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A Brief Introduction to the Geometry of Gauge Fields

  • M. E. Mayer
  • A. Trautman
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 23/1981)

Abstract

In view of the common background required for the understanding of the lectures of both authors, and in order to avoid unnecessary duplications, we have decided to present jointly this brief introduction to the language and properties of fiber bundles. By now the advantages of the fiber-bundle formulation of gauge field theories have led to a widespread acceptance of this language, and a number of reviews of the subject have appeared or are in course of publication. These, together with a number of standard textbooks are listed in the references to this introduction. Nevertheless, we felt that it would be convenient for the reader of these proceedings to have at his disposal a summary of the basic facts. We also tried to clarify a number of concepts and propose an acceptable terminology wherever a standard has not been established in the literature. This refers, in particular, to the terms gauge transformation, pure gauge transformation, and the related (infinite-dimensional) groups as well as to the concepts of extension, prolongation, restriction, and reduction of bundles, which are used with slightly varying meaning in different texts.

Keywords

Gauge Theory Gauge Transformation Base Space Parallel Transport Principal Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

Textbooks and Lecture Note Volumes

  1. M.F. Atiyah, Geometry of Yang-Mills Fields, Academia Nazionale del Lincei, Pisa, 1979.Google Scholar
  2. R.L. Bishop, R.J. Crittenden, Geometry of Manifolds, Academic Press, NY 1964.MATHGoogle Scholar
  3. D.D. Bleecker, Gauge Theory and Variational Principles, Addison-Wesley, to appear.Google Scholar
  4. N. Bourbaki, Varietes différentielles et analytiques, El. de math. XXXIII and XXXIV, Hermann, Paris, 1967, 1971.Google Scholar
  5. J. Dieudonné, Treaties on Analysis, vols. III and IV, Acdemic Press, NY 1972, 1974 (French originalGoogle Scholar
  6. J. Dieudonné, Treaties on Analysis, vols. III and IV, Traite d’Analyse, Gauthier-Villars 1974, 1971).Google Scholar
  7. Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick, Analysis, Manifolds, and Physics, North-Holland, 1977.MATHGoogle Scholar
  8. W. Drechsler, M.E. Mayer, Fiber-Bundle Techniques in Gauge Theory, Lect. Notes in Physics, vol. 67, Springer Verlag, Berlin-Heidelberg-New York, 1977.Google Scholar
  9. F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1966.MATHGoogle Scholar
  10. D. Husèmoller, Fibre Bundles, 2nd Edition, Springer Verlag, 1975.MATHGoogle Scholar
  11. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, 2 vols. Wiley-Interscience, NY, 1963, 1969.MATHGoogle Scholar
  12. A. Lichnerowicz, Global Theory of Connections and Holonomy Groups, Noordhoff, 1976.MATHCrossRefGoogle Scholar
  13. M.E. Mayer, Gauge Theory, Vectur Bundles, and the Index Theorem, Springer, to appear.Google Scholar
  14. I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer Verlag, New York, Heidelberg, Berlin, 1976.MATHGoogle Scholar
  15. M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1971.Google Scholar
  16. S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, 1964.MATHGoogle Scholar
  17. R. Sulanke, P. Wintgen, Differentialgeometrie und Faserbündel, Birkhaüser, Basel, 1972.Google Scholar
  18. Various summer and winter school lecture notes (Cargese, Erice, Schladming, etc.).Google Scholar

Articles

  1. P.G. Bergmann, E.J. Jr Flaherty, J. Math. Phys. 19 (1978) 212.ADSCrossRefGoogle Scholar
  2. M. Daniel, C.M. Viallet, Rev. Mod. Phys. 52 (1980) 1–75.MathSciNetCrossRefGoogle Scholar
  3. T. Eguchi, P.B. Gilkey, and A.J. Hanson, Phys. Reports 66 (1980) 213–393.MathSciNetADSCrossRefGoogle Scholar
  4. M.E. Mayer, Ann. Israel Phys. Soc. 3 (1979) 80–99Google Scholar
  5. M.E. Mayer, Hadronic J. 4 (1981) 108–152.Google Scholar
  6. A. Trautman, Rep. Math. Phys. (Toruń) 1 (1970) 29.MathSciNetADSMATHCrossRefGoogle Scholar
  7. A. Trautman, Rep. Math. Phys. (Toruń) 10 (1976) 297.MathSciNetADSMATHCrossRefGoogle Scholar
  8. Czech, J. Phys. B29 (1979) 107–116.Google Scholar
  9. Czech, J. Bull. Acad. Polon. Sci., ser. sci. phys. et astron. 27 (1979) 7.Google Scholar
  10. Fiber Bundles, Gauge Fields, and Gravitation, in General Relativity and Gravitation, vol. 1, edited by A. Held, Plenum Press, New York 1980.Google Scholar
  11. Fiber Bundles, Gauge Fields, and Gravitation J. Phys. A13 (1980) L1.Google Scholar
  12. Additional references can be found in the lectures of the two authors in this volume.Google Scholar
  13. S.S. Chern, Geometry of Characteristic Classes, Proc. 13th Biennial Seminar, Canadian Mathematical Congress, vol. I, 1972, p.1–40.MathSciNetGoogle Scholar
  14. M.S. Narasimhan, S. Ramanan, Existence of Universal Connections, Amer. J. Math. 83 (1961) 563–572.Google Scholar
  15. J. Nowakowski, A. Trautman, Natural Connections on Stiefel Bundles are Sourceless Gauge Fields, J. Math, Phys. 19 (1978) 1100.MathSciNetADSMATHCrossRefGoogle Scholar
  16. R. Schlafly, Universal Connections, Inventiones math. 59 (1980) 59–65.MathSciNetADSMATHCrossRefGoogle Scholar
  17. V.A. Rokhlin, D.B. Fuks, Nachal’nyi kurs topologii (Introductory course in topology), Nauka, 1977Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • M. E. Mayer
    • 1
  • A. Trautman
    • 2
  1. 1.Dept. of PhysicsUniv. of CaliforniaIrvineUSA
  2. 2.Inst, of Theoretical PhysicsWarsaw Univ.WarsawPoland

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