David Favero, Daniel Kaplan, Tyler L. Kelly
We prove the existence of a full exceptional collection for the derived category of equivariant matrix factorizations of an invertible polynomial with its maximal symmetry group. This proves a conjecture of Hirano--Ouchi. In the Gorenstein case, we also prove a stronger version of this conjecture due to Takahashi. Namely, that the full exceptional collection is strong.
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We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category which does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.
We prove that the Borcea-Voisin mirror pairs of Calabi-Yau threefolds admit projective birational models that satisfy the Berglund-Hübsch-Chiodo-Ruan transposition rule. This shows that the two mirror constructions provide the same mirror pairs, as soon as both can be defined.
In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.
A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if k < 15, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Mü…
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold.
In this article we survey recent results of joint work with Lutz Hille on exceptional sequences of invertible sheaves on rational surfaces and give examples.
Homological mirror symmetry for invertible polynomials in two variables
针对二元可逆多项式，本文在B侧采用极大对称群，借助Milnor纤维的显式粘合构造证明同调镜像对称，并导出若干含非平凡一般稳定子的节点堆叠曲线间的等价关系。
Let $X$ be a toric Del-Pezzo surface and let $Crit(W) \subset (\mathbb{C}^{\ast})^n$ be the solution scheme of the Landau-Ginzburg system of equations. Denote by $X^{\circ}$ the polar variety of $X$. Our aim in this work is to describe a map $L : Crit(W) \rightarrow Fuk_{trop}(X^{\circ})$ whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.
A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be described in terms of a marginal deformation of an invertible polynomial $W$. In the papers \cite{KS} and \cite{MR} the authors proved a mirror symmetry statement for some particular choices of $W$ and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold $\mathbb{P}^1$s. However, the choice of the polynomial $W$ and its marginal deformation $ϕ_μ$ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family $W+σϕ_μ$. In each case the m…
The notion of the Gamma integral structure for the quantum cohomology of an algebraic variety was introduced by Iritani, Katzarkov-Kontsevich-Pantev. In this paper, we define the Gamma integral structure for an invertible polynomial of chain type. Based on the $Γ$-conjecture by Iritani, we prove that the Gamma integral structure is identified with the natural integral structure for the Berglund-Hübsch transposed polynomial by the mirror isomorphism.
In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hübsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only und…
We study the categories of singularities coming from Landau-Ginzburg models given by the invertible polynomials. Such categories appear on the B-side of the Berglund-Hübsch mirror symmetry. We provide an efficient method of computing morphism spaces in these categories and explicitly construct full strongly exceptional collections in the cases of small dimensions ($n\le 3$). Finally, we use this construction in order to prove Orlov's conjecture stating that such collections can be chosen to have block decompositions of size one more than the number of variables.
In 1977, Orlik--Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles. The purpose of this paper is to prove the algebraic counterpart of the Orlik--Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund--Hübsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we s…
Motivated by mirror symmetry for weighted projective spaces, we give a new characterization of exceptionoal unimodal singularities.
We give a survey on results related to the Berglund-Hübsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
An invertible polynomial in $n$ variables is a quasihomogeneous polynomial consisting of $n$ monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau--Ginzburg models, P.~Berg\-lund, T.~Hübsch and M.~Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold zeta functions of dual pairs $(f,G)$ and $(\widetilde{f}, \widetilde{G})$ and sh…
In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction tec…
We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.
In this paper, we confirm a conjecture of Orlik-Randell from 1977 on the Seifert form of chain type invertible singularities. We use Lefschetz bifibration techniques as developed by Seidel (inspired by Arnold and Donaldson) and take advantage of the symmetries at hand. We believe that our method will be useful in understanding the homological/categorical version of Berglundt-Hubsch mirror conjecture for invertible singularities.
Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak Fano surface a toric system, which defines a new toric surface. In this note we interprete this construction as an instance of mirror symmetry and extend it to a duality on the set toric weak Fano surfaces equiped with a cyclic full strongly exceptional sequence.
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.
To construct mirror symmetric Landau-Ginzburg models, P.Berglund, T.Hübsch and M.Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold Euler characteristics of the Milnor fibres of $f$ and $\widetilde f$ with the actions of the groups $G$ and $\widetilde G$ respectively and show that they coincide up to a sign.
An invertible polynomial is a quasihomogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P.~Berglund and M.~Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. We consider the so-called orbifold E-function of such a pair $(f,G)$ which is a generating function for …
Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections
本文研究Fano三重积分Picard秩为一的Landau-Ginzburg模型，探讨其与例外集合的关系，并验证Homological Mirror Symmetry的预测。
We develop an approach to calculating the cup and cap products on Hochschild cohomology and homology of curved algebras associated with polynomials and their finite abelian symmetry groups. For polynomials with isolated critical points, the approach yields a complete description of the products. We also reformulate the result for the corresponding categories of equivariant matrix factorizations. In an Appendix written jointly with Alexey Basalaev, we apply the formulas to calculate the Hochschild cohomology of a simple but non-trivial class of so-called invertible LG orbifold models. The resul…
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.
Invertible Calabi-Yau Orbifolds over Finite Fields
在 Berglund-Huebsch 镜像对称背景下，计算 Frobenius 自同态在 Borisov 复形 p-adic 版本上的特征值，并猜想有限域上可逆 Calabi-Yau 轨道形 crepant 解的有理点数的显式公式。
Mirror symmetry for Berglund-Hübsch Milnor fibers
该研究通过先计算甚仿射Fukaya范畴再进行形变的方法，计算Berglund-Hübsch可逆多项式的Milnor纤维的Fukaya范畴，大体证明了相关同调镜像对称猜想，核心是确定反常schober的典范延拓。
P. Berglund, T. Hübsch, and M. Henningson proposed a method to construct mirror symmetric Calabi-Yau manifolds. They considered a pair consisting of an invertible polynomial and of a finite (abelian) group of its diagonal symmetries together with a dual pair. A. Takahashi suggested a method to generalize this construction to symmetry groups generated by some diagonal symmetries and some permutations of variables. In a previous paper, we explained that this construction should work only under a special condition on the permutation group called parity condition (PC). Here we prove that, if the p…
In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on $\mathbb P^n$, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection $\mathcal O_{\mathbb P^n}(-n-1),...,\mathcal O_{\mathbb P^n}(-1)$ is mapped to standard Lagrangians in the sense of \cite{nz}. Passing to constructible sheaves, we explicitly compute the quiver structure of these Lagrangians, and find that they match the quiver structure of this exceptional collection of $\mathbb P^n$. In this way, T-duality prov…
Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-Hübsch transpose. This was previously shown for Brieskorn-Pham and $D$-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side.
We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed $A_{\infty}$-categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya-Seidel categories, most notably a generation statement.
This is a report on the recent work "Mirror symmetry for exceptional unimodular singularities" joint with Changzheng Li, Si Li and Yefeng Shen.
We discuss the relation between the graded stable derived category of a hypersurface and that of its hyperplane section. The motivation comes from the compatibility between homological mirror symmetry for the Calabi-Yau manifold defined by an invertible polynomial and that for the singularity defined by the same polynomial.
We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the category ${\rm HMF}^{L_f}(f)$ of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type, and explicitly construct a full strong exceptional collection $E_1$,..., $E_μ$ in ${\rm HMF}^{L_f}(f)$ whose length $μ$ is the Milnor number of the Berglund--Hübsch transpose $\widetilde{f}$ of $f$. This proves a conjecture, which postulates t…
We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory of [W/G] is isomorphic (as a Frobenius algebra) to the orbifolded Milnor ring of [W^T/G^T], associated to the dual polynomial W^T and dual group G^T.
We evaluate half-indices of $\mathcal{N}=(2,2)$ half-BPS boundary conditions in 3d $\mathcal{N}=4$ supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror trans…
Mirror partner for a Klein quartic polynomial
本文研究了Klein四次多项式与Landau-Ginzburg奇点的镜像关系，证明其镜像奇点为Klein四元群的Z/2Z扩张。
A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a "non-abelian" generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition"). An inspect…
We study complete exceptional collections of coherent sheaves over Del Pezzo surfaces, which consist of three blocks such that inside each block all Ext groups between the sheaves are zero. We show that the ranks of all sheaves in such a block are the same and the three ranks corresponding to a complete 3-block exceptional collection satisfy a Markov-type Diophantine equation that is quadratic in each variable. For each Del Pezzo surface, there is a finite number of these equations; the complete list is given. The 3-string braid group acts by mutations on the set of complete 3-block exceptiona…
On the mirrors of low-degree del Pezzo surfaces
本文比较了度数 $d \leq 3$ 的 del Pezzo 曲面的不同镜像构造，通过提取 Lefschetz 纤维化并利用类曲面伪格理论与突变方法，证明了这些纤维化定义了范畴镜像，并建立了相关 Fukaya-Seidel 范畴的等价性。
We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold's strange duality between the 14 exceptional unimodal singularities.
Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries $G$. Duality between $\mathcal{A}$ and $\mathcal{B}$ models has been conjectured for particular choices of $W$ and $G$. These conjectures have been proven in many instances where $W$ is restricted to having the same number of monomials as variables (called $\textit{invertible}$). Some conjectures have been made regarding isomorphisms between $\mathcal{A}$ and …
Full exceptional collections for anticanonical log del Pezzo surfaces
本文受同调镜像对称启发，利用 McKay 对应与极小解消，为 Johnson–Kollár 对数 del Pezzo 曲面对应典范栈构造显式完全例外集，并刻画相关二次 del Pezzo 曲面。
Motivic mirror symmetry and $χ$-independence for Higgs bundles in arbitrary characteristic
本文证明了SL_n和PGL_n-Higgs丛在任意特征下的动机同构性，并建立了GL_n-Higgs丛的动机χ独立性。
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。
The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the exis…
Rabinowitz Fukaya categories as cluster categories
探讨可逆多项式双悬架的米尔诺纤维的拉比诺维茨富卡亚范畴的同调镜像对称，对非卡拉比 - 丘型布里斯克恩 - 法姆多项式证明该对称，可据此计算米尔诺纤维的拉比诺维茨弗洛尔同调。
The number of full exceptional collections modulo spherical twists for extended Dynkin quivers
本文计算了扩展 Dynkin 箭图对应的正欧拉示性数轨形射影线凝聚层范畴的全例外序列模球形扭转的数量，发现其与 Lyashko-Looijenga 映射次数一致，揭示了稳定性条件与镜像对称的关联。
This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when $\dim X \geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKa…
We prove the Dubrovin's conjecture for the Stokes matrices for the quantum cohomology of orbifold projective lines. The conjecture states that the Stokes matrix of the first structure connection of the Frobenius manifold constructed from the Gromov-Witten theory coincides with the Euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves. Our proof is based on the homological mirror symmetry, primitive forms of affine cusp polynomials and the Picard-Lefschetz theory.
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