Galois symmetries of fundamental groupoids and noncommutative geometry

By A. Goncharov. 2002

336 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/af8a9e20f37b1ce1b7e3de46bd4a52b1ab979902

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In Geometry and Topology

1. Claude Duhr
2. Falko Dulat
3. L. Dixon
4. S. Stieberger
5. Li Guo
6. O. Schlotterer
7. Johannes Broedel
8. F. Brown

In Mathematical Physics

1. Claude Duhr
2. F. Brown
3. Li Guo
4. Don Zagier
5. Falko Dulat
6. L. Dixon
7. Herbert Gangl
8. John R. Rhodes

Citations in the corpus

• Evaluation of the multiple zeta values $\zeta(2,\dots,2,3,2,\dots,2)$ (2012, methodology): “) However, the true MZV’s in R are the images under a Q-linear map of certain “motivic” MZV’s which are defined purely algebraically, and it was proved (independently) by Terasoma [10] and Goncharov [6], [3] that the upper bound implied by the conjectural dimension formula held for these, implying i”
• FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, background): “ = y(T 1 ⊔···⊔Tb)−1 = y(T)− 1. This completes the induction. To get a tree representation of the Rota–Baxter bracketed words themselves we introduce the concept of angular decorated trees, similar to [24]. An angular decoration of a tree by a set X is the tree with a decoration by elements from X on”
• Depth-graded motivic multiple zeta values (2013, methodology): “The general quadratic relations between the σ2n+1 have been known explicitly for some time [22, 32, 19, 17] and are re-derived very easily using the formalism we introduce below.”
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, methodology): “[58], augmented by some ideas inspired by a recent paper by Brown [59].”
• Jet substructure at the Large Hadron Collider: A review of recent advances in theory and machine learning (2017, methodology): “To calculate the corresponding NGLs to this order, the authors exploited techniques of the modern amplitudes program, expressing the result in terms of generalized polylogarithm functions [368, 369, 370].”
• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “The algebra of generalised polylogarithm functions admits a coproduct ∆ compatible with multiplication, rendering it a Hopf algebra [70] (see also [74] for a review aimed at”
• New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series (2012, background): “A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math.”
• From polygons and symbols to polylogarithmic functions (2011, methodology): “[3, 4, 5, 6, 7, 8], with Hilbert’s third problem (on scissors congruences), e.”
• Double Zeta values and modular forms (2006, background): “Goncharov, who defined a coproduct structure on (formal) multiple zeta values in [6] and described relations between double zeta values and the cohomology of PSL2(Z) in [5].”
• Single-valued harmonic polylogarithms and the multi-Regge limit (2012, methodology): “The result is a complicated expression involving multiple polylogarithms which can be matched to a combination of SVHPLs and zeta values by means of the symbol [20, 21, 22, 23, 24] and coproduct [59, 60, 61].”
• Mixed Tate motives over $\Z$ (2011, methodology): “[6] A. B. Goncharov: Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math.”
• Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$ (2014, methodology): “Inst. Publ., 16 (1989) [15] P. Deligne: Multizêtas, Séminaire Bourbaki, expos 1048, Astrisque 352 (2013) [16] P. Deligne, A. B. Goncharov: Groupes fondamentaux motiviques de Tate mixte, Ann.”
• Graphical functions and single-valued multiple polylogarithms (2013, neutral): “An explicit formula for Δ is given in [29].”
• Polylogarithms, multiple zeta values and superstring amplitudes (2013, background): “Regularization of those integrals is discussed in detail in references [35, 40].”
• On the decomposition of motivic multiple zeta values (2011, background): “In [5], Goncharov showed how to lift the ordinary iterated integrals I(a0; .”
• Motivic amplitudes and cluster coordinates (2013, background): “The precise mathematical definitions of motivic avatars of polylogarithm-like functions is given in [9].”