Masao Jinzenji

Quantum Cohomology and Free Field Representation

Abstract 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 Visasoro 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 ≠ 0, p ≠ q ) and suggest that the Virasoro condition holds universally for all compact smooth Kahler manifolds.

On the Structure of the Small Quantum Cohomology Rings of Projective Hypersurfaces

Abstract:We give an explicit procedure which computes for degree d≤ 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.

MIRROR SYMMETRY AND AN EXACT CALCULATION OF AN (N–2)-POINT CORRELATION FUNCTION ON A CALABI-YAU MANIFOLD EMBEDDED IN CPN−1

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.

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.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.

On quantum cohomology rings for hypersurfaces in CPN−1

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⩽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.

N=4 supersymmetric Yang-Mills theory on orbifold-T4/Z2

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. curve blowup formula.

CALCULATION OF GROMOV-WITTEN INVARIANTS FOR CP3, CP4AND Gr(2, 4)

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.

COMPLETION OF THE CONJECTURE: QUANTUM COHOMOLOGY OF FANO HYPERSURFACES

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

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 MNk: ∑i = 1NXik = 0 in N−1 for various values of k and N. When 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.

Construction of Free Energy of Calabi–Yau Manifold Embedded in CPN-1 via Torus Actions

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.