Galois symmetries of fundamental groupoids and noncommutative geometry
A. B. Goncharov
TL;DR
The paper develops a comprehensive motivic framework for iterated integrals on the affine line, encoding their coproducts in Hopf algebras and linking these algebraic structures to automorphisms of noncommutative varieties and decorated planar trees. It builds the motivic fundamental groupoid within mixed Tate categories, connecting Hodge and $l$-adic realizations to path algebras and deriving explicit coproducts for motivic multiple polylogarithms, including cyclotomic cases. Through depth filtrations and explicit examples, it ties these motives to double zeta values and modular geometry, while proposing a correspondence principle that relates Feynman diagrams and motivic correlators, suggesting a unified picture for motivic data arising from quantum-field-theoretic constructs. The Appendix formalizes the Tannakian underpinnings of mixed Tate categories and framed objects, grounding the constructions in rigorous categorical language.
Abstract
We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral to be unramified at a prime ideal, and use it to estimate from above the space spanned by the values of iterated integrals. In chapter 7 we discuss some general principles relating Feynman integrals and mixed motives.