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Cohomology and connections on fiber bundles and applications to field theories

Given any connection on a fiber bundle B(M,F,G), we discuss the question which closed differential forms on the fiber F can be extended to closed forms on the whole bundle B in a canonical way such that the horizontal parts of the extended forms are given in terms of the gauge fields. Such questions arise for many models in theoretical physics such as the skyrmion bundle and related theories with non‐Abelian groups G. We introduce the notion of a G‐transgressive form as a sufficient condition for the form to be extendable to any bundle which comes along with a given left action L:G×F→F. Using Lie algebra cohomology we prove that if the structure group G of the bundle is semisimple, then every closed n‐form, n≤2, on the fiber which is invariant under L is G‐transgressive and thus defines a unique de Rham cohomology class on any bundle which comes along with L.