Dimer models and crepant resolutions

TL;DR

This paper investigates how tautological bundles on quiver moduli spaces related to dimer models behave under stability parameter changes, establishing conditions for derived equivalences and connections to crepant resolutions.

Contribution

It proves that derived equivalences induced by tautological bundles persist across generic stability parameters and links all projective crepant resolutions to specific stability conditions.

Findings

Derived equivalence holds for all generic stability parameters.

All projective crepant resolutions can be realized as moduli spaces for some stability parameter.

The results facilitate proofs of the abelian McKay correspondence.

Abstract

We study variations of tautological bundles on moduli spaces of representations of quivers with relations associated with dimer models under a change of stability parameters. We prove that if the tautological bundle induces a derived equivalence for some stability parameter, then the same holds for any generic stability parameter, and any projective crepant resolution can be obtained as the moduli space for some stability parameter. This result is used in 0905.0059 to prove the abelian McKay correspondence without using the result of math/9908027.