{
  "abstract": "The modular data of the doubles of the Haagerup--Izumi categories is a grafting of two finite metric groups (conjectured by Evans--Gannon, proved for untwisted categories by Gannon--Schopieray--Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ $S$-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy $φ$, a word in $L$ and $R$, the two groups are $A_\\mp=\\coker(φ\\mp1)$, with the torsion linking forms of the mapping tori of $φ$ and $-φ$. Over all words of trace $m^2+2$ these forms reproduce the Evans--Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon--Schopieray--Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, $c_{\\eff}=\\#L$, and the handle-gluing values $2^{\\#L}λ^{\\pm1}|A_-||A_+|$ at the identity classes, $λ$ the stretch factor. Each kernel vector of $φ\\mp1$ modulo $|A_\\mp|$ at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of $σV^*$, $c\\equiv\\#L-6\\#R\\pmod{24}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18512",
  "author_total": 1,
  "authors": [
    {
      "name": "David E Evans"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9zkvxaqdyz0b9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18512",
  "fulltext_url": null,
  "id": "9zkvxaqdyz0b9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18512",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221815188",
  "pdf_url": "https://arxiv.org/pdf/2610.18512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9zkvxaqdyz0b9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2610.18512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua",
  "title_text": "Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/9zkvxaqdyz0b9",
  "venue": "arXiv",
  "versions": []
}