# Berry phase and Wilson loop

According to the definition, the Wilson loop is $$$$W[\mathcal{C}] =\operatorname{Tr}\left[\mathcal{P} \exp\left\{i\oint _{\mathcal{C}} A_{\mu } dx^{\mu }\right\}\right]$$$$ where $$\mathcal{P}$$ is the path ordering, $$A_{\mu }$$ is the gauge field.

And closed-path Berry phase is defined as $$\begin{equation*} \gamma _{n} =\oint _{c} A\cdot dR \end{equation*}$$ where A is the Berry connection $$$$A=i\ \langle n(R)| \nabla _{R}| n(R)\rangle$$$$

$$| n(R)\rangle$$ is the eigenstate of Hamiltonian $$\hat{H}(R)$$, which depends on parameter $$R$$.

Since they can both be viewed as holonomy , my question is under what condition could they be identical? Or when could Berry connection be interpreted as gauge field in Yang-Mills theory?

1. Both the usual gauge fields ($$A_\text{QFT}$$) and the Berry connection ($$A_🍓$$) are connections on a principal bundle.
2. In usual gauge theory in QFT you have a spacetime manifold, $$S$$ and a Lie algebra $$\mathfrak{g}$$. The gauge fields are $$\mathfrak{g}$$-valued one-forms on $$S$$, $$A_\text{QFT}\in\Omega^1(S;\mathfrak{g})$$.
3. In the Berry phase business you have a quantum mechanical theory defined on some space with a parameter space, $$P$$. Moreover, the underlying Lie algebra is $$\mathfrak{u}(1)$$. The Berry connection takes values in $$A_🍓 \in\Omega^1(P;\mathfrak{u}(1))$$.
7. Finally, another thing you can do, if you have a map $$f:M\to P$$, and a QFT defined on $$M$$, is that you can pull back the Berry connection along $$f$$, so that $$f^*A_🍓\in\Omega^1(M;\mathfrak{u}(1))$$. Now $$f^*A_🍓$$ behaves like a usual abelian background gauge field, which you can now couple to your physical fields.