GENERALIZING THE O(N)-FIELD THEORY TO N-COLORED MANIFOLDS OF ARBITRARY INTERNAL DIMENSION-D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)- model reduces in the limit N --> 0 to self-avoiding polymers, the N-co lored manifold model leads to self-avoiding tethered membranes. Tn the other limit, for inner dimension D --> 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the mod el at criticality by a one-loop perturbative renormalization group ana lysis around an upper critical line. The freedom to optimize with resp ect to the expansion point on this line allows us to obtain the expone nt v of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to th is model. By comparison of low- and high-temperature expansions, we ar rive at a conjecture for the nature of droplets dominating the 3d Isin g model at criticality, which is satisfied by our numerical results. W e can also construct an appropriate generalization that describes cubi c anisotropy, by adding an interaction between manifolds of the same c olor. The two parameter space includes a variety of new phases and fix ed points, some with Ising criticality, enabling us to extract a remar kably precise value of 0.6315 for the exponent v in d = 3. A particula r limit of the model with cubic anisotropy corresponds to the random b ond Ising problem; unlike the field theory formulation, we find a fixe d point describing this system at 1-loop order. O 1998 Elsevier Scienc e B.V.