{
  "abstract": "Let $G$ be a compact simply connected simple Lie group and $k \\geq 1$. Jump loops give an outer homomorphism $θ_k$ from $G$ to the outer automorphisms of the local factor of the level-$k$ loop group net, which is the hyperfinite type $\\mathrm{III}_1$ factor. We show that its Sutherland obstruction is $k$ times the positive basic class in $H^3_{\\mathrm{Borel}}(G, \\mathbb{T}) \\cong H^{4}(BG, \\mathbb{Z})$ and that $θ_k$ is faithful, so it is a $G$-kernel. Central elements act by the outer automorphisms of the simple currents. In types A, D, E the level-one class is computed by comparing maximal-torus jump loops with the lattice $α$-induction construction of the companion paper on both current and charged generators, followed by integral degree-four Cartan restriction. Higher levels follow by tensor additivity, and the remaining types by restriction along index-one embeddings into simply laced groups. As a consequence, for every compact simply connected semisimple $G$, every class in $H^4(BG, \\mathbb{Z})$ is the obstruction of a $G$-kernel on the hyperfinite type $\\mathrm{III}_1$ factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23321",
  "author_total": 2,
  "authors": [
    {
      "name": "Marcel Bischoff"
    },
    {
      "name": "Pradyut Karmakar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/924swr165wwsg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23321",
  "fulltext_url": null,
  "id": "924swr165wwsg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23321",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7222921746",
  "pdf_url": "https://arxiv.org/pdf/2610.23321",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/924swr165wwsg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:32Z",
      "native_id": "2610.23321",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23321"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Anomalies for compact simple Lie groups via $G$-kernels for loop group nets",
  "title_text": "Anomalies for compact simple Lie groups via G-kernels for loop group nets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/924swr165wwsg",
  "venue": "arXiv",
  "versions": []
}