A Local Approach to Dimensional Reduction: I. General Formalism

We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (“fields”) and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e., to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a “non-canonical” reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space. Subj. Class.: Differential geometry; Geometrical approaches to partial differential equations; Field theory. 2000 MSC: 37J15, 58Z05, 53Z05, 78A25, 81T40.