Construction of Free Energy of Calabi-Yau manifold embedded in CPn−1CP^{n-1} via Torus Actions

We calculate correlation functions of topological sigma model (A-model) on Calabi-Yau hypersurfaces in $CP^{N-1}$ using torus action method. We also obtain path-integral represention of free energy of the theory coupled to gravity.Comment: 30 page

Gauss-Manin System and the Virtual Structure Constants

In this paper, we discuss some applications of Givental's differential equations to enumerative problems on rational curves in projective hypersurfaces. Using this method, we prove some of the conjectures on the structure constants of quantum cohomology of projective hypersurfaces, proposed in our previous article. Moreover, we clarify the correspondence between the virtual structure constants and Givental's differential equations when the projective hypersurface is Calabi-Yau or general type.Comment: 25 pages, Latex, references added, minor errors are correcte

Geometrical Proof of Generalized Mirror Transformation of Projective Hypersurfaces

In this paper, we outline geometrical proof of the generalized mirror transformation of genus 0 Gromov-Witten invariants of degree k hypersurface in CP^{N-1}.Comment: 10 pages, minor errors are corrected, appendix is added, Introduction is expanded, references are adde

Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation

In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss-Manin system.Comment: 19 pages, latex, minor errors are corrected, discussions in Section 4 are refine

Multi-Point Virtual Structure Constants and Mirror Computation of CP^2-model

In this paper, we propose a geometrical approach to mirror computation of genus 0 Gromov-Witten invariants of CP^2. We use multi-point virtual structure constants, which are defined as intersection numbers of a compact moduli space of quasi maps from CP^1 to CP^2 with 2+n marked points. We conjecture that some generating functions of them produce mirror map and the others are translated into generating functions of Gromov-Witten invariants via the mirror map. We generalize this formalism to open string case. In this case, we have to introduce infinite number of deformation parameters to obtain results that agree with some known results of open Gromov-Witten invariants of CP^2. We also apply multi-point virtual structure constants to compute closed and open Gromov-Witten invariants of a non-nef hypersurface in projective space. This application simplifies the computational process of generalized mirror transformation.Comment: 26 pages, Late

Completion of the Conjecture: Quantum Cohomology of Fano Hypersurfaces

In this paper, we propose the formulas that compute all the rational structural constants of the quantum K\"ahler sub-ring of Fano hypersurfaces.Comment: 19pages, Latex, minor changes in English, some formulas are adde

Generalization of Calabi-Yau/Landau-Ginzburg correspondence

We discuss a possible generalization of the Calabi-Yau/Landau-Ginzburg correspondence to a more general class of manifolds. Specifically we consider the Fermat type hypersurfaces $M_N^k$: $\sum_{i=1}^N X_i^k =0$ in ${\bf CP}^{N-1}$ for various values of k and N. When k<N, the 1-loop beta function of the sigma model on $M_N^k$ is negative and we expect the theory to have a mass gap. However, the quantum cohomology relation $\sigma^{N-1}={const.}\sigma^{k-1}$ suggests that in addition to the massive vacua there exists a remaining massless sector in the theory if k>2. We assume that this massless sector is described by a Landau-Ginzburg (LG) theory of central charge $c=3N(1-2/k)$ with N chiral fields with U(1) charge $1/k$. We compute the topological invariants (elliptic genera) using LG theory and massive vacua and compare them with the geometrical data. We find that the results agree if and only if k=even and N=even. These are the cases when the hypersurfaces have a spin structure. Thus we find an evidence for the geometry/LG correspondence in the case of spin manifolds.Comment: 19 pages, Late

Moduli Space of Quasi-Maps from P^{1} with Two Marked Points to P(1,1,1,3) and j-invariant

In this paper, we construct toric data of moduli space of quasi maps of degree $d$ from P^{1} with two marked points to weighted projective space P(1.1,1,3). With this result, we prove that the moduli space is a compact toric orbifold. We also determine its Chow ring. Moreover, we give a proof of the conjecture proposed by Jinzenji that a series of intersection numbers of the moduli spaces coincides with expansion coefficients of inverse function of -log(j(tau)).Comment: 22 pages, minor errors are corrected, references are added, some comments are added in Introductio

On Quantum Cohomology Rings for Hypersurfaces in CPN−1CP^{N-1}

Using the torus action method, we construct one variable polynomial representation of quantum cohomology ring for degree $k$ hypersurface in $CP^{N-1}$ . The results interpolate the well-known result of $CP^{N-2}$ model and the one of Calabi-Yau hypersuface in $CP^{N-1}$. We find in $k\leq N-2$ case, principal relation of this ring have very simple form compatible with toric compactification of moduli space of holomorphic maps from $CP^{1}$ to $CP^{N-1}$.Comment: 32 pages, Revised versio