A Gravity, U(4)× U(4) Yang-Mills and Matter Unification in Clifford Spaces

A brief review of a Conformal Gravity and U(4)×U(4) Yang-Mills Unification model in four dimensions from a Clifford Gauge Field Theory in C-spaces (Clifford spaces) is presented. It based on the (complex) Clifford Cl(4, C) algebra underlying a complexified four dimensional spacetime (8 real dimensions). The 16 fermions of each generation can be arranged into the 16 entries of a 4 × 4 matrix associated with the A = 1, 2, 3, ....., 16 indices corresponding to the dimensions of the Cl(4) gauge algebra. The Higgs sector is also part of the Cl(4)-algebra polyvector valued gauge field in C-space. The Yukawa couplings which furnish masses to the fermions (after symmetry breaking) admit a C-space geometric interpretation as well.