Classification of commutation relation for multi-chordal SLE

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 $\kappa >0$ and BPZ spectral parameter $\lambda \in \mathbb{R}$, we determine the spectrum and Jordan structure of the translation operator. When $\lambda =0$ and $\kappa \in (0,8)$, 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 $\kappa \in (0,8)$, translation-invariant BPZ solutions with $\lambda =0$ 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 $\kappa =8$, the third Ward identity imposes an additional constraint when $N\ge 2$, but the full Möbius-covariant solution space still has Catalan dimension, establishing completeness of the partition functions constructed from uniform spanning trees. Finally, at $\kappa =4$ and $\lambda >0$, 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.