A simple construction of Grassmannian polylogarithms

By A. Goncharov. 2009

138 citations recorded by the source corpus.

Abstract The classical n -logarithm is a multivalued analytic function defined inductively: Li n ( z ) : = ∫ 0 z Li n − 1 ( t ) d log t , Li 1 ( z ) = − log ( 1 − z ) . In this paper we give a simple explicit construction of the Grassmannian n -logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n -dimensional subspaces in C 2 n in generic position to the coordinate hyperplanes by the natural action of the torus ( C ∗ ) 2 n . The classical n -logarithm appears at a certain one dimensional boundary stratum. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms d log f i where f i are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n -logarithm. Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M 0 , n , calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms. Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H • ( X ) . We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H • ( X ) . The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H • ( X ) generated by constant variations, there is a simple way to calculate the coproduct. It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/15fdb081af8f7aaf1b1c3fbc045b27e4dece60c3

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Citations in the corpus

A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “Precise definitions and additional details may be found in [69, 70, 71, 72] (see also [73] for a review), but here it is sufficient to recall the recursive definition according to which fk is called a generalised polylogarithm function of weight (or transcendentality) k if its total differential may”
Motivic multiple zeta values and superstring amplitudes (2012, background): “The symbol of a transcendental function represents a motivic road map encoding all the relevant information about the function without further specifying the latter explicitly in terms of multiple polylogarithms [3,2,31].”
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].”
Scattering Amplitudes and the Positive Grassmannian (2012, methodology): “There has also been extraordinary progress in determining the amplitude itself beyond the integrand, using the technology of symbols of transcendental functions to powerfully constrain and control the polylogarithms occurring in the final results, [19, 20].”
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes (2012, methodology): “re unknown in general, they can often be circumvented in practice by using the so-called symbol, which we will review in the next section. 3. Symbols In this section we give a short review of symbols [43]. Symbols were ﬁrst introduced in physics in Ref. [40] where they were used to simplify the six-”
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Bootstrapping the three-loop hexagon (2011, background): “nts. The deﬁnition (8) includes logarithms and classical polylogarithms, as well as other iterated integrals, such as harmonic polylogarithms of one [58] or more [59, 60, 61, 62] variables. The symbol[37] S(f) of a pure function fis deﬁned recursively with respect to eq. (8), S(f(k)) = X r S(f(k−1) ”