Holomorphic anomaly equation and BPS state counting of rational elliptic surface

We consider the generating function (prepotential) for Gromov-Witten invariants of rational elliptic surface. We apply the local mirror principle to calculate the prepotential and prove a certain recursion relation, holomorphic anomaly equation, for genus 0 and 1. We propose the holomorphic anomaly equation for all genera and apply it to determine higher genus Gromov-Witten invariants and also the BPS states on the surface. Generalizing Gottsche's formula for the Hilbert scheme of g points on a surface, we find precise agreement of our results with the proposal recently made by Gopakumar and Vafa[ll].