THE NO-SCALE NONLINEAR SIGMA-MODEL, MAGNETIC CHARGE, THE COSMOLOGICALCONSTANT, COMPACTIFICATION AND SYMMETRY-BREAKING

'No-scale non-linear sigma models' are considered in three-, four-, an d six-dimensional spacetimes. These are theories with global gauge inv ariance, which here we take to be SO(3) or SL(3,R) and where a homogen eous non-linear constraint is imposed. In contrast with the more stand ard non-linear sigma model, this constraint does not determine a parti cular scale for the strength of the isovector scalar field. In three d imensions, a version of the model is totally equivalent to ordinary el ectrodynamics, while the generalization of this model to 3+1 dimension s leads to a version of relativistic magnetohydrodynamics. Still in 31 dimensions, the constraint in terms of a field strength, which in tu rn is defined in terms of the fundamental scalars, defines a coupling of this field strength to a magnetic source. In this model we also obt ain an additional vector U(1) local gauge invariance associated with t his magnetic charge. In six dimensions the minimal magnetic coupling t o fundamental membranes appears naturally. In six dimensions, it is po ssible to obtain a compactification of two dimensions into a sphere by the presence of a hedgehog configuration of the isovector scalar fiel d, with the resulting four-dimensional effective cosmological constant being zero. A mechanism is discussed for generating breaking of the g auge symmetry, induced by SL(3,R) breaking terms, at a scale much smal ler than the Planck scale. The SL(3,R) symmetry is expected to protect this hierarchy. Also, no massless-'moduli' scalar fields remain after compactification.