{
  "abstract": "Krauel, Shafiq and Wood (\\KSW{}) realized torus \\(1\\)-point functions for the simple affine \\voa{}s \\(L(k,0)\\) built from \\(\\alg{sl}(2)\\) as vector-valued modular forms attached to a cyclic \\(R\\)-module structure and to the modular tensor category \\(\\rep{L(k,0)}\\). We extend this to torus \\(n\\)-point functions, defined as traces of chains of \\(n\\) intertwining operators winding once around the \\(τ\\)-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \\vvmf{}, lying, whenever a mild lowest weight hypothesis holds, in the very \\(R\\)-modules of \\cite{KSW}. Giving a torus-primarity hypothesis verified below for \\(\\alg{sl}(2)\\), the operator product expansion then reduces leading short-distance behaviour to the classified \\(1\\)-point theory. For \\(\\alg{sl}(2)\\) we classify the fusion-chain conformal block spaces, identify torus \\(n\\)-point primary vectors, and treat \\(n=2\\) in detail, obtaining a \\((k+1)\\)-dimensional family of vector-valued Jacobi-type forms generalizing the level-\\(k\\) forms \\(η^{3k/2}\\) of \\cite{KSW}. We also extend \\KSW{} categorical picture, representations of \\(\\Bthree=\\PMod(\\surf{1}{1})\\) built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the \\(n\\)-punctured torus generated by \\(S\\), \\(T\\), and adjacent braidings, with a Verlinde-type dimension formula and categorical \\(S\\)-, \\(T\\)-operators. The \\(\\alg{sl}(2)\\) matrix form of \\(S\\) beyond this abstract construction remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01496",
  "author_total": 1,
  "authors": [
    {
      "name": "A. Zuevsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd8w6ytxb9xxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01496",
  "fulltext_url": null,
  "id": "bd8w6ytxb9xxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01496",
  "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": "W7206150610",
  "pdf_url": "https://arxiv.org/pdf/2609.01496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd8w6ytxb9xxm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.01496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Modular properties of affine \\(\\SLA{sl}{2}\\) torus \\(n\\)-point functions",
  "title_text": "Modular properties of affine SLAsl2 torus n-point functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/bd8w6ytxb9xxm",
  "venue": "arXiv",
  "versions": []
}