# Higher spin sl_2 R-matrix from equivariant (co)homology

## TL;DR

This paper computes the $rak{sl}_2$ $R$-matrix for higher spin representations using equivariant (co)homology of algebraic varieties, extending methods from Nakajima quiver varieties to singular higher spin cases.

## Contribution

It introduces a method to compute the $rak{sl}_2$ $R$-matrix for higher spin representations via equivariant (co)homology of singular algebraic varieties, generalizing Nakajima quiver varieties.

## Key findings

- Explicit formula for the higher spin $rak{sl}_2$ $R$-matrix.
- Extension of geometric methods to singular varieties.
- Connection between algebraic geometry and quantum integrable systems.

## Abstract

We compute the rational $\mathfrak{sl}_2$ $R$-matrix acting in the product of two spin-$\ell\over 2$ (${\ell \in \mathbb{N}}$) representations, using a method analogous to the one of Maulik and Okounkov, i.e., by studying the equivariant (co)homology of certain algebraic varieties. These varieties, first considered by Nekrasov and Shatashvili, are typically singular. They may be thought of as the higher spin generalizations of $A_1$ Nakajima quiver varieties (i.e., cotangent bundles of Grassmannians), the latter corresponding to $\ell=1$.