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

TL;DR

This work establishes a concrete bridge between small crepant resolutions of affine toric Calabi–Yau $3$-folds and quivers with superpotential via a explicit derived equivalence mediated by a tilting bundle. It then develops a comprehensive counting framework for perverse coherent systems, proving a wall-crossing formula for generating functions that interrelates $DT$, $PT$, and noncommutative $DT$ invariants, including explicit product expansions and $BPS$-state data. Mutations of the quiver and corresponding stability conditions are shown to preserve the moduli-space descriptions, with the tilting generator remaining a vector bundle and fixed-point data governed by torus-crystal combinatorics. The results provide structural insights into the DT/PT/NCDT landscape in the toric CY setting, enabling explicit computations and revealing deep links between geometry, representation theory, and wall-crossing phenomena.

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.

Figures (11)

• Figure 1: $\Gamma_\sigma$
• Figure 2: S
• Figure 3: H
• Figure 4: $P_\sigma$ in case Example \ref{['example']}
• Figure 5: Universal representations on ${\mathfrak{M}}^{\sigma}_{\theta_0}(\delta)$ in case Example \ref{['example']}

Theorems & Definitions (119)

• Definition 1.1
• Example 1.2
• Lemma 1.3
• proof
• Example 1.4
• Lemma 1.5
• proof
• Proposition 1.6
• proof
• Proposition 1.7