Berglund-Hübsch mirrors of invertible curve singularities via Floer theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916445467705344 |
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| author | Cho, Cheol-Hyun Choa, Dongwook Jeong, Wonbo |
| author_facet | Cho, Cheol-Hyun Choa, Dongwook Jeong, Wonbo |
| contents | We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror.
There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $Γ$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $Γ$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Berglund-Hübsch mirrors of invertible curve singularities via Floer theory Cho, Cheol-Hyun Choa, Dongwook Jeong, Wonbo Symplectic Geometry Rings and Algebras 53D37, 32S50, 13C14 We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $Γ$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $Γ$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $Γ$. |
| title | Berglund-Hübsch mirrors of invertible curve singularities via Floer theory |
| topic | Symplectic Geometry Rings and Algebras 53D37, 32S50, 13C14 |
| url | https://arxiv.org/abs/2410.14678 |