Generalized pure spinors

Abstract

With every non-zero spinor χ there is associated a totally isotropic subspace N(χ) of the underlying vector space W; the subspace N(χ) consists of all vectors annihilating the spinor. The dimension υ of N(χ)—the nullity of χ—is an invariant of the action of the Clifford group and provides a coarse classification of spinors. According to a terminology introduced by Cartan and Chevalley, a spinor is pure if the space N(χ) is maximal among totally isotropic subspaces of W. In this paper, we consider ‘partially pure’ spinors, i.e. Weyl (= semi-) spinors such that 0< υ< n, where 2 n is the dimension of the vector space W endowed with a neutral quadratic form. All homogeneous polynomial invariants are shown to vanish on Weyl spinors of positive nullity. We also show that there are no Weyl spinors of nullity υ suchD that n−4< υ< n or υ= n−5. We compute the dimensions of spaces of partially pure spinors and show that, for n=4 or >5, generic spinors have nullity 0. The paper contains also a heuristic introduction to the notion of pure spinors, comments about their geometrical and physical significance and remarks on the history of the subject.