R_m,$$ which would generate a VEV in the same way described in section \[sec:wl\] for the extra dimensional components of the gauge fields $$\braket{A_m^a}=\beta^a_m/2\pi R_m,$$ generalizing Eq. \[Wilsonvev\] for more extra dimensions.

In gauge theories, the fields do not live only in the Minkowski manifold but in a fibre bundle whose base is the Minkowski spacetime $\mathcal{M}^M$ and the fibre is a Lie group $G$. A gauge theory is built in such way that all physical observables are blind to the choice of a section (local gauge invariance). This means that the fibre bundle is not a trivial product $\mathcal{M}^M\times G $ since it has a twist defined by $\beta^a_m$. This induces a preferred section of the fiber bundle, which is similar (and in the case where $\alpha$ a linear function, equivalent) to a Spontaneous Symmetry Breaking (SSB). However this SSB happens in the gauge group as well as in the Poincaré group, breaking Lorentz symmetry. 4d Lorentzian symmetry must be unbroken, but in ED, the extra dimensional part of the Poincaré group disappear after compactification and they can be broken.

To summarize, the Wilson line mechanism is equivalent to choosing a twist in the fibre bundle so that the 