Eisenstein Series and String Thresholds

TL;DR

This paper explores the use of Eisenstein series to represent string theory amplitudes that are invariant under duality groups, providing a unified framework for understanding perturbative and non-perturbative effects in M-theory compactifications.

Contribution

It constructs invariant modular functions generalizing Eisenstein series for various duality groups and relates their properties to string amplitude corrections, proposing a non-perturbative U-duality invariant extension.

Findings

Eisenstein series effectively encode string amplitude corrections.

The framework provides T-duality invariant representations of loop amplitudes.

Non-perturbative effects are analyzed through the properties of these series.

Abstract

We investigate the relevance of Eisenstein series for representing certain $G(Z)$-invariant string theory amplitudes which receive corrections from BPS states only. $G(Z)$ may stand for any of the mapping class, T-duality and U-duality groups $Sl(d,Z)$, $SO(d,d,Z)$ or $E_{d+1(d+1)}(Z)$ respectively. Using $G(Z)$-invariant mass formulae, we construct invariant modular functions on the symmetric space $K\backslash G(R)$ of non-compact type, with $K$ the maximal compact subgroup of $G(R)$, that generalize the standard non-holomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincar\'e upper half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one- and $g$-loop string amplitudes, we obtain a manifestly T-duality invariant representation of the latter, conjecture their non-perturbative U-duality invariant extension, and analyze the resulting non-perturbative effects. This includes the $R^4$ and $R^4 H^{4g-4}$ couplings in toroidal compactifications of M-theory to any dimension $D\geq 4$ and $D\geq 6$ respectively.