What can we learn from Knizhnik--Zamolodchikov Equations?

TL;DR

This paper explores the structural parallels between Knizhnik--Zamolodchikov equations and Dyson--Schwinger equations, highlighting their algebraic similarities and potential for organizing complex expansions in quantum field theory.

Contribution

It reveals the algebraic relationship between KZ and Dyson--Schwinger equations, providing a foundation for organizing next-to-leading log expansions.

Findings

Identifies filtration structures in Dyson--Schwinger equations

Shows how Dyson--Schwinger equations generalize KZ equations

Provides a basis for algebraic organization of quantum field theory expansions

Abstract

We discuss structural similarities between Knizhnik--Zamolodchikov equations (in fact, their simplest version needed to introduce the Drinfeld associator) and Dyson--Schwinger equations. We emphasize that the latter allow for a filtration by co-radical degree using quasi-shuffle products and the lower central series filtration of the Lie algebra of Feynman graphs. This clarifies how they are a generalization of the KZ equations. This is a starting point for a algebraic organization of the next-to...-to leading log expansion which has been worked out in collaboration with Olaf Krueger and which will be given elsewhere [1,2].