loop brings us back to the same point, and due to the continuity of the field function this phase acts as a boundary condition $$\Psi(x,y)=U\Psi(x,y+2\pi R)=e^{i\beta^a T_a}\Psi(x,y+2\pi R),
\label{eq:psi2pir}$$ from which we can obtain the explicit dependence of the extra coordinate $y$ $$\Psi(x,y)=e^{-i\beta^a yT_a/2\pi R}\tilde{\Psi}(x,y),\ \ \ \tilde{\Psi}(x,y)=\tilde{\Psi}(x,y+2\pi R),
\label{eq:solpsi}$$ where $\tilde{\Psi}$ are periodic functions in $y$. If we move around a full circle $y\to y+2\pi R$, we recover Eq. \[eq:psi2pir\]. We can apply a gauge transformation throughout $$\Psi(x,y)\to e^{i\beta^a yT_a/2\pi R}\Psi(x,y)=\tilde{\Psi}(x,y),
\label{eq:U}$$ so that all fields are periodic in the extra dimension. This would gauge away its effect from all the fields except the gauge fields that change as $$\begin{split}
A_m^b&\to \tilde{A}_m^b+(\partial_5 e^{i\beta^a yT_a/2\pi R}) e^{-i\beta^a yT_a/2\pi R}
\\ &= \tilde{A}_m^b+\delta_m^5 \beta^b/2\pi R,
\label{Wilsonvev}
\end{split}$$

where $m=0,1,2,3,5$. This adds a constant to the 5th component of the gauge field, generating an effective VEV and breaking the symmetry spontaneously. Note that the VEV can only happen in the 5th component so that 4d Lorentz symmetry is not broken. This is the Wilson line mechanism to break the gauge symmetry [@Candelas:1985en].

Note that the effective VEV is in the adjoint representation and it can’t reduce the rank of the gauge group.

