The new concept of "system" over double fibred manifolds is introduced and systems of vector valued forms and connections are investigated.
A graded universal differential calculus for involutive systems induced by the Frölicher-Nijenhuis bracket is shown. The system of overconnections, which projects on a given system of connections of a fibred manifold and on the system of linear connections of the base space, is also presented.
A direct formulation of gauge theories and a re-formulation of the lagrangian approach are obtained by means of the graded universal calculus.
In the particular case of principal bundles, the standard differential techniques are recovered and new results are shown as well. The present approach, which is based on differential and functorial methods, can provide new hints for field theory.
Each notion and result is expressed both in an intrinsic way and by explicit formulas in local coordinates.
Gauge Theory Vector Field Vector Bundle Principal Bundle Differential Calculus
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