Abstract
The geometric Fierz identities are here employed to generate new emergent fermionic fields on the parallelizable (curvatureless, torsionfull) 7-sphere ($S^7$). Employing recently found new classes of spinor fields on the $S^7$ spin bundle, new classes of fermionic fields are obtained from their bilinear covariants by a generalized reconstruction theorem, on the parallelizable $S^7$. Using a generalized non-associative product on the octonionic bundle on the parallelizable $S^7$, these new classes of algebraic spinor fields, lifted onto the parallelizable $S^7$, are shown to correctly transform under the Moufang loop generators on $S^7$.
Highlights
INTRODUCTION(Classical) spinor fields are well known to be elements in the carrier space of the Spin group irreducible representations on any given spacetime that admits a spin structure, namely, if the second Stiefel–Whitney class vanishes
The bilinear covariants were shown to be the homogeneous part of a multivector Fierz aggregate [1]
II, after briefly reviewing how the geometric Fierz identities are used to derive additional spinor field classes on S7, we propose a reconstruction procedure for obtaining the spinor fields, in these new classes, from the bilinear covariants and the geometric Fierz identities
Summary
(Classical) spinor fields are well known to be elements in the carrier space of the Spin group irreducible representations on any given spacetime that admits a spin structure, namely, if the second Stiefel–Whitney class vanishes. [8] encompassing spinor multiplets as realizations of (non-Abelian) gauge fields In this more general classification, composed flagpoles, dipoles, and flag-dipoles naturally descend within fourteen disjoint classes of spinor fields, under the gauge symmetry SU(2) × U(1). Despite the natural obstructions due to the existence of algebraic and geometric structures on a given spacetime dimension/signature, the Lounesto’s spinor field classification on 4D Minkowski spacetime was successfully generalized to other spacetime dimensions and signatures, of relevance in their applications, as the emergence of fermionic fields in the respective spacetime compactifications. [1] proposed new classes of spinor fields on S7, based on the geometric Fierz identities in Ref. The geometric Fierz identities are used to derive the spinor field classes on S7, that are going to be lifted onto the parallelizable S7, whereon new fermionic fields can be constructed through the introduction of a generalized octonionic law of transformation
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