On the Goncharov depth conjecture and a formula for volumes of orthoschemes

Abstract

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles.Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension 3 to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of points on P 1 . In particular, we derive both formulas from their common generalization.

Cutkosky representation and direct integration

“…Interestingly, square roots of cross-ratios have appeared in the recent work of Rudenko (see ref. [51]). We have…”

Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

“…Further to the notation used in [5] we introduce the following shorthand: Any subpolygon comes equipped with a partition of its internal angles into two subsets (these often correspond in [6] to 'even' and 'odd' polytopes, but we note that our conventions allow successive even or odd indices for lower depth terms). We equip the angles of one of the two sets with a slice of pi(e), indicating that the associated argument is given as the cyclic ratio (already used extensively in [5])…”

“…These findings date back to 2018 at the MPI Bonn and were both communicated to Don Zagier and subsequently presented at a workshop on cluster algebras and the geometry of scattering amplitudes at the Higgs Centre in Edinburgh in March 2020. Our more ambitious goal of finding a bootstrapping procedure that would produce analogous results for general weight has now apparently been superseded by Rudenko's beautiful new preprint [6] pertaining to Goncharov's depth conjecture. As our approach does not seem to take the exact same symmetries into account, it has the big disadvantage of being harder to generalise but on the other hand it may have produced identities that are of a slightly different natureboth potentially in the use of symmetries and of the choice of functions-than the ones that we anticipate to appear eventually in his already announced 'cluster polylogarithm' preprint (with Matveiakin).…”

Cluster Polylogarithms I: Quadrangular Polylogarithms

“…The existence of cluster polylogarithms of depth greater than one is far from obvious. We discovered the concept in attempt to understand quadrangular polylogarithms, which played a fundamental role in [Rud20].…”

“…For m " 6, a new cluster integrable symbol appears in weight four. This function (modulo lower depth corrections) is known in physics literature under the name "A 3 -function", in [GR18] under the name L 1 4 , and in [Rud20] under the name "quadrangular polylogarithm QLi 4 ." In general, symbols of quadrangular polylogarithms generate the space of cluster integrable symbols on CL n pM 0,m`2 q; we can also describe all relations which they satisfy.…”