Automorphic properties of low energy string amplitudes in various dimensions

TL;DR

This paper investigates the automorphic functions governing low-energy string amplitudes in various dimensions, revealing their structure through Laplace equations and Eisenstein series, and connecting string theory with supergravity divergences.

Contribution

It provides a detailed analysis of automorphic functions for string amplitudes up to D^6R^4 order across different dimensions, using consistency conditions and automorphic forms.

Findings

Automorphic functions satisfy Laplace eigenvalue equations with source terms.

Solutions involve Eisenstein series and automorphic functions for U-duality groups.

String theory amplitudes are finite despite supergravity divergences.

Abstract

This paper explores the moduli-dependent coefficients of higher derivative interactions that appear in the low-energy expansion of the four-graviton amplitude of maximally supersymmetric string theory compactified on a d-torus. These automorphic functions are determined for terms up to order D^6R^4 and various values of d by imposing a variety of consistency conditions. They satisfy Laplace eigenvalue equations with or without source terms, whose solutions are given in terms of Eisenstein series, or more general automorphic functions, for certain parabolic subgroups of the relevant U-duality groups. The ultraviolet divergences of the corresponding supergravity field theory limits are encoded in various logarithms, although the string theory expressions are finite. This analysis includes intriguing representations of SL(d) and SO(d,d) Eisenstein series in terms of toroidally compactified one and two-loop string and supergravity amplitudes.