Global and local aspects of spectral actions

Journal of Physics A: Mathematical and Theoretical

Volumn 45, Issue 37, 2012, Pages

  Iochum, B ^a   Levy, C ^b   Vassilevich, D V ^c,d  

^a AIX MARSEILLE UNIVERSITY   (France)

^b UNIVERSITY OF POTSDAM   (Germany)

^c UNIVERSIDADE FEDERAL DO ABC   (Brazil)

^d ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84866324618     PISSN: 17518113     EISSN: 17518121     Source Type: Journal    
DOI: 10.1088/1751-8113/45/37/374020     Document Type: Article

References (81)

1. Andrianov A A, Kurkov M A and Lizzi F 2011 Spectral action, Weyl anomaly and the Higgs-dilaton potential J. High Energy Phys. JHEP10(2011)001;
2. Andrianov A A and Lizzi F 2010 Bosonic spectral action induced from anomaly cancellation J. High Energy Phys. JHEP05(2010)057;
3. Angelescu N, Nenciu G and Bundaru M 1975 On the perturbation of Gibbs semigroups Commun. Math. Phys. 42 29-30;
4. Arendt W, Nittka R, PeterWand Steiner F 2009Weyl's law: Spectral properties of the Laplacian in mathematics and physics Mathematical Analysis of Evolution, Information, and Complexity edWArendt andWSchleich (Weinheim: Wiley);
5. Avramidi I G 2008 Heat kernel asymptotics on homogeneous bundles Int. J. Geom. Methods Mod. Phys. 5 1-23;
6. Banasiak J and Arlotti L 2006 Perturbation of Positive Semigroups with Applications (London: Springer);
7. Barvinsky A O and Vilkovisky G A 1987 Beyond the Schwinger-Dewitt technique: Converting loops into trees and in-in currents Nucl. Phys. B 282 163-88;
8. Barvinsky A O and Vilkovisky G A 1990 Covariant perturbation theory (II): Second order in the curvature. General algorithms Nucl. Phys. B 333 471-511;
9. Chamseddine A and Connes A 1997 The spectral action principle Commun. Math. Phys. 186 731-50;
10. Chamseddine A and Connes A 1996 Universal formula for noncommutative geometry actions: Unification of gravity and the standard model Phys. Rev. Lett. 77 4868-71; (Pubitemid 126627713)
11. Chamseddine A and Connes A 2006 Inner fluctuations of the spectral action J. Geom. Phys. 57 1-21; (Pubitemid 44469291)
12. Chamseddine A and Connes A 2006 Scale invariance in the spectral action J. Math. Phys. 47 063504;
13. Chamseddine A and Connes A 2007 Quantum gravity boundary terms from the spectral action on noncommutative space Phys. Rev. Lett. 99 071302;
14. Chamseddine A and Connes A 2008 Why the standard model J. Geom. Phys. 58 38-47;
15. Chamseddine A and Connes A 2010 The uncanny precision of the spectral action Commun. Math. Phys. 293 867-97;
16. ChamseddineAand ConnesA2010 Noncommutative geometry as a framework for unification of all fundamental interactions including gravity: Part I Fortschr. Phys. 58 553-600;
17. Chamseddine A and Connes A 2010 Noncommutative geometric spaces with boundary: Spectral action, arXiv:1008.3980v1 [hep-th];
18. Chamseddine A and Connes A 2011 Spectral action for Robertson-Walker metrics arXiv:1105.4637v1 [hep-th];
19. Chamseddine A, Connes A and Marcolli M 2007 Gravity and the standard model with neutrino mixing Adv. Theor. Math. Phys. 11 991-1090;
20. Connes A 1988 The action functional in non-commutative geometry Commun. Math. Phys. 117 673-83;
21. Connes A 1994 Noncommutative Geometry (London: Academic);
22. Connes A 2008 On the spectral characterization of manifolds arXiv:0810.2088v1;
23. Connes A 2008 A unitary invariant in Riemannian geometry Int. J. Geom. Methods Mod. Phys. 5 1215-42;
24. Connes A and Marcolli M 2008 Noncommutative Geometry, Quantum Fields and Motives (Colloquium Publications vol 55) (Providence, RI: American Mathematical Society);
25. Dixmier J 1966 Existence de traces non-normales C. R. Acad. Sci., Paris A 262 1107-8;
26. Dowker J S and Critchley R 1976 Effective Lagrangian and energy-momentum tensor in de Sitter space Phys. Rev. D 13 3224-32;
27. Elizalde E 1995 Ten Physical Applications of Spectral Zeta Functions (Berlin: Springer);
28. Elizalde E, Vanzo L and Zerbini S 1998 Zeta function regularization, the multiplicative anomaly and the Wodzicki residue Commun. Math. Phys. 194 613-30 (arXiv:hep-th/9701060);
29. Essouabri D, Iochum B, Levy C and Sitarz A 2008 Spectral action on noncommutative torus Commun. Math. Phys. 2 53-123;
30. Estrada R 1998 The Cesaro behaviour of distributions Proc. R. Soc. A 454 2425-43;
31. Estrada R, Gracia-Bond́ia J M and V́arilly J C 1998 On summability of distributions and spectral geometry Commun. Math. Phys. 191 219-48;
32. Estrada R and Kanwal R P 2002 A Distributional Approach to Asymptotics 2nd edn (Boston, MA: Birkhäuser);
33. Fock V 1937 Proper time in classical and quantum mechanics Phys. Z. Sowjetunion 12 404;
34. Fulling S 1999 Some properties of Riesz means and spectral expansions Electron. J. Diff. Eqns 6 1-39; (Pubitemid 38991267)
35. FursaevDandVassilevichD2011 Operators, Geometry and Quanta: Methods of Spectral Geometry inQuantum Field Theory (Dordrecht: Springer);
36. Gayral V, Gracia-Bond́ia J M, Iochum B, Schücker T and V́arilly J C 2004 Moyal planes are spectral triples Commun. Math. Phys. 246 569-623; (Pubitemid 38651784)
37. Gayral V and Iochum B 2005 The spectral action for Moyal plane J. Math. Phys. 46 043503;
38. Gayral V, Iochum B and Vassilevich D 2007 Heat kernel and number theory on NC-torus Commun. Math. Phys. 273 415-43;
39. Gilkey P B 1995 Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theory (Boca Raton, FL: CRC Press);
40. Gilkey P B and Grubb G 1998 Logarithmic terms in asymptotic expansions of heat operator traces Commun. Partial Diff. Eqns 23 777-92; (Pubitemid 128363410)
41. Gilkey P B 2003 Asymptotic Formulae in Spectral Geometry (London: Chapman and Hall);
42. Gracia-Bond'ia J M, V'arilly J C and Figueroa H 2001 Elements of Noncommutative Geometry (Birkhäuser Advanced Texts) (Boston, MA: Birkhäuser);
43. Hanisch F, Pfäffle F and Stephan C 2010 The spectral action for Dirac operators with skew-symmetric torsion Commun. Math. Phys. 300 877-88;
44. Hawking S W 1977 Zeta function regularization of path integrals in curved spacetime Commun. Math. Phys. 55 133-48;
45. Iochum B and Levy C 2011 Tadpoles and commutative spectral triples J. Noncommut. Geom. 5 299-320;
46. Iochum B and Levy C 2011 Spectral triples and manifolds with boundary J. Funct. Anal. 260 117-34;
47. Iochum B, Levy C and Sitarz A 2009 Spectral action on SUq(2) Commun. Math. Phys. 289 107-55;
48. Iochum B, Levy C and Vassilevich D 2012 Spectral action for torsion with and without boundaries Commun. Math. Phys. 310 367-82;
49. Iochum B, Levy C and Vassilevich D 2011 Spectral action beyond the weak-field approximation, arXiv:1108.3749 [hep-th];
50. Jureit J-H and Stephan C 2008 On a classification of irreducible almost commutative geometries: IV J. Math. Phys. 49 033502;
51. Jureit J-H and Stephan C 2008 Finding the standard model of particle physics, a combinatorial problem Comput. Phys. Commun. 178 230-47;
52. Kato T 1995 Perturbation Theory for Linear Operators (Berlin: Springer);
53. Kirsten K 2001 Spectral Functions in Mathematics and Physics (Boca Raton, FL: Chapman & Hall);
54. Kolodrubetz D and Marcolli M 2010 Boundary conditions of the RGE flow in the noncommutative geometry approach to particle physics and cosmology Phys. Lett. B 693 166-74;
55. Landi G and Rovelli C 1997 General relativity in terms of Dirac eigenvalues Phys. Rev. Lett. 78 3051-4; (Pubitemid 127657285)
56. Lizzi F, Mangano G, Miele G and Sparano G 1996 Constraints on unified gauge theories from noncommutative geometry Mod. Phys. Lett. A 11 2561-72; (Pubitemid 126435302)
57. Marcolli M, Pierpaoli E and Teh K 2011 The spectral action and cosmic topology Commun. Math. Phys. 304 125-74;
58. Marcolli M, Pierpaoli E and Teh K 2012 The coupling of topology and inflation in noncommutative cosmology Commun. Math. Phys. 309 341-60;
59. Marcolli M and Pierpaoli E 2010 Early universe models from noncommutative geometry Adv. Theor. Math. Phys. 14 1373-432;
60. Martin C P, Gracia-Bond́ia J M and V́arilly J C 1998 The standard model as a noncommutative geometry: The low-energy regime Phys. Rep. 294 363-406; (Pubitemid 128176527)
61. NelsonWand Sakellariadou M 2009 Inflation mechanism in asymptotic noncommutative geometry Phys. Lett. B 680 263-6;
62. Novozhilov Y V and Vassilevich D V 1991 Induced classical gravity Lett. Math. Phys. 21 253-71;
63. Osborn T A and Molzahn F H 1991 The Wigner-Weyl transform on tori and connected graph propagator representations Forty More Years of Ramifications: Spectral Asymptotics and Its Applications (Discourses in Mathematics and Its Applications vol 1) ed S A Fulling and F J Narcowich (College Station, TX: Texas A&M University) pp 199-236;
64. Pfäffle F and Stephan C 2012 On gravity, torsion and the spectral action principle J. Funct. Anal. 262 1529-65;
65. Pfäffle F and Stephan C 2011 TheHolst action by the spectral action principle Commun. Math. Phys. 307 261-73;
66. Sakellariadou M 2011 Cosmology within noncommutative spectral geometry arXiv:1101.2174v1 [hep-th];
67. Sakharov A D 1975 Spectral density of eigenvalues of the wave equation and vacuum polarization Theor. Math. Phys. 23 435-44;
68. Schücker T 2005 Forces from Connes' geometry Topology and Geometry in Physics (Lecture Notes in Physics vol 659) ed E Bick and F Steffen (Berlin: Springer) pp 285-350;
69. Strelchenko A V and Vassilevich D V 2007 On space-time noncommutative theories at finite temperature Phys. Rev. D 76 065014 (arXiv:0705.4294 [hep-th]);
70. Uhlenbrock D A 1971 Perturbation of statistical semigroups in quantum statistical mechanics J. Math. Phys. 12 2503-12;
71. van Suijlekom W D 2011 Renormalization of the asymptotically expanded Yang-Mills spectral action arXiv:1104.5199 [math-ph];
72. Vassilevich D V 2003 Heat kernel expansion: User's manual Phys. Rep. 388 279-360; (Pubitemid 37394272)
73. Vassilevich D V 2004 Noncommutative heat kernel Lett. Math. Phys. 67 185-94; (Pubitemid 38945327)
74. Vassilevich D V 2005 Heat kernel, effective action and anomalies in noncommutative theories J. High Energy Phys. JHEP08(2005)085;
75. Vassilevich D V 2007 Heat trace asymptotics on noncommutative spaces SIGMA 3 093 (arXiv:0708.4209;
76. Voros A 1992 Spectral zeta functions Adv. Stud. Pure Math. 21 327-58;
77. Wodzicki M 1984 Local invariants of spectral asymmetry Invent. Math. 75 143-77;
78. Wodzicki M 1987 Noncommutative residue: Chapter 1. Fundamentals K-theory, Arithmetic and Geometry (Lecture Notes in Math. vol 1289) ed Y I Manin (Berlin: Springer) pp 320-99;
79. Zagrebnov V A 1989 Perturbation of Gibbs semigroups Commun. Math. Phys. 120 653-64;
80. Zagrebnov V A 2003 Topics in the Theory of Gibbs Semigroups (Leuven: Leuven University Press);
81. Zeldovich Y B 1967 Cosmological constant and elementary particles JETP Lett. 6 316-7