Eisenstein series for higher-rank groups and string theory amplitudes

TL;DR

This paper explores how Eisenstein series and automorphic functions constrain superstring scattering amplitudes, specifically analyzing higher derivative corrections and their automorphic properties within the E_n series.

Contribution

It explicitly determines the first two higher derivative interactions using Eisenstein series and analyzes the automorphic properties of the third term in the E_8 case.

Findings

Derived expressions for BPS-protected higher derivative interactions.

Identified automorphic functions satisfying inhomogeneous Laplace equations.

Analyzed constant terms in parabolic subgroups for amplitude coefficients.

Abstract

Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, E_n(Z), of simply-laced Lie groups in the E_n series (n<= 8). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein's theory, with coefficients that are automorphic functions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the E_n series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansion of the four-supergraviton amplitude (which is also BPS-protected) in the E_8 case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms.