Manifest calculation and the finiteness of the superstring Feynman diagrams

TL;DR

This paper develops a finite, symmetry-preserving method for calculating multi-loop superstring amplitudes, demonstrating their finiteness and the nullification of certain massless state amplitudes.

Contribution

It introduces a consistent integration region respecting local symmetries and unitarity, and proves the finiteness and nullification of specific superstring amplitudes.

Findings

Amplitudes are finite and divergence-free.

Zero amplitudes for massless states are confirmed.

Amplitudes for longitudinal gauge bosons vanish.

Abstract

The multi-loop amplitudes for the closed, oriented superstring are represented by finite dimensional integrals of explicit functions calculated through the super-Schottky group parameters and interaction vertex coordinates on the supermanifold. The integration region is proposed to be consistent with the group of the local symmetries of the amplitude and with the unitarity equations. It is shown that, besides the SL(2) group, super-Schottky group and modular one, the total group of the local symmetries includes an isomorphism between sets of the forming group transformations, the period matrix to be the same. The singular integration configurations are studied. The calculation of the integrals over the above configurations is developed preserving all the local symmetries of the amplitude, the amplitudes being free from divergences. The nullification of the 0-, 1-, 2- and 3-point amplitudes of massless states is verified. Vanishing the amplitudes for a longitudinal gauge boson is argued.