{
  "abstract": "We classify the linear solution spaces of the chordal Belavin--Polyakov--Zamolodchikov (BPZ) equations, which arise in Dubédat's commutation relations for multiple Schramm--Loewner evolutions (SLE). Without imposing growth conditions, we determine the subspaces selected by successive conformal Ward identities and their exact dimensions. For every $κ>0$ and BPZ spectral parameter $λ\\in\\mathbb{R}$, we determine the spectrum and Jordan structure of the translation operator. When $λ=0$ and $κ\\in(0,8)$, we further determine all admissible scaling exponents and the dimensions of the corresponding translation-invariant solution spaces. A triangular change of derivative coordinates yields a first-order system of rational Knizhnik--Zamolodchikov type, allowing us to represent the Ward operators explicitly and reduce the classification to finite-dimensional linear algebra. For $2N$ boundary points and $κ\\in(0,8)$, translation-invariant BPZ solutions with $λ=0$ and the homogeneity required by Möbius covariance automatically satisfy the third Ward identity and the usual power-law bound. This recovers, under weaker assumptions, the Catalan dimension formula previously established by Flores and Kleban. At $κ=8$, the third Ward identity imposes an additional constraint when $N\\ge2$, but the full Möbius-covariant solution space still has Catalan dimension, establishing completeness of the partition functions constructed from uniform spanning trees. Finally, at $κ=4$ and $λ>0$, we construct an explicit basis of positive BPZ solutions and identify its elements as partition functions for level lines of a Gaussian free field with suitable harmonic means.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04362",
  "author_total": 3,
  "authors": [
    {
      "name": "Mo Chen"
    },
    {
      "name": "Chongzhi Huang"
    },
    {
      "name": "Hao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ebg4qxqy73hs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04362",
  "fulltext_url": null,
  "id": "8ebg4qxqy73hs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04362",
  "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.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7220711510",
  "pdf_url": "https://arxiv.org/pdf/2610.04362",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ebg4qxqy73hs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:28Z",
      "native_id": "2610.04362",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04362"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Classification of commutation relation for multi-chordal SLE",
  "title_text": "Classification of commutation relation for multi-chordal SLE",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8ebg4qxqy73hs",
  "venue": "arXiv",
  "versions": []
}