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Some remarks on the Gribov ambiguity. (English) Zbl 0379.53009

The author studies the quotient map of the set of all connections of a principal bundle over the 4-sphere with compact non-abelian Lie structure group by the action of the gauge group. He proves that this map has no section, showing that the ambiguity exhibited by V. N. Gribov [Instability of non-abelian gauge theories and impossibility of choice of Coloumb gauge. SLAC Translation 176 (1977)] for the Coulomb gauge occurs in all the gauges. The techniques used involve an approximation theorem and computations of homotopy groups of the gauge and related groups.
For Gribov’s article see http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-trans-0176.pdf.

MSC:

53C05 Connections (general theory)
58B05 Homotopy and topological questions for infinite-dimensional manifolds
55Q05 Homotopy groups, general; sets of homotopy classes
53C80 Applications of global differential geometry to the sciences
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58D30 Applications of manifolds of mappings to the sciences
81T99 Quantum field theory; related classical field theories
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References:

[1] Gribov, V. N.: Instability of non-abelian gauge theories and impossibility of choice of Coloumb gauge. SLAC Translation 176 (1977)
[2] Jackiw, R., Muzinick, I., Rebbi, C.: Coloumb gauge description of large Yang-Mills field. Phys. Rev.17, 1576 (1978)
[3] Ebin, D.: The manifold of Riemannian metrics. In: Proceedings of symposia in pure mathematics, Vol. 15, pp. 11–40. Providence, Rhode Island: American Mathematical Society 1970 · Zbl 0205.53702
[4] Palais, R.: Foundations of global non-linear analysis. New York: W. A. Benjamin, Inc. 1968 · Zbl 0164.11102
[5] Fischer, A., Marsden, J.: The space of conformally related metrics. Can. J. Math.29, 193–209 (1977) · Zbl 0358.58006
[6] Bourguignon, J.: Une stratification de l’espace des structures riemanniennes. Compositio Math.30, 1–42 (1975) · Zbl 0301.58015
[7] Munkries, J.R.: Elementary differential topology. In: Annals of mathematics studies, Vol. 54. Princeton, New Jersey: Princeton University Press 1963
[8] Toda, H.: Composition methods in homotopy groups of spheres. In: Annals of mathematics studies, Vol. 49. Princeton, New Jersey: Princeton University Press 1962 · Zbl 0101.40703
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