{
  "abstract": "Evans and Gannon showed that a positive vertex operator algebra of central charge eight whose representation category is the Drinfeld center $Z(\\mathcal{H}_3)$ of the Haagerup fusion category has one of two character vectors, with six or nineteen currents. Call such an algebra, if strongly rational, a Haagerup algebra. We prove that a Haagerup algebra $U$ has six currents, that $U_1 \\cong \\mathfrak{sl}_2 \\oplus \\mathfrak{sl}_2$ embeds into $V_{E_6 \\oplus A_2}$ in one of two explicit ways, with levels $(1,39)$ or $(3,13)$, and we determine the lowest weight spaces of its modules and $U_2$ as $U_1$-modules. For the levels $(3,13)$, $U_2$ is one of four Galois conjugate subspaces, and the algebra it generates is unitary, $C_2$-cofinite and isomorphic to the algebra constructed by Jiao; for the levels $(1,39)$, $U$ contains the algebra announced by Riesen. The associated lattice of $U$ is that of the form $x^2+xy+10y^2$, resp. $3x^2+3xy+4y^2$, the two classes of the principal genus of discriminant $-39$. Exchanging them in the decomposition of $U$ over its Heisenberg commutant gives a Haagerup algebra $U^\\natural$ of the other levels, and $U \\mapsto U^\\natural$ is a bijection between the isomorphism classes of the two cases. Consequently, if either explicit algebra of Jiao or Riesen is positive and strongly rational with representation category braided equivalent to $Z(\\mathcal{H}_3)$, there are exactly two Haagerup algebras up to isomorphism, forming a single hyperbolic genus. The explicit algebras of Jiao and Riesen have rational forms. The proofs combine exact trace relations given by vector-valued modular forms, exact computations in a model of $V_{E_6 \\oplus A_2}$, and genera of vertex operator algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01618",
  "author_total": 1,
  "authors": [
    {
      "name": "Cédric Arhancet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5grn1vawjnrrr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01618",
  "fulltext_url": null,
  "id": "5grn1vawjnrrr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01618",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01618",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5grn1vawjnrrr.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.01618",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01618"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Haagerup vertex operator algebras at central charge eight",
  "title_text": "Haagerup vertex operator algebras at central charge eight",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/5grn1vawjnrrr",
  "venue": "arXiv",
  "versions": []
}