Abstract
We study renormalizability aspects of the spectral action for the Yang–Mills system on a flat 4-dimensional background manifold, focusing on its asymptotic expansion. Interpreting the latter as a higher-derivative gauge theory, a power-counting argument shows that it is superrenormalizable. We determine the counterterms at one-loop using zeta function regularization in a background field gauge and establish their gauge invariance. Consequently, the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action.
This manuscript provides more details than the shorter companion paper, where we have used a (formal) quantum action principle to arrive at gauge invariance of the counterterms. Here, we give in addition an explicit expression for the gauge propagator and compare to recent results in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Avramidi I.: A covariant technique for calculation of one loop effective action. Nucl. Phys. B 355, 712–754 (1991)
Babelon O., Namazie M.: Comment on the ghost problem in a higher derivative Yang-Mills theory. J. Phys. A A 13, L27–L30 (1980)
Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer-Verlag, Berlin (1992)
Brandt F., Dragon N., Kreuzer M.: Lie algebra cohomology. Nucl. Phys. B 332, 250 (1990)
Branson T.P., Gilkey P.B., Ørsted B.: Leading terms in the heat invariants. Proc. Amer. Math. Soc 109, 437–450 (1990)
Chamseddine A.H., Connes A.: Universal formula for noncommutative geometry actions: Unifications of gravity and the standard model. Phys. Rev. Lett. 77, 4868–4871 (1996)
Chamseddine A.H., Connes A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997)
Chamseddine A.H., Connes A., Marcolli M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)
Connes A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. Providence, RI: Amer. Math. Soc., 2008
Connes A., Chamseddine A.H.: Inner fluctuations of the spectral action. J. Geom. Phys 57, 1–21 (2006)
Dixon J.A.: Calculation of BRS cohomology with spectral sequences. Commun. Math. Phys. 139, 495–526 (1991)
Dubois-Violette M., Talon M., Viallet C.M.: BRS algebras: Analysis of the consistency equations in gauge theory. Commun. Math. Phys. 102, 105 (1985)
Dubois-Violette M., Talon M., Viallet C.M.: Results on BRS cohomology in gauge theory. Phys. Lett. B 158, 231 (1985)
Dubois-Violette M., Henneaux M., Talon M., Viallet C.-M.: Some results on local cohomologies in field theory. Phys. Lett. B 267, 81–87 (1991)
Faddeev L., Slavnov A.: Gauge Fields Introduction to Quantum Theory. NewYork, Benjamin Cummings (1980)
Gilkey P.B.: The spectral geometry of the higher order Laplacian. Duke Math. J. 47, 511–528 (1980)
Gilkey, P.B.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Volume 11 of Mathematics Lecture Series. Wilmington, DE: Publish or Perish Inc., 1984
Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Boston: Birkhäuser, 2001
Hawking S.W.: Zeta Function Regularization of Path Integrals in Curved Space-Time. Commun. Math. Phys. 55, 133 (1977)
Iochum, B., Levy, C., Vassilevich, D.: Spectral action beyond the weak-field approximation. http://arxiv.org/abs/1108.3749v1 [hep-th], 2011
Martin C., Ruiz Ruiz F.: Higher covariant derivative Pauli-Villars regularization does not lead to a consistent QCD. Nucl. Phys. B 436, 545–581 (1995)
Martin C., Ruiz Ruiz F.: Higher covariant derivative regulators and nonmultiplicative renormalization. Phys. Lett. B 343, 218–224 (1995)
Pronin P.I., Stepanyantz K.V.: One-loop effective action for an arbitrary theory. Teor. Mat. Fyz. 109, 215–231 (1996)
Pronin P.I., Stepanyantz K.V.: One-loop counterterms for higher derivative regularized Lagrangians. Phys. Lett. B 414, 117–122 (1997)
Slavnov A.A.: Invariant regularization of nonlinear chiral theories. Nucl. Phys. B 31, 301–315 (1971)
Slavnov A.A.: Invariant regularization of gauge theories. Teor. Mat. Fiz. 13, 174–177 (1972)
van Suijlekom W.D.: Perturbations and operator trace functions. J. Funct. Anal 260, 2483–2496 (2011)
van Suijlekom W.D.: Renormalization of the spectral action for the Yang-Mills system. JHEP 1103, 146 (2011)
Vassilevich D.V.: Heat kernel expansion: User’s manual. Phys. Rept. 388, 279–360 (2003)
Acknowledgements
The author would like to thank Olivier Babelon, Dirk Kreimer and Matilde Marcolli for useful correspondence and discussions. We gratefully acknowledge the anonymous referee for helpful comments. Caltech is acknowledged for hospitality and financial support during a visit in April 2011. The ESF is thanked for financial support under the program ‘Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics (ITGP)’. NWO is acknowledged for support under VENI-project 639.031.827.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Connes
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
van Suijlekom, W.D. Renormalization of the Asymptotically Expanded Yang–Mills Spectral Action. Commun. Math. Phys. 312, 883–912 (2012). https://doi.org/10.1007/s00220-012-1464-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1464-5