## The Exceptional E_{8} Geometry of Clifford (16) Superspace and Conformal Gravity Yang-Mills Grand Unification

**Authors:** Carlos Castro

We continue to study the Chern-Simons E_{8} 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 E_{8} Generalized Yang-Mills (GYM) field theory, and is defined
in the 15D boundary of a 16D bulk space. The Exceptional E_{8} Geometry of the 256-dim
slice of the 256 X 256-dimensional flat Clifford (16) space is explicitly constructed
based on a spin connection Ω_{M}^{AB}, that gauges the generalized Lorentz transformations
in the tangent space of the 256-dim curved slice, and the 256 X 256 components of the
vielbein field E_{M}^{A}, that gauge the nonabelian translations. Thus, in one-scoop, the vielbein
E_{M}^{A}
encodes all of the 248 (nonabelian) E_{8} 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 E_{8} ordinary Yang-Mills in 8D, after a sequence of symmetry
breaking processes E_{8} → E_{7} → E_{6} → SO(8, 2), and performing a Kaluza-Klein-Batakis
compactification on CP^{2}, 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.

**Comments:** 33 pages, This article appeared in the IJGMMP vol 6, no. 3 (2009) 385.

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### Submission history

[v1] 22 Aug 2009

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