One-Loop BPS amplitudes as BPS-state sums

Understand

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 \int_F \Gamma_{d+k,d} {\Phi}, with {\Phi} being a weak (almost) holomorphic modular form of weight -k/2.
• We use the fact that any such {\Phi} can be expressed as a linear combination of certain absolutely convergent Poincar\'e 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.