论文标题
Riemann表面和模块化形式的正则积分
Regularized Integrals on Riemann Surfaces and Modular Forms
论文作者
论文摘要
我们介绍了一个简单的程序，将差异形式与黎曼表面上的任意全体形态杆整合在一起。它导致这种奇异积分的固有正规化，这是基本的保形几何形状的。应用于Riemann表面的产物，该正则化方案建立了在配置空间上积分的分析理论，包括由二维手性量子场理论引起的Feynman图积分。我们专门研究椭圆曲线，我们显示这样的正则图积分是几乎是圆形的模块化形式，几何形式可提供相应订购的$ a $ a $ cycle积分的模块化完成。这导致了有序的A-Cycle积分的混合权重模数以及针对所有不同权重的所有组件的新型组合公式的简单几何证明。
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry. Applied to products of Riemann surfaces, this regularization scheme establishes an analytic theory for integrals over configuration spaces, including Feynman graph integrals arising from two dimensional chiral quantum field theories. Specializing to elliptic curves, we show such regularized graph integrals are almost-holomorphic modular forms that geometrically provide modular completions of the corresponding ordered $A$-cycle integrals. This leads to a simple geometric proof of the mixed-weight quasi-modularity of ordered A-cycle integrals, as well as novel combinatorial formulae for all the components of different weights.
