Berglund-H\"ubsch mirrors of invertible curve singularities via Floer theory

TL;DR

This paper introduces a Floer theoretic approach to construct mirror transpose polynomials for invertible curve singularities, linking Lagrangian Floer theory with matrix factorizations and revealing new geometric interpretations of algebraic structures.

Contribution

It provides an intrinsic Floer theoretic method to obtain the transpose polynomial and constructs a canonical A-infinity functor connecting Lagrangians and matrix factorizations, with applications to ADE singularities.

Findings

Constructed mirror transpose polynomials via Floer theory.

Established a functor linking Lagrangians to matrix factorizations.

Connected Auslander-Reiten sequences with Lagrangian surgery triangles.

Abstract

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 $\Gamma$ 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 $\Gamma$ 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 $\Gamma$.