The Vertex Algebras R(p) and V(p).

The vertex algebras V (p) and 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 V (p) (respectively, R (p) ) is a large extension of the simple affine vertex algebra L k (sl 2) (respectively, L k (sl 2) times a Heisenberg algebra), at level k = - 2 + 1 / p for positive integer p. Particularly, the algebra V (2) is the simple small N = 4 superconformal vertex algebra with c = - 9 , and R (2) is L - 3 / 2 (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 V (p) and we decompose V (p) as an L k (sl 2) -module and R (p) as an L k (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 Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of R (p) yields the B (p) -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). 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 (sl 2) -modules at level k = - 2 + 1 / p is a rigid vertex tensor category equivalent to a twist of the category Rep (S U (2)) . This finally completes rigid braided tensor category structures for L k (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 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 2 p) and (A 1 , A 2 p - 3) . [ABSTRACT FROM AUTHOR]