Galois symmetries of fundamental groupoids and noncommutative geometry

We define a Hopf algebra of motivic iterated integrals on the line and prove an explicit formula for the coproduct ∆ 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 ∆ with 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 P1 − ({0,∞}∪μN ), where μN is the group of all N -th roots of unity, and modular varieties for GLm ([G1–2]). In Chapter 7 we discuss some general principles relating Feynman integrals and mixed motives. They are suggested by Chapter 4 and the Feynman integral approach for multiple polylogarithms on curves given in [G2]. Chapter 8 contains background material. 1. The Hopf algebra of motivic iterated integrals. Consider the iterated integral Iγ(a0; a1, ..., an; an+1) := (2πi) −n · ∫ ∆n,γ dt1 t1 − a1 ∧ dt2 t2 − a2 ∧ ... ∧ dtn tn − an (1) Here γ is a path from a0 to an+1 in C − {a1 ∪ ... ∪ an}, and integration is over a simplex ∆n,γ consisting of all ordered n–tuples of points (t1, ..., tn) on γ. We assume no restrictions on the points a1, ..., an ∈ C. So the iterated integral (1) can be divergent, and in this case it has to be regularized.