An Introduction to Noncommutative Spaces and their Geometry

TL;DR

This paper introduces noncommutative geometry concepts to physicists, illustrating applications to gauge theories, gravity, and quantum models, and discusses recent developments in noncommutative lattices and spectral actions.

Contribution

It provides an accessible introduction to noncommutative geometry for physicists, including new insights into spectral actions and noncommutative lattice models.

Findings

Application of spectral action to gravity models

Construction of topologically nontrivial quantum models on noncommutative lattices

Accessible mathematical tools for physicists in noncommutative geometry

Abstract

These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate applications to Yang-Mills, fermionic and gravity models, notably we describe the spectral action recently introduced by Chamseddine and Connes. We also present an introduction to recent work on noncommutative lattices. The latter have been used to construct topologically nontrivial quantum mechanical and field theory models, in particular alternative models of lattice gauge theory. Here is the list of sections: 1. Introduction. 2. Noncommutative Spaces and Algebras of Functions. 3. Noncommutative Lattices. 4. Modules as Bundles. 5. The Spectral Calculus. 6. Noncommutative Differential Forms. 7. Connections on Modules. 8. Field Theories on Modules. 9. Gravity Models. 10. Quantum Mechanical Models on Noncommutative Lattices. Appendices: Basic Notions of Topology. The Gel'fand-Naimark-Segal Construction. Hilbert Modules. Strong Morita Equivalence. Partially Ordered Sets. Pseudodifferential Operators