Pursuing Quantum Difference Equations II: 3D mirror symmetry

Abstract

Abstract 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].