Modular forms for three-loop banana integrals (2502.15325v1)

Abstract: We study periods of multi-parameter families of K3 surfaces, which are relevant to compute the maximal cuts of certain classes of Feynman integrals. We focus on their automorphic properties, and we show that generically the periods define orthogonal modular forms. Using exceptional isomorphisms between Lie groups of small rank, we show how one can use the intersection product on the periods to identify K3 surfaces whose periods can be expressed in terms of other classes of modular forms that have been studied in the mathematics literature. We apply our results to maximal cuts of three-loop banana integrals, and we show that depending on the mass configuration, the maximal cuts define ordinary modular forms or Hilbert, Siegel or hermitian modular forms.

Analysis of Modular Forms for Three-Loop Banana Integrals

The paper explores the analysis of three-loop banana integrals, focusing on their association with modular forms through the lens of algebraic geometry and quantum field theory. Specifically, it investigates periods of multi-parameter families of K3 surfaces and their automorphic properties, aiming to understand the modular transformations associated with Feynman integrals in theoretical physics. This analysis is imperative for the efficient computation of Feynman integrals, which play a significant role in perturbative quantum field theory.

Key Findings and Numerical Illustrations

1. K3 Surface Periods and Quantum Field Theory:
   • The paper demonstrates how periods of K3 surfaces relate to maximal cuts in Feynman integrals, using their orthogonal modular forms.
   • The research identifies the conditions under which these periods define various classes of modular forms, such as ordinary, Hilbert, Siegel, or hermitian modular forms, depending on specific mass configurations.
2. Impact of Lie Group Isomorphisms:
   • The exceptional isomorphisms between low-rank Lie groups are pivotal in expressing K3 periods via different modular forms. This insight provides a bridge for computations traditionally expressed through elliptic curves.
   • The isomorphisms guide the classification of periods into standard, Hilbert, Siegel, or hermitian modular forms, based on the symmetry group transforms.
3. Case Studies: The Three-Loop Banana Integral:
   • Detailed analyses of various mass configurations show that the periods, under these settings, can be reinterpreted in well-known modular contexts.
   • For example, in a case of three masses being equal, the paper reduces the problem to finding a relationship between the maximal cuts and expressions of ordinary modular forms—a significant simplification.
4. Automorphic and Theoretical Implications:
   • The paper extends known methods for elliptic curves to K3 surfaces, aiding in cataloging automorphic properties using orthogonal groups.
   • It presents a methodological framework for speculating future extensions to broader classes of Calabi-Yau varieties in theoretical physics.

Implications for Future AI and Mathematical Physics

This research provides a foundation for employing modern mathematical techniques alongside classical physics problems, emphasizing the role of modular forms in crafting sophisticated computational models. The paper not only facilitates more efficient numerical methods for solving integrals within particle physics but also encourages integrating AI technologies for symbolic mathematical computations in quantum field theories.

While current models and forms harnessed in this research provide clarity in specific integral solutions, future exploration may extend these techniques to multi-loop scenarios involving more complex mass configurations. This could inevitably contribute to the advancement of perturbative computations in both fundamental physics and applied quantum computations.

In sum, this paper signifies a crucial step towards unifying complex algebraic geometry with high-energy physics, leveraging mathematical innovations to tackle longstanding computational challenges. The implications on how AI can assimilate these methodologies point to a promising future in automated theorem proving and symbolic mathematics within theoretical domains.