Journal of High Energy Physics

, 2015:195 | Cite as

Residual Weyl symmetry out of conformal geometry and its BRST structure

  • J. François
  • S. Lazzarini
  • T. Masson
Open Access
Regular Article - Theoretical Physics


The conformal structure of second order in m-dimensions together with the so-called (normal) conformal Cartan connection, is considered as a framework for gauge theories. The dressing field scheme presented in a previous work amounts to a decoupling of both the inversion and the Lorentz symmetries such that the residual gauge symmetry is the Weyl symmetry. On the one hand, it provides straightforwardly the Riemannian parametrization of the normal conformal Cartan connection and its curvature. On the other hand, it also provides the finite transformation laws under the Weyl rescaling of the various geometric objects involved. Subsequently, the dressing field method is shown to fit the BRST differential algebra treatment of infinitesimal gauge symmetry. The dressed ghost field encoding the residual Weyl symmetry is presented. The related so-called algebraic connection supplies relevant combinations found in the literature in the algebraic study of the Weyl anomaly.


Gauge Symmetry Differential and Algebraic Geometry Conformal and W Symmetry BRST Symmetry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J.P. Harnad and R.B. Pettitt, Gauge theories for space-time symmetries, J. Math. Phys. 17 (1976) 1827 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Friedrich, Einstein equations and conformal structureExistence of anti de Sitter type space-times, J. Geom. Phys. 17 (1995) 125 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T.W.B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961) 212 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F.W. Hehl, P. Von Der Heyde, G.D. Kerlick and J.M. Nester, General Relativity with Spin and Torsion: Foundations and Prospects, Rev. Mod. Phys. 48 (1976) 393 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Blagojević and F. Hehl eds., Gauge Theories of Gravitation. A Reader with commentaries, Imperial College Press, London U.K. (2013).Google Scholar
  7. [7]
    D.K. Wise, MacDowell-Mansouri gravity and Cartan geometry, Class. Quant. Grav. 27 (2010) 155010 [gr-qc/0611154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. Lazzarini and C. Tidei, Polyakov soldering and second order frames: the role of the Cartan connection, Lett. Math. Phys. 85 (2008) 27 [arXiv:0802.3772] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D.K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA 5 (2009) 080 [arXiv:0904.1738] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  10. [10]
    K. Ogiue, Theory of Conformal Connections, Kodai Math. Sem. Rep. 19 (1967) 193.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Kobayashi, Classics in Mathematics. Vol. 70: Transformation groups in differential geometry, Spinger-Verlag, Berlin Germany (1972).Google Scholar
  12. [12]
    R. Baston, Almost Hermitian symmetric manifolds. I. Local twistor theory, Duke Math. J. 63 (1991) 81.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Michor and D. Alekseevsky, Differential geometry of Cartan connections, Publ. Math. Debrecen (1995) 349.Google Scholar
  14. [14]
    A. Cap, J. Slovák and V. Soucek, Invariant operators on manifolds with almost Hermitian symmetric structures. II. Normal Cartan connections, Acta Math. Univ. Comenian. (N.S.) 66 (1997) 203.MathSciNetzbMATHGoogle Scholar
  15. [15]
    R. Sharpe, Graduate texts in mathematics. Vol. 166: Differential geometry: Cartan’s generalization of Klein’s erlangen program, Springer, Heidelberg Germany (1997).Google Scholar
  16. [16]
    J.P. Harnad and R.B. Pettitt, Group Theoretical Methods in Physics: Proceedings of the Fifth International Colloquium, Academic Press Inc., New York U.S.A. (1977).Google Scholar
  17. [17]
    J.P. Harnad and R.B. Pettitt, Gauge theories for space-time symmetries II: second order conformal structures, unpublished, CRM-745 (1978).Google Scholar
  18. [18]
    R.A. Coleman and H. Korte, Space-time G Structures and Their Prolongation, J. Math. Phys. 22 (1981) 2598 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Fournel, J. François, S. Lazzarini and T. Masson, Gauge invariant composite fields out of connections, with examples, Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1450016.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    J. François, Reductions of gauge symmetries: a new geometrical approach, Ph.D. Thesis, Aix-Marseille University, Marseille France (2014).Google Scholar
  21. [21]
    C. Becchi, A. Rouet and R. Stora, Renormalization of the Abelian Higgs-Kibble Model, Commun. Math. Phys. 42 (1975) 127 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    C. Becchi, A. Rouet and R. Stora, Renormalization of Gauge Theories, Annals Phys. 98 (1976) 287 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    I.V. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism, arXiv:0812.0580 [INSPIRE].
  24. [24]
    T. Masson and J.-C. Wallet, A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model, arXiv:1001.1176 [INSPIRE].
  25. [25]
    J. François, S. Lazzarini and T. Masson, Nucleon spin decomposition and differential geometry, Phys. Rev. D 91 (2015) 045014 [arXiv:1411.5953] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    A. Čap and A. Gover, Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002) 1511.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Grigoriev and A. Waldron, Massive Higher Spins from BRST and Tractors, Nucl. Phys. B 853 (2011) 291 [arXiv:1104.4994] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. Stora, Continuum gauge theories, in NATO ASI Series B, Vol. 26: New Developments in Quantum Field Theory and Statistical Mechanics, Cargèse 1976, M. Lévy and P. Mitter eds., Plenum Press, New York U.S.A. (1977).Google Scholar
  29. [29]
    R. Stora, Algebraic structure and topological origin of chiral anomalies, in NATO ASI Series B, Vol. 115: Progress in Gauge Field Theory, Cargèse 1983, Plenum Press, New York U.S.A. (1983).Google Scholar
  30. [30]
    P.A.M. Dirac, Gauge-invariant formulation of Quantum Electrodynamics, Canad. J. Phys. 33 (1955) 650.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    P.A.M. Dirac, The principles of Quantum Mechanics, fourth edition, Oxford University Press, Oxford U.K. (1958).Google Scholar
  32. [32]
    M. Korzynski and J. Lewandowski, The Normal conformal Cartan connection and the Bach tensor, Class. Quant. Grav. 20 (2003) 3745 [gr-qc/0301096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    O. Piguet and A. Rouet, Symmetries in Perturbative Quantum Field Theory, Phys. Rept. 76 (1981) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    L. Baulieu, Perturbative Gauge Theories, Phys. Rept. 129 (1985) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    G. Bandelloni, Diffeomorphism Cohomology in Quantum Field Theory Models, Phys. Rev. D 38 (1988) 1156 [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    J.A. Dixon, Calculation of BRS cohomology with spectral sequences, Commun. Math. Phys. 139 (1991) 495 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    O. Piguet and S. Sorella, Lecture Notes in Physics. Vol. 28: Agebraic Renormalization, Spinger-Verlag, Heidelberg Germany (1995).Google Scholar
  39. [39]
    G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Y. Ne’eman, T. Regge and J. Thierry-Mieg, Ghost-Fields, BRS and Extended Supergravity as Applications of Gauge Geometry, in Matter particles, R. Ruffini et al. eds., World Scientific, Singapore (1978), pg. 301.Google Scholar
  41. [41]
    L. Baulieu and J. Thierry-Mieg, The Principle of BRS Symmetry: An Alternative Approach to Yang-Mills Theories, Nucl. Phys. B 197 (1982) 477 [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    M. Dubois-Violette, The Weil-BRS algebra of a Lie algebra and the anomalous terms in gauge theory, J. Geom. Phys. 3 (1986) 525.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    D. Garajeu, R. Grimm and S. Lazzarini, W gauge structures and their anomalies: An Algebraic approach, J. Math. Phys. 36 (1995) 7043 [hep-th/9411125] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    N. Boulanger, A Weyl-covariant tensor calculus, J. Math. Phys. 46 (2005) 053508 [hep-th/0412314] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    N. Boulanger, General solutions of the Wess-Zumino consistency condition for the Weyl anomalies, JHEP 07 (2007) 069 [arXiv:0704.2472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    N. Boulanger, Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions, Phys. Rev. Lett. 98 (2007) 261302 [arXiv:0706.0340] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    R. Stora, Algebraic Structure of Chiral Anomalies, in New Perspectives in Quantum Field Theory, Proceedings of XVI GIFT International Seminar on Theoretical Physics, J. Abad, M. Asorey and A. Cruz eds., Jaca (Huesca), Spain (1985).Google Scholar
  48. [48]
    A.R. Gover, A. Shaukat and A. Waldron, Tractors, Mass and Weyl Invariance, Nucl. Phys. B 812 (2009) 424 [arXiv:0810.2867] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    J. François, S. Lazzarini and T. Masson, in preparation.Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Centre de Physique Théorique, Aix Marseille Université & Université de Toulon & CNRS UMR 7332MarseilleFrance

Personalised recommendations