发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对多弦 SLE 的 BPZ 方程线性解空间进行分类，通过三角坐标变换化为有限维线性代数，确定了平移不变解空间的维数，恢复了 Catalan 维数公式，并构造了高斯自由场水平线的配分函数。
AI 中文摘要
我们对弦型 Belavin--Polyakov--Zamolodchikov (BPZ) 方程的线性解空间进行分类，这些方程出现在 Dubédat 关于多个 Schramm--Loewner 演化 (SLE) 的交换关系中。在不施加增长条件的情况下，我们确定了由连续共形 Ward 恒等式选取的子空间及其精确维数。对于每个 $\kappa>0$ 和 BPZ 谱参数 $\lambda\in\mathbb{R}$，我们确定了平移算子的谱和 Jordan 结构。当 $\lambda=0$ 且 $\kappa\in(0,8)$ 时，我们进一步确定了所有允许的标度指数以及相应平移不变解空间的维数。对导数坐标进行三角变换可得到一阶有理 Knizhnik--Zamolodchikov 型系统，这使我们能够显式表示 Ward 算子并将分类简化为有限维线性代数。对于 $2N$ 个边界点和 $\kappa\in(0,8)$，具有 $\lambda=0$ 且满足 Möbius 协变性所需齐次性的平移不变 BPZ 解自动满足第三 Ward 恒等式和通常的幂律界。这恢复了在较弱假设下 Flores 和 Kleban 先前建立的 Catalan 维数公式。在 $\kappa=8$ 时，当 $N\ge2$ 时第三 Ward 恒等式施加了额外约束，但完整的 Möbius 协变解空间仍具有 Catalan 维数，从而建立了由均匀生成树构造的配分函数的完备性。最后，在 $\kappa=4$ 且 $\lambda>0$ 时，我们构造了正 BPZ 解的显式基，并将其元素识别为具有适当调和均值的高斯自由场水平线的配分函数。
英文摘要
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.
Comments45 pages, 2 figures
