Derived categories of small toric Calabi-Yau 3-folds and counting invariants

TL;DR

This paper establishes a derived equivalence for small crepant resolutions of affine toric Calabi-Yau 3-folds, derives wall-crossing formulas for counting invariants, and relates moduli spaces through quiver mutations and stability conditions.

Contribution

It constructs a new derived equivalence and develops wall-crossing formulas for counting invariants in the context of toric Calabi-Yau 3-folds, linking different moduli spaces via quiver mutations.

Findings

Derived equivalence between crepant resolutions and quivers with superpotential

Wall-crossing formulas for counting invariants of perverse coherent systems

Moduli spaces related through quiver mutations and stability changes

Abstract

We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the stability conditions.