Galois symmetries of fundamental groupoids and noncommutative geometry

Abstract

We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct $\Delta$ in this Hopf algebra. We show that this formula encodes the group law of the automorphism group of a certain noncommutative variety. We relate the coproduct $\Delta$ to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees, which is a plane decorated version of the one defined by Connes and Kreimer [CK]. As an application, we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. These formulas play a key role in the mysterious correspondence between the structure of the motivic fundamental group of ${\mathbb{P}}^1-\left(\left\{0,\infty \right\}\cup {\mu }_N\right)$, where ${\mu }_N$ is the group of all $N$th roots of unity, and modular varieties for ${\mathrm{GL}}_m$ (see [G6], [G7]). In Section 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Section 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G7]. The appendix contains background material.