  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.
