{
  "abstract": "Locally commuting multiple radial Schramm-Loewner evolutions ($\\mathrm{SLE}_κ$) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters $λ,ν\\in\\mathbb{R}$. For $κ>0$ and $λ\\in\\mathbb{R}$, we show that the solution space of the radial BPZ system has dimension $2^n$, where $n$ is the number of variables. We then determine all admissible Ward parameters $ν$ and the exact dimensions of the subspaces selected by the conformal Ward identity, covering both generic and degenerate cases. The classification reveals a parity difference: nonzero rotation-invariant solutions exist for every $λ$ when $n$ is even, but only at finitely many exceptional values when $n$ is odd. When $0<κ\\leq4$, $λ>0$ for odd $n$ or $λ>-3/2$ for even $n$, we construct a basis of positive solutions using multiple SLE, providing global realizations of the locally commuting SLEs. At $κ=4$, we identify a family of explicit solutions as partition functions for level lines of a Gaussian free field with suitable boundary data and interior singularities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19739",
  "author_total": 2,
  "authors": [
    {
      "name": "Chongzhi Huang"
    },
    {
      "name": "Hao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7dydmmttfjbwr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19739",
  "fulltext_url": null,
  "id": "7dydmmttfjbwr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19739",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213589091",
  "pdf_url": "https://arxiv.org/pdf/2609.19739",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7dydmmttfjbwr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:19Z",
      "native_id": "2609.19739",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19739"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Classification of commutation relation for multi-radial SLE",
  "title_text": "Classification of commutation relation for multi-radial SLE",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/7dydmmttfjbwr",
  "venue": "arXiv",
  "versions": []
}