  We calculate the genus-one three- and four-point amplitudes in the 2+2
dimensional closed N=(2,2) worldsheet supersymmetric string within the RNS
formulation. Vertex operators are redefined with the incorporation of spinor
helicity techniques, and the quantum scattering is shown to be manifestly gauge
and Lorentz invariant after normalizing the string states. The continuous spin
structure summation over the monodromies of the worldsheet fermions is carried
out explicitly, and the field-theory limit is extracted. The amplitude in this
limit is shown to be the maximally helicity violating amplitude in pure gravity
evaluated in a two-dimensional setting, which vanishes, unlike the
four-dimensional result. The vanishing of the genus-one N=2 closed string
amplitude is related to the absence of one-loop divergences in dimensionally
regulated IIB supergravity. Comparisons and contrasts between self-dual field
theory and the N=2 string theory are made at the quantum level; they have
different S-matrices. Finally, we point to further relations with self-dual
field theory and two-dimensional models.
