{
  "abstract": "In a companion paper arXiv:2610.26610, we classified the Narain conformal field theories obtained by gluing copies of the $A_3$ root and weight lattices along their $\\mathbb{Z}_4$ discriminant group. For two hyperbolic pairs, this gives twenty-two even self-dual gluing codes, which fall into seven inequivalent theories at $c=\\bar c=6$. In this paper, we develop the physical interpretation of this family and its connections to three-dimensional topological field theory. We show that all twenty-two codes are topological boundary conditions of a single abelian Chern-Simons theory. Its invertible symmetries, which include fractional $B$-field shifts, organize the boundaries into three orbits: free, mixed and torsion. Boundaries in different orbits are related only by gauging. We determine all gaugings of order at most four between the seven theories, by $\\mathbb{Z}_2$, $\\mathbb{Z}_4$ and $\\mathbb{Z}_2\\times\\mathbb{Z}_2$, with explicit twisted partition functions, and follow their effect on the spectrum and the current algebra. We then sum the Chern-Simons partition function over handlebodies at genus one and prove a finite Siegel-Weil identity, valid for any number of pairs. For two pairs, the handlebody sum is the uniform average over the free theories. Recovering the average over all twenty-two theories requires one additional contribution, a solid torus with the torsion lines along its core. For three or more pairs, we find that the handlebody sum is not a positive combination of code boundary states.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30525",
  "author_total": 2,
  "authors": [
    {
      "name": "Rajae Sammani"
    },
    {
      "name": "Abdellah Sebbar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/25930t15j8fv7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30525",
  "fulltext_url": null,
  "id": "25930t15j8fv7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30525",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7223272607",
  "pdf_url": "https://arxiv.org/pdf/2610.30525",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/25930t15j8fv7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.30525",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30525"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "TQFT gravity dual to the orbifold groupoid of Narain code CFTs over $\\mathbb{Z}_4$",
  "title_text": "TQFT gravity dual to the orbifold groupoid of Narain code CFTs over ℤ_4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:34Z",
  "url": "https://research.draconitic.com/paper/25930t15j8fv7",
  "venue": "arXiv",
  "versions": []
}