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Construction of Free Energy of Calabi-Yau manifold embedded in $CP^{n-1}$ via Torus Actions
Masao Jinzenji
Physics , 1995, DOI: 10.1142/S0217751X97003030
Abstract: 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.
Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points
Masao Jinzenji
Mathematics , 2007, DOI: 10.1007/s11005-008-0278-z
Abstract: In this paper, we derive the virtual structure constants used in mirror computation of degree k hypersurface in CP^{N-1}, by using localization computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1} with two marked points. We also apply this technique to non-nef local geometry O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff factorization.
Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two K?hler Forms
Masao Jinzenji
Mathematics , 2010, DOI: 10.1007/s00220-013-1786-y
Abstract: In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and Calabi-Yau hypersurface in weighted projective space P(1,1,2,2,2) as examples. We expect that our results can be easily generalized to arbitrary toric manifold.
Completion of the Conjecture: Quantum Cohomology of Fano Hypersurfaces
M. Jinzenji
Mathematics , 1999, DOI: 10.1142/S0217732300000116
Abstract: In this paper, we propose the formulas that compute all the rational structural constants of the quantum K\"ahler sub-ring of Fano hypersurfaces.
Direct Proof of Mirror Theorem of Projective Hypersurfaces up to degree 3 Rational Curves
Masao Jinzenji
Mathematics , 2009, DOI: 10.1016/j.geomphys.2011.03.014
Abstract: In this paper, we directly derive generalized mirror transformation of projective hypersurfaces up to degree 3 genus 0 Gromov-Witten invariants by comparing Kontsevich localization formula with residue integral representation of the virtual structure constants. We can easily generalize our method for rational curves of arbitrary degree except for combinatorial complexities.
Virtual Gromov-Witten Invariants and the Quantum Cohomology Rings of General Type Projective Hypersurfaces
Masao Jinzenji
Mathematics , 2000, DOI: 10.1142/S0217732300000633
Abstract: In this paper, we propose another characterization of the generalized mirror transformation on the quantum cohomology rings of general type projective hypersurfaces. This characterics is useful for explicit determination of the form of the generalized mirror transformation. As applications, we rederive the generalized mirror transformation up to $d=3$ rational Gromov-Witten invariants obtained in our previous article, and determine explicitly the the generalized mirror transformation for the $d=4, 5$ rational Gromov-Witten invariants in the case when the first Chern class of the hypersurface equals $-H$ (i.e., $k-N=1$).
Gauss-Manin System and the Virtual Structure Constants
Masao Jinzenji
Mathematics , 2001,
Abstract: 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.
Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation
Masao Jinzenji
Mathematics , 2003, DOI: 10.1142/S0217751X05020641
Abstract: 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.
On Quantum Cohomology Rings for Hypersurfaces in $CP^{N-1}$
Masao Jinzenji
Mathematics , 1995, DOI: 10.1063/1.532228
Abstract: 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 $CP^{N-1}$.
On the Quantum Cohomology Rings of General Type Projective Hypersurfaces and Generalized Mirror Transformation
M. Jinzenji
Mathematics , 1998, DOI: 10.1142/S0217751X00000707
Abstract: In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective Calabi-Yau hypersurface has a close relation with the ring of symmetric functions, or with Schur polynomials. With this result in mind, we propose a generalized mirror transformation on the quantum cohomology of a hypersurface with negative first Chern class and construct an explicit prediction formula for three point Gromov-Witten invariants up to cubic rational curves. We also construct a projective space resolution of the moduli space of polynomial maps, which is in a good correspondence with the terms that appear in the generalized mirror transformation.
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