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

TL;DR

This paper extends the theory of elliptic stable envelopes to non-symplectic varieties, relates them to superspin chains via the Bethe/gauge correspondence, and explicitly solves the dynamical Yang-Baxter equation for an elliptic rak{sl}(1|1) spin chain.

Contribution

It generalizes elliptic stable envelopes to new classes of varieties, connects them to superspin chains and R-matrices, and provides explicit solutions for the rak{sl}(1|1) case.

Findings

Solved the dynamical Yang-Baxter equation for elliptic rak{sl}(1|1) spin chain.

Connected elliptic stable envelopes to superspin chains and R-matrices.

Reduced elliptic stable envelopes to K-theoretic and cohomological cases, recovering known results.

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 Rim\'anyi-Rozansky's results that appeared recently in the mathematical literature.