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.
Lectures given at the XX. Internationale Universitätswochen für Kernphysik, Schladming, Austria, February 17–26, 1981.
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References
Textbooks and Lecture Note Volumes
M.F. Atiyah, Geometry of Yang-Mills Fields, Academia Nazionale del Lincei, Pisa, 1979.
R.L. Bishop, R.J. Crittenden, Geometry of Manifolds, Academic Press, NY 1964.
D.D. Bleecker, Gauge Theory and Variational Principles, Addison-Wesley, to appear.
N. Bourbaki, Varietes différentielles et analytiques, El. de math. XXXIII and XXXIV, Hermann, Paris, 1967, 1971.
J. Dieudonné, Treaties on Analysis, vols. III and IV, Acdemic Press, NY 1972, 1974 (French original
J. Dieudonné, Treaties on Analysis, vols. III and IV, Traite d’Analyse, Gauthier-Villars 1974, 1971).
Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick, Analysis, Manifolds, and Physics, North-Holland, 1977.
W. Drechsler, M.E. Mayer, Fiber-Bundle Techniques in Gauge Theory, Lect. Notes in Physics, vol. 67, Springer Verlag, Berlin-Heidelberg-New York, 1977.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 1966.
D. Husèmoller, Fibre Bundles, 2nd Edition, Springer Verlag, 1975.
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, 2 vols. Wiley-Interscience, NY, 1963, 1969.
A. Lichnerowicz, Global Theory of Connections and Holonomy Groups, Noordhoff, 1976.
M.E. Mayer, Gauge Theory, Vectur Bundles, and the Index Theorem, Springer, to appear.
I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer Verlag, New York, Heidelberg, Berlin, 1976.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1971.
S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, 1964.
R. Sulanke, P. Wintgen, Differentialgeometrie und Faserbündel, Birkhaüser, Basel, 1972.
Various summer and winter school lecture notes (Cargese, Erice, Schladming, etc.).
Articles
P.G. Bergmann, E.J. Jr Flaherty, J. Math. Phys. 19 (1978) 212.
M. Daniel, C.M. Viallet, Rev. Mod. Phys. 52 (1980) 1–75.
T. Eguchi, P.B. Gilkey, and A.J. Hanson, Phys. Reports 66 (1980) 213–393.
M.E. Mayer, Ann. Israel Phys. Soc. 3 (1979) 80–99
M.E. Mayer, Hadronic J. 4 (1981) 108–152.
A. Trautman, Rep. Math. Phys. (Toruń) 1 (1970) 29.
A. Trautman, Rep. Math. Phys. (Toruń) 10 (1976) 297.
Czech, J. Phys. B29 (1979) 107–116.
Czech, J. Bull. Acad. Polon. Sci., ser. sci. phys. et astron. 27 (1979) 7.
Fiber Bundles, Gauge Fields, and Gravitation, in General Relativity and Gravitation, vol. 1, edited by A. Held, Plenum Press, New York 1980.
Fiber Bundles, Gauge Fields, and Gravitation J. Phys. A13 (1980) L1.
Additional references can be found in the lectures of the two authors in this volume.
S.S. Chern, Geometry of Characteristic Classes, Proc. 13th Biennial Seminar, Canadian Mathematical Congress, vol. I, 1972, p.1–40.
M.S. Narasimhan, S. Ramanan, Existence of Universal Connections, Amer. J. Math. 83 (1961) 563–572.
J. Nowakowski, A. Trautman, Natural Connections on Stiefel Bundles are Sourceless Gauge Fields, J. Math, Phys. 19 (1978) 1100.
R. Schlafly, Universal Connections, Inventiones math. 59 (1980) 59–65.
V.A. Rokhlin, D.B. Fuks, Nachal’nyi kurs topologii (Introductory course in topology), Nauka, 1977
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Mayer, M.E., Trautman, A. (1981). A Brief Introduction to the Geometry of Gauge Fields. In: Mitter, H., Pittner, L. (eds) New Developments in Mathematical Physics. Acta Physica Austriaca, vol 23/1981. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8642-8_11
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