Donaldson-Thomas invariants of $[\mathbb C^4/\mathbb Z_r]$

TL;DR

This paper calculates zero-dimensional Donaldson-Thomas invariants for the orbifold $[C^4/Z_r]$, confirming a prior conjecture using degeneration formulas and explicit orientation analysis.

Contribution

It introduces an orbifold degeneration formula approach to compute invariants and confirms a conjecture for the invariants of $[C^4/Z_r]$.

Findings

Confirmed the conjecture of Cao-Kool-Monavari.

Computed invariants for $[C^4/Z_r]$ using degeneration techniques.

Explicitly determined orientations of Hilbert schemes on the orbifold.

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 $[\mathbb{C}^4/\mathbb{Z}_r]$.