Gauge Symmetry Reduction from the Extra Space

The study of physics beyond 4-dimensional (4D) space-time traces back to the work by Kaluza and Klein. 1) The exsitence of an extra space is essential in superstring theory 2), 3) and M -theory. 4) Recently, there is a lot of interesting research of the phenomenological and theoretical implications of an extra space with large radii based on several motivations. 5) As a feature in theories with an extra compact space K, symmetries in the system can change after compactification. Here we give typical examples, where K is an n-dimensional torus T . Supersymmetry (SUSY) is, in general, enhanced by a simple dimensional reduction. 6) Also, SUSY can be broken by the choice of different boundary conditions for bosons and fermions in supermultiplets. This mechanism is called the Scherk-Schwarz mechanism. 7) The other example is a gauge symmetry breaking by the Hosotani mechanism. 8), 9) Gauge symmetry is broken by the appearance of non-integrable phase factors which are dynamical degrees of freedom and are related to Wilson loops of gauge fields along compactified directions. The phenomenon of symmetry transition leads to the idea that symmetries in 4D low-energy theory are derived from a high-energy theory with extra dimensions. It is meaningful to study the relation between symmetries and a compact space more carefully, because the origin of symmetries in the Standard Model is not yet known. As a compact space, the one-dimensional orbifold S/Z2 has attracted our attention since the study of heterotic M-theory. 4) Starting from 5-dimensional (5D) SUSY model, 4D theory with N = 1 SUSY is derived through compactification on S/Z2. 10), 11) Bulk fields have a Z2 parity in a compact direction. The reduction of SUSY originates from a non-universal Z2 parity assignment among component fields in supermultiplets. In this paper, we focus on 5D field theories and study a symmetry transition upon compactification on S/Z2. The above-mentioned SUSY reduction mechanism is generalized as follows. Unless all components in a multiplet of some symmetry