Spectral Action in Noncommutative Geometry

  • Michał Eckstein
  • Bruno Iochum

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 27)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Michał Eckstein, Bruno Iochum
    Pages 1-36
  3. Michał Eckstein, Bruno Iochum
    Pages 37-62
  4. Michał Eckstein, Bruno Iochum
    Pages 63-94
  5. Michał Eckstein, Bruno Iochum
    Pages 95-112
  6. Michał Eckstein, Bruno Iochum
    Pages 113-119
  7. Back Matter
    Pages 121-155

About this book


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 serves both 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

Authors and affiliations

  • Michał Eckstein
    • 1
  • Bruno Iochum
    • 2
  1. 1.Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Aix-Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance

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