Α-surfaces for Complex Space-times with Torsion

This paper studies necessary conditions for the existence of α-surfaces in complex space-time manifolds with nonvanishing 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 α-surfaces which does not involve just the self-dual Weyl spinor, as in complexified general relativity, but also the torsion spinor, in a nonlinear way, and its covariant derivative. Interestingly, a particular solution of the integrability condition is given by right-flat and right-torsion-free space-times. PACS 04.20.Cv Fundamental problems and general formalism. PACS 04.50 Unified field theories and other theories of gravitation.