Invertible Polynomial Research Articles

Exceptional collections for mirrors of invertible polynomials

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.

On homological mirror symmetry for chain type polynomials

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.

Orbifold Jacobian algebras for invertible polynomials

Orbifold Jacobian algebras for invertible polynomials

Derived factorization categories of non‐Thom–Sebastiani‐type sums of potentials

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 HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ of maximally graded matrix factorizations of an invertible polynomial f $f$ of chain type, and explicitly construct a full strong exceptional collection E 1 , ⋯ , E μ $E_1,\hdots ,E_{\mu }$ in HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ whose length μ $\mu$ is the Milnor number of the Berglund–Hübsch transpose f ∼ $\widetilde{f}$ of f $f$ . This proves a conjecture, which postulates that for an invertible polynomial f $f$ the category HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ admits a tilting object, in the case when f $f$ is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects E i $E_i$ , we explicitly determine the quiver with relations ( Q , I ) $(Q,I)$ which represents the endomorphism ring of the associated tilting object ⊕ i = 1 μ E i $\oplus _{i=1}^{\mu }E_i$ in HMF L f ( f ) $\operatorname{HMF}^{L_f}(f)$ , and in particular we obtain an equivalence HMF L f ( f ) ≅ D b ( mod k Q / I ) $\operatorname{HMF}^{L_f}(f)\cong \operatorname{D}^{\operatorname{b}}(\operatorname{mod}kQ/I)$ .

Gamma integral structure for an invertible polynomial of chain type

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.

Homological mirror symmetry for invertible polynomials in two variables

In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the B-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and, as an application, we prove derived equivalences between certain nodal stacky curves, some of whose irreducible components have non-trivial generic stabiliser.

Strange Duality Between the Quadrangle Complete Intersection Singularities

There is a strange duality between the quadrangle isolated complete intersection singularities discovered by the first author and Wall. We derive this duality from a variation of the Berglund–Hübsch transposition of invertible polynomials introduced in our previous work about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

Categorical Saito theory, II: Landau-Ginzburg orbifolds

Let W∈C[x1,⋯,xN] be an invertible polynomial with an isolated singularity at origin, and let G⊂SLN∩(C⁎)N be a finite diagonal and special linear symmetry group of W. In this paper, we use the category MFG(W) of G-equivariant matrix factorizations and its associated VSHS to construct a G-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of MFG(W) characterized by GWmax-equivariance. Conjecturally, this G-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of (15(x15+⋯+x55),Z/5Z) with its mirror dual B-model Landau-Ginzburg orbifold (15(x15+⋯+x55),(Z/5Z)4). In the case of the Quintic family W=15(x15+⋯+x55)−ψx1x2x3x4x5, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan [14].

Averages of Products and Ratios of Characteristic Polynomials in Polynomial Ensembles

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

Frameworks for Two-Dimensional Keller Maps

The classical Jacobian Conjecture asserts that every locally invertible polynomial self-map of the complex affine space is globally invertible. A Keller map is a (hypothetical) counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a map between the Picard groups of suitable compactifications of the affine plane, that satisfy a complicated set of conditions. This is essentially a combinatorial problem. Several solutions to it ("frameworks") are described in detail. Each framework corresponds to a large system of equations, whose solution would lead to a Keller map.

Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials

Let f be an invertible polynomial and G a group of diagonal symmetries of f . This note shows that the orbifold Jacobian algebra Jac (f,G) of (f,G) defined by [2] is isomorphic as a \mathbb Z/2\mathbb ZZ -graded algebra to the Hochschild cohomology \mathsf{HH}^*(\mathrm {MF}_G(f)) of the dg-category \mathrm {MF}_G(f) of G -equivariant matrix factorizations of f by calculating the product formula of \mathsf{HH}^*(\mathrm {MF}_G(f)) given by Shklyarov [10]. We also discuss the relation of our previous results to the categorical equivalence.

Maximally-graded matrix factorizations for an invertible polynomial of chain type

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 show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov, whose Euler matrix are calculated in terms of the “zeta function” of the inverse of the polarization.As a corollary, it turns out that the homological mirror symmetry holds at the level of the Grothendieck group in the following sense; the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the (quasi-)Seifert form (with a suitable sign).

Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

P. Berglund, T. H\"ubsch, 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 permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

On the orbifold Euler characteristics of dual invertible polynomials with non-abelian symmetry groups

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hubsch 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 under a special condition on the action of the group $G$: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair $(f,G)$. We show that, for an abelian group $G$, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group $G$ and the dual group $\widetilde{G}$ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

Abstract 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 that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

G-twisted Braces and Orbifold Landau–Ginzburg Models

Given an algebra with group $G$-action, we construct brace structures for its $G$-twisted Hochschild cochains. An an application, we construct $G$-Frobenius algebras for orbifold Landau-Ginzburg B-models and present explicit orbifold cup product formula for all invertible polynomials.

Lattices for Landau–Ginzburg orbifolds

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, Hubsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.

A Version of the Berglund–Hübsch–Henningson Duality With Non-Abelian Groups

Abstract 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 inspection of data on Calabi–Yau three-folds obtained from quotients by non-abelian groups shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers if and only if they satisfy PC.

Enhanced equivariant Saito duality

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring [Formula: see text] of a finite group [Formula: see text] is defined. An element of it is represented by a finite [Formula: see text]-set with a [Formula: see text]-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic [Formula: see text]-manifold with a [Formula: see text]-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of [Formula: see text]. It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier.

Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Let $X_\lambda$ and $X_\lambda'$ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].