Topological Amplitudes in String Theory

We show that certain type II string amplitudes at genus g are given by the topologi-cal 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 multi-plet. 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.