Feynman integrals, elliptic integrals and two-parameter K3 surfaces

TL;DR

This paper explores the deep mathematical structure of certain Feynman integrals, revealing their connection to K3 surfaces and modular forms, and uncovers hidden symmetries in these integrals.

Contribution

It establishes a link between complex Feynman integrals and algebraic geometry, specifically K3 surfaces, and expresses their solutions in terms of modular forms, revealing hidden symmetries.

Findings

Maximal cuts of the three-loop banana integral relate to products of sunrise integrals.

Solutions can be expressed using ordinary modular forms and functions.

Discovered a hidden symmetry exchanging two elliptic curves in the banana integral.

Abstract

The three-loop banana integral with three equal masses and the conformal two-loop five-point traintrack integral in two dimensions are related to a two-parameter family of K3 surfaces. We compute the corresponding periods and the mirror map, and we show that they can be expressed in terms of ordinary modular forms and functions. In particular, we find that the maximal cuts of the three-loop banana integral with three equal masses can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Our computation reveals a hidden symmetry of the banana integral not manifest from the Feynman integral representation, which corresponds to exchanging the two copies of the sunrise elliptic curve.