Narrow Results

It is shown how a conformal gravity and U(4) × U(4) Yang–Mills grand unification model in four dimensions can be attained from a Clifford gauge field theory in C-spaces (Clifford spaces) based on the (complex) Clifford Cl(4, C) algebra underlying a complexified four-dimensional space–time (eight real dimensions). Upon taking a real slice, and after symmetry breaking, it leads to ordinary gravity and the Standard Model in four real dimensions. A brief conclusion about the noncommutative star-product deformations of this Grand Unified Theory of gravity with the other forces of Nature is presented.

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 , 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 , that gauge the nonabelian translations. Thus, in one-scoop, the vielbein 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.

A candidate action for an Exceptional E₈ gauge theory of gravity in 8D is constructed. It is obtained by recasting the E₈ group as the semi-direct product of GL(8,R) with a deformed Weyl–Heisenberg group associated with canonical-conjugate pairs of vectorial and antisymmetric tensorial generators of rank two and three. Other actions are proposed, like the quarticE₈ group-invariant action in 8D associated with the Chern–Simons E₈ gauge theory defined on the 7-dim boundary of a 8D bulk. To finalize, it is shown how the E₈ gauge theory of gravity can be embedded into a more general extended gravitational theory in Clifford spaces associated with the Cl(16) algebra and providing a solid geometrical program of a grand unification of gravity with Yang–Mills theories. The key question remains if this novel gravitational model based on gauging the E₈ group may still be renormalizable without spoiling unitarity at the quantum level.

A Moyal deformation of a Clifford $\mathrm{\text{Cl}}(3,1)$ Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant ${\mathrm{\Theta }}^{\mu \nu }$ parameters). In the very special case when one imposes certain constraints on the fields, there are $no$ first-order contributions in the ${\mathrm{\Theta }}^{\mu \nu }$ parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does $not$ impose constraints on the fields, there are first-order contributions in ${\mathrm{\Theta }}^{\mu \nu }$ to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of $U(2,2),\mathrm{\text{SO}}(4,2),\mathrm{\text{SO}}(2,3)$ can be expressed in terms of the Clifford algebra generators this does $not$ imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein–Hilbert gravitational action with a cosmological constant to first-order in ${\mathrm{\Theta }}^{\mu \nu }$. Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.