Weiqiang He, Si Li, Yefeng Shen, Rachel Webb
We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.
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Nonabelian Landau-Ginzburg orbifolds and Calabi-Yau/Landau-Ginzburg correspondence
本文研究了Landau-Ginzburg轨道褶皱的双阶化向量空间结构。我们证明了可能非交换的Landau-Ginzburg轨道折叠的Hodge数的生成函数的公式。作为应用，我们计算了所有五元非退化五次齐次多项式的Hodge数.这些结果为Calabi-Yau几何与Landau-Ginzburg B模型之间的Calabi-Yau/Landau-Ginzburg对应提供了证据。
Given a Fano complete intersection defined by sections of a collection nef line bundles $L_1,\ldots, L_c$ on a Fano toric manifold $Y$, a construction of Givental/Hori-Vafa provides a mirror-dual Landau-Ginzburg model. This construction depends on a choice of suitable nef partition; that is, a partition of the rays of the fan determined by $Y$. We show that toric Landau-Ginzburg models constructed from different nef partitions representing the same complete intersection are related by a volume preserving birational map. In particular, various Laurent polynomial mirrors which may be obtained fr…
We summarize the description of the open-closed TFT datum for B-type Landau-Ginzburg models with Stein manifold targets and discuss various constructions which lead to large classes of examples of such models.
The Landau-Ginzburg/Calabi-Yau correspondence claims that the Gromov-Witten invariant of the quintic Calabi-Yau 3-fold should be related to the Fan-Jarvis-Ruan-Witten invariant of the associated Landau-Ginzburg model via wall crossings. In this paper, we consider the stack of quasi-maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for Gromov-Witten invariants and the moduli stack for Fan-Jarvis-Ruan-Witten invariants.
For an in invertible quasihomogeneous singularity $w$ we prove an all-genus mirror theorem establishing an isomorphism between two cohomological field theories. On the $B$-side it is the Saito-Givental theory given by a certain choice of a primitive form. On the $A$-side, it is the reduced matrix factorization CohFT for the dual singularity $w^T$ with the maximal diagonal symmetry group.
A Mirror Theorem between Landau-Ginzburg models
本文综述了Landau-Ginzburg模型镜像对称性研究的最新进展。
Dual Polyhedra, Mirror Symmetry and Landau-Ginzburg Orbifolds
对于具有典型超势的模型，本文给出了朗道-金兹伯格轨道褶皱的新的几何特征。我们证明了一些带有$U(1)$电荷$(-1,1)$的$(a,c)$态与对偶多面体上的积分点之间的一一对应关系，这是构造镜像流形的有用工具。依赖于复曲面几何，这些状态对应于来自爆破过程的$(1,1)$形式。根据上述证明，可以证明作者提出的Landau-Ginzburg轨道的单名-因子镜像映射与数学家们得到的Calabi-Yau流形的镜像映射是等价的。
A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic
我们证明了镜像五次函数的Landau-Ginzburg/Calabi-Yau对应。特别地，我们计算亏格为零的FJRW理论对（W，G），其中W是费马五次多项式和G = SL（W）。我们通过显式解析延拓和辛变换将其与镜像五次三重函数的Gromov-Witten理论联系起来。在这个过程中，我们证明了相应的Landau-Ginzburg模型（W，G）的镜像定理。
On the existence of affine Landau-Ginzburg phases in gauged linear sigma models
证明了规范线性sigma模型的Kähler模空间中仿射Landau-Ginzburg点存在的一个简单判据.
Mirror Symmetry and the Web of Landau-Ginzburg String Vacua
我们提出了一些数学方面的朗道-金兹伯格弦真空的复曲面几何。对于典型的超势，给出了Landau-Ginzburg模型中的某些（-1，1）态与复曲面因子的一一对应关系.非复曲面因子的Landau-Ginzburg解释。利用这种解释，我们提出了一种方法来解决所谓的“扭曲扇区问题”的轨道建设。此外，这种构造表明，原来的Landau-Ginzburg弦真空的模空间和它们的轨道是连通的。通过考虑Landau-Ginzburg模型的镜像映射，我们得到了Mori向量与我们轨道化的扭算子之间的关系。这种考虑使我们能够论证塞伯格-威滕曲线嵌入到卡-丘流形的定义方程中，在卡-丘流形上II型弦被紧化。讨论了卡-丘四重和极值跃迁的相关问题。
Landau-Ginzburg Hodge numbers for mirrors of del Pezzo surfaces
我们考虑Katzarkov、Kontsevich和Pantev关于Landau-Ginzburg Hodge数的论文中关于可紧致Landau-Ginzburg模型的推论。我们在二维情况下测试这些假设，验证了一些，并给出了一个反例。
Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds
证明了光滑Fano三重曲面具有复曲面Landau-Ginzburg模型.更精确地说，我们证明了他们的朗道-金兹伯格模型，提出的洛朗多项式，承认紧化家庭的K3表面，我们描述了他们的纤维在无穷大。我们也给出了一个明确的建设Landau-Ginzburg模型的del Pezzo曲面和任何因子。
On Calabi-Yau compactifications of toric Landau-Ginzburg models for Fano complete intersections
复曲面朗道-金斯堡模型的Givental的类型为法诺完全交叉已知有卡拉比-丘紧化。我们给出了这一事实的另一种证明。作为我们的证明的输出，我们得到了紧复曲面朗道-金兹伯格模型的无限远纤维的描述。
Hori--Vafa mirror models for complete intersections in weighted projective spaces and weak Landau--Ginzburg models
证明了加权射影空间中光滑Fano完全交的Hori-Vafa镜像模型可以解释为Laurent多项式.
Singularities of Landau-Ginzburg models for complete intersections and derived categories
本文在数值层面验证了镜像对称中关于Fano完全交的Landau-Ginzburg模型纤维奇点与例外集合对应的预期。
Fibers of Landau-Ginzburg models and rationality
本文通过镜像Landau-Ginzburg模型的纤维单值性刻画Fano三维流形的有理性，证明有理当且仅当所有可约纤维的单值性为幺幂。
We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this conjecture for log Calabi--Yau compactifications of toric Landau--Ginzburg models for smooth Fano threefolds, complete intersections, and some toric varieties.
Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models
我们证明了每一个秩一光滑的Fano三重有一个弱的Landau-Ginzburg模型来自一个环面退化。这些Landau-Ginzburg模型的纤维可以紧化为秩为19的Picard格的K3曲面。我们还证明了在加权射影空间中，任意维数的光滑Fano簇，只要是Cartier因子的完全交，都有一个非常弱的来自环面退化的Landau-Ginzburg模型.
On Landau--Ginzburg models for Fano varieties
我们观察到一种方法找到弱朗道-金斯堡模型的法诺品种，并找到他们光滑法诺三倍的属9，10和12。
We give a new proof of the computation of Hodge integrals we have previously obtained for the quantum singularity (FJRW) theory of chain polynomials. It uses the classical localization formula of Atiyah--Bott and we phrase our proof in a general framework that is suitable for future studies of gauged linear sigma models (GLSM). As a by-product, we obtain the first equivariant version of mirror symmetry without concavity, generalizing the work of Chiodo--Iritani--Ruan on the Landau--Ginzburg side.
This is a review of the theory of toric Landau-Ginzburg models - the effective approach to mirror symmetry for Fano varieties. We mainly focus on the cases of dimensions 2 and 3, as well as on the case of complete intersections in weighted projective spaces and Grassmannians. Conjectures that relate invariants of Fano varieties and their Landau--Ginzburg models, such as Katzarkov-Kontsevich-Pantev conjectures, are also studied.
A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence
我们展示了如何五次三重的朗道-金兹伯格/卡拉比-丘对应可以被铸造成一个全球性的镜像对称框架。然后我们从Berglund-Hübsch镜像对偶构造中得到启发，给出了一个类似的拓扑图，它刻画了加权射影空间中所有的Calabi-Yau超曲面和有限阿贝尔群作用的某些等价物.
Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models
本文证明了Landau-Ginzburg模型的模空间的光滑性。我们制定和证明了田托多罗夫定理的Landau-Ginzburg模型的变形，发展必要的霍奇理论的品种潜力，并证明了双重退化的声明所需的畅通的结果。本文讨论了Landau-Ginzburg型非对易Hodge结构的Hodge数的各种定义及其在镜像对称中所起的作用。我们还解释了由此产生的家庭的de Rham复形吸引到一个潜在的镜像对称的辛Fano流形的一个参数的家庭，并认为，模自然平凡性质的模空间的Landau-Ginzburg模型的正则特殊坐标。
Some remarks on Landau-Ginzburg potentials for odd-dimensional quadrics
我们研究了为奇维二次曲面$Q_{2n+1}$的标准Landau-Ginzburg模型构造Frobenius流形并将其与这些二次曲面的量子上同调的Frobenius流形相匹配的可能性。也就是说，我们表明，量子上同调Frobenius流形的二次曲面的初始条件，可以得到适当修改的标准Landau-Ginzburg模型。
On the arithmetic of Landau-Ginzburg model of a certain class of threefolds
本文证明了一类Fano三重函数的Aesthetics常数可以作为一个特殊的高阶正规函数的值来求出。
We study the Hodge numbers of Landau-Ginzburg models as defined by Katzarkov, Kontsevich and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau-Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold $X$ there is a natural LG mirror $(Y,\mathsf{w})$ so that $h^{p,q}(X) = f^{3-q,p}(Y,\mathsf{w})$.
We consider the Berglund-Hübsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants.
We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group $G^\star$, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials., we prove that this mirror map is an isomorphism between the…
l-adic realization of some aspect of Landau-Ginzburg B-models
K.构造了具有孤立临界点的全纯函数芽的Landau-Ginzburg B-模型。Saito和M.齐藤Douai和Sabbah构造了一些Laurent多项式的Landau-Ginzburg B-模型。这种构造依赖于解析过程，不能指望用纯代数方法来完成。在本文中，我们给出了该构造的代数部分的l-adic实现。
This paper is an introduction to D-branes in Landau-Ginzburg models and Homological Mirror Symmetry. The paper is based on a series of lectures which were given on Second Latin Congress on Symmetries in Geometry and Physics that took place at the University of Curitiba, Brazil in December 2010.
The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map
本文基于内蕴镜像对称定义了光滑对数 Calabi--Yau 对的镜像真正 Landau--Ginzburg 势，证明其为两点相对 Gromov--Witten 不变量的生成函数，并在特定条件下证明了其与相对镜像映射互逆及与开镜像映射的猜想关系。
Non-affine Landau-Ginzburg models and intersection cohomology
我们构造了复曲面流形中数值有效的完全相交的Landau-Ginzburg模型，作为Laurent多项式族的部分紧化。我们显示一个镜像声明说，量子D-模的周围部分的上同调的子流形是同构的相交上同调D-模定义从这个部分紧化，我们推导出霍奇这些微分系统的性质。
Self-Duality for Landau--Ginzburg models
克拉克将镜像对称描述为朗道-金兹伯格模型之间的对偶，因此LG模型的对偶是另一个LG模型。我们描述的例子中，底层空间是一个全空间的向量丛上的射影线，我们表明，自对偶恰好发生在两种情况下：余切丛和解决conifold。
Fractional Calabi-Yau Categories from Landau-Ginzburg Models
给出了规范Landau-Ginzburg模型的导出范畴上Serre函子存在的判别准则.这是用来提供一个一般定理的存在性的一个容许的（分数）的Calabi-Yau子范畴的规范的Landau-Ginzburg模型和几何背景crepant分类决议。我们明确地描述了我们的框架中的复曲面设置。因此，我们推广了Orlov和Kuznetsov的几个定理和例子，最后给出了包含（分数）Calabi-Yau范畴的半正交分解的新例子。
McKay correspondence for Landau-Ginzburg models
本文证明了Landau-Ginzburg模型的McKay对应的一个类似。我们的证明是基于T. Bridgeland，A.金和M。Reid，用导出范畴的语言重新表述和推广了McKay对应，以及J. - C.尘
Compactifications of spaces of Landau-Ginzburg models
本文回顾了作者以前的工作“朗道-金兹伯格模型的辛态群关系和退化”的结果和技术，并将其应用于基本的例子。主要的例子是$A_n$类别，我们观察到的关系，稳定性条件和有向双线性表示。最后，我们简要回顾了德尔佩佐曲面的双有理几何的应用。
Mirror duality of Landau-Ginzburg models via Discrete Legendre Transforms
我们介绍了一个对偶的Landau-Ginzburg模型的概念的基础上给出的Gross-Siebert的离散勒让德变换。它概括了用于构建在最近与格罗斯和Katzarkov的联合工作复曲面品种的完整的交叉点的镜子的对偶。我们讨论了它如何表现为一个极限版本的镜像对称的半平坦Strominger-Yau-Zaslow建设与潜在的障碍，在弗洛尔理论。
Duality for toric Landau-Ginzburg models
我们介绍了一个对偶结构的复曲面朗道-金兹伯格模型，适用于完整的交叉口复曲面品种通过西格玛模型/朗道-金兹伯格模型对应。这种结构被证明是对Batyrev-Borisov、Berglund-H“ubsch、Givental和Hori-Vafa的结构的重构。它可以在更一般的情况下完成，并在上述结构给出奇异镜像时提供部分解决方案。给出了一个推广的例子：Landau-Ginzburg模型对偶于（P^1）^[2]中的椭圆曲线。
A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors
本文描述了Landau-Ginzburg模型LG（2）中sl（2，C）的半单伴随轨道的Szaya-Seidel范畴.我们证明了这个范畴等价于第二Hirzebruch曲面上凝聚层范畴的一个完全三角化子范畴。我们证明了没有投射簇可以镜像到LG（2），并且在紧化之后仍然如此。
Triangulated categories of singularities and equivalences between Landau-Ginzburg models
在本文中，我们证明了存在的三角形范畴之间的一些类型的等价性的奇异品种的不同尺寸。这类等价推广了所谓的克诺尔周期性。因此，我们得到了不同维数的Landau-Ginzburg模型上的B型D-膜范畴之间的等价性。
A proof of the Landau-Ginzburg/Calabi-Yau correspondence via the crepant transformation conjecture
我们建立了扭曲FJRW理论和局部Gromov-Witten理论在所有属中的一种新的关系（MLK对应）。因此，我们证明了亏格为零的Fermat型的crepant变换猜想所隐含的Landau-Ginzburg/Calabi-Yau对应.利用这一结果证明了Fermat型的Landau-Ginzburg/Calabi-Yau对应，推广了A. Chiodo和Y.阮。
We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.
Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras
我们证明了朗道-金斯堡镜像对称猜想的水平（orbifolded）Frobenius代数的一大类可逆奇点，包括任意和的循环和费马与任意对称群。特别地，我们证明了对于拟齐次多项式W和类内的容许群G，在[W/G]的FJRW理论中产生的Frobenius代数同构于（作为Frobenius代数）[W^T/G^T]的轨道折叠Milnor环，并与对偶多项式W^T和对偶群G^T相关联.
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