One-loop BPS amplitudes as BPS-state sums

Recently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure to all BPS-saturated amplitudes of the form ∫FΓd+k,dΦ, with Φ being a weak (almost) holomorphic modular form of weight − k/2. We use the fact that any such Φ can be expressed as a linear combination of certain absolutely convergent Poincaré series, against which the fundamental domain F can be unfolded. The resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge symmetry enhancement, in a chamber-independent fashion. We illustrate our method with concrete examples of interest in heterotic string compactifications.