The Compactified Principal Chiral Model's Mass Gap

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 nonpertubatively generated. As examples, we consider two sigma models compactified on a small circle with twisted boundary conditions. In the compactified SO(3) 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.