Spectral action in noncommutative geometry and global pseudodifferential calculus

TL;DR

This thesis explores the computation of spectral actions in noncommutative geometry, focusing on specific spectral triples like the noncommutative torus and quantum spheres, and develops a pseudodifferential calculus for these settings.

Contribution

It introduces a Diophantine condition for spectral action computation and constructs a global pseudodifferential calculus generalizing the Weyl--Moyal product.

Findings

Diophantine condition is crucial for spectral action with real structure

Existence of tadpoles in commutative geometries studied

Developed a symbolic pseudodifferential calculus for noncommutative settings

Abstract

In this thesis, we studied certain mathematical issues related to the computation of the Chamseddine--Connes spectral action on some fundamental noncommutative spectral triples, such as the noncommutative torus and the quantum 3-sphere SUq(2). We showed in particular that a Diophantine condition on the deformation matrix of the torus is crucial to obtain the spectral action with real structure. We also studied the question of existence of tadpoles (linear terms in the gauge potential of the fluctuation of the metric in the spectral action) for commutative Riemannian geometries, and the construction of a symbolic global pseudodifferential calculus allowing a generalization of the Weyl--Moyal product on a Schwartz space of rapidly decaying sections on a cotangent bundle of a manifold with linearization.