Eisenstein series in string theory

We discuss the relevance of Eisenstein series for representing certain G (icons/Journals/Common/BbbZ" ALT="BbbZ" ALIGN="TOP"/> )-invariant string theory amplitudes which receive corrections from BPS states only. The Eisenstein series are constructed using G (icons/Journals/Common/BbbZ" ALT="BbbZ" ALIGN="TOP"/> )-invariant mass formulae and are manifestly invariant modular functions on the symmetric space K \ G (icons/Journals/Common/BbbR" ALT="BbbR" ALIGN="TOP"/> ) of non-compact type, with K the maximal compact subgroup of G (icons/Journals/Common/BbbR" ALT="BbbR" ALIGN="TOP"/> ). In particular, we show how Eisenstein series of the T-duality group SO (d ,d ,icons/Journals/Common/BbbZ" ALT="BbbZ" ALIGN="TOP"/> ) 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)} (icons/Journals/Common/BbbZ" ALT="BbbZ" ALIGN="TOP"/> ).