We show that certain type II string amplitudes at genus g are given by the topological partition function F g discussed recently by Bershadsky, Cecotti, Ooguri and Vafa.These amplitudes give rise to a term in the four-dimensional effective action of the form g F g W 2g , where W is the chiral superfield of N = 2 supergravitational multiplet. The holomorphic anomaly of F g is related to non-localities of the effective action due to the propagation of massless states. This result generalizes the holomorphic anomaly of the one loop case which is known to lead to non-harmonic gravitational couplings.⋆

We test the recently conjectured duality between N =2 supersymmetric type II and heterotic string models by analysing a class of higher dimensional interactions in the respective low-energy Lagrangians. These are F -terms of the form F g W 2g where W is the gravitational superfield. On the type II side these terms are generated at the g-loop level and in fact are given by topological partition functions of the twisted Calabi-Yau sigma model. We show that on the heterotic side these terms arise at the one-loop level. We study in detail a rank 3 example and show that the corresponding couplings F g satisfy the same holomorphic anomaly equations as in the type II case.Moreover we study the leading singularities of F g 's on the heterotic side, near the enhanced symmetry point and show that they are universal poles of order 2g−2 with coefficients that are given by the Euler number of the moduli space of genus-g Riemann surfaces. This confirms a recent conjecture that the physics near conifold singularity is governed by c=1 string theory at the self-dual point.

We discuss the prepotential describing the effective field theory of N =2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory.