Noncommutative geometry and fundamental interactions

TL;DR

This thesis explores the application of noncommutative geometry to fundamental physics, including gauge theories, topological properties, and quantum aspects, advancing the mathematical framework and physical models in particle physics.

Contribution

It provides a comprehensive study of noncommutative geometry's role in particle physics, extending to gauge theories, index theorems, and quantum gauge theories on noncommutative spaces.

Findings

Application of index theorem and cyclic homology to topologically non-trivial actions

Analysis of Yang-Mills-Higgs models on space-times with discrete components

Development of classical and quantum gauge theories on noncommutative tori

Abstract

This PhD thesis aims at describing the applications of noncommutative geometry to particle physics and quantum field theory. It includes a brief survey of the basic principles and definitions of noncommutative geometry such as spectral triples, differential calculus or gauge theories on projective modules. Within this framework, The index theorem and cyclic homology are applied to the probe of topologically non trivial properties of Chern-Simons and Yang-Mills action functionals in dimension 3 and 4. Following earlier work on the standard model in noncommutative geometry, it also provides a general study of all Yang-Mills-Higgs models based on space-times structures which are the product of usual space-time by a discrete space, with emphasis on the physical properties of the resulting theories. Finally, the last part is devoted to the development of classical and quantum gauge theory on the noncommutative torus in any dimension. It includes a discussion of topological properties and symmetries of the action as well as its perturbative quantization.