Noncommutative Geometry of Gravity, Strings and Fields: A Panoramic Overview

TL;DR

This paper provides an overview of noncommutative geometry and its applications to quantum spacetime, string theory, and quantum field theory, highlighting key mathematical tools and physical models.

Contribution

It introduces fundamental concepts and techniques of noncommutative geometry and demonstrates their relevance to modeling quantum spacetime and physical theories.

Findings

Noncommutative geometry offers a versatile framework for quantum spacetime modeling.

Mathematical tools like operator algebras and K-theory are essential in this approach.

Applications include string theory, quantum field theory, and condensed matter systems.

Abstract

These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key mathematical approaches presented include operator algebras such as $C^*$-algebras, K-theory, spectral geometry, quantum groups, and deformation quantization. Physical application areas considered include string theory, quantum field theory, and the Standard Model, as well as certain condensed matter systems.