{
  "id": "2203.08188",
  "version": "v1",
  "published": "2022-03-15T18:37:42.000Z",
  "updated": "2022-03-15T18:37:42.000Z",
  "title": "Category $\\mathcal O$ for vertex algebras of $\\mathfrak{osp}_{1|2n}$",
  "authors": [
    "Thomas Creutzig",
    "Naoki Genra",
    "Andrew Linshaw"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT",
    "hep-th",
    "math.QA"
  ],
  "abstract": "We show that the affine vertex superalgebra $V^k(\\mathfrak{osp}_{1|2n})$ at generic level $k$ embeds in the equivariant $\\mathcal W$-algebra of $\\mathfrak{sp}_{2n}$ times $4n$ free fermions. This has two corollaries: (1) it provides a new proof that for generic $k$, the coset $\\text{Com}(V^k(\\mathfrak{sp}_{2n}), V^k(\\mathfrak{osp}_{1|2n}))$ is isomorphic to $\\mathcal W^\\ell(\\mathfrak{sp}_{2n})$ for $\\ell = -(n+1) + \\frac{k+n+1}{2k+2n+1}$, and (2) we obtain the decomposition of ordinary $V^k(\\mathfrak{osp}_{1|2n})$-modules into $V^k(\\mathfrak{sp}_{2n}) \\otimes \\mathcal W^\\ell(\\mathfrak{sp}_{2n})$-modules. Next, if $k$ is an admissible level and $\\ell$ is a non-degenerate admissible level for $\\mathfrak{sp}_{2n}$, we show that the simple algebra $L_k(\\mathfrak{osp}_{1|2n})$ is an extension of the simple subalgebra $L_k(\\mathfrak{sp}_{2n}) \\otimes {\\mathcal W}_{\\ell}(\\mathfrak{sp}_{2n})$. Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k(\\mathfrak{osp}_{1|2n})$-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of $\\mathcal W_\\ell(\\mathfrak{sp}_{2n})$-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k(\\mathfrak{sp}_{2n})$-modules are rigid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-15T18:37:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vertex algebras",
      "rigid vertex tensor supercategory",
      "inequivalent simple objects",
      "admissible level",
      "affine vertex superalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}