A. Libgober
We show how Chern classes of a Calabi Yau hypersurface in a toric Fano manifold can be expressed in terms of the holomorphic at a maximal degeneracy point period of its mirror. We also consider the relation between Chern classes and the periods of mirrors for complete intersections in Grassmanian Gr(2,5).
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Recent work on the relation between a special class of Kähler manifolds with positive first Chern class and critical N$=$2 string vacua with c$=$9 is reviewed and extended. (Based on a talk presented at the International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY 93), Boston, MA, March 1993)
We obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
In a recent paper, A. Libgober showed that the multiplicative sequence {Q_i(c_1,...,c_i)} of Chern classes corresponding to the power series Q(z)=1/Gamma(1+z) appears in a relation between the Chern classes of certain Calabi-Yau manifolds and the periods of their mirrors. We show that the polynomials Q_i can be expressed in terms of multiple zeta values.
The Fano classification program proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk is based on the mirror symmetry prediction that the regularized quantum period of a Fano should be equivalent to the classical period of its mirror Landau-Ginzburg potential. We prove that this mirror equivalence follows from versions of the Frobenius structure conjecture of Gross-Hacking-Keel. We also find that the regularized quantum period, which is defined in terms of descendant Gromov-Witten numbers, is in fact given by certain naive curve counts.
We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with th…
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 rat…
Exponential concentration for quantum periods via mirror symmetry
本文研究满足指数集中性质的幂级数，证明修改后的超几何级数保持该性质，并证明当Fano流形具有非负系数的方便弱Landau-Ginzburg模型时，其量子周期也满足该性质。
From wall structures to closed mirror symmetry. The case of $K_{\mathbb{P}^2}$: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration
本文通过墙函数统一恢复 $K_{\mathbb{P}^2}$ 的闭 Gromov-Witten 不变量与重整化周期，建立 Gross-Siebert 构造到经典枚举镜像对称的直接通道，并给出多项式性定理及热带树和，预期推广至一般 Calabi-Yau 镜像对。
Quantum periods, toric degenerations and intrinsic mirror symmetry
本文证明了 Fano 簇 $X$ 的正则化量子周期可由仿射对数 Calabi-Yau 簇 $U$ 的内禀镜像代数元素的经典周期给出，推导出 Fano 镜像对称性的推论，并构造了相关多面体族与 toric 退化，明确了格拉斯曼流形的超势。
Relative mirror symmetry for non-Fano varieties
本文在不假设反典范除子 nef 的条件下，研究对数 Calabi--Yau 对的镜像对称性，指出量子周期与经典周期的差异源于除子中的曲线计数，并给出 proper 势的显式公式，证明其与量子周期加镜像映射信息等价。
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physic…
Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional…
The 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi-Yau surface, given by the complement of an anticanonical pentagon of $(-1)$-curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$\mathbb Q$-factorial) affine Fano 3-fold of type $V_{12}$, and we…
In this note we study several conditions to be imposed on a mirror symmetry candidate to the generic multi-quasihomogeneous Calabi-Yau variety defined in the product of the quasihomogeneous projective spaces. We propose several properties for a Calabi-Yau complete intersection variety so that its period integrals can be expressed by means of quasihomogeneous weights of its mirror symmetry candidate as it has been suggested by Berglund, Candelas et altri. As a corollary, we see certain duality between the monodromy data and the Poincaré polynomials of the Euler characteristic for the pairs of o…
Intrinsic enumerative mirror symmetry: Takahashi's log mirror symmetry for $(\mathbb{P}^2,E)$ revisited
本文重新审视Takahashi对（P²,E）的对数镜像对称性猜想，通过本质镜像构造和周期积分计算，证明该猜想是本质镜像对称性的自然结果，并推广至任意维数的log Calabi-Yau变种。
Because of the existence of rigid Calabi--Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kähler manifolds with positive first Chern class which generalize Calabi--Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class is comprised of a special type of Fano manifolds which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of $(2,2)$--…
Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces
本文研究了通过星形三角化非凸多面体构造非Fano品种的全交集时，Chern特性与Transpolar对的Todd-Hirzebruch恒等式在非代数但光滑的Toric空间中的应用。
Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians, and can be proved for the simple case of $T^2$.
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 e…
In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation for toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of variations of Z-Hodge structure for a mirror pair of Calabi-Yau hypersurfaces (Batyrev's mirror).
This paper discusses the overlap of the Hori-Vafa formulation of mirror symmetry with some other constructions. We focus on compact Calabi-Yau hypersurfaces \mathcal{M}_G = {G = 0} in weighted complex projective spaces. The Hori-Vafa formalism relates a family {\mathcal{M}_G \in WCP^{m-1}_{Q_1,...,Q_m}[s] | \sum_{i=1}^m Q_i = s} of such hypersurfaces to a single Landau-Ginzburg mirror theory. A technique suggested by Hori and Vafa allows the Picard-Fuchs equations satisfied by the corresponding mirror periods to be determined. Some examples in which the variety \mathcal{M}_G is crepantly resol…
We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that the…
We show that refined Chern-Simons theory and large N duality can be used to study the refined topological string with and without branes. We derive the refined topological vertex of hep-th/0701156 and hep-th/0502061 from a link invariant of the refined SU(N) Chern-Simons theory on S^3, at infinite N. Quiver-like Chern-Simons theories, arising from Calabi-Yau manifolds with branes wrapped on several minimal S^3's, give a dual description of a large class of toric Calabi-Yau. We use this to derive the refined topological string amplitudes on a toric Calabi-Yau containing a shrinking P^2 surface.…
In this paper, we will study the connections between the mirror symmetry of Calabi-Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop a method to compute the Deligne's period for a Calabi-Yau threefold in the mirror family of a one-parameter mirror pair. We will give two examples to show how this method works, and we will express the Deligne's period in terms of the classical periods of the threeform. Using this method, we will compute the Deligne's period of a Calabi-Yau threefold studied…
The predictions of the Mirror Symmetry are extended in dimensions n>3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov-Witten invariants of such variety can be expressed in terms of certain invariants of a new generalization of variation of Hodge structures attached to the dual variety. To formulate the general principles of Mirror Symmetry in arbitrary dimension it is necessary to introduce the ``extended moduli space of complex structures'' M. An analog M\to H*(X,C)[n] of the classical period map is de…
We study mirror symmetry of Calabi-Yau manifolds within the framework of the Gauss-Manin system. Applying the flat coordinates to the Gauss-Manin system for the periods, we derive differential equations for the mirror map in addition to the ordinary Picard-Fuchs equations for the periods. These equations are obtained for a class of one-parameter models and a two-parameter model of Fermat type Calabi-Yau manifolds.
Mirror symmetry suggests that on a Calabi-Yau 3-fold moduli spaces of stable bundles, especially those with degree zero and indivisible Chern class, might be smooth (i.e. unobstructed, though perhaps of too high a dimension). This is because smoothly embedded special lagrangian cycles in the mirror have unobstructed deformations. As there does not seem to be a counterexample in the literature we provide one here, showing that such a Tian-Todorov-McLean-type result cannot hold.
For the Fermat Calabi-Yau threefold and the theory of stability conditions [Bri07], there have been two mathematical aims given by physical reasoning. One is that we should define stability conditions by central charges of quintic periods [Hos04,Kon12,KonSoi13], which extend the Gamma class [KKP,Iri09,Iri11]. The other is that for well-motivated stability conditions on a derived Fukaya-type category, each stable object should be a Lagrangian [ThoYau]. We answer affirmatively to these aims with the simplest homological mirror symmetry (HMS for short) of the Fermat Calabi-Yau threefold [Oka09,Fu…
Mirror symmetry for quadric hypersurfaces
本文计算了二次超曲面镜像Landau-Ginzburg模型的Calabi-Yau紧化解消，推导出窄周期与宽周期的Picard-Fuchs方程，并证明了其量子上同调与镜像模型对应的Frobenius流形同构。
Gromov-Witten Invariants and Mirror Symmetry For Non-Fano Varieties Via Tropical Disks
本文通过热带圆盘和散射图中的断线，将非 Fano 簇的镜像对称性推广到瞬子修正势，证明其与开镜像映射及对数 Gromov-Witten 不变量的等价性。
We describe in detail the space of the two Kähler parameters of the Calabi--Yau manifold $¶_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa co…
In this article we study a differential algebra of modular-type functions attached to the periods of a one parameter family of Calabi-Yau varieties which is mirror dual to the universal family of quintic threefolds. Such an algebra is generated by seven functions satisfying functional and differential equations in parallel to the modular functional equations of classical Eisenstein series and the Ramanujan differential equation. Our result is the first example of automorphic-type functions attached to varieties whose period domain is not Hermitian symmetric. It is a reformulation and realizati…
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