Periods of hyperplane arrangements and multiple polylogarithms

TL;DR

This paper identifies conditions under which periods of certain hyperplane arrangements can be expressed as linear combinations of multiple polylogarithm values, extending Brown's method and applying to reflection arrangements.

Contribution

It introduces a combinatorial condition linking hyperplane arrangement intersections to multiple polylogarithm periods, generalizing Brown's approach.

Findings

Periods of specific hyperplane arrangements are expressible via multiple polylogarithms.

The method applies to reflection arrangements of the full monomial group.

Periods at roots of unity are characterized as linear combinations of multiple polylogarithm values.

Abstract

We compute the periods associated with a special class of hyperplane arrangements. In particular, we exhibit a combinatorial condition on the intersection lattice of a hyperplane arrangement that ensures that its associated periods are linear combinations of special values of multiple polylogarithms. Our method generalizes Brown's approach to the periods of moduli spaces of curves of genus zero. We apply this result to the reflection arrangement of the full monomial group, whose periods are shown to be linear combinations of values of multiple polylogarithms at roots of unity.