N=4 topological amplitudes and string effective action

Certain scattering amplitudes in the gravitational sector of type II string theory on K3×T2 are found to be computed by correlation functions of the N=4 topological string. This analysis extends the already known results for K3 by Berkovits and Vafa, which correspond to six-dimensional terms in the effective action, involving four Riemann tensors and 4g−4 graviphotons, R4T4g−4, at genus g. We find two additional classes of topological amplitudes that use the full internal SCFT of K3×T2. One of these string amplitudes is mapped to a 1-loop contribution in the heterotic theory, and is studied explicitly. It corresponds to the four-dimensional term R2(ddΦ)2T2g−4, with Φ a Kaluza-Klein graviscalar from T2. Finally, the generalization of the harmonicity relation for its moduli dependent coupling coefficient is obtained and shown to contain an anomaly, generalizing the holomorphic anomaly of the N=2 topological partition function Fg.