# New algebraic aspects of perturbative and non-perturbative Quantum Field Theory

## TL;DR

This paper reviews recent algebraic and combinatorial advances in perturbative and non-perturbative quantum field theory, focusing on Feynman graphs, Dyson-Schwinger equations, and their connections to algebraic geometry and number theory.

## Contribution

It highlights new algebraic structures and methods for understanding quantum field theory, including Hopf algebras, Hochschild cohomology, and their applications to Dyson-Schwinger equations.

## Key findings

- Algebraic reformulation of perturbative renormalization.
- Progress in solving Dyson-Schwinger equations algebraically.
- Connections between Feynman integrals and algebraic geometry.

## Abstract

In this expository article we review recent advances in our understanding of the combinatorial and algebraic structure of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf algebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the high energy sector using the algebraic machinery. Finally there is a short account on a relation to algebraic geometry and number theory: understanding Feynman integrals as periods of mixed (Tate) motives.