Topological Amplitudes in Heterotic Superstring Theory

We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N=1 spacetime supersymmetric background. The genus g topological partition function $F^g$ corresponds to a term in the effective action of the form $W^{2g}$, where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form $W^{2g}\Pi^n$, where $\Pi$ is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N=1 spacetime supersymmetry Ward identity. We give also a solution of the tree level recursion relations and discuss orbifold examples.