parity. This is the most general breaking for 5d. In higher dimensions, rotations can also be broken as discussed in section \[Wilson\].

This orbifold has two fixed branes (at $y=0,\pi R$) and the gauge tranformations $U,P$ define its boundary conditions which act on the fields as $$\Psi(x,y)=P_0\Psi(x,-y),\ \ \ \Psi(x,y+\pi R)=P_{\pi R} \Psi(x,-y+\pi R).$$ where $P_{0,\pi R}$ are the corresponding boundary conditions at $0,\pi R$. These are uniquely fixed by the gauge transformations which act as boundary conditions from Eq. \[eq:pu\] to be $$P_0=P,\ \ \ P_{\pi R}= PU,$$ where the second equality comes from from applying the reflection and then the translation $$\Psi(x,y+\pi R)=P\Psi(x,-y-\pi R)=PU\Psi(x,-y-\pi R+2\pi R)=PU\Psi(x,-y+\pi R).$$ From this equation we can see that $(PU)^2=1$ implying the condition on Eq. \[eq:pupu\].

The orbifold condition $P$ breaks the gauge symmetry by modding out zero modes. The Wilson line breaking through $U$ induces an effective VEV in the scalar coming from the extra dimension. However the orbifold consistency condition from eq. \[eq:pupu\] restricts the possible VEVs $$1=PUPU=Pe^{i\beta^aT_a}Pe^{i\beta^aT_a}=e^{i\beta^a PT_aP}e^{i\beta^aT_a},$$ where the last equality comes from the fact that $P^2=1$ and it can be inserted anywhere in the expansion of the exponential. This equation is only satisfied if the phase is aligned 