Spectral Action in Noncommutative Geometry

TL;DR

This book provides a comprehensive overview of spectral action in noncommutative geometry, detailing computation methods, key examples, and open problems, aimed at researchers and mathematical physicists.

Contribution

It offers a detailed, structured exploration of spectral action, including computational tools, theoretical insights, and links to physical models, advancing understanding in noncommutative geometry.

Findings

Methodology for computing spectral action

Connections between heat trace asymptotics and spectral zeta functions

Analysis of spectral action behavior under gauge fluctuations

Abstract

What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry \`a la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.