Centre de Physique Théorique


Vendredi 9 décembre 2022

10h00 – 12h30, CPT

Thesis defense : Noncommutative Geometry and Gauge theories on 𝐴𝐹 algebras

Gaston Nieuviarts

Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes.
It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian
framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with
physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle
Physics (SMPP), taking into account general relativity, in a single "geometric" representation, based on Non-
Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to
study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to
show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to
elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite
𝐶∗-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying
differential structure of the Gauge theory. The inductive sequence defining the AF-algebra is used to allow the
construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank 𝑛 + 1 can represent a grand
unified theory of the rank 𝑛.The main advantage of this framework is that it controls, using appropriate conditions,
the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests
a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the