Generating toric noncommutative crepant resolutions

TL;DR

This paper introduces an algorithm to find all toric noncommutative crepant resolutions of 3D Gorenstein singularities, connecting quiver embeddings with dimer models and mutations, with potential extensions to higher dimensions.

Contribution

The paper presents a novel algorithm that systematically finds all toric noncommutative crepant resolutions and relates quiver embeddings to dimer models, advancing understanding of singularity resolutions.

Findings

All resolutions of a finite quotient of the conifold can be obtained by mutating a basic dimer model.

The algorithm embeds quivers in a 3D torus with relations as homotopy relations.

Potential extension of the method to higher-dimensional singularities.

Abstract

We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss how this algorithm might be extended to higher dimensional singularities.