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From polygons and symbols to polylogarithmic functions

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Abstract

We present a review of the symbol map, a mathematical tool introduced by Goncharov and used by him and collaborators in the context of \( \mathcal{N} \) = 4 SYM for simplifying expressions among multiple polylogarithms, and we recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon, and it is indicated how that recipe relates to a similar explicit formula for it previously given by Goncharov. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

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  1. Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE, U.K

    Claude Duhr

  2. Institut für theoretische Physik, ETH Zürich, Wolfgang-Paulistr. 27, CH-8093, Switzerland

    Claude Duhr

  3. Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, U.K

    Herbert Gangl & John R. Rhodes

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Duhr, C., Gangl, H. & Rhodes, J.R. From polygons and symbols to polylogarithmic functions. J. High Energ. Phys. 2012, 75 (2012). https://doi.org/10.1007/JHEP10(2012)075

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