Cosmic Wheels: From integrability to the Galois coaction

TL;DR

This paper explores the connection between Feynman loop integrals, their motivic properties, and the Cosmic Galois Group, proposing a new conjecture related to differential equations for periods in quantum field theory.

Contribution

It introduces a novel link between integrable systems, motivic structures, and Galois coaction in Feynman integrals, supported by fishnet graph analysis.

Findings

Coaction relations derive from Quantum Spectral Curve constructions.

Conjecture of a differential equation governing periods.

Connection between integrability and motivic properties of Feynman integrals.

Abstract

We argue that the description of Feynman loop integrals as integrable systems is intimately connected with their motivic properties and the action of the Cosmic Galois Group. We show how in the case of a family of fishnet graphs, coaction relations between graphs follow directly from iterative constructions of Q-functions in the Quantum Spectral Curve formalism. Using this observation we conjecture a "differential equation for numbers" that enter these periods.