Spinor fields classification in arbitrary dimensions and new classes of spinor fields on 7-manifolds
- 242 Downloads
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.
KeywordsDifferential and Algebraic Geometry M-Theory
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- P. Lounesto, Clifford algebras and spinors, 2nd edition, Cambridge Univeristy Press, Cambridge U.K. (2002).Google Scholar
- B. Biran, F. Englert, B. de Wit and H. Nicolai, Gauged N = 8 supergravity and its breaking from spontaneous compactification, Phys. Lett. B 124 (1983) 45 [Erratum ibid. B 128 (1983) 461] [INSPIRE].
- I. Benn and R. Tucker, An introduction to spinors and geometry with applications in physics, Adam Hilger, Bristol U.K. (1987).Google Scholar
- D.V. Alekseevsky, V. Cortés, Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p, q), math/9511215.