etric connection with totally skew-symmetric torsion requires more local re strictions than the existence of a parallel (+)-spinor. It is proved that e very harmonic spinor with respect to the Dirac operator of this connection on a compact four-dimensional spin Riemannian manifold is parallel with res pect to a naturally arising metric connection with totally skew-symmetric t orsion and all such spaces are classified up to a conformal transformation.