Feynman Amplitudes and Limits of Heights

TL;DR

This paper explores the mathematical connection between Feynman amplitudes and string theory limits, proving integrand convergence through Hodge theory and height pairing asymptotics during Riemann surface degeneration.

Contribution

It introduces a rigorous mathematical framework linking Feynman amplitudes to string theory limits using Hodge theory and height pairings, with proven integrand convergence.

Findings

Proved convergence of integrands in the low-energy limit

Derived asymptotic behavior of height pairings during surface degeneration

Applied nilpotent orbit theorem to analyze degenerations

Abstract

We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic behavior of the height pairing between degree-zero divisors, as a family of Riemann surfaces degenerates. These are obtained by means of the nilpotent orbit theorem in Hodge theory.