Noncommutative geometry, gauge theory and renormalization

TL;DR

This paper explores extending the Grosse-Wulkenhaar renormalization approach from scalar models to gauge theories on noncommutative Moyal space, proposing a new candidate for a renormalizable noncommutative gauge theory.

Contribution

It introduces a novel noncommutative gauge theory related to the Grosse-Wulkenhaar model and analyzes its properties and mathematical structure, aiming for renormalizability.

Findings

Proposed a new noncommutative gauge theory model.

Analyzed vacuum configurations of the model.

Provided a mathematical interpretation via superalgebra calculus.

Abstract

Nowadays, noncommutative geometry is a growing domain of mathematics, which can appear as a promising framework for modern physics. Quantum field theories on "noncommutative spaces" are indeed much investigated, and suffer from a new type of divergence called the ultraviolet-infrared mixing. However, this problem has recently been solved by H. Grosse and R. Wulkenhaar by adding to the action of a noncommutative scalar model a harmonic term, which renders it renormalizable. One aim of this thesis is the extension of this procedure to gauge theories on the Moyal space. Indeed, we have introduced a new noncommutative gauge theory, strongly related to the Grosse-Wulkenhaar model, and candidate to renormalizability. We have then studied the most important properties of this action, and in particular its vacuum configurations. Finally, we give a mathematical interpretation of this new action in terms of a derivation-based differential calculus associated to a superalgebra. This work contains among the results of this PhD, an introduction to noncommutative geometry, an introduction to epsilon-graded algebras, and an introduction to renormalization of scalar (wilsonian and BPHZ point of view) and gauge quantum field theories.