Calculation and modular properties of multi-loop superstring amplitudes

TL;DR

This paper investigates the calculation of multi-loop superstring amplitudes using a gauge with Grassmann moduli in the 2D gravitino, revealing their symmetry properties and the non-modular nature of the measures for higher loops.

Contribution

It introduces a method for calculating multi-loop superstring amplitudes in a specific gauge and analyzes their symmetry and modular properties, especially for loops greater than three.

Findings

Amplitudes exhibit symmetry under modular and supersymmetry transformations.

For more than three loops, measures are not modular forms.

Amplitude expressions involve integrals over the fundamental region boundary.

Abstract

Multi-loop superstring amplitude are calculated in the convenient gauge where Grassmann moduli are carried by the 2D gravitino field. Generally, instead of the modular symmetry, the amplitudes hold the symmetry under modular transformations added by relevant transformations of the 2D local supersymmetry. If a number of loops is larger than 3, the integration measures are not modular forms. In this case the expression for the amplitude contains an integral over the bound of the fundamental region of the modular group.