{
  "id": "2001.08048",
  "version": "v1",
  "published": "2020-01-22T15:05:56.000Z",
  "updated": "2020-01-22T15:05:56.000Z",
  "title": "The vertex algebras $\\mathcal R^{(p)}$ and $\\mathcal V^{(p)}$",
  "authors": [
    "Drazen Adamovic",
    "Thomas Creutzig",
    "Naoki Genra",
    "Jinwei Yang"
  ],
  "categories": [
    "math.RT",
    "hep-th",
    "math.QA"
  ],
  "abstract": "The vertex algebras $V^{(p)}$ and $R^{(p)}$ introduced in [2] are very interesting relatives of the famous triplet algebras of logarithmic CFT. The algebra $V^{(p)}$ (respectively, $R^{(p)}$) is a large extension of the simple affine vertex algebra $L_k(\\mathfrak{sl}_2)$ (respectively, $L_k(\\mathfrak{sl}_2)$ times a Heisenberg algebra), at level $k=-2+1/p$ for positive integer $p$. 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 $L_k(\\mathfrak{sl}_2)$-module and $R^{(p)}$ as an $L_k(\\mathfrak{gl}_2)$-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 [12], while the reduction of $R^{(p)}$ yields the $B^{(p)}$-algebra of [39]. 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 $KL_{k}$ of ordinary $L_k(\\mathfrak{sl}_2)$-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 $L_k(\\mathfrak{sl}_2)$ at all 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 $(A_1, D_{2p})$ and $(A_1, A_{2p-3})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-22T15:05:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum hamiltonian reduction",
      "rigid vertex tensor category equivalent",
      "rigid braided tensor category structures",
      "simple affine vertex algebra",
      "completes rigid braided tensor category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}