W-symmetry, topological vertex and affine Yangian

TL;DR

This paper explores the deep connections between the $ ext{W}$-symmetry algebra, topological vertex, and affine Yangian, revealing new insights into their representation theory and combinatorial structures.

Contribution

It establishes a link between $ ext{W}_{1+ olinebreak} ext{infinity}$ algebra representations, topological vertex characters, and affine Yangian charges, providing new tools for their analysis.

Findings

Characters of degenerate $ ext{W}_{1+ olinebreak} ext{infinity}$ representations relate to topological vertex.

Yangian provides commuting charges diagonalizable in $ ext{W}_{1+ olinebreak} ext{infinity}$ representations.

Complex properties of $ ext{W}_{1+ olinebreak} ext{infinity}$ have simple combinatorial interpretations.

Abstract

We discuss the representation theory of non-linear chiral algebra $\mathcal{W}_{1+\infty}$ of Gaberdiel and Gopakumar and its connection to Yangian of $\hat{\mathfrak{u}(1)}$ whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of $\mathcal{W}_{1+\infty}$ are for generic values of parameters given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in $\mathcal{W}_{1+\infty}$ highest weight representations. Many properties that are difficult to study in $\mathcal{W}_{1+\infty}$ picture turn out to have a simple combinatorial interpretation.