SL₂(ℝ)特征簇富克斯轨迹上Atiyah-Bott-Goldman辛结构的迹函数泊松括号
Poisson bracket of trace functions on the Fuchsian locus and Wolpert's formulas
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中文总结 AI 辅助
该研究针对双曲曲面上莫比乌斯变换乘积的迹，开发了基于SL₂(ℝ)矩阵归一化的系统计算方法，结合Goldman的辛形式描述计算相关迹函数的泊松括号，应用中得到Wolpert的余弦与正弦公式。
中文摘要 AI 辅助
我们开发了一种系统方法，用于计算双曲曲面上与定向测地线相关的莫比乌斯变换乘积的迹。该方法基于SL₂(ℝ)中矩阵的归一化，将迹恒等式用双曲长度、交角及测地线上的有向距离表示。结合这些迹计算与Goldman对特征簇上Atiyah-Bott-Goldman辛形式的描述，我们计算了源自测地线代表元的迹函数的泊松括号。作为应用，我们得到了Wolpert的余弦公式和正弦公式。
英文摘要
We develop a systematic method for computing traces of products of Möbius transformations associated with oriented geodesics on a hyperbolic surface. The method is based on a normalization of matrices in ${SL}_2(\mathbb R)$ which expresses trace identities in terms of hyperbolic lengths, intersection angles, and signed distances along geodesics. Using these trace computations together with Goldman's description of the Atiyah-Bott-Goldman symplectic form on the character variety, we derive explicit geometric formulas for the Poisson brackets of trace functions associated with closed geodesics. More precisely, we express the Poisson bracket of two trace functions and the iterated Poisson bracket of three trace functions in terms of the hyperbolic lengths of the corresponding geodesics, their intersection angles, and the signed distances between intersection points. The results naturally lead to a unified perspective for revisiting Wolpert's cosine and sine formulas and deriving new proofs of them.
发表机构
- Indian Institute of Technology Palakkad（印度理工学院帕拉卡德分校）
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