Title: Fukaya category for Landau-Ginzburg orbifolds and Berglund-H\"ubsch conjecture for invertible curve singularities
Abstract:For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on the wrapped Fukaya category of its Milnor fiber together with monodromy inform...For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on the wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a full version of Berglund-Hubsch homological mirror symmetry and prove it for the case of two variables. Namely, given one of the polynomials $W= x^p+y^q, x^p+xy^q,x^py+xy^q$ and a symmetry group $G$, we use Floer theoretic construction to obtain the transpose polynomial $W^T$ with the transpose symmetry group $G^T$ as well as derived equivalence between the new Fukaya category of $(W,G)$ and the matrix factorization category of $(W^T, G^T)$. In this case, monodromy corresponds to the restriction of LG model to a hypersurface in the mirror. For ADE singularities, Auslander-Reiten quivers for indecomposable matrix factorizations were known from 80's, and we find the corresponding Lagrangians as well as surgery exact triangles.Read More
Publication Year: 2020
Publication Date: 2020-10-19
Language: en
Type: preprint
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Cited By Count: 3
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