On n-ary Algebras, Branes and Polyvector Gauge Theories in Noncommutative Clifford Spaces

Polyvector-valued gauge field theories in noncommutative Clifford spaces are presented. The noncommutative star products are associative and require the use of the Baker-Campbell-Hausdorff formula. Actions for pbranes in noncommutative (Clifford) spaces and noncommutative phase spaces are provided. An important relationship among the n-ary commutators of noncommuting spacetime coordinates [X, X, ......, X] with the poly-vector valued coordinates X in noncommutative Clifford spaces is explicitly derived [X, X, ......, X] = n! X. The large N limit of n-ary commutators of n hyper-matrices Xi1i2....in leads to Eguchi-Schild p-brane actions for p + 1 = n. Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Cliffordspaces would require quantum Hopf algebraic deformations of Clifford algebras. 1 Polyvector Gauge Field Theories in Noncommutative Clifford Spaces Clifford algebras are deeply related and essential tools in many aspects in Physics. The Extended Relativity theory in Clifford-spaces ( C-spaces ) is a natural extension of the ordinary Relativity theory [3] whose generalized polyvectorvalued coordinates are Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in D-dimensional target spacetime backgrounds.