Lorentzian Applications of Pure Spinors

Abstract

It was Cartan who-first observed that certain spinors associated with a metric g on a vector space V can be put into correspondence with maximal isotropic subspaces of V. Such spinors are called pure. Chevalley later gave a thorough treatment of pure spinors. In Cartan’s treatment the vector space was over the complex field. Chevalley allowed an arbitrary field but assumed the quadratic form to be of maximal index (i.e. the totally isotropic subspaces have the largest possible dimension). I will firstly show how the notion of pure spinors may be extended to the case of a real vector space whose metric has a signature with r + 2 (r) plus (minus) signs; the pure spinors being in correspondence with oriented isotropic (r + l)-planes. For r = 1 all spinors are pure. As an example of how this may be used I shall show how to find Lorentzian manifolds for which the Kahler equation admits spinorial solutions. Similarly all Lorentzian manifolds admitting parallel spinors may be found, giving possible background geometries for supersymmetry.