Supersymmetric Lorentz invariant deformations of superspaces

Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor ${\lambda }_a$ and Ramond grassmannian vector ${\psi }_m=-\frac{1}{2}(\bar{\theta }{\gamma }_m\lambda )$ in the spinor realization \cite{VZ} are proposed. The map of supergravity background into composite supercoordinates: $(B^{-1}{mn}, \Psi^{a}{m}, C_{ab}) \leftrightarrow (i\psi_{m}\psi_{n}, \psi_{m}\lambda^{a}, \lambda_{a}\lambda_{b})$ valid up to the second order corrections in deformation parameter $h$ and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector ${\psi }_{1m}=1\mathrm{/}2(\bar{\theta }{\gamma }_m{\gamma }_5\lambda )$ and to vanish for the $\theta$ components with the same chiralities. The deformations in the (super)twistor picture are discussed.