nLab polylogarithm

Idea

Polylogarithms $Li_n(x)$ generalize dilogarithms.

There are also multiple polylogarithms of many variables, where the index $n$ is replaced by a partition of number $n$.

References

• Wikipedia: Polylogarithm

• L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1981 MR618278

• Don Zagier: Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic Algebraic Geometry, Progr. Math. 89, Birkhäuser (1990) 391-430 [MR1085270]

• A. B. Goncharov, Polylogarithms, regulators and Arakelov motivic complexes, JAMS math.AG/0207036; A simple construction of Grassmannian polylogarithms_, arxiv/0908.2238; Multiple polylogarithms, cyclotomy and modular complexes, Math Res. Letters 5, (1998) 497-516, arxiv/1105.2076

• Herbert Gangl, Alexander B. Goncharov, Andrey Levin, Multiple logarithms, algebraic cycles and trees, in “Frontiers in Number Theory, Physics and Geometry”, vol. 2 (Cartier, Julia, Moussa, Vanhove, eds.) math.AG/0504552; Multiple polylogarithms, polygons, trees and algebraic cycles, arXiv:math.NT/0508066

Applications to renormalization in quantum field theory and in relation to motives is in

• Alexander B. Goncharov, Marcus Spradlin, C. Vergu, Anastasia Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys.Rev.Lett.105:151605,2010, arxiv/1006.5703

In relation to spherical infinitesimal braid relations:

• Anton Alekseev, Megan Howarth, Florian Naef, Muze Ren, Pavol Ševera: The kernel of formal polylogarithms [arXiv:2601.19455]