Communications in Mathematical Physics

, Volume 284, Issue 1, pp 93–116 | Cite as

Extended Connection in Yang-Mills Theory



The three fundamental geometric components of Yang-Mills theory –gauge field, gauge fixing and ghost field– are unified in a new object: an extended connection in a properly chosen principal fiber bundle. To do this, it is necessary to generalize the notion of gauge fixing by using a gauge fixing connection instead of a section. From the equations for the extended connection’s curvature, we derive the relevant BRST transformations without imposing the usual horizontality conditions. We show that the gauge field’s standard BRST transformation is only valid in a local trivialization and we obtain the corresponding global generalization. By using the Faddeev-Popov method, we apply the generalized gauge fixing to the path integral quantization of Yang-Mills theory. We show that the proposed gauge fixing can be used even in the presence of a Gribov’s obstruction.


Ghost Gauge Group Gauge Transformation Principal Bundle Universal Family 
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  1. 1.
    Atiyah M.F., Singer I.M.: Dirac operators coupled to vector potentials. Proc. National Acad. Sci. 81, 2597 (1984)MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Baulieu L., Bellon M.: p-forms and Supergravity: Gauge symmetries in curved space. Nucl. Phys. B 266, 75 (1986)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Baulieu L., Singer I.M.: Topological Yang-Mills symmetry. Nucl. Phys. 5(Proc. Suppl.), 12 (1988)MathSciNetGoogle Scholar
  4. 4.
    Baulieu L., Thierry-Mieg J.: The principle of BRST symmetry: An alternative approach to Yang-Mills theories. Nucl. Phys. B 197, 477 (1982)CrossRefADSGoogle Scholar
  5. 5.
    Birmingham D., Blau M., Rakowski M., Thompson G.: Topological Field Theory. Phys. Rep. 209, 129 (1991)CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bonora L., Cotta-Ramusino P.L.: Some remarks on BRS transformations, anomalies and the cohomology of the lie algebra of the group of gauge transformations. Commun. Math. Phys. 87, 589 (1983)MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Choquet-Bruhat, Y., DeWitt-Morette, C.: Analysis, Manifolds and Physics. Part II: 92 Applications. New York: Elsevier Science Publishers B.V., 1989Google Scholar
  8. 8.
    Cordes S., Moore G., Ramgoolam S.: Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories. Nucl. Phys. 41(Proc Suppl.), 184 (1995)MATHMathSciNetGoogle Scholar
  9. 9.
    Donaldson S.K., Kronheimer P.B.: The geometry of four-manifolds. Oxford University Press, Oxford (1990)MATHGoogle Scholar
  10. 10.
    Dubois-Violette M.: The Weil-B.R.S. algebra of a Lie algebra and the anomalous terms in gauge theory. J. Geom. Phys. 3, 525 (1986)MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Faddeev L., Slavnov A.: Gauge fields: An introduction to quantum theory. Second ed., Frontiers in Physics, Cambridge: Perseus Books, 1991Google Scholar
  12. 12.
    Gribov V.: Quantization of non-Abelian gauge theories. Nucl. Phys. B 139, 1 (1978)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Guillemin V., Sternberg S., Guillemin V.W.: Supersymmetry and equivariant de Rham theory. Berlin-Heidelberg-NewYork, Spinger-Verlag (1999)MATHGoogle Scholar
  14. 14.
    Henneaux M.: Hamiltonian form of the path integral for theories with a gauge freedom. Phys. Rep. 126(1), 1 (1985)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Henneaux M., Teitelboim C.: Quantization of gauge systems. Princeton Univ. Press, Princeton, NJ (1994)Google Scholar
  16. 16.
    Kobayashi S., Nomizu K.: Foundations of differential geometry. Vol. I. Wiley, New York (1963)Google Scholar
  17. 17.
    Kriegl, A., Michor, P.: it A convenient setting for global analysis. Mathematical Surveys and Monographs, Vol. 53, Amer. Math. Soc., 1997Google Scholar
  18. 18.
    Michor, P.: Gauge theory for fiber bundles. Monographs and Textbooks in Physical Sciences, Lecture Notes 19, Napoli: Bibliopolis, 1991Google Scholar
  19. 19.
    Narasimhan M.S., Ramadas T.R.: Geometry of SU(2) Gauge Fields. Commun. Math. Phys. 67, 121 (1979)MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Singer I.: Some remarks on the Gribov Ambiguity. Commun. Math. Phys. 60, 7 (1978)MATHCrossRefADSGoogle Scholar
  21. 21.
    Szabo R.: Equivariant cohomology and localization of path integrals. Springer-Verlag, Berlin-Heidelberg-NewYork (2000)MATHGoogle Scholar
  22. 22.
    Thierry-Mieg J.: Geometrical reinterpretation of Faddeev-Popov ghost particles and BRS transformations. J. Math. Phys. 21, 2834 (1980)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Witten E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Witten, E.: Dynamics of Quantum Field Theory. In: Quantum Fields and Strings: A course for mathematicians. Vol(2), Providence, RI: Amer. Math. Soc., (1999), pp. 1119–1424Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Instituto de Astronomia y Fisica del EspacioCasilla de Correo 67Buenos AiresArgentina
  2. 2.Math. Dept. FCENUniversidad de Buenos Aires, Ciudad UniversitariaBuenos AiresArgentina

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