Spinor Fields Classification in Arbitrary Dimensions and New Classes of Spinor Fields on 7-Manifolds

TL;DR

This paper generalizes the classification of spinor fields based on bilinear covariants to arbitrary dimensions, with a focus on 7-manifolds, revealing new classes of spinors with potential applications in supergravity compactifications.

Contribution

It extends Lounesto's 4D spinor classification to higher dimensions and analyzes specific classes of spinors on 7-manifolds, introducing new potential physical applications.

Findings

Classified spinor fields in arbitrary dimensions using bilinear covariants.

Identified one class of Majorana spinors in 7D Riemannian manifolds.

Found four classes of general spinors in 7D, expanding the known taxonomy.

Abstract

A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto's 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null -- with the exception of the current density which is invariably different from zero for physical reasons -- and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields. To obtain an analogous classification in higher dimensions we use the Fierz identities, which constrain the covariant bilinears in the spinor fields and force some of them to vanish. A generalized graded Fierz aggregate is moreover obtained in such a context simply from the completeness relation. We analyze the particular and important case of Riemannian 7-manifolds, where the Majorana spinor fields turn out to have a quite special place. In particular, at variance with spinor fields in 4D Minkowski spacetime that are classified in six disjoint classes, spinors in Riemannian 7-manifolds are shown to be classified, according to the bilinear covariants: (a) in just one class, in the real case of Majorana spinors; (b) in four classes, in the most general case. Much like new classes of spinor fields in 4D Minkowski spacetime have been evincing new possibilities in physics, we think these new classes of spinor fields in seven dimensions are, in particular, potential candidates for new solutions in the compactification of supergravity on a seven-dimensional manifold and its exotic versions.