Pure spinors, intrinsic torsion and curvature in odd dimensions

Abstract

We study the geometric properties of a (2m+1)-dimensional complex manifold M admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m+1,C), the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m-plane distribution Nξ along which g is totally degenerate.We develop a spinor calculus, by means of which we encode the geometric properties of Nξ and of its rank-(m+1) orthogonal complement Nξ⊥ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.Applications to spinorial differential equations are given. Notably, we relate the integrability properties of Nξ and Nξ⊥ to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when (M,g) has prescribed curvature.We discuss applications of this work to the study of real pseudo-Riemannian manifolds.