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From polygons and symbols to polylogarithmic functions

  • Claude DuhrEmail author
  • Herbert Gangl
  • John R. Rhodes
Article

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.

Keywords

Scattering Amplitudes Supersymmetric gauge theory 

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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Institute for Particle Physics PhenomenologyUniversity of DurhamDurhamU.K
  2. 2.Institut für theoretische PhysikETH ZürichSwitzerland
  3. 3.Department of Mathematical SciencesUniversity of DurhamDurhamU.K

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