Noncommutative Geometry of Gravity, Strings and Fields: A Panoramic Overview
arXiv:2511.22672
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 -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.
69 pages; v2: presentation expanded and improved, typos corrected, references added; Based on lectures given at the conference "Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Spacetime", CIRM, Marseille, 7-11 April 2025; to be published in EMS Series of Lectures in Mathematics