AI 中文总结
研究紧致谱曲线上含指数奇点等的广义拓扑递归，通过建立递归留数公式的轮廓变形，为指数分歧谱曲线提供递归框架，该形式体系是Bouchard-Eynard高阶拓扑递归的极限情况，还通过例子进行了说明。
AI 中文摘要
我们研究了在紧致谱曲线上的广义拓扑递归，该谱曲线允许指数奇点，更一般地允许本质奇点作为分歧点。利用广义拓扑递归的全局公式，我们建立了递归留数公式的轮廓变形，用亚纯点处的留数代替这些本质奇点的贡献。这为指数分歧谱曲线提供了一个自然的递归框架，同时完全保留在广义拓扑递归形式体系内。我们的形式体系也可视为Bouchard-Eynard高阶拓扑递归的极限情况，例如当分歧点的阶以收敛方式趋于无穷时（如指数奇点情况，同时dy保持正则且非零）。我们通过几个例子进一步说明所得形式体系，包括超越函数和米尔扎哈尼曲线的x - y对偶。
英文摘要
We investigate generalized topological recursion on compact spectral curves admitting exponential, and more generally essential, singularities as ramification points. Exploiting the global formulation of generalized topological recursion, we establish a contour deformation of the recursive residue formula that replaces contributions from these essential singularities by residues at meromorphic points. This provides a natural recursive framework for exponentially ramified spectral curves while remaining entirely within the generalized topological recursion formalism. Our formalism can also be viewed as a limiting case of the Bouchard-Eynard higher-order topological recursion, obtained when the order of a ramification point tends to infinity in a convergent manner, as occurs, for example, for exponential singularities while $dy$ remains regular and non-vanishing. We further illustrate the resulting formalism through several examples, including transcendental functions and the $x$-$y$ dual of the Mirzakhani curve.
Comments29 pages + reference
