Following the work of Brown, we can canonically associate a family of motivic periods — called the motivic Feynman amplitude — to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown’s “small graphs principle”. This serves as motivation for explicitly computing the motivic Galois action, or, dually, the coaction of the Hopf algebra of functions on the motivic Galois group. In this paper, we study the motivic Galois coaction on the motivic Feynman amplitudes associated to one-loop Feynman graphs. We study the associated variations of mixed Hodge structures, and provide an explicit formula for the coaction on the four-edge cycle graph — the box graph — with non-vanishing generic kinematics, which leads to a formula for all one-loop graphs with non-vanishing generic kinematics in four-dimensional space-time. We also show how one computes the coaction in some degenerate configurations — when defining the motive of the graph requires blowing up the underlying family of varieties — on the example of the three-edge cycle graph.

中文翻译：

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Motivic Galois coaction 和 one-loop Feynman graphs

根据布朗的工作，我们可以规范地将一系列动机周期（称为动机费曼振幅）与任何收敛的费曼积分相关联，被视为运动学变量的函数。动机费曼振幅的动机伽罗瓦理论通过布朗的“小图原理”提供了一个组织原则，以及对一般振幅空间的强约束。这作为显式计算动机伽罗瓦作用的动机，或双重计算动机伽罗瓦群上函数的 Hopf 代数的协同作用。在本文中，我们研究了与单环费曼图相关的动机费曼振幅上的动机伽罗瓦相互作用。我们研究了混合霍奇结构的相关变化，并为四边循环图（箱形图）上的相互作用提供一个明确的公式，具有非消失的通用运动学，这导致了所有单环图在四维空间中具有非消失的通用运动学的公式 -时间。我们还以三边循环图为例，展示了如何计算某些退化配置中的协同作用——当定义图的动机时需要炸毁潜在的变体家族。