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

The vertex algebras $$\mathcal {V}^{(p)}$$ and R^{(p)}$$ introduced in Adamovi? (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet logarithmic CFT. algebra (respectively, {R}^{(p)}$$ ) is a large extension simple affine $$L_{k}(\mathfrak {sl}_{2})$$ times Heisenberg algebra), at level $$k=-2+1/p$$ for positive integer p. Particularly, {V}^{(2)}$$ small $$N=4$$ superconformal with $$c=-9$$ , {R}^{(2)}$$ $$L_{-3/2}(\mathfrak {sl}_3)$$ . In this paper, we derive structural results these prove various conjectures coming from representation theory physics. We show that SU(2) acts as automorphisms on {V}^{({p})}$$ decompose an -module {R}^{({p})}$$ $$L_k(\mathfrak {gl}_2)$$ -module. decomposition shows limit corner appearing context S-duality. also quantum Hamiltonian reduction doublet {A}^{({p})}$$ Milas (Contemp Math 602:23–38, 2013), while yields {B}^{({p})}$$ -algebra Creutzig et al. (Lett Phys 104(5):553–583, 2014). Conversely, realize via procedure deserves to be called inverse reduction. As corollary, obtain category $$KL_{k}$$ ordinary -modules rigid tensor equivalent twist $$\text {Rep}(SU(2))$$ This finally completes braided structures all complex levels k. establish uniqueness result certain operator extensions use both non-principal {W}$$ -algebras type A boundary admissible levels. same chiral Argyres-Douglas theories $$(A_1, D_{2p})$$ A_{2p-3})$$