We consider the generating function (prepotential) for GromovWitten 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]. In this paper we will propose a recursion relation holomorphic anomaly equation as a basic equation for the higher genus Gromov-Witten invariants of rational elliptic surface, and will make explicit predictions for them. We find exquisite agreement of our results with those by Gopakumar and Vafa.To state main results of this paper let us consider a generic rational elliptic surface obtained by blowing up nine base points of two generic cubics in P 2 . Under the assumption for the cubics the surface S has an elliptic fibration over P 1 with exactly twelve singular fibers of Kodaira Ii type. We consider a situation in which the generic rational elliptic surface S appears as a divisor in a Calabi-Yau 3-fold X. Since the normal bundle Mx/s ' IS given by the canonical bundle Ks we can extract the genus g GromovWitten invariants Ng{(3) of class /? G -/^(S', Z) taking a suitable limit of the prepotential of the Calabi-Yau 3-fold X, which is called local mirror principle. Since even for genus zero invariants the determination of Ng = o(P) is technically tedious, in what follows, we will mainly be concerned with the following sum of the invariants ((3,H)=d, (p,F)=nwhere H and F represent the pull back of the hyperplane class of P 2 and the fiber class, respectively. Associated to these invariants we define generating The latter is the genus g prepotential in topological string theory. For g -0 and g -1 we determine it via the local mirror principle applying to X, and find a recursion relation satisfied by Zg ]n (g = 0,1; n = 1,2, • • •) which we generalize for arbitrary g as follows: We derive the same result following the proposal made in [11] for the BPS state counting of the families of genus g curves. From this viewpoint our result 1.4 comes from the following generalization of Gottsche's formula [9] for the Hilbert scheme S^ of g points on a surface S: G(t L , t R , q) = n { (1 _ (t L t a )»-lg») (1 -(t L t R )»+iq")(1.5)We explain our result 1.4 in terms of the above generalization of Gottsche's formula byThis implies that the genus g curves Cg in S' satisfying (Cg, F) = 1 split into irreducible parts, one coming from the Mordell-Weil group and the others from elliptic curves (with possible nodal singularities) in the fiber direction.The readers who are not interested in the derivation and the proofs of the holomorphic anomaly equation may omit the following two sections and may start from the section 4 for ou...

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