Feynman amplitudes, coaction principle, and cosmic Galois group

The first part of a set of notes based on lectures given at the IHES in May 2015 on Feynman amplitudes and motivic periods. 0.1. Some motivation for physicists. Scattering amplitudes are ubiquitous in high energy physics and have been intensively studied from at least three angles: (1) in phenomenology, where amplitudes in quantum field theory are obtained as a sum of Feynman integrals associated to graphs which represent interactions between fundamental particles. This presents a huge computational challenge with important applications to collider experiments. (2) in superstring perturbation theory, where amplitudes are expressed as integrals over moduli spaces of curves with marked points. (3) in various modern approaches, most notably in the planar limit of N = 4 SYM, which avoid the use of Feynman graphs altogether and seek to construct the amplitude directly, either via the bootstrap method, or via geometric approaches such as on-shell diagrams or the amplituhedron. The goal of these notes is to study a new kind of structure which is potentially satisfied by amplitudes in all three situations. To motivate it, consider first the case of the dilogarithm function, defined for |z| < 1 by the sum