Eisenstein Series in String Theory

TL;DR

This work develops a unified automorphic framework for perturbative and non-perturbative string amplitudes by employing generalized Eisenstein series of duality groups acting on symmetric spaces. It shows that one-loop and higher-genus string integrals can be expressed in terms of $SO(d,d,\mathbb{Z})$ Eisenstein series in appropriate representations, with exact non-perturbative completions given by $E_{d+1(d+1)}(\mathbb{Z})$ Eisenstein series in the string/particle/membrane representations, predicting D-brane effects and consistent with supersymmetry eigenvalue constraints. The approach reproduces known perturbative results and provides a principled path to exact non-perturbative couplings such as $f_{R^4}$ and related $R^4 H^{4g-4}$ and $R^4 F^{4g-4}$ terms across toroidal compactifications. Overall, the paper offers a rigorous link between BPS spectra, duality invariance, and automorphic forms to advance the understanding of non-perturbative string dynamics.

Abstract

We discuss the relevance of Eisenstein series for representing certain G(Z)-invariant string theory amplitudes which receive corrections from BPS states only. The Eisenstein series are constructed using G(Z)-invariant mass formulae and are manifestly invariant modular functions on the symmetric space K\G(R) of non-compact type, with K the maximal compact subgroup of G(R). In particular, we show how Eisenstein series of the T-duality group SO(d,d,Z) can be used to represent one- and g-loop amplitudes in compactified string theory. We also obtain their non-perturbative extensions in terms of the Eisenstein series of the U-duality group E_{d+1(d+1)}(Z).