Unimodular transformations of the supermanifolds and the calculation of the multi-loop amplitudes in the superstring theory

TL;DR

This paper investigates the supermodular transformations of supermanifolds in superstring theory, explicitly calculates their dependence on spinor structures, and demonstrates the covariance of multi-loop superstring partition functions under these transformations.

Contribution

It provides explicit formulas for supermodular transformations depending on spinor structures and proves the covariance of superstring partition functions under these transformations.

Findings

Supermodular transformations depend on spinor structures and odd modular parameters.

Partition functions are covariant under supermodular transformations.

Explicit calculations aid in addressing divergence issues in superstring theory.

Abstract

The modular transformations of the $(1|1)$ complex supermanifolds in the like-Schottky modular parameterization are discussed. It is shown that these "supermodular" transformations depend on the spinor structure of the supermanifold by terms proportional to the odd modular parameters. The above terms are calculated in the explicit form. They are urgent for the divergency problem in the Ramond-Neveu-Schwarz superstring theory and for calculating the fundamental domain in the modular space. The supermodular transformations of the multi-loop superstring partition functions calculated by the solution of the Ward identities are studied. The above Ward identities are shown to be covariant under the supermodular transformations. So the partition functions necessarily possess the covariance under the transformations discussed. It is demonstrated explicitly the covariance of the above partition functions at zero odd moduli under those supermodular transformations, which turn a pair of even genus-1 spinor structures to a pair of the odd genus-1 spinor ones. The brief consideration of the cancellation of divergences is given.