Localization formulas of cohomology intersection numbers

TL;DR

This paper develops localization formulas for cohomology intersection numbers related to logarithmic connections, linking them to residues, hypergeometric cases, and stringy integrals, providing new computational tools and theoretical insights.

Contribution

It proves new localization formulas for cohomology intersection numbers, relates Laurent expansion leading terms to Grothendieck residues, and connects stringy integrals to self-cohomology intersection numbers.

Findings

Localization formula expressed via residues.

Leading Laurent term equals Grothendieck residue in hypergeometric case.

Stringy integral's leading term equals self-cohomology intersection number.

Abstract

We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.