34 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
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
Gap Condition and Self-Dualized Super Yang-Mills Theory for ADE Gauge Group on K3
We try to determine the partition function of super Yang-Mills
theoy for ADE gauge group on K3 by self-dualizing our previous ADE partition
function. The resulting partition function satisfies gap condition.Comment: 17 page
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
Open Virtual Structure Constants and Mirror Computation of Open Gromov-Witten Invariants of Projective Hypersurfaces
In this paper, we generalize Walcher's computation of the open Gromov-Witten
invariants of the quintic hypersurface to Fano and Calabi-Yau projective
hypersurfaces. Our main tool is the open virtual structure constants. We also
propose the generalized mirror transformation for the open Gromov-Witten
invariants, some parts of which are proven explicitly. We also discuss possible
modification of the multiple covering formula for the case of higher
dimensional Calabi-Yau manifolds. The generalized disk invariants for some
Calabi-Yau and Fano manifolds are shown and they are certainly integers after
re-summation by the modified multiple covering formula. This paper also
contains the direct integration method of the period integrals for higher
dimensional Calabi-Yau hypersurfaces in the appendix.Comment: 24pages, 5figure
Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points
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.Comment: 10 pages, latex, a minor change in Section 4, English is refined,
Some typing errors in Section 3 are correcte
Prepotentials for local mirror symmetry via Calabi-Yau fourfolds
In this paper, we first derive an intrinsic definition of classical triple
intersection numbers of K_S, where S is a complex toric surface, and use this
to compute the extended Picard-Fuchs system of K_S of our previous paper,
without making use of the instanton expansion. We then extend this formalism to
local fourfolds K_X, where X is a complex 3-fold. As a result, we are able to
fix the prepotential of local Calabi-Yau threefolds K_S up to polynomial terms
of degree 2. We then outline methods of extending the procedure to non
canonical bundle cases.Comment: 42 pages, 7 figures. Expanded, reorganized, and added a theoretical
background for the calculation
-duality in Vafa-Witten theory for non-simply laced gauge groups
Vafa-Witten theory is a twisted N=4 supersymmetric gauge theory whose
partition functions are the generating functions of the Euler number of
instanton moduli spaces. In this paper, we recall quantum gauge theory with
discrete electric and magnetic fluxes and review the main results of
Vafa-Witten theory when the gauge group is simply laced. Based on the
transformations of theta functions and their appearance in the blow-up
formulae, we propose explicit transformations of the partition functions under
the Hecke group when the gauge group is non-simply laced. We provide various
evidences and consistency checks.Comment: 14 page