**Authors:** Carlos Castro

A Clifford Cl(5,C) Unified Gauge Field Theory of Conformal Gravity,
Maxwell and U(4)xU(4) Yang-Mills in 4D is rigorously presented extending
our results in prior work. The Cl(5,C) = Cl(4,C)⊕Cl(4,C) algebraic
structure of the Conformal Gravity, Maxwell and U(4)xU(4) Yang-Mills
unification program advanced in this work is that the group structure
given by the direct products U(2, 2)xU(4)xU(4) = [SU(2, 2)]_{spacetime}x
[U(1) x U(4) x U(4)]_{internal} is ultimately tied down to four-dimensions
and 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) x_{s} C^{4}, the realizations
of twistors and the N = 1 superconformal su(2, 2|1) algebra purely in
terms of Clifford algebras and their plausible role in Witten's formulation
of perturbative N = 4 super Yang-Mills theory in terms of twistor-string
variables.

**Comments:** 22 pages, submitted to Advances in Applied Clifford Algebras (AACA).

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[v1] 12 Jan 2011

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