Quivers from Matrix Factorizations

TL;DR

This paper explores how matrix factorizations can be used to compute quivers and superpotentials for hypersurface singularities, providing geometric insights into D-branes and noncommutative resolutions, especially for complex non-toric cases.

Contribution

It introduces a practical method using matrix factorizations to analyze quivers and superpotentials for hypersurface singularities, including non-toric examples with richer structures.

Findings

Matrix factorizations effectively compute quivers and superpotentials.

Explicit geometric interpretations of D-branes on resolutions are provided.

Non-toric singularities with complex structures are analyzed using this method.

Abstract

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions.