Riemann曲面上的正则化积分与模形式
Regularized Integrals on Riemann Surfaces and Modular Forms
AI总结：
本文提出一种基于共形几何的内禀正则化方法，用于在Riemann曲面上对具有任意全纯极点的微分形式进行积分，并应用于建立位形空间（含Feynman图积分）的解析理论，证明了椭圆曲线上正则化图积分的殆全纯模性质，给出了有序A-圈积分拟模性的几何证明与组合公式。
摘要：
我们引入了一种简单的方法，用于在Riemann曲面上对具有任意全纯极点的微分形式进行积分。该方法基于底层的共形几何，为这类奇异积分提供了一种内禀的正则化。将此正则化方案应用于Riemann曲面的乘积时，它为位形空间上的积分建立了解析理论，其中包括源于二维手征量子场论的Feynman图积分。专门针对椭圆曲线时，我们证明了这种正则化图积分是殆全纯模形式，它们在几何上为相应的有序$A$-圈积分提供了模完备化。这为有序A-圈积分的混合权拟模性质提供了一个简单的几何证明，并给出了所有不同权重组合的新颖组合公式。
英文摘要：
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.
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