Lorentz invariant supersymmetric mechanism for non(anti)commutative deformations of space-time geometry

Abstract

A supersymmetric Lorentz invariant mechanism for superspace deformations is proposed. It is based on an extension of superspace by one $\lambda_{a}$ or several Majorana spinors associated with the Penrose twistor picture. Some examples of Lorentz invariant supersymmetric Poisson and Mojal brackets are constructed and the correspondence: $\theta_{mn}\leftrightarrow i\psi_{m}\psi_{n},\quad C_{ab}\leftrightarrow \lambda_{a}\lambda_{b},\quad \Psi^{a}_{m}\leftrightarrow \psi_{m}\lambda^{a}$ 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 $\psi_{m}=-{1\over 2}(\bar\theta\gamma_{m}\lambda)$ as a covariant measure of space-time coordinate noncommutativity.