Eisenstein Series and String Thresholds

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(...

Eisenstein Series in String Theory

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 noncompact type, with K the maximal compact subgroup of G(R). In particular, we show ...

Eisenstein Series*

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...

Eisenstein Series and Their Applications

The simplest Eisenstein series

We explain some essential aspects of the simplest Eisenstein series for SL2(Z) on the upper half-plane H. There are many different proofs of meromorphic continuation and functional equation of the simplest Eisenstein series for Γ = SL2(Z). We will follow [Godement 1966a] rewriting of a Poisson summation argument that appeared in [Rankin 1939], if not earlier. This argument is the most elementar...