Eisenstein series for higher-rank groups and string theory amplitudes

TL;DR

The paper connects the low energy expansion of maximally supersymmetric string amplitudes to Langlands Eisenstein series for higher rank duality groups $E_{d+1}$, showing that the first two analytic coefficients in the $R^4$ and $\partial^4 R^4$ terms are given by maximal parabolic Eisenstein series with parameters $s=3/2$ and $s=5/2$. By leveraging degeneration limits—decompactification, string perturbation, and M theory limits—the authors derive precise boundary conditions and show these coefficients satisfy appropriate Laplace eigenvalue equations, consistently matching perturbative and semiclassical supergravity data. The $E_8$ case is pushed further by solving the inhomogeneous equation for the $\partial^6 R^4$ coefficient, whose constant terms encode the entire tower of lower rank coefficients, thereby unifying the hierarchy of duality groups through a single analytic framework. Together, these results establish a robust automorphic-form description of key string theory amplitudes and hint at extensions to higher derivative orders and broader duality settings.

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.

Figures (1)

• Figure 1: The Dynkin diagrams of the U-duality groups real split form of rank $d+1$ Lie group $E_{d+1}$ ($0\leq d\leq 7$)

Theorems & Definitions (5)

• Definition 2.1
• Definition 2.5
• Theorem 2.7
• Definition 2.14
• Theorem 2.18