Multi-phases in gauge theories on non-simply connected spaces

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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 is shown to possess three phases: Hosotani, Higgs and coexisting phases. The critical radius and the order of the phase transitions are explicitly determined. 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 a dynamical problem of how to stabilize a radius of a compactified space in close vicinity to a critical radius.