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.
Categories: hep-th math-ph math.MP