Multi-Phases in Gauge Theories on Non-Simply Connected Spaces Hisaki Hatanaka,1,∗) Katsuhiko Ohnishi,2,∗∗) Makoto Sakamoto3,∗∗∗) and

It is pointed out that phase structures of gauge theories compactified on non-simply connected spaces are not trivial. As a demonstration, an SU(2) gauge model on M3 ⊗ S1 is studied, and it is shown to possess three phases: Hosotani, Higgs and coexisting phases. The critical radius and the order of the phase transitions are determined explicitly. A general discussion about phase structures for small and large scales of compactified spaces is given. The appearance of phase transitions suggests a GUT scenario in which the gauge hierarchy problem is replaced by the dynamical problem of the stabilization of the radius of a compactified space in the vicinity of a critical radius. Recently, it has been discovered that field theories with nontrivial backgrounds on extra dimensions have rich phase structures. Magnetic flux passing through a circle S1 can cause spontaneous breakdown of the translational invariance of S1. 1) A kink-like configuration is then generated dynamically as a vacuum configuration