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 D6R4 term that is known to satisfy an inhomogeneous Laplace equation. We exhibit a novel method for solving this equation in terms of a Poincaré 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.
Highlights
Where we have suppressed an overall dimensionful factor
We have outlined a method for solving inhomogeneous automorphic differential equations of the type that appear in string theory in several places
This method relied on making an ansatz for the solution of a Poincare sum form as in (2.7). The advantage of this method is that the resulting differential equation for the seed σ of the Poincare sum is less involved than the original equation and is solved in a Fourier expansion
Summary
The homogeneous Laplace equations in (1.3) for E((0D,0)) and E((1D,0)) can be solved in terms of (combinations of) Eisenstein series E(λ, g) on the symmetric space E11−D/K(E11−D). Such Eisenstein series are invariant under E11−D(Z), can be parametrised by a (complex). As the k-component of g does not enter in the above definition, the Eisenstein series (2.2) descends to a function on the symmetric space E11−D/K(E11−D).
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