Orateur
Description
I shall report on ongoing work with S. Scott and B. Zhang by which we
generalise regularised spectral zeta functions to a generalised
Kontsevich-Vishik trace associated with a Feynman graph. These in turn
generalise Feynman amplitudes on a Riemannian manifold studied by
Dang and Zhang [JEMS 2021] in two ways. Whereas they consider graphs
decorated by a single Riemannian Laplacian on a Riemannian manifold, we
consider a general closed manifold and decorate the edges of the graph
with arbitrary classical pseudo-differential operators. Whereas Dang
and Zhang use complex powers of the Laplacian to regularise, we
consider general holomorphic perturbations of the operators decorating
the edges. Similarly to their approach, our method involves several
complex parameters in the spirit of analytic renormalisation by Speer.
We claim that the resulting regularised Feynman amplitudes admit
analytic continuation as meromorphic germs with linear poles in the
sense of the works of Guo, Paycha and Zhang. We give an explicit
determination of the affine hyperplanes supporting the poles, which only
depends on the Betti number of the graph and the orders of the
operators. Neither the poles nor the method by which we determine them
make use of the underlying geometry of the manifold.
