Multiple polylogarithms and mixed Tate motives

By A. Goncharov. 2001

568 citations recorded by the source corpus.

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.

Subfields: Algebra and Number Theory, Mathematical Physics, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/4995b7e45c1043af6fa45a04dff33e1b16de03d9

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Citations in the corpus

On the periods of some Feynman integrals (2009, background): “(5) Can one compute the motivic coproduct [15] for graphs of matrix type? Certain families of graphs should be highly constrained in the coradical filtration, which would explain the prevalence of certain linear combinations of MZVs in the Feynman integral calculations known to date [2, 21].”
Galois symmetries of fundamental groupoids and noncommutative geometry (2002, background): “orphism c∗ M: GMot(F) → GF It seems that the Feynman integral itself should be upgraded to an inﬁnite dimensional mixed motive. Its matrix coeﬃcients corresponding to diﬀerent framings (see ch. 3 of [G3]) provide us all its correlators. 5. An example: the mixed Tate case. In the particularly interes”
QUASI-SYMMETRIC FUNCTIONS AND MOD p MULTIPLE HARMONIC SUMS (2004, background): “It has been proved by T. Terasoma [28] and A. B. Goncharov [10] that the true dimension of this rational vector space is at most dn.”
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, background): “1) defined recursively via the iterated integral [37, 38] G(a1, .”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, background): “Regularization of those integrals is discussed in detail in references [35, 40].”
Iterated integrals of modular forms and noncommutative modular symbols (2005, background): “ 9 of [Go6] (for convergent integrals), and in 7.5 of [Go6] elaborating the last page of [Go4] (for regularized integrals). Conversely, integral shuﬄe relations can be deduced from harmonic ones: see [Go5], Ch. 2. 28 §5. Iterated Eichler–Shimura and Hecke relations 5.1. Eichler–Shimura relations for”
Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism (2017, background): “rithms alone. Here we consider the simplest obstructions that one can encounter when studying iterated integrals on the Riemann sphere, multiple polylogarithms (MPLs), which are dened recursively by [3] G(c 1;:::;c n;x) = Z x 0 dtf(c 1;t)G(c 2;:::;c n;t); (2.3) with f(c;t) = 1 t c : (2.4) The recurs”
From polygons and symbols to polylogarithmic functions (2011, methodology): “0;0;1;1;x) = H(0;0;1;1;x) are not generic and we would hence obtain divergences in the right-hand side of eq. (3.23). We therefore need to use a regularized version of the dierential equation (3.23) [2], dH(0;0;1;1;x) = H(0;1;1;x)dlogx; (6.16) and so S(H(0;0;1;1;x) = S(H(0;1;1;x)) x: (6.17) { 33 { W”