Spinors and the Weyl Tensor Classification in Six Dimensions
Abstract
A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.
Cite
@article{arxiv.1212.2689,
title = {Spinors and the Weyl Tensor Classification in Six Dimensions},
author = {Carlos Batista and Bruno Carneiro da Cunha},
journal= {arXiv preprint arXiv:1212.2689},
year = {2013}
}
Comments
23 pages; This version matches the published one