N=4 topological amplitudes and string effective action

Certain scattering amplitudes in the gravitational sector of type II string theory on K3 × T 2 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, R 4 T 4g 4 , at genus g. We find two additional classes of topological amplitudes that use the full internal SCFT of K3 × T 2 . 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 �