BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form

TL;DR

This work analyzes BPS-saturated one-loop amplitudes in type II strings on ${\rm K3}\times{\mathbb T}^2$, revealing a tight link between amplitude integrands and the ${\rm K3}$ elliptic genus. Massless 1/2 BPS insertions isolate Appell–Lerch contributions (mock modular pieces) of the elliptic genus, while including massive external states yields the full elliptic genus through derivatives of ${\phi_{K3}}$, organized via ${\cal N}=4$ harmonic superspace. A generating functional for the 1/4 BPS class is shown to be the weight-10 Siegel modular form ${\chi_{10}}$ of ${\rm Sp}(4,\mathbb{Z})$, obtained from a torus integral with a tunable coupling ${\lambda}$, linking to dyon degeneracies and BKM algebras. The results suggest deep algebraic structures governing BPS spectra in ${\cal N}=4$ compactifications and offer a bridge to non-perturbative dyons and ${\mathbb M}_{24}$ symmetries in the ${\rm K3}$ sector.

Abstract

We study BPS saturated one-loop amplitudes in type II string theory compactified on K3 x T^2. The classes of amplitudes we consider are only sensitive to the very basic topological data of the internal K3 manifold. As a consequence, the integrands of the former are related to the elliptic genus of K3, which can be decomposed into representations of the internal N=4 superconformal algebra. Depending on the precise choice of external states these amplitudes capture either only the contribution of the short multiplets or the full series including intermediate multiplets. In the latter case we can define a generating functional for the whole class, which we show is given by the weight ten Igusa cusp form chi_{10} of Sp(4,Z). We speculate on possible algebraic implications of our result on the BPS states of the N=4 type II compactification.

Figures (1)

• Figure 1: Reducible diagram with external fields $\Phi_1$ and $\Phi_2$ pairwise colliding. The intermediate channel can be decomposed into a series of fields of different conformal dimensions, which belong to different multiplets. The latter are not necessarily only short ones but might also include $1/4$ BPS multiplets.