The no-scale non-linear sigma model, magnetic charge, the cosmological constant, compactification and symmetry breaking

'No-scale non-linear sigma models' are considered in three-, four-, and six-dimensional spacetimes. These are theories with global gauge invariance, which here we take to be SO(3) or SL(3,R) and where a homogeneous non-linear constraint is imposed. In contrast with the more standard non-linear sigma model, this constraint does not determine a particular scale for the strength of the isovector scalar field. In three dimensions, a version of the model is totally equivalent to ordinary electrodynamics, while the generalization of this model to 3+1 dimensions leads to a version of relativistic magnetohydrodynamics. Still in 3+1 dimensions, the constraint in terms of a field strength, which in turn is defined in terms of the fundamental scalars, defines a coupling of this field strength to a magnetic source. In this model we also obtain an additional vector U(1) local gauge invariance associated with this magnetic charge. In six dimensions the minimal magnetic coupling to fundamental membranes appears naturally. In six dimensions, it is possible to obtain a compactification of two dimensions into a sphere by the presence of a hedgehog configuration of the isovector scalar field, with the resulting four-dimensional effective cosmological constant being zero. A mechanism is discussed for generating breaking of the gauge symmetry, induced by SL(3,R) breaking terms, at a scale much smaller than the Planck scale. The SL(3,R) symmetry is expected to protect this hierarchy. Also, no massless-'moduli' scalar fields remain after compactification.