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
本文对多径向SLE的局部交换关系进行分类，确定解空间维数及Ward参数，并构造正解基，揭示奇偶性差异，为SLE提供整体实现。
AI 中文摘要
局部交换的多径向Schramm-Loewner演化（$\mathrm{SLE}_{\kappa}$）由满足径向Belavin-Polyakov-Zamolodchikov（BPZ）方程和带谱参数$\lambda,\nu\in\mathbb{R}$的共形Ward恒等式的配分函数编码。对于$\kappa>0$和$\lambda\in\mathbb{R}$，我们证明了径向BPZ系统的解空间维数为$2^n$，其中$n$是变量个数。然后我们确定了所有可容许的Ward参数$\nu$以及由共形Ward恒等式选出的子空间的精确维数，涵盖了一般情形和退化情形。该分类揭示了一个奇偶性差异：当$n$为偶数时，对每个$\lambda$都存在非零旋转不变解；而当$n$为奇数时，仅存在有限多个例外值。当$0<\kappa\leq4$，奇数$n$时$\lambda>0$或偶数$n$时$\lambda>-3/2$，我们利用多SLE构造了正解的一组基，提供了局部交换SLE的整体实现。在$\kappa=4$时，我们识别出一族显式解，作为具有适当边界数据和内部奇异点的高斯自由场的水平线配分函数。
英文摘要
Locally commuting multiple radial Schramm-Loewner evolutions ($\mathrm{SLE}_κ$) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters $λ,ν\in\mathbb{R}$. For $κ>0$ and $λ\in\mathbb{R}$, 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 $0<κ\leq4$, $λ>0$ 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.
Comments40 pages, 4 figures
