星积分、卷积和单纯形
Star Integrals, Convolutions and Simplices
摘要：
我们探讨单圈和多圈共形积分，如在平坦空间的对偶共形理论中出现的。利用梅林振幅，一大类高阶环积分可以写成星积分上的简单积分微分算子：$n$维的单环$n$ -边形积分。这些已知由双曲单形的体积给出。我们显式计算五维五边形积分在充分的一般性使用Schläfli的公式。然后，作为理解更高回路的第一步，我们使用样条技术来明确地构造二维运动学中的$6d$六边形和$8d$八边形积分。然后，完全重六边形和八边形积分分别与双盒积分和三盒积分相关联。我们评论类的功能需要表达这些积分的一般运动学，涉及椭圆函数和超越。
英文摘要：
We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop $n$-gon integrals in $n$ dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schläfli's formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the $6d$ hexagon and $8d$ octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.
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