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

TL;DR

This paper proves Goncharov's conjecture relating multiple polylogarithms to lower-depth polylogarithms and provides explicit formulas for their expressions and volumes of hyperbolic orthoschemes, revealing deep geometric and algebraic connections.

Contribution

It establishes the Goncharov depth conjecture with an explicit summation formula and generalizes the volume formula for hyperbolic orthoschemes to higher dimensions.

Findings

Proof of Goncharov's conjecture on polylogarithm depth reduction

Explicit formula involving tree decompositions for polylogarithm expressions

Generalized volume formula for hyperbolic orthoschemes in arbitrary dimensions

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 $\mathbb{P}^1.$ In particular, we derive both formulas from their common generalization.