156 research outputs found
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
Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two K\"ahler Forms
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.Comment: 45 pages, 2 figures, minor errors are corrected, English is refined.
Section 1 and Section 2 are enlarged. Especially in Section 2, confusion
between the notion of resolution and the notion of compactification is
resolved. Computation under non-zero equivariant parameters are added in
Section
Construction of Free Energy of Calabi-Yau manifold embedded in via Torus Actions
We calculate correlation functions of topological sigma model (A-model) on
Calabi-Yau hypersurfaces in 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
On the Quantum Cohomology Rings of General Type Projective Hypersurfaces and Generalized Mirror Transformation
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.Comment: 32 pages, 3 figures, discussion in section 5 is refined, some minor
errors are correcte
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
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
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 : in for various values of k and N. When k<N, the 1-loop beta function of
the sigma model on is negative and we expect the theory to have a mass
gap. However, the quantum cohomology relation
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 with N chiral fields with U(1) charge . 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
On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces
We give an explicit procedure which computes for degree the
correlation functions of topological sigma model (A-model) on a projective Fano
hypersurface as homogeneous polynomials of degree in the correlation
functions of degree 1 (number of lines). We extend this formalism to the case
of Calabi-Yau hypersurfaces and explain how the polynomial property is
preserved. Our key tool is the construction of universal recursive formulas
which express the structural constants of the quantum cohomology ring of as
weighted homogeneous polynomial functions in the constants of the Fano
hypersurface with the same degree and dimension one more. We propose some
conjectures about the existence and the form of the recursive formulas for the
structural constants of rational curves of arbitrary degree. Our recursive
formulas should yield the coefficients of the hypergeometric series used in the
mirror calculation. Assuming the validity of the conjectures we find the
recursive laws for rational curves of degree 4 and 5.Comment: 32 pages, changed fonts, exact results on quintic rational curves are
added. To appear in Commun. Math. Phy
- …