It is well known that Feynman integrals in dimensional regularization often evaluate to functions of hypergeometric type. Inspired by a recent proposal for a coaction on one-loop Feynman integrals in dimensional regularization, we use intersection numbers and twisted homology theory to define a coaction on certain hypergeometric functions. The functions we consider admit an integral representation where both the integrand and the contour of integration are associated with positive geometries. As in dimensionally- regularized Feynman integrals, endpoint singularities are regularized by means of exponents controlled by a small parameter ϵ. We show that the coaction defined on this class of integral is consistent, upon expansion in ϵ, with the well-known coaction on multiple polylogarithms. We illustrate the validity of our construction by explicitly determining the coaction on various types of hypergeometric p +1 F p and Appell functions.

中文翻译：

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从正几何到超几何函数的协同作用

众所周知，维数正则化中的费曼积分通常评估为超几何类型的函数。受最近关于维度正则化中单环费曼积分协同作用的提议的启发，我们使用交集数和扭曲同源理论来定义某些超几何函数的协同作用。我们考虑的函数允许积分表示，其中被积函数和积分轮廓都与正几何相关。与维数正则化的费曼积分一样，端点奇点通过由小参数 ϵ 控制的指数进行正则化。我们表明，在 ϵ 扩展时，定义在此类积分上的协同作用与众所周知的多个多对数上的协同作用是一致的。