Spinors and the Weyl Tensor Classification in Six Dimensions

TL;DR

This paper develops a spinorial framework for classifying the Weyl tensor in six-dimensional complexified spaces, revealing new geometric insights and connections to integrability and generalizations of the Goldberg-Sachs theorem.

Contribution

It introduces a novel spinorial classification of the Weyl tensor in six dimensions, leveraging SO(6) symmetry and linking tensor properties to isotropic subspace integrability.

Findings

Classification of Weyl tensor as a map from 3-vectors to 3-vectors

Connection between tensor classification and integrability of isotropic subspaces

Framework for generalizing the Goldberg-Sachs theorem 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.