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.
Highlights
W-algebras are extensions of the Virasoro algebra by currents of higher spin [1, 2]
The Yangian picture is very useful for studying the representation theory
The representation theory in the Yangian picture reduces to a large extent to the combinatorics of box-counting
Summary
W-algebras are extensions of the Virasoro algebra by currents of higher spin [1, 2]. The most studied class of these is the family of algebras WN obtained by extending the Virasoro algebra by currents of spin 3, 4, . The original approach of Zamolodchikov [1] for W3 which was extended to W4 in [15, 16] was to solve the OPE associativity conditions directly. There is a construction of WN using the free field representation — the Miura transform [3, 4] — and constructions coming from the affine Lie algebra su(N ): the Casimir construction [17] and its generalization — the coset construction [18] — and the quantized Drinfeld-Sokolov reduction [19,20,21]
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