Combinatorics

Physics

Graph

Operator (biology)

Pure mathematics

Mathematics

Mathematical physics

Transcription factor

Gene

Jordan algebra

Epistemology

Algebra over a field

Philosophy

Algebra representation

Repressor

Vertex operator algebra

Gauge theory

Chemistry

Yangian

Simple (philosophy)

Quantum mechanics

Degenerate energy levels

Vertex (graph theory)

Biochemistry

Journal of High Energy Physics

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.

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects