lines and orbifolds
--------------------------

The simplest setup to study orbifolds together with Wilson lines happens in 5d and the orbifold $S^1/Z_2$. This orbifold is defined by the conditions on the extra dimension $$\begin{split}
y&\sim y+2\pi R,\\
y&\sim -y,
\label{eq:y}
\end{split}$$ where the first equation defines compactification on a circle while the second one is the orbifolding identification. Actually, as we will see in a later section, this is the only consistent orbifolding identification in 5d. Any field $\Psi(x,y)$ propagating through the 5th dimension would be subject to comply with the orbifold conditions from eq. \[eq:y\]. In the case, where there is a gauge symmetry, the field must comply with them up to the arbitrary phases introduced earlier, $$\begin{split}
\Psi(x,y)&=U\Psi(x,y+2\pi R),\\
\Psi(x,y)&=P\Psi(x,-y),
\label{eq:pu}
\end{split}$$ where $U,P$ are equivalent to applying Eqs. \[eq:psi2pir\],\[eq:orcon\] together. They must satisfy the consistency conditions $$P^2=1,\ \ \ PUPU=1,
\label{eq:pupu}$$ coming from the fact that they are related to a translation and a reflection as discussed below (see also [@deAnda:2019anb; @Hebecker:2003jt; @Hebecker:2001jb]).

The Wilson line mechanism and orbifolding break the 5d Poincaré symmetry but respect the 4d part. The non trivial $U$ breaks translation symmetry under the 5th coordinate while a non trivial $P$ breaks the corresponding 