Calculation of multi-loop superstring amplitudes

TL;DR

This paper develops a method to compute multi-loop superstring amplitudes by integrating local amplitudes expressed via super-Schottky parameters, ensuring symmetry preservation and divergence-free results.

Contribution

It provides explicit formulas for local superstring amplitudes using super-Schottky group parameters and addresses the ambiguity in integration while maintaining superstring symmetries.

Findings

Amplitudes are free from divergences.

Vacuum and low-point amplitudes vanish after integration.

Results are consistent with world-sheet symmetries.

Abstract

Multi-loop interaction amplitudes in the theory of the closed, oriented superstrings are obtained by the integration of local amplitudes which are represented by a sum of the spinning string local amplitudes. The last local amplitudes are given explicitly through super-Schottky group parameters and interaction vertex coordinates on the $(1|1)$ complex supermanifold. The integration is ambiguous under those replacements of the integration variables which admix Grassmann variables to the boson ones. So the calculation is guided by a preservation of local symmetries of the superstring. The obtained amplitudes are free from divergences and consistent with the world-sheet symmetries. The vacuum amplitude and 1-, 2- and 3-point amplitudes of massless states vanish once the integration over certin modular variables and interaction vertex coordinates.