Berglund-Hübsch mirrors of invertible curve singularities via Floer theory

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 $Γ$ 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 $Γ$.

Figures (33)

• Figure 1: Milnor fiber for $D_5^T=C_{2,4}$ and Lagrangians for indecomposable MF's
• Figure 2: Split generator $K$ in Fermat cases $F_{p,q}$
• Figure 3: $\mathrm{KS}^\bold b(\Gamma_W)$ for a chain type singularity
• Figure 4: Fundamental domain of $\mathbb{P}^{1}_{a,b,c}$ in $\mathbb{H}$
• Figure 5: Tessellations of $A_{4}$ and $D_{5}$ singularities

Theorems & Definitions (131)

• Definition 1.1
• Conjecture 1.2: Berglund--Hübsch homological mirror symmetry
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['thm:um']}
• Theorem 1.5: Theorem \ref{['thm:ADEAR']}
• Proposition 1.6
• Theorem 1.7
• Theorem 1.8: Theorem \ref{['thm:ksg']}
• Theorem 1.9
• Theorem 1.10: Theorem \ref{['thm:A3']}