We study quantum geometry of Nakajima quiver varieties of two different types - framed A-type quivers and ADHM quivers. While these spaces look completely different we find a surprising connection between equivariant K-theories thereof with a nontrivial match between their equivariant parameters. In particular, we demonstrate that quantum equivariant K-theory of $A_n$ quiver varieties in a certain $n\to\infty$ limit reproduces equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. We analyze the correspondence from the point of view of enumerative geometry, representation theory and integrable systems. We also propose a conjecture which relates spectra of quantum multiplication operators in K-theory of the ADHM moduli spaces with the solution of the elliptic Ruijsenaars-Schneider model.

中文翻译：

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A 型箭袋变种和 ADHM 模空间

我们研究了两种不同类型的 Nakajima 箭袋变种的量子几何 - 框架 A 型箭袋和 ADHM 箭袋。虽然这些空间看起来完全不同，但我们发现其等变 K 理论与它们的等变参数之间的非平凡匹配之间存在惊人的联系。特别是，我们证明了在某个 $n\to\infty$ 极限内的 $A_n$ 箭袋变体的量子等变 K 理论再现了 $\mathbb{C}^2$ 上点的希尔伯特方案的等变 K 理论。我们从枚举几何、表示论和可积系统的角度分析了对应关系。我们还提出了一个猜想，该猜想将 ADHM 模空间的 K 理论中的量子乘法算子的光谱与椭圆 Ruijsenaars-Schneider 模型的解相关联。