\( \mathcal{W} \)-algebra modules, free fields, and Gukov-Witten defects. (English) Zbl 1416.81195
Summary: We study the structure of modules of corner vertex operator algebras arising 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.
MSC:
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
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