One-Loop BPS amplitudes as BPS-state sums

TL;DR

This paper extends a novel method for computing one-loop BPS-saturated amplitudes in String Theory, expressing them as sums over BPS states to reveal gauge symmetry enhancement singularities.

Contribution

It generalizes a previous procedure to all BPS-saturated amplitudes involving weak holomorphic modular forms, using Poincaré series for manifest T-duality invariance.

Findings

The method explicitly displays singularities at gauge enhancement points.

It provides a chamber-independent BPS-state sum representation.

Applications are demonstrated in heterotic string compactifications.

Abstract

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. We illustrate our method with concrete examples of interest in heterotic string compactifications.