Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold [Formula: see text] with potential terms induced from a Killing vector on [Formula: see text].³. One of the well-known fixed-point theorems is the Bott residue formula⁹ which represents the intersection number of Chern classes of holomorphic vector bundles on a Kähler manifold [Formula: see text] as the sum of contributions from fixed point sets of a holomorphic vector field [Formula: see text] on [Formula: see text]. In this paper, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from [Formula: see text] to [Formula: see text], with potential terms induced from the vector field [Formula: see text]. Our strategy is to restrict phase space of path integral to maps homotopic to constant maps. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model. Our potential term and BRST symmetry are closely related to the idea used in the paper by Beasley and Witten² where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.

DOI： 10.1142/s0217751x20501924

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arXiv

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

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 resummation 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.

DOI： 10.1142/s0219887814500054

arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

In this paper, we extend our geometrical derivation of the expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two Kahler forms. In particular, we consider 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 manifolds.

DOI： 10.1007/s00220-013-1786-y

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arXiv

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Other Link： http://link.springer.com/article/10.1007/s00220-013-1786-y/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：International Press of Boston  

DOI： 10.4310/cntp.2013.v7.n3.a2

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Elsevier BV  

DOI： 10.1016/j.geomphys.2011.03.014

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s11005-008-0278-z

arXiv

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Other Link： http://link.springer.com/article/10.1007/s11005-008-0278-z/fulltext.html

We solve the problem of equivariant mirror symmetry for O(-3)->P^2 for the
(three) cases of one independent equivariant parameter. This gives a
decomposition of mirror symmetry for local P^2 into that of three subspaces,
each of which may be considered independently. Finally, we give a new
interpretation of mirror symmetry for O(k)+O(-2-k)->P^1.

arXiv

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Other Link： http://arxiv.org/pdf/0710.0049v2

Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

We provide a straightforward computational scheme for the equivariant local mirror symmetry of curves, i.e. mirror symmetry for [Formula: see text] for k ≥ 1, and detail related methods for dealing with mirror symmetry of non-nef toric varieties, based on the theorems of Refs. 2 and 13. The basic tools are equivariant I functions and their Birkhoff factorization.

DOI： 10.1142/s0217751x0703649x

arXiv

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We continue our study of equivariant local mirror symmetry of curves, i.e.
mirror symmetry for X_k=O(k)+O(-2-k) over P^1 with torus action
(lambda_1,lambda_2) on the bundle. For the antidiagonal action
lambda_1=-lambda_2, we find closed formulas for the mirror map and a rational B
model Yukawa coupling for all k. Moreover, we give a simple closed form for the
B model genus 1 Gromov-Witten potential. For the diagonal action
lambda_1=lambda_2, we argue that the mirror symmetry computation is equivalent
to that of the projective bundle P(O+O(k)+O(-2-k)) over P^1. Finally, we
outline the computation of equivariant Gromov-Witten invariants for A_n
singularities and toric tree examples via mirror symmetry.

arXiv

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Other Link： http://arxiv.org/pdf/math/0609016v3

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1088/1126-6708/2006/03/061

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：AIP Publishing  

DOI： 10.1063/1.1996441

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

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.

DOI： 10.1142/s0217751x05020641

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1088/1126-6708/2002/09/002

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

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.

DOI： 10.1142/s0129167x02001368

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1088/1126-6708/2001/12/002

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

We derive the partition function of N=4 supersymmetric Yang–Mills theory on orbifold-T⁴/Z₂. In classical geometry, K3 surface is constructed from the orbifold-T⁴/Z₂. Along the same way as the orbifold construction, we construct the partition function of K3 surface from orbifold-T⁴/Z₂. The partition function is given by the product of the contribution of the untwisted sector of T⁴/Z₂, and that of the twisted sector of T⁴/Z₂, i.e. [Formula: see text] curve blowup formula.

DOI： 10.1142/s0217732301003565

arXiv

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Publishing type：Research paper (scientific journal)  

In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with nonpositive first 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 good correspondence with the terms that appear in the generalized mirror transformation.

DOI： 10.1142/S0217751X00000707

Scopus

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Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

In this letter, we propose another characterization of the generalized mirror transformation on the quantum cohomology rings of general type projective hypersurfaces. This characteristic is useful for explicit determination of the form of the generalized mirror transformation. As an application, we rederive the generalized mirror transformation up to d=3 rational Gromov–Witten invariants obtained in our previous paper, and determine explicitly 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).

DOI： 10.1142/s0217732300000633

arXiv

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1088/1126-6708/2000/02/028

arXiv

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

In this letter, we propose the formulas that compute all the rational structural constants of the quantum Kahler subring of Fano hypersurfaces.

DOI： 10.1016/S0217-7323(00)00011-6

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER VERLAG  

We give an explicit procedure which computes for degree d less than or equal to 3 the correlation functions of topological sigma model (A-model) on a projective Fano hypersurface X as homogeneous polynomials of degree d 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 structure constants of the quantum cohomology ring of X as weighted homogeneous polynomial functions of 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 laws for the structure 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 four.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

In our previous article we have proposed that the Virasoro algebra controls the quantum cohomology of Fano varieties at all genera. In this paper we construct a free-field description of Virasoro operators and quantum cohomology. We shall show that to each even (odd) homology class of a Kahler manifold we have a free bosonic (fermionic) field and Virasoro operators are given by a simple bilinear form of these fields. We shall show that the Virasoro condition correctly reproduces the Gromov-Witten invariants also in the case of manifolds with non-vanishing non-analytic classes (h(p,q) not equal 0, p not equal q) and suggest that the Virasoro condition holds universally for all compact smooth Kahler manifolds. (C) 1998 Elsevier Science B.V.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER INST PHYSICS  

Using the torus action method, we construct a one-variable polynomial representation of quantum cohomology ring for degree k hypersurface in CPN-1. The results interpolate the well-known result of CpN-2 model and the one of Calabi-Yau hypersuface in CPN-1. We find in the k less than or equal to N-2 case, the principal relation of this ring has a very simple form compatible with toric compactification of moduli space of holomorphic maps from CP1 to CPN-1. (C) 1997 American Institute of Physics.

DOI： 10.1063/1.532228

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

We calculate correlation functions of topological sigma model (A-model) on Calabi-Yau hypersurfaces in CPN-1 using torus action method. We also obtain path-integral representation of free energy of the theory coupled to gravity.

DOI： 10.1142/S0217751X97003030

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

We consider an (N - 2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of the Fermat type) in CPN-1 and its mirror manifold. We introduce an (N - 2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of algebraic geometry. In enumerating the holomorphic curves in the general-dimensional Calabi-Yau manifolds, we extend the method of counting rational curves on the Calabi-Yau three-fold using the Shubert calculus on Gr(2, N). The agreement of the two calculations for the (N-2)-point function establishes ''the mirror symmetry at the correlation function level'' in the general-dimensional case.

DOI： 10.1142/S0217751X96000559

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

Using the associativity relations of the topological sigma models with target spaces, CP3, CP4 and Gr(2,4), we derive recursion relations of their correlation and evaluate them up to a certain order in the expansion over the instantons. The expansion coefficients are regarded as the number of rational curves in CP3, CP4 and Gr(2, 4) which intersect various types of submanifolds corresponding to the choice of BRST-invariant operators in the correlation functions.

DOI： 10.1142/S0217751X96000092

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Language：English   Publisher：素粒子論グループ 素粒子研究編集部  

In this paper, we review recent discovery of connection between Virasoro algebra and topological sigma model coupled to topological gravity.

DOI： 10.24532/soken.97.4_D119

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