Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence

Abstract

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal{N}=2$ quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the $R$-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal{N}=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I\times \mathbb{E}$ for an interval $I$ and an elliptic curve $\mathbb{E}$ compute the elliptic stable envelopes, and in turn the geometric elliptic $R$-matrix, of the anisotropic $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the $R$-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.