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Y-algebras form a four parameter family of vertex operator algebras associated to Y-shaped junctions of interfaces in the N=4 super Yang-Mills theory. One can glue such Y-shaped junctions into the more complicated webs of interfaces. Corresponding vertex operator algebras can be identified with conformal extensions of tensor products of Y-algebras associated to the trivalent junctions of the web by fusions of Y-algebra bi-modules associated to the finite interfaces. At the level of characters, the construction is analogous to the topological vertex like counting of D0-D2-D4 bound states in toric Calabi-Yau manifolds. Gluing construction sheds new light on the structure of vertex operator algebras conventionally constructed by BRST reductions and provides us with a way to construct new algebras.