Torsion parallel spinors on Lorentzian four-manifolds and supersymmetric evolution flows on bundle gerbes

This dissertation is concerned with the geometric study of differential spinors on oriented and spin Lorentzian four-manifolds via the theory of spinorial polyforms. The main results and applications are directed towards the investigation of torsion parallel spinors and the globally hyperbolic evolution flow determined by the globally hyperbolic solutions of the four-dimensional supersymmetric NS-NS system. This differential system, which originates in supergravity and string theory, involves skew-torsion parallel spinors subject to a curvature condition and provides a natural gauge-theoretic interpretation of skew-symmetric torsion as the curvature of a connection on an abelian bundle gerbe - a natural categorification of the notion of principal circle bundle.