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

Adamović D, Creutzig T, Genra N, Yang J (2021)

**Publication Type:** Journal article

**Publication year:** 2021

### Journal

**Book Volume:** 383

**Pages Range:** 1207-1241

**Journal Issue:** 2

**DOI:** 10.1007/s00220-021-03950-1

### Abstract

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).

### Involved external institutions

### How to cite

**APA:**

Adamović, D., Creutzig, T., Genra, N., & Yang, J. (2021). The Vertex Algebras R(p) and V(p). *Communications in Mathematical Physics*, *383*(2), 1207-1241. https://doi.org/10.1007/s00220-021-03950-1

**MLA:**

Adamović, Dražen, et al. "The Vertex Algebras R(p) and V(p)." *Communications in Mathematical Physics* 383.2 (2021): 1207-1241.

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