{
  "abstract": "We present bulk-primary operator-product coefficients for every Virasoro minimal model with an $E_6$, $E_7$, or $E_8$ modular invariant. Dividing by a universal product of chiral generalized-minimal-model OPE coefficients reduces the conformal bootstrap problem to a finite algebraic problem governed by root-of-unity quantum-group $6j$ symbols. The exceptional factors of the OPE coefficients can be chosen independent of the second Kac indices $s$, and their dependence on the central charge reduces to finitely many reference cases and explicit phases. A finite computer-assisted calculation yields candidate algebraic $s=1$ seed data and high-precision numerical evidence for existence and uniqueness modulo primary-field signs. We prove that exact existence and uniqueness for these seeds imply the corresponding result for all Kac labels within the genus-zero bulk locality and crossing equations; the finite seed step remains uncertified. The three families require $18$, $32$, and $107$ representative reduced OPE coefficients for $E_6$, $E_7$, and $E_8$, respectively. We provide reconstruction conventions and coefficient tables for both unitary and nonunitary models. Our calculation assumes only Virasoro symmetry, bulk locality (permutation symmetry) and OPE associativity (crossing equations).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25215",
  "author_total": 3,
  "authors": [
    {
      "name": "Amaury Lhoste"
    },
    {
      "name": "Jiaxin Qiao"
    },
    {
      "name": "Masahito Yamazaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fr55m2n1d7w6g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25215",
  "fulltext_url": null,
  "id": "fr55m2n1d7w6g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25215",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214071014",
  "pdf_url": "https://arxiv.org/pdf/2609.25215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fr55m2n1d7w6g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:36Z",
      "native_id": "2609.25215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Bulk OPE Coefficients of the $E$-Series Virasoro Minimal Models",
  "title_text": "Bulk OPE Coefficients of the E-Series Virasoro Minimal Models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/fr55m2n1d7w6g",
  "venue": "arXiv",
  "versions": []
}