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

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 $\mathrm{3d}$ $\mathcal N=2$ 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 $\mathrm{3d}$ $\mathcal N=2$ 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 $\mathrm{2d}$ and $\mathrm{1d}$ reductions of elliptic stable envelopes and the $R$-matrix. The reduction to $\mathrm{2d}$ gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to $\mathrm{1d}$ produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi–Rozansky's results that appeared recently in the mathematical literature.