$\mathcal{W}$-algebra Modules, Free Fields, 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.

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• Duality of subregular W-algebras and principal W-superalgebras
• Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations
• Instanton R-matrix and W-symmetry
• On Extensions of $\widehat{\mathfrak{gl}(m|n)}$ Kac-Moody algebras and Calabi-Yau Singularities
• A Note on Quiver Quantum Toroidal Algebra
• Liouville reflection operator, affine Yangian and Bethe ansatz
• Higgsed network calculus
• Rectangular W-(super)algebras and their representations
• Quantum toroidal algebras and solvable structures in gauge/string theory
• $q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation
• Mixed network calculus
• Gluing two affine Yangians of ${\mathfrak{gl}}_1$
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• A Note on Quiver Yangians and $\mathcal{R}$-Matrices
• Twisted holography of defect fusions
• Gauge origami and quiver W-algebras
• Quiver Yangians and $\mathcal{W}$-Algebras for Generalized Conifolds
• More on Affine Dynkin Quiver Yangians
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• R-matrices and Miura operators in 5d Chern-Simons theory
• On even spin $W_{\infty }$
• Bispectral duality and separation of variables from surface defect transition
• Higher rank FZZ-dualities
• Integrable $\mathfrak{gl}(n|n)$ Toda field theory and its sigma-model dual
• The Grassmannian VOA
• Correlator correspondences for subregular $\mathcal{W}$-algebras and principal $\mathcal{W}$-superalgebras