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Wolfgang Ebeling

Publications and source records attributed to Wolfgang Ebeling.

At least 19 recordsLinked to original sources
arXiv2022

Mirror symmetry on levels of non-abelian Landau--Ginzburg orbifolds

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.

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arXiv2021

Graded matrix factorizations of size two and reduction

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C.T.C.Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H.Kajiura, K.Saito, and the second author.

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arXiv2021

Indices of vector fields and 1-forms

We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.

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arXiv2021

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 C.T.C.Wall. We derive this duality from the mirror symmetry, the Berglund-Hübsch transposition of invertible polynomials, and our previous results about the strange duality between hypersurface and complete intersection singularities using matrix factorizations of size two.

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arXiv2020

Dual Invertible Polynomials with Permutation Symmetries and the Orbifold Euler Characteristic

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 permutation group is cyclic and satisfies PC, then the reduced orbifold Euler characteristics of the Milnor fibres of dual pairs coincide up to sign.

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arXiv2019

A version of the Berglund-Hübsch-Henningson duality with non-abelian groups

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 threefolds 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.

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arXiv2018

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. 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 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.

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arXiv2018

Lattices for Landau-Ginzburg orbifolds

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, Hübsch, 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.

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arXiv2018

A McKay correspondence for the Poincaré series of some finite subgroups of ${\rm SL}_3(\CC)$

A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that we obtain a similar relation between the group $G$ and the corresponding surface singularity.

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arXiv2017

An algebraic formula for the index of a 1-form on a real quotient singularity

Let a finite abelian group $G$ act (linearly) on the space $\mathbb{R}^n$ and thus on its complexification $\mathbb{C}^n$. Let $W$ be the real part of the quotient $\mathbb{C}^n/G$ (in general $W \neq \mathbb{R}^n/G$). We give an algebraic formula for the radial index of a 1-form on the real quotient $W$. It is shown that this index is equal to the signature of the restriction of the residue pairing to the $G$-invariant part $Ω^G_ω$ of $Ω_ω= Ω^n_{\mathbb{R}^n,0}/ω\wedge Ω^{n-1}_{\mathbb{R}^n,0}$. For a $G$-invariant function $f$, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function $f$, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form $df$ on the preimage $π^{-1}(W)$ of $W$ under the natural quotient map.

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arXiv2017

A note on distinguished bases of singularities

For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter-Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter-Dynkin diagrams like the exceptional unimodal singularities.

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arXiv2017

Orbifold Milnor lattice and orbifold intersection form

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.

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arXiv2015

Orbifold E-functions of dual invertible polynomials

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 the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of $f$. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.

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arXiv2015

Strange duality between hypersurface and complete intersection singularities

C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.

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arXiv2015

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 $\widehat{B}(G)$ of a finite group $G$ is defined. An element of it is represented by a finite $G$-set with a $G$-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 $G$-manifold with a $G$-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of $\widehat{B}(G)$. It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibres of Berglund-Hübsch dual invertible polynomials coincide up to sign and show that this implies the result about orbifold zeta functions of Berglund-Hübsch-Henningson dual pairs obtained earlier.

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