THE EXCEPTIONAL E8 GEOMETRY OF CLIFFORD (16) SUPERSPACE AND CONFORMAL GRAVITY YANG–MILLS GRAND UNIFICATION

Abstract

We continue to study the Chern–Simons E8 Gauge theory of Gravity developed by the author which is a unified field theory (at the Planck scale) of a Lanczos–Lovelock Gravitational theory with a E8 Generalized Yang–Mills (GYM) field theory, and is defined in the 15D boundary of a 16D bulk space. The Exceptional E8 Geometry of the 256-dim slice of the 256 × 256-dimensional flat Clifford (16) space is explicitly constructed based on a spin connection [Formula: see text], that gauges the generalized Lorentz transformations in the tangent space of the 256-dim curved slice, and the 256 × 256 components of the vielbein field [Formula: see text], that gauge the nonabelian translations. Thus, in one-scoop, the vielbein [Formula: see text] encodes all of the 248 (nonabelian) E8 generators and 8 additional (abelian) translations associated with the vectorial parts of the generators of the diagonal subalgebra [Cl(8) ⊗ Cl(8)] diag ⊂ Cl(16). The generalized curvature, Ricci tensor, Ricci scalar, torsion, torsion vector and the Einstein–Hilbert–Cartan action is constructed. A preliminary analysis of how to construct a Clifford Superspace (that is far richer than ordinary superspace) based on orthogonal and symplectic Clifford algebras is presented. Finally, it is shown how an E8 ordinary Yang–Mills in 8D, after a sequence of symmetry breaking processes E8 → E7 → E6 → SO(8, 2), and performing a Kaluza–Klein–Batakis compactification on CP2, involving a nontrivial torsion, leads to a (Conformal) Gravity and Yang–Mills theory based on the Standard Model in 4D. The conclusion is devoted to explaining how Conformal (super) Gravity and (super) Yang–Mills theory in any dimension can be embedded into a (super) Clifford-algebra-valued gauge field theory.