{
  "id": "1701.03344",
  "version": "v1",
  "published": "2017-01-12T14:11:58.000Z",
  "updated": "2017-01-12T14:11:58.000Z",
  "title": "Representations of superconformal algebras and mock theta functions",
  "authors": [
    "Victor G. Kac",
    "Minoru Wakimoto"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\\hat{\\frak{g}}$ span an $SL_2(\\mathbf{Z})$-invariant space. This result extends to admissible $\\hat{\\frak{g}}$-modules, where $\\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\\hat{\\frak{g}}$-modules when $\\frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invariance occurs for boundary level admissible modules, including when $\\frak{g}$ is a basic Lie superalgebra. For example, if $\\frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $\\hat{\\frak{g}}$-modules, whose QHR produces the minimal series modules for the $N=2$ superconformal algebras (resp. a modular invariant family of $N=3$ superconformal algebra modules). However, in the case when $\\frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $\\hat{\\frak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $N=2,3,4$ and big $N=4$ superconformal algebras, whose modified (super)characters span an $SL_2(\\mathbf{Z})$-invariant space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-12T14:11:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "superconformal algebra",
      "mock theta functions",
      "modular invariant family",
      "minimal series modules",
      "basic lie superalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}