{
  "abstract": "We construct an explicit rational \\(\\mathrm{SL}_2(\\mathbb C)\\)-action on the underlying vector space of the rank-two \\(βγ\\)-system. The construction is motivated by Creutzig's branching decomposition with respect to \\(L_{-1}(\\mathfrak{sl}_2)\\) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original \\(βγ\\)-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine \\(L_{-1}(\\mathfrak{sl}_2)\\)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with \\(V_n\\otimes\\mathcal F_{\\mathrm{osc}}\\), where \\(V_n\\) is the irreducible \\(\\mathrm{SL}_2(\\mathbb C)\\)-module of highest weight \\(n\\) and \\(\\mathcal F_{\\mathrm{osc}}\\) is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of \\(V_n\\). The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08000",
  "author_total": 3,
  "authors": [
    {
      "name": "Tomoyuki Arakawa"
    },
    {
      "name": "Xuanzhong Dai"
    },
    {
      "name": "Bailin Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5pfcq40e98a4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08000",
  "fulltext_url": null,
  "id": "d5pfcq40e98a4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08000",
  "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.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
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      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
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  "openalex": "W7221030318",
  "pdf_url": "https://arxiv.org/pdf/2610.08000",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5pfcq40e98a4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.08000",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08000"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "A new $SL_2$-action on the rank-two $βγ$-system",
  "title_text": "A new SL_2-action on the rank-two βγ-system",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/d5pfcq40e98a4",
  "venue": "arXiv",
  "versions": []
}