×

On gravity, torsion and the spectral action principle. (English) Zbl 1238.53030

On compact Riemannian spin manifolds without boundary and equipped with orthogonal connections, the authors investigate the induced Dirac operators and the associated commutative spectral triples. For the case of four dimensions and totally anti-symmetric torsion, they compute the Chamseddine-Connes spectral action, derive the equations of motions, and discuss critical points.
Contents include: Introduction; Orthogonal connections on Riemannian manifolds; Curvature calculations in case of totally anti-symmetric torsion in four dimensions; Dirac operators associated to orthogonal connections; Commutative geometries and the spectral action principle; Appendix; and References (thirty-two items).

MSC:

53C27 Spin and Spin\({}^c\) geometry
53C80 Applications of global differential geometry to the sciences
58B34 Noncommutative geometry (à la Connes)
53C05 Connections (general theory)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agricola, I., The Srní lectures on non-integrable geometries with torsion, Arch. Math. (Brno), 42, 5-84 (2006), with an appendix by M. Kassuba · Zbl 1164.53300
[2] Agricola, I.; Friedrich, T., On the holonomy of connections with skew-symmetric torsion, Math. Ann., 328, 4, 711-748 (2004) · Zbl 1055.53031
[3] Berger, M., Quelques formules de variation pour une structure riemannienne, Ann. Éc. Norm. Sup. (4), 3, 285-294 (1970) · Zbl 0204.54802
[4] Besse, A. L., Einstein Manifolds, Ergeb. Math. Grenzgeb. (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0613.53001
[5] Bleecker, D., Gauge theory and variational principles, (Global Analysis Pure and Applied Series (1981), Addison-Wesley: Addison-Wesley Reading) (2005), Dover: Dover Mineola, unabridged republication:
[6] Cartan, É., Sur les variétés à connexion affine et la théorie de la rélativité généralisée (première partie), Ann. Éc. Norm. Sup., 40, 325-412 (1923) · JFM 49.0542.02
[7] Cartan, É., Sur les variétés à connexion affine et la théorie de la rélativité généralisée (première partie suite), Ann. Éc. Norm. Sup., 41, 1-25 (1924) · JFM 51.0581.01
[8] Cartan, É., Sur les variétés à connexion affine et la théorie de la rélativité généralisée (deuxième partie), Ann. Éc. Norm. Sup., 42, 17-88 (1925) · JFM 51.0582.01
[9] Chamseddine, A.; Connes, A., The spectral action principle, Comm. Math. Phys., 186, 3, 731-750 (1997) · Zbl 0894.58007
[10] Chamseddine, A.; Connes, A., Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I, Fortschr. Phys., 58, 553-600 (2010) · Zbl 1206.81130
[11] Connes, A., Noncommutative Geometry (1994), Academic Press: Academic Press San Diego · Zbl 1106.58004
[12] Connes, A., Gravity coupled with matter and the foundation of noncommutative geometry, Comm. Math. Phys., 183, 1, 155-176 (1996) · Zbl 0881.58009
[13] Connes, A., Brisure de symétrie spontanée et géometrie du point de vue spectral, Séminaire Bourbaki, vol. 1995/96. Séminaire Bourbaki, vol. 1995/96, Astérisque, 241, 313-349 (1997), Exp. No. 816, 5 · Zbl 0942.46044
[14] Connes, A., On the spectral characterization of manifolds · Zbl 1106.58004
[15] Connes, A.; Marcolli, M., Noncommutative Geometry, Quantum Fields and Motives (2008), AMS/Hindustan Book Agency: AMS/Hindustan Book Agency Providence/New Delhi · Zbl 1209.58007
[16] Friedrich, T., Dirac Operators in Riemannian Geometry (2000), AMS: AMS Providence
[17] Friedrich, T.; Sulanke, S., Ein Kriterium für die formale Selbstadjungiertheit des Dirac-Operators, Colloq. Math., 40, 2, 239-247 (1978/1979) · Zbl 0426.58023
[18] Gilkey, P. B., Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (1995), CRC Press: CRC Press Boca Raton · Zbl 0856.58001
[19] Göckeler, M.; Schücker, T., Differential Geometry, Gauge Theories, and Gravity, Cambridge Monogr. Math. Phys. (1987), Cambridge University Press · Zbl 0649.53001
[20] Goldthorpe, W. H., Spectral geometry and \(SO(4)\) gravity in a Riemann-Cartan spacetime, Nuclear Phys. B, 170, 307-328 (1980)
[21] Gracia-Bondía, J. M.; Várilly, J. C.; Figueroa, H., Elements of Noncommutative Geometry (2001), Birkhäuser: Birkhäuser Boston · Zbl 0958.46039
[22] Grensing, G., Induced gravity for nonzero torsion, Phys. Lett. B, 169, 333-336 (1986)
[23] Hanisch, F.; Pfäffle, F.; Stephan, C. A., The spectral action for Dirac operators with skew-symmetric torsion, Comm. Math. Phys., 300, 3, 877-888 (2010) · Zbl 1203.53073
[24] Hehl, F. W.; von der Heyde, P.; Kerlick, G. D.; Nester, J. N., General relativity with spin and torsion: Foundations and prospects, Rev. Modern Phys., 48, 393-416 (1976) · Zbl 1371.83017
[25] Iochum, B.; Levy, C.; Vassilevich, D., Spectral action for torsion with and without boundaries · Zbl 1245.46055
[26] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry (1969), Interscience Publishers: Interscience Publishers New York · Zbl 0175.48504
[27] Lawson, H. B.; Michelsohn, M.-L., Spin Geometry (1989), Princeton University Press: Princeton University Press Princeton · Zbl 0688.57001
[28] Obukhov, Yu. N., Spectral geometry of the Riemann-Cartan space-time, Nuclear Phys. B, 212, 237-254 (1983)
[29] Obukhov, Yu. N.; Vlachynsky, E. J.; Esser, W.; Hehl, F. W., Effective Einstein theory from metric-affine gravity models via irreducible decompositions, Phys. Rev. D, 56, 7769-7778 (1997)
[30] Shapiro, I. L., Physical aspects of the space-time torsion, Phys. Rep., 357, 113-213 (2002) · Zbl 0977.83072
[31] Sitarz, A.; Zajac, A., Spectral action for scalar perturbations of Dirac operators, Lett. Math. Phys., 98, 3, 333-348 (2011) · Zbl 1245.58005
[32] Tricerri, F.; Vanhecke, L., Homogeneous Structures on Riemannian Manifolds, London Math. Soc. Lecture Note Ser., vol. 83 (1983), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0509.53043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.