Webs of W-algebras

Abstract

We associate vertex operator algebras to $(p,q)$-webs of interfaces in the topologically twisted $\mathcal{N}=4$ super Yang-Mills theory. Y-algebras associated to trivalent junctions are identified with truncations of ${\mathcal{W}}_{1+\infty }$ algebra. Starting with Y-algebras as atomic elements, we describe gluing of Y-algebras analogous to that of the topological vertex. At the level of characters, the construction matches the one of counting D0-D2-D4 bound states in toric Calabi-Yau threefolds. For some configurations of interfaces, we propose a BRST construction of the algebras and check in examples that both constructions agree. We define generalizations of ${\mathcal{W}}_{1+\infty }$ algebra and identify a large class of glued algebras with their truncations. The gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions or coset constructions and provides us with a way to construct new algebras. Many well-known vertex operator algebras, such as $U(N{)}_k$ affine Lie algebra, $\mathcal{N}=2$ superconformal algebra, $\mathcal{N}=2$ super-${\mathcal{W}}_{\infty }$, Bershadsky-Polyakov ${\mathcal{W}}_3^{(2)}$, cosets and Drinfeld-Sokolov reductions of unitary groups can be obtained as special cases of this construction.

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