A simple model of dimensional collapse

We consider a simple model of d families of scalar field interacting with geometry in two dimensions. The geometry is locally flat and has only global degrees of freedom. When d < 0 the universe is locally two-dimensional but for d > 0 it collapses to a one-dimensional manifold. The model has some, but not all, of the characteristics believed to be features of the full theory of conformal matter interacting with quantum gravity which has local geometric degrees of freedom.