Geometric Aspects of Spinors

Abstract

Geometric properties of spinors are reviewed in connection with their role in complex and optical geometry. According to Cartan and Chevalley, a Weyl spinor φ ≠0,associated by the Dirac representation γ with a complex, 2m–dimensional vector space W, is called pure if the vector subspace N(φ) consisting of all elements ω of W such that γ(ω)φ = 0 is maximal, i.e. m-dimensional. If W is the complexification of a real space V with a scalar product of signature (2p+ε,2q+ε), where ε =0 or 1 and p+q+ε= m, then the real index of φ, γ = dim$N(\varphi ) \cap \overline {N(\varphi )}$ in the generic case equals γ. Therefore, the direction of a generic pure spinor defines in V a complex (ε = 0) or an optical (ε = 1) structure. These observations are applied to a smooth, orientable 2m-dimensional spin manifold M with a bundle of directions of generic pure spinors. A section of this bundle - if it exists - defines an almost complex or an almost optical geometry, depending on whether γ = 0 or 1. With such a section one associates a bundle N of maximal, totally null subspaces of the complexified tangent spaces toM. Denoting by Z the module of sections of the bundleN, one considers the integrability conditions[Z,Z] ⊂Z In the pseudo-Euclidean case (γ = 0), the condition is equivalent to the vanishing of the Nijenhuis tensor of the almost complex structure; in the Lorentzian, 4-dimensional case, it is related to the geodetic, shear-free properties of the trajectories of the real line bundle Re$(N \cap \overline N )$→M.