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The vertex algebras $\mathcal V^{;;;(p)};;;$ and $ \mathcal R^{;;; (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 Lk(sl2) (respectively, Lk(sl2) 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(sl3). 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 Lk(sl2)-module and R(p) as an Lk(gl2)-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 KLk of ordinary Lk(sl2)-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 Lk(sl2) 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 (A1, D2p) and (A1, A2p−3).

vertex algebras ; logarithmic vertex algebras

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