{
  "abstract": "We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus $g$ surface by iterated self-sewing of $g$ handles onto the Riemann sphere, each governed by a sewing parameter $ρ_i$ in a punctured disc. We construct an explicit complex computing the chiral homology groups $H^{\\mathrm{ch}}_0$ and $H^{\\mathrm{ch}}_1$ of a vertex algebra $V$ on a genus $g$ surface with $n$ marked points, and equip it with a projectively flat connection, with an explicit central-charge anomaly, over the $g$-dimensional space of sewing parameters. We prove a genus $g$ Fourier-space Borcherds identity for the associated modified vertex operators. We show that two finiteness assumptions 1) finite dimensionality of the first Poisson homology $\\HP_1(R_V)$ of the Zhu $C_2$-algebra, and 2) finite generation of a certain Koszul homology of the associated graded algebra, imply finite dimensionality of $H^{\\mathrm{ch}}_1(X,V)$ for every genus $g$ and every vertex algebra $V$. Using the degeneration $ρ_i \\to 0$ together with the factorization theorem of Damiolini-Gibney-Tarasca, we relate the totally degenerate limit of $H_1^{\\mathrm{ch}}$ to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for rational vertex algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24719",
  "author_total": 1,
  "authors": [
    {
      "name": "A. Zuevsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1en822md35y66.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24719",
  "fulltext_url": null,
  "id": "1en822md35y66",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24719",
  "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": "W7171534124",
  "pdf_url": "https://arxiv.org/pdf/2607.24719",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1en822md35y66.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:13Z",
      "native_id": "2607.24719",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24719"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The first chiral homology group in higher genus",
  "title_text": "The first chiral homology group in higher genus",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/1en822md35y66",
  "venue": "arXiv",
  "versions": []
}