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

**Authors:** Nafiz Ishtiaque, Seyed Faroogh Moosavian, Yehao Zhou  
**arXiv:** [2308.12333](https://arxiv.org/abs/2308.12333)  
**Date:** 2023-08-23  
**Summary page:** https://www.summarizepaper.com/en/paper/2308.12333v1/

## Key points

- Authors explore generalization of Aganagic-Okounkov's theory of elliptic stable envelopes
- Focus on classical Higgs branches of 3d N=2 quiver gauge theories
- Establish connection between gauge theories and anisotropic/elliptic superspin chains through Bethe/Gauge Correspondence
- Use stable envelopes to compute R-matrix for spin chain solving dynamical Yang-Baxter equation (dYBE)
- Solve dYBE for elliptic sl(1|1) spin chain using corresponding 3d N=2 SQCD with Lascoux resolution of determinantal variety
- Janus partition functions used to compute stable envelopes and determine geometric R-matrix for sl(1|1) spin chain
- Explore reductions of elliptic stable envelopes and R-matrices to lower dimensions:
- Reduction to 2d yields K-theoretic stable envelopes and trigonometric R-matrix
- Further reduction to 1d produces cohomological stable envelopes and rational R-matrix aligning with Rimányi and Rozansky's findings
- Expands existing theory by applying it to non-symplectic varieties in order to solve dYBE for superspin chains through Bethe/Gauge Correspondence
- Sheds light on connections between gauge theories, spin chains, various types of stable envelopes, and R-matrices in different dimensions.

## Summary

In their paper titled "Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical R-Matrices for Superspin Chains from The Bethe/Gauge Correspondence," authors Nafiz Ishtiaque, Seyed Faroogh Moosavian, and Yehao Zhou explore the generalization of Aganagic-Okounkov's theory of elliptic stable envelopes to varieties that do not possess a holomorphic symplectic structure or polarization. Specifically, they focus on classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. Through the Bethe/Gauge Correspondence, they establish a connection between these gauge theories and anisotropic/elliptic superspin chains. This allows them to use stable envelopes derived from this correspondence to compute the R-matrix which solves the dynamical Yang-Baxter equation (dYBE) for the spin chain. To illustrate their findings, they solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using a corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is known as the Lascoux resolution of a determinantal variety. They demonstrate that certain Janus partition functions of this theory on $I \times \mathbb E$, where $I$ represents an interval and $\mathbb E$ denotes an elliptic curve, can be used to compute stable envelopes and subsequently determine the geometric R-matrix for the $\mathfrak{sl}(1|1)$ spin chain. The authors further explore reductions of elliptic stable envelopes and R-matrices to lower dimensions. The reduction to 2d yields K-theoretic stable envelopes and a trigonometric R-matrix; while further reducing it to 1d produces cohomological stable envelopes and a rational R-matrix which aligns with Rim\'anyi and Rozansky's findings. Overall, this paper expands on existing theory by applying it to non-symplectic varieties in order to solve the dYBE for superspin chains through the Bethe/Gauge Correspondence. It sheds light on connections between gauge theories, spin chains, various types of stable envelopes and R-matrices in different dimensions.