Genus one super-Green function revisited and superstring amplitudes with non-maximal supersymmetry

We reexamine genus one super-Green functions with general boundary conditions twisted by $(\alpha ,\beta )$ for $(\sigma ,\tau )$ directions in the eigenmode expansion and derive expressions as infinite series of hypergeometric functions. Using these, we compute one-loop superstring amplitudes with non-maximal supersymmetry, taking an example of massless vector emissions of open string type I ${\mathcal{Z}}_2$ orbifold.