One-loop Amplitudes as BPS state sums

We review a novel method for evaluating one-loop BPS-saturated amplitudes in string theory. Contrary to traditional techniques of unfolding the fundamental domain F against the Narain lattice, which are only valid in certain regions of the moduli space and which obscure the T- duality invariance of the result, we will describe how the elliptic genus can be represented as a linear combination of certain absolutely convergent Poincar\'e series, against which F can be unfolded. The result can be expressed as a sum of one-loop contributions of perturbative BPS-states in a manifestly T-duality invariant fashion, valid at any point of (the perturbative) moduli space. Within this framework, the singularity structure of amplitudes around points of gauge symmetry enhancement becomes crystal clear and a series of applications is given in order to better illustrate the power of this approach.