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) φ model on M ⊗ S is studied and the novel phase structure is revealed. Talk given at XXXth International Conference on High Energy Physics (ICHEP2000), July 27-August 2, Osaka, Japan. To be published in the Proceedings (World Scientific, Singapore). E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] The parameter space of field theories compactified on non-simply connected spaces is, in general, wider than that of ordinary field theories on the Minkowski space-time, and is spanned by twist parameters specifying boundary conditions[1, 2], in addition to parameters appearing in the actions. Physical consequences caused by twisted boundary conditions turn out to be unexpectedly rich and many of them have not been uncovered yet. The purpose of this talk is to report some of interesting properties of such theories overlooked so far. One of characteristic features of such theories is the appearance of critical radii of compactified spaces, at which some of symmetries are broken/restored[3]. Symmetry breaking patterns are found to be unconventional. A phase transition can occur at the classical level, or can be caused by quantum effects. Radiative corrections would become important when a compactified scale is less than the inverse of a typical mass scale, and then some of broken symmetries could be restored, or conversely some of symmetries could be broken. Another characteristic and probably surprising feature is the spontaneous breakdown of the translational invariance of compactified spaces[4]. Twisted boundary conditions do not allow vacuum expectation values of twisted bosons to be non-vanishing constants. In other words, vacuum expectation values of twisted bosons have to vanish or to be coordinate-dependent if they are non-vanishing. If the minimum of a potential does not lie at the origin, twisted bosons could acquire non-vanishing vacuum expectation values, which should be coordinate-dependent. Then, we have to minimize the total energy, which consists of both the kinetic term and the potential term, to find the vacuum configuration. When non-vanishing vacuum expectation values of twisted bosons are energetically preferable, they should be coordinate-dependent and hence the translational invariance is broken spontaneously. Among other characteristic features, a phenomenologically important observation is that twisted boundary conditions can break supersymmetry spontaneously[5]. This will give a new type of spontaneous supersymmetry breaking mechanisms and it would be of great interest to investigate a possibility to construct realistic supersymmetric models with this supersymmetry breaking mechanism, though this subject will not be treated in this talk. As an illustrative example, we here concentrate on the O(N) φ model on M ⊗ S. ‖ This subject was discussed in the talk given by M. Tachibana at this Conference. ∗∗ The classical analysis of this model has been done in Ref.[3].