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 $$(A_1, A_{2p-3})$$ .