Spectral Action in Noncommutative Geometry

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 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.

171 pages, 3 figures

References in corpus (22)

• The Spectral Action Principle
• Gravity and the standard model with neutrino mixing
• Fundamentals of Quantum Information Theory
• On Noncommutative and semi-Riemannian Geometry
• Inner fluctuations of the spectral action
• The spectral action for Moyal planes
• On summability of distributions and spectral geometry
• Equivariant Lorentzian Spectral Triples
• Spectral Action for Robertson-Walker metrics
• Heat kernel and number theory on NC-torus
• The noncommutative Lorentzian cylinder as an isospectral deformation
• A twisted spectral triple for quantum SU(2)
• Spectral geometry of the Moyal plane with harmonic propagation
• On twisting real spectral triples by algebra automorphisms
• Distributional Asymptotic Expansions of Spectral Functions and of the Associated Green Kernels
• Local index formula and twisted spectral triples
• Spectral action and big desert
• Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori
• Asymptotic and exact expansions of heat traces
• Heat kernel of non-minimal gauge field kinetic operators on Moyal plane
• Quantum Dimension and Quantum Projective Spaces
• A unified approach for summation formulae

Cited by in corpus (7)

• A survey of spectral models of gravity coupled to matter
• Spectral dimensions and dimension spectra of quantum spacetimes
• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• Second Quantization and the Spectral Action
• Local invariants of non-commutative tori
• Clifford-based spectral action and renormalization group analysis of the gauge couplings
• Spectral action and the electroweak $\theta$-terms for the Standard Model without fermion doubling