Fields and connections in a fiber bundle over spacetime arising from the reduction of the multidimensional linear connection
One of the developments in the field theory for multidimensional space is the study of models on a manifold S, where a compact Lie group G acts with a single orbit-type G/H, H being a closed subgroup of G. The orbit space S/G is identified with the four-dimensional spacetime M, and the gauge group with the group \(K=N/H\) (N is the normalizer of H in G), which is the group of automorphisms of G/H via its canonical right action. Every G-invariant connection in the fiber bundle \(S=G/H\times_ KS_ 0\) \((S_ 0\subset S\) consists of the H-invariant points and is a principal K-bundle over M) allows one to associate with the bundle of linear frames over S a principal bundle \(F_ 0\) with base M, isomorphic to the fibered product \(S_ 0\times_ MF(M)\), where F(M) is the bundle of linear frames over M. One finds that each G-invariant linear connection on S admits a dimensional reduction in the sense that it corresponds uniquely to (i) a \(K\times GL(\dim M,R)\) connection in the principal bundle \(F_ 0\), (ii) a vector field on M, i.e. a one-form on M with values in a vector bundle over M, and (iii) a scalar field on M, i.e. a cross-section of a vector bundle over M associated to \(F_ 0\).
scientific article; zbMATH DE number 3994490