Finite Temperature Effect on Wilson Loop Mechanism

We evaluate the energy splitting of vacua appearing in the gauge theory in the space $M_4\times S^N\mathrm{/}{Z}_2$ ($N=2,3,4,5,6$ and $7$). One-loop quantum effects which come from scalar and gauge fields are considered. We calculate them at zero temperature as well as in high-temperature limit. We find that in these situations there is no breakdown of the gauge symmetry.