The zero set of a twistor spinor in any metric signature

Abstract

Using tractor methods, we exhibit the local structure of the zero set of a twistor spinor in any metric signature. It is given as the image under the exponential map of a distinguished totally lightlike subspace. Based on these results we are able to construct starting from a conformal structure admitting a twistor spinor with zero a projective structure on its zero set in a natural way. The construction is presented both explicitly, using a fixed metric in the conformal class, and more abstractly on the level of associated parabolic Cartan geometries. Finally, we use known results about pseudo-Riemannian extensions of 2-dimensional projective structures to \((2,2)\)-ASD conformal structures to find first examples of non-conformally flat, non Riemannian manifolds admitting a twistor spinor with nontrivial zero set.