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\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 Poincaré 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 ≥ 4 and D ≥ 6 respectively.