Cancellation of divergences in the superstring perturbation theory

Abstract The calculation of the multi-loop superstring amplitudes is developed using the explicit partition functions and the explicit vacuum correlators given in terms of the super-Schottky modular parameters. The multi-loop superstring amplitude is presented by the integrals over super-Schottky parameters that include the Grassmann integrations in addition to the ordinary ones. The integrals are ill defined due to the partition function becomes infinite when the supermanifold is degenerated in a few lower genus supermanifolds. For the calculation of the integrals the regularization procedure is proposed preserving the super-modular group and SL(2)-symmetry as well. In this case the divergences due to a dividing of the genus-n > 1 supermanifold in a few the lower genus supermanifolds are found to disappear since the Grassmann integrations are performed. It is similar to QED, as example, where the photon mass disappears, if a gauge invariant regularization scheme is used. The divergences due to a degenera...