ar X iv : h ep - t h / 05 06 12 7 v 3 1 9 Ju n 20 05 Lorentz invariant supersymmetric mechanism for non ( anti ) commutative deformations of space - time geometry

A supersymmetric Lorentz invariant mechanism for superspace deformations is proposed. It is based on an extension of superspace by one λ a or several Majorana spinors associated with the Penrose twistor picture. Some examples of Lorentz invariant su-persymmetric Poisson and Moyal brackets are constructed and the correspondence: θ mn ↔ iψ m ψ n , C ab ↔ λ a λ b , Ψ a m ↔ ψ m λ a mapping the brackets depending on the constant background into the Lorentz covariant supersymmetric brackets is established. The correspondence reveals the role of the composite anticommuting vector ψ m = − 1 2 (¯ θγ m λ) as a covariant measure of space-time coordinate noncommutativity.