Abstract
AbstractRecently, 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 toallBPS-saturated amplitudes of the form ∫FΓd+k,dΦ, withΦbeing a weak (almost) holomorphic modular form of weight − k/2. We use the fact that any suchΦcan 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.
Highlights
The moduli space of conformal metrics on the torus is the Poincare upper half plane H, parameterised by the complex structure parameter τ = τ1 + iτ2, modulo the action of the modular group SL(2, Z)
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
We extend this procedure to all BPS-saturated amplitudes of the form F Γd+k,d Φ, with Φ being a weak holomorphic modular form of weight −k/2
Summary
We introduce the Niebur-Poincare series F(s, κ, w), a modular invariant regularisation of the naıve Poincare series of negative weight. We present its Fourier expansion for general values of s, and analyse its limit as. N where n is any non-negative integer. We explain how to represent any weak almost holomorphic modular form of negative weight as a suitable linear combinations of such Poincare series
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