Quivers from Matrix Factorizations

TL;DR

Aspinwall and Morrison develop a practical framework to compute quivers and superpotentials for isolated 3D hypersurface singularities using matrix factorizations and noncommutative resolutions. By encoding the geometry in End_R(R⊕M) and employing Grassmannian Grassmann blowups, they connect derived categories of coherent sheaves on resolutions to quiver representations, including explicit examples with length-two exceptional fibers. They analyze universal and concrete length-two flops (Morrison–Pinkham, Laufer) through explicit matrix factorizations, A∞-structures, and Gröbner-basis methods, obtaining detailed quivers and superpotentials and establishing derived equivalences. The work also links these noncommutative resolutions to Landau–Ginzburg D-branes, proposing a conceptual bridge between inside-LG matrix factorizations and outside-sigma-model descriptions, with implications for brane physics and extremal transitions.

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.

Theorems & Definitions (6)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5