Instanton R-matrix and W $$ \mathcal{W} $$ -symmetry

Abstract

Abstract We study the relation between W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ 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.

Keywords

• Conformal and W Symmetry
• Integrable Field Theories
• Conformal Field Theory
• Higher Spin Symmetry