A research programme which I have been developing in parallel with Joan Simón since 2001 is the construction of string and M-theory backgrounds by quotients. So far we have written a number of (rather long and somewhat technical) papers where we classify and analyse the possible Kaluza-Klein reductions of a number of supergravity backgrounds and study the associated discrete quotients by cyclic subgroups.
The initial motivation for this work was to understand the then novel sector of string theory populated by fluxbranes. Fluxbranes are the stringy avatar of the Melvin universe: a solution of Einstein-Maxwell theory which describes a universe where the electromagnetic repulsion of lines of flux precisely balance their gravitational attraction. A dilatonic version of the Melvin universe can be obtained as the Kaluza-Klein reduction of a flat five-dimensional spacetime. In more geometrical language, the dilatonic Melvin universe is the base of a principal ℝ-bundle whose total space is five-dimensional Minkowski spacetime.
The dilatonic Melvin universe can be lifted to a ten-dimensional supergravity background, the lines of flux now filling only a submanifold of the spacetime: the so-called fluxbrane. As before, the fluxbrane can be obtained by quotienting eleven-dimensional Minkowski spacetime by the free action of an ℝ-subgroup of the Poincaré group. At the time we started thinking about these things many such constructions were known, but no global picture had emerged, so we set out to systematically study these quotients.