Two-loop superstrings V Gauge slice independence of the N-point function

A systematic construction of superstring scattering amplitudes for N massless NS bosons to two loop order is given, based on the projection of supermoduli space onto superperiod matrices used earlier for the superstring measure in the first four papers of this series. The one important new difficulty arising for the N-point amplitudes is the fact that the projection onto superperiod matrices introduces corrections to the chiral vertex operators for massless NS bosons which are not pure (1,0) differential forms. However, it is proved that the chiral amplitudes are closed differential forms, and transform by exact differentials on the worldsheet under changes of gauge slices. Holomorphic amplitudes and independence of left from right movers are recaptured after the extraction of terms which are Dolbeault exact in one insertion point, and de Rham closed in the remaining points. This allows a construction of GSO projected, integrated superstring scattering amplitudes which are independent of the choice of gauge slices and have only physical kinematical singularities.