Eisenstein Series in String Theory

TL;DR

This paper explores how Eisenstein series can be used to represent and analyze string theory amplitudes that are invariant under certain duality groups, incorporating both perturbative and non-perturbative effects.

Contribution

It demonstrates the construction of Eisenstein series for G(Z)-invariant string amplitudes and their application to T-duality and U-duality groups, including non-perturbative extensions.

Findings

Eisenstein series effectively represent one- and g-loop amplitudes.

The approach captures non-perturbative corrections via Eisenstein series of U-duality groups.

The constructed series are invariant modular functions on symmetric spaces.

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).