{
  "abstract": "In this paper, we study modular fusion categories whose mapping class group representations are trivial on the Torelli groups, with particular emphasis on the congruence properties of the resulting representations of $\\mathrm{Sp}(2g, \\mathbb{Z})$. We prove that, for any $g \\ge 3$, the Torelli group is contained in the kernel of the genus-$g$ mapping class group representation $ρ_g$ associated with a modular fusion category $\\mathcal{C}$ if and only if $\\mathcal{C}$ is pointed. This refines the corresponding result in \\cite{MW25}. We also characterize the modular fusion categories for which the genus-$2$ Torelli group is contained in $\\ker ρ_2$; in particular, categories with isotropic adjoint subcategories belong to this class. Finally, we give a complete classification of modular fusion categories with isotropic adjoint subcategories, and determine the groups $G$ satisfying $\\mathcal{Z}(\\mathrm{Vect}_G)$ is 2-Torelli trivial but $\\mathcal{Z}(\\mathrm{Vect}_G)_{\\mathrm{ad}}$ is not isotropic are precisely 2-Engel groups of nilpotent class 3.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04462",
  "author_total": 3,
  "authors": [
    {
      "name": "Liang Chang"
    },
    {
      "name": "Siu-Hung Ng"
    },
    {
      "name": "Yilong Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cdb4npqfkwm90.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04462",
  "fulltext_url": null,
  "id": "cdb4npqfkwm90",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04462",
  "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.GT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7196932644",
  "pdf_url": "https://arxiv.org/pdf/2608.04462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cdb4npqfkwm90.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2608.04462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Modular fusion categories with trivial Torelli group actions",
  "title_text": "Modular fusion categories with trivial Torelli group actions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/cdb4npqfkwm90",
  "venue": "arXiv",
  "versions": []
}