Strebel Differentials and stable Matrix Factorizations

TL;DR

This paper explores the relationship between Strebel differentials on punctured surfaces and the construction of moduli spaces of matrix factorizations for dimer models, revealing a geometric correspondence with stability conditions.

Contribution

It establishes a novel link between quadratic Strebel differentials and stable matrix factorizations in dimer models, providing explicit formulas and a geometric interpretation.

Findings

Horizontal trajectories correspond to stable matrix factorizations

Vertical trajectories match the arrows of the dimer quiver

Categories of matrix factorizations can be glued from local data

Abstract

We study the connection between quadratic Strebel differentials on punctured surfaces and the construction of moduli spaces of matrix factorizations for dimer models using GIT-quotients. We show that for each consistent dimer model and each nondegenerate stability condition $\theta$ we can find a Strebel differential for which the horizontal trajectories correspond to the $\theta$-stable matrix factorizations and the vertical trajectories correspond to the arrows of the dimer quiver. We give explicit expressions for the $\theta$-stable matrix factorizations that can be deduced from these horizontal trajectories. Following ideas by Pascaleff and Sybilla we show that each nondegenerate stability condition gives rise to a sheaf of curved algebras coming from consistent dimer models. The corresponding categories of matrix factorizations can be glued together to form the category of matrix factorizations of the original dimer.