General Dyson-Schwinger equations and systems

TL;DR

This paper classifies combinatorial Dyson-Schwinger equations that generate Hopf subalgebras of Feynman graph Hopf algebras, revealing their algebraic structures and providing a comprehensive framework for understanding their solutions.

Contribution

It provides a classification of Dyson-Schwinger equations based on the Hopf algebraic structures they generate, including isomorphisms to known algebraic objects.

Findings

Identifies conditions under which solutions generate Faà di Bruno or symmetric function Hopf algebras.

Describes duals of enveloping algebras of specific Lie algebras arising from Dyson-Schwinger equations.

Classifies systems with multiple equations and insertion operators based on their algebraic properties.

Abstract

We classify combinatorial Dyson-Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary number (eventually infinite) of insertion operators. we distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Fa\`a di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.