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 stable envelopes, and in turn the geometric R-matrix, of the $\mathfrak{sl}(1|1)$ 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 the results of Rim\'anyi and Rozansky.