# Instanton R-matrix and W-symmetry

## TL;DR

This paper explores the deep connection between $ ext{W}_{1+ ext{infinity}}$ algebra and the Arbesfeld-Schiffmann-Tsymbaliuk Yangian via the Maulik-Okounkov R-matrix, providing explicit formulas and a conformal field theory perspective.

## Contribution

It offers an explicit formula for the mixed R-matrix, identifies generators between two algebraic structures, and constructs conserved quantities in Calogero-Sutherland models using the Miura transformation.

## Key findings

- Derived an explicit formula for the mixed R-matrix on two Fock spaces.
- Proposed an explicit identification of algebra generators via free field representation.
- Constructed conserved quantities and ladder operators in Calogero-Sutherland models.

## Abstract

We study the relation between $\mathcal{W}_{1+\infty}$ algebra and Arbesfeld-Schiffmann-Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts.