A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) × U(4) Yang-Mills in 4D

Abstract

A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) × U(4) Yang-Mills in 4D is rigorously presented extending our results in prior work. The \({Cl(5, C) = Cl(4, C) \oplus Cl(4, C)}\) algebraic structure, behind the 4D (complexified) Conformal Gravity-Maxwell and U(4) × U(4) Yang-Mills unification program advanced in this work, is such that it encodes the direct group product U(2, 2) × U(4) × U(4) = [SU(2, 2)] spacetime × [U(1) × U(4) × U(4)] internal and which does not violate the Coleman-Mandula theorem because the spacetime symmetries (conformal group SU(2, 2) in the absence of a mass gap, Poincare group when there is mass gap) do not mix with the internal symmetries. Similar considerations apply to the supersymmetric case when the symmetry group structure is given by the direct product of the superconformal group (in the absence of a mass gap) with an internal symmetry group so that the Haag-Lopuszanski-Sohnius theorem is not violated. A generalization of the de Sitter and Anti de Sitter gravitational theories based on the gauging of the Cl(4, 1, R), Cl(3, 2, R) algebras follows. We conclude with a few remarks about the complex extensions of the Metric Affine theories of Gravity (MAG) based on GL(4, C) × s C 4, the realizations of twistors and the \({\mathcal{N}}\) = 1 superconformal su(2, 2|1) algebra purely in terms of Clifford algebras and their plausible role in Witten’s formulation of perturbative \({\mathcal{N}}\) = 4 super Yang- Mills theory in terms of twistor-string variables.