The Vertex Algebras $\mathcal {R}^{(p)}$ and $\mathcal {V}^{({p})}$

Abstract

The vertex algebras $\mathcal {V}^{(p)}$ and $\mathcal R^{(p)}$ introduced in Adamović (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra $\mathcal {V}^{(p)}$ (respectively, $\mathcal {R}^{(p)}$) 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. Particularly, the algebra $\mathcal {V}^{(2)}$ is the simple small $N=4$ superconformal vertex algebra with $c=-9$, and $\mathcal {R}^{(2)}$ is $L_{-3/2}(\mathfrak {sl}_3)$. 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 $\mathcal {V}^{({p})}$ and we decompose $\mathcal {V}^{({p})}$ as an $L_{k}(\mathfrak {sl}_{2})$-module and $\mathcal {R}^{({p})}$ as an $L_k(\mathfrak {gl}_2)$-module. The decomposition of $\mathcal {V}^{({p})}$ shows that $\mathcal {V}^{({p})}$ 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 $\mathcal {V}^{({p})}$ is the logarithmic doublet algebra $\mathcal {A}^{({p})}$ introduced in Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of $\mathcal {R}^{({p})}$ yields the $\mathcal {B}^{({p})}$-algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize $\mathcal {V}^{({p})}$ and $\mathcal {R}^{({p})}$ from $\mathcal {A}^{({p})}$ and $\mathcal {B}^{({p})}$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $KL_{k}$ 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 $\text {Rep}(SU(2))$. This finally completes rigid braided tensor category structures for $L_{k}(\mathfrak {sl}_{2})$ at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $\mathcal {R}^{({p})}$ and $\mathcal {B}^{({p})}$ are certain non-principal $\mathcal {W}$-algebras of type A at boundary admissible levels. The same uniqueness result also shows that $\mathcal {R}^{({p})}$ and $\mathcal {B}^{({p})}$ are the chiral algebras of Argyres-Douglas theories of type $(A_1, D_{2p})$ and $(A_1, A_{2p-3})$.