Let $\textsf {X}$ and $\textsf {X}^{!}$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $K$-theoretic limit of the elliptic duality interface is an equivariant $K$-theory class $\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $K$-theoretic stable envelopes to the $K$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $K(\textsf {X})$, such as action of quantum groups, quantum dynamical Weyl groups, $R$-matrices, etc., to those for $K(\textsf {X}^{!})$. In particular, we relate the wall $R$-matrices of $\textsf {X}$ to the $R$-matrices of the dual variety $\textsf {X}^{!}$. As an example, we apply our results to $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$—the Hilbert scheme of $n$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].

中文翻译：

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追求量子差分方程 II：3D 镜像对称

令 $\textsf {X}$ 和 $\textsf {X}^{!}$ 是一对关于 3D 镜像对称对偶的辛簇。椭圆对偶接口的$K$-理论极限是等变的$K$-理论类$\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$。我们证明这个类提供了对应 $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end {align*}$$ 将 $K$ 理论稳定包络映射到 $K$ 理论稳定包络。这种构造允许我们将 $K(\textsf {X})$ 的各种表示理论对象，例如量子群的作用、量子动力学外尔群、$R$-矩阵等，与 $K(\ textsf {X}^{!})$。特别是，我们将 $\textsf {X}$ 的墙 $R$-矩阵与对偶变量 $\textsf {X}^{!}$ 的 $R$-矩阵联系起来。例如，我们将结果应用于 $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$——$n$ 点的希尔伯特方案复平面。在这种情况下，我们从 [10] 中得出 Gorsky 和 ​​Negut 的猜想。