The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$

Abstract

The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k({\mathfrak{sl}}_2)$ (respectively, $L_k({\mathfrak{sl}}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $V^{(p)}$ and we decompose $V^{(p)}$ as an $L_k({\mathfrak{sl}}_2)$-module and $R^{(p)}$ as an $L_k({\mathfrak{gl}}_2)$-module. The decomposition of $V^{(p)}$ shows that $V^{(p)}$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $V^{(p)}$ is the logarithmic doublet algebra $A^{(p)}$ introduced in [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of [39]. Conversely, we realize $V^{(p)}$ and $R^{(p)}$ from $A^{(p)}$ and $B^{(p)}$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $K{L}_k$ of ordinary $L_k({\mathfrak{sl}}_2)$-modules at level $k=-2+1/p$ is a rigid vertex tensor category equivalent to a twist of the category Rep$(SU(2))$. This finally completes rigid braided tensor category structures for $L_k({\mathfrak{sl}}_2)$ at all levels $k$. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $R^{(p)}$ and $B^{(p)}$ are certain non-principal W-algebras of type $A$ at boundary admissible levels. The same uniqueness result also shows that $R^{(p)}$ and $B^{(p)}$ are the chiral algebras of Argyres-Douglas theories of type $(A_1,D_{2p})$ and $(A_1,A_{2p-3})$.

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• Tensor categories for vertex operator superalgebra extensions
• Trialities of $\mathcal{W}$-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• A slow review of the AGT correspondence
• On ribbon categories for singlet vertex algebras
• Direct limit completions of vertex tensor categories
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Subregular W-algebras of type A
• Characterizing braided tensor categories associated to logarithmic vertex operator algebras
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Cosets of free field algebras via arc spaces
• On the structure of W-algebras in type A
• Tensor Categories arising from the Virasoro Algebra
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• $N=1$ super Virasoro tensor categories
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Feigin-Semikhatov conjecture and related topics