On webs, polylogarithms and cluster algebras

Abstract

In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose rank is 'As Maximal as Possible' (AMP) and show that many webs associated either to polylogarithmic functional equations or to cluster algebras are of this type. In particular, we prove a few results and state some conjectures about cluster webs associated to (pairs of) Dynkin diagrams. Along the way, we show that many of the classical functional equations satisfied by low-order polylogarithms (such as Spence-Kummer's equation of the trilogarithm or the tetralogarithmic one of Kummer) are of cluster type.

Unmixing the Wilson line defect CFT. Part II. Analytic bootstrap

“…For log this is of course straightforward, since naturally the single-valued extension of log z is log(z z), as already discussed around (3.15). For the ordinary polylogs Li n (z), various single-valued extensions are possible, due to various authors [114,[152][153][154][155], see section 2 of [156] for a nice review. We are particularly interested in the definition given by Wojtkowiak, who in [153] introduced the functions…”