{
  "abstract": "Let $V$ be a vertex operator algebra (VOA), not necessarily rational or $C_2$-cofinite, such that the category $\\mathcal C_1(V)$ of $C_1$-cofinite grading-restricted generalized $V$-modules is semisimple and contains the generalized Verma modules associated to finite-dimensional modules over the Zhu algebra $A(V)$. Such VOAs include affine VOAs at admissible and generic levels. We prove that the bottom degree of Huang-Lepowsky's $P(w)$-tensor product module $M^1\\boxtimes_{P(w)}M^2$ is isomorphic to the left $A(V)$-module $M^1\\odot_{w,0}M^2$ arising from the restricted conformal blocks on $(¶^1,\\infty,w,0)$. By shifting the marked points and using the parallel transport of the restricted conformal blocks, we construct an explicit braiding isomorphism on the category of finite-dimensional $A(V)$-modules, which is the restriction of the braiding isomorphism of Huang-Lepowsky-Zhang to the bottom degrees. For the affine VOA $L_k(\\mathfrak g)$ at a positive integral level, it is given by the Casimir tensor of $\\mathfrak g$ on certain quotients of tensor products of finite-dimensional $\\mathfrak g$-modules, and at generic levels it recovers Drinfeld's commutativity isomorphism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41605",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhiguang Cui"
    },
    {
      "name": "Jianqi Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d2dcvmqmsp7cq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41605",
  "fulltext_url": null,
  "id": "d2dcvmqmsp7cq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41605",
  "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.QA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d2dcvmqmsp7cq.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Restricted conformal blocks and the braiding on Zhu algebra modules",
  "title_text": "Restricted conformal blocks and the braiding on Zhu algebra modules",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/d2dcvmqmsp7cq",
  "venue": "arXiv",
  "versions": []
}