From polygons and symbols to polylogarithmic functions

Duhr, Claude; Gangl, Herbert; Rhodes, John R.

doi:10.1007/JHEP10(2012)075

From polygons and symbols to polylogarithmic functions

Claude Duhr
(
- Durham U., IPPP and
- Zurich, ETH
)
,
Herbert Gangl
(
- Durham U., Dept. of Math.
)
,
John R. Rhodes
(
- Durham U., Dept. of Math.
)

Oct, 2011

75 pages

Published in:

JHEP 10 (2012) 075

e-Print:

1110.0458 [math-ph]

DOI:

10.1007/JHEP10(2012)075

Report number:

IPPP-11-56,
DCPT-11-112

View in:

pdf

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Abstract: (arXiv)

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and 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. 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.

Note:

75 pages. Mathematica files with the expression of all HPLs up to weight 4 in terms of the spanning set are included

space: vector
algebra: tensor
algebra: graded
duality
integrability
Feynman graph

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