论文标题
DLOG形式的双重积分，不是多属刻度的
A double integral of dlog forms which is not polylogarithmic
论文作者
论文摘要
Feynman积分是扰动量子场理论中所有计算的核心。它们通常会引起具有代数参数的DLOG形式的迭代积分，在许多情况下，可以通过多种多种聚类来评估。这导致了社区中某些民间传说的信念，表明所有这些积分都对小聚集体进行了评估。在这里，我们讨论了两个DLOG形式的双重迭代积分的具体示例，该积分将其评估为尖端形式的时期。这些积分的动机版本被证明与代数参数评估的所有多个多种聚集体在代数上独立。 From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.
Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.
