Pure spinors, intrinsic torsion and curvature in odd dimensions

Abstract

We study the geometric properties of a $(2m+1)$-dimensional complex manifold $\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group $P\subset \mathrm{Spin}(2m+1,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $\mathcal{M}$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $\xi$ up to scale. The defining property of $\xi$ is that it determines an almost null structure, i.e.\ an $m$-plane distribution ${\mathcal{N}}_{\xi }$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of ${\mathcal{N}}_{\xi }$ and of its rank-$(m+1)$ orthogonal complement ${\mathcal{N}}_{\xi }^{\perp }$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $\nabla$ of $g$ to be compatible with the $P$-structure. In a similar way, we examine the algebraic properties of the curvature of $\nabla$. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of ${\mathcal{N}}_{\xi }$ and ${\mathcal{N}}_{\xi }^{\perp }$ 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 $(\mathcal{M},g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Odd-dimensional version of arXiv:1212.3595 v2: Presentation improved. A number of corrections made: diagrams describing the curvature and intrinsic torsion classification; Geometric interpretation of spinorial equations; some errors in formulae now fixed. Some material regarding parallel spinors removed (to be including in a separate article) v3: as published

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Cited by in corpus (7)

• Geometry and symmetries of null G-structures
• On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties
• Twistor Geometry of Null Foliations in Complex Euclidean Space
• A Projective-to-Conformal Fefferman-Type Construction
• Almost Robinson geometries
• Conformal geometry and half-integrable spacetimes
• Summary of session A4: Complex and conformal methods in classical and quantum gravity