Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups

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.