Abstract:Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Abstract:We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton-corrected Yukawa couplings, and the topological one-loop partition function to the case of complete intersections with higher-dimensional moduli spaces. We will develop a new method of obtaining the instanton-corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the Kähler moduli fields induced from the ambient space for all complete intersections in non singular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models that are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.

The lattice definition of a two-dimensional topological field theory (TFT) is given generically, and the exact solution is obtained explicitly. In particular, the set of all lattice topological field theories is shown to be in one-to-one correspondence with the set of all associative algebras R, and the physical Hilbert space is identified with the center Z(R) of the associative algebra R. Perturbations of TFT's are also considered in this approach, showing that the form of topological perturbations is automatically determined, and that all TFT's are obtained from one TFT by such perturbations. Several examples are presented, including twisted N = 2 minimal topological matter and the case where R is a group ring. *

Abstract:We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gr6bner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up to h 1'1 = 3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.

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...