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

Abstract

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 [X1, X2, …, Xn] with the poly-vector-valued coordinates X123⋅⋅⋅n in noncommutative Clifford spaces given by [X1, X2, …, Xn] = n!X123⋅⋅⋅n. The large N limit of n-ary commutators of n hyper-matrices 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.