Spectral Geometry

TL;DR

This paper introduces the fundamentals of noncommutative geometry through its spectral approach, focusing on spectral triples, noncommutative tori, and the spectral action motivated by physics.

Contribution

It reviews the spectral tools and generalizes classical geometric objects to noncommutative settings, including detailed analysis of the noncommutative torus and Moyal plane.

Findings

Spectral triples extend geometric concepts to noncommutative spaces.

The spectral action can be computed for noncommutative geometries like the torus.

Analysis of heat kernel asymptotics and noncommutative residues in these contexts.

Abstract

The goal of these lectures is to present the few fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein--Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics of different ingredients will be presented and studied like, Dirac operators, heat equation asymptotics, zeta functions and then, how to get within the framework of operators on Hilbert spaces, the notion of noncommutative residue, Dixmier trace, pseudodifferential operators etc. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus. Its non-compact generalization, namely the Moyal plane, is also investigated.