Eisenstein Series and String Thresholds

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 $K\mathrm{\backslash }G(R)$ 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 $D\ge 4$ and $D\ge 6$ respectively.