Abstract
We establish renormalizability of the full spectral action for the Yang-Mills system on a flat 4-dimensional background manifold. Interpreting the spectral action as a higher-derivative gauge theory, we find that it behaves unexpectedly well as far as renormalization is concerned. Namely, a power counting argument implies that the spectral action is superrenormalizable. From BRST-invariance of the one-loop effective action, we conclude that it is actually renormalizable as a gauge theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Avramidi, Covariant techniques for computation of the heat kernel, Rev. Math. Phys. 11 (1999) 947 [hep-th/9704166] [SPIRES].
F. Brandt, N. Dragon and M. Kreuzer, Lie algebra cohomology, Nucl. Phys. B 332 (1990) 250 [SPIRES].
A.H. Chamseddine and A. Connes, Universal formula for noncommutative geometry actions: Unification of gravity and the standard model, Phys. Rev. Lett. 77 (1996) 4868 [SPIRES].
A.H. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186 (1997) 731 [hep-th/9606001] [SPIRES].
A.H. Chamseddine and A. Connes, Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I, Fortsch. Phys. 58 (2010) 553 [arXiv:1004.0464] [SPIRES].
A.H. Chamseddine and A. Connes, Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part II, to appear.
A.H. Chamseddine, A. Connes and M. Marcolli, Gravity and the standard model with neutrino mixing, Adv. Theor. Math. Phys. 11 (2007) 991 [hep-th/0610241] [SPIRES].
A. Connes, Noncommutative Geometry, Academic Press, San Diego U.S.A. (1994).
A. Connes and M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, AMS, Providence U.S.A. (2008).
J.A. Dixon, Calculation of BRS cohomology with spectral sequences, Commun. Math. Phys. 139 (1991) 495 [SPIRES].
M. Dubois-Violette, M. Talon and C.M. Viallet, BRS algebras: Analysis of the consistency equations in gauge theory,, Commun. Math. Phys. 102 (1985) 105 [SPIRES].
M. Dubois-Violette, M. Talon and C.M. Viallet, Results on BRS cohomology in gauge theory, Phys. Lett. B 158 (1985) 231 [SPIRES].
M. Dubois-Violette, M. Henneaux, M. Talon and C.-M. Viallet, Some results on local cohomologies in field theory, Phys. Lett. B 267 (1991) 81 [SPIRES].
R. Estrada, J.M. Gracia-Bondía and J.C. Várilly, On summability of distributions and spectral geometry, Commun. Math. Phys. 191 (1998) 219 [SPIRES].
L. Faddeev and A. Slavnov, Gauge Fields. Introduction to Quantum Theory, Benjaming Cummings, Menlo Park U.S.A. (1980).
P.B. Gilkey, Mathematics Lecture Series. Vol. 11: Invariance theory, the heat equation, and the Atiyah-Singe index theorem, Publish or Perish Inc., Wilmington U.S.A. (1984).
M. Marcolli, E. Pierpaoli and K. Teh, The spectral action and cosmic topology, arXiv:1005.2256 [SPIRES].
P.I. Pronin and K.V. Stepanyantz, One-loop effective action for an arbitrary theory, Theor. Math. Phys. 109 (1996) 215.
P.I. Pronin and K. Stepanyantz, One-loop counterterms for higher derivative regularized Lagrangians, Phys. Lett. B 414 (1997) 117 [hep-th/9707008] [SPIRES].
A.A. Slavnov, Invariant regularization of nonlinear chiral theories, Nucl. Phys. B 31 (1971) 301 [SPIRES].
A.A. Slavnov, Invariant regularization of gauge theories, Theor. Math. Phys. 13 (1972) 174.
W.D. van Suijlekom, Perturbations and operator trace functions, J. Funct. Anal. 260 (2011) 2483 [arXiv:1012.3306].
J .C. Várilly, An Introduction to Noncommutative Geometry, EMS Series of Lectures in Mathematics, European Math. Soc. Publishing House, Zürich Switzerland (2006) [physics/9709045].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1101.4804
Rights and permissions
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.
About this article
Cite this article
van Suijlekom, W.D. Renormalization of the spectral action for the Yang-Mills system. J. High Energ. Phys. 2011, 146 (2011). https://doi.org/10.1007/JHEP03(2011)146
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2011)146