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We study the structure of modules of corner vertex operator algebras arrisingat junctions of interfaces in $\mathcal{N}=4$ SYM. In most of the paper, weconcentrate on truncations of $\mathcal{W}_{1+\infty}$ associated to thesimplest 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, wepropose a simple parametrization of their generic modules, generalizing theYangian generating function of highest weight charges. Parameters of thegenerating function can be identified with exponents of vertex operators in thefree field realization and parameters associated to Gukov-Witten defects in thegauge theory picture. Finally, we discuss some aspect of degenerate modules. Inthe last section, we sketch how to glue generic modules to produce modules ofmore complicated algebras. Many properties of vertex operator algebras andtheir modules have a simple gauge theoretical interpretation.

$\mathcal{W}$-algebra Modules, Free Fields, and Gukov-Witten Defects