Canonical form of matrix factorizations from Fukaya category of surface

TL;DR

This paper explores the homological mirror symmetry between a pair-of-pants surface and a singularity, providing a geometric interpretation of matrix factorizations and classifying indecomposable modules via geometric objects.

Contribution

It establishes a canonical form for matrix factorizations related to Cohen-Macaulay modules and links higher-multiplicity modules to local systems on the surface, extending previous work.

Findings

Higher-multiplicity modules correspond to local systems over geodesics

Explicit canonical form of matrix factorizations for xyz

Geometric interpretation of algebraic operations on modules

Abstract

This paper concerns homological mirror symmetry for the pair-of-pants surface (A-side) and the non-isolated surface singularity $xyz=0$ (B-side). Burban-Drozd classified indecomposable maximal Cohen-Macaulay modules on the B-side. We prove that higher-multiplicity band-type modules correspond to higher-rank local systems over closed geodesics on the A-side, generalizing our previous work for the multiplicity one case. This provides a geometric interpretation of the representation tameness of the band-type maximal Cohen-Macaulay modules, as every indecomposable object is realized as a geometric object. We also present an explicit canonical form of matrix factorizations of $xyz$ corresponding to Burban-Drozd's canonical form of band-type maximal Cohen-Macaulay modules. As applications, we give a geometric interpretation of algebraic operations such as AR translation and duality of maximal Cohen-Macaulay modules as well as certain mapping cone operations.