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Gauge theory for fiber bundles. (English) Zbl 0953.53001

Monographs and Textbooks in Physical Science. Lecture Notes. 19. Naples: Bibliopolis. iv, 109 p. (1991).
This is a self-contained exhibition of a gauge theory for fiber bundles. It develops the same programme like it has been established for principal connections on principal fiber bundles. General connections in the sense of Ehresmann are considered and the group of self-diffeomorphisms \(\text{Diff}(S)\) on the typical fiber \(S\) plays the role of the gauge group. Classifying spaces for this group are investigated and a sort of Chern-Weil construction is described which may serve as a replacement of characteristic classes. The usual construction of characteristic classes does not work, because the Lie algebra of \(\text{Diff} (S)\) does not admit any invariants.
To start with, the booklet presents a short description of the analysis in infinite dimension based on the article of A. Kriegl and P. W. Michor in [Acta Mathematica 165, No. 1/2, 105-159 (1990; Zbl 0738.46024)] and the monograph of A. Kriegl and P. W. Michor with the title “The convenient setting of Global Analysis” [Mathematical Surveys and Monographs 53 (1997; Zbl 0889.58001)]. The key sections on gauge theory provide a detailed version of results published in a previous paper of the author [NATO ASI Ser., Ser. C 250, 345-371 (1988; Zbl 0657.53059)].

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53C05 Connections (general theory)
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