Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities

TL;DR

This paper establishes a homological mirror symmetry correspondence between indecomposable Cohen-Macaulay modules over certain degenerate cusp singularities and geometric objects like closed geodesics on a pair of pants, using Fukaya categories and matrix factorizations.

Contribution

It constructs an explicit geometric $A_{ abla}$-functor linking Cohen-Macaulay modules to the Fukaya category of a pair of pants, and computes canonical matrix factorizations for these modules.

Findings

One-to-one correspondence between geodesics and Cohen-Macaulay modules of multiplicity one.

Explicit Macaulayfications and canonical matrix factorizations for modules.

Mirror images of modules over $W = x^{3} + y^{2} - xyz$ as loops in an orbifold sphere.

Abstract

Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for $xyz$. Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of a hyperbolic pair of pants have a one-to-one correspondence with indecomposable Cohen-Macaulay modules for $xyz$ with multiplicity one that are locally free on the punctured spectrum. In particular, this correspondence is established first by a geometric $A_{\infty}$-functor from the Fukaya category of the pair of pants to the matrix factorization category of $xyz$, and next by the correspondence between Cohen-Macaulay modules and matrix factorizations due to Eisenbud. For the latter, we compute explicit Macaulayfications of modules from Burban-Drozd's classification and find a canonical form of the corresponding matrix factorizations. In the sequel, we will show that indecomposable modules with higher multiplicity correspond to twisted complexes of closed geodesics. We also find mirror images of rank $1$ indecomposable Cohen-Macaulay modules (of band type) over the singularity $W = x^{3} + y^{2} - xyz$ as closed loops in the orbifold sphere $\mathbb{P}^1_{3,2,\infty}$.