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Mirror symmetry of invariants of the Berglund-Hübsch-Henningson-dual non-commutative orbifolds and its distribution among levelsтезисы доклада Тезисы
- Автор: Gusein-Zade S.M.
- Сборник: Geometry, Groups, Operator Algebras and Integrability, Abstracts of Talks
- Тезисы
- Год издания: 2022
- Место издания: Lomonosov Moscow State University Москва
- Первая страница: 10
- Последняя страница: 10
- Аннотация: The Berglund-H\"ubsch-Henningson duality was the first systematic attempt to construct mirror symmetric Landau-Ginzburg models. The initial data for an (orbifold) Landau-Ginzburg model is a pair $(f,G)$ consisting of a quasihomogeneous polynomial $f$ in several variables and a finite group of linear transformations preserving $f$. In the Berglund-H\"ubsch-Henningson construction $f$ was a so-called invertible polynomial and $G$ was a group of its diagonal symmetries. (A quasihomogeneous polynomial $f$ is called invertible if the number of monomials in it is equal to the number $n$ of variables, i.e.\ $f(\overline{x})=\sum_{i=1}^n a_i\prod_{j=1}^n x_j^{E_{ij}}$, $a_i\ne 0$ and $\det(E_{ij})\ne 0$. Without loss of generality one may assume that $a_i=1$.) From a pair $(f,G)$ of the described type one constructs a dual pair $(\widetilde{f},\widetilde{G})$. (One has $\widetilde{f}(\overline{x})=\sum_{i=1}^n \prod_{j=1}^n x_j^{E_{ji}}$.) Dual pairs $(f,G)$ and $(\widetilde{f},\widetilde{G})$ possess a number of ``mirror symmetry'' properties (for example, a symmetry of a number of orbifold invariants, the simplest of whom is the orbifold Euler characteristic). This duality was extended to pairs $(f,\hat{G})$, where $f$ is an invertible polynomial and $\hat{G}$ is the semidirect product of a group $G$ of diagonal symmetries of $f$ and a group $S$ of permutations of the coordinates preserving $f$ and $G$. The construction is based on ideas of A.Takahashi and therefore is called the Berglund-H\"ubsch-Henningson-Takahashi duality. It appears that dual pairs can pretend to be mirror symmetric only under a special restriction on the grpoup $S$ called the parity condition. For some dual pairs satisfying the parity condition, there were proved symmetries of such invariants as the orbifold Euler characteristic, orbifold monodromy zeta-function, orbifold E-function. One has the conjecture that the invariants under consideration are split into summands (or factors) corresponding to the conjugacy classes of elements of $S$ which possess the same symmetries (if the parity condition holds). This conjecture was verified for some cases.
- Добавил в систему: Гусейн-Заде Сабир Меджидович