On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces

In this paper, poly-vector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative ( but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of the noncommuting spacetime coordinates [X-1, X-2,..., X-n] with the poly-vector-valued coordinates X-123...n in noncommutative Clifford spaces given by [X-1, X-2,..., X-n] = n!X-123...n. The large N limit of n- ary commutators of n hyper- matrices X-i1i2...in leads to Eguchi-Schild p-brane actions for p + 1 = n. A noncomutative n-ary . product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu- Poisson brackets.