Möbius 不变的二聚体模型与 Miquel 动力学
Möbius Invariant Dimers and Miquel Dynamics
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中文总结 AI 辅助
本文建立 Möbius 变换群 PO(3,1) 下不变的圆模式与二聚体模型新对应，证明 Miquel 动力学保持配分函数，并揭示隐藏对称群 PO(3,3) 及若干应用。
中文摘要 AI 辅助
已知圆模式与二聚体模型之间存在对应关系，该对应在欧几里得变换下不变。我们建立了一个新的对应关系，它在 Möbius 变换群 PO(3,1) 下不变，该群是圆模式的自然对称群。我们证明 Miquel 动力学保持配分函数，并且 t-嵌入的凸性保证了面权重的正性。我们还证明该对应关系具有一个略显隐藏的对称群 PO(3,3)，它与 t-嵌入的 Lorentz 提升相关。最后，我们考虑了等半径图、Doyle 螺线、单穿孔环面和等温圆模式。
英文摘要
As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.
