Anatomy of a gauge theory

TL;DR

This paper explores the role of Hochschild cohomology in gauge theories, linking algebraic structures like Hopf and Lie algebras to quantum equations of motion and identities, revealing new algebraic insights into quantum field theory.

Contribution

It demonstrates a novel connection between Hochschild cohomology, Hopf sub-algebras, and gauge theory identities, advancing the algebraic understanding of quantum field equations.

Findings

Hochschild cohomology relates to gauge theory structures.

Hopf sub-algebras encode Dyson-Schwinger equations.

Slavnov-Taylor identities correspond to algebraic sub-structures.

Abstract

We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the Slavnov-Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams.