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The topology of the Yang-Mills theory over torus. (English) Zbl 0575.58012

With the physical motivation of cut-offs for Yang-Mills theory, the authors study the topology of principal bundles over a three-torus. Their main results are that isomorphism classes of principal G-bundles over \(T^ 3\), with G a compact, connected Lie group are labelled by \(\pi_ 1(G)\times \pi_ 1(G)\times \pi_ 1(G)\) and that the homotopy class of an automorphism of a principal G-bundle is determined by an element of \(\pi_ 1(G)\times \pi_ 1(G)\times \pi_ 1(G)\times \pi_ 3(G)\). This latter result gives the fundamental group of the moduli space of connections modulo gauge transformations which are the identity at a point. The method is an inductive use of the theorem of Steenrod that the isomorphism class of a principal bundle on \(M\times S^ 1\) is determined by a principal bundle on M and the homotopy class of an automorphism.
Reviewer: N.Hitchin

MSC:

58D30 Applications of manifolds of mappings to the sciences
57R22 Topology of vector bundles and fiber bundles
81T08 Constructive quantum field theory
53C80 Applications of global differential geometry to the sciences
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