Locality and renormalization: Universal properties and integrals on trees

Abstract

The purpose of this paper is to build an algebraic framework suited to regularise branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularisation of integrals indexed by rooted forests. We study their renormalisation, along the lines of Kreimer's toy model for Feynman integrals.

Double shuffle relations for arborified zeta values

“…• This statement was proven in [CGPZ2] in the more general framework of locality structures. This Theorem can be seen as the case where the independence relation is complete: ⊤ = Ω × Ω.…”

“…This example enjoys a universal property as will be recalled below. This universal property was originally shown in [KP], and formulated in the present form in [G1] and an alternative proof of this result can be found in [CGPZ2]. In order to state this property and derive some of its consequences we need to define the notion of morphism between operated structures.…”

“…This can be carried out in various ways, and we introduce here those that will be used later on. Such branchings were introduced in [CGPZ1] and further used in [CGPZ2].…”

“…Following [CGPZ2], we can now define a map from forests to words. It follows from the observation that (W Ω , ∅, ⊔⊔ λ ) is a commutative algebra operated by Ω through the natural map…”

“…ii. The application of the algebraic tools of [CGPZ1] and [CGPZ2] to study the relations obeyed by arborified zeta values.…”