Small volume expansion of almost supersymmetric large N theories

We consider the small-volume dynamics of nonsupersymmetric orbifold and orientifold field theories defined on a three-torus, in a test of the claimed planar equivalence between these models and appropriate supersymmetric ``parent models". We study one-loop effective potentials over the moduli space of flat connections and find that planar equivalence is preserved for suitable averages over the moduli space. On the other hand, strong nonlinear effects produce local violations of planar equivalence at special points of moduli space. In the case of orbifold models, these effects show that the "twisted" sector dominates the low-energy dynamics.