In the first part of this paper we study the coaction dual to the action of the cosmic Galois group on the motivic lift of the sunrise Feynman integral with generic masses and momenta, and we express its conjugates in terms of motivic lifts of Feynman integrals associated to related Feynman graphs. Only one of the conjugates of the motivic lift of the sunrise, other than itself, can be expressed in terms of motivic lifts of Feynman integrals of subquotient graphs. To relate the remaining conjugates to Feynman integrals we introduce a general tool: subdividing edges of a graph. We show that all motivic lifts of Feynman integrals associated to graphs obtained by subdividing edges from a graph $G$ are motivic periods of $G$ itself. This was conjectured by Brown in the case of graphs with no kinematic dependence. We also look at the single-valued periods associated to the functions on the motivic Galois group, i.e. the ‘de Rham periods’, which appear in the coaction on the sunrise, and show that they are generalisations of Brown’s non-holomorphic modular forms with two weights. In the second part of the paper we consider the relative completion of the torsor of paths on a modular curve and its periods, the theory of which is due to Brown and Hain. Brown studied the motivic periods of the relative completion of $\mathcal{M}_{1,1}$ with respect to the tangential basepoint at infinity, and we generalise this to the case of the torsor of paths on any modular curve. We apply this to reprove the claim that the sunrise Feynman integral in the equal-mass case can be expressed in terms of Eichler integrals, periods of the underlying elliptic curve defined by one of the associated graph hypersurfaces, and powers of $2\pi i$.

中文翻译：

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宇宙伽罗瓦群、日出费曼积分以及$\Gamma^1(6)$的相对完备


在本文的第一部分中，我们研究了宇宙伽罗瓦群对具有通用质量和动量的日出费曼积分的动机升力的作用的对偶相互作用，并且我们用与以下相关的费曼积分的动机升力来表达其共轭：相关的费曼图。日出的动机升力的共轭之一（除了其本身）可以用次商图的费曼积分的动机升力来表达。为了将剩余的共轭与费曼积分联系起来，我们引入了一个通用工具：细分图的边。我们证明，与通过从图 $G$ 中细分边获得的图相关的费曼积分的所有动机提升都是 $G$ 本身的动机周期。这是布朗在没有运动学依赖性的情况下猜想的。我们还研究了与动机伽罗瓦群上的函数相关的单值周期，即“德拉姆周期”，它们出现在日出的相互作用中，并表明它们是布朗非全纯模形式的推广两个重量。在本文的第二部分中，我们考虑模曲线及其周期上路径扭转的相对完备性，其理论源自 Brown 和 Hain。布朗研究了 $\mathcal{M}_{1,1}$ 相对于无穷远切向基点的相对完成的动机周期，并且我们将其推广到任何模曲线上路径的扭转的情况。我们应用这一点来反驳这样的说法：等质量情况下的日出费曼积分可以用艾希勒积分、由相关图超曲面之一定义的基础椭圆曲线的周期以及 $2\pi i$ 的幂来表示。