Let $\mathcal{A}_n \to \mathbb{C}^2 / \mathbb{Z}_{n+1}$ be the minimal resolution of $\mathcal{A}_n$-singularity and $X = \mathcal{A}_n \times \mathbb{C}^2$ be the associated toric Calabi–Yau $4$-fold. In this note, we study curve counting on $X$ from both Donaldson–Thomas and Gromov–Witten perspectives. In particular, we verify conjectural formulae relating them proposed by the author, Maulik and Toda.

中文翻译：

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$ \ mathcal {A} _n \ times \ mathbb {C} ^ 2 $上的曲线计数

设$ \ mathcal {A} _n \ to \ mathbb {C} ^ 2 / / mathbb {Z} _ {n + 1} $为$ \ mathcal {A} _n $奇异性和$ X = \的最小分辨率mathcal {A} _n \ times \ mathbb {C} ^ 2 $是相关的复曲面Calabi–Yau $ 4 $倍。在本说明中，我们从唐纳森·托马斯（Donaldson–Thomas）和格罗莫夫·威滕（Gromov–Witten）的角度研究基于$ X $的曲线计数。特别是，我们验证了作者Maulik和Toda提出的与它们有关的猜想公式。