$D^6R^4$ curvature corrections, modular graph functions and Poincar\'e series

TL;DR

This paper develops a new method using Poincaré series to solve for U-duality invariant functions related to higher curvature corrections in string theory, providing new results across various dimensions.

Contribution

It introduces a novel approach to solving inhomogeneous Laplace equations for curvature correction functions using Poincaré series, extending known results and applying to modular graph functions.

Findings

Recovered known results in 10 dimensions

Derived new results in lower dimensions

Applied method to modular graph functions

Abstract

In this note we study the U-duality invariant coefficient functions of higher curvature corrections to the four-graviton scattering amplitude in type IIB string theory compactified on a torus. The main focus is on the $D^6R^4$ term that is known to satisfy an inhomogeneous Laplace equation. We exhibit a novel method for solving this equation in terms of a Poincar\'e series ansatz and recover known results in $D=10$ dimensions and find new results in $D<10$ dimensions. We also apply the method to modular graph functions as they arise from closed superstring one-loop amplitudes.