Classification of commutation relation for multi-radial SLE
Abstract
Locally commuting multiple radial Schramm-Loewner evolutions (SLEκ) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters λ,ν∈ℝ. For κ>0 and λ∈ℝ, we show that the solution space of the radial BPZ system has dimension 2n, 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≤4, 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.
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BibTeX
@misc{huang2026classification,
title = {{Classification of commutation relation for multi-radial SLE}},
author = {Chongzhi Huang and Hao Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19739},
eprint = {2609.19739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19739},
}
RIS
TY - UNPB
TI - Classification of commutation relation for multi-radial SLE
AU - Chongzhi Huang
AU - Hao Wu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19739
UR - https://arxiv.org/abs/2609.19739
AB - Locally commuting multiple radial Schramm-Loewner evolutions (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∈ℝ. For κ>0 and λ∈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≤4, 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.
ER -