Fukaya category for Landau-Ginzburg orbifolds and Berglund-H\"ubsch homological mirror symmetry for curve singularities

Fukaya category for Landau-Ginzburg orbifolds and Berglund-H\"ubsch homological mirror symmetry for curve singularities

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Cheol-Hyun Cho, Seoul National University

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For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on 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 complete version of Berglund-H\"ubsch homological mirror symmetry and prove it for two variable cases. Namely, given one of the polynomials $f= x^p+y^q, x^p+xy^q,x^py+xy^q$ and a symmetry group $G$, we use Floer theoretic construction to obtain the transpose polynomial $f^t$ with the transpose symmetry group $G^t$ as well as an explicit A-infinity equivalence between the new Fukaya category of $(f,G)$ to the matrix factorization category of $(f^t, G^t)$. In this case, monodromy is mirror to the restriction of LG model to a hypersurface.  For ADE singularities, Auslander-Reiten quiver for indecomposable matrix factorizations were known from 80's, and we find the corresponding Lagrangians as well as surgery exact sequences. 

This is a joint work with Dongwook Choa and Wonbo Jung.