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

曲线奇点的同调 Berglund-Hübsch-Henningson 镜像对称

Homological Berglund-Hübsch-Henningson mirror symmetry for curve singularities

本文证明曲线奇点的同调 Berglund-Hübsch-Henningson 镜像对称，通过拉回超势实现等变性，并证明矩阵分解范畴存在倾斜对象，长度等于 FJRW 状态空间维数。

Polytope duality for families of $K3$ surfaces associated to transpose duality

We consider whether or not transpose-dual pairs, which is a Berglund--H$\ddot{\textnormal{u}}$bsch mirror studied by Ebeling and Ploog \cite{EbelingPloog}, extend to a polytope duality that has a potential to be lattice dual.

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 …

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.

Orbifold Euler characteristics for dual invertible polynomials

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.

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 …