{
  "abstract": "We study the representation theory of the vertex operator algebras (VOAs) associated with rank-one 4d $\\mathcal{N} = 3$ superconformal field theories. For the $\\mathbb{Z}_3$ S-fold theory, whose VOA $\\mathcal{W}_{\\mathbb{Z}_3}$ has central charge $c_{\\mathrm{2d}} = -15$, we use the $\\mathcal{N} = 1$ Lagrangian description to obtain the unflavored Schur index in terms of Dedekind eta functions, while Wilson-loop indices yield the unflavored non-vacuum characters. These characters all solve a modular linear differential equation (MLDE) whose solution space also contains a logarithmic character. Combining flavored MLDEs from null states with Zhu's associative algebra and a free-field realization, we study four highest-weight modules of $\\mathcal{W}_{\\mathbb{Z}_3}$ and their flavored characters in closed form. A parallel analysis applies to the $\\mathcal{N} = 3$ theories obtained by gauging a discrete $\\mathbb{Z}_n$ flavor subgroup of $\\mathcal{N} = 4$ $U(1)$ and $SU(2)$ super-Yang--Mills, for which we also obtain closed-form Schur indices and a new free-field realization of the VOA of the $\\mathbb{Z}_4$ quotient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10685",
  "author_total": 4,
  "authors": [
    {
      "name": "Zhaoting Guo"
    },
    {
      "name": "Satoshi Nawata"
    },
    {
      "name": "Yiwen Pan"
    },
    {
      "name": "Qituan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/950nmnp2m1d08.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10685",
  "fulltext_url": null,
  "id": "950nmnp2m1d08",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10685",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7212288928",
  "pdf_url": "https://arxiv.org/pdf/2609.10685",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/950nmnp2m1d08.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:26Z",
      "native_id": "2609.10685",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10685"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Rank-one 4d $\\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity",
  "title_text": "Rank-one 4d 𝒩=3 SCFTs: Schur index, VOA modules, and modularity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/950nmnp2m1d08",
  "venue": "arXiv",
  "versions": []
}