Motivic renormalization and singularities

Understand

We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals.

• We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities.
• In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs.
• We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundaries and, like dimensional regularization, replaces a divergent integral with a Laurent series in a complex parameter.