  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.
