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 𝒩 = 2 quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an isotropic/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 𝔰𝔩(1|1) spin chain with fundamental representations using the corresponding 3d 𝒩 = 2 SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on 𝐼 × 𝔼 for an interval 𝐼 and an elliptic curve 𝔼 compute the elliptic stable envelopes, and in turn the geometric elliptic 𝑅-matrix, of the anisotropic 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.