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Modular forms for three-loop banana integrals

Claude Duhr

TL;DR

The work develops a program to understand Feynman integrals associated with Calabi–Yau K3 surfaces depending on multiple moduli by studying their periods as automorphic objects under orthogonal groups. By exploiting exceptional isomorphisms among small-rank Lie groups, the authors show that K3 periods can be reinterpreted as ordinary, Hilbert, Siegel, or hermitian modular forms, depending on the transcendental lattice, and they apply this framework to classify the maximal cuts of the three-loop banana integral across all mass configurations. The analysis unifies the elliptic-case intuition with higher-genus modular structures, providing a map from geometric data (transcendental lattices, period maps) to explicit modular objects. The results offer a structured route to express banana-maximal-cuts in terms of known modular forms, with potential implications for analytic expressions and differential equations for multi-loop Feynman integrals.

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.

Modular forms for three-loop banana integrals

TL;DR

The work develops a program to understand Feynman integrals associated with Calabi–Yau K3 surfaces depending on multiple moduli by studying their periods as automorphic objects under orthogonal groups. By exploiting exceptional isomorphisms among small-rank Lie groups, the authors show that K3 periods can be reinterpreted as ordinary, Hilbert, Siegel, or hermitian modular forms, depending on the transcendental lattice, and they apply this framework to classify the maximal cuts of the three-loop banana integral across all mass configurations. The analysis unifies the elliptic-case intuition with higher-genus modular structures, providing a map from geometric data (transcendental lattices, period maps) to explicit modular objects. The results offer a structured route to express banana-maximal-cuts in terms of known modular forms, with potential implications for analytic expressions and differential equations for multi-loop Feynman integrals.

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.

Paper Structure

This paper contains 39 sections, 7 theorems, 186 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Calabi-Yau varieties and their periods
  3. Brief review of Calabi-Yau varieties
  4. Automorphic properties of periods: a motivational example
  5. K3 surfaces and their periods
  6. Lattices and their orthogonal groups
  7. Lattices
  8. Orthogonal groups
  9. The cohomology and periods of K3 surfaces
  10. The middle cohomology and the K3 lattice
  11. The moduli space of K3 surfaces
  12. The period domain and the period map.
  13. Orthogonal transformations as conformal transformations.
  14. The moduli space of lattice-polarised K3 surfaces.
  15. The monodromy group and automorphic properties of families of K3 surfaces
  16. ...and 24 more sections

Key Result

Lemma 1

Let $K\subseteq H\subseteq G$ be groups, and assume that $K$ has finite index in $G$. Then $K$ has finite index in $H$, and $H$ has finite index in $G$.

Figures (1)

  • Figure 1: The three-loop banana integral.

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Corollary 1
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof