Abstract
We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SU q (2).
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Barth N.H.: Heat kernel expansion coefficient. I. An extension. J. Phys. A Math. Gen. 20, 857–874 (1987)
Chamseddine A., Connes A.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, 121 (2006)
Connes, A.: On the spectral characterization of manifolds. arXiv:0810.2088.
Connes A.: Noncommutative Geometry and Reality. J. Math. Phys. 36, 619 (1995)
Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2) (1995)
Connes A., Marcolli M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, New York (2007)
Da̧abrowski L., Landi G., Sitarz A., van Suijlekom W., Várilly J.C.: The Dirac operator on SU q (2). Commun. Math. Phys. 259, 729–759 (2004)
Donnelly, H.: Heat equation asymptotics with torsion. Indiana Univ. Math. J. 34(1) (1985)
Essouabri D., Iochum B., Levy C., Sitarz A.: Spectral action on noncommutative torus. J. Noncommut. Geom. 2, 53123 (2008)
Gracia-Bondía J.M., Várilly J.C., Figueroa H.: Elements of Noncommutative Geometry. Birkhäuser, Boston (2001)
Gusynin V.P., Gorbar E.V., Romankov V.V.: Heat kernel expansion for nonminimal differential operations and manifolds with torsion. Nucl. Phys. B 362, 449–471 (1991)
Hanisch F., Pfffle F., Stephan C.: The Spectral Action for Dirac Operators with Skew-Symmetric Torsion. Commun. Math. Phys. 300(3), 877–888 (2010)
Iochum B., Levy C., Sitarz A.: Spectral action on SU q (2). Commun. Math. Phys. 289, 107–155 (2009)
Iochum, B., Levy, C., Vassilevich, D.: Spectral action for torsion with and without boundaries. arXiv:1008.3630
Kalau W., Walze M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327–344 (1995)
Kastler D.: The Dirac operator and gravitation. Commun. Math. Phys. 166(3), 633–643 (1995)
Marcolli, M., Pierpaoli, E., Teh, K.: The spectral action and cosmic topology. arXiv:1005.2256
Nieh H.T., Yan M.L.: Quantized Dirac field in curved Riemann-Cartan background. Ann. Phys. 138, 237 (1982)
Vassilevich D.V.: Heat kernel expansion: user’s manual. Phys. Rep. 388C, 279–360 (2003)
Yajima S.: Mixed anomalies in 4 and 6 dimensional space with torsion. Prog. Theor. Phys. 79(2), 535–554 (1988)
Yajima S.: Evaluation of the heat kernel in Riemann-Cartan space. Class. Quantum Gravity 13(9), 2423–2435 (1996)
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A. Sitarz was partially supported by MNII grants 189/6.PRUE/2007/7 and N 201 1770 33.
A. Zając supported by a project operated within the Foundation for Polish Science IPP Programme “Geometry and Topology in Physical Models” co-financed by the EU European Regional Development Fund, Operational Program Innovative Economy 2007–2013.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sitarz, A., Zając, A. Spectral Action for Scalar Perturbations of Dirac Operators. Lett Math Phys 98, 333–348 (2011). https://doi.org/10.1007/s11005-011-0498-5
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DOI: https://doi.org/10.1007/s11005-011-0498-5