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

Communications in Mathematical Physics Springer Verlag (Germany)

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

University of Zagreb / Sveučilište u Zagrebu

Croatia (HR) University of Alberta

Canada (CA)

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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