Spectral Action in Noncommutative Geometry

1. Front Matter

   Pages i-xiv

   PDF

2. The Dwelling of the Spectral Action

   • Michał Eckstein, Bruno Iochum

   Pages 1-36

3. The Toolkit for Computations

   • Michał Eckstein, Bruno Iochum

   Pages 37-62

4. Analytic Properties of Spectral Functions

   • Michał Eckstein, Bruno Iochum

   Pages 63-94

5. Fluctuations of the Spectral Action

   • Michał Eckstein, Bruno Iochum

   Pages 95-112

6. Open Problems

   • Michał Eckstein, Bruno Iochum

   Pages 113-119

7. Back Matter

   Pages 121-155

   PDF

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 à 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 servesboth as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.

• Spectral triples
• Mellin transforms
• action functional
• noncommutative geometry
• almost-commutative geometries
• noncommutative integral
• operator heat trace
• Poisson summation formula
• spectral functions
• Heat operator
• Noncommutative tori
• Podles sphere
• heat Kernel expansion
• Sobolev space

“The book is a valuable review of the methods and applications of spectral action, especially in that it not only combines results scattered over many research papers but also adds new material and provides detailed proofs of many statements which were omitted in the original publications.” (Andrzej Sitarz, Mathematical Reviews, November, 2019)

“The purpose of the book is to provide a rigid first course in the spectral action and to charm the reader with the marvelous interaction between mathematics and physics encapsulated in the notion of the spectral action.” (Vida Milani, zbMath 1416.81008, 2019)

 

• Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Kraków, Poland

  Michał Eckstein

• Aix-Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

  Bruno Iochum

 

Policies and ethics