From polygons and symbols to polylogarithmic functions

By Claude Duhr, Herbert Gangl, John R. Rhodes. 2011

341 citations recorded by the source corpus.

A bstractWe 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.

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

View Paper: https://www.semanticscholar.org/paper/7592fde349bd19a908f2887937be4f207067e690

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Citations in the corpus

Lectures on differential equations for Feynman integrals (2014, background): “For these purposes, the notion of the “symbol” of iterated integrals is useful [51, 68].”
Bootstrapping an NMHV amplitude through three loops (2014, background): “We characterize the functions in terms of their weight-five {5, 1} coproduct components [63, 64], which are essentially their first derivatives.”
Mathematical aspects of scattering amplitudes (2014, background): “Similar conjectures can be made for higher weights [66], and also to decide whether a function can be expressed through product of lower weight functions only [34, 67, 68].”
Polylogarithms, multiple zeta values and superstring amplitudes (2013, background): “(23)Our definitions for polylogarithms agree with Goncharovs paper [35] and references [12] and [36].”
Hexagon functions and the three-loop remainder function (2013, neutral): “[80, 82] for a review).”
Uniqueness of two-loop master contours (2012, neutral): “This enables us to obtain the symbols [59, 60] of each of the chiral double boxes.”
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].”
Motivic amplitudes and cluster coordinates (2013, background): “We hope that motivic amplitudes and cluster coordinates will be found to be useful in these and other endeavors, just as general ‘motivic’ methods based on the symbol calculus of polylogarithm functions are finding ever wider applications to physical computations in quantum field theory, including F”
Color-kinematics duality for QCD amplitudes (2015, methodology): “Modern methods have been essential for many aspects of this progress, such as the unitarity method [3–5], on-shell recursion [6, 7], more recent unitarity-based one-loop methods [8– 19], and advanced integration techniques [20–26], to mention a few.”
Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph (2017, neutral): “05 46”
Two-loop planar master integrals for Higgs → 3 partons with full heavy-quark mass dependence (2016, background): “Note that no assumptions were made on the rationality of the alphabet letters, so that the above steps generalize the algorithm of [32] to algebraic cases.”
Soft triple-real radiation for Higgs production at N3LO (2013, neutral): “[123]) that the symbol can only have the following entries: [xk], [R 1 k] , [ −R(1) k/Q(1)k ] = [R(1) k]− [Q(1)l ] , [ 1 + x1Q 1 k R1 k ] = [R(1) k +Q 1 k x1]− [R(1) k] = [Pk]− [R(1) k] , [ 1 + Q(1)k R 1 l R1 kQ 1 l ] = [Q(1)k R 1 l −R(1) kQ(1)l ]− [R(1) k]− [Q(1)l ] .”
The four-loop six-gluon NMHV ratio function (2015, background): “Equivalently, there are nine different {n − 1, 1} elements of the coproduct [49, 50], corresponding to nine letters in the symbol [51–54] of the function.”