Spectral Action in Noncommutative Geometry

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.

[1]  E. Schrohe.  Complex powers of elliptic pseudodifferential operators , 1986 .

[2]  Peter B. Gilkey,et al.  Asymptotic Formulae in Spectral Geometry , 2003 .

[3]  Stuart White,et al.  On spectral triples on crossed products arising from equicontinuous actions , 2011, 1103.6199.

[4]  The spectral action for Moyal planes , 2004, hep-th/0402147.

[5]  E Bergshoeff,et al.  Ten Physical Applications of Spectral Zeta Functions , 1996 .

[6]  SINGULAR TRACES AND RESIDUES OF THE ζ-FUNCTION , 2017 .

[7]  B. Iochum,et al.  Heat asymptotics for nonminimal Laplace type operators and application to noncommutative tori , 2017, Journal of Geometry and Physics.

[8]  E. Minguzzi.  Compactification of closed preordered spaces , 2012, 1209.1839.

[9]  T. Apostol.  Introduction to analytic number theory , 1976 .

[10]  Michał Eckstein,et al.  Asymptotic and Exact Expansions of Heat Traces , 2014, 1412.5100.

[11]  H. Moscovici.  LOCAL INDEX FORMULA AND TWISTED SPECTRAL TRIPLES , 2009, 0902.0835.

[12]  B. Iochum,et al.  Spectral Action for Torsion with and without Boundaries , 2010, 1008.3630.

[13]  B. Iochum,et al.  Heat Trace and Spectral Action on the Standard Podleś Sphere , 2013, 1307.4188.

[14]  A. Connes,et al.  The Uncanny Precision of the Spectral Action , 2008, 0812.0165.

[15]  B. Iochum,et al.  Spectral action on noncommutative torus , 2007, 0704.0564.

[16]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[17]  A. Connes.  Geometry from the spectral point of view , 1995 .

[18]  A. Rennie.  Summability for Nonunital Spectral Triples , 2004 .

[19]  R. Senior,et al.  A twisted spectral triple for quantum SU(2) , 2011, 1109.2326.

[20]  G. Grubb,et al.  Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems , 1995 .

[21]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[22]  A. Zemanian.  The Distributional Laplace and Mellin Transformations , 1966 .

[23]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[24]  G. Landi,et al.  On Twisting Real Spectral Triples by Algebra Automorphisms , 2016, 1601.00219.

[25]  Spectral action and big desert , 2006, hep-ph/0605166.

[26]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[27]  Philippe Flajolet,et al.  Mellin Transforms and Asymptotics: Harmonic Sums , 1995, Theor. Comput. Sci..

[28]  A. Connes,et al.  Spectral action for Robertson-Walker metrics , 2011, 1105.4637.

[29]  G. Hardy,et al.  The General Theory Of Dirichlets Series , .

[30]  A. Zemanian.  Distribution Theory and Transform Analysis; An Introduction to Generalized Functions, With Applications , 1965 .

[31]  Twisted spectral triples and quantum statistical mechanical systems , 2014 .

[32]  G. Dijk.  Distribution Theory: Convolution, Fourier Transform, And Laplace Transform , 2013 .

[33]  Rudolf Haag,et al.  Local quantum physics : fields, particles, algebras , 1993 .

[34]  A. Connes,et al.  Mathematical Physics the Action Functional in Non-commutative Geometry , 2022 .

[35]  N. Higson,et al.  Analytic K-Homology , 2000 .

[36]  Steven G. Krantz,et al.  Invariance Theory Heat Equation and Atiyah Singer Index Theorem , 1995 .

[37]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[38]  B. Iochum,et al.  Heat Kernel and Number Theory on NC-Torus , 2006, hep-th/0607078.

[39]  W. D. Suijlekom.  The noncommutative Lorentzian cylinder as an isospectral deformation , 2003, math-ph/0310009.

[40]  Peter B. Gilkey,et al.  The Atiyah-Singer Index Theorem* , 2010 .

[41]  Ali H. Chamseddine,et al.  Scale invariance in the spectral action , 2005 .

[42]  R. Paris,et al.  Asymptotics and Mellin-Barnes Integrals , 2001 .

[43]  K. Athreya,et al.  Measure Theory and Probability Theory , 2006 .

[44]  A. Connes,et al.  The Spectral Action Principle , 1996, hep-th/9606001.

[45]  B. Iochum,et al.  Crossed product extensions of spectral triples , 2014, 1406.4642.

[46]  Marco Matassa.  Quantum Dimension and Quantum Projective Spaces , 2014, 1405.5396.

[47]  A. Rennie.  Smoothness and Locality for Nonunital Spectral Triples , 2003 .

[48]  On Summability of Distributions and Spectral Geometry , 1997, funct-an/9702001.

[49]  Christopher J Howls,et al.  ASYMPTOTICS AND MELLIN–BARNES INTEGRALS (Encyclopedia of Mathematics and its Applications 85) By R. B. P ARIS and D. K AMINSKI : 422 pp., £65.00 (US$95.00), ISBN 0 521 79001 8 (Cambridge University Press, 2001). , 2003 .

[50]  Gravity and the standard model with neutrino mixing , 2006, hep-th/0610241.

[51]  Alain Connes,et al.  The Local Index Formula in Noncommutative Geometry , 1995 .

[52]  R. Wulkenhaar,et al.  Spectral geometry of the Moyal plane with harmonic propagation , 2011, 1108.2184.

[53]  A. Erdélyi.  Asymptotic Expansions of Fourier Integrals Involving Logarithmic Singularities , 1956 .

[54]  E. Pierpaoli,et al.  The Spectral Action and Cosmic Topology , 2010, 1005.2256.

[55]  F. Strocchi.  An introduction to the mathematical structure of quantum mechanics : a short course for mathematicians : lecture notes , 2005 .

[56]  Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds , 2004, hep-th/0412233.

[57]  Raphael Ponge,et al.  Index map, $\sigma$-connections, and Connes-Chern character in the setting of twisted spectral triples , 2013, 1310.6131.

[58]  Moyal Planes are Spectral Triples , 2003, hep-th/0307241.

[59]  Distributional Asymptotic Expansions of Spectral Functions and of the Associated Green Kernels , 1997, funct-an/9710003.

[60]  M. Keyl.  Fundamentals of quantum information theory , 2002, quant-ph/0202122.

[61]  Victor Ivrii,et al.  100 years of Weyl’s law , 2016, Microlocal Analysis, Sharp Spectral Asymptotics and Applications V.

[62]  D. L. Cohn.  Measure Theory: Second Edition , 2013 .

[63]  Ricardo Estrada,et al.  A Distributional Approach to Asymptotics: Theory and Applications , 2002 .

[64]  Joseph C. Várilly,et al.  Elements of Noncommutative Geometry , 2000 .

[65]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[66]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[67]  F. Jean-christophe.  A unified approach for summation formulae , 2016, 1604.05578.