  We study the structure of modules of corner vertex operator algebras arrising
at junctions of interfaces in $\mathcal{N}=4$ SYM. In most of the paper, we
concentrate on truncations of $\mathcal{W}_{1+\infty}$ associated to the
simplest trivalent junction. First, we generalize the Miura transformation for
$\mathcal{W}_{N_1}$ to a general truncation $Y_{N_1,N_2,N_3}$. Secondly, we
propose a simple parametrization of their generic modules, generalizing the
Yangian generating function of highest weight charges. Parameters of the
generating function can be identified with exponents of vertex operators in the
free field realization and parameters associated to Gukov-Witten defects in the
gauge theory picture. Finally, we discuss some aspect of degenerate modules. In
the last section, we sketch how to glue generic modules to produce modules of
more complicated algebras. Many properties of vertex operator algebras and
their modules have a simple gauge theoretical interpretation.
