dimensional components of the gauge fields, which may play the role of flavons.

Wilson lines {#sec:wl}
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In four dimensional gauge theories one defines a gauge invariant Wilson loop as the trace of a path ordered (P) exponential of the gauge fields $A^a_{\mu}$ transported along a closed line $C$, $$W_C=P \exp i \oint_C T_aA^a_{\mu}dx^{\mu}
\label{W}$$ where $T_a$ are the generators of the gauge group. In the vacuum the associated field strength $F_{\mu \nu}$ can be zero. But this does not necessarily mean that the gauge fields $A^a_{\mu}(x)$ in the exponent can be set equal to zero at all points along the path $C$ by a gauge transformation. This is the Aharonov-Bohm observation.

Recall that in four dimensional gauge theories one can apply a general spacetime dependent gauge transformations on any field $\Psi(x)$, without affecting the physics $$T_aA^a_{\mu}\rightarrow \mathcal{U}(x) T_aA^a_{\mu}\mathcal{U}^{\dagger}(x) -i(\partial_{\mu} \mathcal{U}(x)) \mathcal{U}^\dagger(x), \ \ 
 \Psi(x)\rightarrow \mathcal{U}(x)\Psi(x), \ \ 
\mathcal{U}(x)=e^{i\alpha^a(x)T_a},
\label{G}$$ where we absorb the gauge coupling constant into $A_\mu$. In the vacuum, if the field strength is equal to zero, it is possible to work in a gauge where the gauge fields $A^a_{\mu}$ are also equal to zero or a pure gauge. Here lies the usefulness of the Wilson line 