definition, where, for a pure gauge $ T_aA^a_{\mu}=-i(\partial_{\mu} \mathcal{U}(x))\mathcal{U}^{\dagger}(x)=\partial_\mu\alpha^a T_a$, it becomes $$W_C=P \exp iT_a \oint_C \partial_{\mu}\alpha^a(x) dx^{\mu},
\label{WC}$$ where the the integral is identically zero. However, this is true for only smooth functions $\alpha^a(x)$. When these functions are ill defined at some point, the loop can’t be contracted (due to the existence of a singular point) and it won’t vanish. Therefore it can generate physical field strengths.

In extra dimensions the same effect can happen even when the gauge transformations are smooth. This happens if the extra dimensions are not a simply connected space. For simplicity let us assume a single compact extra dimension which is a circle $S^1$ with radius $R$. In this case, the gauge transformation $$\mathcal{U}(y)=e^{i\alpha^a(y)T_a},$$ which only depends on the extra dimension $y$, becomes physical. This is measured through the phase (c.f. Eq. \[WC\]), $$U=e^{i\oint\partial_y\alpha^a(y)T_a dy}=e^{iT_a \int_0^{2\pi R}\partial_y\alpha^a(y) dy}=e^{i\beta^a T_a},$$ which arises from an integration through a non contractible loop and can’t be trivialized [@Hall:2001tn]. In this case, the loop is non contractible because the space is not simply connected (a circle) even with smooth functions. Any field transported around this non contractible loop will pick up the phase $U$. However, going around the whole 