On C $$\varvec{\otimes }$$ H $$\varvec{\otimes }$$ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory

Abstract

We review briefly how R $$\otimes $$ C $$\otimes $$ H $$\otimes $$ O-valued gravity (real-complex-quaterno-octonionic gravity) naturally can describe a grand unified field theory of Einstein’s gravity with a Yang–Mills theory containing the Standard Model group $$SU(3) \times SU(2) \times U(1)$$ . The algebra of left actions associated with the composite algebras involving the Division algebras, and the Sedenions $$\mathbf{S}$$ , and acting on themselves, all lead to complex Clifford algebras (complex matrix algebras). The complex Cl(16) algebra is the most appealing one since it is the one corresponding to the algebra of left actions of $$\mathbf{C} \otimes \mathbf{H} \otimes \mathbf{O} \otimes \mathbf{S}$$ acting on itself, and containing the $$\mathbf{e_8 \oplus e_8}$$ algebra of the anomaly free 10D Heterotic String. An analysis of $$ C \otimes H \otimes O$$ -valued gravity reveals that it bears a connection to Nonsymmetric Kaluza–Klein theories and complex Hermitian Matrix Geometry. The key behind these connections is in finding the relation between $$ C \otimes H \otimes O$$ -valued metrics in two complex dimensions with complex metrics in higher dimensional real manifolds ( $$ D = 32 $$ real dimensions in particular). It is desirable to extend these results to hypercomplex, quaternionic manifolds and Exceptional Jordan Matrix Models.