Duke Mathematical Journal

Galois symmetries of fundamental groupoids and noncommutative geometry

A. B. Goncharov

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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 $\mathbb{P}^1 - (\{0, \infty\}\cup \mu_N)$, where $\mu_N$ is the group of all $N$th roots of unity, and modular varieties for ${\GL_m}$ (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.

Article information

Duke Math. J. Volume 128, Number 2 (2005), 209-284.

First available in Project Euclid: 2 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11R32: Galois theory 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]


Goncharov, A. B. Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128 (2005), no. 2, 209--284. doi:10.1215/S0012-7094-04-12822-2. http://projecteuclid.org/euclid.dmj/1117728416.

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  • A. A. Beilinson and P. Deligne, Motivic polylogarithms and Zagier's conjecture, unpublished manuscript, version of 1992.
  • A. A. Beilinson, A. B. Goncharov, V. V. Schechtman, and A. N. Varchenko, ``Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane'' in The Grothendieck Festschtrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 135--172.
  • P. Belkale and P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116 (2003), 147--188.
  • A. Borel, Cohomologie de $\rm SL\sbn$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 613--636.; Errata, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 373. ;
  • D. J. Broadhurst, J. A. Gracey, and D. Kreimer, Beyond the triangle and uniqueness relations: Non-zeta counterterms at large $N$ from positive knots, Z. Phys. C 75 (1997), 559--574.
  • K. T. Chen, Iterated path integrals. Bull. Amer. Math. Soc. 83 (1977), 831--879.
  • A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203--242.
  • P. Deligne, ``Le groupe fondamental de la droite projective moins trois points'' in Galois Groups over $\mathbbQ$ (Berkeley, 1987), Math. Sci Res. Inst. Publ. 16, Springer, New York, 1989, 79--297.
  • --. --. --. --., ``Catégories tannakiennes'' in The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser, Boston, 1990, 111--195.
  • P. Deligne and A. B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm Sup. (4), 38 (2005), 1--56.
  • V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\Gal(\overline\mathbbQ/\mathbbQ)$ (in Russian), Algebra i Analiz 2, no. 4 (1990), 149--181.; English translation in Leningrad Math. J. 2 (1991), 829--860.
  • A. B. Goncharov, ``Polylogarithms and motivic Galois groups'' in Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., Providence, 1994, 43--96.
  • --. --. --. --., ``Polylogarithms in arithmetic and geometry'' in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 374--387.
  • --. --. --. --., ``Mixed elliptic motives'' in Galois representations in Arithmetic Algebraic Geometry, (Durham, 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998, 147--221.
  • --. --. --. --., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998), 497--516.
  • --. --. --. --., Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. 12 (1999), 569--618.
  • --. --. --. --., The dihedral Lie algebras and Galois symmetries of $\pi_1^(l)(\mathbbP^1 - (\0, \infty\\cup \mu_N)$, Duke Math. J. 110 (2001), 397--487.
  • --. --. --. --., ``Multiple $\zeta$-values, Galois groups, and geometry of modular varieties'' in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math. 201, Birkhäuser, Basel, 2001, 361--392.
  • --------, Multiple $\zeta$-numbers, hyperlogarithms and mixed Tate motives, preprint, June 1993, Mathematical Sciences Research Institute, Berkeley, no. 058-93.
  • --------, Galois groups, geometry of modular varieties and graphs, Proceedings of Arbeitstagung, June 1999, preprint, no. MPIM1999-50f, http://www.mpim-bonn.mpg.de
  • --------, Multiple polylogarithms and mixed Tate motives, preprint.
  • --------, Periods and mixed motives, preprint.
  • D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), 303--334.
  • M. Levine, ``Tate motives and the vanishing conjectures for algebraic $K$-theory'' in Algebraic $K$-Theory and Algebraic Topology (Lake Louise, Canada, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407, Kluwer, Dordrecht, 1993, 167--188.
  • J.-L. Loday, ``Dialgebras'' in Dialgebras and Related Operads, Lecture Notes in Math. 1763, Springer, Berlin, 2001, 7--66.
  • --. --. --. --., Arithmetree, J. Algebra 258 (2002), 275--309.
  • D. Zagier, ``Periods of modular forms, traces of Hecke operators, and multiple zeta values'' in Research into Automorphic Forms and $L$ Functions (Kyoto, 1992) (in Japanese), Sūrikaisekikenkyūsho Kōkyūroku 843, Kyoto Univ., Kyoto, 1993, 162--170.