Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of the analytic renormalization by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit an analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo-Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth. As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov-Todorov-Stora in the case of flat space-times.

中文翻译：

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通过谱 zeta 正则化和爆炸对流形上的费曼振幅进行重整化

我们在本文中的目标是提出一般费曼振幅的谱 zeta 正则化的推广。我们的方法使用椭圆算子的复数幂，但本着 Speer 的解析重整化的精神涉及几个复杂的参数，为微扰相互作用量子场论的重整化建立数学基础。我们的主要结果表明，谱正则化的费曼振幅允许作为具有线性极点的亚纯胚芽的解析延拓，在郭佩查和第二作者的作品意义上。我们还明确确定了支持极点的仿射超平面。我们的证明依赖于热核乘积的奇点的合适分辨率，以使其平滑。作为解析延拓结果的应用，我们使用全纯胚芽上具有线性极点的亚纯胚芽的通用投影来构建重整化图，该图减去欧几里德场的费曼振幅奇点。在平坦时空的情况下，我们的重整化映射被证明满足之前在 Nikolov-Todorov-Stora 的工作中引入的一致性条件。