Rokkodai, Nada, Kobe 657-8501, Japan*]{}\
${}^{(c)}$ [*Department of Physics, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan*]{}\
${}^{(d)}$ [*Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan*]{}

Introduction
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Recently, physics with extra dimensions has been studied extensively in connection with the long standing problems, namely, new mechanism and/or the origin of (gauge, super) symmetry breaking in elementary particle physics. It can provide us new insight and understanding for low-energy physics. In fact, it has been pointed out[@SUSY] that new mechanism of spontaneous supersymmetry breaking is possible in a certain class of models as a consequence of the breakdown of the translational invariance for the extra dimension $S^1$[@translation; @O(N); @model]. Furthermore, one of our authors (M.S) and his collaborators have shown[@monopole] that the rotational invariance of $S^2$ is spontaneously broken in a monopole background above some critical radius due to the appearance of vortex configuration as vacuum configuration.

When one considers gauge-Higgs models on space-time with some of the space directions being compactified on a multiply connected space, one should take account of gauge symmetry breaking through the Hosotani mechanism[@Hosotani]. The mechanism essentially occurs due to quantum corrections in the extra dimension, reflecting the topology of the compactified space. It is 