Semi-steady non-commutative crepant resolutions via regular dimer models

TL;DR

This paper introduces semi-steady non-commutative crepant resolutions (NCCRs) and characterizes them via homotopy equivalence to regular dimer models, expanding the understanding of NCCRs in toric singularities.

Contribution

It defines semi-steady NCCRs and establishes their equivalence to homotopy classes of regular dimer models, generalizing previous results on steady NCCRs.

Findings

Semi-steady NCCRs correspond to homotopy equivalence to regular dimer models.

Regular dimer models produce semi-steady NCCRs.

Extension of the class of NCCRs beyond steady cases.

Abstract

A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a $3$-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a nice class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we introduce the notion of semi-steady NCCRs, and show a consistent dimer model gives a semi-steady NCCR if and only if it is homotopy equivalent to a regular dimer model.