Global conformal anomaly in $\mathcal{N}=2\mathrm{}$ string

We show the existence of a global anomaly in the one-loop graphs of $\mathcal{N}=2\mathrm{}$ string theory, defined by sewing tree amplitudes, unless spacetime supersymmetry is imposed. The anomaly is responsible for the non-vanishing maximally helicity violating amplitudes. The supersymmetric completion of the $\mathcal{N}=2\mathrm{}$ string spectrum is formulated by extending the previous cohomological analysis with an external spin factor; the target space-time spin-statistics of these individual fields in a self-dual background are compatible with previous cohomological analysis as fields of arbitrary spin may be bosonized into one another. We further analyze duality relations between the open and closed string amplitudes and demonstrate this in the supersymmetric extension of the target space-time theory through the insertion of zero-momentum operators.