We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi–Yau $3$-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via “raising operators”. Conjecturally the COHA action extends to an action of the Drinfeld double by adding the “lowering operators”. In this paper, we show that the Drinfeld double is a generalization of the notion of the Cartan doubled Yangian defined earlier by Finkelberg and others. We extend this “$3d$ Calabi–Yau perspective” on the Lie theory furthermore by associating a root system to certain families of $X$. We formulate a conjecture that the above-mentioned action of the Drinfeld double factors through a shifted Yangian of the root system. The shift is explicitly determined by the moduli problem and the choice of stability conditions, and is expressed explicitly in terms of an intersection number in $X$. We check the conjectures in several examples, including a special case of an earlier conjecture of Costello.

中文翻译：

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上同调霍尔代数和环面 Calabi-Yau 上的反常相干滑轮 $3$-folds

我们研究 Kontsevich 和 Soibelman 意义上的（等变球面）上同调霍尔代数的 Drinfeld 双，与平滑复曲面 Calabi-Yau $3$-fold $X$ 相关。出于一般原因，COHA 通过“提升算子”作用于 $X$ 上某些反相干系统模空间的上同调。据推测，COHA 动作通过添加“降低运算符”扩展为 Drinfeld 双精度动作。在本文中，我们证明了德林菲尔德双值是芬克尔伯格等人之前定义的嘉当双值杨量概念的推广。我们通过将根系统与 $X$ 的某些家族相关联，进一步扩展了李理论的“$3d$ Calabi-Yau 视角”。我们提出这样一个猜想：Drinfeld 的上述作用是通过根系的杨根转移来实现的。该偏移由模量问题和稳定性条件的选择明确确定，并以 $X$ 中的交集数明确表示。我们在几个例子中检查猜想，包括科斯特洛早期猜想的一个特例。