Some Aspects of Deformations of Supersymmetric Field Theories

We investigate some aspects of Moyal-Weyl deformations of superspace andtheir compatibility with supersymmetry. For the simplest case, when only bosonic coordinates are deformed, we consider a four dimensional supersymmetric field theory which is the deformation of the Wess-Zumino renormalizable theory of a chiral superfield. We then consider the deformation of a free theory of an abelian vector multiplet, which is a noncommutative version of the rank oneYang-Mills theory. We finally give the supersymmetric version of the$\alpha'\mapsto 0$ limit of the Born-Infeld action with a B-field turned on, which is believed to be related to the noncommutative U(1) gauge theory.