W-symmetry, topological vertex and affine Yangian

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 $\widehat{\mathfrak{u}(1)}$ 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.

JHEP version

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