Calabi-Yau meets Gravity: A Calabi-Yau three-fold at fifth post-Minkowskian order

TL;DR

This paper identifies a Calabi-Yau three-fold geometry in a four-loop Feynman integral relevant to gravitational wave physics, revealing new mathematical functions in the fifth post-Minkowskian order of black hole scattering.

Contribution

It is the first to connect Calabi-Yau three-folds with high-order post-Minkowskian gravitational calculations, advancing the understanding of geometric structures in quantum gravity.

Findings

Discovery of a Calabi-Yau three-fold in a four-loop integral

Indication of new special functions in gravitational scattering results

Extension of geometric structures known from lower orders

Abstract

We study geometries occurring in Feynman integrals that contribute to the scattering of black holes in the post-Minkowskian expansion. These geometries become relevant to gravitational-wave production during the inspiralling phase of binary black hole mergers through the classical conservative potential. At fourth post-Minkowskian order, a K3 surface is known to occur in a three-loop integral, leading to elliptic integrals in the result. In this letter, we identify a Calabi-Yau three-fold in a four-loop integral, contributing at fifth post-Minkowskian order. The presence of this Calabi-Yau geometry indicates that completely new functions occur in the full analytical results at this order.