Phase structure of renormalizable four-fermion models in spacetimes of constant curvature

A number of 2d and 3d four-fermion models which are renormalizable ---in the $1/N$ expansion--- in a maximally symmetric constant curvature space, are investigated. To this purpose, a powerful method for the exact study of spinor heat kernels and propagators on maximally symmetric spaces is reviewed. The renormalized effective potential is found for any value of the curvature and its asymptotic expansion is given explicitly, both for small and for strong curvature. The influence of gravity on the dynamical symmetry breaking pattern of some U(2) flavor-like and discrete symmetries is described in detail. %It is seen explicitly that the effect of a %negative curvature is similar to that of a magnetic field. The phase diagram in $S^2$ is constructed and it is shown that, for any value of the coupling constant, a curvature exists above which chiral symmetry is restored. For the case of $H^2$, chiral symmetry is always broken. In three dimensions, in the case of positive curvature, $S^3$, it is seen that some values of the coupling constants lead to a gap equation which has no solutions. Both for $H^3$ and $S^3$ the configuration given by the auxiliary fields equated to zero is not a solution of the gap equation.