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The internal symmetry group of a connection on a principal fiber bundle with applications to gauge field theories. (English) Zbl 0638.53039

Let P(M,G) be a principal fiber bundle with structure group G over a connected manifold M. We denote by AUT(P) the group of automorphisms of P, and Aut(P) the normal subgroup of automorphisms of P covering the identity of M. For a connection form \(\omega\) on P we denote by \(AUT_{\omega}(P)=\{F\in AUT(P)| \quad F^*\omega =\omega \}\) the symmetry group of \(\omega\), and by \(I_{\omega}(P)=Aut(P)\cap AUT_{\omega}(P)\) the internal symmetry group of \(\omega\). For a Riemannian metric g on M we denote by \(I_{(g,\omega)}(P)\) the group of \(F\in AUT(P)\) such that \(F^*\omega =\omega\) and \(g=f^*(g)\), where f is the diffeomorphism of M induced by the automorphism F.
The author studies the Lie group structure of \(I_{\omega}(P)\) and \(I_{(g,\omega)}(P)\) and proves, among other things, that \(I_{\omega}(P)\) and \(I_{(g,\omega)}(P)\) become Lie transformation groups and the orbit of these groups are closed submanifolds of P. Moreover, taking the orbit space B we get a principal fiber bundle \(P(B,I_{\omega}(P))\) with structure group \(I_{\omega}(P)\). The author asserts that in Lagrangean gauge field theories, the Lie group \(I_{(g,\omega)}(P)\) plays an important role as the generator of global conservation laws for matter fields coupled to the fixed parameter fields (g,\(\omega)\).
Reviewer: A.Morimoto

MSC:

53C05 Connections (general theory)
57S99 Topological transformation groups
53C80 Applications of global differential geometry to the sciences
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[1] Abraham, R., Marsden, J.: Foundation of mechanics, second edition. Reading, MA: Benjamin/Cummings 1978 · Zbl 0393.70001
[2] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. A362, 425-461 (1978) · Zbl 0389.53011
[3] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97-118 (1978) · Zbl 0387.55009 · doi:10.1007/BF01609489
[4] Bleecker, D.: Gauge Theory and Variational Principles. Reading, MA: Addison-Wesley 1981 · Zbl 0481.58002
[5] Daniel, M., Viallet, C.M.: The geometrical setting of gauge theories of the Yang-Mills type. Rev. Mod. Phys.52, 175-197 (1980) · doi:10.1103/RevModPhys.52.175
[6] Fischer, A.: A unified approach to conservation laws in general relativity, gauge theories, and elementary particle physics. Gen. Relativ. Gravitation14, 683-689 (1982) · Zbl 0491.53057
[7] 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
[8] Fischer, A.: A geometric approach to gauge field theories. Lecture Notes in Physics. Berlin, Heidelberg, New York: Springer (to appear)
[9] Forgacs, P., Manton, N.S.: Space-time symmetries in gauge theories. Commun. Math. Phys.72, 15-35 (1980) · doi:10.1007/BF01200108
[10] Greenberg, M., Harper, J.: Algebraic topology, a first course. Reading, MA: Benjamin/Cummings 1981 · Zbl 0498.55001
[11] Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962 · Zbl 0111.18101
[12] Jackiw, R., Manton, N.S.: Symmetries and conservation laws in gauge theories. Nucl. Phys.B 158, 141 (1979)
[13] Kobayashi, S.: Transformation groups in differential geometry. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0246.53031
[14] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. I. New York: Intersceince 1963 · Zbl 0119.37502
[15] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. II. New York: Interscience 1969 · Zbl 0175.48504
[16] 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
[17] Myers, S.B., Steenrod, N.: The group of isometries of a Riemannian manifold. Ann. Math.40, 400-416 (1939) · Zbl 0021.06303 · doi:10.2307/1968928
[18] Poor, W.A.: Differential Geometric Structures. New York: McGraw-Hill 1981 · Zbl 0493.53027
[19] Rawnsley, J.H.: Differential geometry of instantons. Communications of the Dublin Institute for Advanced Studies, Series A (Theoretical Physics), No. 25, 1978 · Zbl 0389.53032
[20] Singer, I.M.: Some remarks on the Gribov ambiguity. Commun. Math. Phys.60, 7-12 (1978) · Zbl 0379.53009 · doi:10.1007/BF01609471
[21] Spanier, F.: Algebraic topology. New York: McGraw-Hill 1966 · Zbl 0145.43303
[22] 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 · Zbl 0447.53061
[23] Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Glenview, Illinois: Scott, Foresman 1971 · Zbl 0241.58001
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