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

TL;DR

This paper explores BPS saturated one-loop amplitudes in type II string theory on K3 x T^2, linking them to the elliptic genus of K3 and the Igusa cusp form, revealing new algebraic structures related to BPS states.

Contribution

It establishes a connection between BPS amplitudes, the elliptic genus of K3, and the Igusa cusp form chi_{10}, providing a novel generating functional for these amplitudes.

Findings

Amplitudes are related to the elliptic genus of K3.

The generating functional is given by the Igusa cusp form chi_{10}.

Results suggest new algebraic structures in BPS state counting.

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.