{
  "abstract": "We generalize the geometric construction of vertex operator algebra (VOA) bundles and their associated automorphic forms from the elliptic modular curve to arbitrary Fuchsian groups $ Γ\\subset \\mathrm{PSL}_2(\\mathbb{R})$. A sharp topological dichotomy emerges regarding the existence of a holomorphic weight-$2$ quasi-automorphic generator $E_2^Γ$. When $Γ$ has a cusp, we construct $E_2^Γ$ via the analytic continuation of parabolic Eisenstein series and prove that the space of quasi-automorphic forms is a free polynomial extension, allowing the algebraic setup of the genus one theory, including the quasi-VOA structure and the characterization of strict automorphic forms via a lowering operator. Conversely, when $Γ$ is cocompact of genus $g\\ge 2$, Atiyah theorem on holomorphic connections rigorously obstructs the existence of $E_2^Γ$. For this obstructed case, we provide dimension formulas that link the shortage in lifting quasi-automorphic forms to the failure of quasi-primarity within the VOA, fully resolving the torsion-free case and conjecturing the extension to groups with elliptic points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10483",
  "author_total": 1,
  "authors": [
    {
      "name": "A. Zuevsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x6kfbhb3wjfh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10483",
  "fulltext_url": null,
  "id": "0x6kfbhb3wjfh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.10483",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7168267231",
  "pdf_url": "https://arxiv.org/pdf/2607.10483",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x6kfbhb3wjfh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2607.10483",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10483"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups",
  "title_text": "Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/0x6kfbhb3wjfh",
  "venue": "arXiv",
  "versions": []
}