Fibre bundles and generalized dimensional reductions

We study some geometrical and topological aspects of the generalized dimensional reduction of supergravities in D = 11 and D = 10 dimensions, which give rise to massive theories in lower dimensions. In these reductions, a global symmetry is used in order to allow some of the fields to have a non-trivial dependence on the compactifying coordinates. Global consistency in the internal space imposes topological restrictions on the parameters of the compactification as well as the structure of the space itself. Examples that we consider include the generalized reduction of the type IIA and IIB theories on a circle, and also the massive ten-dimensional theory obtained by the generalized reduction of D = 11 supergravity.