Donaldson-Thomas invariants of $[\mathbb C^4/\mathbb Z_r]$
Xiaolong Liu
TL;DR
This work determines the zero-dimensional DT4 invariants of the orbifold $[\mathbb C^4/\mathbb Z_r]$, confirming the Cao–Kool–Monavarí conjecture by combining an orbifold degeneration framework with DT4 data for the crepant resolution $\mathcal A_{r-1}\times\mathbb C^2$ and a careful orientation analysis of Hilbert schemes. It develops a robust fourfold vertex formalism for both commutative and noncommutative Hilbert schemes, derives canonical sign rules, and employs relative/rubber DT4 theory and degeneration to reduce the global computation to tractable local pieces. The main result expresses the invariant as a MacMahon-type product with explicit refinements in terms of torus weights, and shows that, upon specialization, it reduces to the 3-fold DT theory, thereby validating a DT crepant resolution type conjecture in this orbifold setting. The methods fuse orbifold vertex calculus, quiver moduli, and square-root virtual pullbacks to control signs and pole structure, offering a blueprint for future DT4 computations on broader classes of orbifolds and their crepant resolutions.
Abstract
We compute the zero-dimensional Donaldson-Thomas invariants of the quotient stack $[\mathbb{C}^4/\mathbb{Z}_r]$, confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for $\mathcal{A}_{r-1}\times\mathbb{C}^2$ and an explicit determination of orientations of Hilbert schemes of points on $[\mathbb{C}^4/\mathbb{Z}_r]$.