Subregular W-algebras of type A

Abstract

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the ${\mathfrak{sl}}_{n+1}$ subregular W-algebra can be realised in terms of the ${\mathfrak{sl}}_{n+1}$ regular W-algebra and the half lattice vertex algebra $\Pi$. This generalises the realisations found for $n=1$ and $2$ in [arXiv:1711.11342, arXiv:2007.00396] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realisation to explore the representation theory of ${\mathfrak{sl}}_{n+1}$ subregular W-algebras. Much of the structure encountered for ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$ is also present for ${\mathfrak{sl}}_{n+1}$. Particularly, the simple ${\mathfrak{sl}}_{n+1}$ subregular W-algebra at nondegenerate admissible levels can be realised purely in terms of the ${\mathsf{W}}_{n+1}$ minimal model vertex algebra and $\Pi$.

arXiv comment28 pages, 1 figure

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• Modularity of Bershadsky-Polyakov minimal models
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• On the structure of W-algebras in type A
• Orthosymplectic Feigin-Semikhatov duality
• Connecting affine $\mathcal{W}$-algebras: A case study on ${\mathfrak{sl}}_4$
• Feigin-Semikhatov conjecture and related topics