Alpha-surfaces for complex space-times with torsion

This paper studies necessary conditions for the existence of alpha-surfaces in complex space-time manifolds with non-vanishing torsion. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for alpha-surfaces which does not involve just the self-dual Weyl spinor, as in complexified general relativity, but also the torsion spinor, in a non-linear way, and its covariant derivative. Interestingly, a particular solution of the integrability condition is given by right-flat and right-torsion-free space-times.