Temperature inversion symmetry in gauge-Higgs unification models

We study the temperature inversion symmetry $R\to1/T$ for the finite-temperature effective potential of the $N{=}1$, $d{=}5$ supersymmetric $SU(3)_{\mathrm c}{\times}SU(3)_{\mathrm w}$ model on the orbifold $S^1/Z_2$. For the value of the Wilson line parameter $\alpha=1$ ($SU(2)_{\mathrm L}$ breaks to $U'(1))$, we show that the effective potential contains a symmetric part and an antisymmetric part under $\xi\to1/\xi$, $\xi=RT$. For $\alpha=0$ $(SU(2)_{\mathrm L}$ is preserved in this case), we find that the only contribution to the effective potential that breaks the temperature inversion symmetry comes from the fermions in the fundamental representation of the gauge group with the $Z_2$ parities $(+,+)$ or $(-,-)$. This is interesting because it implies that the bulk effective potential corresponding to models with fundamental fermions localized at a fixed point in the orbifold (and models with no bulk fundamental fermions) has the temperature inversion symmetry.