Orbifolds
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In 5d the simplest orbifold is $S^1/Z_2$. The orbifolding happens through the identification $$y\sim-y.$$ This forces boundary conditions on the fields which have to comply with the identification, up to a phase

$$\Psi(x,y)=P\Psi(x,-y)=e^{i\alpha^aT_a}\Psi(x,-y),
\label{eq:orcon}$$

where $P=e^{i\alpha^aT_a}$ is an arbitrary non-Abelian phase satisfying $P^2=1$, since it accompanies the parity transformation $\mathcal{P}_5$. The phase $P$ belongs to the gauge group and we assume that $\Psi$ is in the fundamental representation.

This phase $P$ affects the gauge fields $A_m=A_m^a T_a$ in the adjoint representation as $$A_\mu(x,y)=A_\mu(x,-y)^a\ PT_a P,\ \ \ A_5(x,y)=-A_5(x,-y)^a\ PT_a P,$$ where $T_a$ are the generators of the gauge group. The 5th component receives an extra minus sign since it is part of a vector field and we are applying a parity transformation on the 5th coordinate. After compactification all fields are decomposed into an infinite tower of KK modes. Only the fields that have trivial boundary condition have a massless zero mode and survive at low energies.

In the case where $P=1$, the unit matrix, after compactification, the 5d gauge vector is separated into the 4d gauge vector $A_\mu$ that has a positive eigenvalue and therefore a massless zero mode. The remaining component $A_5$ behaves like a 4d 