Regularization of divergent integrals

TL;DR

This paper investigates the regularization of divergent integrals with singularities, providing formulas for their finite parts and exploring applications in complex geometry and string theory.

Contribution

It introduces new formulas for the finite part of divergent integrals and relates them to local residue maps, with detailed cases for complex hypersurfaces and boundary components.

Findings

Formulas for the dependence of the finite part on regularization choices

Explicit expressions involving local residue maps

Application to complex hypersurfaces and string theory boundary components

Abstract

We study the Hadamard finite part of divergent integrals of differential forms with singularities on submanifolds. We give formulae for the dependence of the finite part on the choice of regularization and express them in terms of a suitable local residue map. The cases where the submanifold is a complex hypersurface in a complex manifold and where it is a boundary component of a manifold with boundary, arising in string perturbation theory, are treated in more detail.