o as to be a symmetry of the complete action, and then verify whether the constructed nonlinear transformation satisfies the supersymmetry algebra. We find that the supersymmetry algebra holds, up to the equations of motion and gauge transformation, only except for a nonlinear transformation. It is shown, however, that this extra transformation can also be absorbed into the gauge transformation up to the equations of motion at the linearized level. Under the assumption that the asymptotic condition holds also for the string field theory, this implies, at least perturbatively, that the constructed transformation acts as space-time supersymmetry on the physical states defined by the asymptotic string fields. This guarantees that supersymmetry is realized on the physical S-matrix.[^3]

The rest of the paper is organized as follows. In section 2, we summarize the known results on the complete action for the WZW-like open superstring field theory. In addition, restricting the background to the flat space-time, we introduce the GSO projection operator, which is essential to make the physical spectrum supersymmetric. For later use, some basic ingredients, such as the Maurer-Cartan equations and the covariant derivatives, are extended to those based on general derivations of the string product which can be noncommutative. After this preparation, the space-time supersymmetry transformation is constructed in section 3. Using the first-quantized supercharge, a linear transformation is first defined so as to be consistent with the restriction in the Ramond sector. Since this transformation is only a symmetry of the free theory, we first construct the nonlinear transformation perturbatively by requiring it to keep the complete action invariant. Based on some lower-order results, we suppose the full nonlinear transformation $\delta_{\mathcal{S}}$ in a closed form, and prove that it is actually a symmetry of the action. In section 4, the commutator of two transformations is calculated explicitly. We show that it provides the sp