Classification of commutation relation for multi-chordal SLE
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∈ℝ, we determine the spectrum and Jordan structure of the translation operator. When λ=0 and κ∈(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∈(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≥2, 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.
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BibTeX
@misc{chen2026classification,
title = {{Classification of commutation relation for multi-chordal SLE}},
author = {Mo Chen and Chongzhi Huang and Hao Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04362},
eprint = {2610.04362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04362},
}
RIS
TY - UNPB
TI - Classification of commutation relation for multi-chordal SLE
AU - Mo Chen
AU - Chongzhi Huang
AU - Hao Wu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.04362
UR - https://arxiv.org/abs/2610.04362
AB - 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∈ℝ, we determine the spectrum and Jordan structure of the translation operator. When λ=0 and κ∈(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∈(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≥2, 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.
ER -