## Pure spinors, intrinsic torsion and curvature in even dimensions    [PDF]

Arman Taghavi-Chabert
We develop a spinor calculus on a $2m$-dimensional complex Riemannian manifold $(\mcM,g)$ equipped with a preferred projective pure spinor field. Such a projective spinor, $[\xi]$ say, defines a distribution $\mcN$ by totally null $m$-planes on $\mcM$, and the structure group of the frame bundle of $(\mcM,g)$ is reduced to $P$, the Lie parabolic subgroup of $\SO(2m,\C)$ stabilising $[\xi]$. This leads to a higher-dimensional notion of a principal spinor, and on this basis, we give an algebraic classification of curvature tensors of the Levi-Civita connection of $g$, which, for the Weyl tensor, generalises the Petrov-Penrose classification of the (anti-)self-dual Weyl tensor from four to higher even dimensions. In analogy to the Gray-Hervella classification of almost Hermitian manifolds, we classify the intrinsic torsion of the $P$-structure of the frame bundle in terms of irreducibles, thereby measuring the failure of the Levi-Civita connection to preserve $[\xi]$. The classification thus obtained encodes the geometric properties of $\mcN$, and also constitutes a complex Riemannian analogue of the notion of shearfree congruences of null geodesics in four-dimensial Lorentzian geometry. We then study the relation between spinorial differential equations, such as the twistor equation, on pure spinor fields and the geometric properties of their associated null distributions. In particular, we give necessary and sufficient conditions for the null distribution of a pure twistor-spinor to be integrable. We finally conjecture a refined version of the complex Goldberg-Sachs theorem in higher dimensions. Much of this work can be applied to the study of real smooth pseudo-Riemannian manifolds of signature $(m,m)$ equipped with a preferred projective real pure spinor field.
View original: http://arxiv.org/abs/1212.3595