Calabi-Yau Meets Gravity: A Calabi-Yau Threefold at Fifth Post-Minkowskian Order

Area of Science:

• Theoretical physics
• String theory
• Quantum gravity

Background:

• Feynman integrals are essential for calculating scattering amplitudes in quantum field theory.
• The post-Minkowskian (PM) expansion is a theoretical framework used to describe the dynamics of binary systems, particularly black hole mergers.
• At the 4PM order, K3 surfaces have been identified in three-loop integrals relevant to gravitational wave emission.

Purpose of the Study:

• To investigate the mathematical structures, specifically geometries, appearing in Feynman integrals relevant to black hole scattering.
• To extend the understanding of these geometries beyond the 4PM order, specifically to the 5PM order.
• To identify new mathematical functions arising from these higher-order calculations.

Main Methods:

• Analysis of Feynman integrals at the four-loop level.
• Identification of specific geometric structures within these integrals.
• Relating these geometric structures to the post-Minkowskian expansion of black hole scattering.

Main Results:

• A Calabi-Yau threefold geometry has been identified within a four-loop integral.
• This geometry contributes to the scattering of black holes at the 5PM order.
• The presence of this Calabi-Yau threefold implies the emergence of novel mathematical functions in the analytical results.

Conclusions:

• The identification of a Calabi-Yau threefold at 5PM signifies a new level of mathematical complexity in black hole scattering calculations.
• These findings suggest that advanced mathematical functions, beyond elliptic integrals, are necessary for a complete description of gravitational wave generation from binary mergers.
• This research opens new avenues for exploring the interplay between quantum field theory, general relativity, and advanced geometry in the context of gravitational waves.