hand, nor is any alignment potential required. However a realistic model would require additional flavons with independent vacuum alignments. This requires some further discussion about Wilson lines and orbifolds, and this is the subject of the next section.

Wilson Lines and Orbifolds Revisited {#Wilson}
====================================

We now present a general definition of the Wilson Lines together with the orbifold conditions where we assume a spacetime with $M$ dimensions, where $m=0,...,M$. Such a formalism may be necessary for a realistic model where several flavons with independent vacuum alignments are required.

Wilson lines and fibre bundles 
-------------------------------

In compact 5d periodic spaces the Wilson line is defined by the phase in Eq. \[eq:psi2pir\]. This can be rewritten with the translation operator in the 5th dimension $$\Psi(x,y+2\pi R)=e^{2\pi R \partial_5}\Psi(x)=e^{i2\pi R P_5}\Psi(x)=e^{i\beta^a T_a}\Psi(x),$$ where $P_5$ is the momentum operator which is the generator of translations. We can see that defining the Wilson line gauge transformation is equivalent to changing the momentum operator to include an element of the gauge algebra $$P_5\to P_5-\beta^a T_a /2\pi R.$$

The Wilson line mechanism can be implemented in any number of dimensions by generalizing the momentum operator to include an element of the gauge algebra $$P_m\to P_m-\beta^a_m T_a/2\pi 