Symmetry Breaking/Restoration in a Non-Simply Connected Space-Time

Field theories compactified on non-simply connected spaces, which in general allow to impose twisted boundary conditions, are found to unexpectedly have a rich phase structure. One of characteristic features of such theories is the appearance of critical radii, at which some of symmetries are broken/restored. A phase transition can occur at the classical level, or can be caused by quantum effects. The spontaneous breakdown of the translational invariance of compactified spaces is another characteristic feature. As an illustrative example,the O(N) ${\varphi }^4$ model on $M^3\otimes S^1$ is studied and the novel phase structure is revealed.