Differential spinors and Kundt three-manifolds with skew-torsion

Abstract

We develop the theory of spinorial polyforms associated with bundles of irreducible Clifford modules of non-simple real type, obtaining a precise characterization of the square of an irreducible real spinor in signature $(p-q) = 1\,\mathrm{mod(8)}$ as a polyform belonging to a semi-algebraic real set. We use this formalism to study differential spinors on Lorentzian three-manifolds, proving that in this dimension and signature, every differential spinor is equivalent to an isotropic line preserved in a given direction by a metric connection with prescribed torsion. We apply this result to investigate Lorentzian three-manifolds equipped with a skew-torsion parallel spinor, namely a spinor parallel with respect to a metric connection with totally skew-symmetric torsion. We obtain several structural results about this class of Lorentzian three-manifolds, which are necessarily Kundt, and in the compact case, we obtain an explicit differential condition that guarantees geodesic completeness. We further elaborate on these results to study the supersymmetric solutions of three-dimensional NS-NS supergravity, which involve skew-torsion parallel spinors whose torsion is given by the curvature of a curving on an abelian bundle gerbe. In particular, we obtain a correspondence between NS-NS supersymmetric solutions and null coframes satisfying an explicit exterior differential system that we solve locally.