  We construct a manifestly SO(4) x SO(4) invariant, supersymmetric extension
of the closed string cubic interaction vertex and dynamical supercharges in
light-cone string field theory on the plane wave space-time. We find that the
effective vertex for states built out of bosonic creation oscillators coincides
with the one previously constructed in the SO(8) formalism and conjecture that
in general the two formulations are physically equivalent. Further evidence for
this claim is obtained from the discrete Z_2-symmetry of the plane wave and by
computing the mass-shift of the simplest stringy state using perturbation
theory. We verify that the leading non-planar correction to the anomalous
dimension of the dual gauge theory operators is correctly recovered.
