Matrix Factorizations and Representations of Quivers I

TL;DR

This paper develops a mathematical framework connecting matrix factorizations, quiver representations, and D-branes in Landau-Ginzburg models, establishing equivalences and stability conditions within derived categories.

Contribution

It introduces a new categorical definition for D-branes in Landau-Ginzburg orbifolds and proves equivalences with quiver representation categories, also constructing a fundamental stability condition.

Findings

Category for polynomial x^{n+1} is equivalent to derived category of A_n quiver representations

Established a special stability condition in the sense of Bridgeland

Unified descriptions of matrix factorizations and quiver representations

Abstract

This paper introduces a mathematical definition of the category of D-branes in Landau-Ginzburg orbifolds in terms of $A_\infty$-categories. Our categories coincide with the categories of (graded) matrix factorizations for quasi-homogeneous polynomials. After setting up the necessary definitions, we prove that our category for the polynomial $x^{n+1}$ is equivalent to the derived category of representations of the Dynkin quiver of type $A_{n}$. We also construct a special stability condition for the triangulated category in the sense of T. Bridgeland, which should be the "origin" of the space of stability conditions.