Dimensional Reduction of Invariant Fields and Differential Operators. I. Reduction of Invariant Fields

Abstract

Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group (“reducing group”) and its orbits (“external dimensions”) are compact, and this is essential in models where the volume of the orbits is related to physical quantities. However, this case is only a part of the natural problems related to dimensional reduction. In the present paper, we consider an action of a (generally non-compact) Lie group on a vector bundle, construct a formalism of reduced bundles for description of all invariant sections of the original bundle, and study the algebraic structures that occur in the reduced bundle. We show that in the case of a non-compact reducing group it is possible that the reduction is non-standard (“non-canonical”), and construct an explicit obstruction for canonical reduction in terms of cohomology of groups. We consider in detail the reduction of tangent and cotangent bundles, and show that, in general, the duality between the two is violated in the process of reduction. The reduction of the tensor product of tangent and cotangent bundles is also discussed. We construct examples of non-canonical dimensional reduction and of violation of duality between the tangent and cotangent bundles in the reduction.