(Anti) de Sitter Geometry, Complex Conformal Gravity-Maxwell Theory from a Cl(4, C) Gauge Theory of Gravity and Grand Unification

Abstract

We present the deep connections among (Anti) de Sitter geometry, and complex conformal gravity-Maxwell theory, stemming directly from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). This is attained by simply promoting the de (Anti) Sitter algebras so(4, 1), so(3, 2) to the real Clifford algebras Cl(4, 1, R), Cl(3, 2, R), respectively. This interplay between gauge theories of gravity based on Cl(4, 1, R), Cl(3, 2, R) , whose bivector-generators encode the de (Anti) Sitter algebras so(4, 1), so(3, 2), respectively, and 4D conformal gravity based on Cl(3, 1, R) is reminiscent of the \(AdS_{ D+1}/CFT_D\) correspondence between \(D+1\)-dim gravity in the bulk and conformal field theory in the D-dim boundary. Although a plausible cancellation mechanism of the cosmological constant terms appearing in the real-valued curvature components associated with complex conformal gravity is possible, it does not occur simultaneously in the imaginary curvature components. Nevertheless, by including a Lagrange multiplier term in the action, it is still plausible that one might be able to find a restricted set of on-shell field configurations leading to a cancellation of the cosmological constant in curvature-squared actions due to the coupling among the real and imaginary components of the vierbein. We finalize with a brief discussion related to \(U(4) \times U(4)\) grand-unification models with gravity based on \( Cl (5, C) = Cl(4,C) \oplus Cl(4,C)\). It is plausible that these grand-unification models could also be traded for models based on \( GL (4, C) \times GL(4, C) \).