Abstract
The internal symmetry group of a connection on a principal fiber bundleP is studied. It is shown that this group is a smooth proper Lie transformation group ofP, which, ifP is connected, is also free. Moreover, this group is shown to be isomorphic to the centralizer of the holonomy group of the connection. Several examples and applications of these results to gauge field theories are given.
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Abraham, R., Marsden, J.: Foundation of mechanics, second edition. Reading, MA: Benjamin/Cummings 1978
Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. A362, 425–461 (1978)
Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97–118 (1978)
Bleecker, D.: Gauge Theory and Variational Principles. Reading, MA: Addison-Wesley 1981
Daniel, M., Viallet, C.M.: The geometrical setting of gauge theories of the Yang-Mills type. Rev. Mod. Phys.52, 175–197 (1980)
Fischer, A.: A unified approach to conservation laws in general relativity, gauge theories, and elementary particle physics. Gen. Relativ. Gravitation14, 683–689 (1982)
Fischer, A.: Conservation laws in gauge field theories. In: Differential topology, global analysis on manifolds, and their applications. Rassias, G.M., Rassias, T. (eds.). Berlin, Heidelberg, New York: Springer 1985
Fischer, A.: A geometric approach to gauge field theories. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer (to appear)
Forgacs, P., Manton, N.S.: Space-time symmetries in gauge theories. Commun. Math. Phys.72, 15–35 (1980)
Greenberg, M., Harper, J.: Algebraic topology, a first course. Reading, MA: Benjamin/Cummings 1981
Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962
Jackiw, R., Manton, N.S.: Symmetries and conservation laws in gauge theories. Nucl. Phys.B 158, 141 (1979)
Kobayashi, S.: Transformation groups in differential geometry. Berlin, Heidelberg, New York: Springer 1972
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. I. New York: Intersceince 1963
Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. II. New York: Interscience 1969
Mitter, P. K.: Geometry of the space of gauge orbits and the Yang-Mills dynamical system. In: Recent developments in gauge theories (Cargese Lectures, 1979). Hooft, G. et al. (eds.), New York: Plenum Press 1980
Myers, S.B., Steenrod, N.: The group of isometries of a Riemannian manifold. Ann. Math.40, 400–416 (1939)
Poor, W.A.: Differential Geometric Structures. New York: McGraw-Hill 1981
Rawnsley, J.H.: Differential geometry of instantons. Communications of the Dublin Institute for Advanced Studies, Series A (Theoretical Physics), No. 25, 1978
Singer, I.M.: Some remarks on the Gribov ambiguity. Commun. Math. Phys.60, 7–12 (1978)
Spanier, F.: Algebraic topology. New York: McGraw-Hill 1966
Trautman, A.: On groups of gauge transformations. In: Geometrical and topological methods in gauge theories. Harnad, J.P., Shnider, S. (eds.). Lecture Notes in Physics, Vol. 129. Berlin, Heidelberg, New York: Springer 1980
Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Glenview, Illinois: Scott, Foresman 1971
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Communicated by S. W. Hawking
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Fischer, A.E. The internal symmetry group of a connection on a principal fiber bundle with applications to gauge field theories. Commun.Math. Phys. 113, 231–262 (1987). https://doi.org/10.1007/BF01223513
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DOI: https://doi.org/10.1007/BF01223513