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If the space of minima of the effective potential of a weakly coupled 2d quantum field theory is not connected, then a mass-gap will be nonperturbatively generated. As examples, we consider two ? models compactified on a small circle with twisted boundary conditions. In the compactified CP1 model, the vacuum manifold consists of two points and the mass-gap is nonperturbative. In the case of the compactified SU(2) principal chiral model, the vacuum manifold is a single circle and the mass-gap is perturbative.