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

TL;DR

This paper explores modules of corner vertex operator algebras from junctions in $ ext{N}=4$ SYM, generalizing transformations, parametrizations, and linking algebraic structures to gauge theory defects.

Contribution

It introduces a generalized Miura transformation for truncations of $ ext{W}_{1+ ext{infinity}}$, proposes a new parametrization of modules, and connects algebraic parameters to gauge theory defects.

Findings

Generalized Miura transformation for $ ext{W}_{N_1}$ truncations.

Parametrization of modules via free field exponents.

Connection between module parameters and Gukov-Witten defects.

Abstract

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.